Patent Publication Number: US-8121791-B2

Title: Spectral shaping inversion and migration of seismic data

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is the National Stage entry under 35 U.S.C. 371 of PCT/US2008/085674 that published as WO 2009/088602 and was filed on 5 Dec. 2008, which claims the benefit of U.S. Provisional Application No. 61/010,407, filed on 8 Jan. 2008, which is incorporated by reference, in its entirety, for all purposes. 
    
    
     TECHNICAL FIELD 
     This description relates generally to the field of geophysical exploration. Specifically, this description relates to seismic reflection imaging based on inversion and migration to estimate subsurface physical properties, e.g., impedance, and/or to generate geophysical models of a subsurface region(s). 
     BACKGROUND 
     Relationships among seismic, electrical and reservoir properties are often exploited in geophysical exploration to model geophysical properties of subsurface regions, e.g., where data from seismic and/or electromagnetic surveys are used to predict a range of features of a subsurface region. The predicted geophysical features are then used for various exploration decisions, e.g., the number of wells to drill, the type of well(s) to drill, and optimal well location to recover resource(s) from a reservoir. 
     Seismic properties of a subsurface region are those properties that directly determine the reflection and transmission of seismic waves by the subsurface, and together define at least the compressional wave velocity, shear wave velocity, and density of the subsurface region. It is often more convenient to express the seismic properties of a subsurface in terms of elastic properties, such as bulk modulus and shear modulus (also called the elastic moduli). Various functions of the velocities and density of the subsurface can also be equivalently used to express seismic properties, including: bulk modulus, Poisson&#39;s ratio, Vp/Vs ratio, P-wave modulus, impedance, and Lamé parameters. Seismic properties may also include, for example, anisotropy and attenuation. Seismic wave velocities may also vary with the frequency of the seismic wave, a phenomenon called dispersion. 
     Among the seismic properties, impedance is the product of seismic velocity and the density. Impedance, also called acoustic impedance and often symbolized by I P , will typically vary among different rock layers, e.g., opposing sides of an interface will have different impedances. The reflection coefficient of an interface generally depends on the contrast in acoustic impedance of the rock on either side of the interface. Specifically, the difference in acoustic impedance between rock layers affects the reflection coefficient. One geophysical modeling process for determining the impedance structure of a subsurface region based on recorded seismic reflection data is seismic inversion. 
     Seismic inversion techniques rely upon seismic reflection data, typically obtained through a seismic survey and analysis of the seismic data from the survey. Seismic reflection techniques are typically based on the generation of seismic waves in the earth&#39;s surface, through the use of one or more seismic sources, e.g., dynamite, air guns, vibrators, and the recording and analysis of the portions of these waves that get reflected at the boundaries between the earth&#39;s layers.  FIGS. 1A-1B  are views of convolutional models for seismograms generated from primary reflections at one or more boundaries between two or more media. Referring to  FIG. 1A , a single boundary model  100  shows that at a given boundary between two media, the amplitude (strength) of the reflected wave is proportional to the amplitude of the incident wave and a quantity called a reflection coefficient. The value of the reflection coefficient depends on the elastic parameters of the two media, and for normal incidence it is given by equation (1). The seismic trace for this case contains a single pulse, whose shape is that of the seismic wavelet. 
     The reflection coefficient, for normal incidence (rays perpendicular to the reflecting interface), is defined as:
 
 R =( I   P2   −I   P1 )/( I   P2   +I   P1 )  (1)
 
In equation (1), R is the reflection coefficient and the quantities I P1  and I P2  are called compressional impedances.
 
     The terms P-impedance and acoustic impedance are also commonly used to describe the same quantities. For example, compressional impedance is defined as the product of density and compressional (P-wave) velocity:
 
 I   P   =ρV   P   (2)
 
In this equation ρ is density and V P  is the P-wave velocity. In equation (1), I P1  and I P2  are the compressional impedances of the layers above and below the reflective interface, respectively. For a large number of reflecting boundaries, the recorded seismic reflection response is the sum of the responses for the different boundaries.
 
     Referring to  FIG. 1B , the multiple boundary model  150  shows that a reflection event is typically recorded on every seismic trace at any given time. The recorded seismogram for the multiple boundary reflection configuration can then be thought of as a reflectivity time series, e.g., that is symbolized by r(t) and based on an impedance profile I P (t). If multiple reflections are ignored, and the pulse generated by the seismic acquisition system is a simple spike, the recorded seismic trace is composed of a sequence of reflectivity spikes, with the size of each of them calculated based on equations (1) and (2). 
     However, the incident seismic wave is typically not a simple spike, but a broader waveform, called the seismic wavelet w(t). In this case, the recorded seismogram is not be r(t). Instead, every spike is replaced by an appropriately scaled version of the seismic wavelet, and the results added. When the reflecting medium contains multiple reflecting boundaries, the resulting seismic trace is further evaluated by calculating the convolution of the seismic wavelet and the reflectivity time series. The reflectivity time series is a sequence of spikes, each of them generated by a single boundary, according to equation (1). The mathematical operation that combines the reflectivity time series r(t) and the wavelet w(t) in the manner just described is convolution:
 
 s ( t )= r ( t )* w ( t )  (3)
 
where the symbol * denotes the operation of convolution in equation (3). In equation (3), the recorded seismogram s(t) is calculated as the convolution of the reflectivity series r(t) and the wavelet w(t). Equation (3) expresses what is typically referred to as the convolutional model of reflection seismology.
 
     Assuming continuous recording of seismic reflections, the equation for calculating the normal-incidence reflection coefficient (equation (1)), can be generalized to the following expression:
 
 r ( t )=( dI   P ( t )/ dt )/(2 I   P ( t ))  (4)
 
     In equation (4), I P (t) represents the impedance value for a layer at a depth such that the reflection from the layer is recorded at a time t. The operator d/dt represents the derivative with respect to time. An exemplary seismic inversion problem from normal-incidence seismic data amounts to solving equations (3) and (4) to determine the impedance function I P (t), and assuming knowledge of the recorded seismic data s(t) and the seismic wavelet w(t). In the limit when the time interval between recorded spikes is very small, one can consider the reflectivity series as a continuous function of time, whose relationship to impedance, for normal incidence, is given by equation (4). For non-normal incidence the calculation of the reflection coefficients is modified, but the convolutional model, as described here for primary reflections only, remains valid. 
     Estimation of the seismic wavelet w(t) can be achieved by making use of well log data. When a well is available and appropriate sonic and density well logs have been collected, the impedance I P (t) and reflectivity r(t) are known. Equation (3) can then be used to solve for w(t), given r(t) and the seismic trace s(t). For this estimation to work adequately, an accurate correlation usually needs to be established between subsurface information at the well and the seismic events. The term “well tie” is commonly used to describe the process of establishing this correlation. Accordingly, accurate well ties are a prerequisite for most inversion methods. 
     The aforementioned concepts can also be generalized to the case where the recorded reflections correspond to larger angles between the incident and reflected wave propagation paths, e.g., oblique or non-normal incidence case. For such situations the convolutional model equation (3) is still valid, but the expression for the reflection coefficient equation (4) is replaced with a more complicated expression, e.g., containing additional elastic parameters, such as shear-wave velocity. 
     Various seismic inversion techniques based on the convolutional models have been applied in common practice. Two recently developed seismic inversion techniques that are implemented as simple modifications of the frequency spectrum are Coloured inversion and Spectral Shaping inversion. These seismic inversion techniques are further described in Lancaster, S., and Whitcombe, D., 2000, “Fast Track “Coloured” Inversion,” Expanded Abstracts, 70th SEG Annual Meeting, Calgary, 1572-1575; and Lazaratos, S., 2006, “Spectral Shaping Inversion For Elastic And Rock Property Estimation,” Research Disclosure, Issue 511, November 2006. 
     Referring to  FIG. 2 , while the two techniques differ in their implementation, both inversion techniques are similar conceptually. For example, impedance estimation is performed by the combination of a phase rotation (−90°) and a spectral shaping operation applied to seismic data. Prior to the application of the phase rotation and the spectral shaping operation, the seismic data is typically converted to zero-phase, e.g., for zero-phase data all frequency components of the seismic wavelet are synchronized and combined to produce a wavelet that is symmetric around the wavelet peak. Coloured inversion assumes a log amplitude spectra follows an exponential law, while spectral shaping inversion (Lazaratos) does not require this assumption. In addition, coloured inversion is strictly a zero-offset inversion. Spectral shaping inversion also provides added benefits of being useful in generating estimates of both elastic and rock properties. 
     The spectral shaping operation is implemented by the application of a filter that reshapes the original seismic spectrum to make the seismic spectrum similar to the average spectrum of well logs recorded at wells in the subsurface region. Referring to  FIG. 2 , a graphical view  200  demonstrates how the spectral shaping filters significantly amplify the energy in the low-frequency part of the seismic spectrum. Average local well log  220  and original seismic frequency  240  spectra are significantly different even over the range of frequencies for which the signal-to-noise ratio of the data is positive. Spectral shaping reshapes the original spectrum to make it similar to the log spectrum. The resulting frequency spectrum is the shaped seismic spectrum  260 . The shaping operation implies significant amplification of the low-frequency energy, as seen in  FIG. 2 . 
     Lazaratos [2006] provides a mathematical derivation demonstrating that, under assumptions that are commonly satisfied, the spectral shaping procedure highlighted above provides an estimate of the impedance, solving equations (3) and (4). For example, based on the convolutional model established above, a seismic trace can be expressed by the convolution equation (5): 
     
