# Patent

## Title
Nonlinear imaging with dual band pulse complexes

## Abstract

The invention presents methods and instrumentation for measurement or imaging of a region of an object with waves of a general nature, for example electromagnetic (EM) and elastic (EL) waves, where the material parameters for wave propagation and scattering in the object depend on the wave field strength. The invention specially addresses suppression of 3rd order multiple scattering noise, referred to as pulse reverberation noise, and also suppression of linear scattering components to enhanced signal components from nonlinear scattering. The pulse reverberation noise is divided into three classes where the invention specially addresses Class I and Class II 3rd order multiple scattering that are generated from the same three scatterers, but in opposite sequence.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
The application claims priority from U.S. Provisional Patent Application Ser. No. 61/542,511, filed Oct. 3, 2011, the contents of which are incorporated by reference herein.


1. FIELD OF THE INVENTION
The present invention is directed towards real time ultrasound imaging of the heart and visualization and analysis of the images for detection of regional deficiencies in myocardial (MC) contraction and relaxation function.
2. BACKGROUND OF THE INVENTION
Material properties that determine electromagnetic (EM) or elastic (EL) wave propagation and scattering in the materials often show a variation with the field strength in the waves. Such materials are termed non-linear and give rise to nonlinear wave propagation and nonlinear scattering of both EM and EL waves. Measurements or imaging of nonlinear scattering sources are in many situations useful to identify properties of such materials.
Both the forward wave propagation and local scattering of EM or EL waves have mathematical similarities, and methods and instrumentation for imaging therefore have similar structures. Examples of uses of EL waves are material testing both with shear waves and compression waves, ultrasound medical imaging with compression waves, SONAR sub-sea and geological measurements, and seismic imaging. EM waves have similar uses, where particularly new developments of EM technology appear useful for both geological and medical imaging, providing added information to elastic wave images. EM imaging in the infra-red and optical frequency ranges also provides useful information both for material testing and medical imaging.
The nonlinear scattering can for EM or EL waves be separated into a parametric and a resonant scattering type. For EL waves, the parametric scattering originates from a nonlinear variation of the local elasticity parameters with the amplitude of the local elastic wave field, where spatial variations of the nonlinear variation produce the nonlinear scattering. For EM waves, the parametric scattering originates from a nonlinear variation of the local dielectric constant or magnetic permeability with the amplitude of the local EM wave field, where spatial variations of the nonlinear variation produce the nonlinear scattering. With elastic compression waves, referred to as acoustic waves, one for example gets strong nonlinear parametric scattering at the interface between soft materials and hard materials, for example as found with ultrasound nonlinear scattering from micro calcifications in soft tissue or acoustic scattering from hard objects in soil like mines or other objects. One also gets strong nonlinear scattering at the interface between harder materials and much softer materials, for example as found with ultrasound scattering from gas micro-bubbles in blood or gas filled swim-bladders of fish and the like in water.
Resonant nonlinear scattering has a time lag involved, which in some situations can be used to separate signal components from local nonlinear scattering and forward propagation distortion of the incident waves. However, the current invention provides further advantages for imaging of local resonant nonlinear scattering sources.
For acoustic waves, gas micro-bubbles show resonant scattering, for example, where the resonance originates from the energy exchange between the nonlinear elasticity of the bubble with shell and gas, and a co-oscillating fluid mass around the bubble with a volume approximately 3 times the bubble volume. As both the elasticity and the mass vary with bubble compression, the resonance frequency is nonlinearly affected by the incident acoustic wave field, producing a particularly strong nonlinear scattering with a large amount of harmonic components of the incident frequency (n-times the incident frequency) and even sub-harmonic components of the incident frequency (a fraction of the incident frequency) in the scattered field, and supra-harmonic components (bands around the harmonic components) of the incident frequency. However, for imaging at frequencies well above the bubble resonance frequency, the nonlinear scattering is much lower, and the present invention provides solutions for enhanced imaging of micro-bubbles at frequencies above the resonance frequency.
Resonant nonlinear EM scattering originates in the interaction between the wavefield and the atoms and molecules, which is best described within the realm of quantum physics. An example of EM resonant scattering is fluorescence which has similarities to sub-harmonic acoustic scattering. When for example the incident frequency is in the ultraviolet range, the scattered frequency can be in the visible range. The scattered frequency can also be the same as the incident frequency which is termed “resonant fluorescence”. Another example is two-photon quantum scattering that is similar to 2nd harmonic parametric scattering, but includes detailed atomic dynamics with time lags in the process.
There is also found a nonlinear interaction between EM and EL waves in materials, where for example EL compression waves change the EM material parameters in the process called the acousto-optic effect. Absorption of EM waves in materials produces a rapid, local heating of the material that generates acoustic waves in a process called the photo-acoustic effect. The invention hence addresses both EM and EL waves, and combinations of these, where the waves referred to in the description and claims can be both EM and/or EL waves.
With a single frequency band incident wave, the parametric nonlinear scattering produces harmonic components of the incident frequency band in the scattered wave. With dual band incident waves that interact locally, the parametric nonlinear scattering produces bands around convolutions of the incident frequency bands, resulting in bands around sums and differences of the incident frequencies. However, the nonlinear variation of the material parameters also produces an accumulative nonlinear distortion of the forward propagating wave. When the pulse length of the high frequency pulse increases above approximately a period of the low frequency pulse, the linear scattering from the nonlinear forward propagation distortion has a similar signature to the local nonlinear scattering, and it is in this case difficult to distinguish the signal components that arises from linear scattering of the nonlinear propagation distortion of the incident wave, and the signal components that occur from local nonlinear scattering. This is for example the situation with current harmonic imaging with medical ultrasound imaging.
On the contrary, when the pulse length of the high frequency pulse becomes shorter than approximately a half period of the low frequency pulse, it is possible to highly suppress the linear scattering components to measure or image the nonlinear scattering components, for example as shown in U.S. Pat. No. 8,038,616 and U.S. patent application Ser. Nos. 12/351,766, 12/500,518, and 13/213,965. Multiple scattering of EL and EM waves from strong scatterers often produce strongly disturbing noise in the measurements and images. When the pulse length of the high frequency pulse becomes shorter than half the wave length of the low frequency pulse, it is possible to highly suppress this multiple scattering noise, where some methods are presented in the cited US Patent applications. Based on this background, the object of the current invention is to present improved methods and instrumentation for measurement and imaging of nonlinear scattering components and suppression of the multiple scattering noise, both with elastic and electromagnetic waves.
3. SUMMARY OF THE INVENTION
An overview of the invention is presented. The overview is meant for illustration purposes only, and by no means represents limitations of the invention, which in its broadest aspect is defined by the claims appended hereto.
The invention presents methods and instrumentation for measurement or imaging of a region of an object with waves of a general nature, for example electromagnetic (EM) or elastic (EL) waves, where the material parameters for wave propagation and scattering in the object depend on the wave field strength. Such materials are termed non-linear, and wave propagation in such materials defines the concepts of nonlinear wave propagation and scattering. The invention specially addresses strong suppression of 3rd order multiple scattering noise, referred to as pulse reverberation noise, and also suppression of linear scattering components to enhance signal components from nonlinear scattering. The pulse reverberation noise is divided into three classes where the invention specially addresses Class I and II 3rd order multiple scattering that are generated from the same three scatterers, but in opposite sequence. One specially addresses methods to achieve close to similar Class I and II pulse reverberation noise in the measurements, which simplifies the suppression of both classes combined, and methods for combined suppression of Class I and II noise that compensate for a difference in Class I and II noise in the measurements.
The methods are based on transmission of dual band pulse complexes composed of a low frequency (LF) pulse and a high frequency (HF) pulse, where the LF pulse is used to nonlinearly manipulate the object material parameters observed by the co-propagating HF pulse. One or both of scattered and transmitted components from the HF pulse are picked up and gives HF receive signals that are further processed to suppress pulse reverberation noise and enhance nonlinear scattering components.
One typically wants the HF pulse to propagate along the crest or trough of the co-propagating LF pulse. The nonlinear manipulation of the object properties by the LF pulse produces a change in the wave propagation velocity for the co-propagating HF pulse. This change in propagation velocity introduces a nonlinear propagation delay given by the LF field strength and polarity at the center of the HF pulse, where said nonlinear propagation delay accumulates with propagation distance. For a given polarity of the LF pulse one can get a negative nonlinear propagation delay, which represents propagation advancement. We shall however use the generalized concept of delay, which can be both positive and negative. A variation of the LF field strength along the co-propagating HF pulse produces a nonlinear pulse form distortion of the HF pulse that also accumulates with propagation distance. Spatial variations of the nonlinear object parameters, produces a scattering of the HF pulse that depends on the local value of the co-propagating LF pulse.
In addition to the nonlinear manipulation by the LF pulse of the propagation velocity and scattering of the co-propagating HF pulse, the nonlinearity of the object parameters introduces a self-distortion of the HF pulse by its own field strength. This self-distortion of the HF pulse introduces harmonic bands in the HF pulse around harmonic components of the fundamental, transmitted HF band, and is the basis for current methods of harmonic imaging. The transfer of energy into harmonic bands are observed as a nonlinear attenuation of the fundamental HF band that is compensated for in the model based methods of estimation of noise suppression according to the invention.
At the first scattering of the pulse complex, the amplitude of the both the LF and HF pulses generally drops so much that the nonlinear effect on HF pulse propagation and scattering can be neglected after this first scattering. This opens for the use of the methods to suppress multiple scattering noise in the HF receive signal, as the nonlinear effects on 1st order and multiple order scattering of the waves at the same propagation lag are different.
To observe multiple scattering noise, the first scatterer must be inside the transmit beam and the last scatterer must be inside the receive beam. With single, fairly narrow transmit beams, this implies that we with back-scattering measurements mainly observe odd order scattering. As the pulse amplitude drops for each scattering, we mainly observe 3rd order multiple scattering noise (i.e. three scatterers), where the 1st scatterer is inside the transmit beam, and the 3rd scatterer is inside the receive beam. With multiple transmits of very wide transmit beams where the receive signals are reconstructed for narrow, synthetic transmit beams, the situation is the same. However, with very wide transmit beams, or parallel transmit beams, overlapped with multiple parallel, highly focused receive beams, it is in principle possible to observe 2nd order multiple scattering, where the 1st scatterer is inside a transmit beam at a distance from the 2nd scatterer inside a highly focused receive beam. However, the main disturbing multiple scattering noise originates from fatty layers inside muscle or parenchyma tissue, where the transmit and receive beams have close to normal incidence to the layers. Such anatomy makes such 2nd order scattering low. Ribs or other similar structures can however have spatial relations to the beams that make 2nd order scattering noise visible in special situations.
With a sequence of 3rd order scatterers at positions r1=>r2=>r3, where r1 is inside the transmit beam and r3 is inside the receive beam, we will also have 3rd order scattering from the same scatterers in the opposite sequence, i.e. r3=>r2=>r1, where r3 is inside the transmit beam and r1 is inside the receive beam. We therefore divide the 3rd order scattering noise into three classes, where for Class I the sequence of scatterers is r1=>r2=>r3 where for the 1st scatterer r1<r/2 for r=ct/2 as the depth of the image point with t being the time of signal arrival (fast time) and c is the wave propagation velocity. For the Class I noise to occur at r=ct/2 we have for the 3rd scatterer that r3>r/2. For Class II noise the situation is opposite where for the 1st scatterer r3>r/2 and for the 3rd scatterer r1<r/2. For Class III noise, both the 1st and 3rd scatterers are close to r, i.e. deeper than r/2. Class III noise can with the model based estimation of noise suppression be suppressed in combination with Class I and Class II noise. However Class III noise is in many situations weaker than the combined Class I and II noise.
At a defined fast time arrival of echoes, t, the HF receive signal is composed of several overlapping pulses, both from 1st and 3rd order scattered components of the transmitted HF pulses. These several overlapping pulses will interfere both constructively and destructively depending on the detailed arrival time of the pulses, and this interference hence produces a depth variation in the envelope of the HF receive signal that is typical for coherent imaging systems, and which we refer to as speckle. The LF pulse will generally introduce both a nonlinear propagation delay/advancement of the HF receive signal, and also an LF pulse dependent variation in the HF receive signal speckle.
A central part of the methods according to the invention is hence

    
    
        processing of said HF receive signal to form measurement or image signals wherein said processing comprises correcting at least one of said HF receive signals by at least one of
        
            i) delay corrections using correction delays in a fast time (depth-time), and
            ii) speckle corrections using speckle corrections in the fast time, to form corrected HF signals, and
        
        
        further processing said corrected HF signals to form at least one of a noise suppressed HF signal with suppression of pulse reverberation noise and a nonlinear scattering signal with suppression of linear scattering components.
    
    


For HF receive signals where the bandwidth is wide compared to the frequency of the LF pulse, the main effect of the pulse form distortion in the HF receive signal is found at the edges of the HF receive band. One gets interesting correction for the pulse form distortion by reducing the band wdth of the HF receive signal through band pass filtering.
For the 1st order scattered signal, the nonlinear propagation delay is given by the accumulated change in propagation velocity by the LF pulse at the site of the co-propagating HF pulse up to the image depth r=ct/2. For the combined Class I and II pulse reverberation noise, the nonlinear noise delay is given by an average of the nonlinear noise delay for Class I and Class II, which both are averages of the nonlinear propagation delays for the distribution of 1st scatterers that contribute to the Class I and II noise at r=ct/2. The relationship between the LF pulse and the combined Class I and II nonlinear noise delay is hence more complex than for 1st order scattering. However, for HF transmit and receive beams that are close to equal and a linear variation of the nonlinear propagation delay t(t) with fast time arrival t of the HF receive signal, the noise correction delay can be approximated to t(t)/2.
The dependency of the 1st order scattered signal speckle on the LF pulse is mainly given by the pulse form distortion of the transmitted HF pulse by the co-propagating LF pulse that produces the 1st order scattered signal at fast time t. The phase of a focused LF pulse will slide 90 deg from the near field to the diffraction limited focal region, and the co-propagating HF pulse will then at certain depth ranges observe a non-negligible gradient of the LF field along the HF pulse, producing a HF pulse compression/expansion depending on the LF field gradient and its polarity along the HF pulse. With wide LF transmit beams where diffraction plays a negligible role, one can position the HF pulse at the crest or trough of the co-propagating LF pulse for the whole measurement or image depth, reducing the pulse form distortion of the HF pulse by the LF pulse. However, if the HF pulse has a non-negligible pulse length compared to the LF pulse oscillation period, one will in this case get a non-negligible HF pulse form distortion of the HF pulse up to the first scattering.
A variation of the nonlinear propagation delay for the 1st order scattered pulses that overlap at a fast time t of the HF receive signal, will also produce a LF pulse dependent speckle of the HF receive signal from 1st order scattered components. A main reason for such a variation is that the HF receive beam is so weakly focused that the nonlinear propagation delay varies across the receive beam. A remedy to reduce this effect is stronger HF receive beam focusing with lower F-number. However, this can introduce a difference between the HF transmit and receive beams that complicates the estimation of the combined Class I and II nonlinear noise delay. Several solutions to these problems is given in this invention, such as the use of multiple transmit foci, or transmit of broad LF and HF transmit beams in different directions with synthetic transmit focusing in a large group of image points, or using model based estimation of the corrections for both 1st order nonlinear signal delay and speckle. The effect on signal speckle from variations of the nonlinear propagation delay along the beam axis, is reduced by using short HF pulses.
The dependency of the combined Class I and Class II noise speckle on the LF pulse is as for the nonlinear noise delay more complex than for the 1st order scattering components. The reason for this is that a large number of combined Class I and II noise pulses with a wide distribution of positions of 1st scatterers can overlap at the same fast time point t=2r/c of the HF receive signal. The pulses with different 1st scatterers have different nonlinear propagation delays and pulse form distortion due to different accumulative propagation lengths to the 1st scatterer along different propagation paths, and hence have different variations with the field strength and polarity of the LF pulse, making the interference and noise speckle much more sensitive to variations in the LF pulse compared to for the 1st order scattered components in the HF receive signal at a defined fast time t. A special situation is found when the HF transmit and receive beams are equal, with negligible pulse form distortion of the transmitted HF pulse, and linear variation with fast time t of the nonlinear propagation delay, where the combined Class I and II noise speckle is independent on the polarity of the LF pulse, that is useful for suppression of the combined Class I and II pulse reverberation noise. In other situations we show how to obtain corrections for both nonlinear noise delay and noise speckle of combined Class I and II pulse reverberation noise through model based estimation.
In a first embodiment according to the invention the HF receive and transmit apertures and HF receive and transmit foci are selected so that a difference between Class I and II pulse reverberation noise in the HF receive signal is substantially minimized. In a further specialized embodiment of this method, the HF transmit and receive apertures and foci are set equal. To obtain adequate beam focusing over a depth range, the invention devices to transmit multiple groups of pulse complexes along each image line with different HF transmit and receive focal depths for each group of pulse complexes, to obtain focus at more than one depth range in the image. The invention also devices to use transversal filtering of the HF receive signal for multiple, neighboring image lines to obtain a depth variable focusing of the combined HF transmit and receive beams with fixed focus. The invention alternatively proposes to transmit multiple pulse complexes with wide transmit beams with different directions, and receive and store all HF receive element signals from a HF receiver array, and carry through synthetic image reconstruction for a large group of image points providing synthetic HF transmit and receive beams that for the image points of said group can be selected to provide substantially similar Class I and II pulse reverberation noise, that greatly simplifies the processing for combined suppression of Class I and II pulse reverberation noise in said group of image points. Substantially similar Class I and II pulse reverberation noise can in many situations be obtained in an image point by choosing the synthetic transmit and receive beams to be equal in said image point. With this method it is desirable, but not necessary, to use transmit wave fronts that are close to plane, at least in one direction, often referred to as plane wave imaging. With little difference between the Class I and II pulse reverberation noise in the HF receive signal, it is an advantage to select the LF and HF transmit apertures so that the nonlinear propagation delay t(t) for the transmitted HF pulse has a substantially linear variation with the fast time t, and for combined suppression of Class I and II pulse reverberation noise, said correction delays are set equal to t(t)/2.
In a further modification of the methods, one do not search to minimize a difference between Class I and II pulse reverberation noise in the HF receive signal, where instead at least one of said correction delays and said speckle corrections are estimated to compensate for a difference between Class I and II noise in said HF receive signal.
In the estimation of correction delays and speckle corrections one frequently makes use of simulation of the nonlinear wave propagation in the object utilizing defined material parameters. Said defined material parameters can be adjusted for suppression in a processed HF receive signal of at least one of i) pulse reverberation noise, and ii) 1st order linear scattering components to provide nonlinear scattering components, through one or both of

    a) manual adjustment of said object material parameters based on observations in a displayed image, and
    b) computer based adjustment of said object material parameters to minimize the total power in said processed HF receive signal in a region of an image.


For advanced suppression of combined Class I and II noise with differences between the Class I and II noise in the HF receive signal, the invention devices:

    obtaining the nonlinear propagation delay and pulse form distortions for the transmitted HF pulses as a function of position of the 1st scatterers through simulations of a nonlinear wave equation for the actual LF/HF pulse complexes with defined parameters, and
    obtaining estimates of parameters relating to relative reflection coefficients of strong reflectors as a function of depth along the beam from the measured HF receive signal, and
    obtaining estimates of noise correction delays and speckle corrections in a process that includes combining in a mathematical model for pulse reverberation noise, said nonlinear propagation delays and pulse form distortions up to the 1st scatterer obtained through simulations, and said estimates of parameters relating to relative reflection coefficients obtained from the measured HF receive signal. Said method can also be extended to suppress Class I/II/III pulse reverberation noise combined.


In this situation one would typically carry through the process in steps where each step applies the method as in the previous paragraph, and for each step said defined parameters are adjusted to increase suppression of pulse reverberation noise, where said adjustment is carried through as one or both of

    a) manual adjustment based on observation of a displayed image, and
    b) automatic adjustment to minimize the total power in a processed HF receive signal in a region of an image.


The invention further devices to carry through said simulations of a nonlinear wave equation for different sets of parameters before the measurements, and electronically store the results of said simulations for said set of parameters, and in the estimation process the simulations for defined parameters is retrieved from memory based on a memory address related to said defined parameters.
The invention further devices to obtain and store from simulations of a nonlinear wave equation for a typical set of material parameters, both the nonlinear propagation delay and pulse form distortion and also gradients of these so that nonlinear propagation delay and pulse form distortions for different material parameters can be obtained through a Taylor or Fourier series representation of these as a function of variations in the material parameters.
For combined suppression of Class I and II pulse reverberation noise in echo weak regions, the same HF receive signal obtained with a non-zero LF pulse is delay corrected with two different correction delays to provide two corrected HF signals, and said two corrected HF signals are combined to provide a noise suppressed HF signal. This method reduces the need for speckle corrections.
The methods further comprises combined suppression of pulse reverberation noise and enhancement of nonlinear scattering components, wherein said step of transmitting includes transmitting at least 3 pulse complexes with different LF pulses, and wherein said step of correcting includes correcting and combining the HF receive signals to form at least two noise suppressed HF signals, and said at least the two noise suppressed HF signals are further processed by at least one of

    
    
        i) delay corrections using correction delays in a fast time (depth-time), and
        ii) speckle corrections using speckle corrections in the fast time,

to form at least two corrected intermediate HF signals, and said corrected intermediate HF signals are combined to suppress linear scattering components to form nonlinear measurement or imaging HF signals.

    
    


The invention further defines instruments for carrying through the methods of the invention, where said instruments comprises

    transmit means comprising a low frequency (LF) transmit aperture and a high frequency (HF) transmit aperture for transmitting at least two pulse wave complexes towards said region, each of said pulse complexes being composed of an LF pulse with frequencies in an LF band transmitted from the LF aperture and an HF pulse with frequencies in an HF band transmitted from the HF aperture, the HF pulse propagating spatially so close to the LF pulse that the HF pulse observes a manipulation of the object material parameters by the co-propagating LF pulse at least for a part of a propagation depth of the HF pulse, and at least the transmitted LF pulse varies for each transmitted pulse complex to produce different manipulations of the object, and
    receive and beam forming means comprising a HF receive aperture that picks up HF receive signals from one or both of scattered and transmitted HF components from the at least two transmitted pulse complexes, and
    correction means for processing said HF receive signal to form measurement or image signals wherein said processing comprises correcting at least one of said received HF signals by at least one of
    
        i) delay corrections using correction delays in a fast time (depth-time), and
        ii) speckle corrections using speckle corrections in the fast time,

to form corrected HF signals, and

    
    
    means for further processing said corrected HF signals to form at least one of a noise suppressed HF signal with suppression of pulse reverberation noise and a nonlinear scattering signal with suppression of linear scattering components.


Embodiments of the instrument comprises means for selecting HF transmit and receive apertures and HF transmit and receive foci to minimize a difference between Class I and II pulse reverberation noise in the HF receive signal. To obtain good focusing in a depth range, embodiments of the instrument can comprise means for transmitting multiple pulse complexes with different HF focal depths along each image line, and means for selecting HF receive apertures and foci for each transmit pulse focus to minimize a difference between Class I and II pulse reverberation noise. In many situations it is satisfactory to choose equal HF transmit and receive apertures and HF transmit and receive foci.
Embodiments of the instrument further comprises means for transmitting more than one pulse complex with wide beams and different directions towards said object, and said receive means comprises a HF receive aperture comprising an array of more than one HF receive element that allows receiving and storing HF receive element signals for each HF receive element and each transmitted pulse complex, and means for image reconstruction for a large group of image points that comprises means for combining said HF receive element signals from said transmitted pulse complexes for each image point of said group to provide synthetic HF transmit and receive beams so that a difference between Class I and II pulse reverberation noise is substantially minimized in the reconstructed HF receive signal for said group of image points. This can in many situations be obtained by choosing the image reconstruction for an image point so that the synthetic HF transmit and receive beams become substantially equal. In one embodiment said transmit means allows transmit of LF and HF pulse waves with wave fronts that are substantially plane in at least one direction.
The instrument further comprises estimation means for estimation of at least one of said correction delays and speckle corrections either according to known methods, or new methods according to this invention where said means for estimation of said correction delays and said speckle corrections are designed to compensate for a difference in Class I and II pulse reverberation noise in said HF receive signal, for combined suppression of Class I and II pulse reverberation noise. Said estimation methods can be further extended to also suppress Class I/II/III pulse reverberation noise combined.
In further embodiments of the invention said transmit means allow the LF and HF transmit apertures to be selected so that the nonlinear propagation delay t(t) for the transmitted HF pulse has a substantially linear variation with the fast time t=2r/c. When the HF transmit and receive apertures and foci, both real and synthetic, are selected so that a difference between the Class I and II noise is minimized, said correction delays for suppression of pulse reverberation noise at fast time t can be set close to t(t)/2.
In further embodiments according to the invention said estimation means comprises means for simulations of nonlinear wave propagation in the object utilizing defined material parameters, where said nonlinear propagation delays and pulse form distortions up to the 1st scatterer are obtained from the simulated HF pulse wave. Said means for simulation can comprise means for storage of a set of simulations with different defined parameters where said set of simulations are carried through before the measurements, and in the estimation process the simulation for defined parameters is retrieved from memory based on a memory address related to said defined parameters.
Said means for simulation can also comprise means for storage of the nonlinear propagation delay and pulse form distortion and gradients of the same, simulated for a given set of material parameters, so that the estimates of the nonlinear propagation delay and pulse form distortion for other material parameters can be obtained through a Taylor series representation of their variation with the material parameters.
Said estimation means further can comprise means for adjusting said defined parameters to increase the suppression of one or both of i) pulse reverberation noise and ii) 1st order linear scattering components in a processed HF receive signal, where said adjusting is carried through as one or both of

    
    
        a) manual adjusting based on observation of a displayed image, and
        b) automatic adjusting to minimize the total power in the processed HF receive signal in a region of an image.
    
    


In a particular model based estimation for suppression of pulse reverberation noise, said means for simulation provides estimates of at least one of the nonlinear propagation delays and pulse form distortions of the propagating HF pulses up to the 1st scatterer, and said means for estimation further comprises means for obtaining estimates of parameters relating to the relative reflection coefficients of strong reflectors as a function of depth along the beam from the measured HF receive signal, and means for obtaining estimates of noise correction delays and speckle corrections in a process that includes combining in a mathematical model for pulse reverberation noise, said nonlinear propagation delays and pulse form distortions obtained through simulations, and said estimates of parameters relating to relative reflection coefficients obtained from the measured HF receive signal.
To provide combined suppression of pulse reverberation noise and enhancement of nonlinear scattering components in the HF receive signal, said transmit means comprises means for transmitting at least 3 pulse complexes with different LF pulses towards the object, and said correction means comprises correcting and combining the HF receive signals to form at least two noise suppressed HF signals, and means for further processing of said at least two noise suppressed HF signals including at least one of

    
    
        i) delay corrections using correction delays in a fast time (depth-time), and
        ii) speckle corrections using speckle corrections in the fast time,

to form at least two corrected intermediate HF signals, and means for combining said corrected intermediate HF signals to suppress linear scattering components to enhance nonlinear scattering components in the HF receive signals.

    
    





4. SUMMARY OF THE DRAWINGS
FIG. 1 Shows Prior Art dual band pulse complexes for suppression of multiple scattering noise and/or 1st order scattering to show nonlinear scattering with pulse wave measurements or imaging, and
FIG. 2 Shows a classification of different types of multiple scattering noise that is found in a typical ultrasound image of an organ, and
FIG. 3 Illustrates coordinates and volumes for calculating the received 1st and 3rd order scattered signals, and
FIG. 4 Shows and illustration of image reconstruction from multi component transmit of broad beams, and
FIG. 5 Shows example plane reflectors that generate strong 3rd order pulse reverberation noise, and
FIG. 6 Shows coordinates for 1st to 3rd reflectors in the fast time domain, and
FIG. 7 Shows interference between Class I and II pulse reverberation noise when Q>1, and
FIG. 8 Shows illustration of the effect on noise correction delay for nonlinear variation with fast time of the nonlinear propagation delay, and
FIG. 9 Shows a block diagram of an instrument according to the invention.



