Patent Description:
<CIT> discloses a method of providing marine seismic data in which an interpolating filter is applied to multicomponent seismic measurements to obtain pressure data and/or vertical particle velocity component related data corresponding to source locations additional to the source locations at which a source was actuated.

This disclosure presents computational systems and methods for interpolating a pressure wavefield based on pressure wavefield and particle motion wavefield data recorded in a marine seismic survey as defined by the appended claims.

The pressure and particle motion wavefields are sampled at a sample rate that is less than a lower bound for sample rates typically used to interpolate the pressure wavefield from the pressure wavefield samples alone. In embodiments of the invention the particle motion wavefield is an acceleration wavefield.

<FIG> show side-elevation and top views, respectively, of a marine seismic data acquisition system composed of an exploration seismology survey vessel <NUM> towing a source <NUM> and six separate streamers <NUM>-<NUM> beneath a free surface <NUM> of a body of water. The body of water can be a region of an ocean, a sea, a lake, or a river. In this example, each streamer is attached at one end to the survey vessel <NUM> via a streamer-data-transmission cable. The streamers <NUM>-<NUM> form a planar horizontal receiver acquisition surface with respect to the free surface <NUM>. However, in practice, the receiver acquisition surface may be smoothly varying due to active sea currents and weather conditions. In other words, although the streamers <NUM>-<NUM> are illustrated in <FIG> and <FIG> as being straight, in practice, the towed streamers may undulate as a result of dynamic conditions of the body of water in which the streamers are submerged. It should be noted that a receiver acquisition surface is not limited to having a planar horizontal orientation with respect to the free surface <NUM>. The streamers may be towed at depths that orient the receiver acquisition surface at an angle with respect to the free surface <NUM> or so that one or more of the streamers are towed at different depths. It should also be noted that a receiver acquisition surface is not limited to six streamers as shown in <FIG>. In practice, the number of receivers used to form a receiver acquisition surface can range from as few as one streamer to as many as <NUM> or more streamers.

<FIG> includes an x<NUM>x<NUM>-plane <NUM> and <FIG> includes an x<NUM>x<NUM>-plane <NUM> of the same Cartesian coordinate system having three orthogonal, spatial coordinate axes labeled x<NUM>, x<NUM> and x<NUM>. The coordinate system is used to specify orientations and coordinate locations within a body of water. The x<NUM>-direction specifies the position of a point in a direction parallel to the length of the streamers and is referred to as the "in-line" direction. The x<NUM>-direction specifies the position of a point in a direction perpendicular to the x<NUM>-direction and substantially parallel to the free surface <NUM> and is referred to as the "cross-line" direction. The x<NUM>-direction specifies the position of a point perpendicular to the x<NUM>x<NUM>-plane (i.e., perpendicular to the free surface <NUM>) with the positive x<NUM>-direction pointing downward away from the free surface <NUM>. Streamer depth below the free surface <NUM> can be estimated at various locations along the streamers using depth measuring devices attached to the streamers. For example, the depth measuring devices can measure hydrostatic pressure or utilize acoustic distance measurements. The depth measuring devices can be integrated with depth controllers, such as paravanes or water kites, that control and maintain the depth and position of the streamers as the streamers are towed through a body of water. The depth measuring devices are typically placed at intervals (e.g., about <NUM> meter intervals) along each streamer. Note that in other embodiments buoys may be used to maintain the orientation and depth of the streamers below the free surface <NUM>.

In <FIG>, shaded rectangles <NUM> represent receivers or sensors that are spaced-apart along the length of each streamer. The streamers <NUM>-<NUM> are long cables containing power and data-transmission lines that connect the receivers <NUM> to seismic acquisition equipment located on board the survey vessel <NUM>. Each receiver is a dual sensor including a particle motion sensor that detects vertical displacement within the body of water over time by detecting particle motion accelerations, and a pressure sensor that detects variations in water pressure over time.

