Methods and systems for reconstruction of low frequency particle velocity wavefields and deghosting of seismic streamer data

Computational methods and systems for deghosting marine seismic streamer data are described. In particular, an exploration-seismology vessel tows a number of streamers that form a data acquisition surface located beneath a free surface. The methods computationally deghost or substantially remove receiver ghost signals from seismic data recorded by steamer receivers. The deghosting methods include low frequency compensation to recover vertical velocity wavefield information that is typically lost due to a low signal-to-noise ratio over a low frequency range independent of the free surface conditions or the shape of the data acquisition surface.

BACKGROUND

In the past few decades, the petroleum industry has invested heavily in the development of marine seismic survey techniques that yield knowledge of subterranean formations beneath a body of water in order to find and extract valuable mineral resources, such as oil. High-resolution seismic images of a subterranean formation are essential for quantitative seismic interpretation and improved reservoir monitoring. For a typical marine seismic survey, an exploration-seismology vessel tows one or more seismic sources and one or more streamers below the surface of the water and over a subterranean formation to be surveyed for mineral deposits. The vessel contains seismic acquisition equipment, such as navigation control, seismic source control, seismic receiver control, and recording equipment. The seismic source control causes the one or more seismic sources, which are typically air guns, to produce acoustic impulses at selected times. Each impulse is a sound wave that travels down through the water and into the subterranean formation. At each interface between different types of rock, a portion of the sound wave is refracted, a portion of the sound wave is transmitted, and another portion is reflected back toward the body of water to propagate toward the surface. The streamers towed behind the vessel are elongated cable-like structures. Each streamer includes a number of seismic receivers or sensors that detect pressure and/or particle motion changes in the water created by the sound waves reflected back into the water from the subterranean formation.

The sounds waves that propagate upwardly from the subterranean formation are referred to as “up-going” wavefields that are detected by the receivers and converted into seismic signals that are recorded by the recording equipment and processed to produce seismic images that characterize the geological structure and properties of the subterranean formation being surveyed. However, seismic signals may also include “source ghost” produced by sound waves that are first reflected from the sea surface before the waves travel into the subsurface to produce scattered wavefields detected by the receivers. Source ghosts are time delayed relative to sound waves that travel directly from the source to the subterranean formation. As a result, source ghosts can amplify some frequencies and attenuate other frequencies and are typically manifest as spectral notches in the recorded seismic waveforms, which make it difficult to obtain accurate high-resolution seismic images of the subterranean formation. In addition to the “source ghosts,” the seismic signal may also include “receiver ghosts” produced by scattered sound waves that are first reflected from the sea surface before reaching the receivers. The receiver ghosts can also amplify some frequencies and attenuate other frequencies and are typically manifested as receiver ghost notches. As a result, those working in the petroleum industry continue to seek systems and methods to remove the effects of ghost reflections, or “deghost” seismic signals.

DETAILED DESCRIPTION

Computational methods and systems for receiver deghosting marine seismic streamer data are described. In particular, an exploration-seismology vessel tows a number of streamers that form a data acquisition surface located beneath an air/fluid surface referred to as the “free surface.” The streamers include receivers that measure pressure and particle motion wavefields that are digitally encoded and stored. The methods computationally deghost or substantially remove the receiver ghost signals from the seismic data recorded by the receivers independent of the free surface conditions or the shape of the data acquisition surface. In other words, the methods described below computationally remove receiver ghost signals from the seismic data without assuming restrictions on the free surface or assuming restrictions on the shape of the data acquisition surface, such as assuming a “frozen” (i.e., stationary) flat free surface and assuming a “frozen” flat and horizontal data acquisition surface. The deghosting methods include low frequency compensation to recover vertical velocity wavefield information that is typically lost due to the low signal-to-noise ratio over a low frequency range.

The following discussion includes two subsections: in subsection (I), an overview of exploration seismology is provided; and in subsection (II) a discussion of computational processing methods for receiver deghosting seismic signal data as an example of computational processing methods and systems to which this disclosure is directed. Reading of the first subsection can be omitted by those already familiar with marine exploration seismology.

I. An Overview of Marine Exploration Seismology

FIG. 1shows a domain volume of the earth's surface. The domain volume102comprises a solid volume of sediment and rock104below the solid surface106of the earth that, in turn, underlies a fluid volume of water108within an ocean, an inlet or bay, or a large freshwater lake. The domain volume shown inFIG. 1represents an example experimental domain for a class of exploration-seismology observational and analytical techniques and systems referred to as “marine exploration seismology.”

FIG. 2shows subsurface features of a subterranean formation in the lower portion of the domain volume shown inFIG. 1. As shown inFIG. 2, for exploration-seismology purposes, the fluid volume108is a relatively featureless, generally homogeneous volume overlying the solid volume104of interest. However, while the fluid volume108can be explored, analyzed, and characterized with relative precision using many different types of methods and probes, including remote-sensing submersibles, sonar, and other such devices and methods, the volume of solid crust104underlying the fluid volume is comparatively far more difficult to probe and characterize. Unlike the overlying fluid volume108, the solid volume104is significantly heterogeneous and anisotropic, and includes many different types of features and materials of interest to exploration seismologists. For example, as shown inFIG. 2, the solid volume104may include a first sediment layer202, a first fractured and uplifted rock layer204, and a second, underlying rock layer206below the first rock layer. In certain cases, the second rock layer206may be porous and contain a significant concentration of liquid hydrocarbon208that is less dense than the second-rock-layer material and that therefore rises upward within the second rock layer206. In the case shown inFIG. 2, the first rock layer204is not porous, and therefore forms a lid that prevents further upward migration of the liquid hydrocarbon, which therefore pools in a hydrocarbon-saturated layer208below the first rock layer204. One goal of exploration seismology is to identify the locations of hydrocarbon-saturated porous strata within volumes of the earth's crust underlying the solid surface of the earth.

FIGS. 3A-3Cshow an exploration-seismology method by which digitally encoded data is instrumentally acquired for subsequent exploration-seismology processing and analysis in order to characterize the structures and distributions of features and materials of a subterranean formation.FIG. 3Ashows an example of an exploration-seismology vessel302equipped to carry out a continuous series of exploration-seismology experiments and data collections. In particular, the vessel302tows one or more streamers304-305across an approximately constant-depth plane generally located a number of meters below the free surface306. The streamers304-305are long cables containing power and data-transmission lines to which receivers, also referred to as “sensors,” are connected at regular intervals. In one type of exploration seismology, each receiver, such as the receiver represented by the shaded disk308inFIG. 3A, comprises a pair of seismic receivers including a geophone that detects vertical displacement within the fluid medium over time by detecting particle motion, velocities or accelerations, and a hydrophone that detects variations in pressure over time. The streamers304-305and the vessel302include sophisticated sensing electronics and data-processing facilities that allow receiver readings to be correlated with absolute positions on the free surface and absolute three-dimensional positions with respect to an arbitrary three-dimensional coordinate system. InFIG. 3A, the receivers along the streamers are shown to lie below the free surface306, with the receiver positions correlated with overlying surface positions, such as a surface position310correlated with the position of receiver308. The vessel302also tows one or more acoustic-wave sources312that produce pressure impulses at spatial and temporal intervals as the vessel302and towed streamers304-305move across the free surface306. Sources312may also be towed by other vessels, or may be otherwise disposed in fluid volume108.