5. DETAILED DESCRIPTION OF THE INVENTION
5.1 Background of 1st and 3rd Order Scattering
Example embodiments according to the invention, are presented in the following. The methods and structure of the instrumentation are applicable to both electromagnetic (EM) and elastic (EL) waves, and to a wide range of frequencies with a wide range of applications. For EL waves one can apply the methods and instrumentation to both shear waves and compression waves, both in the subsonic, sonic, and ultrasonic frequency ranges. We do in the embodiments describe by example ultrasonic measurements or imaging, both for technical and medical applications. This presentation is meant for illustration purposes only, and by no means represents limitations of the invention, which in its broadest aspect is defined by the claims appended hereto.
The most damaging noise in pulse echo wave measurements and imaging has its origin in multiple scattering of the transmitted pulse. We refer to this noise as multiple scattering noise, or pulse reverberation noise. As the amplitude of the scattered pulse drops for each scattering, the 2nd and 3rd order scattering dominates the noise. With back scatter imaging we mainly have 3rd order scattering noise, where the 1st scatterer is inside the transmit beam, and the last (3rd) scatterer is inside the receive beam. With very wide transmit beams, like with plane wave imaging and similar, we can also observe 2nd order scattering noise. However, the major type of strong scatterers are layers of fat and muscles in the body wall, that are fairly parallel, which requires 3rd order scattering also with wide transmit beams. Bone structures and air in lungs and intestines can produce 2nd order scattering noise with synthetic multi beam imaging (e.g. plane wave imaging)
The invention presents methods and instrumentation for suppression of such multiple scattering noise, and also for measurement and imaging of nonlinear scattering of waves. Such methods and instruments are also presented in U.S. Pat. No. 8,038,616 and U.S. patent application Ser. Nos. 12/351,766, 12/500,518, and 13/213,965, where the current invention presents new and more advanced methods and instrumentation, utilizing transmission of similar dual band pulse complexes as in said applications.
An example of such a dual band transmitted pulse complex is shown as 101 in FIG. 1a, composed of a low frequency (LF) pulse (102) and a high frequency (HF) pulse (103). The HF pulse is typically shorter than 1⁄2 the period of the LF pulse, and the methods utilize the nonlinear manipulation of the object material parameters (e.g. object elasticity) by LF pulse as observed by the co-propagating HF pulse. One typically transmits at least two pulse complexes with differences in the LF pulse, and processes HF receive signals picked up from one or both of scattered and transmitted HF pulses of the at least two transmitted pulse complexes, to suppress the pulse reverberation noise and/or measure or image nonlinear scattering from the object.
In FIG. 1a is for example shown two pulse complexes (101, 104) with opposite polarities of the LF pulse, shown as 102 versus 105. The same is shown in FIG. 1b with the two polarities (108, 110) of the LF pulse. In FIG. 1a the HF pulse 103 is found at the crest of the LF pulse 102 with a temporal pulse length, TpH, so short in relation to the LF pulse oscillation period, TL, that it mainly observes a constant manipulation of the material parameters by the LF pulse observed by the co-propagating HF pulse, and hence both the HF pulse wave propagation velocity and the HF pulse scattering from spatial variations in the parameters. When switching polarity of the LF pulse to 105 so that the HF pulse 106 is found at the trough of the LF pulse, the HF pulse observes a different manipulation of the propagation velocity and scattering by the LF pulse. In FIG. 1b the HF pulses 107 and 109 are found at gradients of the LF pulses 108 and 110 so that the LF manipulation of the propagation velocity varies along the HF pulse, producing a pulse form distortion of the co-propagating HF pulses that accumulates with propagation distance. Due to diffraction of the LF pulse, the position of the HF pulse along the LF pulse oscillation can slide with propagation distance, indicated as the different positions 111, 112, 113 of the HF pulse along the LF pulse oscillation in FIG. 1c, providing mainly a pulse form distortion at positions 111 and 113, and a propagation advancement/delay at position 112.
For more detailed analysis of the reverberation noise we refer to FIG. 2, which in FIG. 2a schematically illustrates the situation for 1st order scattering, and in FIG. 2b-c schematically illustrates 3 different classes of 3rd order multiple scattering noise, defined as Class I-III. The Figures all show an ultrasound transducer array 200 with a front transmit/receive and reflecting surface 201. The pulse propagation path and direction is indicated with the lines and arrows 202, where the 1st scatterer is indicated by the dot 203, and in FIG. 2a with 1st order scattering, the transmitted pulse complex propagates to the 1st scatterer 203 and back to the transducer where it is transformed to an electrical signal and further processed to form measurement or image signals.
In FIG. 2b-c the backscattered signal from the 1st scatterer 203 is reflected/scattered by a 2nd scatterer 204 that can have multiple positions as indicated in the Figure. The pulse complex then propagates to a a 3rd scatterer 205 where it is reflected back to the transducer 200 where the signal is picked up and further processed to provide measurement or image signals with strong suppression of pulse reverberation nois and/or suppression of linear scattering to enhance HF receive signal components from nonlinear scattering. Due to the multiple positions of the 2nd scatterers the two 1st and 3rd scatterers 203 and 205 then generates a tail of pulse reverberation noise 206 following the deepest of the 1st and 3rd scatterers. 207 shows for illustrative example a low echogenic region, for example in medicine the cross section of a vessel or a cyst, with a scattering surface 208 that we want to image with high suppression of pulse reverberation noise. The object has so low internal scattering that the pulse reverberation noise 206 generated in the different classes produces disturbances in the definition of the scattering surface 208.
To obtain visible, disturbing reverberation noise in the image, the 1st-3rd scatterers must be of a certain strength, where the 1st scatterer must be inside the transmit beam, and the 3rd scatterer must be inside the receive beam. In medical ultrasound applications this is often found by fat layers in the body wall, while in technical applications one can encounter many different structures, depending on the application. In ultrasound applications, the strongest 2nd scatterer is often the ultrasound transducer array surface itself, as this can have a reflection coefficient that is  ̃10 dB or more higher than the reflection coefficient from other soft tissue structures. The pulse reverberation noise that involves the transducer surface as the 2nd scatterer, is therefore particularly strong indicated by 209. The definition of Class I-III pulse reverberation noise is in more detail
Class I (FIG. 2.1b) is the situation where the 1st scatterer (203) is closest to the transducer at fast time (depth time) of t1=2z1/c, where z1 is the depth of the 1st scatterer in the 3rd order scattering sequence of scatterers, and c is the elastic wave propagation velocity. For Class I pulse reverberations we have t1<t/2 and t=2z/c is the fast time (depth time) for the reception of the 1st order back scattered signal from a scatterer 703 at depth z in FIG. 2a. The 2nd scatterer (204) is at the depth z2 and fast time t2=2z2/c. The distance between the 3rd scatterer 205 and the noise point, which in the Figure coincides with the object surface 208, is the same as the distance d=t1−t2 between the 1st and the 2nd scatterers. With varying positions of the 2nd scatterer 204, we get a tail 206 of pulse reverberation noise. When the transducer surface 201 is the strongest 2nd scatterer, we get particularly strong noise at 209.
Class II (FIG. 2.1c) Class II pulse reverberation noise has the same three scatterers as the Class I noise, with 1st and 3rd scatterers interchanged. Class I and II pulse reverberation noise hence always co-exist, and for Class II noise we have a fast time for the 1st scatterer 203 t1>t/2=z/c. The reasoning for the tail of pulse reverberation noise 206 is the same as that for Class I, where a strongly reflecting transducer surface gives the noise 209. When the 2nd scatterer is the ultrasound transducer array itself, we can define as follows:

    Class I reverberations: 1st scatterer z1<z/2 and 3rd scatterer z3>z/2, z3=z−z1 
    Class II reverberations: 1st scatterer z3<z/2 and 3rd scatterer z1>z/2, z1=z−z3 


Harmonic imaging will mainly suppress Class I pulse reverberation noise where z1 is close to the transducer, where we in the following will show that methods according to this invention suppresses Class I and II pulse reverberation noise combined, regardless of the value of z1, as the nonlinear propagation delay and pulse form distortion of the combined Class I and II noise is quite different from that for the 1st oreder scattering
Class III (FIG. 2.1d) This is a situation where Class I and Class II merges, i.e. the 1st and the 3rd scatterers are at the same depth, and the 1st scatterer is close to z, i.e. all 3 scatterers are close to z. The nonlinear propagation delay and pulse form distortion for the Class III is therefore similar to that for 1st order scattering at the depth the noise is observed, and it is then difficult to suppress Class III noise relative to 1st order scattering. When Class III noise co-exists with Class I and II noise and interferes at the same depth, the model based estimation of correction delays and speckle corrections for the noise is able to suppress Class I, II and III noise combined.
We shall first start by analyzing the situation with linear wave propagation.
5.2 Signal Models for 1st and 3rd Order Scattering in a Linear Material
A. Models for 1st Order Scattering
We first analyze the HF receive signal from the 1st order scattered wave components within the realm of pressure independent elasticity of the object, i.e. linear elasticity. With reference to FIG. 3 we get the linearly transmitted field as [1]




  
    
      
        
          
        
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              r
              t
            
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where w is the angular frequency, k=w/c is the wave number with c as the wave propagation velocity, Pt(w) is the Fourier transform of the transmitted pressure waveform at the transducer surface, tt(r0,r1) is the transmit delays for focusing the transmit beam at rt in a homogeneous material, G(r0,r1;w) is the Greens function for wave propagation from a source at r0 to a field point at r1 in the linear, heterogeneous material (i.e. spatial variation in wave propagation velocity), wt(r0) is an apodization function, and Habc(r0,w;rt) is an aberration correction filter for spatial variations in the wave propagation velocity of the heterogeneous material for focusing at rt. In a homogeneous material where the wave propagation velocity parameters are constant in space, we have Habc(r0,w;rt)=1. If Habc(r0,w;rt) is set equal to unity in a heterogeneous material, the beam focusing will be sub-optimal, but the theory presented below will still be valid.
The contribution to the 1st order scattered field from scatterer density u(r1) of the infinitesimal volume element d3r1 at location r1 in R1 in the object (See FIG. 3) is [1]

dPs1(r,w;r1)=−k2Pt(w)ikG(r,r1;w)Ht(r1,w;rt)u(r1)d3r1  (2)

For back scattering we have u(r1)=s1(r1)−g(r1), where s1 and g are the relative spatial variation of the material compressibility and mass density [1]. The total 1st ordered scattered field is then found by integration of dPs1(r,w;r1) over r1 across R1. The 1st order received signal for transducer element or sub aperture located at r on the receiver transducer

dS1(r,w;r1)=−k2U(w)G(r,r1;w)Ht(r1,w;rt)u(r1)d3r1  (3)

where U(w)=Hrt(w)ikPt(w) is the received pulse for the element at r from a unit plane reflector [1]. Beam forming this signal over the array with receive beam focus at rr, we get the following received 1st order scattered signal from u(r1) as [BA]




  
    
      
        
          
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            ⁢
            
              
            
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                  ,
                  
                    
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                )
              
            
          
          =
          
            
              r
              r
            
            -
            
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                    _
                  
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        (
        4
        )
      
    
  




where r4 is the integration coordinate over the receive array surface Sr, and tr(r4,rr) is the beam forming delays for the receiver beam focused at the receiver beam focus rr, calculated on the assumption of constant propagation velocity c in the tissue, wr(r4) is an apodization function, and Habc(r4,w;rr) is the aberration correction filter that compensates for material heterogeneity to focus the receive beam at rr. As for the transmit beam, we have Habc(r4,w;rr)=1 for the homogeneous material. Inserting Eq.(3) we can express the received 1st order scattered signal as




  
    
      
        
          
        
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where Hr(r1,w;rr) is the aberration corrected receive beam at the field point r1 when the beam is focused at rr.
B. Models for 3rd Order Scattering
Multiple scattering is produced because the 1st order scattered field in Eq.(2) is in turn scattered from a scatterer at r2 in R2 to produce a 2nd order scattered field




  
    
      
        
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                s
                ⁢
                
                  
                
                ⁢
                2
              
            
            ⁡
            
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                  _
                
                ,
                
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                  ;
                  
                    
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              )
            
          
        
        =
        
          
            ∫
            
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              2
            
          
          ⁢
          
            
          
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                3
              
              ⁢
              
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                2
              
            
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              ⁡
              
                (
                
                  
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                    _
                  
                  ,
                  
                    
                      
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                        _
                      
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              ⁡
              
                (
                
                  
                    
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                      _
                    
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                )
              
            
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                    s
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
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                        _
                      
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where s(r2,w) typically is either a strong volume scatterer distribution u(r2), for example structured fatty layers, where we have

s(r2,w)=−k2u(r2)  (7)

or reflection from a layer surface or the transducer surface as

s(r2,w)=ik2R(r2,w)d(SR(r2))  (8)

where R(r2,w) is the reflection coefficient of the reflecting surface SR(r2)=0. In technical applications one often have strongly reflecting surfaces at a distance from the transducer, for example a surface of a metal layer, which can be modeled as Eq.(8) where SR(r2)=0 now defines the layer surface. Strong volume scatterers are modeled as in Eq.(7).
The 2nd order scattered field is then scattered a 3rd time from the volume scatterer distribution u(r3) and produces a received signal for transducer element or sub aperture located at r4 on the receiver transducer

dS3(r4,w;r1,r3)=k2Hrt(r4,w)2G(r4,r3;w)dPs2(r3,w;r1)u(r3)  (9)

where Hrt(w) is defined in relation to Eq.(3). Introducing the expression for Ps1(r2,w) and Ps2(r3,w) and from Eqs.(2,6) we can express the received 3rd order reverberation element signal as




  
    
      
        
          
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              4
            
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            ⁡
            
              (
              
                
                  
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                  ;
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              )
            
          
          =
          
            
              ∫
              
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                2
              
            
            ⁢
            
              
            
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                  3
                
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                  2
                
              
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                ⁡
                
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                        _
                      
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              ⁢
              
                s
                ⁡
                
                  (
                  
                    
                      
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                        _
                      
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              ⁢
              
                G
                ⁡
                
                  (
                  
                    
                      
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                        _
                      
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                  )
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where R2 is the region of the second scatterers U(w)=Hrt(w)ikPt(w) is the received pulse for the element at r for a reflected wave front conformal to the array surface, defined in relation to Eq.(3). Beam forming this signal over the array with receive beam focus at rr, we get the following received reverberation beam signal for the reverberations as




  
    
      
        
          
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                Y
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              ⁡
              
                (
                
                  
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                    ;
                    
                      
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                        _
                      
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                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              ∫
              
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                r
              
            
            ⁢
            
              
            
            ⁢
            
              
                
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                  2
                
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                  r
                  4
                
              
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                  ⁢
                  
                    
                  
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                      r
                    
                    ⁡
                    
                      (
                      
                        
                          
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                            _
                          
                          4
                        
                        ,
                        
                          
                            r
                            _
                          
                          r
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  w
                  r
                
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    4
                  
                  )
                
              
              ⁢
              
                
                  H
                  abc
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      4
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        r
                      
                    
                  
                  )
                
              
              ⁢
              
                d
                
                  
                    S
                    3
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        4
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      ,
                      
                        
                          r
                          _
                        
                        3
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            c
            ⁢
            
              
            
            ⁢
            
              
                t
                r
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    4
                  
                  ,
                  
                    
                      r
                      _
                    
                    r
                  
                
                )
              
            
          
          =
          
            
              r
              r
            
            -
            
              
              
                
                  
                    r
                    _
                  
                  4
                
                -
                
                  
                    r
                    _
                  
                  r
                
              
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where r4 is the integration coordinate over the receive array surface Sr, and the other functions are as defined in relation to Eqs.(4,5). Inserting Eq.(10) we can express the 3rd order reverberation signal as

dY3(w;r1,r3)=k4U)(w)Hr(r3,w;rr)u(r3)d3r3Hrev(r3,r1;w)Ht(r1w;rt)u(r1)d3r1  (12)

where Hr(r3,w;rr) is the aberration corrected receive beam at the field point r3 when the beam is focused at rr as defined in Eq.(5)
5.3 Nonlinear Propagation and Scattering
A. Wave Equation for 2nd Order Nonlinear Elasticity
Nonlinear material parameters influences both the wave propagation velocity and the scattering from spatial variations in the nonlinear parameters. For acoustic waves in fluids and solids, the bulk elasticity can generally be approximated to the 2nd order in the pressure, i.e. the volume compression dV of a small volume DV, is related to the pressure p




  
    
      
        
          
            d
            ⁢
            
              
            
            ⁢
            V
          
          
            D
            ⁢
            
              
            
            ⁢
            V
          
        
        =
        
          
            -
            
              y
              _
            
          
          =
          
            
              
                (
                
                  1
                  -
                  
                    
                      
                        b
                        p
                      
                      ⁡
                      
                        (
                        
                          r
                          _
                        
                        )
                      
                    
                    ⁢
                    p
                  
                
                )
              
              ⁢
              
                k
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
              ⁢
              p
            
            +
            
              
                h
                ab
              
              ⁢
              
                
                  □
                  ~
                
                t
              
              ⁢
              
                k
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
              ⁢
              p
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




where y(r,t) is the acoustic particle displacement vector, p(r,t) is the acoustic pressure, k(r) is the linear bulk compressibility of the material, and bp(r)=bn(r)k(r) where bn(r)=1+B(r)/2A(r) is a nonlinearity parameter of the bulk compressibility, and hab(r,t) is a convolution kernel that represents absorption of wave energy to heat. The 2nd order approximation holds for soft tissue in medical imaging, fluids and polymers, and also rocks that show special high nonlinear bulk elasticity due to their granular micro-structure. Gases generally show stronger nonlinear elasticity, where higher order terms in the pressure often might be included. Micro gas-bubbles with diameter much lower than the acoustic wavelength in fluids, also show a resonant compression response to an oscillating pressure, which is discussed above. When the material parameters can be approximated to 2nd order in the wave fields, we can for example for ultrasonic pressure waves formulate a wave equation that includes nonlinear forward propagation and scattering phenomena as [2,3]




  
    
      
        
          
            
              
                
                  
                
                2
              
              ⁢
              
                p
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    ,
                    t
                  
                  )
                
              
              -
              
                
                  1
                  
                    
                      c
                      0
                      2
                    
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                
                ⁢
                
                  
                    
                      ¶
                      2
                    
                    ⁢
                    
                      p
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ¶
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      2
                    
                  
                
              
            
            
              }
              
                
                  (
                  1
                  )
                
                ⁢
                Linear
                ⁢
                
                  
                
                ⁢
                propagation
              
            
          
          +
          
            
              
                
                  
                    b
                    p
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
                
                  
                    c
                    0
                    2
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
              
              ⁢
              
                
                  
                    ¶
                    2
                  
                  ⁢
                  
                    
                      p
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                    2
                  
                
                
                  ¶
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    2
                  
                
              
            
            
              }
              
                
                  (
                  2
                  )
                
                ⁢
                Nonlinear
                ⁢
                
                  
                
                ⁢
                propagation
              
            
          
          -
          
            
              
                
                  
                    h
                    ab
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
                ⁢
                
                  ~
                  t
                
                ⁢
                
                  1
                  
                    c
                    0
                    2
                  
                
              
              ⁢
              
                
                  
                    ¶
                    2
                  
                  ⁢
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ,
                        t
                      
                      )
                    
                  
                
                
                  ¶
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    2
                  
                
              
            
            
              }
              
                
                  (
                  3
                  )
                
                ⁢
                Absorption
              
            
          
        
        =
        
          
            
              
                
                  
                    
                      s
                      l
                    
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                  
                    
                      c
                      0
                      2
                    
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                
                ⁢
                
                  
                    
                      ¶
                      2
                    
                    ⁢
                    
                      p
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ¶
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      2
                    
                  
                
              
              +
              
                (
                
                  
                    g
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ,
                        t
                      
                      )
                    
                  
                
                )
              
            
            
              }
              
                
                  (
                  4
                  )
                
                ⁢
                Linear
                ⁢
                
                  
                
                ⁢
                scattering
                ⁢
                
                  
                
                ⁢
                source
                ⁢
                
                  
                
                ⁢
                terms
              
            
          
          -
          
            
              
                
                  
                    s
                    n
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
                
                  
                    c
                    0
                    2
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
              
              ⁢
              
                
                  
                    ¶
                    2
                  
                  ⁢
                  
                    
                      p
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                    2
                  
                
                
                  ¶
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    2
                  
                
              
            
            
              }
              
                
                  
                    (
                    5
                    )
                  
                  ⁢
                  Nonlinear
                  ⁢
                  
                    
                  
                  ⁢
                  scattering
                
                ⁢
                
                  

                
                ⁢
                
                  source
                  ⁢
                  
                    
                  
                  ⁢
                  term
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




where c0(r) is the linear wave propagation velocity for low field amplitudes, s1(r) and g(r) are linear scattering parameters given by the relative spatial variation of the material compressibility and mass density, and sn(r) is a nonlinear scattering parameter. The left side propagation parameters vary with r on a scale>approximately the wavelength, while the right side scattering parameters vary with r on a scale<approximately the wave length. A similar equation for shear waves and electromagnetic waves can be formulated that represents similar nonlinear propagation and local scattering phenomena for the shear and EM waves.
B. Nonlinear Interaction in Dual Band Pulse Complexes
The different terms of Eq.(14) have different effects on the wave propagation and scattering: The linear propagation terms (1) guide the forward spatial propagation of the incident wave without addition of new frequency components. The linear scattering source terms (4) produce local scattering of the incident wave without addition of new frequency components, i.e. the linearly scattered wave has the same frequency components as the incident wave, with an amplitude modification  ̃w2 produced by 2nd order differentiation in the scattering terms.
For materials with adequately high nonlinearity in the material parameters relative to the wave field amplitude, the nonlinearity affects both the propagation and local scattering of the wave. A slow variation (close to constant on scale> ̃wave length) of the nonlinear parameters give a value to bp(r) that provides a nonlinear forward propagation distortion of the incident waves that accumulates in amplitude with propagation distance through term (2). A rapid variation of the nonlinear material parameters (on scale< ̃wavelength) gives a value to sn(r) that produces a local nonlinear scattering of the incident wave through term (5).
We study the situation where the total incident wave is the sum of the LF and HF pulses, i.e. p(r,t)=pL(r,t)+pH(r,t). The nonlinear propagation and scattering are in this 2nd order approximation both given by




  
    
      
        
          ∼
          
            
              p
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  ,
                  t
                
                )
              
            
            2
          
        
        =
        
          
            
              (
              
                
                  
                    p
                    L
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
                +
                
                  
                    p
                    H
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
              
              )
            
            2
          
          =
          
            
              
                
                  
                    p
                    L
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
                2
              
              
                }
                
                  
                    nonlin
                    ⁢
                    
                      
                    
                    ⁢
                    self
                  
                  ⁢
                  
                    

                  
                  ⁢
                  distortion
                
              
            
            +
            
              
                2
                ⁢
                
                  
                
                ⁢
                
                  
                    p
                    L
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
                ⁢
                
                  
                    p
                    H
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
              
              
                }
                
                  nonlin
                  ⁢
                  
                    

                  
                  ⁢
                  interaction
                
              
            
            +
            
              
                
                  
                    p
                    H
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
                2
              
              
                }
                
                  
                    nonlin
                    ⁢
                    
                      
                    
                    ⁢
                    self
                  
                  ⁢
                  
                    

                  
                  ⁢
                  distortion
                
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




A multiplication of two functions in the temporal domain produces a convolution of the functions temporal Fourier transforms (i.e. temporal frequency spectra) introducing frequency components in the product of the functions that are sums and differences of the frequency components of the factors of the multiplication. The squares pL2 and pH2 then produces a self-distortion of the incident pulses that pumps energy from the incident fundamental bands into harmonic components of the fundamental bands, and is hence found as a nonlinear attenuation of the fundamental bands of both the LF and HF waves. However, the wave amplitude determines the self-distortion per wave length, and the LF pulse typically propagates only up to  ̃30 LF wavelengths into the object which allows us to neglect both nonlinear self distortion and nonlinear attenuation for the LF wave. The HF wave, there-against, propagates up to  ̃250 HF wavelengths into the object, so that both the nonlinear self-distortion an the nonlinear attenuation becomes observable. We return to these phenomena for simulations of the wave propagation in Section 5.8B.
C. Nonlinear Propagation and Scattering for Dual Band Pulse Complexes
When the temporal HF pulse length TpH is much shorter than half the period of the LF pulse, TL/2, i.e. the bandwidth of the HF pulse BH>wL/2, where wL=2p/TL is the angular frequency of the LF wave, the sum and difference frequency spectra overlaps with each other and the fundamental HF spectrum. This is the situation illustrated in FIG. 1. For the further analysis we assume that |2bp(r)pL(r,t)|=|x|<<1 which allows the approximation 1−x≈1/(1+x) and the propagation terms of the left side of Eq.(14) can for the manipulation of the HF pulse by the co-propagating LF pulse be approxiomated as




  
    
      
        
          
            
              
                
                  
                
                2
              
              ⁢
              
                
                  p
                  H
                
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    ,
                    t
                  
                  )
                
              
              -
              
                
                  1
                  
                    
                      c
                      0
                      2
                    
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                
                ⁢
                
                  
                    
                      ¶
                      2
                    
                    ⁢
                    
                      
                        p
                        H
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ¶
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      2
                    
                  
                
              
            
            
              }
              
                
                  (
                  1
                  )
                
                ⁢
                Linear
                ⁢
                
                  
                
                ⁢
                propagation
              
            
          
          +
          
            
              
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        b
                        p
                      
                      ⁡
                      
                        (
                        
                          r
                          _
                        
                        )
                      
                    
                    ⁢
                    
                      
                        p
                        L
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    
                      c
                      0
                      2
                    
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                
                ⁢
                
                  
                    
                      ¶
                      2
                    
                    ⁢
                    
                      
                        p
                        H
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          t
                        
                        )
                      
                    
                  
                  
                    ¶
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      2
                    
                  
                
              
              
                }
                
                  
                    (
                    2
                    )
                  
                  ⁢
                  Nonlinear
                  ⁢
                  
                    
                  
                  ⁢
                  propagation
                
              
            
            ⁢
            
              
            
            ⁢
            »
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              p
              H
              
              
                
                2
            
            ⁡
            
              (
              
                
                  r
                  _
                
                ,
                t
              
              )
            
          
          -
          
            
              1
              
                
                  
                    c
                    0
                    2
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
                ⁢
                
                  (
                  
                    1
                    +
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          b
                          p
                        
                        ⁡
                        
                          (
                          
                            r
                            _
                          
                          )
                        
                      
                      ⁢
                      
                        
                          p
                          L
                        
                        ⁡
                        
                          (
                          
                            
                              r
                              _
                            
                            ,
                            t
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
            ⁢
            
              
                
                  ¶
                  2
                
                ⁢
                
                  
                    p
                    H
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      t
                    
                    )
                  
                
              
              
                ¶
                ⁢
                
                  
                
                ⁢
                
                  t
                  2
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




The numerator in front of the temporal derivative in this propagation operator is the square propagation velocity, and we hence see that the LF pulse pressure pL modifies the propagation velocity for the co-propagating HF pulse pH as

c(r,pL)=√{square root over (c02(r)(1+2bp(r)pL))}{square root over (c02(r)(1+2bp(r)pL))}>>c0(r)(1+bp(r)pL)  (17)

where pL is the actual LF field variable along the co-propagating HF pulse. The orthogonal trajectories of the HF pulse wave fronts are paths of energy flow in the HF pulse propagation. The propagation lag of the HF pulse along the orthogonal trajectories of the HF pulse wave-fronts, G(r), can then be approximated as




  
    
      
        
          
            t
            ⁡
            
              (
              
                r
                _
              
              )
            
          
          =
          
            
              
                
                  ∫
                  
                    G
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      d
                      s
                    
                    
                      c
                      ⁡
                      
                        (
                        
                          
                            s
                            _
                          
                          ,
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                p
                                L
                              
                              ⁡
                              
                                (
                                
                                  s
                                  _
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  »
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        G
                        ⁡
                        
                          (
                          
                            r
                            _
                          
                          )
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        s
                      
                      
                        c
                        0
                      
                    
                  
                
              
              -
              
                p
                ⁢
                
                  
                    ∫
                    
                      G
                      ⁡
                      
                        (
                        
                          r
                          _
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        d
                        s
                      
                      
                        c
                        0
                      
                    
                    ⁢
                    
                      
                        b
                        p
                      
                      ⁡
                      
                        (
                        
                          s
                          _
                        
                        )
                      
                    
                    ⁢
                    
                      
                        p
                        L
                      
                      ⁡
                      
                        (
                        
                          s
                          _
                        
                        )
                      
                    
                  
                
              
            
            =
            
              
                
                  t
                  0
                
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
              +
              
                p
                ⁢
                
                  
                
                ⁢
                
                  t
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              t
              0
            
            ⁡
            
              (
              
                r
                _
              
              )
            
          
          =
          
            
              ∫
              
                G
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                d
                s
              
              
                
                  c
                  0
                
                ⁡
                
                  (
                  
                    s
                    _
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            p
            ⁢
            
              
            
            ⁢
            
              t
              ⁡
              
                (
                
                  r
                  _
                
                )
              
            
          
          =
          
            
              -
              p
            
            ⁢
            
              
                ∫
                
                  G
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    d
                    s
                  
                  
                    
                      c
                      0
                    
                    ⁡
                    
                      (
                      
                        s
                        _
                      
                      )
                    
                  
                
                ⁢
                
                  
                    b
                    p
                  
                  ⁡
                  
                    (
                    
                      s
                      _
                    
                    )
                  
                
                ⁢
                
                  
                    p
                    L
                  
                  ⁡
                  
                    (
                    
                      s
                      _
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




where p pL(s) is the LF pressure at the center of gravity at s of the co-propagating HF pulse, and p is a scaling/polarity variable for the LF pulse that is extracted to more clearly show results of scaling the amplitude and changing the polarity of the LF pulse. The propagation lag without manipulation of the propagation velocity by the LF pulse is t0(r). t(r) is the added nonlinear propagation delay produced by the nonlinear manipulation of the propagation velocity for the HF pulse by the LF pressure pL(s) . We note that when the 1st scattering/reflection occurs, the LF pressure pL drops so much that the LF modification of the propagation velocity is negligible for the scattered HF wave. This means that we only get contribution in the integral for t(r) up to the 1st scattering, an effect that we will use to suppress multiple scattered waves in the received signal and to enhance the nonlinear scattering components.
Variations of the LF pressure along the co-propagating HF pulse produces a variation of the propagation velocity along the HF pulse, that in turn produces a nonlinear pulse from distortion of the HF pulse that accumulates with propagation distance. When BHF>wLF/2 the nonlinear propagation distortion can be described by a filter, as shown below in Eqs.(19,23).
In summary, the nonlinear propagation term, i.e. term (2) in Eq.(14) produces:
i) A nonlinear propagation delay t(r) produced by the LF pressure at the center of gravity of the co-propagating HF pulse according to Eq.(18), and is accumulated up to the 1st scattering, where the amplitude of the LF pulse drops so much that the effect of the LF pulse can be neglected after this point.
ii) A nonlinear propagation distortion of the HF pulse produced by the variation of the LF pulse field along the co-propagating HF pulse that accumulates with propagation distance up to the 1st scattering, where the amplitude of the LF pulse drops so much that the effect of the LF pulse can be neglected after this point.
iii) A nonlinear self-distortion of the HF pulse which up to the 1st scattering transfers energy from the fundamental HF band to harmonic components of the fundamental HF band, and is hence found as a nonlinear attenuation of the fundamental band of the HF pulse.
When BHF>wLF/2 the transmitted HF pulse field Pt(r1,w;rt) hence observes a nonlinear propagation distortion as described by point i)-iii) above and takes the form Ptp(r1,w;rt)




  
    
      
        
          
            
              P
              tp
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  w
                  ;
                  
                    
                      r
                      _
                    
                    t
                  
                
              
              )
            
          
          =
          
            
              
                V
                p
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
            ⁢
            
              
                P
                t
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              V
              p
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  w
                  ;
                  
                    
                      r
                      _
                    
                    t
                  
                
              
              )
            
          
          =
          
            
              
                
                  P
                  tp
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
              
                
                  P
                  t
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
            
            =
            
              
                
                  
                    V
                    ~
                  
                  p
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        1
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




where the subscript p designates the amplitude/polarity/phase of the LF pulse. Pt(r1,w;rt) is given in Eq.(1), and Vp includes all nonlinear forward propagation distortion, where the linear phase component of Vp is separated out as the nonlinear propagation delay pt(r1) up to the point r1. {tilde over (V)}p hence represents the nonlinear pulse form distortion of the HF pulse by the co-propagating LF pulse, and also the nonlinear attenuation produced by the nonlinear self distortion of the HF pulse.
The nonlinear scattering term (5) in Eq.(14) can for the nonlinear interaction term be approximated within the same regime as




  
    
      
        
          
            
              
                
                  s
                  n
                
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
              
                
                  c
                  0
                  2
                
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
            
            ⁢
            
              
                
                  ¶
                  2
                
                ⁢
                
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ,
                        t
                      
                      )
                    
                  
                  2
                
              
              
                ¶
                ⁢
                
                  
                
                ⁢
                
                  t
                  2
                
              
            
          
          
            }
            
              
                
                  (
                  5
                  )
                
                ⁢
                Nonlinear
                ⁢
                
                  
                
                ⁢
                scattering
              
              ⁢
              
                

              
              ⁢
              
                source
                ⁢
                
                  
                
                ⁢
                term
              
            
          
        
        ⁢
        
          
        
        ⁢
        »
        ⁢
        
          
        
        ⁢
        
          
            2
            ⁢
            
              
            
            ⁢
            
              
                s
                n
              
              ⁡
              
                (
                
                  r
                  _
                
                )
              
            
            ⁢
            p
            ⁢
            
              
            
            ⁢
            
              
                p
                L
              
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  ,
                  t
                
                )
              
            
          
          
            
              c
              0
              2
            
            ⁡
            
              (
              
                r
                _
              
              )
            
          
        
        ⁢
        
          
            
              ¶
              2
            
            ⁢
            
              
                p
                H
              
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  ,
                  t
                
                )
              
            
          
          
            ¶
            ⁢
            
              
            
            ⁢
            
              t
              2
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




We hence see that when the temporal pulse length of the HF pulse TpH<TL/2, the local LF pulse pressure at the co-propagating HF pulse exerts an amplitude modulation of the scattered wave, proportional to p 2sn(r)pL(r,t). For the nonlinear self-distortion, term (5) produces harmonic components in the scattered wave, and is not noticed in the fundamental HF band.
The effect of nonlinearity of material parameters on the scattered signal, can hence be split into two groups:
Group A originates from the linear scattering, i.e. term (4) of Eq.(14), of the forward, accumulative nonlinear propagation distortion of the incident wave, i.e. combination of term (2) and term (4) in Eq.(14), which is split into a nonlinear propagation delay according to Eq.(18) (i above), a nonlinear pulse form distortion (ii above), and a nonlinear attenuation of the HF pulse (iii above), and Group B originates directly in the local nonlinear scattering, i.e. term (5), of the original frequency components in the incident wave, i.e. interaction between term (1), the nonlinear propagation delay of term (2) and term (5), where the local LF pulse pressure at the co-propagating HF pulse exerts an amplitude modulation of the scattered wave, proportional to p 2sn(r)pL(r,t).
There is also a Group C found as local nonlinear scattering from term (5) of the forward accumulative nonlinear propagation distortion components in the incident wave, i.e. interaction between the pulse distortion component of term (2) and term (5) in Eq.(14), but typical nonlinear material parameters are so low that this group is negligible.
5.4 Signal Models for Nonlinear Propagation and Scattering
A. Models for 1st Order Scattering
The 1st order linear and nonlinear scattered wave that hits the receiver array at r is then from Eq.(2,19,20)

dPl1(r,w;r1)=−k2G(r,r1;w)Ptp(r1,w;rt)u(r1)d3r1

dPn1(r,w;r1)=−k2G(r,r1;w)Ptp(r1,w;rt)2pL(r1)sn(r1)d3r1  (21)