       
         
           
             
               
                 
                   
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     In the above expression, and hereinafter, the following notation convention is used to describe one or more of the following features:
         s(t), S(f) seismic trace and its Fourier transform   S quad (f) Fourier transform of quadrature trace   w(t), W(f) wavelet and its Fourier transform   r(t) reflectivity   I P (t), I P (f) P-impedance and its Fourier transform     I P    lowpass filtered P-impedance   Δt the sampling rate       

     The term I P (t) in the denominator can be replaced by a very slowly changing function, which just contains the trend in I P . In practice, such a function can be generated by lowpass filtering I P , to maintain frequencies at the very low end of the spectrum (e.g. 0-2 Hz). This low-frequency term can then be treated as a simple multiplier and moved to the left of the convolution operator. The convolution equation then becomes (equation (6)): 
     
       
         
           
             
               
                 
                   
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     A weak-scattering assumption, stated as follows, is relied upon to mathematically show the ability to transform the convolution equation from its original form to the one given in equation (6). P-impedance can be decomposed into a slowly changing background part, e.g., low-frequency trend, frequencies well below the seismic bandwidth, and a higher-frequency perturbation part including changes in the seismic bandwidth and above. Accordingly, (i) the perturbation should be weak relative to the background, and (ii) the background is essentially constant within the length of the seismic wavelet. Based on numerous observations supporting these conclusions, transforming equation (6) to the frequency domain results in equation (7): 
     
       
         
           
             
               
                 
                   
                     
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     Averaging for several wells (using &lt; &gt; to signify the averaging operation), results in equation (8): 
                     〈       S   quad     ⁡     (   f   )       〉     =         πΔ   ⁢           ⁢   t         I   P     _       ⁢     fW   ⁡     (   f   )       ⁢     〈       I   P     ⁡     (   f   )       〉               (   8   )               
where it is assumed that the seismic wavelet is constant for the area within which the wells are located.
 
     By definition, the shaping filter&#39;s frequency response is the ratio of the average log spectrum and the average seismic spectrum, as seen in equation (9): 
                     Shaping   ⁢           ⁢   Filter   ⁢     :     ⁢           ⁢     H   ⁡     (   f   )         =         〈       I   P     ⁡     (   f   )       〉       〈       S   quad     ⁡     (   f   )       〉       =           I   P     _       πΔ   ⁢           ⁢   t       ⁢     1     fW   ⁡     (   f   )                     (   9   )               
and applying this to the seismic data results in equation (10):
 
Shaped Seismic= H ( f ) S   quad ( f )= I   P ( f )  (10)
 