<FIG> shows a cross-sectional view of the survey vessel <NUM> towing the source <NUM> and streamers above a subterranean formation <NUM>. Curve <NUM> represents a surface of the subterranean formation <NUM>. The subterranean formation <NUM> is composed of a number of subterranean layers of sediment and rock. Curves <NUM>, <NUM>, and <NUM> represent interfaces between subterranean layers of different compositions. A shaded region <NUM>, bounded at the top by a curve <NUM> and at the bottom by a curve <NUM>, represents a hydrocarbon-rich subterranean deposit, the depth and positional coordinates of which may be determined by analysis of seismic data collected during a marine seismic survey. As the survey vessel <NUM> moves over the subterranean formation <NUM> the source <NUM> produces an acoustic impulse at spatial and temporal intervals. In other embodiments, the source <NUM> may be towed by a separate survey vessel or a number of sources may be towed by a number of different vessels. Source <NUM> may be an air gun, marine vibrator, or an array of air guns and/or marine vibrators. <FIG> illustrates an acoustic impulse expanding outward from the source <NUM> as a pressure wavefield <NUM> represented by semicircles of increasing radius centered at the source <NUM>. The wavefronts are, in effect, shown in vertical plane cross section in <FIG>. The outward and downward expanding portion of the pressure wavefield <NUM> is called the "primary wavefield," which eventually reaches the surface <NUM> of the subterranean formation <NUM>, at which point the primary wavefield is partially reflected from the surface <NUM> and partially transmitted downward into the subterranean formation <NUM>, becoming elastic waves within the subterranean formation <NUM>. In other words, in the body of water, the acoustic impulse is composed of compressional pressure waves, or P-waves, while in the subterranean formation <NUM>, the waves include both P-waves and transverse waves, or S-waves. Within the subterranean formation <NUM>, 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 a result, each point of the surface <NUM> and each point of the interfaces within the underlying subterranean formation <NUM> becomes a potential secondary point source from which acoustic and elastic wave energy, respectively, may emanate upward toward the receivers <NUM> in response to the acoustic impulse generated by the source <NUM> and downward-propagating elastic waves generated from the pressure impulse. As shown in <FIG>, secondary waves of significant amplitude are generally emitted from points on or close to the surface <NUM>, such as point <NUM>, and from points on or very close to interfaces in the subterranean formation <NUM>, such as points <NUM> and <NUM>. Tertiary waves called "receiver ghosts" are produced by secondary waves that are reflected from the free surface <NUM> back towards the streamers <NUM>-<NUM> and the subterranean formation <NUM>.

The secondary waves are generally emitted at different times within a range of times following the initial acoustic impulse. A point on the surface <NUM>, such as the point <NUM>, receives a pressure disturbance corresponding to the initial acoustic impulse more quickly than a point within the subterranean formation <NUM>, such as points <NUM> and <NUM>. Similarly, a point on the surface <NUM> directly beneath the source <NUM> receives the acoustic impulse sooner than a more distant-lying point on the surface <NUM>. Thus, the times at which secondary and higher-order waves are emitted from various points within the subterranean formation <NUM> are related to the distance, in three-dimensional space, of the points from the source <NUM>.

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 primary wavefield and secondary wavefield emitted in response to the primary wavefield are functions of distance from the source <NUM> as well as the materials and physical characteristics of the materials through which the primary wave travels. In addition, the secondary expanding wavefronts may be altered as the wavefronts cross interfaces and as the velocity of sound varies in the media are traversed by the wave. The superposition of waves emitted from within the subterranean formation <NUM> in response to the primary wavefield is a generally complicated wavefield that includes information about the shapes, sizes, and material characteristics of the subterranean formation <NUM>, including information about the shapes, sizes, and locations of the various reflecting features within the subterranean formation <NUM> of interest to exploration seismologists.