The received signal from the 1st order linear and nonlinear scattering of the forward nonlinear distorted pressure pulse for the focused and aberration corrected receive beam is hence

dYl1(w;r1)=−k2Hr(r1,w;rr)Up(r1,w;rt)u(r1)d3r1

dYn1(w;r1)=−k2Hr(r1,w;rr)Up(r1,w;rt)2pL(r1)sn(r1)d3r1

Up(r1,w;rt)=Hrt(w)Ptp(r1,w;rt)=Vp(r1,w;rt)U(w)Ht(r1,w;rt)  (22)

where we have assumed that the pressure to signal transfer function Hrt(w) of the transducer elements, is the same for all receiver array elements. Introducing the scaling variable p for the LF field, pL(r1)fi p pL(r1), the total 1st order scattered wave takes the form




  
    
      
        
          
        
        ⁢
        
          
            
              d
              
                
                  Y
                  
                    1
                    ⁢
                    
                      
                    
                    ⁢
                    p
                  
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  )
                
              
            
            =
            
              
                
                  V
                  p
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
              ⁢
              
                {
                
                  
                    d
                    
                      
                        X
                        l
                      
                      ⁡
                      
                        (
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      pdX
                      n
                    
                    ⁡
                    
                      (
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      )
                    
                  
                
                }
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              d
              
                
                  X
                  l
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  )
                
              
            
            =
            
              
                -
                
                  k
                  2
                
              
              ⁢
              
                U
                ⁡
                
                  (
                  w
                  )
                
              
              ⁢
              
                
                  H
                  r
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        r
                      
                    
                  
                  )
                
              
              ⁢
              
                
                  H
                  t
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
              ⁢
              
                u
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    1
                  
                  )
                
              
              ⁢
              
                
                  d
                  3
                
                ⁢
                
                  r
                  1
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              d
              
                
                  X
                  n
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  )
                
              
            
            =
            
              
                -
                
                  k
                  2
                
              
              ⁢
              
                U
                ⁡
                
                  (
                  w
                  )
                
              
              ⁢
              
                
                  H
                  r
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        r
                      
                    
                  
                  )
                
              
              ⁢
              
                
                  H
                  t
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              
                
                  s
                  n
                
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    1
                  
                  )
                
              
              ⁢
              
                
                  p
                  L
                
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    1
                  
                  )
                
              
              ⁢
              
                
                  d
                  3
                
                ⁢
                
                  r
                  1
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                V
                p
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
            =
            
            ⁢
            
              
                
                  
                    U
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        1
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
                
                  
                    U
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                  ⁢
                  
                    
                      H
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          1
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
              =
              
                
                  
                    V
                    ~
                  
                  p
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    w
                    ⁢
                    
                      
                    
                    ⁢
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          1
                        
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




The magnitude of the nonlinear scattering component is generally much lower than the linear scattering component. To estimate this nonlinear scattering component one could typically transmit two pulse complexes with opposite polarities of the LF pulse, giving the HF receive signals

dY+(w;r1)=V+(r1,w;rt)dXl(w;r1)+V+(r1,w;rt)dXn(w;r1)

dY−(w;r1)=V−(r1,w;rt)dXl(w;r1)−V−(r1,w;rt)dXn(w;r1)  (24)

The nonlinear scattering component is then obtained as

2dXn(w;r1)=V+(r1,w;rt)−1dY+(w;r1)−V−(r1,w;rt)−1dY−(w;r1)  (25)

where V−(r1,w;rt)−1={tilde over (V)}−(r1,w;rt)−1eiwpt(r1) are the inverse filters of V−(r1,w;rt) in Eq.(24). The inverse filter is hence composed of a delay correction ∓pt(r1), and a pulse distortion correction {tilde over (V)}−(r1,w;rt)−1.
The signal is received as a function of the fast time (or depth-time) t, where we assign a depth z in the image as z=ct/2. When one have a distribution of scatterers, one must sum over the scatterer coordinate r1, and the LF pressure can in addition to the non-linear propagation delay produce a LF pressure dependent speckle in the received signal that we discuss in Section 5.7. One then needs to generalize the correction for nonlinear propagation delay to correction with a correction delay that is an average nonlinear propagation delay, and generalize the pulse form correction to a speckle correction, as further discussed in Sections 5.7 and 5.8.
C. Models for 3rd Order Scattering
For 3rd order multiple scattering, the nonlinear propagation delay and pulse form distortion will be a function of the first scatterer coordinate r1,3, as both the LF and HF pulse amplitudes drop so much in this first scattering that we get linear propagation after the 1st scattering. The nonlinear effect with dual band pulse complexes as above, hence modifies the 3rd order scattered signals in Eq.(12) to

dYIp(w;r1,r3)=k4Hr(r3,w;rr)u(r3)d3r3Hrev(r3,r1;w)Up(r1,w;rt)u(r1)d3r1

dYIIp(w;r1,r3)=k4Hr(r1,w;rr)u(r1)d3r1Hrev(r1,r3;w)Up(r3,w;rt)u(r3)d3r3  (26)

where dYIp(w;r1,r3) and dYIIp(w;r1,r3) are Class I and II multiple scattering noise with non-zero LF pulse, Up is the HF receive pulse from a plane reflector. The sum of the Class I and Class II pulse reverberation noise can then be written as




  
    
      
        
          d
          
            
              Y
              p
            
            ⁡
            
              (
              
                
                  w
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
                ,
                
                  
                    r
                    _
                  
                  3
                
              
              )
            
          
        
        =
        
          
            
              d
              
                
                  Y
                  Ip
                
                ⁡
                
                  (
                  
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      3
                    
                  
                  )
                
              
            
            +
            
              d
              
                
                  Y
                  IIp
                
                ⁡
                
                  (
                  
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      3
                    
                  
                  )
                
              
            
          
          =
          
            
              {
              
                
                  
                    
                      
                        H
                        r
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            3
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        U
                        p
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        H
                        r
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              r
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        H
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            3
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      U
                      ⁡
                      
                        (
                        w
                        )
                      
                    
                  
                
                +
                
                  
                    
                      U
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          3
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                  
                    
                      
                        H
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            3
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      U
                      ⁡
                      
                        (
                        w
                        )
                      
                    
                  
                
              
              }
            
            ⁢
            
              d
              
                
                  Y
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    II
                  
                
                ⁡
                
                  (
                  
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      3
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




where dY3II(w;r1,r3) is Class II multiple scattering with zero LF pulse, i.e. linear wave propagation as given from Eq.(12) with r3>r1

dY3II(w;r1,r3)=k4U(w)Hr(r1,w;rr)u(r1)d3r1Hrev(r1,r3;w)Ht(r3,w;rt)u(r3)d3r3  (28)

Comparing with Eqs. (19,22,23) we can rewrite Eq.(27) as




  
    
      
        
          d
          
            
              Y
              p
            
            ⁡
            
              (
              
                
                  w
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
                ,
                
                  
                    r
                    _
                  
                  3
                
              
              )
            
          
        
        =
        
          
            
              
                K
                p
              
              ⁡
              
                (
                
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
            ⁢
            
              d
              
                
                  Y
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    II
                  
                
                ⁡
                
                  (
                  
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      3
                    
                  
                  )
                
              
            
            ⁢
            
              

            
            ⁢
            
              
                K
                p
              
              ⁡
              
                (
                
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              
                
                  Q
                  ⁡
                  
                    (
                    
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      ,
                      
                        
                          r
                          _
                        
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  
                    V
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        1
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  
                    V
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        3
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
                ⁢
                
                  

                
                ⁢
                
                  Q
                  ⁡
                  
                    (
                    
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      ,
                      
                        
                          r
                          _
                        
                        3
                      
                    
                    )
                  
                
              
            
            =
            
              
                
                  
                    
                      H
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          1
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      H
                      r
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          3
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            r
                          
                        
                      
                      )
                    
                  
                
                
                  
                    
                      H
                      r
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          1
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            r
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      H
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          3
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
              =
              
                
                  
                    
                      
                        Q
                        ⋒
                      
                      ⁡
                      
                        (
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            3
                          
                        
                        )
                      
                    
                    
                      
                        Q
                        ⋒
                      
                      ⁡
                      
                        (
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        )
                      
                    
                  
                  ⁢
                  
                    

                  
                  ⁢
                  
                    
                      Q
                      ⋒
                    
                    ⁡
                    
                      (
                      
                        w
                        ;
                        
                          r
                          _
                        
                      
                      )
                    
                  
                
                =
                
                  
                    
                      H
                      r
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            r
                          
                        
                      
                      )
                    
                  
                  
                    
                      H
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




The nonlinear modification by the LF pulse of the HF pulse Up from the linear form UHt, is hence given by Vp that is defined in Eqs.(19,22,23), while Q represents the difference between Class I and Class II pulse reverberations in the linear propagation regime. Note that when r1 fi r3, we have Q fi 1 and Vp3 fi Vp1. The linear phase component of Vp represents the nonlinear propagation delay pt(ri) up to the point ri, i=1,3, of the 1st scatterer where the amplitude of both the LF and HF pulses drops so much that one have basically linear wave propagation after this point. The nonlinear pulse form distortion due to variations of the LF pulse along the co-propagating HF pulse and also the nonlinear attenuation of the fundamental band of the HF pulse due to self distortion of the HF pulse, is included in {tilde over (V)}p of Eqs. (19,23). This nonlinear attenuation reduces the magnitude of {tilde over (V)}p(r,w;rt) with depth up to the 1st scatterer, hence reducing |{tilde over (V)}p(r3,w;rt)| compared to |{tilde over (V)}p(r1,w;rt)| in Eq. (29)
In Eq. (29) the total pulse reverberation noise with nonzero LF pulse is hence a filtered version of the Class II reverberation with zero LF pulse, dY3II(w;r1,r3), where the filter depends on the LF pulse in relation to the co-propagating HF pulse and has the form




  
    
      
        
          
        
        ⁢
        
          
            
              
                
                  
                    
                      K
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        ,
                        
                          
                            r
                            _
                          
                          3
                        
                      
                      )
                    
                  
                  =
                  
                  ⁢
                  
                    
                      
                        Q
                        ⁡
                        
                          (
                          
                            
                              w
                              ;
                              
                                
                                  r
                                  _
                                
                                1
                              
                            
                            ,
                            
                              
                                r
                                _
                              
                              3
                            
                          
                          )
                        
                      
                      ⁢
                      
                        
                          V
                          ~
                        
                        p
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          w
                          ⁢
                          
                            
                          
                          ⁢
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          
                            t
                            ⁡
                            
                              (
                              
                                
                                  r
                                  _
                                
                                1
                              
                              )
                            
                          
                        
                      
                    
                    +
                  
                
              
            
            
              
                
                  
                  ⁢
                  
                    
                      
                        V
                        ~
                      
                      p
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          3
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        p
                        ⁢
                        
                          
                        
                        ⁢
                        
                          t
                          ⁡
                          
                            (
                            
                              
                                r
                                _
                              
                              3
                            
                            )
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  =
                  
                  ⁢
                  
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            p
                            ⁡
                            
                              [
                              
                                
                                  t
                                  ⁡
                                  
                                    (
                                    
                                      
                                        r
                                        _
                                      
                                      1
                                    
                                    )
                                  
                                
                                +
                                
                                  t
                                  ⁡
                                  
                                    (
                                    
                                      
                                        r
                                        _
                                      
                                      3
                                    
                                    )
                                  
                                
                              
                              ]
                            
                          
                          /
                          2
                        
                      
                    
                    ⁢
                    
                      
                        K
                        ~
                      
                      p
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        ,
                        
                          
                            r
                            _
                          
                          3
                        
                      
                      )
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                
                  K
                  ~
                
                p
              
              ⁡
              
                (
                
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
            =
            
              
                
                  Q
                  ⁡
                  
                    (
                    
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      ,
                      
                        
                          r
                          _
                        
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      V
                      ~
                    
                    p
                  
                  ⁡
                  
                    (
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      
                        
                          i
                          ⁢
                          
                            
                          
                          ⁢
                          w
                          ⁢
                          
                            
                          
                          ⁢
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          
                            t
                            ⁡
                            
                              (
                              
                                
                                  r
                                  _
                                
                                3
                              
                              )
                            
                          
                        
                        -
                        
                          t
                          ⁡
                          
                            (
                            
                              
                                r
                                _
                              
                              1
                            
                            )
                          
                        
                      
                      ]
                    
                    /
                    2
                  
                
              
              +
              
                
                  
                    
                      V
                      ~
                    
                    p
                  
                  ⁡
                  
                    (
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        3
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    w
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        p
                        ⁡
                        
                          [
                          
                            
                              t
                              ⁡
                              
                                (
                                
                                  
                                    r
                                    _
                                  
                                  3
                                
                                )
                              
                            
                            -
                            
                              t
                              ⁡
                              
                                (
                                
                                  
                                    r
                                    _
                                  
                                  1
                                
                                )
                              
                            
                          
                          ]
                        
                      
                      /
                      2
                    
                  
                
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




The effect of this filter for a distribution of scatterers is discussed in Section 5.7B below.
5.5 Synthetic Transmit and Receive Beams
A. 1st Order Scattering with Multi-Component Transmit
With synthetic transmit and receive beams it is possible to obtain Q=1 in a large group of image points without loosing frame rate. With such methods, one transmits a set of wide transmit beams {Htj(r,w); j=1, . . . , J } with different directions. With a linear array shown as 400 in FIG. 4, Htj(r,w) can for example be beams with close to plane wave fronts with different directions in the azimuth direction, while the elevation variation of the beams are the same as with standard fixed focusing. With two-dimensional arrays, Htj(r,w) can have close to plane wave fronts in both the azimuth and elevation directions. An example of such a plane wave front in the azimuth direction is shown in 401 of FIG. 4. The plane wave is transmitted at an angle jj to the array, and we define t=0 so that the center element transmits at t=0. For each transmit beam direction, the HF receive signal is picked up in parallel at all array elements.
The signal is received on all elements for each transmit beam that gives the following set {Yjk(w); j=1, . . . , J; k=1, . . . , K} HF receive element signals from element #k with transmit beam #j. The 1st order scattered HF receive signal is in its linear approximation given from Eq.(5) as

dYjk(w;r1)=−k2U(w)Hrk(r1,w)Htj(r1,w)u(r1)d3r  (31)

where Hrk(r1,w) is the receive beam of element #k at the location r1 (405) of a scatterer. For the image reconstruction we use the time signal of the HF receive element signal obtained as the inverse Fourier transform of Eq. (31)




  
    
      
        
          
            dy
            jk
          
          ⁡
          
            (
            
              t
              ;
              
                
                  r
                  _
                
                1
              
            
            )
          
        
        =
        
          
            c
            
              -
              2
            
          
          ⁢
          
            
              
                u
                 ̈
              
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              
                h
                rk
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  t
                
                )
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              
                h
                tj
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  t
                
                )
              
            
          
          ⁢
          
            u
            ⁡
            
              (
              
                
                  r
                  _
                
                1
              
              )
            
          
          ⁢
          
            d
            3
          
          ⁢
          
            r
            1
          
        
      
    
    
      
        (
        32
        )
      
    
  




where hrk(r1,t) is the spatial impulse response of array element #k, and htj(r1,t) is the spatial impulse response for transmit beam #j, both obtained by inverse Fourier transform of the spatial frequency responses Hrk(r1,w) and Htj(r1,w) in Eq.(31).
We reconstruct an image signal in a spatial location r (402) of the image with a synthetic receive beam and a synthetic transmit beam that are both focused at r as




  
    
      
        
          
            d
            
              z
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
                )
              
            
          
          =
          
            
              d
              
                y
                ⁡
                
                  (
                  
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      
                        r
                        /
                        c
                      
                    
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  )
                
              
            
            =
            
              
                ∑
                
                  j
                  ,
                  k
                
              
              ⁢
              
                
                  
                    a
                    k
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
                ⁢
                
                  
                    b
                    j
                  
                  ⁡
                  
                    (
                    
                      r
                      _
                    
                    )
                  
                
                ⁢
                
                  d
                  
                    
                      y
                      jk
                    
                    ⁡
                    
                      (
                      
                        
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            
                              r
                              /
                              c
                            
                          
                          -
                          
                            
                              t
                              rk
                            
                            ⁡
                            
                              (
                              
                                r
                                _
                              
                              )
                            
                          
                          -
                          
                            
                              t
                              tj
                            
                            ⁡
                            
                              (
                              
                                r
                                _
                              
                              )
                            
                          
                        
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            c
            ⁢
            
              
            
            ⁢
            
              
                t
                rk
              
              ⁡
              
                (
                
                  x
                  ,
                  z
                
                )
              
            
          
          =
          
            z
            -
            
              
                
                  
                    (
                    
                      
                        x
                        k
                      
                      -
                      x
                    
                    )
                  
                  2
                
                +
                
                  z
                  2
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            c
            ⁢
            
              
            
            ⁢
            
              
                t
                tj
              
              ⁡
              
                (
                
                  x
                  ,
                  z
                
                )
              
            
          
          =
          
            
              x
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              
                j
                j
              
            
            +
            
              z
              ⁡
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      j
                      j
                    
                  
                  -
                  1
                
                )
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




where r=|r| and trk(r) is the focusing delay for HF receive element #k (404) for focusing the receive beam centered around x onto r=xex+zez, ak(r) is a receive beam apodization function, ttj(r) is a delay for component transmit beam #j to provide a synthetic transmit beam focused at r, and bj(r) is a transmit beam apodization function. Introducing the HF receive element signals from Eq. (32), the reconstructed image signal is




  
    
      
        
          
            
              
                
                  d
                  
                    
                      z
                      1
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      )
                    
                  
                
                =
                
                ⁢
                
                  
                    c
                    
                      -
                      2
                    
                  
                  ⁢
                  
                    
                      
                        u
                         ̈
                      
                      ⁡
                      
                        (
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            /
                            c
                          
                        
                        )
                      
                    
                    ⁢
                    
                      ⊗
                      t
                    
                    ⁢
                    
                      
                        ∑
                        k
                      
                      ⁢
                      
                        
                          a
                          k
                        
                        ⁢
                        
                          
                            
                              h
                              rk
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    r
                                    _
                                  
                                  1
                                
                                ,
                                
                                  
                                    r
                                    /
                                    c
                                  
                                  -
                                  
                                    
                                      τ
                                      rl
                                    
                                    ⁡
                                    
                                      (
                                      
                                        r
                                        _
                                      
                                      )
                                    
                                  
                                
                              
                              )
                            
                          
                          ⁢
                          
                            ⊗
                            t
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                ⁢
                
                  
                    ∑
                    j
                  
                  ⁢
                  
                    
                      b
                      j
                    
                    ⁢
                    
                      
                        h
                        j
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            
                              r
                              /
                              c
                            
                            -
                            
                              
                                τ
                                tj
                              
                              ⁡
                              
                                (
                                
                                  r
                                  _
                                
                                )
                              
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      υ
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          1
                        
                        )
                      
                    
                    ⁢
                    
                      
                        d
                        3
                      
                      ⁢
                      
                        r
                        1
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                ⁢
                
                  
                    c
                    
                      -
                      2
                    
                  
                  ⁢
                  
                    
                      
                        u
                         ̈
                      
                      ⁡
                      
                        (
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            /
                            c
                          
                        
                        )
                      
                    
                    ⁢
                    
                      ⊗
                      t
                    
                    ⁢
                    
                      
                        h
                        r
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            t
                            ;
                            
                              r
                              _
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      ⊗
                      t
                    
                    ⁢
                    
                      
                        h
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            t
                            ;
                            
                              r
                              _
                            
                          
                        
                        )
                      
                    
                  
                  ⁢
                  
                    υ
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      d
                      3
                    
                    ⁢
                    
                      r
                      1
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              h
              r
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  t
                  ;
                  
                    r
                    _
                  
                
              
              )
            
          
          =
          
            
              ∑
              k
            
            ⁢
            
              
                a
                k
              
              ⁢
              
                
                  h
                  rk
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      t
                      -
                      
                        
                          τ
                          rk
                        
                        ⁡
                        
                          (
                          
                            r
                            _
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              h
              t
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  t
                  ;
                  
                    r
                    _
                  
                
              
              )
            
          
          =
          
            
              ∑
              j
            
            ⁢
            
              
                b
                j
              
              ⁢
              
                
                  h
                  tj
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      t
                      -
                      
                        
                          τ
                          tj
                        
                        ⁡
                        
                          (
                          
                            r
                            _
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




ht(r1,t;r) and hr(r1,t;r) are the spatial impulse responses of the synthetic transmit and receive beams for a linear material. Out of the total array width, Darr, one can hence operate with a synthetic receive aperture Dr determined by how many elements one sums over, and a synthetic transmit aperture determined by how many transmit beam components one sums over. With a non-zero LF pulse we get the synthetic linearly and nonlinearly scattered signals as in Eq.(23), with the synthetic transmit and receive beams of Eq.(34). We should note that the LF pressure that produces the nonlinear propagation delay and the pulse form distortion is the actual LF pressure for each transmitted component Htj(r,w).
For dual frequency pulse complexes in a nonlinear material, the HF receive signal from component transmit pulse #j at HF receive array element #k is




  
    
      
        
          
            d
            
              
                y
                jk
              
              ⁡
              
                (
                
                  t
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                2
              
            
            ⁢
            
              
                
                  h
                  rk
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    t
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  h
                  tj
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    t
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  
                    u
                     ̈
                  
                  pj
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    t
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                1
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              u
              pj
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                t
              
              )
            
          
          =
          
            
              
                v
                pj
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  t
                
                )
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              u
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              
                h
                tj
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  t
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              v
              pj
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                t
              
              )
            
          
          =
          
            
              
                v
                ~
              
              p
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  t
                  -
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          1
                        
                        )
                      
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




upj(r1,t) is the forward propagating HF pressure pulse #j convolved with the receive element impulse response, similar to Eq.(23). upj(r1,t) has undergone a nonlinear propagation delay and pulse form distortion by the co-propagating LF pulse, and also the nonlinear self distortion.
Applying the synthetic image reconstruction as in Eq. (33,34) we get




  
    
      
        
          
            d
            
              
                y
                l
              
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      /
                      c
                    
                  
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                2
              
            
            ⁢
            
              
                
                  h
                  r
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          r
                          /
                          c
                        
                      
                      ;
                      
                        r
                        _
                      
                    
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  
                    u
                     ̈
                  
                  pt
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          r
                          /
                          c
                        
                      
                      ;
                      
                        r
                        _
                      
                    
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                1
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            d
            
              
                y
                n
              
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      /
                      c
                    
                  
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                2
              
            
            ⁢
            
              
                
                  h
                  r
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          r
                          /
                          c
                        
                      
                      ;
                      
                        r
                        _
                      
                    
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  
                    u
                     ̈
                  
                  pt
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          r
                          /
                          c
                        
                      
                      ;
                      
                        r
                        _
                      
                    
                  
                  )
                
              
            
            ⁢
            2
            ⁢
            
              
            
            ⁢
            
              
                s
                n
              
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                p
                L
              
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                1
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              u
              pt
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      r
                      /
                      c
                    
                  
                  ;
                  
                    
                      r
                      _
                    
                    1
                  
                
              
              )
            
          
          =
          
            
              ∑
              j
            
            ⁢
            
              
                u
                
                  p
                  ,
                  j
                
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    t
                    -
                    
                      
                        t
                        tj
                      
                      ⁡
                      
                        (
                        
                          r
                          _
                        
                        )
                      
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




This expression has the same form as for the focused transmit and receive beams in Eq.(22), and We hence also get the same form for the total 1st order HF receive signal with transmitted dual frequency pulse complexes in a nonlinear material as in Eq.(23). Signal processing for the multi component transmit beam imaging hence takes the same form as for focused transmit and receive beams.
B. 3rd Order Scattering with Multi-Component
The image reconstruction is hence based on the assumption of 1st order scattering. However, each HF receive element signal will contain multiple scattering noise, where the 3rd order scattering noise given by the formula in Eq.(26) is the most dominating component. The HF receive element signal for element #k and component transmit beam #j for the 3rd order scattered pulse reverberations of Class I and II with nonlinear propagation of the transmit pulse up to the 1st scatterer, is hence




  
    
      
        
          
            d
            
              
                y
                
                  Ip
                  ,
                  jk
                
              
              ⁡
              
                (
                
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                4
              
            
            ⁢
            
              
                h
                rk
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    3
                  
                  ,
                  t
                
                )
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  3
                
                )
              
            
            ⁢
            
              
                
                  d
                  3
                
                ⁢
                
                  r
                  3
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  h
                  rev
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      3
                    
                    ,
                    
                      
                        
                          r
                          _
                        
                        1
                      
                      ;
                      t
                    
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  u
                  
                    p
                    ,
                    j
                  
                  
                    (
                    4
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    t
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                1
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            d
            
              
                y
                
                  IIp
                  ,
                  jk
                
              
              ⁡
              
                (
                
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                4
              
            
            ⁢
            
              
                h
                rk
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  t
                
                )
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                
                  d
                  3
                
                ⁢
                
                  r
                  1
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  h
                  rev
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      3
                    
                    ,
                    
                      
                        
                          r
                          _
                        
                        1
                      
                      ;
                      t
                    
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  u
                  
                    p
                    ,
                    j
                  
                  
                    (
                    4
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    t
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  3
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                3
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            d
            
              
                y
                
                  
                    3
                    ⁢
                    
                      
                    
                    ⁢
                    p
                  
                  ,
                  jk
                
              
              ⁡
              
                (
                
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              d
              
                
                  y
                  
                    Ip
                    ,
                    jk
                  
                
                ⁡
                
                  (
                  
                    
                      t
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      3
                    
                  
                  )
                
              
            
            +
            
              d
              
                
                  y
                  
                    IIp
                    ,
                    jk
                  
                
                ⁡
                
                  (
                  
                    
                      t
                      ;
                      
                        
                          r
                          _
                        
                        1
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      3
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




where upj(4) is the transmitted nonlinear HF pulse component #j at r1, scattered twice from a point scatterer and received through the transducer element. Reconstructing an image signal at r according to the lines of Eqs.(33,34), the reconstructed signal will contain a 3rd order pulse reverberation component according to




  
    
      
        
          
            
              
                d
                
                  
                    y
                    
                      1
                      ⁢
                      
                        
                      
                      ⁢
                      p
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          
                            r
                            /
                            c
                          
                        
                        ;
                        
                          
                            r
                            _
                          
                          1
                        
                      
                      ,
                      
                        
                          r
                          _
                        
                        3
                      
                    
                    )
                  
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    j
                    ,
                    k
                  
                
                ⁢
                
                  d
                  
                    
                      y
                      
                        p
                        ,
                        jk
                      
                    
                    ⁡
                    
                      (
                      
                        
                          
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              
                                r
                                /
                                c
                              
                            
                            -
                            
                              
                                τ
                                rk
                              
                              ⁡
                              
                                (
                                
                                  r
                                  _
                                
                                )
                              
                            
                            -
                            
                              
                                τ
                                tj
                              
                              ⁡
                              
                                (
                                
                                  r
                                  _
                                
                                )
                              
                            
                          
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        ,
                        
                          
                            r
                            _
                          
                          3
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  d
                  
                    
                      y
                      Ip
                    
                    ⁡
                    
                      (
                      
                        
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            
                              r
                              /
                              c
                            
                          
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        ,
                        
                          
                            r
                            _
                          
                          3
                        
                      
                      )
                    
                  
                
                +
                
                  d
                  
                    
                      y
                      IIp
                    
                    ⁡
                    
                      (
                      
                        
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            
                              r
                              /
                              c
                            
                          
                          ;
                          
                            
                              r
                              _
                            
                            1
                          
                        
                        ,
                        
                          
                            r
                            _
                          
                          3
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




where the Class I and Class II components are




  
    
      
        
          
            d
            
              
                y
                Ip
              
              ⁡
              
                (
                
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                4
              
            
            ⁢
            
              
                h
                r
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    3
                  
                  ,
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  3
                
                )
              
            
            ⁢
            
              
                
                  d
                  3
                
                ⁢
                
                  r
                  3
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  h
                  rev
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      3
                    
                    ,
                    
                      
                        
                          r
                          _
                        
                        1
                      
                      ;
                      t
                    
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  u
                  p
                  
                    (
                    4
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      t
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                1
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            d
            
              
                y
                IIp
              
              ⁡
              
                (
                
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      1
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    3
                  
                
                )
              
            
          
          =
          
            
              c
              
                -
                4
              
            
            ⁢
            
              
                h
                r
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      r
                    
                  
                
                )
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  1
                
                )
              
            
            ⁢
            
              
                
                  d
                  3
                
                ⁢
                
                  r
                  1
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  h
                  rev
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      3
                    
                    ,
                    
                      
                        
                          r
                          _
                        
                        1
                      
                      ;
                      t
                    
                  
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  u
                  p
                  
                    (
                    4
                    )
                  
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      3
                    
                    ,
                    
                      t
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
            
            ⁢
            
              u
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  3
                
                )
              
            
            ⁢
            
              
                d
                3
              
              ⁢
              
                r
                3
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              u
              p
            
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  1
                
                ,
                
                  t
                  ;
                  
                    r
                    _
                  
                
              
              )
            
          
          =
          
            
              ∑
              j
            
            ⁢
            
              
                b
                k
              
              ⁢
              
                
                  u
                  pj
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      1
                    
                    ,
                    
                      t
                      -
                      
                        
                          t
                          tj
                        
                        ⁡
                        
                          (
                          
                            r
                            _
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




The pulse reverberation noise of the reconstructed HF receive signals hence has the same form as for the focused transmit and receive beams as in Eq.(26). By matching the number of transmit beam components and apodization weights that participates to the synthetic transmit beam to the aperture and apodization weights of the synthetic receive beam we can obtain

Hr(ri,w;r)=Ht(ri,w;r)i=1,3 {circumflex over (Q)}(w;r)=1 Q(w;r1,r3)=1  (40)

in Eq.(29), and the difference between the Class I and II pulse reverberation noise is found in the nonlinear pulse form distortion produced by the LF pulse and the nonlinear self distortion attenuation of the fundamental HF band, produced by the HF pulse itself, both given by {tilde over (V)}p(r1,w) and {tilde over (V)}p(r3,w). The estimation schemes of Sections 5.7 to suppress pulse reverberation noise and estimate nonlinear scattering, are hence fully applicable to the reconstructed synthetic signals above.
C. Synthetic Dynamic Focusing with Transversal Filter
We choose the z-axis along the beam axis with z=0 at the center of the array aperture. This gives a coordinate representation r=zez+r^=z+r^ where ez is the unit vector along the beam axis and r^ is the coordinate in the plane orthogonal to the z-axis, i.e. z r^=0. The image is obtained by lateral scanning of the beam, where the 1st order scattered image signal can from Eq.(23) be modeled as a convolution in the transversal plane




  
    
      
        
          d
          
            
              Y
              
                l
                ⁢
                
                  
                
                ⁢
                1
              
            
            ⁡
            
              (
              
                
                  z
                  i
                
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
                ,
                w
              
              )
            
          
        
        =
        
          
            -
            
              d
              
                zk
                2
              
            
          
          ⁢
          
            
              V
              p
            
            ⁡
            
              (
              
                
                  z
                  i
                
                ,
                w
              
              )
            
          
          ⁢
          
            U
            ⁡
            
              (
              w
              )
            
          
          ⁢
          
            
              ∫
              
                Z
                i
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  d
                  2
                
                ⁢
                
                  r
                  
                    ⋀
                    1
                  
                
              
              ⁢
              
                
                  H
                  b
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      -
                      
                        
                          r
                          _
                        
                        
                          ⋀
                          1
                        
                      
                    
                    ,
                    w
                  
                  )
                