     Seismic migration of seismic data is a correction technique involving rearrangement of seismic events, so that reflections are plotted at a true representation of their subsurface locations. Referring to  FIG. 3 , a graphical model  300  shows, on the original recorded data, reflections from dipping interfaces are recorded at surface positions that are not directly above the subsurface locations where the reflections take place. In addition, isolated point-like discontinuities in the subsurface (point scatterers) generate seismic events (diffractions) recorded over a large range of receivers, that can make the interpretation of seismic data confusing. Seismic velocity variations are one more reason the original recorded data provide only a distorted view of the subsurface geology. The seismic migration technique addresses the above issues and is therefore utilized in many seismic data processing sequences to accurately depict the structures and geometric configurations observed in seismic recordings as an analog of the geologic layers that gave rise to the seismic reflections. 
     The need to correctly position dipping reflectors is best seen in  FIG. 3 . The reflection pulse from point A generated from a source at S 1  and recorded at a receiver also at S 1  is plotted on the trace under S 1 , at point A′, which is selected such that the lengths of S 1 A and S 1 A′ are equal (assuming a constant-velocity subsurface for simplicity). Similarly, the reflection pulse from point B is plotted on the trace under S 2 , at point B′. The reflector segment AB is plotted at the erroneous lateral position A′B′ and has a dip smaller than AB&#39;s true dip. Migration is the correction technique that corrects such distortions. Before migration, the structures and geometric configurations observed in seismic recordings are typically not an accurate description of the geologic layers that gave rise to the seismic reflections. 
     Seismic inversion has traditionally been limited to applications where seismic inversion has been applied after migration as accurate well ties are typically required to estimate the seismic wavelet. Since the original “un-migrated” data forms an inaccurate structural image of the sub-surface, accurate well ties are typically established after migration. The present inventors have determined that there is a need for a seismic inversion technique that can be applied at various stages in a modeling process while still being computationally efficient and accurate when used in conjunction with a migration correction technique to model impedance of a subsurface region. 
     SUMMARY 
     In one general aspect, a method for generating a geophysical model of a subsurface region based on seismic data includes receiving seismic data. Inversion is applied to the seismic data, e.g., the inversion process changes (shapes) the frequency spectrum of the seismic data. The inverted seismic data is then migrated. 
     Implementations of this aspect may include one or more of the following features. For example, receiving seismic data may include obtaining seismic reflection data. Applying inversion to the seismic data may include applying spectral shaping inversion to the seismic data. For example, spectral shaping inversion may include applying coloured inversion or Lazaratos spectral shaping inversion. The application of spectral shaping inversion to the seismic data may include applying a spectral shaping filter to an original seismic data spectrum to generate a shaped seismic data spectrum. An average frequency spectrum of available well log data and an average frequency spectrum of the seismic data may be obtained. The application of spectral shaping inversion to the seismic data may include applying a spectral shaping filter to an original seismic data spectrum to generate a shaped seismic data spectrum. 
     For inversion methods other than spectral shaping inversion, an estimate of the seismic wavelet may be necessary and the estimate may be obtained based on sonic and density well log data. An estimate of a seismic wavelet based on sonic and density well log data is not needed for spectral shaping inversion, and therefore may not be obtained prior to migrating the shaped seismic data. The migrated data may be stacked and/or a phase rotation may be applied to the stacked data to generate an estimate of subsurface impedance. The phase rotation may be a −90 degree phase rotation of the migrated seismic data and the estimate may be of bandlimited P-Impedance. The seismic reflection data received may be converted to zero phase prior to applying inversion, and a phase rotation may be applied to the migrated seismic data to generate an estimate of impedance. 
     The method may be utilized to generate an estimate(s) of one or more of the following seismic or physical properties, including bandlimited P-Impedance, bandlimited S-Impedance, Vp/Vs, bulk modulus, shear modulus, compressional wave velocity, shear wave velocity, Vp/Vs ratio, a Lamé constant, an anisotropy parameter. 
     In another general aspect, a method for generating a geophysical model of a subsurface region based on seismic data includes receiving migrated seismic data and demigrating the migrated data with a migration algorithm and a simple velocity model for the subsurface region. Spectral shaping inversion is applied to the demigrated seismic data. The shaped seismic data is migrated with the migration algorithm and the simple velocity model for the subsurface region. 
     Implementations of this aspect may include one or more of the following features. For example, the simple velocity model for the subsurface region may include a constant velocity model for the subsurface region. The migration algorithm and the simple velocity model for the subsurface region may include a constant-velocity Stolt migration model for the subsurface region. The simple velocity model for the subsurface region may include a laterally invariant model for the subsurface region. The migrated seismic data may include seismic reflection data. The seismic reflection data may be converted to zero phase prior to applying inversion, and/or a phase rotation may be applied to the migrated seismic data to generate an estimate of impedance. Spectral shaping inversion applied to the seismic data may include applying a spectral shaping filter to the demigrated seismic data spectrum to generate a shaped seismic data spectrum. 
     A phase rotation may be applied to the remigrated data to generate an estimate of subsurface impedance. The application of a phase rotation may include applying a −90 degree phase rotation of the migrated seismic data and the estimate may be of bandlimited P-Impedance. The seismic data may be stacked prior to and/or after inversion or migration of the data. A phase rotation may be applied to the stacked seismic data to generate an estimate of impedance. 
     The method may be utilized to generate an estimate(s) of one or more of the following seismic or physical properties, including generating estimates of one or more of bandlimited P-Impedance, bandlimited S-Impedance, Vp/Vs, bulk modulus, shear modulus, compressional wave velocity, shear wave velocity, Vp/Vs ratio, a Lamé constant, and an anisotropy parameter. 
     In another general aspect, a method for generating a geophysical model of a subsurface region based on seismic data includes receiving seismic reflection data. The seismic data is migrated. A spectral shaping inversion filter is applied to the migrated seismic reflection data. A phase rotation is applied to the stacked seismic data to generate an estimate of impedance of the subsurface region. 
     Implementations of this aspect may include one or more of the following features. For example, applying the spectral shaping inversion filter to the migrated seismic reflection data may include calculating a multi-dimensional spectral shaping operator, performing a multi-dimensional Fourier transform of the migrated data, multiplying the calculated multi-dimensional spectral shaping operator with the multi-dimensional Fourier transform of the migrated data, and applying a multi-dimensional inverse Fourier transform. Calculating the multi-dimensional spectral shaping operator may include a 2-D or 3-D Fourier transform. 
     A 2-D or 3-D Fourier transform may be performed of a migration impulse response based on the spectrum of seismic reflection data and a 2-D or 3-D Fourier transform may be performed of a migration impulse response based on the shaped spectrum of the seismic reflection data. The application of the spectral shaping inversion filter to the seismic reflection data may include demigrating the migrated seismic data with a migration algorithm and a simple velocity model for the subsurface region prior to applying spectral shaping inversion to the seismic reflection data; applying spectral shaping inversion to the demigrated seismic data; and/or remigrating the shaped seismic data with the migration algorithm and the simple velocity model for the subsurface region. 