FIG. 3Bshows an expanding, spherical acoustic wavefront, represented by semicircles of increasing radius centered at the acoustic source312, such as semicircle316, following an acoustic pulse emitted by the acoustic source312. The wavefronts are, in effect, shown in vertical plane cross section inFIG. 3B. As shown inFIG. 3C, the outward and downward expanding acoustic wavefield, shown inFIG. 3B, eventually reaches the solid surface106, at which point the outward and downward expanding acoustic waves partially reflect from the solid surface and partially refract downward into the solid volume, becoming elastic waves within the solid volume. In other words, in the fluid volume, the waves are compressional pressure waves, or P-waves, the propagation of which can be modeled by the acoustic-wave equation while, in a solid volume, the waves include both P-waves and transverse waves, or S-waves, the propagation of which can be modeled by the elastic-wave equation. Within the solid volume, at each interface between different types of materials or at discontinuities in density or in one or more of various other physical characteristics or parameters, downward propagating waves are partially reflected and partially refracted, as at solid surface106. As a result, each point of the solid surface and within the underlying solid volume104becomes a potential secondary point source from which acoustic and elastic waves, respectively, may emanate upward toward receivers in response to the pressure impulse emitted by the acoustic source312and downward-propagating elastic waves generated from the pressure impulse.

As shown inFIG. 3C, secondary waves of significant amplitude are generally emitted from points on or close to the solid surface106, such as point320, and from points on or very close to a discontinuity in the solid volume104, such as points322and324. Tertiary waves may be emitted from the free surface306back towards the solid surface106in response to secondary waves emitted from the solid surface and subsurface features.

FIG. 3Calso shows the fact that secondary waves are generally emitted at different times within a range of times following the initial pressure impulse. A point on the solid surface106, such as point320, receives a pressure disturbance corresponding to the initial pressure impulse more quickly than a point within the solid volume104, such as points322and324. Similarly, a point on the solid surface directly underlying the acoustic source receives the pressure impulse sooner than a more distant-lying point on the solid surface. Thus, the times at which secondary and higher-order waves are emitted from various points within the solid volume are related to the distance, in three-dimensional space, of the points from the acoustic source.

Acoustic and elastic waves, however, travel at different velocities within different materials as well as within the same material under different pressures. Therefore, the travel times of the initial pressure impulse and secondary waves emitted in response to the initial pressure impulse are complex functions of distance from the acoustic source as well as the materials and physical characteristics of the materials through which the acoustic wave corresponding to the initial pressure impulse travels. In addition, as shown inFIG. 3Cfor the secondary wave emitted from point322, the shapes of the expanding wavefronts may be altered as the wavefronts cross interfaces and as the velocity of sound varies in the media traversed by the wave. The superposition of waves emitted from within the domain volume102in response to the initial pressure impulse is a generally very complicated wavefield that includes information about the shapes, sizes, and material characteristics of the domain volume102, including information about the shapes, sizes, and locations of the various reflecting features within the subterranean formation of interest to exploration seismologists.

The complicated wavefield that ensues in response to the initial pressure impulse is sampled, over time, by the receivers positioned along the streamers towed by an exploration-seismology vessel.FIGS. 4A-4Bshow processed waveforms generated by a hydrophone and a geophone, respectively. As shown inFIG. 4A, the waveform recorded by the hydrophone represents the pressure at times following the initial pressure impulse, with the amplitude of the waveform at a point in time related to the pressure at the hydrophone at the point in time. Similarly, as shown inFIG. 4B, the geophone provides an indication of the fluid particle motion or velocity or acceleration, in a vertical direction, with respect to time. The pressure and particle motion signals represented by the waveforms inFIGS. 4A and 413can be utilized to generate pressure wavefields and vertical velocity wavefields that, in turn, can be used for seismic processing, such as attenuation of multiples in marine seismic data. However, because the recorded particle motion signals are often contaminated with low frequency noise due to vibrations of the towed streamers, the signal-to-noise ratio over a low frequency range for the combined signals is poor.FIG. 4Cshows a plot of relative amplitude versus a range of acoustic frequencies of the example pressure and particle motion signals represented inFIGS. 4A and 4B. Solid curve401and dotted curve402represent the recorded pressure and motion signals, respectively.FIG. 4Creveals a portion403of the particle motion signals with relative amplitudes greater than zero, which corresponds to a low signal-to-noise ratio over a low frequency range from 0 Hz to an acoustic threshold frequency, fth, which in the example ofFIG. 4Cis about 20 Hz. As a result, the particle motion signal below the threshold frequency fthtypically cannot be used to perform seismic processing. On the other hand, the pressure wavefield below the threshold frequency fthcan be used to recreate certain information lost by the particle motion sensor when the spectrum of the pressure signal has a satisfactory signal-to-noise ratio and when the depth of the pressure and particle motion sensors is known. As a result, the depth of the streamer is selected so that the frequency of the first spectral notch in the pressure sensor signal caused by surface reflections is higher than the threshold frequency fth.

The free surface acts as a near perfect acoustic reflector, causing “ghost” effects in recorded seismic data. At the source location of each impulse, a time delayed reflection from the free surface, called a “source ghost,” trails the seismic wavefield that travels directly from the source into the subsurface of the earth. As a result, both low and high frequency information is penalized, and the earth subsurface cannot be accurately imaged with all seismic frequencies. In addition, at each receiver location along a streamer, a time delayed reflection from the free surface, called a “receiver ghost,” interferes in a continuous and undesirable manner with the seismic wavefield scattered directly from the earth subsurface to the receiver.FIG. 5shows an elevation view of an exploration-seismology vessel502towing a source504and a streamer506located below a free surface508.FIG. 5includes four ray paths that represent four ways in which wavefields associated with the same impulse can travel from the source504to a receiver510located along the streamer506. Dash line514represents a primary ray or direct path in which an impulse output from the source504travels directly into the subsurface512and associated scattered wavefields travel directly from the subsurface512to the receiver510. Solid line516represents a ray path that includes a source ghost517produced by sound waves that are first reflected from the free surface508before entering the subsurface512to produce scattered wavefields that travel directly from the subsurface512to the receiver510. Long dash line518represents a ray path that includes a receiver ghost519produced by sound waves that travel directly from the source504to the subsurface512to produce scattered wavefields that are reflected from the free surface508before being detected by the receiver510. Dot dash line520represents a ray path that includes both a source ghost521and a receiver ghost522.

The scattered wavefields associated with primary ray paths are ideally the desired seismic wavefields to be detected by the receivers. In practice, however, source ghosts and receiver ghosts are also detected, and because the ghosts are time delayed, the ghosts can interfere constructively and destructively with the recorded waveforms of the seismic wavefield. As a result, ghosts can lead to inaccurate seismic images of the subterranean formation located beneath the fluid volume. Computational methods and systems described below are directed to receiver deghosting seismic signal data for a time-varying, arbitrarily-rough free surface and without restrictions on the shape of the data acquisition surface defined by the steamers. The methods are also not limited to using pressure and vertical velocity wavefields above the threshold frequency fth. In other words, receiver deghosting of seismic signal data can be accomplished for pressure and vertical velocity wavefields over the full frequency range.

II. Methods for Receiver Deghosting as an Example of Computational Processing

Methods and Systems

FIG. 6shows a control-flow diagram600associated with a computational method for receiver deghosting seismic signal data with low frequency compensation. The flow diagram600summarizes an overall computational process for receiver deghosting seismic data. Each block of the diagram600is described below in a separate subsection.