<FIG> shows a side-elevation view of the marine seismic data acquisition system with a magnified view <NUM> of receiver <NUM>. The magnified view <NUM> reveals that the receiver <NUM> is a multi-component sensor composed of a pressure sensor <NUM> and three particle motion sensors <NUM>. The pressure sensor may be a hydrophone. The pressure sensor <NUM> measures changes in pressure over time and records pressure wavefield data denoted by <MAT>, where the subscript r is a receiver index, <MAT> are the coordinates of the receiver, and t represents time. The motion sensors <NUM> are responsive to water motion in different directions. For example, as shown in magnified view <NUM>, three particle motion sensors <NUM> are collocated in the streamer <NUM> with the pressure sensor <NUM> to record particle motion wavefields in the x<NUM>-, x<NUM>-, and x<NUM>-directions. According to the invention the particle motion sensors are responsive to the acceleration of the particles in the water. In comparative examples, when the motion sensors <NUM> are responsive to position, the motion sensor data may be differentiated to obtain velocity wavefield data. When the motion sensors are responsive to acceleration (i.e., MEMS accelerometers), the particle acceleration data may be integrated to obtain velocity wavefield data. In the comparative example of <FIG>, the resulting wavefield data produced by the motion sensors <NUM> are orthogonal velocity wavefields denoted by <MAT>, and <MAT>, which correspond to particle motion velocity wavefields in the x<NUM>-, x<NUM>-, and x<NUM>-directions, respectively, at the receiver r and are collectively denoted by (Vx<NUM>, Vx<NUM>, Vx<NUM>)r. The streamers <NUM>-<NUM> and the survey vessel <NUM> may include sensing electronics and data-processing facilities that allow measurements from each receiver to be correlated with absolute positions on the free surface <NUM> and absolute three-dimensional positions with respect to an arbitrary three-dimensional coordinate system. The pressure data and particle motion data represent pressure and particle motion wavefields.

In <FIG>, directional arrow <NUM> represents the direction of an up-going wavefield detected by a receiver <NUM> and dashed arrows <NUM> represents a down-going wavefield produced by the up-going wavefield reflection from the free surface <NUM> before reaching the receiver <NUM>. In other words, the pressure wavefield <MAT> measured at a receiver is composed of an up-going pressure wavefield component and a down-going pressure wavefield component, and the x<NUM>-component or vertical velocity wavefield <MAT> measured at a particle motion sensor is composed of an up-going velocity wavefield component and a down-going velocity wavefield component. The down-going wavefield is called a receiver ghost that contaminates pressure and particle motion velocity data and creates notches in the spectral domain. Filtering is done to remove the down-going wavefields from the pressure and particle motion velocity data leaving the up-going wavefields which are used to generate images of the subterranean formation.

Each pressure sensor and particle motion sensor generates seismic data in the form of a time series that consist of a number of consecutive measured values called amplitudes separated in time by a sample period. The time series recorded by a pressure or motion sensor is called a "trace," which may consist of thousands of samples with a sample rate of about <NUM> to <NUM> samples/ms. A trace is a recording of a subterranean formation response to acoustic energy that passes from the source <NUM>, through subterranean layers, and is ultimately recorded by a sensor. A trace records variations in a time-dependent amplitude that represents acoustic energy in the portion of the secondary wavefield measured by the sensor. A secondary wavefield typically arrives first at the receivers located closest to the source <NUM>. The distance from the source <NUM> to a receiver is called the source-receiver offset, which creates a delay in the arrival time of a secondary wavefield from a substantially horizontal interface within the subterranean formation.

<FIG> shows ray paths <NUM>-<NUM> that represent paths of an acoustic impulse <NUM> output from the source <NUM> to the interface <NUM>. Rays <NUM>-<NUM> represent the paths of acoustic energy reflected from the interface <NUM> (i.e., secondary wavefields) to the receivers located along the streamer <NUM>. Each pressure sensor measures the pressure wavefield <MAT> and the motion sensors at each receiver measure particle motion velocity wavefields <MAT>, and <MAT> of the acoustic energy reflected from the interface <NUM>. The pressure wavefield <MAT> and velocity wavefields (Vx<NUM>, Vx<NUM>, Vx<NUM>)r measured at each receiver r are time sampled, as described in greater detail below, and recorded as four separate traces. A number of traces taken together either from the same streamer or across the streamers can be used to form a gather, which represents a portion of a measured wavefield. For example, the traces associated with the pressure <MAT> wavefield measured at the five pressure sensors located along the streamer <NUM> can be used to form a shot-receiver gather that represents an in-line pressure wavefield, and the traces associated with the velocity wavefield <MAT> measured at the five particle motion sensors located along the streamer <NUM> can be used to form a shot-receiver gather that represents an in-line vertical velocity wavefield. Other types of gathers can be formed from the pressure and velocities including common-receiver gathers and common-midpoint gathers.