              
              ⁢
              
                u
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      
                        ⋀
                        1
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




where Zi is an interval around zi of scatterers that all interfere at zi, and

Hb(z,r^,w)=Hr(z,r^,w)  (42)

is the composite beam spatial frequency response as defined in detail in Appendix B. We seek a whitening filter in the transversal coordinates with an impulse response w that minimizes the lateral point spread function for each depth z as




  
    
      
        
          
            
              
                d
                
                  
                    
                      Y
                      ^
                    
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        i
                      
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
              =
              
              ⁢
              
                ∫
                
                  
                    
                      d
                      2
                    
                    ⁢
                    
                      r
                      
                        ⋀
                        2
                      
                    
                  
                  ⁢
                  
                    w
                    ⁡
                    
                      (
                      
                        
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          -
                          
                            
                              r
                              _
                            
                            
                              ⋀
                              2
                            
                          
                        
                        ,
                        
                          w
                          ;
                          
                            z
                            i
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    
                      
                        Y
                        
                          l
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          
                            z
                            i
                          
                          ,
                          
                            
                              r
                              _
                            
                            
                              ⋀
                              2
                            
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  -
                  
                    d
                    
                      zk
                      2
                    
                  
                
                ⁢
                
                  
                    V
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        i
                      
                      ,
                      w
                    
                    )
                  
                
                ⁢
                
                  U
                  ⁡
                  
                    (
                    w
                    )
                  
                
                ⁢
                
                  
                    ∫
                    
                      Z
                      i
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        d
                        2
                      
                      ⁢
                      
                        r
                        
                          ⋀
                          1
                        
                      
                    
                    ⁢
                    
                      H
                      w
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  (
                  
                    z
                    ,
                    
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      -
                      
                        
                          r
                          _
                        
                        
                          ⋀
                          1
                        
                      
                    
                    ,
                    
                      w
                      ;
                      
                        z
                        i
                      
                    
                  
                  )
                
                ⁢
                
                  u
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        
                          ⋀
                          1
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




where we have defined

Hw(z,{circumflex over (r)}^,w;zi)=∫d2r^2w(r^−r^2,w;zi)Hb(z,r^2,w)

{tilde over (H)}w(z,k^,w;zi)=W(k^,w;zi){tilde over (H)}b(z,k^,w)  (44)

where the tilde indicates Fourier transform in the transversal coordinate. We can now determine W as a whitening, inverse filter as




  
    
      
        
          W
          ⁡
          
            (
            
              
                
                  k
                  _
                
                ⋀
              
              ,
              
                w
                ;
                
                  z
                  i
                
              
            
            )
          
        
        =
        
          
            1
            
              
                
                  H
                  ~
                
                b
              
              ⁡
              
                (
                
                  
                    z
                    i
                  
                  ,
                  
                    
                      k
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          ⁢
          
            1
            
              1
              +
              
                
                  m
                  N
                
                /
                
                  
                    
                    
                      
                        
                          H
                          ~
                        
                        b
                      
                      ⁡
                      
                        (
                        
                          
                            z
                            i
                          
                          ,
                          
                            
                              k
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                    
                  
                  2
                
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




where mN: 10−1−10−3 is a noise parameter that avoids large amplitude gains in W when the amplitude of |{tilde over (H)}b| gets low. We note in particular that the following phase relation

−W(k^,w;zi)=−−{tilde over (H)}b(zi,k^,w)  (46)

which implies that the whitening filter renders a zero phase Hw(zi,r^,w;zi), that gives the narrowest point spread function at zi for a given amplitude spectrum, as analyzed in detail in Appendix B. The increased gain of W when |{tilde over (H)}b| gets low, reduces the width of the beam main lobe, but increases the side lobes, so there is a point in not making mN to low. In many situations it can be advantageous to just use the phase filtering as

W(k^,w;zi)=e−i−{tilde over (H)}b(zi,k^,w)  (47)

to avoid raising the side lobes in the filtering process. We hence see that with this method it is possible to obtain a synthetic depth variable focus for the 1st order scattering as in Appendix B. Comparing with Eqs.(26-29) for 3rd order scattering we see that when Hr=Ht, this statement also holds for the 3rd order scattering. To obtain Q=1 it is important that the aperture and apodization are the same for the transmit and the receive beams. In the imaging process one can however operate with a dynamic focusing of the receive beam, and use filtered focusing of the transmit beam as described in Appendix B. The transversal filtering in Eqs.(43-47) is two-dimensional, i.e. both in the azimuth and the elevation direction. This requires that we have a 3D data set. However, the methods are directly applicable to transversal filtering only in the azimuth direction of a 2D image. The object must be stationary for the time it requires to cover a scatterer with the whole transmit beam cross section, and there are some practical limitations on the synthetic F-numbers that is possible to achieve, due to lateral sampling density in the image. It can be advantageous to use a couple of depth zones with actual beam foci within each zone, and obtain synthetic depth variable focusing within each zone through transversal filtering as described here.
5.6 Signal Models for Plane Reflectors
A. Structure of Beams and Reflecting Planes
Strong pulse reverberations are generally caused by reflecting layers in front of the object, such as fatty layers in the body wall. Plane layers normal to the array axis can be modeled as u(r)=R(z) where z is the coordinate along the beam axis, normal to the fat layers. We continue with reference to FIG. 5 where 500 shows the radiating surface of an ultrasound transducer array. 501 stylistically (i.e. diffraction neglected) shows the geometric boundaries of a transmit beam with transmit aperture Dt (502) and focus at zt. 503 stylistically shows the geometric boundaries of a receive beam with receive aperture Dr (504), and focus at zr. We assume that the receive beam is dynamically focused at the image depth zr=ct/2, where t is the echo arrival time, referred to as the fast time. Slow time is the time-scale on which the object changes, where such changes are observed by repeated transmission of pulse complexes. Due to the dynamic focusing, the receive beam can have a lower F-number FNr=zr/Dr, than the transmit beam which in this example has a fixed focus and a transmit F-number FNt=zt/Dt. The Figure shows 4 reflecting planes 505, 506, 507, and 508, at the positions z1, z2, z3, and z with reflection coefficients R(z1), R(z2;w), R(z3), and R(z). As the 2nd plane 506 also can be the transducer surface, we have allowed R(z2;w) to be frequency dependent. We split the coordinates as r=zez+r^, where r^ is the transversal coordinate in the planes and z is the coordinate along the beam axis. At each reflection, we integrate the signals r^ for each plane, where the scattering densities u(r)=R(z) and sn(r)=Rn(z) are independent of r^.
B. Signal Models for 1st Order Scattering
With dynamic focusing of the receive beam, the 1st order scattering occurs from the plane 508 in the focus of the receive beam. We modify Eq.(23) for the linearly and nonlinearly scattered signals as

dYlp(w;z,r^)=dz R(z)U(w)Vp(z,r^,w;rt)ikHr(z,r^,w;rt)ikHr(z,r^,w;rr)ikHt(z,r^,w;rt)a)

dYnp(w;z,r^)=dz 2pL(z,r^)Rn(z)U(w)·Vp(z,r^,w;rt)ikHr(z,r^,w;rr)ikHt(z,r^,w;rt)b)  (48)

To obtain the signal model for the 1st order scattered signal from the plane 508 at zr, we integrate across the transversal coordinate r^ of the plane. With reference to Eq.(B3) of Appendix B, it is natural to define the following expressions by the transversal integration




  
    
      
        
          
            
              
                
                  H
                  tr
                
                ⁡
                
                  (
                  
                    
                      z
                      _
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      r
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∫
                  
                    R
                    2
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      d
                      2
                    
                    ⁢
                    
                      r
                      ⋀
                    
                  
                  ⁢
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      kH
                      r
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            r
                          
                        
                      
                      )
                    
                  
                  ⁢
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      kH
                      t
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∫
                  
                    S
                    r
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      d
                      2
                    
                    ⁢
                    
                      r
                      4
                    
                  
                  ⁢
                  
                    
                      w
                      r
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        4
                      
                      )
                    
                  
                  ⁢
                  
                    
                      H
                      abc
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          4
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            r
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    w
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        t
                        r
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            4
                          
                          ,
                          
                            
                              r
                              _
                            
                            r
                          
                        
                        )
                      
                    
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                
                  
                    kH
                    t
                  
                  ⁡
                  
                    (
                    
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          
                            z
                            _
                          
                        
                        +
                        
                          
                            r
                            _
                          
                          4
                        
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        49
        )
      
    
  




This expression allows us to write the received 1st order scattered signal from a layer at depth z as

dYlp(w;z)dz=dz R(z)U(w) Vp(z,w;rt)Htr(z,w;rt,rr)a)

dYnp(w;z)dz=dz 2 pL(z)Rn(z)U(w) Vp(z,w;rt)Htr(z,w;rt,rr)b)  (50)

where we have defined




  
    
      
        
          
            
              
                
                  
                    
                  
                  ⁢
                  a
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        
                          
                            
                              V
                              _
                            
                            p
                          
                          ⁡
                          
                            (
                            
                              
                                z
                                _
                              
                              ,
                              
                                w
                                ;
                                
                                  
                                    r
                                    _
                                  
                                  t
                                
                              
                            
                            )
                          
                        
                        =
                        
                        ⁢
                        
                          
                            
                              
                                V
                                ⋒
                              
                              p
                            
                            ⁡
                            
                              (
                              
                                z
                                ,
                                w
                              
                              )
                            
                          
                          ⁢
                          
                            e
                            
                              
                                -
                                i
                              
                              ⁢
                              
                                
                              
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                              ⁢
                              
                                
                              
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                              ⁢
                              
                                
                              
                              ⁢
                              
                                t
                                ⁡
                                
                                  (
                                  z
                                  )
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        =
                        
                        ⁢
                        
                          
                            
                              
                            
                            ⁢
                            
                              
                                
                                  
                                    
                                      ∫
                                      
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                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        
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                                          2
                                        
                                        ⁢
                                        
                                          r
                                          ⋀
                                        
                                      
                                      ⁢
                                      
                                        
                                          V
                                          p
                                        
                                        ⁡
                                        
                                          (
                                          
                                            z
                                            ,
                                            
                                              
                                                r
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                                              w
                                              ;
                                              
                                                
                                                  r
                                                  _
                                                
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                                          )
                                        
                                      
                                    
                                  
                                
                              
                              
                                
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        kH
                                        r
                                      
                                      ⁡
                                      
                                        (
                                        
                                          z
                                          ,
                                          
                                            
                                              r
                                              _
                                            
                                            ⋀
                                          
                                          ,
                                          
                                            w
                                            ;
                                            
                                              
                                                r
                                                _
                                              
                                              r
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              
                                
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      
                                        kH
                                        t
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            z
                                            1
                                          
                                          ,
                                          
                                            
                                              r
                                              _
                                            
                                            ⋀
                                          
                                          ,
                                          
                                            w
                                            ;
                                            
                                              
                                                r
                                                _
                                              
                                              t
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                          
                          
                            
                              H
                              tr
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    z
                                    _
                                  
                                  ;
                                  
                                    
                                      r
                                      _
                                    
                                    t
                                  
                                
                                ,
                                
                                  
                                    r
                                    _
                                  
                                  r
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              ⁢
              b
            
            )
          
          ⁢
          
            
          
          ⁢
          
            
              
                p
                _
              
              L
            
            ⁡
            
              (
              z
              )
            
          
        
        =
        
          
            
              
                
                  
                    ∫
                    
                      R
                      2
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        d
                        2
                      
                      ⁢
                      
                        r
                        ⋀
                      
                    
                    ⁢
                    
                      
                        p
                        L
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                    ⁢
                    
                      
                        V
                        p
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                  
                
              
            
            
              
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      kH
                      r
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            r
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      kH
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          z
                          1
                        
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    ∫
                    
                      R
                      2
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      
                        d
                        2
                      
                      ⁢
                      
                        r
                        ⋀
                      
                    
                    ⁢
                    
                      
                        V
                        p
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                    ⁢
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        kH
                        r
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              r
                            
                          
                        
                        )
                      
                    
                  
                
              
            
            
              
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  k
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      H
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          z
                          1
                        
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        51
        )
      
    
  




We note that in the linear material we Vp=1 and Rn=0. With zero LF pulse in the nonlinear material (pL=0), we will still have an accumulative nonlinear self-distortion of the LF pulse which pumps power from the fundamental HF band into harmonic components of the fundamental HF band. This is seen as a nonlinear attenuation of the fundamental HF-band, that can be included in Vp. However, the nonlinear self-scattering is low and found as components in the harmonic bands.
We should note that if the local nonlinear propagation delay pt(r) varies across the reflector within the receive beam, this variation will produce a LF pressure dependent modification of the pulse form received from the plane, and is in this model included in Vp(z,w), while the average of pt(r) across the receive beam is included in pt(z). This pulse form modification is similar to the LF pressure dependent speckle discussed in Section 5.7A below, with a difference that the plane reflector is not random across the plane. The phenomenon comes in addition to the local pulse form distortion of the transmitted pulses, given by the variation of the LF pulse along the co-propagating HF pulse, accumulated along the pulse propagation path up to r and included in {tilde over (V)}p(r,w) of Eq.(19). To minimize this pulse form modification from the integrated nonlinear propagation delay, one must design HF and LF beams where the nonlinear propagation delay is close to constant across the HF receive beam focus cross section for each depth z. We should note that it is the magnitude variation of pt(r) that produces this pulse form modification, not the relative variation. This implies that this pulse form modification increases with depth together with pt(z). Synthetic transmit beam with for example plane wave imaging do here have an advantage, as the variation of the LF pulse transversal to the wave front of the component HF transmit pulse can be made low, and so also the LF beam diffraction with low sliding between the LF phase and HF pulse position with propagation depth.
The fast time representation of the received 1st order scattered signal is found as the inverse Fourier transform of Eq. (50)




  
    
      
        
          d
          
            
              y
              lp
            
            ⁡
            
              (
              
                t
                ;
                z
              
              )
            
          
        
        =
        
          
            d
            z
          
          ⁢
          
            
          
          ⁢
          
            R
            ⁡
            
              (
              z
              )
            
          
          ⁢
          
            
              u
              ⁡
              
                (
                t
                )
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              
                
                  v
                  ⋒
                
                p
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    t
                    -
                    
                      
                        p
                        k
                      
                      ⁢
                      
                        t
                        ⁡
                        
                          (
                          z
                          )
                        
                      
                    
                  
                
                )
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              
                h
                tr
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    t
                    ;
                    
                      z
                      t
                    
                  
                  ,
                  
                    z
                    r
                  
                
                )
              
            
          
        
      
    
    
      
        (
        52
        )
      
    
  




where each term is the inverse Fourier transform of the corresponding terms in capital letters. In the practical situation there will be strong reflecting planes in the neighborhood of zr. Due to the received pulse length from these planes, a certain thickness Dz=cTpH/2 of reflecting planes around zr, will interfere and give a LF pressure dependent speckle in the received signal. TpH is the HF temporal pulse length. The received pulse length is given by the convolved temporal lengths of the u(t), the transmit and receive beam impulse responses htr(z,t;rt,rr), and the pulse form distortion filter {circumflex over (v)}p(z,t). As we only observe the location of strong reflecting planes, it is convenient to define their positions from the fast time arrival of the signal as z=ct1/2, that gives




  
    
      
        
          
            
              
                
                  
                    a
                    )
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    
                      
                        y
                        
                          l
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁡
                      
                        (
                        
                          t
                          ;
                          
                            t
                            1
                          
                        
                        )
                      
                    
                  
                
                =
                
                  
                    d
                    
                      t
                      1
                    
                  
                  ⁢
                  
                    R
                    ⁡
                    
                      (
                      
                        t
                        1
                      
                      )
                    
                  
                  ⁢
                  
                    
                      u
                      ⁡
                      
                        (
                        t
                        )
                      
                    
                    ⁢
                    
                      ~
                      t
                    
                    ⁢
                    
                      
                        
                          v
                          ⋒
                        
                        p
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            1
                          
                          ,
                          
                            t
                            -
                            
                              
                                p
                                k
                              
                              ⁢
                              
                                t
                                ⁡
                                
                                  (
                                  
                                    t
                                    1
                                  
                                  )
                                
                              
                            
                          
                        
                        )
                      
                    
                    ⁢
                    
                      ~
                      t
                    
                    ⁢
                    
                      
                        h
                        tr
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            1
                          
                          ,
                          
                            t
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                          ,
                          
                            
                              r
                              _
                            
                            r
                          
                        
                        )
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              b
            
            )
          
          ⁢
          
            
          
          ⁢
          
            d
            
              
                y
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              ⁡
              
                (
                
                  t
                  ;
                  
                    t
                    1
                  
                
                )
              
            
          
        
        =
        
          
            d
            
              t
              1
            
          
          ⁢
          2
          ⁢
          
            
          
          ⁢
          
            
              
                p
                _
              
              L
            
            ⁡
            
              (
              
                t
                1
              
              )
            
          
          ⁢
          
            
              R
              n
            
            ⁡
            
              (
              
                t
                1
              
              )
            
          
          ⁢
          
            
              
                u
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  
                    v
                    ⋒
                  
                  p
                
                ⁡
                
                  (
                  
                    
                      t
                      1
                    
                    ,
                    
                      t
                      -
                      
                        
                          p
                          k
                        
                        ⁢
                        
                          t
                          ⁡
                          
                            (
                            
                              t
                              1
                            
                            )
                          
                        
                      
                    
                  
                  )
                
              
            
            ⁢
            
              ~
              t
            
            ⁢
            
              
                h
                tr
              
              ⁡
              
                (
                
                  
                    t
                    1
                  
                  ,
                  
                    t
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    r
                  
                
                )
              
            
          
        
      
    
    
      
        (
        53
        )
      
    
  




C. Signal Models for 3rd Order Scattering
For the Class I type pulse reverberation noise, the transmitted pulse is first reflected from the 1st plane 505 at z1, then re-reflected from the 2nd plane 506 at z2 and finally reflected at the 3rd plane 507 at z3 back to the transducer for reception. As we integrate across the planes, Hrev will interact with one of the transmit and receive beams, Ht or Hr, to provide extended transmit and receive beams mirrored around the plane 2nd reflector. As the effect of the LF pulse on the HF pulse is negligible after the first reflection, it is convenient to extend the receive beam from the 1st reflection and through two more reflections back to the receiver transducers. We then note the following:
For Class I pulse reverberation noise, the receive beam propagates from plane 505 at z1, back to 506 at z2, forward to 507 at z3, and back to the transducer, a total propagation distance of zrec=z1−z2+z3−z2+z3=z1+2(z3−z2). From FIG. 5 we have z−z1=z3−z2, where z=ct/2, where t is the fast time arrival of the center of the 3rd order scattered pulse. Combed we get zrec=2z−z1. The total propagation distance for both transmit and receive is z1+zrec=2z, i.e. independent of the position of the planes at z1 and z2 because z3=z−(z1−z2). The receiver observes the signal reflected from the plane 505 at z1=z−(z3−z2) as if it occurs at a plane 509 symmetrically around the dynamic receive focus at position z4=z+(z3−z2)=2z−z1.
For Class II pulse reverberation noise, the receive beam propagates from plane 507 at z3, back to 506 at z2, forward to 505 at z1 and back to the transducer, a total propagation distance of zrec=z3−z2+z1−z2+z1=z3+2(z1−z2). From above we have z1−z2=z−z3=which gives zrec=2z−z3, i.e. the receiver observes the signal reflected from the plane 505 at z3=z−(z1−z2) as if it occurs at a plane 510 symmetrically around the dynamic receive focus at position z5=z+(z1−z2).
From Eq. (26) we get the following expressions for the Class I and II pulse reverberation noise

dYIp(w;z,r1,z2)=dz1dz2R(z1)R(z−(z1−z2))U(w)·Vp(z1,r^1,w;rt)ikHr(2z−z1,r^1,w;rr)ikHt(z1,r^1,w;rt)a)

dYIIp(w;z,z2,r3)=dz2dz3R(z−(z3−z2))R(z2;w)R(z3)U(w)·Vp(z3,r^3,w;rt)ikHr(2z−z3,r^3,w;rr)ikHt(z3,r^3,w;rt)b)  (54)

R(z2;w) is allowed to be a function of frequency that would occur when the transducer array is the 2nd reflector for z2=0. For the transversal integration across the first reflecting plane, the receiver beam is not by far so limiting to max contributing r^j, j=1,3, as for the 1st order scattering, because z1+2(z3−z2)=2−z1 and z3+2(z1−z2)=2z−z3 is generally far outside the focus of the receive beam as seen in FIG. 5. For the pulse reverberation noise it is therefore not as useful to separate out of the nonlinear manipulation of the HF pulse propagation by the co-propagating LF pulse, as for the 1st order scattering in Eqs.(49,50), and we get a different type of averaging for Vp. Integration across the transversal coordinate of Eq.(54a,b) then gives the total 3rd order pulse reverberation noise as from Appendix B




  
    
      
        
          
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We note that for a fully linear material, Vp=1, and in this case Class I and Class II reverberations are equal for the plane reflectors, even if the transmit and receive beams are different which would make Class I and Class II reverberations different for non-planar reflectors as discussed in relation to Eq.(29). For the nonlinear material, we note as discussed in relation to Eq.(55) that even for zero LF will nonlinear self distortion of the HF pulse produce a nonlinear attenuation of the HF fundamental frequency band, as power is pumped up into harmonic components of the HF band where the absorption is higher. This nonlinear attenuation can be included into a Vp≠1, also for zero LF.
Summing over the planes along the whole depth down to z1=z, we can include Class I-III reverberations into the following integral




  
    
      
        
          
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        (
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        )
      
    
  




where ZpH(z) is the spatial pulse length of the received HF pulse from depth z. Class III noise is obtained when z3=z1 in the integral. We shall use these expressions as a basis for model based estimation of correction delays and speckle corrections in Section 5.8C.
From Eqs. (49,53,55,56) we get the total HF receive signal in and interval Ti ̃>TpH(ti) around the fast time ti=2zi/c, where TpH(ti) is the temporal pulse length of the received HF pulse at ti, as

Yi(w;p)=e−iwpt(ti){circumflex over (V)}pi(w;ti)Xli(w)+e−iwpt(ti){circumflex over (V)}ni(w;ti)Xni(w;p)+e−iwptni{circumflex over (N)}i(w;p)  (57)

where {circumflex over (N)}i(w;p)={circumflex over (N)}(w,cti/2;p), ptni=ptn(cti/2), Xli is the linearly scattered signal and Xni is the nonlinearly scattered signal that is given with the general nonlinear dependency on p that includes nonlinear scattering from resonant scatterers like acoustic micro bubbles. For parametric nonlinear scattering, as ultrasound scattering from fluids and tissue, we can approximate Xni(w;p)>>pXni(w). {circumflex over (V)}pi and {circumflex over (V)}ni are the average pulse form distortion across the HF receive beam defined in Eq.(51). For non-resonant scatterers we have {circumflex over (V)}ni>>{circumflex over (V)}pi.
5.7 Challenges and Methods for Suppression of Multiple or Linear Scattering
A. Delay and Speckle for 1st Order Scattering
Eqs.(23, 36,43) gives signal models for 1st order scattering from a point scatterer u(r1)d3r1, freely positioned in 3D space at r1. In this situation Eq.(25) shows how the linear scattering components can be suppressed to enhance the signal from the nonlinear scattering at r1, assuming the nonlinear pulse distortion filter Vp is known. However, we do generally have a distribution of scatterers which complicates the suppression of the linear scattering. The HF receive signal is picked up versus fast time t, where the depth coordinate in the image along the receive beam axis is calculated as z=ct/2.
The 1st order HF receive signal at each fast time point t=2z/c is generally composed of several overlapping radio frequency (RF) HF pulses from different scatterers that interfere with each other. We define R1(t) as the region of scatterers r1. R1(t) that produces 1st order scattered HF receive pulses that overlap at the fast time t. These HF receive pulses interfere both constructively and destructively, depending on the relative arrival time of the individual HF pulses, which results in a fairly random variation of the HF signal envelope, which we refer to as speckle.
The variation of the HF receive signal with the LF pulse, can hence be separated into two components:
i) An average nonlinear propagation delay pt(t) for the HF receive signal at fast time t that is a weighted average of the nonlinear propagation delays pt(r1) for all HF receive pulses from scatterers at r1. R1(t) that overlap and interfere at fast time t. This average, local nonlinear propagation delay can for example be estimated through correlation techniques between the 1st order HF receive signal for a transmitted pulse complex with zero LF pulse and the 1st order HF receive signal for a transmitted pulse complex with the actual transmitted LF pulse.
ii) A variation in the speckle of the 1st order HF receive signal due to variations in the LF pulse. This 1st order speckle varies due to the LF pulse through two physical effects:
iia) The nonlinear pulse form distortion produced by variations in the LF pulse along the co-propagating HF pulse changes the HF pulses with the LF pulse, and hence also the interference between the HF pulses that overlap in time. This effect makes the HF receive signal speckle vary with the LF pulse, and the effect is contained in {tilde over (V)}p.
iib) Spatial variations in the local nonlinear propagation delay pt(r1) over r1. R1(t), will result in a variation in the interference between the different overlapping pulses that will depend on the LF pulse. This interference has a random nature because of the random nature of the scattering cross section and the position of the scatterers at r1. We can however operate with an average dependency that can be included in {tilde over (V)}p, but the random variation around this average value is difficult to correct for in the image reconstruction. We should note that it is the magnitude of the variation in pt(r1) that produces the effect, not the relative variation, and as pt(r1) accumulates with depth according to Eq.(18), the effect generally increases with depth. The dependency of pt(r1) on the LF pulse, is conveniently further broken down into two components:
iiba) Variation of pt(r1) along the HF receive beam direction, which we call an axial variation. This variation is inherent in the method where pt(r1) from Eq.(18) is obtained by accumulation along the paths of energy flow (orthogonal trajectories) of the transmit beam. To reduce the LF dependency of the interference due to the axial variation of pt(r1), we should use as short HF transmit pulses as possible, i.e. wide HF bandwidth.
iibd) Variation of pt(r1) transverse to the HF receive beam at the depth z=ct/2 we call a transverse variation. The transverse variation depends on the detailed design (aperture and focus) of the combined LF and HF transmit beams, and can be minimized by good beam designs. The sensitivity to this variation can further be reduced by a narrow HF receive beam, which is achieved by a wide HF receive aperture, i.e. low HF receive F-number, and dynamic focus of the HF receive beam. This can make the HF receive beam different from the HF transmit beam, for which we often want a long focus which requires a smaller HF transmit aperture with higher F-number. The combination of low F-number HF receive beam and higher F-number HF transmit beam makes Q>1 in Eqs.(29,30), which we shall see gives challenges for good suppression of pulse reverberation noise and/or linear scattering as we return to in Section 5.7C below.
For the plane wave reflections presented in Section 5.6, the variation of the scatterers transverse to the receive beam axis is not random but well defined in the definition of the planes. The transverse variation of both pt(r1) and {tilde over (V)}p(r1,w) across the receive beam is then included in the deterministic versions of pt(z) and {circumflex over (V)}p(z,w). Axial variations of pt(z) and {circumflex over (V)}p(z,w) along the HF receive pulses that overlaps at fast time t=2z/c, can still contribute to an LF pulse variation of the HF receive signal speckle as discussed under point iiba) above.
B. Delay and Speckle for 3rd Order Scattering
The 3rd order scattering noise at fast time t, occurs through a sequence of scatterings at the three locations r1,r2,r3. R3(t), defined so that the HF receive pulses from this sequence of scatterings overlap at the fast time t. This implies that the total propagation lag from the HF transmit transducer to r1, from r1 to r2, from r2 to r3, and from r3 back to the HF receive transducer, is so close to t that the HF receive pulses for r1,r2,r3. R3(t) overlap. The relationship between the fast time reception and the scatterer positions is hence more complex for 3rd order scattering noise compared to the 1st order scattering signal. We note that the HF receive noise pulse lengths are given by a convolution between the transmitted HF pulse length, the length of the HF transmit and receive beam impulse responses, and the length of the HF transducer receive impulse response.
The width of the HF receive beam is for 3rd order scattering much less limiting to the region of interfering scatterers than for 1st order scattering, where the example in FIG. 5 can be used for understanding this statement. The receive beam focus is generally deeper than the last scatterer in the 3rd order chain, because |r3|<ct/2=zr(t), where zr(t) is the dynamic receive beam focus located at the depth of the 1st order scatterers from which the pulses are received at fast time t. For Class I noise where the 1st scatterer is at the position r1 where |r1|<zr(t)/2=ct/4, the receive beam has generally so weak limitation to the region of 1st scatterers that participates to the Class I noise, that scatterers at r1 across the whole transmit beam generally participates in the 3rd order chain of scatterers r1,r2,r3, R3(t). For Class II noise where zr(t)/2<|r3|<zr(t), the receive beam also generally has a weak limitation to the positions of 1st scatterers within the transmit beam at r3, albeit the limitation can be stronger than for Class I scatterers, as seen from FIG. 5.
To dwell further into these complex relations, we start by analysis of the situation for Q={tilde over (V)}p=1. From Eq. (30) we get

Kp(w;r1,r3)=e−iwp(t(r1)+t(r3))/2{tilde over (K)}p(w;r1,r3)

{tilde over (K)}p(w;r1,r3)=2 cos wp(t(r3)−t(r1))/2  (58)

i.e. the filter Kp(w;r1,r3) represents a nonlinear propagation delay of p(t(r1)+t(r3)/2 for Class I and II noise combined, with a filtering of the noise amplitude by {tilde over (K)}p. This amplitude filtering represents the speckle dependency of the 3rd order Class I and II noise on the amplitude of the LF pulse. We note that in this special situation of Q={tilde over (V)}p=1 and only two point scatterers at r1 and r3, the combined Class I and II noise speckle is independent of a change in polarity of the LF pulse, as {tilde over (K)}p is unchanged by a change in polarity of p. This phenomenon can be utilized for combined suppression of Class I and II noise as discussed in relation to Eq. (62) below.
The reason for the reverberation noise speckle dependency on the LF pulse, is that the noise is composed of two components from r1 and r3 with different nonlinear propagation delays pt(r1) and pt(r3) that depend on both the amplitude and polarity of the LF pulse. This represents a spatial variation of the nonlinear propagation delay similar to that discussed for 1st order scattering under point iib) of Section 5.7A above. However, for the 3rd order scattering we are not only considering a variation of pt(r1) along the HF receive pulse length and the width of the highly focused HF receive beam, but for a larger region of allowable r1 and r3 that satisfies r1,r2,r3, R3(t).
When Q(r1,r3){tilde over (V)}p(r1)n{tilde over (V)}p(r3), where in the practical situation we find |Q(r1,r3){tilde over (V)}p(r1)|>|{tilde over (V)}p(r3)|, {tilde over (K)}p from Eq.(30) has a nonzero phase as shown in FIG. 7a. This phase can be separated into a component that is linear in frequency, producing an added nonlinear propagation delay to p(t(r1)+t(r3)/2, and a component that is nonlinear in frequency that together with the amplitude variation of |{tilde over (K)}p| with frequency produces a LF pulse dependency of the combined Class I and II noise speckle. The reason for the delay component of {tilde over (K)}p is that the Class I noise is now stronger than the Class II noise, making the average nonlinear propagation delay closer to pt(r1) than to pt(r3), reducing the average delay/advancement because |pt(r1)|<|pt(r3)|. FIG. 7b shows in the temporal domain how a change in polarity of the LF pulse changes the relative location between the Class I and II reverberation pulses, changing the speckle as the interference between the two pulses when the Class II noise pulse is lower in amplitude than the Class I pulse, i.e. |Q|>1. The average nonlinear propagation delay is closer to that for Class I pulse due to its larger amplitude. We note that if the Class I and II components were equal, the noise speckle would be independent of the polarity of the LF pulse, and the average nonlinear propagation delay would be pt(r1)+t(r3)/2. This implies that when r1 and r3 move close to the mid distance r/2=ct/4 of the image point, we get QVp1>>Vp3, i.e. dYIp>>dYIIp, and the speckle dependency on the LF pulse polarity is low for scatterers in this mid region.
The Fourier transform of the total pulse reverberation noise in an interval of length Ti around ti, can be expressed as