Estimating Streamer Depth

In block601ofFIG. 6, the streamer depth is estimated.FIGS. 7A-7Dintroduce coordinates and terminology associated with a computational process for receiver deghosting.FIG. 7Ashows a top or xy-plane view of an example exploration-seismology vessel702towing a source704and four separate streamers706-709located beneath a free surface. Each streamer is attached at one end to the vessel702and at the opposite end to a buoy, such as a buoy710attached to the steamer708. The streamers706-709ideally form a planar horizontal acquisition surface located beneath the free surface. However, in practice, the acquisition surface can be smoothly varying in the z-direction due to active sea currents and weather conditions. In other words, the towed streamers may also undulate as a result of dynamic conditions of the fluid.FIG. 7Bshows an elevation or xz-plane view of the streamer708located beneath a free surface712.FIG. 7Brepresents a snapshot, at an instant in time, of the undulating free surface712and corresponding smooth wave-like shape in the streamer708.FIG. 7Aincludes xy-plane714andFIG. 7Bincludes a xz-plane of the same Cartesian coordinate system used to specify coordinate locations within the fluid volume with respect to three orthogonal, spatial coordinate axes labeled x, y and z. The x coordinate uniquely specifies the position of a point in a direction parallel to the length of the streamers, and the y coordinate uniquely specifies the position of a point in a direction perpendicular to the x axis and substantially parallel to the free surface712, and the z coordinate uniquely specifies the position of a point perpendicular to the xy-plane. As shown inFIG. 7B, the streamer708is at a depth, zr, which can be estimated at various locations along the streamers from hydrostatic pressure measurements made by depth controllers (not shown) attached to the streamers. The depth controllers are typically placed at about 300 meter intervals along each streamer. The estimated streamer depths are then used to calculate a two-dimensional interpolated streamer shape that approximates the wave-like shape of an actual streamer at an instant in time. Alternatively, the estimated streamer depths can be used to calculate a three-dimensional interpolated surface approximation of the acquisition surface. The depth zrand the elevation of the free-surface profile are estimated with respect to the geoid, which is represented inFIG. 7Bby dotted line718. The geoid is the hypothetical surface of the earth that coincides everywhere with mean sea level and is used to define zero elevation (i.e., z=0). Shaded disks, such as shaded disk720, represent receivers spaced at regular intervals, Δx. The coordinates of the receiver720are given by (xr, yr, zr), where the depth zrcan be an interpolated value.

Measuring Pressure and Velocity Wavefields

Returning toFIG. 6, in block602, pressure and velocity wavefields are measured at the receivers.FIG. 8also shows an elevation view of the vessel702and the streamer708and includes a magnified view802of a receiver804located along the streamer708. The magnified view802reveals that the receivers are actually dual sensors that include a pressure sensor806, such as a hydrophone, and a motion sensor808, such as a geophone. Each pressure sensor measures a pressure wavefield denoted by p(xr, yr, zr, t) and each motion sensors measures a velocity wavefield denoted by v{right arrow over (n)}(xr, yr, zr, t), where xr=mΔx and y represent the x- and y-spatial coordinate of the sensor, and t represents time. The index m is a positive integer used to identify the receiver and is also referred to as a “channel” index. The acquisition surface, as shown inFIG. 7A, has 4 streamers706-709and each streamer includes 14 receivers. The receivers can be indexed 0 through 13. For example, a receiver810of the streamer708located closest to the vessel702can be assigned channel index m=0, and a receiver812located along the same streamer708farthest from the vessel702can be assigned channel index m=13, with dual sensors in between indexed consecutively from 1 to 12. The motion sensors may be mounted within a gimbal in order to orient the motion sensors to detect particle motion in a direction normal to the streamer as represented by unit normal vector814in magnified view802. Thus, the motion sensors measure a velocity wavefield v{right arrow over (n)}directed normal to the smoothly varying streamer708, where the subscript vector {right arrow over (n)} represents a normal unit vector that points in the direction814normal to the streamer608in the xz-plane. However for particle motion measured below the threshold frequency fth, the signal-to-noise ratio is typically low as described above with reference toFIG. 4C. As a result, the vertical velocities below the threshold frequency fthcannot typically be accurately determined. Methods include low frequency compensation described below to calculate the vertical velocity wavefields at the streamer receivers below the threshold frequency fth.

Returning toFIG. 6, in block603wavefield decomposition is performed on the pressure and velocity wavefields for frequencies greater than the threshold frequency fthdescribed above with reference toFIG. 4C. In the following description the y-spatial component is ignored in order to simplify the description. Note that in practice the y-spatial component is included. In other words, in the discussion that follows, the three spatial coordinates of the measured pressure wavefield p(xr, yr, zr, t) are reduced to two spatial coordinates in pressure wavefield representation, p(xr, zr, t), and the three spatial coordinates of the measured velocity wavefield v{right arrow over (n)}(xr, yr, zr, t) are reduced to two spatial coordinates in velocity wavefield representation, v{right arrow over (n)}(xr, zr, t). The reduction to two spatial coordinates gives clear insight while preserving the main features of the method.

The pressure and velocity wavefields can be decomposed into up-going and down-going pressure and vertical velocity components. InFIG. 8, directional arrow816represents the direction of an up-going wavefield detected by a receiver818and dashed line directional arrow820represents the direction of a down-going wavefield reflected from the free surface712and detected by the receiver818. In other words, the pressure wavefield p(xr, zr, t) is composed of an up-going pressure component and a down-going pressure component, and the velocity wavefield v{right arrow over (n)}(xr, zr, t) is also composed of an up-going vertical velocity component and a down-going vertical velocity component. The down-going pressure wavefield and the down-going vertical velocity wavefield are receiver ghosts wavefields, and the up-going pressure wavefield and the up-going vertical velocity wavefield are receiver deghosted wavefields.

FIGS. 9A-9Ceach show an example plot that represents an aspect of wavefield decomposition. InFIGS. 9A-9C, vertical axis902is a z-coordinate axis that represents the depth or z-spatial dimension, and inFIGS. 9A-9B, horizontal axis904is an x-coordinate axis that represents the x-spatial dimension. The “0” z-spatial coordinate corresponds to a plane tangent at x=0 to the geoid718described above with reference toFIG. 7B.FIG. 9Ashows an example plot of an interpolated streamer906based on the estimated depths obtained from depth controllers attached to the actual streamer as described above with reference toFIG. 7. The shape of the interpolated streamer906substantially matches the shape of the streamer708shown inFIG. 7B. Shaded disks, such as shaded disk908, represent the locations of receivers along the interpolated streamer906. The pressure wavefield p(mΔx, zr,m, t) and velocity wavefield v{right arrow over (n)}(mΔx, zr,m, t) associated with each receiver is transformed from the space-time domain to a pressure wavefield P(mΔx, zr,m, ω) and a velocity wavefield V{right arrow over (n)}(mΔx, zr,m, ω) in the space-frequency domain, where ω=2πf is the angular frequency for the acoustic frequencies f detected by the dual sensors. For example, the pressure wavefield and the velocity wavefield associated with a receiver can be transformed using Fourier transforms:
P(mΔx,zr,m,ω)=∫−∞∞p(mΔx,zr,m,t)e−jωtdt(1)
V{right arrow over (n)}(mΔx,zr,m,ω)=∫−∞∞v{right arrow over (n)}(mΔx,zr,m,t)e−jωtdt(2)
where j is the imaginary unit √{square root over (−1)}. In practice, the transformation can be carried out using a discrete Fast-Fourier Transform (“FFT”) for computational efficiency. Note that lower-case letters p and v are used to represent quantities in the space-time domain while upper-case letters P and V are used to represent quantities in the space-frequency or wavenumber-frequency domain.FIG. 9Bshows an example plot of the interpolated streamer906with the pressure and velocity wavefields transformed to the space-frequency domain.