<FIG> shows a plot of a source-receiver gather of example traces <NUM>-<NUM> of the acoustic energy reflected from the interface <NUM> and recorded by the five receives located along the streamer <NUM> shown in <FIG>. Vertical axis <NUM> represents time and horizontal axis <NUM> represents trace numbers with trace "<NUM>" representing the seismic data generated by the receiver located closest to the source <NUM> and trace "<NUM>" representing the seismic data generated by the receiver located farthest from the source <NUM>. The traces <NUM>-<NUM> can represent variation in the amplitude of the pressure wavefield <MAT> or variation in amplitude of any one of the velocity wavefield components <MAT>, and <MAT>. The example traces include wavelets or pulses <NUM>-<NUM> that represent the acoustic energy reflected from the interface <NUM>. Peaks, colored black, and troughs of each trace represent changes in the amplitude measured by the pressure sensors or motion sensors. The distances along the traces <NUM>-<NUM> from the trace number axis <NUM> to the wavelets <NUM>-<NUM> represents the travel time of the acoustic energy output from the source <NUM> to the interface <NUM> and ultimately to the receivers. The amplitude of the peak or trough of the wavelets <NUM>-<NUM> indicates the magnitude of acoustic energy recorded by the pressure sensor or motion sensor. Note that the arrival times versus source-receiver offset is longer with increased source-receiver offset and, in this example, has a hyperbolic shape <NUM>.

The traces from different source-receiver pairs may be corrected during seismic data processing to remove the effects of different source-receiver offsets in a process called "normal moveout" ("NMO"). <FIG> shows a gather of the traces <NUM>-<NUM> after NMO has been applied to align the pulses in time as represented by horizontal line <NUM>. After NMO corrections, traces from different shot records with a common reflection point can be stacked to form a single trace during seismic data processing. Stacking improves the signal-to-noise ratio, reduces noise, improves seismic data quality, and reduces the amount of data.

A typical trace does not represent a single reflection from a single interface, as represented in <FIG>. In practice, a trace represents the time-dependant amplitude of acoustic energy associated with numerous reflections of acoustic energy from within the subterranean formation. <FIG> shows a gather of <NUM> traces recorded over a period of time. Each trace, such as trace <NUM>, varies in amplitude over time and represents acoustic energy reflected from a number of different interfaces within a subterranean formation as measured by a pressure sensor or a motion sensor. The gather shown in <FIG> can represent a pressure wavefield or a velocity wavefield and can be a source-receiver gather, a common-receiver gather, or a common-midpoint gather.

<FIG> shows a time-domain plot of an example time-dependent continuous signal denoted by x(t). Horizontal axis <NUM> represents time denoted by t, vertical axis <NUM> represents the signal amplitude, and curve <NUM> represents the signal x(t). The signal x(t) can represent time-dependent variations in the pressure wavefield P(t), the particle motion wavefield, or time-dependent variations in the velocity wavefields Vx<NUM>(t), Vx<NUM>(t), and Vx<NUM>(t) measured at a receiver r, where in the following description the receiver coordinates xr are suppressed. The signal x(t) is typically sampled and recorded at regularly spaced times separated by time intervals called "sample periods" denoted by Ts. <FIG> includes a magnified view <NUM> of a segment of the signal x(t). Dashed line <NUM> represents the continuous signal x(t) and solid dots <NUM> represent the amplitude of the signal x(t) recorded at regularly spaced time intervals equal to the sample period Ts. The reciprocal or inverse of the sample period Ts is the sample rate or sample frequency denoted by fs (i.e., fs = <NUM>/Ts), which is the number of samples recorded per unit of time of the continuous signal x(t) to form a discrete signal denoted by x(nTs), where n is an integer time sample index. The notation x(nTs) can also be used to represent to the samples comprising the discrete signal. For example, magnified view <NUM> includes time samples x((n - <NUM>)Ts), x(nTs), and x((n + <NUM>)Ts) that are recorded at regularly spaced times (n - <NUM>)Ts, nTs, and (n + <NUM>)Ts separated by the same sample period Ts. Because the discrete signal x(nTs) can represent a time-sample pressure wavefield recorded by a pressure sensor or a velocity wavefield recorded by a motion sensor, the discrete signal x(nTs) is a trace as described above.