     In another general aspect, a computer program product is tangibly embodied in a machine-readable storage device, the computer program product including instructions that, when executed, cause a hardware system, e.g., a display or other output device, to generate geophysical model(s) of a subsurface region based on seismic data by receiving seismic reflection data, migrating the seismic reflection data, and applying a spectral shaping inversion filter to the seismic reflection data. A phase rotation can be applied to the stacked seismic data spectrum to generate an estimate of impedance of the subsurface region. The spectral shaping inversion filter can be applied prior to migrating the seismic data. Alternatively, the spectral shaping filter can be applied after migrating the seismic data, e.g., a multi-dimensional spectral shaping operator can be calculated and multiplied with a Fourier transform of the migrated data, followed by a multi-dimensional inverse Fourier transform, and/or the migrated data can be demigrated, shaped, and then remigrated after an initial migration process. 
     For example, a tangible computer-readable storage medium includes, embodied thereon, a computer program configured to, when executed by a processor, generate a geophysical model of a subsurface region based on seismic data, the medium comprising one or more code segments configured to receive seismic reflection data; to migrate the seismic data; to apply a spectral shaping inversion filter to the seismic reflection data; to stack the seismic data; and to apply a phase rotation to the stacked seismic data spectrum to generate an estimate of a geophysical property of the subsurface region. The spectral shaping inversion filter is applied to reduce amplification of dipping energy by at least one of (i) applying the spectral shaping inversion filter prior to migrating the data; (ii) demigrating migrated data prior to applying the spectral shaping inversion filter and remigrating the inverted data; and (iii) calculating a multi-dimensional spectral shaping operator and multiplying the multi-dimensional spectral shaping operator with a Fourier transform of the migrated data. 
     In another general aspect, an exemplary hardware system for generating estimates of geophysical properties is configured to generate a geophysical model of a subsurface region based on seismic data, e.g., obtained through hydrophones and/or geophones, to receive seismic reflection data, to migrate the seismic reflection data, and to apply a spectral shaping inversion filter to the seismic reflection data. A phase rotation can be applied to the stacked seismic data spectrum to generate an estimate of impedance of the subsurface region, e.g., that may be displayed through a display component of the system. The spectral shaping inversion filter can be applied prior to migrating the seismic data. Alternatively, the spectral shaping filter can be applied after migrating the seismic data, e.g., a multi-dimensional spectral shaping operator can be calculated and multiplied with a Fourier transform of the migrated data, or the migrated data can be demigrated, shaped, and then remigrated after an initial migration process. The geophysical model can be displayed on a display component of the hardware system. 
     In another general aspect, a method for producing hydrocarbons from a subsurface region includes generating a geophysical model of a subsurface region based on seismic data. Generating the geophysical model further includes receiving seismic reflection data; migrating the seismic data; applying a spectral shaping inversion filter to the seismic reflection data; stacking the seismic data; and applying a phase rotation to the stacked seismic data spectrum to generate an estimate of a geophysical property of the subsurface region. The spectral shaping inversion is applied to reduce amplification of dipping energy by at least one of (i) applying the spectral shaping inversion filter prior to migrating the data; (ii) demigrating migrated data prior to applying the spectral shaping inversion filter and remigrating the inverted data; and (iii) calculating a multi-dimensional spectral shaping operator and multiplying the multi-dimensional spectral shaping operator with a Fourier transform of the migrated data. A well is drilled to a formation interpreted in the generated geophysical model as potentially hydrocarbon bearing. Hydrocarbons are produced from the well. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a convolutional model for a seismogram of the background art generated from a primary reflection at a single boundary between two media. 
         FIG. 1B  is a convolutional model for a seismogram of the background art generated from primary reflections at multiple boundaries between media. 
         FIG. 2  is a graphical view of seismic and log spectra of the background art compared in terms of amplitude and frequency. 
         FIG. 3  is a schematic view of plotted reflection pulses of the background art showing migrated reflector segments and distorted reflector segments. 
         FIG. 4  is a graphical view of a migration impulse response in terms of time versus trace number. 
         FIG. 5A  is a view of an unshaped migration impulse response for a bandlimited wavelet without spectral shaping. 
         FIG. 5B  is a view of a result achieved by applying a spectral shaping filter to the migration impulse response of  FIG. 5A . 
         FIG. 5C  is a view of a result achieved by applying the spectral shaping filter to an input pulse which generated the impulse response of  FIG. 5A  and migrating the shaped input pulse. 
         FIG. 6A  is a flowchart of a process for estimating a physical property of a subsurface which includes migration and inversion. 
         FIG. 6B  is a flowchart of a process for estimating a physical property of a subsurface based on spectral shaping inversion applied after migration. 
         FIG. 7  is a flowchart of a process for estimating a physical property of a subsurface based on applying spectral shaping inversion prior to migration. 
         FIG. 8  is a flowchart of a process for estimating a physical property of a subsurface based on migration, de-migration with a simple velocity model, spectral shaping inversion, and re-migration with the simple velocity model. 
         FIG. 9A  is a graphical view showing a correct relative amplitude variation along the migration impulse response of  FIG. 5A  and a relative amplitude variation along the migration impulse response of  FIG. 5B  (spectral shaping post-migration). 
         FIG. 9B  is a graphical view showing amplitude variation with respect to the migration impulse response of  FIG. 5A  and over a range of migration velocities. 
         FIG. 10  is a comparative flowchart showing a frequency-wavenumber (F-K) spectrum generated by applying spectral shaping inversion before and after migration. 
         FIG. 11A  is a view of a process for generating a shaped frequency-wavenumber (F-K) spectrum of a migration impulse response (post-migration). 
         FIG. 11B  is a view of a process for generating a shaped frequency-wavenumber (F-K) spectrum of a migration impulse response (before migration). 
         FIG. 12  is a flowchart of a process for constructing a two-dimensional (frequency-wavenumber) shaping operator from a one-dimensional (frequency only) shaping operator. 
         FIG. 13  is a flowchart of an alternative process for constructing a two-dimensional shaping operator. 
         FIG. 14  is a flowchart of a process for estimating a physical property of the subsurface which includes applying a multi-dimensional spectral shaping filter for performing seismic inversion. 
         FIG. 15A  is a screenshot of test seismic data. 
         FIG. 15B  is a screenshot of test seismic data after application of an exemplary demigration/spectral shaping/remigration process. 
         FIG. 15C  is a screenshot of test seismic data after application of a post-migration spectral shaping filter. 
         FIG. 16A  is a screenshot of a migrated CDP gather and a corresponding velocity semblance panel. 
         FIG. 16B  is a screenshot of a migrated CDP gather and a corresponding velocity semblance panel with spectral shaping inversion applied after migration. 
         FIG. 16C  is a screenshot of a migrated CDP gather and a corresponding velocity semblance panel with spectral shaping applied before migration. 
         FIG. 17  is a flowchart of an exemplary process for generating a geophysical model of one or more properties based on applying spectral shaping inversion prior to migration. 
         FIG. 18  is a flowchart of an exemplary process for generating a geophysical model of one or more properties based on a demigration/shaping/remigration technique. 
         FIG. 19  is a flowchart of an exemplary process for generating a geophysical model of one or more properties based on applying a 3-D or 2-D spectral shaping filter after stacking. 
         FIG. 20  is a flowchart of an exemplary process for generating a geophysical model of one or more properties based on applying a 3-D or 2-D spectral shaping filter before stacking. 
     