After the pressure and velocity wavefields associated with each receiver have been transformed from the space-time domain to the space-frequency domain, the pressure wavefields and the velocity wavefields are combined to produce an up-going pressure component at the geoid (i.e., z=0) in the wavenumber-frequency domain. Pressure sensors alone cannot distinguish between the opposing polarity of the seismic wavefield scattered up from the subterranean formation and the time-delayed seismic wavefield reflected down from the sea surface (i.e., receiver ghost), but the information obtained from the particle motion sensors can be used to distinguish the polarity of the seismic wavefield. A mathematical expression relating the pressure and velocity wavefields to an up-going pressure component of the pressure wavefield at the geoid is given by:

kz=(ωc)2-kx2with c the speed of sound in the fluid;kxis the horizontal wavenumber in the x-spatial directionm is the dual sensor or channel index;M is the total number of dual sensors located along the streamer;ρ is the density of the fluid;zr,mis the interpolated depth of the streamer at the mthdual sensor;nxis the x-component of the normal vector {right arrow over (n)};nzis the z-component of the normal vector {right arrow over (n)}; andV{right arrow over (n)}th(mΔx, zr,m, ω) is the velocity wavefield for angular frequencies greater than an angular threshold frequency ωth(i.e., ωth=2πfth).
Analogously, the down-going pressure component in the wavenumber-frequency domain at z=0 is calculated in a similar manner by:

FIG. 9Cshows an example plot of the wavenumber-frequency domain. Horizontal axis910represents the wavenumber kxcoordinate axis. Note that the angular frequency axis w is perpendicular to the kxz-plane, which is not represented inFIG. 9C. Shaded disk912represents a point in the wavenumber-frequency domain located along the kxcoordinate axis910. The wavenumber-frequency domain point912has associated up-going and down-going pressure plane wavefield components Pup(kx, z=0, ω)914and Pdown(kx, z=0, ω)916.

After the up-going and down-going pressure wavefield components at the geoid have been determined in the wavenumber-frequency domain, the pressure wavefield components are shifted to an observation level at a depth, zobs, between the geoid and the streamer. InFIG. 9C, dashed line918represents the kxaxis at an observation level located between the geoid and the streamer indicated at the z-axis by zr920. The depth, zobs, of the observation level918lies between the geoid and the depth of the streamer920. The up-going pressure component at a point (kx, zobs)921along the observation level zobs918is calculated from Pup(kx, z=0, ω) by:
Pup(kx,zobs,ω)=Pup(kx,z=0,ω)ejkzzobs(5)
Likewise, the down-going pressure component at the observation level zobs918is calculated by:
Pdown(kx,zobs,ω)=Pdown(kx,z=0,ω)e−jkzzobs(6)

The up-going vertical velocity component at the point (kx, zobs)921is calculated from the up-going pressure component Pup(kx, zobs, ω) by:

The down-going vertical velocity component at the point (kx, zobs)921is calculated from the down-going pressure component Pdown(kx, zobs, ω) by:

Vzdown⁡(kx,zobs,ω)=kzρω⁢Pdown⁡(kx,zobs,ω)(8)
In other embodiments, calculation of the free-surface profile can also be accomplished using the up-going and down-going vertical velocity components.

In summary, wavefield decomposition is the process of transforming the pressure wavefield p(mΔx, zr,m, t) and velocity wavefield v{right arrow over (n)}(mΔx, zr,m, t) measured at the corresponding dual sensors of each receiver in the space-time domain, as shown inFIG. 9A, into a down-going pressure component Pdown(kx, zobs, ω)922, an up-going pressure component Pup(kx, zobs, ω)923, a down-going vertical velocity component vzdown(kx, zobs, ω)924and an up-going vertical velocity component Vzup(kx, zobs, ω)925in the wavenumber-frequency domain, as shown inFIG. 9C. The Pup, Pdown, Vzup, and Vzdownare computed from the pressure wavefield P and the velocity wavefield V{right arrow over (n)}thabove the threshold frequency ωthdescribed above with reference toFIG. 4C.

Note that a three-spatial-dimensional version of the up-going and down-going pressure components and the up-going and down-going vertical velocity components can be obtained by replacing the vertical wavenumber kzby:

kz=(ωc)2-ky2-kx2where kyis the horizontal wavenumber in the y-spatial direction.

Computing a Free-Surface Profile

Returning toFIG. 6, after wavefield decomposition has been accomplished in block603, an imaged free-surface profile is calculated in block604.FIGS. 10A-10Cshow three different plots in a process of computing an imaged free-surface profile of an actual free-surface. An imaged free-surface profile is calculated by first computing up-going and down-going pressure components, or the up-going and down-going vertical velocity components, for each point of a series of depth levels that extend upward from the observation level to beyond the geoid or z=0 axis910.FIG. 10Ashows an example of a series of points represented by a column of open circles1002that extend from the point (kx, zobs)921on the observation level zobs918to beyond the z=0 axis910. Each open circle in the series1002represents a point with coordinates (kx, z), where the kx-coordinate is the same for each point and the z-coordinates are increased incrementally in a process called extrapolation. The up-going pressure wavefield is calculated at each point in the series1002by:
Pup(kx,z,ω)=Pup(kx,zobs,ω)ejkz(zobs−z)(9a)
where z is a coordinate value in the series of points1002. The down-going pressure wavefield is calculated at each point in the series1002by:
Pdown(kx,z,ω)=Pdown(kx,zobs,ω)e−jkz(zobs−z)(9b)

In another embodiment, the up-going vertical velocity component at each point in the series1002is calculated by:
Vzup(kx,z,ω)=Vzup(kx,zobs,ω)ejkz(zobs−z)(10a)
The down-going vertical velocity component at each point in the series1002is calculated by:
Vzdown(kx,z,ω)=Vzdown(kx,zobs,ω)e−jkz(zobs−z)(10b)

After the extrapolated up-going and down-going pressure components, or the extrapolated up-going and down-going vertical velocity components, have been calculated in the wavenumber-frequency domain, an inverse Fourier transform is used to transform the extrapolated up-going and down-going pressure components and/or extrapolated up-going and down-going vertical velocity components into the space-frequency domain.FIG. 10Bshows a series of points1010in the space-frequency domain described above with reference toFIGS. 9A-9B. An inverse Fourier transform is used to transform the pressure and vertical velocity components associated with the points in the series1002inFIG. 10Ato obtain corresponding pressure and vertical velocity components of the points1010inFIG. 10B. The pressure wavefields transformed to the space-frequency domain are given by:

Pup⁡(xr,z,ω)=12⁢π⁢∫-∞∞⁢⁢ⅆkx⁢Pup⁡(kx,z,ω)⁢ⅇj⁢⁢kx⁢xr(11⁢a)Pdown⁡(xr,z,ω)=12⁢π⁢∫-∞∞⁢⁢ⅆkx⁢Pdown⁡(kx,z,ω)⁢ⅇj⁢⁢kx⁢xr(11⁢b)
The vertical velocity wavefields transformed to the space-frequency domain are given by:

An imaging condition is used to calculate an image value I(xr, z) at each point in the series of points in the space-frequency domain. For example inFIG. 10C, an imaging condition is applied to each point in the series of points1010to obtain an image value, such as an image value I(xr, z) associated with the point (xr, z)1018. The imaging condition can be a cross correlation of the extrapolated up-going and down-going pressure, or vertical velocity, components in the space-frequency domain. In one embodiment, the imaging condition that represents a free-surface image value for a selected receiver position x and extrapolation depth z is calculated by applying the following cross-correlation equation:

I⁡(xr,z)=∑ω⁢⁢U⁡(xr,z,ω)⁢D⁡(xr,z,ω)_∑ω⁢⁢U⁡(xr,z,ω)⁢U⁡(xr,z,ω)_(14)
The z-coordinate value or extrapolation depth associated with the maximum image value Imax(xr, z) for a given channel position x corresponds to a free-surface elevation z at the receiver position x. In the example ofFIG. 10C, points in the series1010associated with the receiver908are represented as dashed-line open circles except for the point1022which corresponds to a point (xr, z0) that yields a maximum cross-correlation Imax(xr, z0). As a result, the point (xr, z0) is an image point of a free-surface profile. Embodiments are not limited to the imaging conditions described above. Other types of imaging conditions can also be used. Other image points represented by open circles, such as open circles1024, are calculated in the same manner by applying the same imaging condition to each point in a series of points as described above. Interpolation over the collection of image points forms a time stationary approximate image of the free-surface profile above the streamer908corresponding to a selected time window of the input data. For example, inFIG. 10Ca curve1026is an imaged free-surface profile of the hypothetical free-surface profile represented by dotted curve1028in a selected time window. An imaged free-surface profile above a streamer can be interpolated using spline interpolation, Lagrange interpolation, polynomial interpolation, or another suitable interpolation technique. In other embodiments, the image points associated with two or more streamers can be used to calculate an approximate three-dimensional free surface above the streamers using multi-dimensional interpolation techniques, such as Barnes interpolation, Bezier interpolation, bicubic interpolation, and bilinear interpolation. Points along the image free-surface profile1026are represented by [x, ƒ(x)], where x represents a horizontal coordinate along the x-axis904and ƒ(x) is the interpolated function value that represents the free-surface height. A time-varying, free-surface estimation is obtained by continuously moving the input data time window and combining the computed “frozen” free-surface profiles for all the time windows from a start recording time to an end recording time.

FIG. 11shows a control-flow diagram of a computational method for calculating the free-surface profile of block604. In block1101, up-going and down-going pressures and vertical velocities are calculated for points in the wavenumber-frequency domain, as described above with reference toFIG. 10A. In block1102, the pressures and vertical velocities associated with the points are transformed from the wavenumber-frequency domain to the space-frequency domain. In block1103, an imaging condition is applied as described above with reference toFIG. 10C. In block1104, the point associated with the maximum imaging condition for a given receiver channel is calculated, as described above with reference toFIG. 10C. In block1105, the image points associated with a streamer are used to calculate a “frozen” image profile of the free-surface.

Extending the Free-Surface Profile

Returning toFIG. 6, in block605, the free-surface profile is extended to cover the source location.FIG. 12Ashows a plot of a hypothetical free surface represented by dashed curve1028and the imaged free-surface profile1026that approximates the free surface1028above the streamer906. Circle1202represents the coordinate location of a source towed by an exploration-seismology vessel (not shown). As shown inFIG. 12A, and inFIGS. 5, 7B and 8described above, the source1202is not located over the streamers but is instead located between the vessel and the streamers. Because the imaged free-surface profile1026is computed from subsurface reflections, the imaged free-surface profile1026is limited to the streamer spread as indicated by dashed lines1204and1206. As a result, the imaged free-surface profile1026does not cover the source1202and can only be used to deghost ghosts that are reflected from the free surface directly above the streamer and cannot be used to deghost ghosts that are reflected from the free surface above the source1202. For example, directional arrows1208and1210represent two sound impulse paths output from the source1208and1210. Both paths result in ghosted direct down-going wavefields and source ghost reflections as described above with reference toFIG. 5. The computation of the direct down-going wavefield including its ghost are used for low frequency vertical velocity wavefield reconstruction and deghosting of seismic data with sources above the streamer level. The imaged free surface1026can be used to deghost seismic data with ghosts that follow the path1210, but because the imaged free surface1026does not extend to cover the region above the source1202, the image free surface1026cannot be used to deghost seismic data that follows the path1208.

In order to account for ghosts associated with reflections from the free surface above the source, the imaged free-surface profile is extended by calculating a free-surface profile extension of the imaged free-surface profile that covers the region above the coordinate location of the source.FIG. 12Bshows a plot of the hypothetical free surface1028and an extended imaged free-surface profile. The extended imaged free-surface profile is the image free surface1026extended to include a free-surface profile extension1212. The free-surface profile extension1212can be calculated based on parameters associated with a realistic physical model of a fully developed sea state determined at the time the seismic data is measured. For example, a realistic physical model can be a Pierson-Moskowitz model that provides parameters such as wavelength, wave-heights, and the velocity of the free surface waves. The Pierson-Moskowitz model of the free surface can be used to calculate the free-surface profile extension1212that covers the source1202.

The Pierson-Moskowitz model assumes that when the wind blows steadily for a long period of time over a large area, the waves eventually reach a state of equilibrium with the wind. This condition is referred to as a “fully developed sea.” The Pierson-Moskowitz spatial roughness spectrum for a fully developed sea in one-dimension is given by:

W⁡(K)=[α4⁢K3]⁢exp⁡(-β⁢⁢g2/K2⁢Uw4)(15)where K is the free surface-wave spatial wavenumber;Uwis the wind speed measured at a height of about 19 meters;α is 8.0×10−3;β is 0.74; andg is the acceleration due to gravity.
The free-surface height function ƒ(x) is generated at a point x as follows:

f⁡(x)=1L⁢∑i=0N-1⁢⁢F⁡(Ki)⁢ⅇj⁢⁢Ki⁢x(16)where for the index i≧0,

F⁡(Ki)=2⁢π⁢⁢L⁢⁢W⁡(Ki)⁢{[N⁡(0,1)+j⁢⁢N⁡(0,1)]/2for⁢⁢i≠0,N/2N⁡(0,1)for⁢⁢i=0,N/2(17)and for i<0, F(Ki)=F(K−i)*.
In Equations (16) and (17), the spatial wavenumber for component i is given by Ki=2πi/L, where L is the length of free surface. The random number N(0,1) can be generated from a Gaussian distribution having zero mean and a unit variance. As a result, the free surface is formed by adding each wavenumber component imposing random phase shifts. A frozen in time Pierson-Moskowitz free surface can be computed from Equation (16) using a fast Fourier Transform for computational efficiency.

Free-surface waves are dispersive and in deep water, the frequency and wavenumber are related by a dispersion relation given by:
Ω(Ki)=√{square root over (gKi)}  (18)
Equation (18) implies that each spatial harmonic component of the surface may move with a definite phase velocity. As a result, in general, free surface waves of longer wavelengths travel faster relative to waves with shorter wavelengths. Combining Equations (16) and (18) gives a time-varying Pierson-Moskowitz free surface:

f⁡(x,t)=1L⁢∑i=0N-1⁢⁢F⁡(Ki)⁢ⅇj⁡(Ki⁢x-Ω⁡(Ki)⁢t)(19)
where t is the instantaneous time. Equation (19) characterizes a one-dimensional rough free surface moving in the positive x-direction and can be used to compute the free-surface extension at earlier or later time instances.

In general, a frozen free-surface for earlier or later times can be computed as follows. Consider a free surface shape at an instant in time t with wave heights given by ƒ(x, t), the wavenumber spectrum F(Ki) of the free-surface is computed and an arbitrary known dispersion relation Ω(Ki) can be used to calculate the frozen free surface at earlier (t−Δt) or later (t+Δt) instances of time by:

Returning toFIG. 6, in block606the free-surface wavefield reflectivity (i.e., the response of a unit point source at the receiver position {right arrow over (r)}r) can be computed over the extended free-surface using:

Hn(1)⁡(x)≅2π⁢⁢x⁢exp⁡(j⁡(x-π4-n⁢⁢π2))is the asymptotic form of the first-order Hankel function with n=0 and 1. The parameters of Equation (21) are represented inFIG. 13as follows:k is the wavenumber of the propagating wavefield;ƒ(x) is the free-surface height described above;[x0, ƒ(x0)] is a coordinate position1302of a running scattering point on the surface;{right arrow over (r)}0is a vector1304from the origin to the running scattering point1302;{right arrow over (r)}ris a vector1306from the origin to a fixed receiver position1308with spatial coordinates (xr, yr, zr);{right arrow over (r)}0−{right arrow over (r)}ris a vector1310from the running scattering point1302to the receiver position1408;{right arrow over (r)}sis a vector1312from the origin to a source position1314;{right arrow over (r)}dis a vector1316from the source position1314to the receiver1308;{right arrow over (ρ)} is a vector1318from the source1314to the scattering point1302; andη(x′)={circumflex over (n)}·{circumflex over (ρ)}=cos θ is the obliquity factor with normal vectors {circumflex over (n)}1320and {circumflex over (ρ)}1322corresponding to the free surface normal and the unit vector direction of the incident field at [x0, ƒ(x0)]1302and θ is the angle between the vectors {circumflex over (n)}1320and {circumflex over (ρ)}1322.