The signal x(t) is effectively band-limited because of various environmental, source and recording factors that limit the frequency content of the signal. The signal x(t) is called B-band limited when its associated energy is finite and its Fourier transform X(ω) vanishes outside the interval (- B, B): <MAT> <MAT>.

In other words, the signal x(t) and its Fourier transform X(ω) are square-integrable functions with an inverse Fourier transform: <MAT>.

The band limit of a trace is revealed when the trace is transformed to the spectral domain. For example, a Fourier transform ("FT") can be used to transform the trace x(nTs) to the spectral domain: <MAT> where.

In practice, a fast Fourier transform ("FFT") can be used for computational efficiency and speed. <FIG> shows a spectral domain plot of an example frequency spectrum resulting from transforming the trace x(nTs) represented in <FIG> to the spectral domain. The horizontal axis represents the frequency ω, vertical axis <NUM> represents the spectral amplitude, and jagged-curve <NUM> represents the B-band limited frequency spectrum X(ωk) of the trace x(nTs). Point B along the frequency axis represents a finite frequency beyond which the spectrum X(ωk) is essentially zero. Any signal that has a Fourier transform, or frequency spectrum, that is essentially zero for frequencies greater than a finite frequency B is called "band limited" and the frequency B is called the "band limit.

The signal x(t) can be interpolated or reconstructed from its samples x(nTs) when each sample is considered as specifying the scaling and location of a sine function. In other words, from sampling theory, interpolation can be used to interpolate the signal x(t) in terms of the time samples x(nTs) as follows: <MAT> where
Ts = π/B; and <MAT>.

Equation (<NUM>) is called "the sampling theorem. " In order to avoid temporal aliasing (i.e., losing signal information as a result of sampling), the signal x(t) must be band limited to less than half the sample rate fs. In other words, there can be no energy in the signal x(t) at frequencies greater than fN/<NUM>, which is the band limited frequency B (i.e., B = fN/<NUM>) and fN is the minimum sample rate called the "Nyquist rate. " Therefore, as long as the sample rate fs is greater than or equal to the Nyquist rate, fN = <NUM>B, of the signal x(t) can be interpolated from the samples x(nTs) using Equation (<NUM>). The Nyquist rate fN is a lower bound on the sample rate fs to avoid temporal aliasing.

A signal can also be interpolated at sample rates approaching half the Nyquist rate when the samples and the time derivatives of the samples are known using: <MAT> where
∂t represents the time derivative; and <MAT>.

In other words, given the samples x(nT) and their associated time derivatives ∂tx(nT), the signal x(t) can be interpolated using Equation (<NUM>) with a sample rate greater than or equal to half the Nyquist rate (i.e., fN/<NUM>).

<FIG> shows a horizontal axis <NUM> that represents frequencies greater than zero, and <FIG> shows a horizontal axis <NUM> that represents time greater than zero. Dashed directional arrow <NUM> represents sample rates that are greater than or equal to the Nyquist rate, fN, and dashed directional arrow <NUM> represents corresponding sample periods between zero and a Nyquist period denoted by TN = <NUM>/fN. The signal x(t) can be interpolated using Equation (<NUM>) with just the samples x(nTs) produced at a sample rate fs greater than the Nyquist rate and sample periods Ts that lie between zero and the Nyquist period TN. Dot-dashed directional arrow <NUM> represents sample rates fR that are greater than or equal to half the Nyquist rate, fN/<NUM>, and dot-dashed directional arrow <NUM> represents corresponding sample periods T between zero and twice the Nyquist period TN. In this case, the signal x(t) can be interpolated using Equation (<NUM>) with samples x(nT) and time derivatives of the samples ∂tx(nT) at a sample rate fR greater than fN/<NUM> (i.e., fR > fN/<NUM>). In other words, Equation (<NUM>) allows for reconstruction of the signal x(t) with samples x(nT) and time derivatives of the samples ∂tx(nT) recorded at a sample rate fR between fN/<NUM> and fN (i.e., fN > fR > fN/<NUM>) and sample period T between TN and <NUM>TN (i.e., TN < T < <NUM>TN).