    
    
     The invention will be described in connection with its preferred embodiments. However, to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention, this is intended to be illustrative only, and is not to be construed as limiting the scope of the invention. On the contrary, it is intended to cover all alternatives, modifications and equivalents that may be included within the spirit and scope of the invention, as defined by the appended claims. 
     DETAILED DESCRIPTION 
     In one general aspect, the present inventors have determined that spectral shaping is mathematically equivalent to other methods of inversion achieving the same objective. In addition, the present inventors have also discovered various computational advantages to spectral shaping inversion which have not previously been appreciated by others utilizing traditional seismic inversion techniques. Accordingly, the behavior of spectral shaping inversion can be generalized for other types of inversion procedures, as long as these inversion procedures are based on the convolutional model. For example, one advantage of spectral shaping inversion is that, unlike other inversion methods, spectral shaping inversion does not require an estimate of the seismic wavelet w(t), and, therefore, spectral shaping inversion does not rely on accurate well ties. Therefore, knowledge of the average frequency spectrum of available well log data and the average frequency spectrum of the seismic data is sufficient for performing the inversion. 
     The present inventors have determined that spectral shaping inversion does not require an estimate of the seismic wavelet and, therefore, is an inversion technique that can be advantageously performed before or after migration. For example, assuming there are no other considerations, inversion techniques are typically applied after migration, for the following reasons. First, migration of large modern 3D seismic data sets is typically a very time-consuming and expensive process. Since a migrated version of the original recorded data is generally required, generating a migrated version of spectrally shaped inversion data normally adds to the cost of an additional migration run. Therefore, if an inversion technique is applied after migration, one only needs to migrate the data once. Second, any change in the design of the spectral shaping filter requires additional migration runs to generate a final product, and this further increases the cost of the process. For these reasons, the standard practice in the background art has been to apply an inversion technique, particularly spectral shaping applications, after migration correction techniques have been applied to the data. 
     However, as discussed further below, changing the order of application of migration and inversion techniques produces very different final results that can be utilized advantageously in various ways. In addition, the present inventors have also determined that applying a seismic inversion process that assumes a convolutional model with a single, time-independent wavelet to migrated seismic data distorts the relative amplitudes of the seismic events, e.g., artificially amplifying the steep dips. 
     A typical migration impulse response  400  for a constant-velocity medium in terms of time versus trace number is shown in  FIG. 4 . Referring to  FIG. 4 , the effect of inversion on the migration impulse response is useful in demonstrating the impact of applying inversion after migration. The output of the migration process is defined by impulse response  400  when the input is a localized impulse. Since the seismic data input to migration can be thought of as just a superposition of such impulses, understanding what happens to a single impulse fully characterizes the behavior of migration for any given input. As seen in  FIG. 4 , the wavelet is dip-dependent after migration, with lower-frequency wavelets corresponding to high dips  440 . For example, the relationship seen in  FIG. 4  is further described in Levin, S. A., 1998, “Resolution In Seismic Imaging: Is It All A Matter Of Perspective?,” Geophysics, 63, 743-749; and Tygel, M., Schleicher, J., and Hubral, P., 1994, “Pulse Distortion in Depth Migration: Geophysics,” 59, 1561-1569. A zero-dip wavelet  420  has the same frequency as the input. 
     Despite the fact that it has been appreciated that migration distorts the seismic wavelet, the implications for seismic inversion have not been fully understood. Since the wavelet is dip-dependent after migration, the convolutional model (equation (3)) is generally not valid after migration, e.g., equation (3) assumes a dip-independent wavelet. The present inventors have determined that the consequences of ignoring this fact affects inversion algorithms applied after migration, and are described in greater detail hereinafter. 
       FIG. 5A  is a view of an unshaped migration impulse response  501 A for a bandlimited wavelet without spectral shaping.  FIG. 5B  is a view of a result achieved by applying a spectral shaping filter to the migration impulse response of  FIG. 5A .  FIG. 5C  is a view of a result achieved by applying the spectral shaping filter to an input pulse which generated the impulse response of  FIG. 5A , and migrating the shaped input pulse. Referring to  FIGS. 5A and 5B , a migration impulse response before  501  and after  502  the application of a spectral shaping filter is shown, e.g., a spectral shaping filter typically applied in Coloured or Spectral Shaping inversion. The impulse response  502  exhibits large amplification of the steeply-dipping flanks  502 A, e.g., compared to original response  501 A. As described with respect to  FIG. 2 , the spectral shaping filters used for inversion significantly amplify the energy in the low-frequency part of the seismic spectrum. However, the fact that the steeply-dipping parts of the impulse response have lower-frequency wavelets than the flatter parts is not a result of spectral shaping. As discussed in greater detail hereinafter, the steeply-dipping parts of the migration impulse response have lower-frequency wavelets than the flatter parts. The consequence of the behavior observed here is that application of spectral shaping after migration causes steeply-dipping energy, signal or noise, to be excessively amplified. 
     However, referring to  FIG. 5C , the present inventors have determined that applying the same spectral shaping filter, and then migrating, leads to an impulse response  503  with correctly-preserved relative amplitudes between the flat and steeply-dipping parts  503 A. Accordingly, the relative amplitude variation along the impulse response in  FIG. 5C  is very similar to that of  FIG. 5A , while the relative amplitude variation seen in  FIG. 5B  has significantly changed. 
       FIGS. 6A-6B  are flowcharts of processes for estimating a physical property of a subsurface which include migration correction techniques and inversion. Specifically, the generalized flowcharts depict two inversion practices  600 ,  650 . Referring to  FIG. 6A , an inversion step  620  is applied after a migration step  610  in process  600 . The inversion process  620  is typically applied to the migrated data to generate an estimate of subsurface impedance, and/or one or more other seismic or physical properties, such as the compressional wave velocity, shear wave velocity, density of the subsurface region, bulk modulus, and/or shear modulus (also called the elastic moduli). Alternatively, or in addition to, the processes  600 ,  650  may be used to generate estimates of one or more of bandlimited P-Impedance, bandlimited S-Impedance, Vp/Vs, bulk modulus, shear modulus, compressional wave velocity, shear wave velocity, Vp/Vs ratio, a Lamé constant, and an anisotropy parameter. 
     Referring to  FIG. 6B , a spectral shaping inversion step  670  is applied after a migration step  660  in process  650 . In addition, a phase rotation, e.g., −90°, is applied to the shaped and migrated data in step  680  to generate an estimate of the subsurface impedance and/or one or more other seismic or physical properties. An exemplary process may contain several additional processing steps, but in both processes  600 ,  650 , inversion  620 ,  670  has been applied after migration. As discussed above, current inversion practice can be summarized, at a very general level, by the flowchart of  FIG. 6A , showing migration  610  followed by inversion  620  to estimate impedance and/or other rock properties. In process  650 , a spectral shaping inversion step  670 , such as applying a spectral shaping filter (Lazaratos) or Coloured inversion previously described, is applied to the migrated data  660 . Referring to  FIG. 6B , spectral shaping  670  is typically applied after migration  660 , followed by a −90° phase rotation  680  and/or further processing to estimate impedance and/or other rock properties, e.g. as outlined by Lazaratos (2006). 
     However, the present inventors have determined that current approaches, such as process  600 ,  650 , ignore the dip-dependence of the wavelet after migration, and, as a consequence, excessively amplify steeply-dipping energy, signal or noise. Accordingly, the present inventors have developed a technique for performing seismic inversion that avoid the amplification of dipping energy while optimizing computational efficiency and/or accuracy. 
       FIG. 7  is a flowchart of a process for estimating a physical property of a subsurface based on applying spectral shaping inversion prior to migration. Referring to  FIG. 7 , a first process  700  for performing seismic inversion relative to migration includes applying spectral shaping inversion  710  to seismic data, followed by migrating the shaped data  720 , and additional processing step(s)  730 , e.g., applying a phase rotation of −90° to estimate subsurface impedance and/or other seismic and physical properties. Another optional step may include stacking the data prior to, after, or concurrent with one or more of the other steps of process  700 . Although a typical seismic processing workflow may, in general, contain several additional processing steps, process  700  is particularly advantageous if the inversion  710  is performed before the migration  720 . 
     The present inventors have determined that the application of a spectral shaping filter, e.g., Lazaratos spectral shaping inversion or Coloured inversion, before migration, further optimizes results. Unlike other inversion techniques that typically require an estimate of the seismic wavelet, e.g., usually obtained through a well tie, an estimate of the seismic data spectrum is all that is needed for spectral shaping. Further, the estimate of the seismic data spectrum can be obtained reliably even when the geometric configurations of the recorded reflectors are inaccurately imaged, e.g., as the geometric configurations may be prior to any correction, e.g., through migration  720 . Any problems associated with amplification of steeply-dipping energy are reduced and/or eliminated if the seismic wavelet is independent of reflector dip before migration  720 . 
     After spectral shaping  710  and migration  720 , a −90° phase rotation is applied and/or additional processing is applied. For example, Lazaratos (2006) describes additional processing techniques and/or properties that may be applied or determined in combination with a spectral shaping inversion technique. Alternatively, or in addition, one of ordinary skill in the art will appreciate that one or more standard seismic processing steps may be applied before and/or after spectral shaping and migration. For example, other processing techniques may include one or more of the following processes, such as a de-signature process, de-ghosting process, random noise attenuation, multiple attenuation, a deconvolution process, and/or estimation of stacking and migration velocities. With respect to migration  720 , the process  700  exhibits favorable results across a wide range of migration algorithms, and the process  700  is therefore not limited to any particular migration technique. 
     As discussed earlier, seismic inversion, particularly spectral shaping, is routinely applied after migration in current practice. Specifically, inversion is typically applied after migration to avoid multiple data runs of the time-consuming migration process. However, the present inventors have determined that the accepted practice of applying inversion after migration can result in one or more limitations. Specifically, as described with respect to  FIGS. 5A-5C  and  6 A- 6 B, current geophysical techniques ignore the dip-dependence of the wavelet after migration, and, as a consequence, excessively amplify steeply-dipping energy, signal and/or noise. Accordingly, the process  700  described in connection with  FIG. 7 , reverses the order of spectral shaping inversion and migration in a manner that increases the ability of the overall process to estimate subsurface properties, e.g., such as impedance. 
