WO2014150897A2 - Methods and systems for deghosting variable depth seismic data - Google Patents

Abstract

Methods, apparatuses, and systems are disclosed for deghosting variable depth seismic data. In one example of such a method, a deghosting operator is applied to a plurality of plane wave components of seismic data for a plurality of defined reference depths to obtain a plurality of sets of deghosted plane wave components. Respective sets of deghosted plane wave components are transformed to respective sets of deghosted traces, and a first deghosted output trace corresponding to one of the seismic receivers at a first actual depth is generated from the sets of deghosted traces.

Description

METHODS AND SYSTEMS FOR DEGHOSTING

VARIABLE DEPTH SEISMIC DATA

CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to and the benefit of United States provisional application number 61/798,577 entitled "Deghosting Variable Depth Seismic Data," which was filed on March 15, 2013, and which is hereby incorporated by reference in its entirety for all purposes.

TECHNICAL FIELD

[0002] This disclosure relates generally to geophysical exploration systems, and more particularly to methods of deghosting variable depth seismic data obtained in geophysical surveys.

BACKGROUND

[0003] Petrochemical products such as oil and gas are ubiquitous in society and can be found in everything from gasoline to children's toys. Because of this, the demand for oil and gas remains high. In order to meet this high demand, it is important to locate oil and gas reserves in the Earth. Scientists and engineers conduct "surveys" utilizing, among other things, seismic and other wave exploration techniques to find oil and gas reservoirs within the Earth. These seismic exploration techniques often include controlling the emission of seismic energy into the Earth with a seismic source of energy (e.g., dynamite, air guns, vibrators, etc.), and monitoring the Earth's response to the seismic source with one or more receivers in order to create an image of the subsurface of the Earth. By observing the reflected seismic wave detected by the receiver(s) during the survey, the geophysical data pertaining to reflected signals may be acquired and these signals may be used to form an image of the Earth near the survey location.

[0004] In marine-based acquisitions, the receiver(s) may measure the seismic wave after it is reflected from the sub-surface of the earth. The reflection from the sub-surface may, however continue upwards to the surface of the water, where it may again be reflected by the boundary between the water and the air above the water. Because the water-air boundary is a near perfect reflector, the seismic wave is reflected from the water-air boundary and propagates back towards the sub-surface. The downwardly reflecting seismic wave is detected by the receivers and is commonly known as a receiver-side "ghost." In some cases, the ghost may again reflect off of the sub-surface, and again reflect off of the water-air boundary, thus creating multiple reflections. Also, a source-side ghost may be present, which is similar to the receiver-side ghost except the source-side ghost is the seismic signal that propagates from the source upwards to the water-air boundary, reflects off of the water-air boundary, and then reflects off of the sub-surface back up to the receivers. [0005] Both receiver and source ghosts limit the amount of energy in the seismic wavelet at frequencies determined by the streamer depth. Also, the phase of the seismic data is distorted around the ghost notch frequency. Marine towed streamer surveys have historically been designed with sources and receivers towed at a single, relatively shallow depth in order to effectively capture the higher frequencies desired for the targets. Towing sources and streamers at a relatively shallow depth, however, can lead to distortion of low frequencies because of the increased susceptibility to noise generated by waves at the sea surface. Furthermore, when the receivers on a streamer are towed at a single depth, the ghost interferes with the same frequencies of the seismic signal for all of the receivers, thereby limiting the ability of subsequent processing to remove the effects of the ghost by stacking traces from different receivers together. SUMMARY

[0006] As described in more detail below, some of the disclosed embodiments provide methods and systems for deghosting seismic data, such as seismic data acquired using variable depth streamers. In one example method of deghosting seismic data, a plurality of reference depths are defined that span a range of actual depths over which a plurality of seismic receivers acquire seismic data. The seismic data is transformed into a plurality of plane wave components, and a deghosting operator is applied to each of the plurality of plane wave components for each of the plurality of reference depths to obtain a plurality of sets of reference deghosted plane wave components, one set of reference deghosted plane wave components for each of the plurality of reference depths. Respective sets of reference deghosted plane wave components are transformed to respective sets of reference deghosted traces, and a first deghosted output trace corresponding to one of the seismic receivers at a first actual depth is generated by interpolating from the sets of reference deghosted traces.

[0007] In some embodiments, each set of reference deghosted traces may correspond to one of the plurality of reference depths and may include a respective reference trace corresponding to each respective one of the plurality of seismic receivers. The first actual depth may correspond to a depth at which the first seismic receiver acquired a portion of the seismic data. The seismic data may be decomposed into the plurality of plane wave components using a 3D Fourier transform over time and two spatial dimensions. The seismic data may be decomposed into the plurality of plane wave components using a tau-p transform followed by a temporal Fourier transform. The plurality of reference depths may be defined such that a maximum phase difference of deghosting operators for adjacent reference depths is less than a predetermined value. [0008] In some embodiments, the first seismic receiver may be associated with a first offset in a first shot record, and the method may further include generating a second deghosted output trace corresponding to a second seismic receiver associated with a second offset in a second shot record. The second deghosted output trace may be generated for a second actual depth corresponding to a depth at which the second seismic receiver acquired a portion of the seismic data, and the method may further include stacking the first and second deghosted output traces to form a stacked trace. The deghosting operator applied to the plurality of plane wave components for the plurality of reference depths may be a phase-only deghosting operator, and the method may further include applying an amplitude correction to the stacked trace. In some embodiments, the method may further include storing a list of amplitudes associated with each of the phase-only deghosting operators that would provide a full deghosted spectrum in the first and/or second deghosted output traces if applied contemporaneously with respective phase-only deghosting operators, and deterministically applying the amplitude correction to the stacked trace using the stored list of amplitudes as a function of time and frequency. The amplitude correction may include a statistical spectral whitening that fills in spectral notches of the stacked trace. The deghosting operator applied to the plurality of plane wave components may be a phase-only deghosting operator, and the method may further include migrating the first and second deghosted traces to form a seismic image, and statistically applying an amplitude correction to the migrated seismic image to fill in spectral notches of the seismic image.