  
    
      
        
          
            Y
            
              3
              ⁢
              i
            
          
          ⁡
          
            (
            
              w
              ,
              p
            
            )
          
        
        =
        
          
            ∑
            
              m
              ,
              
                M
                
                  3
                  ⁢
                  i
                
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              d
              
                Y
                
                  iII
                  ,
                  m
                
              
            
            ⁢
            
              e
              
                
                  -
                  i
                
                ⁢
                
                  
                
                ⁢
                w
                ⁢
                
                  
                
                ⁢
                
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          t
                          
                            1
                            ⁢
                            m
                          
                        
                        +
                        
                          t
                          
                            3
                            ⁢
                            m
                          
                        
                      
                      )
                    
                  
                  /
                  2
                
              
            
            ⁢
            2
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            w
            ⁢
            
              
            
            ⁢
            
              
                p
                ⁡
                
                  (
                  
                    
                      t
                      
                        3
                        ⁢
                        m
                      
                    
                    -
                    
                      t
                      
                        1
                        ⁢
                        
                          
                        
                        ⁢
                        m
                      
                    
                  
                  )
                
              
              /
              2
            
          
        
      
    
    
      
        (
        59
        )
      
    
  




where M3i labels the group of combinations of scatterers at r1m,r2m,r3m, R3(ti) that contributes to the Class I+II noise in Ti. This expression is no longer in general independent of a change in polarity of the LF pulse. Introducing the fast time arrival of the signals, we can rewrite Eq.(59) as




  
    
      
        
          
            N
            i
          
          ⁡
          
            (
            
              w
              ,
              p
            
            )
          
        
        =
        
          
            ∑
            
              m
              ,
              
                M
                
                  3
                  ⁢
                  i
                
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              d
              
                N
                
                  iII
                  ,
                  m
                
              
            
            ⁢
            
              e
              
                
                  -
                  i
                
                ⁢
                
                  
                
                ⁢
                w
                ⁢
                
                  
                
                ⁢
                
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          t
                          ⁡
                          
                            (
                            
                              t
                              
                                1
                                ⁢
                                m
                              
                            
                            )
                          
                        
                        +
                        
                          t
                          ⁡
                          
                            (
                            
                              t
                              
                                3
                                ⁢
                                m
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  /
                  2
                
              
            
            ⁢
            2
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            w
            ⁢
            
              
            
            ⁢
            
              
                p
                ⁡
                
                  (
                  
                    
                      t
                      ⁡
                      
                        (
                        
                          t
                          
                            3
                            ⁢
                            m
                          
                        
                        )
                      
                    
                    -
                    
                      t
                      ⁡
                      
                        (
                        
                          t
                          
                            1
                            ⁢
                            m
                          
                        
                        )
                      
                    
                  
                  )
                
              
              /
              2
            
          
        
      
    
    
      
        (
        60
        )
      
    
  




For the transducer as the main 2nd reflector, we have t2=0 and t3=t−t1. With a nonlinear propagation delay that is linear in the fast time as t(t)=at, we get

(t(t3)+t(t1)/2=a(t−t1+t1)/2=at/2=t(t)/2 a)

(t(t3)−t(t1)/2=a(t−t1−t1)/2=a(t−2t1)/2=t(t)/2−t(t1)b)  (61)

i.e. the delay of the combined Class I and II pulse reverberation noise is independent of the location of plane 505 at t1 and plane 507 at t3=t−t1. Eq.(60) now changes to




  
    
      
        
          
            N
            i
          
          ⁡
          
            (
            
              w
              ;
              p
            
            )
          
        
        =
        
          
            e
            
              
                -
                i
              
              ⁢
              
                
              
              ⁢
              w
              ⁢
              
                
              
              ⁢
              p
              ⁢
              
                
              
              ⁢
              
                
                  t
                  ⁡
                  
                    (
                    
                      t
                      i
                    
                    )
                  
                
                /
                2
              
            
          
          ⁢
          
            
              ∑
              
                m
                ,
                
                  M
                  i
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                d
                
                  
                    N
                    
                      iII
                      ,
                      m
                    
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              w
              ⁢
              
                
              
              ⁢
              
                
                  p
                  ⁡
                  
                    (
                    
                      
                        t
                        ⁡
                        
                          (
                          
                            t
                            i
                          
                          )
                        
                      
                      -
                      
                        2
                        ⁢
                        
                          t
                          ⁡
                          
                            (
                            
                              t
                              
                                1
                                ⁢
                                m
                              
                            
                            )
                          
                        
                      
                    
                    )
                  
                
                /
                2
              
            
          
        
      
    
    
      
        (
        62
        )
      
    
  




The basis for this equation is that Q={tilde over (V)}p=1, t2=0, and a linear variation of t(t) with fast time t. Q≈1 is obtained with plane reflectors as discussed in Section 5.6, and can be achieved for more irregular scatterers with equal transmit and receive beams. To obtain image sharpness in a depth range with equal transmit and receive beams requires multi-focus transmissions which reduces frame rate, or one can use transversal filtering as described in Section 5.5C, or a combination of both, or multi-component transmissions with synthetic transmit and receive beams reconstruction as described in Section 5.5A, B. In this last case it is easier to avoid pulse form distortion, i.e. we get {tilde over (V)}p>>1, as the diffraction sliding of the unfocused LF pulse component is low, and we can place the HF pulse close to the crest or trough of the LF pulse for the whole image range.
It can also in some situation be advantageous to select a region of the LF aperture to be non-transmitting, so that manipulation of the object material parameters by the LF pulse observed by the co-propagating HF pulse is very low in a near field region of the HF array aperture, for example as described in U.S. Patent application 12/500,518. Delay corrections or speckle corrections are then not necessary for suppression of the multiple scattering noise components where the scatterers are in this near-field range. One can then concentrate on estimation of delay corrections or speckle corrections for scatterers that are outside this near field range.
For the 3rd order Class I and II scattering we then conclude:
i) The average nonlinear propagation delay for combined Class I and II noise is more dependent on spatial variations in the nonlinear propagation delay pt(r) compared to 1st order scattering, as we at fast time t get overlapping contributions of scattered signal for a wider range of scatterer locations r1 and r3 that satisfies r1,r2,r3, R3(t). However, for a linear variation of the nonlinear propagation delay with fast time, t2=0, and Q={tilde over (V)}p=1, the combined Class I and II noise gets an average nonlinear propagation delay equal to ptn(t)=pt(t)/2, and the noise speckle is independent of a change in polarity of the LF pulse. This can through Eq. (62) be used to suppress combined Class I and II noise as in Eq.(67,68) below.
When the variation of the nonlinear propagation delay with the fast time deviates from linear variation in t, we get ptn(t)′′pt(t)/2. For a variation with fast time that is upwards convex, which is often found, the average nonlinear propagation delay for the combined Class I and II noise is ptn(t)<pt(t)/2 as illustrated in FIG. 8. The speckle is in this case no longer independent of the polarity of the LF pulse, even for Q={tilde over (V)}p=1.
When |Q(r1,r3){tilde over (V)}p(r1)|>|{tilde over (V)}p(r3)| we get a non-zero phase of {tilde over (K)}p as illustrated in FIG. 7a. The component of this phase that is linear in frequency produces a nonlinear propagation delay added to p(t(r1)+t(r3)/2, that depends both on the amplitude and the polarity of the LF pulse. The combined Class I and II noise delay is an average of pt(r1) and pt(r3), weighted with the amplitude of the Class I and Class II noise components at the different scatterer locations r1 and r3 that contribute to the composite noise in an interval, i.e. r1,r2,r3, R3(t). When |Q(r1,r3){tilde over (V)}p(r1)|>|{tilde over (V)}p(r3)| we get stronger weight on pt(r1) than pt(r3), reducing the average nonlinear propagation delay below p(t(r1)+t(r3)/2, i.e. ptn(t)<pt(t)/2.
ii) The combined Class I and II noise speckle has a larger sensitivity to variation in the LF pulse than the 1st order scattering signal. This variation can be related to two different physical effects:
iia) At fast time t we get overlapping contributions of scattered signal for a wide range of scatterer locations r1 and r3 that satisfies r1,r2,r3, R3(t). These different scatterer locations have different nonlinear propagation delays pt(r1) and pt(r3) that depends on the amplitude and polarity of the LF pulse and hence produces a larger sensitivity to LF pulse changes for the combined Class I and II noise speckle compared to the 1st order scattering speckle. When Q={circumflex over (V)}p=1, t2=0, and a linear variation in the nonlinear propagation delay with fast time, the combined Class I and II noise speckle do not change with the polarity of the LF pulse, albeit it depends on the amplitude of the LF pulse.
iib) The nonlinear pulse form distortion of the transmitted HF pulse, produced by variations of the LF pressure along the co-propagating HF pulse and given by {tilde over (V)}p in Eqs.(19, 29, 30), changes the HF pulses that interfere at a fast time t, and hence also the interference speckle. The pulse form distortion is given by the nonlinear phase component of {tilde over (V)}p in and also the spectral amplitude modification by |{tilde over (V)}p|, and depends on both the amplitude and the polarity of the LF pulse.
C. Estimation of Linear and Nonlinear Scattering Components with Noise Suppression
Based on the HF receive signal model in Eq.(57), we study the following set of equations obtained as the HF receive signal from transmission of three pulse complexes with amplitude +p, 0, −p of the LF pulse

Yi(w;p)=e−iwpti{{circumflex over (V)}pl(w;ti)Xli(w)+p{circumflex over (V)}pn(w;ti)Xni(w)}+e−iwptni{circumflex over (N)}i(w;p)a)

Yi(w;0)=Xli(w)+Ni(w;0)b)

Yi(w;−p)=eiwpti{{circumflex over (V)}−pl(w;ti)Xli(w)−p{circumflex over (V)}−pn(w;ti)Xni(w)}+iwptni{circumflex over (N)}i(w;−p)c)  (63)

where we have made the approximation Xni(w;p)>>pXni(w) for the nonlinear scattering. In the same token we generally have {circumflex over (V)}pn(w;ti)=(w;ti) except in situations of strong, resonant nonlinear scattering. We have here three linear equations with the five unknowns Xli(w), Xni(w), {circumflex over (N)}i(w;p), Ni(w;0) and {circumflex over (N)}i(w;−p). We can relate {circumflex over (N)}i(w;p) to Ni(w;0) by the following filter




  
    
      
        
          
        
        ⁢
        
          
            
              
                N
                i
              
              ⁡
              
                (
                
                  w
                  ;
                  0
                
                )
              
            
            =
            
              
                
                  M
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    p
                  
                  )
                
              
              ⁢
              
                
                  
                    N
                    ⋒
                  
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    p
                  
                  )
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                M
                i
              
              ⁡
              
                (
                
                  w
                  ;
                  p
                
                )
              
            
            =
            
              
                
                  
                    
                      N
                      i
                    
                    ⁡
                    
                      (
                      
                        w
                        ;
                        0
                      
                      )
                    
                  
                  
                    
                      
                        N
                        ⋒
                      
                      i
                    
                    ⁡
                    
                      (
                      
                        w
                        ;
                        p
                      
                      )
                    
                  
                
                ⁢
                
                  1
                  
                    1
                    +
                    
                      
                        m
                        N
                      
                      ⁢
                      
                        
                          
                          
                            
                              
                                N
                                i
                              
                              ⁡
                              
                                (
                                
                                  w
                                  ;
                                  0
                                
                                )
                              
                            
                            /
                            
                              
                                
                                  N
                                  ⋒
                                
                                i
                              
                              ⁡
                              
                                (
                                
                                  w
                                  ;
                                  p
                                
                                )
                              
                            
                          
                          
                        
                        2
                      
                    
                  
                
              
              =
              
                
                  
                    
                      M
                      ⋒
                    
                    i
                  
                  ⁡
                  
                    (
                    
                      w
                      ;
                      p
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    w
                    ⁢
                    
                      
                    
                    ⁢
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      ni
                    
                  
                
              
            
          
        
      
    
    
      
        (
        64
        )
      
    
  




where mN:10−3−10−2 is a noise parameter. Delay correction and filtering according to Eq.(63) and combination of Eqs.(63a,b) and Eqs.(63b,c) gives us two equations with noise highly suppressed

a11Xli(w)+a12Xni(w)=b1

a21Xli(w)+a22Xni(w)=b2

a11=eiwp(tni−ti){circumflex over (M)}i(w;p){circumflex over (V)}pl(w;ti)−1

a21=e−iwp(tni−ti){circumflex over (M)}i(w;−p){circumflex over (V)}pl(w;ti)−1

a12=peiwp(tni−ti){circumflex over (M)}i(w;p){circumflex over (V)}pn(w;ti)

a22=−pe−iwp(tni−ti){circumflex over (M)}i(w;−p){circumflex over (V)}−pn(w;ti)

b1={circumflex over (M)}i(w;p)eiwp(tni−ti)Yi(w;p)−Yi(w;0)

b2={circumflex over (M)}i(w;−p)e−iwp(tni−ti)Yi(w;−p)−Yi(w;0)  (65)

with the solution for the linear and nonlinear scattering




  
    
      
        
          
            
              
                
                  
                    a
                    )
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      X
                      li
                    
                    ⁡
                    
                      (
                      w
                      )
                    
                  
                
                =
                
                  
                    
                      
                        a
                        22
                      
                      ⁢
                      
                        b
                        1
                      
                    
                    -
                    
                      
                        a
                        12
                      
                      ⁢
                      
                        b
                        2
                      
                    
                  
                  
                    
                      
                        a
                        11
                      
                      ⁢
                      
                        a
                        22
                      
                    
                    -
                    
                      
                        a
                        12
                      
                      ⁢
                      
                        a
                        21
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              b
            
            )
          
          ⁢
          
            
          
          ⁢
          
            
              X
              ni
            
            ⁡
            
              (
              w
              )
            
          
        
        =
        
          
            
              
                -
                
                  a
                  21
                
              
              ⁢
              
                b
                1
              
            
            -
            
              
                a
                11
              
              ⁢
              
                b
                2
              
            
          
          
            
              
                a
                11
              
              ⁢
              
                a
                22
              
            
            -
            
              
                a
                12
              
              ⁢
              
                a
                21
              
            
          
        
      
    
    
      
        (
        66
        )
      
    
  




Even if we can not make the approximation Xni(w;p)>>pXni(w), Eq. (66b) gives a signal where both the noise and the linear scattering is highly suppressed, and hence represents a nonlinear scattering estimate. As the linear scattering is generally much stronger than the nonlinear scattering, Eq. (66a) is an estimate for the linear scattering with noise suppressed, or we could alternatively use the approximate estimate in Eq.(68) below for the linear scattering.
The nonlinear scattering term is generally low relative to the linear scattering term in Eq.(63) and can be neglected to estimate the linear scattering term with strong suppression of Class I and II pulse reverberation noise. We then get the starting equations as

Yi(w;p)=e−iwpti{circumflex over (V)}pl(w;ti)Xli(w)+e−iwptni{circumflex over (N)}i(w;p)a)

Yi(w;−p)=eiwpti{circumflex over (V)}−pl(w;ti)Xli(w)+eiwptni{circumflex over (N)}i(w;−p)b)  (67)

Delay correcting and filtering as in Eqs. (62-65) we get




  
    
      
        
          
            X
            li
          
          ⁡
          
            (
            w
            )
          
        
        =
        
          
            
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    ni
                  
                
              
              ⁢
              
                
                  
                    M
                    ⋒
                  
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    p
                  
                  )
                
              
              ⁢
              
                
                  Y
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    p
                  
                  )
                
              
            
            -
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    ni
                  
                
              
              ⁢
              
                
                  
                    M
                    ⋒
                  
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      -
                      p
                    
                  
                  )
                
              
              ⁢
              
                
                  Y
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      -
                      p
                    
                  
                  )
                
              
            
          
          
            
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          t
                          ni
                        
                        -
                        
                          t
                          i
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  
                    M
                    ⋒
                  
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    p
                  
                  )
                
              
              ⁢
              
                
                  
                    V
                    ⋒
                  
                  pl
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      t
                      i
                    
                  
                  )
                
              
            
            -
            
              

            
            ⁢
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  
                    p
                    ⁡
                    
                      (
                      
                        
                          t
                          ni
                        
                        -
                        
                          t
                          i
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                
                  
                    M
                    ⋒
                  
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      -
                      p
                    
                  
                  )
                
              
              ⁢
              
                
                  
                    V
                    ⋒
                  
                  
                    -
                    pl
                  
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      t
                      i
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        68
        )
      
    
  




When the noise speckle is independent of the polarity of the LF pulse, we can obtain a linear scattering signal with strong suppression of the 3rd order scattering noise through only delay correction of Eqs. (67a,b) (i.e. no speckle corrections) and subtracting as




  
    
      
        
          
            Z
            i
          
          ⁡
          
            (
            w
            )
          
        
        =
        
          
            
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                      t
                      ^
                    
                    ni
                  
                
              
              ⁢
              
                
                  Y
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    p
                  
                  )
                
              
            
            -
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                      t
                      ^
                    
                    ni
                  
                
              
              ⁢
              
                
                  Y
                  i
                
                ⁡
                
                  (
                  
                    w
                    ;
                    
                      -
                      p
                    
                  
                  )
                
              
            
          
          =
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                [
                
                  
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                t
                                i
                              
                              -
                              
                                
                                  t
                                  ^
                                
                                ni
                              
                            
                            )
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          V
                          ⋒
                        
                        pl
                      
                      ⁡
                      
                        (
                        
                          w
                          ;
                          
                            t
                            i
                          
                        
                        )
                      
                    
                  
                  -
                  
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        
                          p
                          ⁡
                          
                            (
                            
                              
                                t
                                i
                              
                              -
                              
                                
                                  t
                                  ^
                                
                                ni
                              
                            
                            )
                          
                        
                      
                    
                    ⁢
                    
                      
                        
                          V
                          ⋒
                        
                        
                          -
                          pl
                        
                      
                      ⁡
                      
                        (
                        
                          w
                          ;
                          
                            t
                            i
                          
                        
                        )
                      
                    
                  
                
                ]
              
              ⁢
              
                
                  X
                  li
                
                ⁡
                
                  (
                  w
                  )
                
              
            
            +
            
              [
              
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    w
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        p
                        1
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              i
                            
                          
                          -
                          
                            
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                              ^
                            
                            
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                              ⁢
                              
                                
                              
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                        )
                      
                    
                  
                
                ⁢
                ⁢
                
                  
                    
                      N
                      ⋒
                    
                    i
                  
                  ⁡
                  
                    (
                    
                      w
                      ;
                      
                        -
                        p
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        69
        )
      
    
  




We see from Eq. (62) above that for Q={circumflex over (V)}p=1 and a linear depth dependency of the non-linear propagation delay, the speckle of combined Class I and II pulse reverberation noise is independent on the polarity of the LF pulse, i.e. {circumflex over (N)}i(w;−p)={circumflex over (N)}i(w;p)m, and tni=t(ti)/2. Limited suppression of the noise in the last term in Eq.(69) is caused by improper estimation of tni, i.e. {circumflex over (t)}ni′′tni, and variations in the speckle of combined Class I and II pulse reverberation noise with the polarity of the LF pulse, i.e. {circumflex over (N)}i(w;−p)′′{circumflex over (N)}i(w;p). If the nonlinear propagation delay deviates from being linear in the fast time t, we can not take the delay term outside the sum in Eq. (62), and the signal speckle is no longer independent of the polarity of the LF pulse. In addition, if any of Q or V%p′′1, the signal speckle will be highly dependent on both the amplitude and the polarity of the LF pulse, which further reduces the suppression of the noise term in Eq.(69).
The limitation of noise suppression produced by variations in the noise speckle with the polarity of the LF pulse can be reduced in a low echogene region, for example a cyst or a blood vessel, using the following algorithm




  
    
      
        
          
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              [
              
                
                  
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        (
        70
        )
      
    
  




Inside the cyst we have Xli(w)=0 and {circumflex over (t)}ni>>tni>>ti>>{circumflex over (t)}i so that the the last term, i.e. the noise term, disappears because the speckle, {circumflex over (N)}i(w;p1), is the same for both components of the noise term. Outside the cyst we have Xli(w)′′0 and {circumflex over (t)}ni>>ti/2 so that the 1st term is non-zero.
By the comments following Eq.(86) below we note that we can obtain some correction for the pulse form distortion by reduction of the bandwidth of the received HF signal through band pass filtering. This band pass filtering provides corrections of the effects of pulse form distortion both on the 1st order and multiple order scattering signals.
The signals with strong suppression of multiple scattering noise are also very useful for estimations of corrections for wave front aberrations due to heterogeneous spatial variation of propagation velocity in the material, for example according to the methods described in U.S. Pat. Nos. 6,485,423 and 6,905,465.
D. Limitations in Suppression Due to Delay and Speckle Uncertainty
In Eqs.(66,68) we have compensated for both the nonlinear propagation delay, pulse form distortion, and the LF pulse dependency of the combined Class I and II noise speckle. The success of the suppression of the combined Class I and Class II noise and the linear scattering depends on how accurately we can estimate t(ti) {circumflex over (V)}pl(w;ti), tni, and  (w;p). If nonlinear pulse form distortion is negligible we can avoid estimating the full Vpi(w;ti) and only its linear phase components that gives the nonlinear propagation delay t(ti).
In Eq.(69) we do not compensate for potential variations in the combined Class I and II noise speckle with a change in polarity of the LF pulse. As shown in Section 5.7B, this is achieved for a linear variation of the nonlinear propagation delay with fast time, t2=0, and Q={circumflex over (V)}p=1. Deviations from these requirements will reduce the suppression of the 3rd order scattering noise in Eq.(69).
In Eq.(70) there is a single noise term where the nonlinear propagation delay and noise speckle is unchanged for the two signals which are combined. The degree of suppression of the pulse reverberation noise does in this expression only depend on the accuracy of the estimate of {circumflex over (t)}ni. However, if we have 1st order scattering signals together with the combined Class I and II pulse reverberation noise, where the 1st order signals are much weaker than the noise, the 1st order signal will be suppressed together with the noise.
Limited suppression of combined Class I and II noise and also linear scattering components to obtain nonlinear scattering components, including remedies to enhance the suppression, is hence in more detail:
i) Improper estimation of tni for example through using {circumflex over (t)}ni=ti/2 from Eq.(61a). Reasons for fallacy of this estimate can be that
ia) −{tilde over (K)}pi of Eq.(30) depends on the magnitude/polarity of p, for example because Q>1 or V1>V3. Q>1 implies that Class II reverberations has lower amplitude than Class I reverberations, as illustrated in FIG. 7 where FIG. 7a shows an example in the frequency domain that this situation provides −{tilde over (K)}pi=−(Qi{tilde over (V)}pi1eiwp(t3−t1)+{tilde over (V)}pi3e−iwp(t3−t1))′′0 even with {tilde over (V)}pi1,3=1. The same situation is illustrated in the time domain in FIG. 7b, showing that the Class I and II reverberations have different relation to each other for p>0 (mid panel) and p<0 (lower panel). The nonlinear attenuation of the HF pulse due to self-distortion increases this phenomenon because it gives |{tilde over (V)}pi3|<|{tilde over (V)}pi1|.
A 1st remedy to solve this situation is to design the transmit and receive beams of close to the same form so that −{tilde over (K)}pi>>0. Sharpness through a depth region of the image can be obtained for each image line by multiple transmits with different transmit/receive foci, which reduces the frame rate, or transversal filtering as described in Section 5.5C. Another solution is to use multiple, wide transmit beam components with synthetic transmit and receive beams as described in Section 5.5A,B. A 2nd remedy is to use a model based estimation of {circumflex over (t)}ni, for example as given in Section 5.8 below.
We note that when t1 approaches t3 because t1 approaches (ti+t2/2, we generally get Q≈1 and {tilde over (V)}pi1>>{tilde over (V)}pi3. This implies good suppression of the noise term in Eq. (68) as t1 approaches t3 with delay correction with {circumflex over (t)}ni=ti/2 for t(t)>>at, i.e. the nonlinear propagation delay is linear in t.
ib) Nonlinear depth variation of t(t) as illustrated in FIG. 8 implies that {circumflex over (t)}ni′′ti/2. A 1st remedy to solve this situation is to design the HF and LF beams so that adequate linearity of t(t) is obtained. A 2nd remedy is to use a model based estimation of {circumflex over (t)}ni, for example as given in Section 5.8 below, which is also useful for the situation in ia) above. To only use delay correction {circumflex over (t)}ni=ti/2 requires both that −{tilde over (K)}pi>>0 as discussed above, and model based estimation of tni hence solves both these situations.
ii) Variation in noise speckle with the LF pulse. This variation has three reasons:
iia) Spatial variation of the nonlinear propagation delay of the HF pulse over the cross section of the HF transmit beam at t1 and t3. A remedy to improve this situation is to design HF and LF beams so that one gets minimal variation of the nonlinear propagation delay of the HF pulse over the cross section of the HF transmit beam at t1 and t3.
iib) Class I and II pulse reverberation noise is different which implies Q and |V|′′1, similar to the illustration in FIG. 7. A remedy to solve this situation is to use the same transmit and receive beam with multiple transmits at different focus depths for each image line or transversal filtering as described in Section 5.5C, or a combination of both, or use synthetic transmit and receive beams as discussed in Section 5.5A, B above.
iic) Nonlinear pulse form distortion as given by a frequency dependency of the module and phase of {tilde over (V)}pi. With synthetic transmit beams one have less sliding of the LF pulse phase because the full image depth is within the near field of the transmit beam components. One can then keep the HF pulse at the crest or trough of the LF pulse for the full image depth.
Remedies for best suppression of the combined Class I and II pulse reverberation noise and also 1st order linear scattering components are hence:
A) Design HF and LF transmit apertures and foci so that one gets as linear variation with fast time t as possible of the nonlinear propagation delay, and
B) Use equal HF transmit and receive apertures and foci, either through using multi-focus transmits for each image line, or synthetic focusing in a large group of image points as described in Section 5.5, and
C) Use model based estimation of t(ti){circumflex over (V)}pl(w;ti), tni, and {circumflex over (M)}i(w;p), for example as presented in the next section.
5.8 Model Based Estimation of Delay and Speckle Corrections
A. Simulation Model for Wave Propagation
When the nonlinear propagation delay is not linear in the fast time, and/or t2>0, and/or Q>1, and/or one have disturbing nonlinear pulse form distortion Vpi(w;r)′′0, improved methods of estimating ti, tni, Vpi(w;r), and Wi(w;p) are needed. For given material parameters we simulate the 3D expressions for Ht(r1,w;rt) in Eq.(1), Hr(r3,w;rr) in Eq.(4), Up(r1,w;rt) in Eq.(22), Vp(w;r) and pt(r) in Eqs.(18,19). Integrating these 3D expressions across a transversal plane in (x,y) at z, we get Ht(z,w;zt), Hr(z,w;z), Vp(z,w), {circumflex over (V)}p(z,w), and pt(z) in Eqs.(49,50).
For simulation of the nonlinear (and also linear) wave propagation we neglect the left side scattering terms in Eq.(14) that gives




  
    
      
        
          
            
              
            
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        =
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        (
        71
        )
      
    
  




For numerical simulations of the forward propagating pulse, we introduce retarded time t as

t=t−z/c0 t=t+z/c0 p(z,r^,t)fi {circumflex over (p)}(z,r^,t)  (72)

The differentials of Eq.(71) take the following form in retarded time




  
    
      
        
          
            
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        (
        73
        )
      
    
  




Changing to the retarded time variables then changes Eq.(71) to




  
    
      
        
          
            
              
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        (
        74
        )
      
    
  




For collimated beams we can neglect ¶2{circumflex over (p)}/¶z2 referred to as the paraxial approximation, and introducing {circumflex over (p)}={circumflex over (p)}L+{circumflex over (p)}H as in Eq.(15), we can separate Eq.(74) into separate equations for the LF and HF bands, as




  
    
      
        
          
        
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                    ⁢
                    
                      t
                      2
                    
                  
                
              
              +
              
                
                  
                    b
                    p
                  
                  
                    c
                    0
                  
                
                ⁢
                
                  
                    
                      ¶
                      2
                    
                    ⁢
                    
                      
                        p
                        ⋒
                      
                      L
                    
                    ⁢
                    
                      
                        p
                        ⋒
                      
                      H
                    
                  
                  
                    ¶
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      2
                    
                  
                
              
              +
              
                
                  
                    b
                    p
                  
                  
                    2
                    ⁢
                    
                      c
                      0
                    
                  
                
                ⁢
                
                  
                    
                      ¶
                      2
                    
                    ⁢
                    
                      
                        p
                        ⋒
                      
                      H
                      2
                    
                  
                  
                    ¶
                    ⁢
                    
                      
                    
                    ⁢
                    
                      t
                      2
                    
                  
                
              
            
          
        
        ⁢
        
          
        
      
    
    
      
        (
        75
        )
      
    
  




As discussed after Eq.(15) the propagation distance of the LF wave is generally so short ( ̃30 1LF) that one can neglect the right hand nonlinear propagation term in Eq.(75a) and simulate the LF wave propagation in the linear scheme, which allows us to obtain the LF pulse along the co-propagating HF pulse, independent of the HF pulse as

{circumflex over (p)}L(z,r^,t)={circumflex over (p)}L(z,r^,0)cos wL(t+tL(z,r^))  (76)

where tL(z,r^) is the location of the center of the co-propagating HF pulse within the LF pulse. We define

a(z,r^)=bp(z,r^){circumflex over (p)}L(z,r^,0)  (77)

This allows us to write a self contained equation for the HF pulse from Eq.(75b) that takes into account both the nonlinear interaction distortion from the LF pulse and the nonlinear self distortion, as defined in Eq.(15), as




  
    
      
        
          
            
              ¶
              2
            
            ⁢
            
              
                p
                ⋒
              
              H
            
          
          
            ¶
            ⁢
            
              
            
            ⁢
            z
            ⁢
            
              
            
            ⁢
            ¶
            ⁢
            
              
            
            ⁢
            t
          
        
        =
        
          
            
              
                c
                0
              
              2
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                     
                    ⋀
                    2
                  
                  ⁢
                  p
                
                ⋒
              
              H
            
          
          -
          
            
              
                h
                ab
              
              ⁢
              
                ~
                t
              
              ⁢
              
                1
                
                  2
                  ⁢
                  
                    c
                    0
                  
                
              
            
            ⁢
            
              
                
                  ¶
                  2
                
                ⁢
                
                  
                    p
                    ⋒
                  
                  H
                
              
              
                ¶
                ⁢
                
                  
                
                ⁢
                
                  t
                  2
                
              
            
          
          +
          
            
              a
              
                c
                0
              
            
            ⁢
            
              
                
                  ¶
                  2
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  
                    w
                    L
                  
                  ⁡
                  
                    (
                    
                      t
                      +
                      
                        
                          t
                          L
                        
                        ⁡
                        
                          (
                          
                            z
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    p
                    ⋒
                  
                  H
                
              
              
                ¶
                ⁢
                
                  
                
                ⁢
                
                  t
                  2
                
              
            
          
          +
          
            
              
                b
                p
              
              
                2
                ⁢
                
                  c
                  0
                
              
            
            ⁢
            
              
                
                  ¶
                  2
                
                ⁢
                
                  
                    p
                    ⋒
                  
                  H
                  2
                
              
              
                ¶
                ⁢
                
                  
                
                ⁢
                
                  t
                  2
                
              
            
          
        
      
    
    
      
        (
        78
        )
      
    
  




The first term on the right side represents diffraction, the 2nd term absorption, the 3rd term the nonlinear interaction distortion of the HF pulse by the LF pulse, and the last term the nonlinear self distortion of the HF pulse,
Temporal Fourier transform of the LF-HF interaction term in Eq.(78) gives




  
    
      
        
          
            F
            t
          
          ⁢
          
            {
            
              
                a
                
                  c
                  0
                
              
              ⁢
              
                
                  
                    ¶
                    2
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      w
                      L
                    