       FIG. 8  is a flowchart of a process  800  for estimating a physical property of a subsurface based on migration, demigration with a simple velocity model, spectral shaping inversion, and remigration with the simple velocity model. Alternative process  800  also addresses the dip-dependence of the wavelet after migration, and therefore does not excessively amplify steeply-dipping energy, e.g., signal and/or noise. In process  800 , seismic data is first migrated  810 . After migration  810 , a demigration technique  820 , spectral shaping inversion technique  830 , and remigration technique  840  are applied to the previously migrated data  810 . In addition, phase rotation and/or other computational and/or imaging processes  850  may be applied after the demigration  820 , shaping  830 , and remigration  840  techniques are applied. In contrast to process  700 , which applies spectral shaping before migration to achieve highly accurate results, process  800  provides a way of improving the accuracy of current processing techniques while offering an alternative technique which is less computational intensive than process  700 . Specifically, process  700  may be considered impractical for some applications as an additional migration of the seismic data set is typically required. For example, if a migrated version of the original unshaped recorded data is always required, one would have to migrate the original data set, and also shape the data set and migrate the shaped data set again. In process  800 , a substantially equivalent result is achieved at a small fraction of the computational load and cost of an additional migration. 
     Process  800  includes demigrating  820  the migrated input data, e.g., using a relatively fast and inexpensive migration technique. For example, a migration technique that assumes an extremely simple velocity model, e.g., constant-velocity Stolt migration, or a laterally invariant model, will typically produce a fast and inexpensive migration technique. The publication “Migration By Fourier Transform: Geophysics,” 43, 23-48, by Stolt, R. H., 1978, further describes an exemplary Stolt migration. Specifically, demigration is the inverse of migration. Therefore, a demigration process receives a migrated version of a seismic data set as an input, and outputs a close approximation of the original data set. In addition, the process of demigration is well understood for several classes of migration algorithms, e.g., including the Stolt migration mentioned earlier. 
     The cost of migration and demigration algorithms largely depends on the velocity model used, e.g., with simple models leading to relatively fast computation times and reduced costs. A velocity model is a model of the subsurface under investigation, in which values representing the velocity of propagation of seismic waves are assigned at different locations across the region. Accordingly, a simple model, e.g., constant velocity or a laterally invariant model, applies a velocity model with relatively isotropic velocity values across the subsurface region. In step  830 , a spectral shaping filter is applied to the demigrated data. In step  840 , the spectrally shaped and demigrated data is remigrated, using the same migration algorithm and velocity model used in the demigration step  820 . Accordingly, if a Stolt migration algorithm and constant velocity model were used in step  820 , the data is remigrated after shaping with the Stolt migration algorithm and constant velocity model. 
     The demigration/shaping/remigration process  800  generates a result that is a very close approximation to the estimate, e.g., subsurface impedance, achieved in process  700 . Even if the migration velocity used for the demigration and remigration processes are significantly different from the true earth velocity across the actual region, the present inventors have determined that process  800  demonstrates favorable accuracy coupled with computational efficiency. Therefore, the technique of demigration/shaping/remigration process  800 , produces an enhancement over the prior practice of applying a spectral shaping operation after migration, e.g., even when performed with a velocity model which has previously been regarded being a relatively inaccurate and/or simple velocity model, e.g., a constant velocity model. 
       FIG. 9A  is a graphical view showing a correct relative amplitude variation along the migration impulse response of  FIG. 5A  and a relative amplitude variation along the migration impulse response of  FIG. 5B  (spectral shaping post-migration).  FIG. 9B  is a graphical view showing amplitude variation with respect to the migration impulse response of  FIG. 5A  and over a range of migration velocities. Referring to  FIG. 9A , a graphical view  900  of migration applied with a correct velocity, e.g., known velocity is shown having correct relative amplitudes along migration impulse response  910  of  FIG. 5A  and relative amplitudes along the response of  FIG. 5B  corresponding to post-migration application of spectral shaping  920 . Referring to  FIG. 9B , a graphical view  950  showing demigration/shaping/remigration, e.g., such as process  800 , across a range of velocities, demonstrates various curves corresponding to the relative amplitude variation along the migration impulse response. The input data to the process  800  was the migration impulse response of  FIG. 5A . Each curve shown in  FIG. 9B  corresponds to a different velocity, e.g., a set of curves  960  ranging from approximately 75% to 150% of the correct (actual) velocity  970 . The result corresponding to the correct velocity  910 ,  970  is shown with an arrow in  FIGS. 9A and 9B , respectively. The variation described by the set of curves  960  in  FIG. 9B  is more robust than the equivalent result achieved after post-migration shaping, shown in  FIG. 9A . 
     Therefore, even when the demigration and remigration steps are performed with velocities that are very different from the actual value, the amplitude variation of the migration operator much more closely approximates the correct result than what is achieved when spectral shaping is applied after migration. This relative insensitivity of the results to the migration velocity values is one of the observations that supports the demigration/shaping/remigration process  800  described hereinabove. Since the demigration/shaping/remigration process  800  may be applied with even very simple migration algorithms, e.g., with only constant-velocity or laterally invariant models, a robust and computational process is achieved with process  800 . Due to the computational efficiency of such algorithms, the demigration/shaping/remigration process  800  may be orders of magnitude faster and less expensive than the process  700  (spectral shaping before migration). 
       FIG. 10  is a comparative flowchart showing a frequency-wavenumber (F-K) spectrum generated by applying spectral shaping inversion before and after migration by process  1000 . If one assumes a constant-velocity subsurface, the demigration/shaping/remigration process  800  can be equivalently performed with a single operator applied after migration. Referring to  FIG. 2 , an analogy with spectral shaping is an operator transforming the frequency spectrum of the original data. The spectral shaping concept is extended to modify not just the frequency (temporal), but also the wavenumber (spatial) spectrum of the data, e.g., as seen in process  1000  ( FIG. 10 ). Instead of shaping the one-dimensional spectra of the seismic data, e.g., generated with a Fourier transform of the seismic traces along the time dimension, the two-dimensional spectra or three-dimensional spectra is shaped. For 2-D seismic data, the spectra are generated with a two-dimensional Fourier transform of the seismic traces along the time and horizontal distance, e.g., along the x-axis. For 3-D seismic data, the spectra are generated with a three-dimensional Fourier transform of the seismic traces along the time and two horizontal dimensions, e.g., along the x and y axes. 
     Referring to  FIG. 10 , the migration impulse responses  501 - 503  of  FIGS. 5A-5C , e.g., unshaped  501 , shaping after migration  502  and shaping before migration  503  are shown with corresponding two-dimensional spectra  1010 ,  1040 ,  1050 . The vertical axis for the spectra is frequency (F) and the horizontal axis is wavenumber (K). Wavenumber is a measure of the variation in space, similar to the way frequency is a measure of the variation in time. After spectral shaping, low-frequency energy is boosted considerably. However, there is a significant difference between the pre-migration shaped spectra  1050  generated with spectral shaping before migration  1030 , and the post-migration shaped spectra  1040  generated with spectral shaping after migration  1020 . Shaping after migration  1020  enhances low-frequency energy for all values of wavenumber, and the large boost for large wavenumber values corresponds to the brightening of the steeply-dipping flanks of the impulse response. In contrast, with shaping before migration  1030 , only the small-wavenumber, low-frequency part of the two-dimensional spectrum is boosted. 
       FIG. 11A  is a view of a process for generating a frequency-wavenumber (F-K) spectrum of a migration impulse response with shaping applied post-migration.  FIG. 11B  is a view of a process for generating a frequency-wavenumber (F-K) spectrum of a migration impulse response with shaping applied before migration. Referring to  FIGS. 11A-11B , the effect of spectral shaping after migration  1100  and before migration  1150  is expressed in the two-dimensional Fourier (F-K) domain. In both cases, the migration impulse response is shaped by multiplying the spectrum of the original response  1110 ,  1160  with the spectrum of the shaping operator  1120 ,  1170 . 
     However, the operators for the post-migration spectral shaping and pre-migration spectral shaping cases are very different. Referring to  FIG. 11A , post-migration spectral shaping is essentially one-dimensional, e.g., the shaping operator is only dependent on frequency, and is the same for all wavenumbers. Referring to  FIG. 11B , pre-migration spectral shaping is two-dimensional, e.g., the value of the shaping operator is variable with changes in either frequency or wavenumber. For constant velocity, pre-migration shaping with a 1-D (frequency only) shaping operator is equivalent to post-migration shaping with the 2-D (frequency-wavenumber) operator shown in  FIGS. 11A-11B . 
     The shaping of the migration impulse response  1110 ,  1160  can be implemented in the two-dimensional (F-K) Fourier domain by multiplying the spectrum of the original response with the spectrum of the shaping operator to achieve the shaped responses  1130 ,  1180 . The difference between the F-K spectra  1120 ,  1170  of the operators for the two cases is also apparent. Post migration spectral shaping  1120  has a F-K spectra response that is the same for all wavenumbers. In fact, the wavenumber axis is ignored and the operator is designed on the basis of the frequency axis only, e.g., in effect a one-dimensional operator. Premigration shaping  1170 , in effect amounts to a two dimensional operator, whose values depend not only on frequency, but also on wavenumber. While one-dimensional spectral shaping boosts small and large wavenumbers for low frequencies  1130 , two-dimensional spectral shaping will only boost the small-wavenumber, low-frequency part of the spectrum  1180 . 
     For the constant-velocity case described, spectral shaping can be applied correctly in two ways: (i) apply one-dimensional (frequency only) spectral shaping and then migrate; (ii) migrate and then apply two-dimensional (frequency-wavenumber) spectral shaping. For the constant-velocity case, the frequency-wavenumber spectrum is independent of the location, e.g., apex time, of the operator, and so the constant-velocity approach is feasible. However, when the velocity is variable, these assumptions are not true any more. The equivalence (in the case of constant velocity) of one-dimensional premigration spectral shaping and two-dimensional post migration spectral shaping can be easily explained. It is well known that, for constant-velocity migration, the F-K spectrum P M  of the migrated data is related to the F-K spectrum P U  of the unmigrated data, through the relationship:
 