The pressure wavefield P({right arrow over (r)}r, ω) over the entire frequency range can be obtained from pressure measurements at the pressure sensors. However, as explained above with reference toFIG. 4C, the recorded particle motion signal may be contaminated with low frequency noise due to vibrations of the towed streamers. As a result, particle motion signals recorded below the threshold frequency ωthmay not be useful for determining vertical velocity wavefields at receiver locations along the streamer below the threshold frequency fth. Instead, only the vertical velocity wavefield V{right arrow over (n)}th({right arrow over (r)}r, ω) obtained from particle motion measurements over a range of frequencies greater than ωthare typically reliable. A vertical velocity wavefield for frequencies below the threshold frequency ωth, V{right arrow over (n)}cal({right arrow over (r)}r, ω), can be calculated from the normal derivate of the pressure wavefield, {right arrow over (n)}·∇rP, in a process called “low frequency compensation.” Wavefield decomposition can then be performed to calculate receiver deghosted pressure and velocity wavefields.

Computing Gradient of Pressure Wavefield

In block607, the gradient of the pressure wavefield, ∇rP, is computed. Computation of the gradient of the pressure wavefield, ∇rP, is determined by whether the depth of the source is greater than or less than the depth of the streamer.FIGS. 14A-14Bshow two-dimensional xz-plane views of an example model space associated with a fluid volume. InFIGS. 14A-14B, the space includes a representation of a streamer1402, denoted by Sr, a source1404, that produces sound impulses. The streamer1402and the source1404can be towed by an exploration-seismology vessel (not shown) within the fluid volume below the free surface that can be imaged as described above to produce an imaged free-surface profile represented by curve1406and denoted by S0. The imaged free-surface S0includes a free-surface profile extension1408above the source1404. Vectors such as vector {right arrow over (r)}s1410represent source spatial coordinates (xs, ys), vectors such as vector {right arrow over (r)}1412represent coordinates (x, y) in the fluid volume, and vectors such as vector {right arrow over (r)}r1414represent receiver coordinates (xr, zr) along the streamer1402. When the depth of the source1404is less than the depth of the streamer, as shown inFIG. 14A, the expression used to calculate the gradient of the pressure wavefield over the entire frequency range is give by:
∫SrdSr{right arrow over (n)}·R({right arrow over (r)}r,{right arrow over (r)})∇rP({right arrow over (r)}r,ω)=a(ω)R({right arrow over (r)}s,{right arrow over (r)})+∫SrdSr{right arrow over (n)}·P({right arrow over (r)}r,ω)∇rR({right arrow over (r)}r,{right arrow over (r)})  (22)where α(ω) is the Fourier transform of the source-time function for the source at the coordinate location {right arrow over (r)}s; andR({right arrow over (r)},{right arrow over (r)}s) is the free-surface wavefield reflectivity given by Equation (21).
In order to compute the free-surface wavefield reflectivity with a hypothetical source at {right arrow over (r)} below the receiver surface and a hypothetical receiver position at {right arrow over (rs)} knowledge of an extended free surface profile is needed. Equation (22) is a Fredholm integral equation of the first kind for the gradient of the pressure wavefield ∇rP({right arrow over (r)}r, ω) where the right-hand side of Equation (22) contains only known parameters such as the field pressure wavefield P({right arrow over (r)}r, ω) and the free-surface wavefield reflectivity R({right arrow over (r)}r, {right arrow over (r)}). The free-surface wavefield reflectivity with a hypothetical source at r below the receiver surface and a hypothetical receiver at {right arrow over (r)}ris computed from the imaged free-surface profile. The gradient of the reflectivity, ∇rR({right arrow over (r)}r, {right arrow over (r)}), can be computed using numerical gradient techniques. On the other hand, when the source is located at a depth below the streamer Sr, as shown inFIG. 14B, the expression used to calculate the gradient of the pressure wavefield ∇rP({right arrow over (r)}r, ω) over the entire frequency range is give by:
∫SrdSr{right arrow over (n)}·R({right arrow over (r)}r,{right arrow over (r)})∇rP({right arrow over (r)}r,ω)=∫SrdSr{right arrow over (n)}·P({right arrow over (r)}r,ω)∇rR({right arrow over (r)}r,{right arrow over (r)})  (23)
In Equation (23), the source function a(ω) is not used to calculate ∇rP({right arrow over (r)}r, ω). Note that the solutions of Equations (22) and (23) become unstable when the spectrum of the pressure wavefield has very small values (e.g., close to receiver ghost notches). These spectral notches occur generally for typical towing depths at frequencies ω>ωth. As a result, the gradient of the pressure wavefield can be computed for the frequencies below the threshold frequency ωthfor a time-varying, arbitrarily rough free-surface. A derivation of Equations (22) and (23) is provided in an APPENDIX below.

Depending on whether the source is at a depth above or below the streamer, as shown inFIGS. 14A and 14B, respectively, Equations (22) and (23) can be solved numerically for the gradient of the pressure wavefield, ∇rP, at receiver locations along the streamer using quadrature or expansion methods. For quadrature methods, the integrals are approximated by quadrature formulas and the resulting system of algebraic equations is solved. For expansion methods, the solution is approximated by an expansion in terms of basis functions.

Computing Normal Velocity Wavefield

Returning toFIG. 6, in block608the normal component of the vertical velocity wavefield below the threshold frequency fthat each receiver location along a streamer can be calculated by:

Vn⇀cal⁡(xr,zr,ω)=-jρ⁢⁢ω⁢n⇀·∇r⁢P⁡(xr,zr,ω)(24)where ρ is the density of the fluid; and{right arrow over (n)} is a normal vector to a receiver.
FIG. 15shows a segment of a streamer1502located beneath an imaged free surface1604in the xz-plane of a fluid volume. A normal vector1508to the streamer1502at the receiver1506is given by:

n⇀=(nx,nz)=(-sin⁢⁢ϕ,cos⁢⁢ϕ)=(-ⅆzrⅆq,ⅆxrⅆq)where ∇rP is the gradient of the pressure wavefield at the receiver1606; and{right arrow over (n)}·∇rP is the normal derivative of the pressure wavefield P at the receiver1606.