                    ⁡
                    
                      (
                      
                        t
                        +
                        
                          
                            t
                            L
                          
                          ⁡
                          
                            (
                            
                              z
                              ,
                              
                                
                                  r
                                  _
                                
                                ⋀
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      p
                      ⋒
                    
                    H
                  
                
                
                  ¶
                  ⁢
                  
                    
                  
                  ⁢
                  
                    t
                    2
                  
                
              
            
            }
          
        
        =
        
          
            -
            
              
                a
                ⁢
                
                  
                
                ⁢
                
                  w
                  2
                
              
              
                2
                ⁢
                
                  c
                  0
                
              
            
          
          ⁢
          
            {
            
              
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      w
                      L
                    
                    ⁢
                    
                      
                        t
                        L
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                  
                
                ⁢
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      
                        w
                        -
                        
                          w
                          L
                        
                      
                    
                    )
                  
                
              
              +
              
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      w
                      L
                    
                    ⁢
                    
                      
                        t
                        L
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                  
                
                ⁢
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      
                        w
                        +
                        
                          w
                          L
                        
                      
                    
                    )
                  
                
              
            
            }
          
        
      
    
    
      
        (
        79
        )
      
    
  




Full temporal Fourier transform of Eq.(78) then takes the form




  
    
      
        
          
            d
            
              
                
                  p
                  ⋒
                
                H
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          
            d
            z
          
        
        =
        
          
            
              -
              i
            
            ⁢
            
              
                c
                0
              
              
                2
                ⁢
                w
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                    
                       
                      ⋀
                      2
                    
                    ⁢
                    p
                  
                  ⋒
                
                H
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          -
          
            i
            ⁢
            
              w
              
                2
                ⁢
                
                  c
                  0
                
              
            
            ⁢
            
              
                H
                ab
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
            ⁢
            
              
                
                  p
                  ⋒
                
                H
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          +
          
            
              i
              2
            
            ⁢
            
              w
              
                c
                0
              
            
            ⁢
            
              a
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            
              {
              
                
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        w
                        L
                      
                      ⁢
                      
                        
                          t
                          L
                        
                        ⁡
                        
                          (
                          
                            z
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        p
                        ⋒
                      
                      H
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          -
                          
                            w
                            L
                          
                        
                      
                      )
                    
                  
                
                +
                
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        w
                        L
                      
                      ⁢
                      
                        
                          t
                          L
                        
                        ⁡
                        
                          (
                          
                            z
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      
                        p
                        ⋒
                      
                      H
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          +
                          
                            w
                            L
                          
                        
                      
                      )
                    
                  
                
              
              }
            
          
          +
          
            
              i
              2
            
            ⁢
            
              w
              
                c
                0
              
            
            ⁢
            
              
                a
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                  
                  )
                
              
              
                
                  
                    p
                    ⋒
                  
                  L
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                    ,
                    0
                  
                  )
                
              
            
            ⁢
            
              
                
                  
                    p
                    ⋒
                  
                  H
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                    ,
                    w
                  
                  )
                
              
              ⁢
              
                ~
                w
              
              ⁢
              
                
                  
                    p
                    ⋒
                  
                  H
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                    ,
                    w
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        80
        )
      
    
  




where the last term represents convolution in the Fourier domain, introducing harmonic components of the fundamental HF band as




  
    
      
        
          
            
              
                p
                ⋒
              
              H
            
            ⁡
            
              (
              
                z
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
                ,
                w
              
              )
            
          
          ⁢
          
            ~
            w
          
          ⁢
          
            
              
                p
                ⋒
              
              H
            
            ⁡
            
              (
              
                z
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
                ,
                w
              
              )
            
          
        
        =
        
          
            1
            
              2
              ⁢
              p
            
          
          ⁢
          
            ∫
            
              
                d
                w
              
              ⁢
              
                
                  
                    p
                    ⋒
                  
                  H
                
                ⁡
                
                  (
                  
                    w
                    -
                    w
                  
                  )
                
              
              ⁢
              
                
                  
                    p
                    ⋒
                  
                  H
                
                ⁡
                
                  (
                  w
                  )
                
              
            
          
        
      
    
    
      
        (
        81
        )
      
    
  




In the receiver one would typically band pass filter the received signal around the fundamental frequency. The last term hence represents a nonlinear absorption attenuation of the fundamental HF band by pumping energy into the harmonic and sub-harmonic components of the HF band.
Eq.(80) lends itself well for integration of the HF pressure pulse along the beam axis, z. We get overlap between the shifted spectra {circumflex over (p)}H(z,r^,w−wL) and {circumflex over (p)}H(z,r^,w+wL) when the bandwidth BH of the HF pulse and the angular frequency wL of the LF pulse have the following relation




  
    
      
        
          
            w
            L
          
          <
          
            
              B
              /
              2
            
            ⁢
            
              
            
            ⁢
            
              T
              
                p
                ⁢
                
                  
                
                ⁢
                H
              
            
          
        
        =
        
          
            
              
                2
                ⁢
                p
              
              B
            
            <
            
              
                2
                ⁢
                p
              
              
                2
                ⁢
                
                  w
                  L
                
              
            
          
          =
          
            
              T
              L
            
            /
            2
          
        
      
    
    
      
        (
        82
        )
      
    
  




TpH is the pulse length of the HF pulse and TL is the period of the LF pulse,. In this case it makes sense to further modify Eq.(80) as




  
    
      
        
          
            d
            
              
                
                  p
                  ⋒
                
                H
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          
            d
            z
          
        
        =
        
          
            
              
                -
                i
              
              ⁢
              
                
                  c
                  0
                
                
                  2
                  ⁢
                  w
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      
                         
                        ⋀
                        2
                      
                      ⁢
                      p
                    
                    ⋒
                  
                  H
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                    ,
                    w
                  
                  )
                
              
            
            
              }
              diffraction
            
          
          +
          
            
              
                w
                
                  2
                  ⁢
                  
                    c
                    0
                  
                
              
              ⁢
              
                
                  H
                  abi
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                    ,
                    w
                  
                  )
                
              
              ⁢
              
                
                  
                    p
                    ⋒
                  
                  H
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                    ,
                    w
                  
                  )
                
              
            
            
              }
              absorption
            
          
          -
          
            
              
                i
                ⁢
                
                  w
                
                ⁢
                
                  
                    H
                    abr
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
                ⁢
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
              }
            
            dispersion
          
          +
          
            
              
                i
                ⁢
                
                  w
                  
                    c
                    0
                  
                
                ⁢
                
                  a
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                    
                    )
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  w
                  L
                
                ⁢
                
                  
                    t
                    L
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                    
                    )
                  
                
                ⁢
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
              }
            
            
              nonlinear
              ⁢
              
                
              
              ⁢
              propagation
              ⁢
              
                
              
              ⁢
              delay
            
          
          +
          
            
              
                i
                2
              
              ⁢
              
                w
                
                  c
                  0
                
              
              ⁢
              
                a
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                  
                  )
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                w
                L
              
              ⁢
              
                
                  t
                  L
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        r
                        _
                      
                      ⋀
                    
                  
                  )
                
              
              ⁢
              
                {
                
                  
                    
                      
                        p
                        ⋒
                      
                      H
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          -
                          
                            w
                            L
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      
                        p
                        ⋒
                      
                      H
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        
                          w
                          +
                          
                            w
                            L
                          
                        
                      
                      )
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                
                }
              
            
            
              }
              
                pulse
                ⁢
                
                  
                
                ⁢
                form
                ⁢
                
                  
                
                ⁢
                distirtion
                ⁢
                
                  
                
                ⁢
                related
                ⁢
                
                  
                
                ⁢
                to
                ⁢
                
                  
                
                ⁢
                curvature
                ⁢
                
                  
                
                ⁢
                of
                ⁢
                
                  
                
                ⁢
                LF
                ⁢
                
                  
                
                ⁢
                pressure
                ⁢
                
                  
                
                ⁢
                oscialltion
              
            
          
          -
          
             
            
              
            
            ⁢
            
              
                
                  
                    1
                    2
                  
                  ⁢
                  
                    w
                    
                      c
                      0
                    
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                      
                      )
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    w
                    L
                  
                  ⁢
                  
                    
                      t
                      L
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                      
                      )
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            p
                            ⋒
                          
                          H
                        
                        ⁡
                        
                          (
                          
                            z
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            
                              w
                              -
                              
                                w
                                L
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          
                            p
                            ⋒
                          
                          H
                        
                        ⁡
                        
                          (
                          
                            z
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            
                              w
                              +
                              
                                w
                                L
                              
                            
                          
                          )
                        
                      
                    
                    }
                  
                
                
                  }
                  
                    pulse
                    ⁢
                    
                      
                    
                    ⁢
                    form
                    ⁢
                    
                      
                    
                    ⁢
                    distirtion
                    ⁢
                    
                      
                    
                    ⁢
                    related
                    ⁢
                    
                      
                    
                    ⁢
                    to
                    ⁢
                    
                      
                    
                    ⁢
                    gradient
                    ⁢
                    
                      
                    
                    ⁢
                    of
                    ⁢
                    
                      
                    
                    ⁢
                    LF
                    ⁢
                    
                      
                    
                    ⁢
                    pressure
                    ⁢
                    
                      
                    
                    ⁢
                    oscialltion
                  
                
              
              +
              
                
                  
                    i
                    2
                  
                  ⁢
                  
                    w
                    
                      c
                      0
                    
                  
                  ⁢
                  
                    
                      a
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                    
                      
                        
                          p
                          ⋒
                        
                        L
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          0
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                    ⁢
                    
                      ~
                      w
                    
                    ⁢
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                
                
                  }
                  
                    H
                    ⁢
                    
                      
                    
                    ⁢
                    F
                    ⁢
                    
                      
                    
                    ⁢
                    pulse
                    ⁢
                    
                      
                    
                    ⁢
                    self
                    ⁢
                    
                      
                    
                    ⁢
                    distortion
                  
                
              
            
          
        
      
    
    
      
        (
        83
        )
      
    
  




where we have separated the absorption kernel Hab=Habr+iHabi into its imaginary component Habi that produces the absorption and its real component Habr that introduces a frequency dependent phase velocity, i.e. dispersion. Habr′′0 because the absorption kernel hab in Eq.(13) represents a causal physical process. When we measure the frequency dependent attenuation due to absorption, we measure only Habi(w) and Habr(w) can be obtained from the Hilbert transform as




  
    
      
        
          
            H
            abr
          
          ⁡
          
            (
            w
            )
          
        
        =
        
          
            1
            p
          
          ⁢
          Pr
          ⁢
          
            
              ∫
              
                -
                ¥
              
              ¥
            
            ⁢
            
              
            
            ⁢
            
              
                d
                w
              
              ⁢
              
                
                  
                    H
                    abi
                  
                  ⁡
                  
                    (
                    w
                    )
                  
                
                
                  w
                  -
                  w
                
              
            
          
        
      
    
    
      
        (
        84
        )
      
    
  




The dispersion of the phase velocity given by Habr(w) produces a linear pulse form distortion that is independent of the LF pulse, but similar to the nonlinear pulse form distortion given by the last three terms in Eq.(83). We note that for wLtL=0 or p, the HF pulse is at the crest or trough of the LF oscillation. These are the HF pulses 103 and 106 in FIG. 1a where the LF pulse produces a nonlinear propagation delay given by the LF pressure at the peak of the HF pulse, and a pulse form distortion given by the curving of the LF oscillation along the HF pulse. In this situation we get cos wLtL=−1 and sin wLtL=0, which shows that the 3rd term on the right side represents the nonlinear propagation delay of the center of the HF pulse due to the LF pulse as in Eq.(18), while the 4th term represents pulse form distortion of the HF pulse due to the curvature of the LF pulse along the HF pulse which produces a nonlinear phase distortion.
For wLtL=−p/2, the HF pulse is at the zero crossing of the LF pulse, shown as 111 and 113 in FIG. 1c where the LF pulse produces a nonlinear time compression/elongation of the HF pulse with no contribution to the nonlinear propagation delay. In this situation we have sin wLtL=−1 and cos wLtL=0, which shows that the 5th term represents pulse form distortion of the HF pulse due to the gradient of the LF pulse around the center of the HF pulse which produces a spectral amplitude distortion. These effects are more clearly depicted if we divide Eq.(83) by {circumflex over (p)}H(z,r^, w), neglecting the nonlinear self-distortion term




  
    
      
        
          
            d
            
              
                
                  p
                  ⋒
                
                H
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          
            
              
                p
                ⋒
              
              H
            
            ⁡
            
              (
              
                z
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
                ,
                w
              
              )
            
          
        
        =
        
          
            
              {
              
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                      c
                      0
                    
                    
                      2
                      ⁢
                      w
                    
                  
                  ⁢
                  
                    
                      
                        
                          
                            
                               
                              ⋀
                              2
                            
                            ⁢
                            p
                          
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                
                +
                
                  
                    w
                    
                      2
                      ⁢
                      
                        c
                        0
                      
                    
                  
                  ⁢
                  
                    
                      H
                      abi
                    
                    ⁡
                    
                      (
                      
                        z
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        ,
                        w
                      
                      )
                    
                  
                
              
              }
            
            ⁢
            
              d
              z
            
          
          -
          
            i
            ⁢
            
              w
              
                2
                ⁢
                
                  c
                  0
                
              
            
            ⁢
            
              
                H
                abr
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          +
          
            i
            ⁢
            
              w
              
                c
                0
              
            
            ⁢
            
              a
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              w
              L
            
            ⁢
            
              
                t
                L
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            
              d
              z
            
          
          +
          
            i
            ⁢
            
              w
              
                c
                0
              
            
            ⁢
            
              a
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              w
              L
            
            ⁢
            
              
                t
                L
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          
                            w
                            -
                            
                              w
                              L
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          
                            w
                            +
                            
                              w
                              L
                            
                          
                        
                        )
                      
                    
                  
                  
                    2
                    ⁢
                    
                      
                        
                          p
                          ⋒
                        
                        H
                      
                      ⁡
                      
                        (
                        
                          z
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                
                -
                1
              
              }
            
            ⁢
            
              d
              z
            
          
          -
          
            i
            ⁢
            
              w
              
                c
                0
              
            
            ⁢
            
              a
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              w
              L
            
            ⁢
            
              
                t
                L
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                
                )
              
            
            ⁢
            
              
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      
                        w
                        -
                        
                          w
                          L
                        
                      
                    
                    )
                  
                
                -
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      
                        w
                        +
                        
                          w
                          L
                        
                      
                    
                    )
                  
                
              
              
                2
                ⁢
                
                  
                    
                      p
                      ⋒
                    
                    H
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
            
            ⁢
            
              d
              z
            
          
        
      
    
    
      
        (
        85
        )
      
    
  




Integrating from depth point #(i−1) to #i, gives a recursive integral equation




  
    
      
        
          
            
              p
              ⋒
            
            H
          
          ⁡
          
            (
            
              
                z
                i
              
              ,
              
                
                  r
                  _
                
                ⋀
              
              ,
              w
            
            )
          
        
        =
        
          
            
              
                p
                ⋒
              
              H
            
            ⁡
            
              (
              
                
                  z
                  
                    i
                    -
                    1
                  
                
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
                ,
                w
              
              )
            
          
          ⁢
          exp
          ⁢
          
            
              {
              
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                      c
                      0
                    
                    
                      2
                      ⁢
                      w
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        z
                        
                          i
                          -
                          1
                        
                      
                      
                        z
                        i
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        s
                      
                      ⁢
                      
                        
                          
                            
                              
                                
                                   
                                  ⋀
                                  2
                                
                                ⁢
                                p
                              
                              ⋒
                            
                            H
                          
                          ⁡
                          
                            (
                            
                              s
                              ,
                              
                                
                                  r
                                  _
                                
                                ⋀
                              
                              ,
                              w
                            
                            )
                          
                        
                        
                          
                            
                              p
                              ⋒
                            
                            H
                          
                          ⁡
                          
                            (
                            
                              s
                              ,
                              
                                
                                  r
                                  _
                                
                                ⋀
                              
                              ,
                              w
                            
                            )
                          
                        
                      
                    
                  
                
                +
                
                  
                    w
                    
                      2
                      ⁢
                      
                        c
                        0
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        z
                        
                          i
                          -
                          1
                        
                      
                      
                        z
                        i
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      d
                      
                        
                          sH
                          abi
                        
                        ⁡
                        
                          (
                          
                            s
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            w
                          
                          )
                        
                      
                    
                  
                
              
              }
            
            ·
            exp
          
          ⁢
          
            
              {
              
                
                  
                    -
                    i
                  
                  ⁢
                  
                    w
                    
                      2
                      ⁢
                      
                        c
                        0
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        z
                        
                          i
                          -
                          1
                        
                      
                      
                        z
                        i
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        s
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          H
                          abr
                        
                        ⁡
                        
                          (
                          
                            s
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            w
                          
                          )
                        
                      
                    
                  
                
                -
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  
                    d
                    
                      
                    
                    ⁢
                    
                      t
                      ⁡
                      
                        (
                        
                          
                            z
                            i
                          
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                  
                
              
              }
            
            ·
            exp
          
          ⁢
          
            
              {
              
                i
                ⁢
                
                  w
                  
                    c
                    0
                  
                
                ⁢
                
                  
                    ∫
                    
                      z
                      
                        i
                        -
                        1
                      
                    
                    
                      z
                      i
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      d
                      s
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      a
                      ⁡
                      
                        (
                        
                          s
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      w
                      L
                    
                    ⁢
                    
                      
                        t
                        L
                      
                      ⁡
                      
                        (
                        
                          s
                          ,
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        )
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            
                              
                                
                                  p
                                  ⋒
                                
                                H
                              
                              ⁡
                              
                                (
                                
                                  s
                                  ,
                                  
                                    
                                      r
                                      _
                                    
                                    ⋀
                                  
                                  ,
                                  
                                    w
                                    -
                                    
                                      w
                                      L
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                
                                  p
                                  ⋒
                                
                                H
                              
                              ⁡
                              
                                (
                                
                                  s
                                  ,
                                  
                                    
                                      r
                                      _
                                    
                                    ⋀
                                  
                                  ,
                                  
                                    w
                                    +
                                    
                                      w
                                      L
                                    
                                  
                                
                                )
                              
                            
                          
                          
                            2
                            ⁢
                            
                              
                                
                                  p
                                  ⋒
                                
                                H
                              
                              ⁡
                              
                                (
                                
                                  s
                                  ,
                                  
                                    
                                      r
                                      _
                                    
                                    ⋀
                                  
                                  ,
                                  w
                                
                                )
                              
                            
                          
                        
                        -
                        1
                      
                      )
                    
                  
                
              
              }
            
            ·
            exp
          
          ⁢
          
            {
            
              
                -
                
                  w
                  
                    c
                    0
                  
                
              
              ⁢
              
                
                  ∫
                  
                    z
                    
                      i
                      -
                      1
                    
                  
                  
                    z
                    i
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    s
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    a
                    ⁡
                    
                      (
                      
                        s
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                      
                      )
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    w
                    L
                  
                  ⁢
                  
                    
                      t
                      L
                    
                    ⁡
                    
                      (
                      
                        s
                        ,
                        
                          
                            r
                            _
                          
                          ⋀
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        
                          
                            p
                            ⋒
                          
                          H
                        
                        ⁡
                        
                          (
                          
                            s
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            
                              w
                              -
                              
                                w
                                L
                              
                            
                          
                          )
                        
                      
                      -
                      
                        
                          
                            p
                            ⋒
                          
                          H
                        
                        ⁡
                        
                          (
                          
                            s
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            
                              w
                              +
                              
                                w
                                L
                              
                            
                          
                          )
                        
                      
                    
                    
                      2
                      ⁢
                      
                        
                          
                            p
                            ⋒
                          
                          H
                        
                        ⁡
                        
                          (
                          
                            s
                            ,
                            
                              
                                r
                                _
                              
                              ⋀
                            
                            ,
                            w
                          
                          )
                        
                      
                    
                  
                
              
            
            }
          
        
      
    
    
      
        (
        86
        )
      
    
  




where the term with the nonlinear propagation delay is defined through




  
    d
    
      t
      ⁡
      
        (
        
          
            z
            i
          
          ,
          
            
              r
              _
            
            ⋀
          
        
        )
      
    
  
  =
  
    
      -
      
        1
        
          c
          0
        
      
    
    ⁢
    
      
        ∫
        
          z
          
            i
            -
            1
          
        
        
          z
          i
        
      
      ⁢
      
        
      
      ⁢
      
        
          d
          s
        
        ⁢
        
          
        
        ⁢
        
          a
          ⁡
          
            (
            
              s
              ,
              
                
                  r
                  _
                
                ⋀
              
            
            )
          
        
        ⁢
        cos
        ⁢
        
          
        
        ⁢
        
          w
          L
        
        ⁢
        
          
            t
            L
          
          ⁡
          
            (
            
              s
              ,
              
                
                  r
                  _
                
                ⋀
              
            
            )
          
        
      
    
  




With this form of Eq.(83) we see that the 3rd term clearly takes the form of a nonlinear propagation delay, while the 4th and 5th terms represent pulse form distortion.
We note that the inclusion of {circumflex over (p)}H(s,r^, wmwL) in the two lowest terms is responsible for the pulse form distortion. {circumflex over (p)}H is fairly flat around the HF center frequency and as from Eq.(82) wL<B/2, the major effect on the pulse form distortion is found at the edges of the HF pulse spectrum. One hence obtains some correction for the effect of pulse form distortion in the received HF signal by reducing the bandwidth of the received HF signal in a band pass filter.
B. Model Simulation Procedure
The strongest pulse reverberation noise is generated by reflections from fatty layers in muscular or parenchyma tissue. Typical acoustic parameters of such tissues are given in Table I. bn=1+B/2A is the bulk compressibility nonlinearity parameter, k is the bulk compressibility, bp=bn*k is the nonlinearity parameter used for simulations. r is the tissue mass density, Z0=rc is the characteristic acoustic impedance, and R is the reflection coefficient at planar interfaces between neighbor materials in the table.







TABLE I







Acoustic parameters for typical tissues.
















κ 10−12 
βp
ρ
c
Z0



Tissue
βn
Pa−1
10−9 Pa−1
kg/m3
m/s
MRayl
R

















Fat
6
508
3.05
950
1440
1.37
0.1


Muscle
4.7
380
1.79
1070
1568
1.68
−0.1


Liver
4.4
385
1.67
1050
1573
1.65
0.01


Parenchyma









For the linear HF transmit beam and receive beams we carry through a simulation of Eq.(83) with a=0, where we use aperture boundary conditions as




  
    
      
        
          a
          )
        
        ⁢
        
          
        
        ⁢
        Transmit
        ⁢
        
          
        
        ⁢
        beam
        ⁢
        
          
        
        ⁢
        excitation
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      
                        P
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            0
                          
                          ,
                          w
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            t
                            t
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  r
                                  _
                                
                                0
                              
                              ,
                              
                                
                                  r
                                  _
                                
                                t
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      r
                      0
                    
                    ,
                    
                      S
                      t
                    
                  
                
              
              
                
                  0
                
                
                  else
                
              
            
            ⁢
            
              

            
            ⁢
            b
          
          )
        
        ⁢
        
          
        
        ⁢
        Receive
        ⁢
        
          
        
        ⁢
        beam
        ⁢
        
          
        
        ⁢
        excitation
        ⁢
        
          :
        
        ⁢
        
          
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      P
                      r
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          0
                        
                        ,
                        w
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      w
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          t
                          r
                        
                        ⁡
                        
                          (
                          
                            
                              
                                r
                                _
                              
                              0
                            
                            ,
                            
                              
                                r
                                _
                              
                              r
                            
                          
                          )
                        
                      
                    
                  
                
              
              
                
                  
                    r
                    0
                  
                  ,
                  
                    S
                    r
                  
                
              
            
            
              
                0
              
              
                else
              
            
          
        
      
    
    
      
        (
        87
        )
      
    
  




where the variation of Pt and Pr with r0 opens for apodization of the HF transmit and receive apertures. These excitations at the transducer surface generates the pressure transmit fields Pt(r1,w;rt) and Pr(r1,w;rr) by Eq.(1) for the transmit and receive beams. The spatial frequency responses of the transmit and receive beams are then obtained from Eqs.(1,5) as




  
    
      
        
          
            
              
                
                  
                    a
                    )
                  
                  ⁢
                  
                    
                  
                  ⁢
                  Transmit
                  ⁢
                  
                    
                  
                  ⁢
                  beam
                  ⁢
                  
                    :
                  
                  ⁢
                  
                    
                  
                  ⁢
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  k
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      H
                      t
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          1
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
                =
                
                  
                    
                      
                        P
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            1
                          
                          ,
                          
                            w
                            ;
                            
                              
                                r
                                _
                              
                              t
                            
                          
                        
                        )
                      
                    
                    
                      
                        P
                        t
                      
                      ⁡
                      
                        (
                        
                          0
                          ,
                          w
                        
                        )
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    

                  
                  ⁢
                  b
                
              
              )
            
            ⁢
            
              
            
            ⁢
            Receive
            ⁢
            
              
            
            ⁢
            beam
            ⁢
            
              :
            
            ⁢
            
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            k
            ⁢
            
              
            
            ⁢
            
              
                H
                r
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      r
                    
                  
                
                )
              
            
          
          =
          
            
              
                P
                r
              
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    1
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      r
                    
                  
                
                )
              
            
            
              
                P
                r
              
              ⁡
              
                (
                
                  0
                  ,
                  w
                
                )
              
            
          
        
        ⁢
        
          
        
      
    
    
      
        (
        88
        )
      
    
  




To simulate the effect of the nonlinear manipulation of the HF pulse by the LF pulse, we start by simulating the LF beam from Eq.(75a) in the linear regime, i.e. bp=0, which for the temporal Fourier transform has the same structure as Eq.(83) with a=0. Since we only need the LF pressure within an LF period, we can carry through the simulation at the LF center frequency wL only. The simulation starts at z=0 with PL(r0,WL)e−iwLtL(r0,rL), where the variation of PL(r0,wL) with r0 opens for apodization of the LF transmit aperture. The simulation gives Pt(z,r^,wL;rL), from which one can derive




  
    
      
        
          
            
              
                p
                ⋒
              
              L
            
            ⁡
            
              (
              
                z
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
                ,
                0
              
              )
            
          
          =
          
            
            
              
                P
                L
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  
                    
                      w
                      L
                    
                    ;
                    
                      
                        r
                        _
                      
                      L
                    
                  
                
                )
              
            
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              t
              L
            
            ⁡
            
              (
              
                z
                ,
                
                  
                    r
                    _
                  
                  ⋀
                
              
              )
            
          
          =
          
            
              1
              
                w
                L
              
            
            -
            
              
            
            ⁢
            
              
                P
                L
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  
                    
                      w
                      L
                    
                    ;
                    
                      
                        r
                        _
                      
                      L
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        89
        )
      
    
  




This allows calculation of a(z,r^) from Eq.(77) to simulate the HF transmit beam with nonlinear manipulation from Eq.(83) with a boundary condition at the transmit aperture of z=0 as in Eq.(87a). This gives Ptp(r1,w;rt) of Eq.(19) and hence also Vp(r,w;rt).
We are basing our estimation of the linear and nonlinear scattering on the total receive signal as a function of fast time. As we do not know the details of the 3D scatterer distribution, we assume that we have plane reflectors where the signal models are given in Section 5.6. We then calculate

 Ht(z,w;zt), Hr(z,w;zr), and Vp(z,w) from Eq.(50).  (90)

The linear phase of Vp gives the nonlinear propagation delay from the linear component of the phase as shown, while {circumflex over (V)}p represents nonlinear pulse form distortion, nonlinear self distortion attenuation, and spatial variation of the nonlinear propagation delay across the planes, as discussed in Section 5.7. For the plane reflectors we have U(w)=Hrt(w)ikPt(0,w), where Hrt(w) is the receiver transducer transfer function as given after Eq.(3) [1].
We are now in position of estimation of the combined Class I and II pulse reverberation noise filters from the 1st order HF receive signal with zero LF pulse. We first estimate the ultrasound reflection coefficient for strong plane reflectors. For zero LF pulse the 1st order received signal is from Eq.(53)




  
    
      
        
          
            y
            
              l
              ⁢
              
                
              
              ⁢
              1
            
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            ∫
            0
            ¥
          
          ⁢
          
            
          
          ⁢
          
            
              d
              
                t
                1
              
            
            ⁢
            
              R
              ⁡
              
                (
                
                  t
                  1
                
                )
              
            
            ⁢
            
              
                u
                ⁡
                
                  (
                  t
                  )
                
              
              ⁢
              
                ~
                t
              
              ⁢
              
                
                  h
                  tr
                
                ⁡
                
                  (
                  
                    
                      t
                      1
                    
                    ,
                    
                      t
                      ;
                      
                        t
                        t
                      
                    
                    ,
                    
                      t
                      r
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        91
        )
      
    
  




We apply a whitening filter so that




  
    
      
        
          
            u
            ⁡
            
              (
              t
              )
            
          
          ⁢
          
            ~
            t
          
          ⁢
          
            
              h
              tr
            
            ⁡
            
              (
              
                
                  t
                  1
                
                ,
                
                  t
                  ;
                  
                    t
                    t
                  
                
                ,
                
                  t
                  r
                
              
              )
            
          
        
        ⁢
        fi
        ⁢
        
          
        
        ⁢
        approx
        ⁢
        
          
        
        ⁢
        
          
            U
            a
          
          ⁡
          
            (
            t
            )
          
        
        ⁢
        
          d
          
            (
            
              t
              -
              
                t
                1
              
            
            )
          
        
      
    
    
      
        (
        92
        )
      
    
  




This gives the estimate of the reflection coefficient




  
    
      
        
          
            R
            ^
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              y
              w
            
            ⁡
            
              (
              t
              )
            
          
          
            
              U
              a
            
            ⁡
            
              (
              t
              )
            
          
        
      
    
    
      
        (
        93
        )
      
    
  




where yw(t) is the whitened form of yl1(t). However, as the yl1(t) is a band-pass signal, the whitening will also give a band-pass signal, and we use the envelope of the estimate of the reflection coefficient |{tilde over (R)}(t)| where {tilde over (R)}(t) is the complex envelope of {circumflex over (R)}(t). For the transducer as the 2nd reflector we have z2=0 and R(z2;w)=Rt(w). With a 1st reflector at z1 we will for a fast time t=2z/c get contributions from 3rd reflectors within the interval (z−z1−ZpH/2, z−z1+ZpH/2) where ZpH is the HF pulse length. However, for our estimation of the noise correction delay and speckle correction it is adequate to take contribution from z3=z−z1 only. Writing z=ct/2, we get from Eq.(56) an estimate of the Class I and II noise in this case as




  
    
      
        
          
        
        ⁢
        
          
            
              
                N
                a
              
              ⁡
              
                (
                
                  w
                  ,
                  
                    z
                    ;
                    p
                  
                
                )
              
            
            =
            
              
                
                  
                    N
                    ⋒
                  
                  a
                
                ⁡
                
                  (
                  
                    w
                    ,
                    
                      z
                      ;
                      p
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      t
                      n
                    
                    ⁡
                    
                      (
                      z
                      )
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                N
                a
              
              ⁡
              
                (
                
                  w
                  ,
                  
                    z
                    ;
                    p
                  
                
                )
              
            
            =
            
              
                U
                ⁡
                
                  (
                  w
                  )
                
              
              ⁢
              
                
                  R
                  t
                
                ⁡
                
                  (
                  w
                  )
                
              
              ⁢
              
                
                  H
                  tr
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      r
                    
                  
                  )
                
              
              ⁢
              
                
                  ∫
                  0
                  z
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    
                      z
                      1
                    
                  
                  ⁢
                  
                    
                      
                        R
                        ~
                      
                      a
                    
                    ⁡
                    
                      (
                      
                        
                          z
                          1
                        
                        ,
                        z
                      
                      )
                    
                  
                  ⁢
                  
                    
                      