<FIG> illustrate how Equation (<NUM>) can be used to reduce the number of samples to less than half the number of samples recorded with a typical sample rate used to record seismic data provided the sample data and the time derivative of the sample data are known. For example, suppose the Nyquist rate for recording a pressure wavefield P(t) is about <NUM> samples/ms. In order to avoid temporal aliasing in the trace P(nTs), a typical sample rate of about <NUM> samples/ms can be used and the pressure wavefield P(t) can be interpolated from just the pressure samples P(nTs) using Equation (<NUM>). By contrast, according to Equation (<NUM>), the sample rate can be reduced to a sample rate greater than half the Nyquist rate of <NUM>, and the pressure signal P(t) can be interpolated provided the pressure samples P(nT) and time derivatives of the pressure samples ∂tP(nT) are known for each time sample. In other words, Equation (<NUM>) enables interpolation of a time-dependent pressure wavefield with about half the number of samples typically used to record pressure data, provided pressure samples and corresponding time derivatives of the pressure samples are recorded for each time sample.

The time derivatives of the pressure wavefield can be calculated from the velocity wavefields. Consider the basic wave equations, which comprise the equation of motion and the equation of deformation, respectively, given by: <MAT> <MAT> where.

When the volume of water is without sources, q = <NUM> and fj = <NUM> in Equations (7a) and (7b), and the time derivative of the pressure wavefield P(t) can be expressed in terms of the divergence of the velocity wavefield using Equation (7b) as follows: <MAT>.

From Equation (7b), the time derivative of the pressure wavefield can also be expressed in terms of the velocity wavefield as follows: <MAT>.

The measurements of P(t) and Vj(t) are recorded by sensors in a streamer with known properties that can be considered as a locally homogenous time-invariant medium, or is dragged through a water layer with similar properties that are known, or can be measured independently. The time derivative ∂j/∂t in Equation (<NUM>) is proportional to the velocity of sound in water c, which is given by c = (ρκ)-<NUM>/<NUM>. As a result, from Equation (<NUM>), the time derivative of the pressure in a homogeneous medium reduces to: <MAT>.

In other words, when the multi-component velocity samples Vx<NUM>(nT), Vx<NUM>(nT), and Vx<NUM>(nT) and the pressure samples P(nT) are known are known for a receiver r, an expression for interpolating the pressure wavefield at the receiver r can be obtained by substituting Equation (<NUM>) into Equation (<NUM>) to give: <MAT> where.

Typically, the seismic reflection data of interest are the up-going or vertically travelling wavefields (i.e., x<NUM>-direction). As a result, the time derivative of the pressure wavefield in Equation (<NUM>) can be approximated by: <MAT> where nh represents noise that depends on the horizontal particle velocity components Vx<NUM>(t) and Vx<NUM>(t).

When the noise nh is small (i.e., nh ≪ <NUM>), the pressure wavefield P(t) can be interpolated by substituting Equation (<NUM>) into Equation (<NUM>) to obtain: <MAT> where.

As shown above, Equation (<NUM>) depends on time derivatives of the velocities V<NUM>(nT), V<NUM>(nT), and V<NUM>(nT) and Equation (<NUM>) depends on time derivatives of the velocity V<NUM>(nT), which are particle motion accelerations. For example, the time derivative of the vertical velocity, ∂tV<NUM>(nT), is also the particle motion acceleration given by a<NUM>(nT) = ∂tV<NUM>(nT). The particle motion acceleration can be determined in a number ways. For example, the particle motion sensors deployed at the receivers are MEMS accelerometers that output sampled acceleration a<NUM>(nT), which is time sampled at substantially the same sample rate fR as the pressure wavefield. Alternatively, the acceleration can be calculated from the slope of the recorded vertical velocity samples surrounding the sample V<NUM>(nT). <FIG> shows a plot of three consecutive example vertical velocity samples. Horizontal axis <NUM> represents time, vertical axis <NUM> represents velocity, and shaded dots <NUM>-<NUM> represent vertical velocity samples V<NUM>((n - <NUM>)T), V<NUM>(nT), and V<NUM>((n + <NUM>)T), respectively. The acceleration a<NUM>(nT) at time sample nT is the slope of a tangent line <NUM> at the point <NUM>. The acceleration a<NUM>(nT) can be approximated from the neighboring vertical velocity samples V<NUM>((n - <NUM>)T), V<NUM>(nT), and V<NUM>((n + <NUM>)T) as follows: <MAT> <MAT> <MAT>.