 P   M ( F,K )= P   U (√{square root over ( F   2   +K   2 ( v/ 2π) 2 )}, K )  (11)
 
where F is frequency, K is wavenumber, and v is the migration velocity. This relationship implies that the energy in the F-K spectrum moves to a lower frequency after migration, but remains at the same wavenumber. Equation (11) shows how F-K spectra get transformed by migration. The F-K spectrum of a one-dimensional (frequency-only) shaping filter S U  (such as the one shown in  FIG. 11A  for the post-migration case), does not depend on K, and, according to equation (5), after migration the filter gets transformed to a truly two-dimensional F-K filter S M , according to the equation (12):
 
 S   M ( F,K )= S   U (√{square root over ( F   2   +K   2 ( v/ 2π) 2 )})  (12)
 
       FIG. 12  is a flowchart of a process  1200  for constructing a two-dimensional (frequency-wavenumber) shaping operator  1225  from a one-dimensional (frequency only) shaping operator  1215 . Referring to  FIG. 12 , the construction  1220  of a two-dimensional (frequency-wavenumber) shaping operator  1225  is based on a one-dimensional (frequency only) shaping operator  1215  designed in step  1210 . For constant velocity, either the one-dimensional operator is applied before migration, and then migrated, or the two-dimensional operator is applied after migration to achieve the same results. In either case, the results will not suffer from a steep-dip amplification problem. 
     Referring to  FIG. 13 , an alternative process  1300  for constructing a two-dimensional shaping operator  1380  requires an estimate of the seismic spectrum  1310 . Assuming constant velocity, the operator  1380  can be applied after migration to produce spectrally shaped data with the correct amplitude variation as a function of dip, e.g., without suffering from the steep-dip amplification problem. First, the migration impulse response is constructed  1330 , using the original estimated seismic spectrum. The migration impulse response is also constructed  1340  after applying spectral shaping  1320  to the spectrum  1310 , e.g., using a conventional one-dimensional (frequency only) spectral shaping operator. The two-dimensional (frequency (F)-wavenumber (K)) spectra for each of these two impulse responses is calculated  1350 ,  1360 , using a two-dimensional Fourier transform. The ratio of these two-dimensional spectra is obtained  1370 , e.g., the ratio  1370  defines the frequency response of a two-dimensional shaping operator. In order to spectrally shape the migrated seismic data, the two-dimensional Fourier transforms are calculated, and the transform is multiplied with the frequency response of the two-dimensional shaping operator derived hereinabove. Spectrally shaped migrated data, not suffering from the steep-dip amplification problem, is generated with an inverse transform back. 
       FIG. 14  is a flowchart of a process  1400  for estimating a physical property of the subsurface which includes applying a multi-dimensional spectral shaping filter  1430  for performing seismic inversion. Referring to  FIG. 14 , an exemplary seismic inversion process  1400  based on multi-dimensional spectral shaping is shown. A migrated set of data is created in step  1410 . The seismic spectrum is estimated  1415  and the spectrum of the multi-dimensional, spectral shaping operator is calculated  1425 . A multi-dimensional Fourier transform, e.g., 2-D or 3-D, is performed on the migrated data  1420 . In step  1430 , the multi-dimensional spectral shaping operator  1425  is multiplied  1430  with the result of step  1420 . A multi-dimensional (2-D or 3-D) inverse Fourier transform is performed in step  1440 , and a phase rotation, e.g., −90°, is applied  1450  along with any additional processing to estimate one or more physical or seismic properties of the subsurface region, such as impedance. 
     The process  1400  can be applied to two dimensional data (horizontal distance and time) and/or can be easily generalized for 3-D data. The main difference for the three dimensional case is that the three-dimensional (frequency (F)-X wavenumber (Kx)-Y wavenumber (Ky)) spectra is calculated, e.g., instead of the two-dimensional (F-K) spectra. If one assumes a constant-velocity subsurface, the multi-dimensional spectral shaping approach is even more computationally efficient than a demigration/shaping/remigration process  800 , described hereinabove. 
       FIG. 15A  is a screenshot  1500  of test seismic data.  FIG. 15B  is a screenshot  1510  of test seismic data after application of an exemplary demigration/spectral shaping/remigration process.  FIG. 15C  is a screenshot  1520  of test seismic data after application of a post-migration spectral shaping filter. Referring to  FIGS. 15A-15C , the same shaping filter was applied in  1510  and  1520 . However, the result  1510  shown in  FIG. 15B  demonstrates significant improvement of the signal-to-noise ratio of the original test data  1500 . Further, the result  1510  is clearly superior to the result  1520  achieved in  FIG. 15C . The improvement in the signal-to-noise ratio of the data is evident, e.g., the result  1520  achieved in  FIG. 15C  with simple post-migration spectral shaping is inferior to the result  1510  shown in  FIG. 15B . Specifically, the noise in  FIG. 15C  has a vertical appearance, e.g., the term “curtain effect” is sometimes used in practice to describe this type of noise, as the noise mostly includes steeply-dipping components that were enhanced with post-migration spectral shaping. 
       FIG. 16A  is a screenshot  1600  of a migrated common depth point (CDP) gather  1605  and a corresponding velocity semblance panel  1608 .  FIG. 16B  is a screenshot  1610  of a migrated CDP gather  1615  and a corresponding velocity semblance panel  1618  with spectral shaping inversion applied after migration.  FIG. 16C  is a screenshot  1620  of a migrated CDP gather  1625  and a corresponding velocity semblance panel  1628  with spectral shaping applied before migration. Referring to  FIGS. 16A-16B , the clarity of the seismic events  1625  and the semblance peaks  1628  is clearly superior when spectral shaping is applied before migration, e.g.,  1620 ,  1628  of  FIG. 16C . Referring to  FIG. 16A , a migrated common-depth-point (CDP) gather  1605  and the associated velocity semblance panel  1608  are shown. A CDP gather is a collection of seismic traces corresponding to reflections from the same subsurface points, but at different angles of incidence. Velocity semblances measure the coherence of seismic events for different times. The horizontal locations of semblance peaks (bright amplitudes) within the semblance panels provide measures of seismic velocities to be used for flattening and stacking, e.g., summing, the seismic events within the CDP gather from which they were produced. In general, the brighter and better defined the semblances, the easier it becomes to determine velocities. Referring to  FIGS. 16B-16C , the effect on the gather and the associated semblance panel of applying spectral shaping after migration, and the equivalent results with spectral shaping applied before migration are shown, respectively. The clarity of the seismic events and the semblance peaks is superior when spectral shaping is applied before migration, e.g.,  FIG. 16C  ( 1625 ,  1628 ). 
     A number of embodiments have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, although the foregoing implementations have described the application of a spectral shaping inversion filter, alternative inversion algorithms can be applied after migration, e.g., the same amplification of steeply-dipping energy observed with the spectral shaping filter would occur if another inversion algorithm was applied after migration. Although the methods described here were presented within the context of seismic inversion, this is not the only potential application of the technology. Applying the methods to CDP gathers, as in the last example, often improves the signal-to-noise ratio of the gathers significantly. The improved gathers can then be used for more accurate velocity estimation and Amplitude-Versus-Offset (AVO) analysis. Although the foregoing processes were presented in association with the spectral shaping filters encountered in seismic inversion, the same concept can be easily extended to improve the amplitude-preservation characteristics, e.g., ratio of flat to dipping events, of any filtering process applied to migrated seismic data, e.g., bandpass filtering, spectral decomposition. 
     One or more additional processing techniques may be applied to the data, e.g., before and/or after spectral shaping and/or migration has been performed in one of the aforementioned techniques. For example, one or more additional, exemplary processing techniques that may be incorporated into one or more of the aforementioned process include a de-signature process, de-ghosting process, random noise attenuation, multiple attenuation, a deconvolution process, estimation of stacking and migration velocities, or other processing techniques further described in “Spectral Shaping Inversion for Elastic and Rock Property Estimation,” by Lazaratos, 2006. One or more additional processing techniques may be performed before, after, or intermediate to the processing steps described hereinabove, e.g., between the obtaining of seismic data and prior to converting the data to zero phase. The data is typically converted to zero phase prior to the application of any migration and/or inversion technique, e.g., spectral shaping inversion. Stacking velocities of migrated data, if necessary, may be refined using shaped seismic data and the one or more stacks generated prior to the application of a phase rotation, e.g., angle stacks may be generated and a −90° phase rotation and appropriate linear combinations may be applied to the generated angle stacks to generate estimates of bandlimited P-Impedance and S-Impedance, Vp/Vs, and/or other seismic or physical properties. 