                        V
                        _
                      
                      p
                    
                    ⁡
                    
                      (
                      
                        
                          z
                          1
                        
                        ,
                        
                          w
                          ;
                          
                            
                              r
                              _
                            
                            t
                          
                        
                      
                      )
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                
                  R
                  ~
                
                a
              
              ⁡
              
                (
                
                  
                    z
                    1
                  
                  ,
                  z
                
                )
              
            
            =
            
              
                
                
                  
                    R
                    a
                  
                  ⁡
                  
                    (
                    
                      z
                      1
                    
                    )
                  
                
                
              
              ⁢
              
                
                  ∫
                  
                    z
                    -
                    
                      z
                      1
                    
                    -
                    
                      
                        Z
                        
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          H
                        
                      
                      /
                      2
                    
                  
                  
                    z
                    -
                    
                      z
                      1
                    
                    +
                    
                      
                        Z
                        
                          p
                          ⁢
                          
                            
                          
                          ⁢
                          H
                        
                      
                      /
                      2
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    
                      z
                      3
                    
                  
                  ⁢
                  
                    
                    
                      
                        R
                        ~
                      
                      ⁡
                      
                        (
                        
                          z
                          3
                        
                        )
                      
                    
                    
                  
                
              
            
          
        
      
    
    
      
        (
        94
        )
      
    
  




With zero LF pulse we have Vp=V0 which is the attenuation due to self distortion of the HF pulse. We are from Eq.(94) now in position to estimate the noise delay correction and speckle correction filter for 3rd order pulse reverberation noise where the transducer is the 2nd reflector as




  
    
      
        
          
            
              M
              a
            
            ⁡
            
              (
              
                w
                ,
                
                  z
                  ;
                  p
                
              
              )
            
          
          =
          
            
              
                
                  
                    N
                    a
                  
                  ⁡
                  
                    (
                    
                      w
                      ,
                      
                        z
                        ;
                        0
                      
                    
                    )
                  
                
                
                  
                    N
                    a
                  
                  ⁡
                  
                    (
                    
                      w
                      ,
                      
                        z
                        ;
                        p
                      
                    
                    )
                  
                
              
              ⁢
              
                1
                
                  1
                  +
                  
                    
                      m
                      N
                    
                    ⁢
                    
                      
                        
                        
                          
                            
                              N
                              a
                            
                            ⁡
                            
                              (
                              
                                w
                                ,
                                
                                  z
                                  ;
                                  0
                                
                              
                              )
                            
                          
                          /
                          
                            
                              N
                              a
                            
                            ⁡
                            
                              (
                              
                                w
                                ,
                                
                                  z
                                  ;
                                  p
                                
                              
                              )
                            
                          
                        
                        
                      
                      2
                    
                  
                
              
            
            =
            
              
                
                  
                    M
                    ⋒
                  
                  a
                
                ⁡
                
                  (
                  
                    w
                    ,
                    
                      z
                      ;
                      p
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      t
                      n
                    
                    ⁡
                    
                      (
                      z
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              
                N
                a
              
              ⁡
              
                (
                
                  w
                  ,
                  
                    z
                    ;
                    0
                  
                
                )
              
            
            
              
                N
                a
              
              ⁡
              
                (
                
                  w
                  ,
                  
                    z
                    ;
                    p
                  
                
                )
              
            
          
          =
          
            
              
                ∫
                0
                z
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    z
                    1
                  
                
                ⁢
                
                  
                    
                      R
                      ~
                    
                    a
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      z
                    
                    )
                  
                
              
            
            
              
                ∫
                0
                z
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    z
                    1
                  
                
                ⁢
                
                  
                    
                      R
                      ~
                    
                    a
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      z
                    
                    )
                  
                
                ⁢
                
                  
                    
                      V
                      _
                    
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        95
        )
      
    
  




where the correction delay ptn(z) is found from the linear phase component of Ma(w,z;p). We note that U(w)Rt(w) is shorted out in this expression, and can be neglected. With extra strong volume scatterers we should also include the pulse reverberation noise components with z2>0 that takes the form




  
    
      
        
          
            
              N
              b
            
            ⁡
            
              (
              
                w
                ,
                
                  z
                  ;
                  p
                
              
              )
            
          
          =
          
            
              U
              ⁡
              
                (
                w
                )
              
            
            ⁢
            
              
                H
                tr
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    r
                  
                
                )
              
            
            ⁢
            
              
                ∫
                0
                z
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    z
                    1
                  
                
                ⁢
                
                  
                    
                      R
                      ~
                    
                    b
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      z
                    
                    )
                  
                
                ⁢
                
                  
                    
                      V
                      _
                    
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              
                R
                ~
              
              b
            
            ⁡
            
              (
              
                
                  z
                  1
                
                ,
                z
              
              )
            
          
          =
          
            
              
              
                
                  R
                  ~
                
                ⁡
                
                  (
                  
                    z
                    1
                  
                  )
                
              
              
            
            ⁢
            
              
                ∫
                0
                
                  z
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    z
                    2
                  
                
                ⁢
                
                  
                  
                    
                      R
                      ~
                    
                    ⁡
                    
                      (
                      
                        z
                        2
                      
                      )
                    
                  
                  
                
                ⁢
                
                  
                    ∫
                    
                      z
                      -
                      
                        (
                        
                          
                            z
                            1
                          
                          -
                          
                            z
                            2
                          
                        
                        )
                      
                      -
                      
                        
                          Z
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            H
                          
                        
                        /
                        2
                      
                    
                    
                      z
                      -
                      
                        (
                        
                          
                            z
                            1
                          
                          -
                          
                            z
                            2
                          
                        
                        )
                      
                      +
                      
                        
                          Z
                          
                            p
                            ⁢
                            
                              
                            
                            ⁢
                            H
                          
                        
                        /
                        2
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      d
                      
                        z
                        3
                      
                    
                    ⁢
                    
                      
                      
                        
                          R
                          ~
                        
                        ⁡
                        
                          (
                          
                            z
                            3
                          
                          )
                        
                      
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        96
        )
      
    
  




The total 3rd order scattered noise is then




  
    
      
        
          
            N
            ⁡
            
              (
              
                w
                ,
                
                  z
                  ;
                  p
                
              
              )
            
          
          =
          
            
              
                
                  AN
                  a
                
                ⁡
                
                  (
                  
                    w
                    ,
                    
                      z
                      ;
                      p
                    
                  
                  )
                
              
              +
              
                
                  N
                  b
                
                ⁡
                
                  (
                  
                    w
                    ,
                    
                      z
                      ;
                      p
                    
                  
                  )
                
              
            
            =
            
              
                
                  N
                  ⋒
                
                ⁡
                
                  (
                  
                    w
                    ,
                    
                      z
                      ;
                      p
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      t
                      n
                    
                    ⁡
                    
                      (
                      z
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            N
            ⁡
            
              (
              
                w
                ,
                
                  z
                  ;
                  p
                
              
              )
            
          
          =
          
            
              U
              ⁡
              
                (
                w
                )
              
            
            ⁢
            
              
                H
                tr
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    r
                  
                
                )
              
            
            ⁢
            
              
                ∫
                0
                z
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    
                      z
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              AR
                              t
                            
                            ⁡
                            
                              (
                              w
                              )
                            
                          
                          ⁢
                          
                            
                              
                                R
                                ~
                              
                              a
                            
                            ⁡
                            
                              (
                              
                                
                                  z
                                  1
                                
                                ,
                                z
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              R
                              ~
                            
                            b
                          
                          ⁡
                          
                            (
                            
                              
                                z
                                1
                              
                              ,
                              z
                            
                            )
                          
                        
                      
                      ]
                    
                  
                
                ⁢
                
                  
                    
                      V
                      _
                    
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        97
        )
      
    
  




The noise correction filter is then




  
    
      
        
          
            M
            ⁡
            
              (
              
                w
                ,
                
                  z
                  ;
                  p
                
              
              )
            
          
          =
          
            
              
                
                  N
                  ⁡
                  
                    (
                    
                      w
                      ,
                      
                        z
                        ;
                        0
                      
                    
                    )
                  
                
                
                  N
                  ⁡
                  
                    (
                    
                      w
                      ,
                      
                        z
                        ;
                        p
                      
                    
                    )
                  
                
              
              ⁢
              
                1
                
                  1
                  +
                  
                    
                      m
                      N
                    
                    ⁢
                    
                      
                        
                        
                          
                            N
                            ⁡
                            
                              (
                              
                                w
                                ,
                                
                                  z
                                  ;
                                  0
                                
                              
                              )
                            
                          
                          /
                          
                            N
                            ⁡
                            
                              (
                              
                                w
                                ,
                                
                                  z
                                  ;
                                  p
                                
                              
                              )
                            
                          
                        
                        
                      
                      2
                    
                  
                
              
            
            =
            
              
                
                  M
                  ⋒
                
                ⁡
                
                  (
                  
                    w
                    ,
                    
                      z
                      ;
                      p
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      t
                      n
                    
                    ⁡
                    
                      (
                      z
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              N
              ⁡
              
                (
                
                  w
                  ,
                  
                    z
                    ;
                    0
                  
                
                )
              
            
            
              N
              ⁡
              
                (
                
                  w
                  ,
                  
                    z
                    ;
                    p
                  
                
                )
              
            
          
          =
          
            
              
                ∫
                0
                z
              
              ⁢
              
                
              
              ⁢
              
                d
                
                  
                    z
                    1
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            AR
                            t
                          
                          ⁡
                          
                            (
                            w
                            )
                          
                        
                        ⁢
                        
                          
                            
                              R
                              ~
                            
                            a
                          
                          ⁡
                          
                            (
                            
                              
                                z
                                1
                              
                              ,
                              z
                            
                            )
                          
                        
                      
                      +
                      
                        
                          
                            R
                            ~
                          
                          b
                        
                        ⁡
                        
                          (
                          
                            
                              z
                              1
                            
                            ,
                            z
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
            
              
                ∫
                0
                z
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    
                      z
                      1
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              AR
                              t
                            
                            ⁡
                            
                              (
                              w
                              )
                            
                          
                          ⁢
                          
                            
                              
                                R
                                ~
                              
                              a
                            
                            ⁡
                            
                              (
                              
                                
                                  z
                                  1
                                
                                ,
                                z
                              
                              )
                            
                          
                        
                        +
                        
                          
                            
                              R
                              ~
                            
                            b
                          
                          ⁡
                          
                            (
                            
                              
                                z
                                1
                              
                              ,
                              z
                            
                            )
                          
                        
                      
                      ]
                    
                  
                
                ⁢
                
                  
                    
                      V
                      _
                    
                    p
                  
                  ⁡
                  
                    (
                    
                      
                        z
                        1
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          t
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        98
        )
      
    
  




where A is a parameter that we want to adjust to balance the z2=0 reflection from the transducer surface with the z2>0 reflection from fatty layers. We could use an a priori value of A when the magnitude of the reflection coefficient is well known, or we could adjust A for maximal suppression of the pulse reverberation noise. The expression for zero pressure Nb(w,z;0) is found by setting Vp=V0 in Eq.(96), and the noise filter takes the same form as in Eq.(97) substituting Na for N.
C. Estimation of Linear and Nonlinear Scattering with Noise Suppression
From the simulations in Sectiopn 5.8B, Eqs.(90) provides estimates of both the nonlinear propagation delay and the pulse form distortion, while Eqs.(95,98) provides estimates of the noise correction delay and noise speckle filter with the material parameters used in the simulation of the wave equations. These parameters can then be used in Eqs.(63-66) to provide estimates of the linear and nonlinear scattering signals with strong suppression of the combined Class I and II pulse reverberation noise, or in Eqs.(67,68) to provide estimates of only the linear scattering with suppression of the combined Class I and II pulse reverberation noise.
The simulations in Section 5.8B can be done with an a priori assessment of the acoustic material parameters as c0(z) and bp(z). An a priori assessment of bp(z) can be obtained from experience, but also from estimation of the nonlinear propagation delay t(z) as for example given in U.S. Pat. No. 8,038,616 or U.S. patent application Ser. No. 12/500,518. We then can estimate bp(z) by Eq.(18) as




  
    
      
        
          
            b
            p
          
          ⁡
          
            (
            z
            )
          
        
        =
        
          
            -
            
              
                c
                0
              
              
                
                  
                    
                      p
                      ⋒
                    
                    L
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      0
                    
                    )
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                
                  w
                  L
                
                ⁢
                
                  
                    t
                    L
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      0
                    
                    )
                  
                
              
            
          
          ⁢
          
            
              ¶
              ⁢
              
                
              
              ⁢
              t
            
            
              ¶
              ⁢
              
                
              
              ⁢
              z
            
          
        
      
    
    
      
        (
        99
        )
      
    
  




Based on the signal estimates from this 1st stage of estimation, one then obtains first estimates of the linear scattering signal Xli(w) in Eqs.(66a) and (68) and nonlinear scattering signal Xni(w) in Eq.(66b).
For improved suppression one can then for example obtain an improved estimate of the nonlinear propagation delay t(z) by correlation techniques between two noise suppressed signals, for example b1 and b2 in Eq.(65). One can then by obtain an improved estimate of bp(z) using Eq.(99), and carry through the procedure again as described above.
The estimate of bp(z), and possibly also c0(z), can be further improved by adjustment in steps to minimize the total energy in for example the linear scattering signal Xli(w) in Eqs.(66a) and (68) and nonlinear scattering signal Xni(w) in Eq.(66b) for maximal suppression of pulse reverberation noise, and also maximizing suppression of leak-through of the linear scattering component Xli(w) in the estimate of the nonlinear scattering component Xni(w). The adjustment of the material parameters can be carried through manually, or automatically by optimizing a functional on the estimated linear and nonlinear scattering components, for example by minimizing the total energy in the linear scattering component Xli(w) of Eq.(66a,68) or the nonlinear scattering component Xni(w) in Eq.(66b).
To improve the speed of the estimation, one can for example carry through the simulations of the wave pulses (Eq.(83) for many values of the material parameters and transmitted amplitudes before measurements, and pre-store in an electronic memory the nonlinear propagation delay and pulse form distortion for these values, and in the estimation process simulations for defined parameters are retrieved from memory based on an address related to said defined parameters.
The storage requirements can be reduced by expressing the functional relationship of the nonlinear propagation delay and pulse form distortion to the material parameters and transmit amplitudes in series representations, and then pre-store in an electronic memory only the coefficients of said series representations. Said series representation can for example be of the Taylor series or Fourier series form in relation to the material parameters and transmitted amplitudes. Estimate values of the nonlinear propagation delay and pulse form distortion can then be obtained from such series representation for given values of the material parameters and transmit amplitudes.
By the comments following Eq.(86) we note that we can obtain some correction for the pulse form distortion by reduction of the bandwidth of the received HF signal through band pass filtering. This band pass filtering provides corrections of the effects of pulse form distortion both on the 1st order and multiple order scattering signals.
5.9 Instrument
The methods according to the invention are conveniently implemented in an instrument where an example embodiment of such an instrument is shown in FIG. 9. The instrument sub-units are set up by a controller unit 901, that communicates with the sub-units via a controller bus 902. The controller takes input from a user via a user interface 903, that is composed of a keyboard, generally composed of control keys and alpha-numeric keys, a track-ball, a mouse, a touch-pad, or alike.
The pulse complexes are transmitted from a transmit radiation unit 904 that have different implementations for the different type of waves. For example, for EL waves the transmit unit typically comprises an EL transducer array, for EM waves the transmit unit typically comprises an EM antenna array in the sub-optical frequency range, while in the optical frequency range the transmit unit can comprise lenses and/or optical fibers, etc., all according to known methods. The transmit radiation unit is driven by a transmit drive unit 907 that generates the LF and HF pulses that composes the transmit pulse complex. For EL and EM waves at sub-optical frequencies, the transmit drive unit is typically implemented with transistors or electron tubes, while at optical frequencies the drive unit would typically comprises one or more lasers.
The transmit radiation unit 904 is often composed of different radiation components for the LF and HF pulses, due to bandwidth limitations in the LF and HF radiation components, where the Figure shows one radiation component 905 for transmission of the LF pulses, and one radiation component 906 for transmission of the HF pulses. The size of the LF and HF apertures are also generally different. However, if wide bandwidth radiation components are available, the same component can be used for transmission of both LF and HF pulses.
The scattered or transmitted HF pulses are picked by a similar HF radiation receive unit, where the Figure shows that the HF radiation receive unit is the same as the HF radiation transmit unit 906. However, the invention also covers the use of different radiation components for HF radiation transmit and receive. Different radiation components for HF transmit and receive can for example be found with an ultrasound ring array that encompasses the object, for example a female breast, to be imaged. With arrays one also generally selects a wider HF receive aperture than HF transmit aperture when the HF receive beam is dynamically focused.
The HF transmit beam necessarily has a fixed focus for each transmit, and in order to have adequate length of the transmit focus region, the HF transmit aperture is often chosen smaller than the HF receive aperture (higher F-number) that operates with dynamic focusing. However, to simplify combined suppression of Class I and II pulse reverberation noise, the invention devices to use close to similar HF transmit and receive beams. Multiple focal points for each image line can then for example be obtained through multiple transmit foci per image line, which allows for wider HF transmit apertures where the methods also have solutions with equal HF transmit and receive apertures and foci to simplify combined suppression of Class I and II pulse reverberation noise. With a given HF transmit and receive focus the instrument can provide synthetic HF receive and transmit focus for at least one more depth through transversal filtering of HF receive signals for many neighboring HF receive beams for said at least one more depth.
The instrument can also comprise means for transmission of pulse complexes with wide HF and LF beams in many directions towards the object, and storing HF receive element signals from each pulse complex transmit. The stored HF receive element signals for a set of transmitted pulse components are then processed to reconstruct HF receive signals in many image points where for each said image points the HF transmit and receive beams are selected close to equal, to simplify the combined suppression of Class I and II pulse reverberation noise.
When the same radiation component is used for HF transmit and receive, a transmit/receive switch 908 connects the HF radiation receive unit 906 to a receiver unit 909. The receiver unit amplifies and digitizes the individual array element signals that are used for further processing to generate measurement or image signals in image points. With synthetic HF transmit and receive beams, the individual HF receive element signals are stored in receiver unit 909 until the HF receive element signals for all transmit pulse complex components that are the basis for the image reconstruction, are received.
The HF receive element signals are then transferred to the HF receive beam former unit 910 that generates the HF receive signal for a focused HF receive beam for each image line. The receive beam former unit can for example generate a dynamically focused HF receive beam for each image line, or generate a HF receive beam with the same aperture and focus as the HF transmit beam, or present a synthetic reconstructed HF receive signal for image points with multi-component transmit pulse complexes.
The output of the beam former unit is then passed to a correction unit 911 that carries through one or both of a i) delay correction with correction delays and ii) a pulse speckle correction with speckle corrections, according to known methods or the methods described above, before a final processing in the processing unit 912, to form measurement or image signals with strong suppression of multiple scattering noise, for the linear and/or the nonlinear scattering components. The processing unit carries through further processing according to known methods to obtain measurement or image signals for display on the unit 913, for example related to the amplitude of the back scattered signal for imaging of structures, displacement or displacement strain or strain rate of the object for example obtained through Doppler phase comparison between RF signals, speckle tracking techniques, imaging of shear wave propagation, etc. according to known methods.
The output of the beam former unit is also taken into an estimation unit 914 that estimates at least one of said correction delays and pulse distortion corrections, according to known methods or the methods described above. The outputs of these estimations are transferred to the correction unit 911 to be applied in the pulse form and/or delay corrections of the HF receive signals in the correction unit.
Many of the components of the instrument are typically implemented as software components in the same computer, where typically after the signal digitization in unit 909 the digital signal would be transferred to a computer with multiple multi-core CPUs (Central Processing Units) and also GPUs (Graphic Processor Units). The simulation of the wave equation as part of the model based estimation scheme described above, would conveniently be implemented in one or more GPUs.
Thus, while there have shown and described and pointed out fundamental novel features of the invention as applied to preferred embodiments thereof, it will be understood that various omissions and substitutions and changes in the form and details of the devices illustrated, and in their operation, may be made by those skilled in the art without departing from the spirit of the invention.
It is also expressly intended that all combinations of those elements and/or method steps which perform substantially the same function in substantially the same way to achieve the same results are within the scope of the invention. Moreover, it should be recognized that structures and/or elements and/or method steps shown and/or described in connection with any disclosed form or embodiment of the invention may be incorporated in any other disclosed or described or suggested form or embodiment as a general matter of design choice. It is the intention, therefore, to be limited only as indicated by the scope of the claims appended hereto.
Appendix A. Synthetic Dynamic Focus Through Transversal Filtering
The composite beam spatial frequency response is the product of the transmit and receive beam spatial frequency responses, i.e.




  
    
      
        
          
            
              
                
                  a
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  
                    H
                    b
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
              =
              
                
                  
                    H
                    r
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
                ⁢
                
                  
                    H
                    t
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          r
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
            
          
        
        
          
            
              
                
                  b
                  )
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      H
                      ~
                    
                    b
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          k
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
              =
              
                
                  
                    
                      H
                      ~
                    
                    r
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          k
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
                ⁢
                
                  ~
                  
                    
                      k
                      _
                    
                    ⋀
                  
                
                ⁢
                
                  
                    
                      H
                      ~
                    
                    t
                  
                  ⁡
                  
                    (
                    
                      z
                      ,
                      
                        
                          k
                          _
                        
                        ⋀
                      
                      ,
                      w
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        
          A
          ⁢
          
            
          
          ⁢
          1
        
        )
      
    
  




where the tilde indicates the Fourier transform that for the composite beam is obtained as a convolution of the transversal Fourier transforms for the transmit and receive beams. To develop an equation of propagation for the composite beam spatial frequency response, we calculate




  
    
      
        
          H
          b
          
          
            
            2
        
        =
        
          
            
               
              2
            
            ⁢
            
              (
              
                
                  H
                  r
                
                ⁢
                
                  H
                  t
                
              
              )
            
          
          =
          
            
              
                2
                ⁢
                
                  
                
                ⁢
                
                  H
                  r
                
                ⁢
                
                  
                
                ⁢
                
                  H
                  t
                
              
              +
              
                
                  
                    
                      H
                      r
                    
                    ⁢
                    
                      
                    
                  
                  2
                
                ⁢
                
                  H
                  t
                
              
              +
              
                
                  
                    
                      H
                      t
                    
                    ⁢
                    
                      
                    
                  
                  2
                
                ⁢
                
                  H
                  t
                
              
            
            =
            
              
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      H
                      r
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      H
                      t
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      k
                      1
                      2
                    
                    ⁢
                    
                      H
                      r
                    
                    ⁢
                    
                      H
                      t
                    
                  
                
                ⇒
                
                  
                    
                      H
                      
                        
                          
                        
                        ⁢
                        b
                      
                      
                      
                        
                        2
                    
                    ⁡
                    
                      (
                      
                        
                          r
                          _
                        
                        ,
                        w
                      
                      )
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      k
                      1
                      2
                    
                    ⁢
                    
                      
                        H
                        b
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                  -
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        H
                        r
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          w
                        
                        )
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        H
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            r
                            _
                          
                          ,
                          w
                        
                        )
                      
                    
                  
                
              
              =
              0
            
          
        
      
    
    
      
        (
        
          A
          ⁢
          
            
          
          ⁢
          2
        
        )
      
    
  




We note from Eq.(1) that in the homogeneous medium, i.e. Habc=1, the spatial frequency responses of the transmit and receive beams take the form




  
    
      
        
          
            
              H
              i
            
            ⁡
            
              (
              
                
                  r
                  _
                
                ,
                w
              
              )
            
          
          =
          
            
              
                ∫
                
                  S
                  i
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    d
                    2
                  
                  ⁢
                  
                    r
                    0
                  
                
                ⁢
                
                  
                    e
                    
                      
                        
                          -
                          i
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          k
                          1
                        
                        ⁢
                        
                          
                          
                            
                              r
                              _
                            
                            -
                            
                              
                                r
                                _
                              
                              0
                            
                          
                          
                        
                      
                      -
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        w
                        ⁢
                        
                          
                        
                        ⁢
                        
                          t
                          ⁡
                          
                            (
                            
                              
                                r
                                _
                              
                              0
                            
                            )
                          
                        
                      
                    
                  
                  
                    2
                    ⁢
                    p
                    ⁢
                    
                      
                      
                        
                          r
                          _
                        
                        -
                        
                          
                            r
                            _
                          
                          0
                        
                      
                      
                    
                  
                
                ⁢
                
                  w
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      0
                    
                    )
                  
                
              
            
            ⪢
            
              
                
                
                  
                    H
                    i
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      ,
                      w
                    
                    )
                  
                
                
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    k
                    1
                  
                  ⁢
                  
                    f
                    ⁡
                    
                      (
                      
                        r
                        _
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            i
            =
            t
          
          ,
          r
        
      
    
    
      
        (
        A3
        )
      
    
  




where k1f(r) is the phase front at r. The phase represents the fastest spatial variation of Hi(r,w) and the gradient can be approximated as [1]




  
    
      
        
          
            H
            i
          
          ⁡
          
            (
            
              
                r
                _
              
              ,
              w
            
            )
          
        
        ⪢
        
          
            -
            i
          
          ⁢
          
            
          
          ⁢
          
            k
            1
          
          ⁢
          
            
          
          ⁢
          
            f
            ⁡
            
              (
              
                r
                _
              
              )
            
          
          ⁢
          
            
            
              
                H
                i
              
              ⁡
              
                (
                
                  
                    r
                    _
                  
                  ,
                  w
                
                )
              
            
            
          
          ⁢
          
            e
            
              
                -
                i
              
              ⁢
              
                
              
              ⁢
              
                k
                1
              
              ⁢
              
                f
                ⁡
                
                  (
                  
                    r
                    _
                  
                  )
                
              
            
          
        
        ⪢
        
          
            -
            i
          
          ⁢
          
            
          
          ⁢
          
            k
            1
          
          ⁢
          
            
              
                e
                _
              
              ni
            
            ⁡
            
              (
              
                r
                _
              
              )
            
          
          ⁢
          
            
              H
              i
            
            ⁡
            
              (
              
                
                  r
                  _
                
                ,
                w
              
              )
            
          
        
      
    
    
      
        (
        A4
        )
      
    
  
  
    
      
        
          i
          =
          t
        
        ,
        r
      
    
    
      
        
      
    
  




where eni is the unit vector normal to the phase fronts. When the transmit and receive beams have close to the same phase fronts, i.e. ent=enr, Eq.(A2) gives a Helmholz propagation equation for the composite beam spatial frequency response as

2Hb(r,w)−4k12Hb(r,w)=0  (A5)

We get (2k1)2 as the propagation factor for the composite beam due to the double propagation distance 2r from the transducer to the scatterer and back to the transducer again. The Fourier transform across the transversal plane gives




  
    
      
        
          
            
              
                ¶
                2
              
              ⁢
              
                
                  
                    H
                    ~
                  
                  b
                
                ⁡
                
                  (
                  
                    z
                    ,
                    
                      
                        k
                        _
                      
                      ⋀
                    
                    ,
                    w
                  
                  )
                
              
            
            
              ¶
              ⁢
              
                
              
              ⁢
              
                z
                2
              
            
          
          +
          
            
              (
              
                
                  4
                  ⁢
                  
                    k
                    1
                    2
                  
                
                -
                
                  k
                  ⋀
                  2
                
              
              )
            
            ⁢
            
              
                
                  H
                  ~
                
                b
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      k
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
        
        =
        0
      
    
    
      
        (
        A6
        )
      
    
  




with the solution for forward propagating waves as




  
    
      
        
          
            
              H
              ~
            
            b
          
          ⁡
          
            (
            
              z
              ,
              
                
                  k
                  _
                
                ⋀
              
              ,
              w
            
            )
          
        
        =
        
          
            
              
                
                  H
                  ~
                
                b
              
              ⁡
              
                (
                
                  0
                  ,
                  
                    
                      k
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
            ⁢
            
              e
              
                
                  -
                  i2
                
                ⁢
                
                  
                
                ⁢
                
                  k
                  1
                
                ⁢
                z
                ⁢
                
                  
                    1
                    -
                    
                      
                        
                          k
                          ⋀
                          2
                        
                        /
                        4
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        k
                        1
                        2
                      
                    
                  
                
              
            
          
          =
          
            
              
                
                  H
                  ~
                
                b
              
              ⁡
              
                (
                
                  
                    z
                    t
                  
                  ,
                  
                    
                      k
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
            ⁢
            
              e
              
                
                  -
                  i2
                
                ⁢
                
                  
                
                ⁢
                
                  
                    k
                    1
                  
                  ⁡
                  
                    (
                    
                      z
                      -
                      
                        z
                        1
                      
                    
                    )
                  
                
                ⁢
                
                  
                    1
                    -
                    
                      
                        
                          k
                          ⋀
                          2
                        
                        /
                        4
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        k
                        1
                        2
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          A
          ⁢
          
            
          
          ⁢
          7
        
        )
      
    
  




For a common focal depth zr=zt for the transmit and the receive beam, the composite field in the focus is from Eq.(A1)




  
    
      
        
          
            H
            b
          
          ⁡
          
            (
            
              
                z
                i
              
              ,
              
                
                  r
                  _
                
                ⋀
              
              ,
              w
            
            )
          
        
        =
        
          
            
              
                H
                r
              
              ⁡
              
                (
                
                  
                    z
                    t
                  
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
            ⁢
            
              
                H
                t
              
              ⁡
              
                (
                
                  
                    z
                    t
                  
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  w
                
                )
              
            
          
          =
          
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    k
                    1
                  
                  ⁢
                  
                    z
                    t
                  
                
              
              
                
                  (
                  
                    2
                    ⁢
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    
                      z
                      t
                    
                  
                  )
                
                2
              
            
            ⁢
            
              
                ∫
                
                  S
                  r
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    x
                    0
                  
                
                ⁢
                
                  d
                  
                    y
                    0
                  
                
                ⁢
                
                  
                    w
                    r
                  
                  ⁡
                  
                    (
                    
                      
                        x
                        0
                      
                      ,
                      
                        y
                        0
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    i
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          k
                          1
                        
                        ⁡
                        
                          (
                          
                            
                              xx
                              0
                            
                            +
                            
                              yy
                              0
                            
                          
                          )
                        
                      
                      /
                      
                        z
                        t
                      
                    
                  
                
                ⁢
                
                  
                    ∫
                    
                      S
                      t
                    
                    
                      
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      d
                      
                        x
                        0
                      
                    
                    ⁢
                    
                      d
                      
                        y
                        0
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        w
                        t
                      
                      ⁡
                      
                        (
                        
                          
                            x
                            0
                          
                          ,
                          
                            y
                            0
                          
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        i
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            
                              k
                              1
                            
                            ⁡
                            
                              (
                              
                                
                                  xx
                                  0
                                
                                +
                                
                                  yy
                                  0
                                
                              
                              )
                            
                          
                          /
                          
                            z
                            t
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        A8
        )
      
    
  




The composite focal field is hence the product of two Fourier transforms for the transmit and receive apertures across the transversal plane. This implies that the focal field is the Fourier transform of the convolution of the transmit and receive apertures, and the transversal Fourier transform of the composite beam spatial frequency response is hence




  
    
      
        
          
            
              H
              ~
            
            b
          
          ⁡
          
            (
            
              
                z
                t
              
              ,
              
                
                  k
                  _
                
                ⋀
              
              ,
              w
            
            )
          
        
        =
        
          
            
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    k
                    1
                  
                  ⁢
                  
                    z
                    t
                  
                
              
              (
              
                
                  
                    l
                    2
                  
                  ⁢
                  
                    z
                    t
                  
                
                
                  2
                  ⁢
                  p
                
              
              )
            
            2
          
          ⁢
          
            
              
                w
                r
              
              ⁡
              
                (
                
                  
                    
                      k
                      _
                    
                    ⋀
                  
                  ⁢
                  
                    
                      z
                      t
                    
                    /
                    
                      k
                      1
                    
                  
                
                )
              