At this point the pressure wavefield P(xr, zr, ω) over the entire frequency range is known from pressure sensor measurements, the vertical velocity wavefield V{right arrow over (n)}th(xr, zr, ω) over a frequency range above the low frequency threshold ωthis obtained from particle motion measurements and the vertical velocity wavefield below the threshold frequency ωthis calculated from Equation (24). The vertical velocity wavefield V{right arrow over (n)}(xr, zr, ω) over the entire frequency range can be calculated by:

Vn⇀⁡(xr,Zr,ω)={Vn⇀cal⁡(xr,zr,ω)for⁢⁢0≤ω≤ωth(1-w⁡(ω))Vn⇀cal⁡(xr,zr,ω)+w⁡(ω)⁢Vn⇀th⁡(xr,zr,ω)for⁢⁢ωth<ω≤ω1Vn⇀th⁡(xr,zr,ω)for⁢⁢ω1<ω(25)where w(ω) is a weight function with the property that w(ω)=0 when ω=ωthand w(ω)=1 when ω=ω1to smoothly transition V{right arrow over (n)}from V{right arrow over (n)}calto V{right arrow over (n)}thas ω increases from ωthto ω1, andω1is less than πc/zr(i.e., fth<c/2zr) in order to maintain a good signal-to-noise ratio of the hydrophone signal.
For example, the weight function w(ω) can be a linear weight function given by:

w⁡(ω)=ω-ωthω1-ωth⁢⁢for⁢⁢ωth<ω≤ω1
In other embodiments, other types of linear weight functions as well as non-linear weight functions can be used such as a harming weight function.

In block609, the recorded pressure wavefield and the vertical velocity wavefield given by Equation (25) are used to calculate the receiver side deghosted pressure wavefield (i.e., the up-going pressure wavefield Pup) over the entire frequency range in the wavenumber-frequency domain using:

Vzup⁡(kx,zr,ω)=-kzρω⁢Pup⁡(kx,zr,ω)(28)
Note that the deghosted pressure wavefield Pup(kx, zr, ω) and the deghosted vertical velocity wavefield Vzup(kx, zr, ω) are the up-going pressure and vertical velocity wavefields over the entire frequency range.

The deghosted pressure wavefield can then be transformed from the wavenumber-frequency domain into the space-frequency domain to obtain:

Pup⁡(xr,zr,ω)=1M⁢∑m=0M-1⁢⁢Pup⁡(kx,zr,ω)⁢ⅇj⁢⁢xr⁢m⁢⁢Δ⁢⁢kx(29)
followed by a transformation from the space-frequency domain to the space-time domain:

pup⁡(xr,zr,t)=12⁢π⁢∫-∞∞⁢⁢ⅆω⁢⁢Pup⁡(xr,zr,ω)⁢ⅇjω⁢⁢t(30)
Likewise, the deghosted vertical velocity wavefield can be transformed from the wavenumber-frequency domain to the space-frequency domain to obtain:

Vzup⁡(xr,zr,ω)=1M⁢∑m=0M-1⁢⁢Vzup⁡(kx,zr,ω)⁢ⅇj⁢⁢xr⁢m⁢⁢Δ⁢⁢kx(31)
followed by a transformation from the vertical velocity wavefield from the space-frequency domain to the space-time domain:

vzup⁡(xr,zr,t)=12⁢π⁢∫-∞∞⁢⁢ⅆω⁢⁢Vzup⁡(xr,zr,ω)⁢ⅇj⁢⁢ω⁢⁢t(32)
The inverse Fourier transformations of Equations (29)-(32) can be carried out using inverse fast Fourier transformations for computational efficiency. In summary, the receiver deghosted, or up-going, wavefields pupand vzupat the receivers are used to calculate images of the subterranean formations located beneath the fluid volume.

FIG. 16shows one illustrative example of a generalized computer system that executes an efficient method for source deghosting a scattered wavefield and therefore represents a seismic-analysis data-processing system to which the description is directed. The internal components of many small, mid-sized, and large computer systems as well as specialized processor-based storage systems can be described with respect to this generalized architecture, although each particular system may feature many additional components, subsystems, and similar, parallel systems with architectures similar to this generalized architecture. The computer system contains one or multiple central processing units (“CPUs”)1602-1605, one or more electronic memories1608interconnected with the CPUs by a CPU/memory-subsystem bus1610or multiple busses, a first bridge1612that interconnects the CPU/memory-subsystem bus1610with additional busses1614and1616, or other types of high-speed interconnection media, including multiple, high-speed serial interconnects. These busses or serial interconnections, in turn, connect the CPUs and memory with specialized processors, such as a graphics processor1618, and with one or more additional bridges1620, which are interconnected with high-speed serial links or with multiple controllers1622-1627, such as controller1627, that provide access to various different types of computer-readable media, such as computer-readable medium1628, electronic displays, input devices, and other such components, subcomponents, and computational resources. The electronic displays, including visual display screen, audio speakers, and other output interfaces, and the input devices, including mice, keyboards, touch screens, and other such input interfaces, together constitute input and output interfaces that allow the computer system to interact with human users. Computer-readable medium1628is a data-storage device, including electronic memory, optical or magnetic disk drive, USB drive, flash memory and other such data-storage device. The computer-readable medium1628can be used to store machine-readable instructions associated with the source deghosting computational methods described above and can be used to store encoded data, during store operations, and from which encoded data can be retrieved, during read operations, by computer systems, data-storage systems, and peripheral devices.

APPENDIX

A derivation of integral equations that can be used to compute the gradient of the pressure wavefield at a receiver of a streamer, ∇rP, is described with reference toFIGS. 17A-17E(See also “Extraction of the normal component of the particle velocity from marine pressure data,” L. Amundsen et al.,Geophysics, Vol. 60, No. 1, pp 212-222 (January-February 1995)).

FIG. 17Ashows a two-dimensional xz-plane view of an example model space1700associated with a fluid volume. The space1700includes a representation of a streamer1702, denoted by Sr, and a source1704that produces sound impulses. The streamer1702and the source1704can be towed by an exploration-seismology vessel (not shown) within the fluid volume below an imaged free-surface profile represented by curve1706and denoted by S0. The imaged free-surface S0includes a free-surface profile extension1708above the source1704. Vectors such as vector {right arrow over (r)}s1704represent the spatial coordinates (xs, ys) of the source, vectors such as vector {right arrow over (r)}1710represent coordinates (x, y) in the space1700, and vectors such as vector {right arrow over (r)}r1712represent receiver coordinates (xr, zr) located along the streamer1702. A shaded region1714inFIG. 17A, and in subsequentFIGS. 17B-17E, represents a scattering region beneath the streamer1702. The speed of sound in the region1714is given by:

c⁡(r⇀)=c01-α⁡(r⇀)(A⁢-⁢1)where c0is the speed of sound in a fluid; and0≦α({right arrow over (r)})<1 is the index of refraction over the region1714.

The pressure field generated by the source1704at the point1710is p({right arrow over (r)}, {right arrow over (r)}s, t) and can be transformed from the space-time domain to the space-frequency domain using a Fourier transform to obtain a pressure wavefield P({right arrow over (r)}, {right arrow over (r)}s, ω) at the point1710. In practice the transformation from the space-time domain to the space-frequency domain can be accomplished using a Fast-Fourier Transform for computational efficiency.