The accelerations in Equations (14a)-(14c) are represented by slopes of dotted lines <NUM>-<NUM>, respectively. Similar equations can be used to calculate approximate accelerations a<NUM>(nT) and a<NUM>(nT) in the in-line and cross-line directions in Equation (<NUM>).

<FIG> shows a flow diagram of a method for interpolating a pressure wavefield in the time domain. In block <NUM>, pressure wavefield samples P(nT) sampled with a sample rate fR greater than or equal to half the Nyquist rate fN are received. The samples P(nT) are pressure sensors located at a receiver r along a streamer towed by a survey vessel as described above. In block <NUM>, one or more particle motion wavefields that are sampled at the sample rate fR are received. The particle motion wavefields are recorded by particle motion sensors collocated with the pressure sensor. When the motion sensors are responsive to position, the particle motion wavefields may be differentiated to convert the wavefield data into velocity wavefields V<NUM>(nT), V<NUM>(nT), and V<NUM>(nT), and when the motion sensors are responsive to acceleration (i.e., MEMS accelerometers), the particle motion wavefields are acceleration wavefields ax<NUM>(nT), ax<NUM>(nT), and ax<NUM>(nT). In block <NUM>, when the particle motion wavefields are acceleration wavefields, control flow to block <NUM>, otherwise, control flows to block <NUM>. In block <NUM>, acceleration wavefields are calculated from the velocity wavefields as described above with reference to Equations (14a)-(14c). In block <NUM>, a time derivative for the pressure sample ∂tP(nT) is calculated according to Equation (<NUM>) or Equation (<NUM>). In block <NUM>, the pressure wavefield is interpolated from the pressure samples P(nT) and the approximate time derivatives for the pressure sample ∂tP(nT) using Equation (<NUM>) or Equation (<NUM>).

<FIG> shows an example of a generalized computer system that executes efficient methods for interpolating the pressure wavefield in the time domain and therefore represents a geophysical-analysis data-processing system. 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") <NUM>-<NUM>, one or more electronic memories <NUM> interconnected with the CPUs by a CPU/memory-subsystem bus <NUM> or multiple busses, a first bridge <NUM> that interconnects the CPU/memory-subsystem bus <NUM> with additional busses <NUM> and <NUM>, or other types of high-speed interconnection media, including multiple, high-speed serial interconnects. The busses or serial interconnections, in turn, connect the CPUs and memory with specialized processors, such as a graphics processor <NUM>, and with one or more additional bridges <NUM>, which are interconnected with high-speed serial links or with multiple controllers <NUM>-<NUM>, such as controller <NUM>, that provide access to various different types of computer-readable media, such as computer-readable medium <NUM>, 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 medium <NUM> is 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 medium <NUM> can be used to store machine-readable instructions that encode the 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.

Embodiments described above are not intended to be limited to the descriptions above. For example, any number of different computational-processing-method implementations that carry out the interpolation methods described above 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. The systems and methods for interpolating seismic data can be executed in near-real time while conducting a marine survey of a subterranean formation. The term "near-real time" refers to a time delay due to data transmission and data processing that is short enough to allow timely use of the processed data during further data acquisition. For example, near-real time can refer to a situation in which the time delay due to transmission and processing is insignificant. In other words, near-real time approximates real time when the time for data transmission and data processing appears imperceptible. Near-real time can also refer to a perceptible time delay for data transmission and data processing but the time delay is not so long that quality control cannot be executed.

Claim 1:
A method for interpolating seismic data obtained from a marine seismic survey using a programmable computer programmed to at least perform the operations of:
receiving (<NUM>) pressure wavefield samples of a pressure wavefield measured with a pressure sensor (<NUM>) at a sample rate;
characterized by receiving (<NUM>) acceleration wavefield samples of an acceleration wavefield measured with a MEMS accelerometer at the sample rate, the MEMS accelerometer collocated with the pressure sensor in a streamer; and
interpolating (<NUM>) a pressure wavefield in the time domain based on the pressure wavefield samples and the acceleration wavefield samples, wherein the sample rate is less than a Nyquist rate of the pressure wavefield and is greater than half the Nyquist rate.