            
            ⁢
            
              ~
              
                
                  r
                  _
                
                ⋀
              
            
            ⁢
            
              
                w
                t
              
              ⁡
              
                (
                
                  
                    
                      k
                      _
                    
                    ⋀
                  
                  ⁢
                  
                    
                      z
                      t
                    
                    /
                    
                      k
                      1
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        A9
        )
      
    
  




We hence see that this expression complies with the general theorem that the transversal width of the beam is lowest when the transversal phase is zero. To move the focus of the composite beam from a depth zt to zf can be achieved by the transversal filtering as




  
    
      
        
          
            
              H
              ~
            
            b
          
          ⁡
          
            (
            
              z
              ,
              
                
                  k
                  _
                
                ⋀
              
              ,
              
                w
                ;
                
                  z
                  f
                
              
            
            )
          
        
        =
        
          
            
              
                H
                ~
              
              b
            
            ⁡
            
              (
              
                z
                ,
                
                  
                    k
                    _
                  
                  ⋀
                
                ,
                
                  w
                  ;
                  
                    z
                    t
                  
                
              
              )
            
          
          ⁢
          
            e
            
              i
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              
                
                  k
                  1
                
                ⁡
                
                  (
                  
                    
                      z
                      f
                    
                    -
                    
                      z
                      t
                    
                  
                  )
                
              
              ⁢
              
                
                  1
                  -
                  
                    
                      
                        k
                        ⋀
                        2
                      
                      /
                      4
                    
                    ⁢
                    
                      k
                      1
                      2
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          A
          ⁢
          
            
          
          ⁢
          10
        
        )
      
    
  




Eqs.(A9,10) are based on a common focus for the transmit and receive beams. If there are different apertures of the transmit and receive beams, we can still have ent≈enr within the narrowest of the beams. Eqs.(A5-10) are still valid, but the achievable focus is determined by the narrowest aperture. We are however interested in the filtered synthetic focusing to obtain dynamic focusing with Q=1 as defined in Eqs.(29,40), i.e. Ht=Hr, which requires wr(r^)=wt(r^), i.e. also Sr=St.
Another interesting situation is that we have a fixed focus transmit beam and a dynamic focus receive beam. The transversal convolution in Eq.(A10b) will then give a phase filter that can be used to obtain a synthetic depth focusing of the transmit beam also.
Appendix B. Plane Reflectors
B1. 1st Order Scattering
We can now apply this result to the transversal integration across the plane at z in the expression for the 1st order scattered signal from the plane at z in Eq.(48). For a linear material we have Vp=1 and the transversal integral takes the form




  
    
      
        
          
            ∫
            
              R
              2
            
            
              
            
          
          ⁢
          
            
              
                d
                2
              
              ⁢
              
                r
                ⋀
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            
              
                kH
                r
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      r
                    
                  
                
                )
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            
              
                kH
                t
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
          
        
        =
        
          
            ∫
            
              R
              2
            
            
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                d
                2
              
              ⁢
              
                r
                ⋀
              
            
            ⁢
            
              
                ∫
                
                  S
                  r
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    d
                    2
                  
                  ⁢
                  
                    r
                    ⋀
                  
                
                ⁢
                
                  
                    w
                    r
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      4
                    
                    )
                  
                
                ⁢
                
                  
                    H
                    abc
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        4
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          r
                        
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    w
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        t
                        r
                      
                      ⁡
                      
                        (
                        
                          
                            
                              r
                              _
                            
                            4
                          
                          ,
                          
                            
                              r
                              _
                            
                            r
                          
                        
                        )
                      
                    
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                k
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  
                    G
                    ⁡
                    
                      (
                      
                        
                          
                            z
                            _
                          
                          +
                          
                            
                              r
                              _
                            
                            ⋀
                          
                        
                        ,
                        
                          
                            r
                            _
                          
                          4
                        
                        ,
                        w
                      
                      )
                    
                  
                  ·
                  
                    
                      ∫
                      
                        S
                        t
                      
                      
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          d
                          2
                        
                        ⁢
                        
                          r
                          0
                        
                      
                      ⁢
                      
                        
                          w
                          t
                        
                        ⁡
                        
                          (
                          
                            
                              r
                              _
                            
                            0
                          
                          )
                        
                      
                      ⁢
                      
                        
                          H
                          abc
                        
                        ⁡
                        
                          (
                          
                            
                              
                                r
                                _
                              
                              0
                            
                            ,
                            
                              w
                              ;
                              
                                
                                  r
                                  _
                                
                                t
                              
                            
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          w
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              t
                              t
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    r
                                    _
                                  
                                  0
                                
                                ,
                                
                                  
                                    r
                                    _
                                  
                                  t
                                
                              
                              )
                            
                          
                        
                      
                      ⁢
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        G
                        ⁡
                        
                          (
                          
                            
                              
                                z
                                _
                              
                              +
                              
                                
                                  r
                                  _
                                
                                ⋀
                              
                            
                            ,
                            
                              
                                r
                                _
                              
                              0
                            
                            ,
                            w
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        B1
        )
      
    
  
  
    
      
        
          
            We
            ⁢
            
              
            
            ⁢
            note
            ⁢
            
              
            
            ⁢
            that
            ⁢
            
              
            
            ⁢
            
              G
              ⁡
              
                (
                
                  
                    
                      r
                      _
                    
                    a
                  
                  ,
                  
                    
                      r
                      _
                    
                    b
                  
                  ,
                  w
                
                )
              
            
          
          =
          
            G
            ⁡
            
              (
              
                
                  
                    r
                    _
                  
                  b
                
                ,
                
                  
                    r
                    _
                  
                  a
                
                ,
                w
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              ∫
              
                R
                2
              
              
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  d
                  2
                
                ⁢
                
                  r
                  ⋀
                
              
              ⁢
              i
              ⁢
              
                
              
              ⁢
              k
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                G
                ⁡
                
                  (
                  
                    
                      
                        z
                        _
                      
                      +
                      
                        
                          r
                          _
                        
                        ⋀
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      4
                    
                    ,
                    w
                  
                  )
                
              
              ⁢
              i
              ⁢
              
                
              
              ⁢
              k
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                
              
              ⁢
              
                G
                ⁡
                
                  (
                  
                    
                      
                        z
                        _
                      
                      +
                      
                        
                          r
                          _
                        
                        ⋀
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      0
                    
                    ,
                    w
                  
                  )
                
              
            
          
          =
          
            
              
                ∫
                
                  R
                  2
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    d
                    2
                  
                  ⁢
                  
                    r
                    ⋀
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                k
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  
                
                ⁢
                
                  G
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        4
                      
                      ,
                      
                        
                          z
                          _
                        
                        +
                        
                          
                            r
                            _
                          
                          ⋀
                        
                      
                      ,
                      w
                    
                    )
                  
                
                ⁢
                i
                ⁢
                
                  
                
                ⁢
                k
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  G
                  ⁡
                  
                    (
                    
                      
                        
                          
                            r
                            _
                          
                          ⋀
                        
                        +
                        
                          z
                          _
                        
                      
                      ,
                      
                        
                          r
                          _
                        
                        0
                      
                      ,
                      w
                    
                    )
                  
                
              
            
            =
            
              i
              ⁢
              
                
              
              ⁢
              k
              ⁢
              
                
              
              ⁢
              2
              ⁢
              
                G
                ⁡
                
                  (
                  
                    
                      
                        2
                        ⁢
                        
                          z
                          _
                        
                      
                      +
                      
                        
                          r
                          _
                        
                        4
                      
                    
                    ,
                    
                      
                        r
                        _
                      
                      0
                    
                    ,
                    w
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        B2
        )
      
    
  
  
    
      
        
          
            
              which
              ⁢
              
                
              
              ⁢
              inserted
              ⁢
              
                
              
              ⁢
              into
              ⁢
              
                
              
              ⁢
              
                (
                
                  B
                  ⁢
                  
                    
                  
                  ⁢
                  3
                
                )
              
              ⁢
              
                
              
              ⁢
              gives
            
            =
            
              
                ∫
                
                  S
                  r
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    d
                    2
                  
                  ⁢
                  
                    r
                    4
                  
                
                ⁢
                
                  
                    w
                    r
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      4
                    
                    )
                  
                
                ⁢
                
                  
                    H
                    abc
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        4
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          r
                        
                      
                    
                    )
                  
                
                ⁢
                
                  
                    e
                    
                      
                        -
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      w
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          t
                          r
                        
                        ⁡
                        
                          (
                          
                            
                              
                                r
                                _
                              
                              4
                            
                            ,
                            
                              
                                r
                                _
                              
                              r
                            
                          
                          )
                        
                      
                    
                  
                  ·
                  
                    
                      ∫
                      
                        S
                        t
                      
                      
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          d
                          2
                        
                        ⁢
                        
                          r
                          0
                        
                      
                      ⁢
                      
                        
                          w
                          t
                        
                        ⁡
                        
                          (
                          
                            
                              r
                              _
                            
                            0
                          
                          )
                        
                      
                      ⁢
                      
                        
                          H
                          abc
                        
                        ⁡
                        
                          (
                          
                            
                              
                                r
                                _
                              
                              0
                            
                            ,
                            
                              w
                              ;
                              
                                
                                  r
                                  _
                                
                                t
                              
                            
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          w
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              t
                              t
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    r
                                    _
                                  
                                  0
                                
                                ,
                                
                                  
                                    r
                                    _
                                  
                                  t
                                
                              
                              )
                            
                          
                        
                      
                      ⁢
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        G
                        ⁡
                        
                          (
                          
                            
                              
                                2
                                ⁢
                                
                                  z
                                  _
                                
                              
                              +
                              
                                
                                  r
                                  _
                                
                                4
                              
                            
                            ,
                            
                              
                                r
                                _
                              
                              0
                            
                            ,
                            w
                          
                          )
                        
                      
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            and
            ⁢
            
              
            
            ⁢
            finally
          
        
        ⁢
        
          
        
      
    
    
      
        (
        
          B
          ⁢
          
            
          
          ⁢
          3
        
        )
      
    
  
  
    
      
        
          
            ∫
            
              R
              2
            
            
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                d
                2
              
              ⁢
              
                r
                ⋀
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            k
            ⁢
            
              
            
            ⁢
            
              
                H
                r
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      r
                    
                  
                
                )
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            
              
                kH
                t
              
              ⁡
              
                (
                
                  z
                  ,
                  
                    
                      r
                      _
                    
                    ⋀
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
          
        
        =
        
          
            
              H
              tr
            
            ⁡
            
              (
              
                
                  z
                  _
                
                ,
                
                  w
                  ;
                  
                    
                      r
                      _
                    
                    t
                  
                
                ,
                
                  
                    r
                    _
                  
                  r
                
              
              )
            
          
          =
          
            
              ∫
              
                S
                r
              
              
                
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                  d
                  2
                
                ⁢
                
                  r
                  4
                
              
              ⁢
              
                
                  w
                  r
                
                ⁡
                
                  (
                  
                    
                      r
                      _
                    
                    4
                  
                  )
                
              
              ⁢
              
                
                  H
                  abc
                
                ⁡
                
                  (
                  
                    
                      
                        r
                        _
                      
                      4
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        r
                      
                    
                  
                  )
                
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  w
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      t
                      r
                    
                    ⁡
                    
                      (
                      
                        
                          
                            r
                            _
                          
                          4
                        
                        ,
                        
                          
                            r
                            _
                          
                          r
                        
                      
                      )
                    
                  
                
              
              ⁢
              i
              ⁢
              
                
              
              ⁢
              
                
                  kH
                  t
                
                ⁡
                
                  (
                  
                    
                      
                        2
                        ⁢
                        
                          z
                          _
                        
                      
                      +
                      
                        
                          r
                          _
                        
                        4
                      
                    
                    ,
                    
                      w
                      ;
                      
                        
                          r
                          _
                        
                        t
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        B4
        )
      
    
  




For linear materials the 1st order reflected field is obtained through a combination of Eqs.(B1,4) as

dYl0(w;z)=dzR(z)U(w)Htr(z,w;rt,rr)  (B5)

B2. 3rd Order Scattering
For a fully linear material we have Vp=1 and the transversal integral in Eq.(49) takes the form




  
    
      
        
          
            ∫
            
              R
              2
            
            
              
            
          
          ⁢
          
            
              
                d
                2
              
              ⁢
              
                r
                
                  ⋀
                  j
                
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            k
            ⁢
            
              
            
            ⁢
            
              
                H
                r
              
              ⁡
              
                (
                
                  
                    
                      2
                      ⁢
                      z
                    
                    -
                    
                      z
                      j
                    
                  
                  ,
                  
                    
                      r
                      _
                    
                    
                      ⋀
                      
                        
                      
                      ⁢
                      j
                    
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      r
                    
                  
                
                )
              
            
            ⁢
            i
            ⁢
            
              
            
            ⁢
            
              
                kH
                t
              
              ⁡
              
                (
                
                  
                    z
                    j
                  
                  ,
                  
                    
                      r
                      _
                    
                    
                      ⋀
                      j
                    
                  
                  ,
                  
                    w
                    ;
                    
                      
                        r
                        _
                      
                      t
                    
                  
                
                )
              
            
          
        
        =
        
          
            ∫
            
              R
              2
            
            
              
            
          
          ⁢
          
            
              d
              2
            
            ⁢
            
              r
              ⋀
            
            ⁢
            
              
                ∫
                
                  S
                  r
                
                
                  
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    d
                    2
                  
                  ⁢
                  
                    r
                    4
                  
                
                ⁢
                
                  
                    w
                    r
                  
                  ⁡
                  
                    (
                    
                      
                        r
                        _
                      
                      4
                    
                    )
                  
                
                ⁢
                
                  
                    H
                    abc
                  
                  ⁡
                  
                    (
                    
                      
                        
                          r
                          _
                        
                        4
                      
                      ,
                      
                        w
                        ;
                        
                          
                            r
                            _
                          
                          r
                        
                      
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
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which is modified to




  
    
      
        
          
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For the linear material, Eq.(54) then takes the form

dYIp(w;z,z1,z2)=dz1dz2R(z1)R(z2;w)R(z−(z1−z2))U(w)Htr(z,w;rt,rr)a)

dYIIp(w;z,z2,z3)=dz2dz3R(z−(z3−z2))R(z2;w)R(z3)U(w)Htr(z,w;rt,rr)b)  (B9)

As z3=z−(z1−z2) and z−(z3−z2) we see that for the linear material the Class I and Class II pulse reverberation noise is equal.
REFERENCES


    [1] Bjorn A. J. Angelsen, “Ultrasound Imaging—Waves, Signals, and Signal Processing”, Emantec, Trondheim, December 2001. http://www.ultrasoundbook.com
    [2] Bjorn A. J. Angelsen and Rune Hansen, “Surf Imaging—a New Method for Ultrasound Contrast Agent Imaging”, IEEE Ultrasonics Symp 2007, New York N.Y.
    [3] Bjorn A. J. Angelsen and Thor Andreas Tangen, “Method for imaging of nonlinear interaction scattering” United Kingdom Patent Application No. 1018933.0

## Claims

1. A method for measurement or imaging of a region of an object with waves where object material parameters for wave propagation and scattering are modified by the wave field strength, comprising:
a) transmitting at least two pulse wave complexes towards said region, each of said pulse complexes comprising
a low frequency (LF) pulse with frequencies in an LF band transmitted from an LF array aperture with LF array elements, and
a high frequency (HF) pulse with frequencies in an HF band transmitted from an HF array aperture with HF array elements, where
the pulse length of the transmitted HF pulse is less than half an oscillation period of the transmitted LF pulse, and where
the transmitted HF pulse propagates spatially so close to the transmitted LF pulse that the HF pulse observes a manipulation of the object material parameters by the co-propagating LF pulse at least for a part of a propagation depth of the HF pulse, and where
at least the transmitted LF pulse varies for each transmitted pulse complex to produce different manipulations of the object, where the transmitted LF pulse for at least one pulse complex can be zero, and the transmitted LF pulse for at least one pulse complex is non-zero, and

b) picking up for each transmitted pulse complex HF receive element signals from one or both of scattered and transmitted HF components from the transmitted pulse complexes using HF receive array elements, where the HF receive element signals are combined into a HF receive signal with a HF receive aperture and HF receive focus, and
c) correcting said HF receive signals at a fast time interval by
i) delay corrections with average correction delays in a fast time where said average correction delays are obtained as a weighted average of an estimate of the nonlinear propagation delay at a 1st reflection for the each multiple scattered HF pulses that are found in the composite multiple scattered HF receive noise signal in said fast time interval, where said weighting is obtained from the received HF signal as a function of depth, and selectably in addition also
ii) speckle corrections in the fast time that compensates for the LF pressure variations of the speckle of the composite multiple scattered HF receive noise signal in said fast time interval,

where one or both of said delay corrections and speckle corrections can be zero for at least one HF receive signal, to form corrected HF signals for said fast time interval, and
d) combining said corrected HF signals from at least two transmitted pulse complexes to form at least one noise suppressed HF signal with suppression of pulse reverberation noise.

2. A method according to claim 1, wherein
steps a) and b) includes transmitting at least 3 pulse complexes with different LF pulses towards the object producing at least 3 HF receive signals, and wherein steps c) and d) includes,
correcting and combining the HF receive signals to form at least two noise suppressed HF signals, and
said at least two noise suppressed HF signals are further corrected by at least one of
i) delay corrections using correction delays in a fast time, and
ii) speckle corrections using speckle corrections in the fast time,

to form at least two corrected noise suppressed HF signals, and
combining said at least two corrected noise suppressed HF signals to form at least one nonlinear scattering HF signal with linear scattering components and pulse reverberation noise substantially suppressed.

3. A method according to claim 2, where said speckle corrections are done with a filter in the fast time.

4. A method according to claim 2, where at least one of said correction delays and said speckle corrections are estimated in a process that uses a mathematical model of LF pulse generated variations in at least one of i) HF receive signal delays, and ii) speckle of the multiple scattered noise signal components in the HF receive signals.

5. A method according to claim 1, where said speckle corrections are done with a filter in the fast time.

6. A method according to claim 5, where said filter is a bandpass filter that reduces the bandwidth of the signals.

7. A method according to claim 1, where the HF transmit and receive beams are matched so that Class I and Class II 3rd order scattering noise for zero LF pulse is close to equal.

8. A method according to claim 1, where in addition
multiple transmit beams with matched HF receive beams are transmitted in different beam directions to form HF receive signals for said different beam directions, and one of
i) processing of the HF receive signals according to point c) and d) of claim 1 to form noise suppressed HF receive signals for said different beam directions, followed by
transversal filtering of the noise suppressed HF receive signals for said different beam directions for a set of depths to obtain synthetic noise suppressed HF receive signals with synthetic HF image beam foci for said set of depths, and

ii) transversal filtering of the HF receive signals from said different beam directions for a set of depths to obtain synthetic HF receive signals with synthetic HF image beam foci for said set of depths, followed by
processing of the synthetic HF receive signals according to point c) and d) of claim 1 to form synthetic noise suppressed HF receive signals with synthetic HF image beam foci for said set of depths.

9. A method according to claim 7, comprising the image of more than one depth region where each depth region is imaged with different HF receive signals from different transmitted pulse complexes with HF transmit and receive beam focus selected within said each depth region.

10. A method according to claim 1, comprising
a) transmitting more than one pulse complex with different directions towards the object, and
b) receiving and storing HF receive array element signals for each HF receive array element and each transmitted pulse complex, and
c) combining said HF receive array element signals from said transmitted pulse complexes for a group of image points to provide synthetic HF receive signals with synthetic HF transmit and receive beams for each image point, said synthetic HF transmit and receive beams are constructed so that a difference between Class I and Class II 3rd order scattered noise-signals in said synthetic HF receive signals is substantially minimized in each image point for pulse complexes with zero LF pulse.

11. A method according to claim 10, where each said more than one transmitted pulse complex have wave fronts that are substantially plane in at least one direction.

12. A method according to claim 1, where at least one of said correction delays and said speckle corrections are estimated in a process that uses a mathematical model of LF pulse generated variations in at least one of i) HF receive signal delays, and ii) speckle of the multiple scattered noise signal components in the HF receive signals.

13. A method according to claim 12, said process comprising
a) obtaining the nonlinear propagation delay and pulse form distortion for the forward propagating HF pulse at multiple depths through simulation of a nonlinear wave equation for the forward propagating HF pulse with the actual transmitted LF/HF pulse complex and defined object material parameters, and
b) obtaining from the measured HF receive signal, parameters relating to relative reflection coefficients as a function of depth along the beam, and
c) estimating said correction delays and speckle corrections in processes that include combining one or both of said nonlinear propagation delays and pulse form distortions for the HF pulse obtained under point a) with said parameters relating to relative reflection coefficients as a function of depth obtained under point b) in a mathematical model for multiple scattered HF noise signals.

14. A method according to claim 13, where the process is carried through in steps where each step applies the method according to claim 13, and for each step said defined object material parameters are adjusted to increase the suppression of one or both of i) pulse reverberation noise and ii) 1st order linear scattering components in the processed HF receive signal, where said adjustment is carried through as one or both of
a) manual adjustment based on observation of a displayed image, and
b) automatic adjustment to optimize a functional on a processed HF receive signal in a region of an image.

15. A method according to claim 14, where said functional is the total power of a processed HF receive signal in a region of an image.

16. A method according to claim 14, where the simulation of a nonlinear wave equation for the forward propagating HF pulse with an actual transmitted LF/HF pulse complex and defined object material parameters is carried through for a set of multiple defined parameters before the measurements and pre-stored in an electronic memory, and in the estimation process the simulation for defined object material parameters is retrieved from memory based on a memory address related to said defined parameters.

17. A method according to claim 13, where for point a) coefficients of series expansions of the nonlinear propagation delay and pulse form distortions dependency on the LF pulse and defined object material parameters are calculated and stored, so that for a given transmitted LF pulse and defined object material parameters, estimates of the nonlinear propagation delays and pulse form distortions can be calculated by said series expansions utilizing said stored coefficients of the series expansions.

18. A method according to claim 17, where said series expansions are one or both of Taylor series and Fourier series expansions.

19. A method according to claim 1, where the LF and HF transmit apertures are defined so that the nonlinear propagation delay τ(t) for the transmitted HF pulse has a substantially linear variation with the fast time t, and for suppression of pulse reverberation noise said correction delays are approximated by τ(t)/2.

20. A method according to claim 1, where for suppression of pulse reverberation noise in echo weak regions, the same HF receive signal obtained with a non-zero LF pulse is delay corrected with two different correction delays to provide two corrected HF signals, and said two corrected HF signals are combined to provide a noise suppressed HF signal.

21. A method according to claim 1, where transmitting LF array elements are selected so that said LF array aperture has a region around the center with no transmission of the LF pulse, so that manipulation of the object material parameters by the LF pulse observed by the co-propagating HF pulse is very low in a near field region of the HF array aperture.

22. A method according to claim 1, where said noise suppressed HF signals are used in the process of estimating corrections for wave front aberrations in materials with spatial variations in the wave propagation velocity.

23. A method according to claim 1, where in addition
multiple transmit beams with matched HF receive beams are transmitted in different beam directions to form HF receive signals for said different beam directions, further including transversal filtering of one of
i) the HF receive signals from said different beam directions for a set of depths to obtain synthetic HF receive signals with synthetic HF image beam foci for said set of depths, where said set of synthetic HF receive signals is further processed according to c) and d) of claim 1 and
ii) the noise suppressed HF receive signals for said different beam directions for a set of depths to obtain synthetic noise suppressed HF receive signals with synthetic HF image beam foci for said set of depths.

24. An instrument for measurement or imaging of a region of an object with waves where the object material parameters for wave propagation and scattering are modified by the wave field strength, comprising:
a) transmit means comprising
a selectable low frequency (LF) transmit array aperture with LF array elements and a selectable LF transmit focus and LF transmit beam direction, and
a selectable high frequency (HF) transmit array aperture with HF array elements and a selectable HF transmit focus and HF transmit beam direction, that are used for
transmitting at least two pulse wave complexes towards said region, each of said pulse complexes being composed of an LF pulse with frequencies in an LF band, and an HF pulse with frequencies in an HF band, where
the transmitted HF pulse propagates spatially so close to the transmitted LF pulse that the HF pulse observes a manipulation of the object material parameters by the co-propagating LF pulse at least for a part of a propagation depth of the HF pulse, and
at least the transmitted LF pulse varies for each transmitted pulse complex to produce different manipulations of the object, where the transmitted LF pulse for at least one pulse complex can be zero, and the transmitted LF pulse for at least one pulse complex is non-zero, and

b) HF receive and receive and HF beam forming means comprising HF receive array elements that for each transmitted pulse complex picks up HF receive element signals from one or both of scattered and transmitted HF components from the transmitted pulse complexes, where the beam forming means combines said HF receive element signals into HF receive signals with a HF receive aperture and a HF receive focus, and
c) correction means for correcting said HF receive signals by
i) delay corrections with average correction delays in a fast time where said average correction delays are obtained as a weighted average of an estimate of the nonlinear propagation delay at the 1st reflection for the each multiple scattered HF pulses that are found in the composite multiple scattered HF receive noise signal in said fast time interval, where said weighting is obtained from the received HF signal as a function of depth, and selectably in addition also
ii) speckle corrections in the fast time that compensates- for the LF pressure variations of the speckle of the composite multiple scattered HF receive noise signal in said fast time interval,

where one or both of said delay corrections and speckle corrections can be zero for at least one HF receive signal, to form corrected HF signals for said fast time interval, and
d) processing means for further combining said corrected HF signals to form at least one noise suppressed HF signal with suppression of pulse reverberation noise.

25. An instrument according to claim 24, where said transmit means comprises means for transmitting at least 3 pulse complexes with different LF pulses towards the object, and wherein step c) includes means for
a) correcting and combining the HF receive signals according to claim 21 to form at least two noise suppressed HF signals, and
b) means for further processing of said at least two noise suppressed HF signals including at least one of
i) delay corrections using correction delays in a fast time (depth-time), and
ii) speckle corrections in the fast time,

to form at least two corrected noise suppressed HF signals, and
c) means for combining said corrected noise suppressed HF signals to suppress linear scattering components to form nonlinear measurement or imaging HF signals.

26. An instrument according to claim 25, including means for speckle corrections in the form of a filter in the fast time.

27. An instrument according to claim 25, comprising estimation means for estimation of at least one of said correction delays and speckle corrections.

28. An instrument according to claim 24, including means for speckle corrections in the form of a filter in the fast time.

29. An instrument according to claim 28, where said filter in the fast time is a bandpass filter that reduces the bandwidth of the signals.

30. An instrument according to claim 24, where said HF transmit, HF receive and HF beam forming means comprise means for matching HF transmit and receive beams so that Class I and Class II 3rd order scattering noise for zero LF pulse is close to equal.

31. An instrument according to claim 24, where said transmit, HF receive and HF beam forming means comprise
means for transmitting multiple pulse complexes with different transmit beam directions with matched HF receive beams to form HF receive signals in said different directions, and one of
i) means for processing of the HF receive signals according to point c) and d) of claim 1 to form noise suppressed HF receive signals for said different beam directions, followed by
means for transversal filtering of the noise suppressed HF receive signals for said different beam directions for a set of depths to obtain synthetic noise suppressed HF receive signals with synthetic HF image beam foci for said set of depths, and

ii) means for transversal filtering of the HF receive signals from said different beam directions for a set of depths to obtain synthetic HF receive signals with synthetic HF image beam foci for said set of depths, followed by
means for processing of the synthetic HF receive signals according to point c) and d) of claim 1 to form synthetic noise suppressed HF receive signals with synthetic HF image beam foci for said set of depths.

32. An instrument according to claim 24, where
a) said transmit means allows transmitting more than one pulse complex with different directions towards said object, and
b) said HF receive means comprises means for receiving and storing HF receive array element signals for each HF receive array element and each transmitted pulse complex, and
c) said HF beam forming means comprises means for combining said HF receive array element signals from said transmitted pulse complexes for a group of image points to provide synthetic HF receive signals with synthetic HF transmit and receive beams for each image point, said synthetic HF transmit and receive beams are constructed so that a difference between Class I and Class II 3rd order scattered noise signals in said synthetic HF receive signals is substantially minimized in each image point of said group of image points for pulse complexes with zero LF pulses.

33. An instrument according to claim 32, where said transmit means allows transmit of LF and HF pulse waves with wave fronts that are substantially plane in at least one direction.

34. An instrument according to claim 24, comprising estimation means for estimation of at least one of said correction delays and speckle corrections.

35. An instrument according to claim 34, where said estimation means uses a mathematical model of LF pulse generated variations in at least one of i) HF receive signal delays, and ii)) speckle of the multiple scattered noise signal components in the HF receive signals.

36. An instrument according to claim 35, where said estimation means comprises
a) means for simulation of a nonlinear wave equation for the forward propagating HF pulse with an actual transmitted LF/HF pulse complex with defined object material parameters, and obtaining the nonlinear propagation delay and pulse form distortion for the simulated HF pulsed wave, and
b) means for obtaining from the measured HF receive signal, parameters relating to relative reflection coefficients as a function of depth along the beam, and
c) means for estimation of said correction delays and speckle corrections in a process that includes combining one or both of said nonlinear propagation delays and pulse form distortions for the simulated HF pulsed wave obtained under point a) with said parameters relating to relative reflection coefficients as a function of depth obtained under point b) in a mathematical model for multiple scattered HF noise signals.

37. An instrument according to claim 36, where said estimation means comprises means for adjusting said defined object material parameters to increase the suppression of one or both of i) pulse reverberation noise and ii) 1st order linear scattering components in the processed HF receive signal, where said adjusting is carried through as one or both of
i) manual adjusting based on observation of the displayed image, and
ii) automatic adjusting to optimize a functional HF receive signal in a region of an image.

38. An instrument according to claim 36, where said means for simulation of a nonlinear wave equation for the forward propagating HF pulse with an actual transmitted LF/HF pulse complex with defined object material parameters, comprises means for storage of a set of simulations with different defined object material parameters where said set of simulations are carried through before the measurements, and in the estimation process the simulation for defined object material parameters is retrieved from memory based on a memory address related to said defined parameters.

39. An instrument according to claim 36, where said means for simulation comprises means for calculation and storage of coefficients of series expansions of the nonlinear propagation delay and pulse form distortions dependency on the LF pulse and defined object material parameters, so that for a given transmitted LF pulse and defined object material parameters, estimates of the nonlinear propagation delays and pulse form distortions can be calculated by said series expansions utilizing said stored coefficients of the series expansions.

40. An instrument according to claim 24, where said transmit means allow the LF and HF transmit apertures to be defined so that the nonlinear propagation delay τ(t) for the transmitted HF pulse has a substantially linear variation with the fast time t, so that for suppression of pulse reverberation noise at fast time t, said correction delays can be approximated by τ(t)/2.

41. An instrument according to claim 24, where said correction means delay corrects the same HF receive signal obtained with a non-zero LF pulse with two different correction delays to provide two corrected HF signals, and said two corrected HF signals are combined to provide a noise suppressed HF signal.

42. An instrument according to claim 24, including means for selecting the transmitting LF array aperture to obtain an LF array aperture that has a region around the centre with no transmission of the LF pulse, so that manipulation of the object material parameters by the LF pulse observed by the co-propagating HF pulse is very low in a near field region of the HF array aperture.

43. An instrument according to claim 24, including
means for estimation of corrections for wave front aberrations in materials with spatial variations in the wave propagation velocity, and
means for corrections of the HF receive element signals for wave front aberrations.

44. An instrument according to claim 24, where said transmit, HF receive and HF beam forming means comprise
means for transmitting multiple pulse complexes with different transmit beam directions with matched HF receive beams to form HF receive signals in said different directions, and further including means for transversal filtering of one of
i) the HF receive signals from said different beam directions for a set of depths to obtain synthetic HF receive signals with synthetic HF image beam foci for said set of depths, where said set of synthetic HF receive signals is further processed according to c) and d) of claim 1 and
ii the noise suppressed HF receive signals for said different beam directions for a set of depths to obtain synthetic noise suppressed HF receive signals with synthetic HF image beam foci for said set of depths.