The constant-density acoustic wave equation that characterizes a pressure wavefield P({right arrow over (r)}, {right arrow over (r)}s, ω) caused by a single source that generates sound impulses, such as the source1704, located at the spatial coordinates {right arrow over (r)}sis given by:

(∇2⁢+ω2c2⁡(r⇀))⁢P⁡(r⇀,r⇀s,ω)=a⁡(ω)⁢δ⁡(r⇀-r⇀s)(A⁢-⁢2)where ∇2is the Laplacian;ω is the angular frequency;δ({right arrow over (r)}−{right arrow over (r)}fs) is the three-dimensional Dirac delta function that represents the impulse of sound output from the source1704; anda(ω) is the Fourier transform of the source-time function for the source at the coordinate location {right arrow over (r)}s.
The acoustic wave Equation (A-2) characterizes the propagation of the acoustic pressure wavefield P({right arrow over (r)}, {right arrow over (r)}s, ω) in the fluid volume, where the acoustic pressure waves originate from the impulse sound source1704. Substituting the Equation (A-1) for the speed of sound c({right arrow over (r)}) into the acoustic wave Equation (A-2) and rearranging gives:
(∇2+k02)P({right arrow over (r)},{right arrow over (r)}s,ω)=k02α({right arrow over (r)})P({right arrow over (r)},{right arrow over (r)}s,ω)+a(ω)δ({right arrow over (r)}−{right arrow over (r)}s)  (A-3)

Now consider Green's second identity from vector calculus:
∫Vdr[B∇2C−C∇2B]=∫SdS{right arrow over (n)}·[B∇C−C∇B](A-4)
where B and C are both twice differentiable scalar fields in a volume V bounded by a closed surface S with an outward pointing vector {right arrow over (n)}.FIG. 17Bshows a volume V in the space1700. The volume V lies within a closed surface S, which is the streamer surface Sr1702and a hemispherical cap SR1716of radius R=|{right arrow over (r)}′|. The volume V encloses the imaged free surface S0but does not include the scattering region1714.FIG. 17Bshows two examples of outward normal vectors {right arrow over (n)}1718and1720. An integral equation for the pressure wavefield, P, and the gradient of the pressure wavefield, ∇P, can be obtained by setting B=P and C=Gk0in Green's second identity (A-4) to give:

∫V⁢⁢ⅆr⇀′⁡[P⁡(r⇀′,ω)⁢∇′2⁢Gk0⁡(r⇀′,r⇀)-Gk0⁡(r⇀′,r⇀)⁢∇′2⁢P⁡(r⇀′,ω)]=∫S⁢⁢ⅆS⁢⁢n⇀·[P⁡(r⇀′,ω)⁢∇′⁢Gk0⁡(r⇀′,r⇀)-Gk0⁡(r⇀′,r⇀)⁢∇′⁢P⁡(r⇀′,ω)](A⁢-⁢5)
where {right arrow over (r)}′ is restricted to points within the volume V. The Green's function Gk0({right arrow over (r)}′,{right arrow over (r)}) characterizes reflections from the free surface and is a solution of the acoustic wave equation (21) for a unit impulse source within V and is given by:
(∇2+k02)Gk0({right arrow over (r)}′,{right arrow over (r)})=δ({right arrow over (r)}′−{right arrow over (r)})  (A-6)where k0=ω/c0; and{right arrow over (r)}′ represents a point in the space1700and {right arrow over (r)} the source location of the Green's function.
For example, the reflectivity R({right arrow over (r)}′, {right arrow over (r)}) given by Equation (17) can be used as the Green's function Gk0({right arrow over (r)}′, {right arrow over (r)}) in Equations (A-5) and (A-6) (i.e., Gk0({right arrow over (r)}′, {right arrow over (r)})=R({right arrow over (r)}′, {right arrow over (r)})). Substituting the Equations (A-3) and (A-6) into Equation (A-5) gives:

∫V⁢⁢ⅆr⇀′⁡[P⁡(r⇀′,r⇀s,ω)⁢δ⁡(r⇀′,r⇀)-Gk0⁡(r⇀′,r⇀)⁢a⁡(ω)⁢δ⁡(r⇀′-r⇀s)-Gk0⁡(r⇀′,r⇀)⁢k02⁢α⁡(r⇀′)⁢P⁡(r⇀′,r⇀s,ω)]=∫S⁢⁢ⅆS⁢⁢n⇀·[P⁡(r⇀′,r⇀s,ω)⁢∇′⁢Gk0⁡(r⇀′,r⇀)-Gk0⁡(r⇀′,r⇀)⁢∇′⁢P⁡(r⇀′,r⇀s,ω)](A⁢-⁢7)
By letting the radius R of the hemispherical cap SRgo to infinity, SRapproaches an infinite hemispherical shell, as shown inFIG. 17C. The surface integral over the surface S in Equation (A-7) reduces to an integral equation over the streamer surface Sr. In addition, the first term in Equation (A-7) goes to zero when the source of the reflectivity represented by the Green's function is outside the volume V. Because the scattering region α1714is located outside the volume V, the third term in Equation (A-7) also goes to zero leaving:

Equation (A-8) represents a functional relationship between the pressure wavefield P({right arrow over (r)}r, ω), which is obtained from measurements taken at pressure sensors along the streamer Sr, and the gradient of the pressure wavefield ∇rP({right arrow over (r)}r, ω) at the pressure sensors along the streamer Sr. This functional relationship between the pressure wavefield and the gradient of the pressure wavefield given by Equation (A-8) can be used to compute the gradient of the pressure wavefield ∇rP({right arrow over (r)}r, ω) when only the pressure wavefield P({right arrow over (r)}r, ω) is known.

Equation (A-8) can also be applied to address two cases in which the source1704lies above the level of the streamer1702or lies below the level of the streamer1702. When the source1704is located within the volume V at a depth between the free surface and the receiver surface as shown in the example representation ofFIG. 17D, according to the shifting property associated with the Dirac delta function the integral over the volume Vin the left-hand side of Equation (A-8) reduces to:
a(ω)Gk0({right arrow over (r)}s,{right arrow over (r)})=−∫SrdSr{right arrow over (n)}·[P({right arrow over (r)}r,ω)∇rGk0({right arrow over (r)}r,{right arrow over (r)})−Gk0({right arrow over (r)}r,{right arrow over (r)})∇rP({right arrow over (r)}r,ω)]  (A-9)
The terms of Equation (A-9) can be rearranged to give:

∫Sr⁢⁢ⅆSr⁢n⇀·Gk0⁡(r⇀r,r⇀)⁢∇r⁢P⁡(r⇀r,ω)=a⁡(ω)⁢Gk0⁡(r⇀s,r⇀)+∫Sr⁢⁢ⅆSr⁢n⇀·P⁡(r⇀r,ω)⁢∇r⁢Gk0⁡(r⇀r,r⇀)(A⁢-⁢10)
Equation (A-10) is a Fredholm integral equation of the first kind for the gradient of the pressure wavefield ∇rP({right arrow over (r)}r, ω) where the right-hand side of Equation (A-10) contains only known parameters such as the field pressure wavefield P({right arrow over (r)}r, ω) and the Green's function Gk0({right arrow over (r)},{right arrow over (r)}r) described above. On the other hand, when the source is located outside the volume Vat a depth below the streamer Sr, as shown inFIG. 17E, Equation (A-7) reduces to
∫SrdSr{right arrow over (n)}·Gk0({right arrow over (r)}r,{right arrow over (r)})∇rP({right arrow over (r)}r,ω)=∫SrdSr{right arrow over (n)}·P({right arrow over (r)}r,ω)∇rGk0({right arrow over (r)}r,{right arrow over (r)})  (A-11)
for the gradient of the pressure ∇rP({right arrow over (r)}r, ω). In Equation (A-11), the source function a(ω) is not used to determine ∇rP({right arrow over (r)}r, ω).

Although the present invention has been described in terms of particular embodiments, it is not intended that the invention be limited to these embodiments. Modifications within the spirit of the invention will be apparent to those skilled in the art. For example, any number of different computational-processing-method implementations that carry out efficient source and receiver deghosting may be designed and developed using various different programming languages and computer platforms and by varying different implementation parameters, including control structures, variables, data structures, modular organization, and other such parameters. The computational representations of wavefields, operators, and other computational objects may be implemented in different ways. Although the efficient wavefield extrapolation method discussed above can be carried out on a single two-dimensional sampling, source and receiver deghosting can be carried out using multiple two-dimensional samplings or three-dimensional sampling obtained experimentally for different depths, can be carried out from greater depths to shallower depths, and can be applied in many other contexts.