     The data may be stacked before or after the spectral shaping of any data, e.g., stacking may be performed post-migration and post-inversion, post-migration and pre-inversion, and/or at other points in the overall data processing routine. For example,  FIG. 17  is a flowchart of an exemplary process  1700  for generating a geophysical model of one or more properties based on applying spectral shaping inversion prior to migration.  FIG. 18  is a flowchart of an exemplary process  1800  for generating a geophysical model of one or more properties based on a demigration/shaping/remigration technique.  FIG. 19  is a flowchart of an exemplary process  1900  for generating a geophysical model of one or more properties based on applying a 3-D or 2-D spectral shaping filter after stacking  FIG. 20  is a flowchart of an exemplary process  2000  for generating a geophysical model of one or more properties based on applying a 3-D or 2-D spectral shaping filter before stacking. 
     Referring to  FIG. 17 , process  1700  generates estimates of one or more of Bandlimited P-Impedance, Bandlimited S-Impedance, Vp/Vs and/or other elastic or other rock properties. In general, process  1700  is based upon acoustic and elastic inversion through the application of spectral shaping before migration. Seismic data is obtained  1710 , and any other desired processing techniques are performed  1720  on the seismic data. For example, other processing techniques may include one or more of the following processes, such as a de-signature process, de-ghosting process, random noise attenuation, multiple attenuation, a deconvolution process, and/or estimation of stacking and migration velocities. The data is next converted to zero phase  1730  and an estimate of the seismic spectrum is generated  1740 . A spectral shaping filter is applied to the data  1750  and the shaped data is migrated  1760 . In step  1770 , the stacking velocities are refined, if necessary, using shaped seismic data. Depending upon the desired geophysical model, the data is stacked  1780 ,  1785 . For example, if estimates of Bandlimited P-Impedance, Bandlimited S-Impedance, Vp/Vs, and/or other elastic or other rock properties are desired, angle stacks are generated in step  1785 , and a −90° phase rotation and appropriate linear combinations are applied to generate the desired estimates. If Bandlimited P-Impedance is the desired estimate, the data is stacked  1780  and a −90° phase rotation is applied to generate an estimate of Bandlimited P-Impedance. Stacking  1780 ,  1785  produces stacked data sections based on some common criteria between seismic traces. For example, seismic data can be stacked, e.g., combined, according to seismic traces having the same or similar angles, common source-receiver midpoint, common subsurface imaging location, and/or some other common criteria. 
     Referring to  FIG. 18 , process  1800  also generates estimates of one or more of Bandlimited P-Impedance, Bandlimited S-Impedance, Vp/Vs and/or other elastic or other rock properties. However, process  1800  is based upon acoustic and elastic inversion through the application of demigration/shaping/remigration. In process  1800 , the application of demigration/shaping/remigration occurs prior to any stacking, e.g., stacking  1890  or generating angle stacks  1895 . Specifically, seismic data is obtained  1810 , and any other desired processing techniques are performed  1820  on the seismic data. The data is next converted to zero phase  1830  and then migrated  1840 . The migrated data is then demigrated with a simple velocity model  1850 , an estimate of the seismic spectrum is generated  1860 , and a spectral shaping filter is applied to the data  1870 . The shaped data is then remigrated  1880  with the same simple velocity model used in the demigration step  1850 . If necessary, the stacking velocities are refined using shaped seismic data  1885 . Depending upon the desired geophysical model, the data is stacked  1890 ,  1895  and a −90° phase rotation  1896 ,  1898  and appropriate linear combinations  1898  are applied to generate the desired estimates. 
     Referring to  FIGS. 19 and 20 , processes  1900  and  2000  both generate estimates of one or more of Bandlimited P-Impedance, Bandlimited S-Impedance, Vp/Vs and/or other elastic or other rock properties through acoustic and elastic inversion based on the application of a multi-dimensional spectral shaping filter, e.g., a 2-pass 2-D (F-K) or 3-D (F-Kx-Ky) spectral shaping filter. In process  1900 , the spectral shaping filter is applied after any stacking steps, e.g., stacking  1945  or generating angle stacks  1950 . In process  2000 , the spectral shaping filter is applied prior to any stacking steps  2085 ,  2090 . 
     In process  1900 , the data is obtained  1910 , optionally processed  1920 , and converted to zero phase  1930 . The data is migrated  1940 , and the migrated data is then stacked  1945 ,  1950 . If angle stacks are generated  1950 , the seismic spectrum is estimated for each angle stack  1960 . A frequency domain response of a multi-dimensional spectral shaping filter is derived  1970 , e.g., a 3-D (F-Kx-Ky) or 2-pass 2-D (e.g., F-K) spectral shaping filter, that converts an unshaped migration impulse response to a spectrally shaped migration impulse response when the multi-dimensional filter is applied  1980 . For example, the migration impulse response is constructed using appropriate, e.g., average, constant velocity. A −90° phase rotation and appropriate linear combinations are applied  1990  to generate estimates of one or more of Bandlimited P-Impedance, Bandlimited S-Impedance, Vp/Vs, or other elastic or other rock properties. If an estimate of Bandlimited P-Impedance is generated  1985 , the stacked data  1945  is used to estimate a seismic spectrum  1955 , the frequency-domain response of the multi-dimensional shaping filter is derived  1965 , the filter applied  1975 , and the −90° phase rotation is applied to generate the estimate. 
     In process  2000 , seismic data is obtained  2010 , optionally processed  2020 , converted to zero phase  2030 , and migrated  2040 . In contrast to process  1900 , the seismic spectrum is estimated for each common-offset or common-angle data set  2050 . A frequency-domain response of a multi-dimensional spectral shaping filter, e.g., 3-D (F-Kx-Ky) or 2-pass 2-D (e.g., F-K) spectral shaping filters, is derived for each offset or angle data set  2060 . The spectral shaping filter is applied for each common-offset or common-angle data set  2070  to convert the unshaped migration impulse response to a spectrally shaped migration impulse for each offset or angle. The migration impulse response is constructed using appropriate constant velocity, e.g., average velocity. The stacking velocities are refined  2080 , if necessary, using the shaped seismic data. The data is then stacked  2085 ,  2090  and a −90° phase rotation (and appropriate linear combinations, if necessary) applied to generate one or more estimates of Bandlimited P-Impedance, Bandlimited S-Impedance, Vp/Vs, or other elastic or other rock properties  2095 ,  2096 . 
     One or more of the aforementioned processes and/or techniques, e.g., such as the application of a shaping filter, can be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in any combination thereof. Any of the aforementioned functionality may be implemented as a computer program product, e.g., a computer program tangibly embodied in an information carrier, e.g., in a machine-readable storage device or in a propagated signal, for execution by, or to control the operation of, data processing apparatus, e.g., a programmable processor, a computer, or multiple computers. A computer program can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program can be deployed to be executed on one computer or on multiple computers at one site or distributed across multiple sites and interconnected by a communication network. 
     One or more process steps of the invention can be performed by one or more programmable processors executing a computer program to perform functions of the invention by operating on input data and generating output. One or more steps can also be performed by, and an apparatus or system can be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application-specific integrated circuit). In addition, data acquisition and display may be implemented through a dedicated data collection and/or processing system, e.g., containing data acquisition hardware, such as hydrophones and/or geophones, a processor(s), and various user and data input and output interfaces, such as a display component for graphically displaying one or more of the generated estimates obtained through any of the aforementioned process steps or processes. 
     Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read-only memory or a random access memory or both. The essential elements of a computer are a processor for executing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, or optical disks. Information carriers suitable for embodying computer program instructions and data include all forms of non-volatile memory, including by way of example semiconductor memory devices, e.g., EPROM (erasable programmable read-only memory), EEPROM (electrically erasable programmable read-only memory), and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; and CD-ROM (compact disk read-only memory) and DVD-ROM (digital versatile disk read-only memory) disks. The processor and the memory can be supplemented by, or incorporated in special purpose logic circuitry. 
     All such modifications and variations are intended to be within the scope of the present invention, as defined in the appended claims. Persons skilled in the art will also readily recognize that in preferred embodiments, at least some of the method steps method are performed on a computer, e.g., the method may be computer implemented. In such cases, the resulting model parameters may either be downloaded or saved to computer memory.