[0009] In some embodiments, the deghosting operator may be substantially of the form

^ i ¹

Cr = __2fe , where r is a reflectivity of a water-air boundary, z is a respective reference

1 + r e

depth, and k_z is a vertical wavenumber. The vertical wave number k_z may be estimated using a far field dispersion approximation, such that the deghosting operator is substantially of the form

G ¹ (k_r , & co) = w v y w - u y, z is a respective reference depth, k_x is a horizontal wave number in a first lateral dimension, k_y is a horizontal wave number in a second lateral dimension, v is a velocity of an acoustic medium, and co is a frequency, or is substantially of the form , where r is

a reflectivity of a water-air boundary, z is a respective reference depth, p_x is a ray parameter in a first dimension, p_y is a ray parameter in a second dimension, v is a velocity of an acoustic medium, and ω is a frequency. [0010] In some embodiments, the method may further include interpolating the seismic data to a dense receiver x-y grid prior to decomposing the seismic data into the plurality of plane wave components, and storing residual information of the seismic data not represented by the interpolated traces for later processing. The method may further include estimating one or more of a reflectivity of an water-air boundary proximate the seismic receivers or a velocity of water proximate the seismic receivers by minimizing noise in the first deghosted output trace after applying the deghosting operator using the estimated parameters, where the one or more of the reflectivity of the water-air boundary, the velocity of water, or the first actual depth is estimated for a subset of the seismic data in either time or space.

[0011] In some embodiments, the method may further include applying receiver motion correction prior to decomposing the seismic data into the plurality of plane wave components, where the receiver motion correction includes applying one or both of a differential normal move out and/or a coordinate transform process. Each seismic receiver may include a hydrophone and/or a particle motion sensor, and the plurality of seismic receivers may be positioned on a plurality of variable depth streamers towed behind a vessel together with one or more seismic sources. Alternatively, the plurality of seismic receivers may positioned on a single streamer towed behind a vessel, or may be positioned on a non-towed, variable depth cable in a water column.

[0012] In some embodiments, interpolating from the sets of reference deghosted traces may include performing a piecewise linear interpolation. The interpolating from the sets of reference deghosted traces may include selecting one of the sets of reference deghosted traces whose corresponding reference depth is closest to the first actual depth of the first seismic receiver. The method may further include iterating over the plurality of seismic receivers and generating a respective deghosted output trace for each of the plurality of seismic receivers from the sets of reference deghosted traces. [0013] In some embodiments, the respective sets of reference deghosted plane wave components may be spatially transformed to the respective sets of reference deghosted traces, the respective sets of reference deghosted traces still decomposed into pluralities of temporal frequency components when generating the first deghosted output trace. In these embodiments, the method may further include transforming the first deghosted output trace to a time domain to obtain a time-series of pressure measurements in the time-space domain. In some embodiments, the respective sets of reference deghosted plane wave components may be spatially and temporally transformed to the respective sets of reference deghosted traces, the respective sets of deghosted traces including time-series of pressure measurements in the time-space domain. Also, in some embodiments, the seismic data transformed into a plurality of plane wave components may be from a single shot record.

[0014] Another example of a method of deghosting seismic data includes the acts of applying a deghosting operator to a plurality of plane wave components of seismic data for a plurality of defined reference depths to obtain a plurality of sets of reference deghosted plane wave components, transforming respective sets of reference deghosted plane wave components to respective sets of reference deghosted traces, and generating a first deghosted output trace corresponding to one of the seismic receivers at a first actual depth from the sets of reference deghosted traces.

[0015] Another example of a method of deghosting seismic data includes the acts of defining a plurality of reference depths spanning a range of actual depths over which a plurality of seismic receivers acquire respective seismic input traces, determining a plurality of deghosting operators for the plurality of reference depths, transforming a plurality of the seismic input traces into a set of temporal frequency domain traces, and applying a first of the plurality of deghosting operators to the set of temporal frequency domain traces to generate a first deghosted output trace. [0016] In some embodiments, the plurality of seismic input traces transformed into the set of temporal frequency domain traces may include seismic input traces within an aperture of the first of the plurality of deghosting operators. The first deghosted output trace may be in a temporal frequency domain, and the method may further include the act of transforming the first deghosted output trace to the time-space domain. The seismic input traces may each include a plurality of pressure measurements in the time-space domain.

[0017] In some embodiments, the first seismic input trace may have been acquired by a first of the plurality of seismic receivers at a first actual depth. The first of the plurality of deghosting operators may thus be selected from the plurality of deghosting operators by selecting a reference depth associated with the first of the plurality of deghosting operators closest to the first actual depth, or the first of the plurality of deghosting operators may be selected from the plurality of deghosting operators by interpolating between deghosting operators associated with two or more of the plurality of reference depths close to the first actual depth. [0018] In some embodiments, each of the plurality of deghosting operators may include a plurality of filters for a plurality of temporal frequency components of the temporal frequency domain traces, and each filter may be applied to the temporal frequency components of the temporal frequency domain traces within the aperture of the respective filter. Each of the plurality of filters may be defined by a respective set of coefficients, and/or a plurality of respective sets of coefficients corresponding to respective ones of the plurality of filters may be stored in a table. The plurality of deghosting operators may be explicit finite difference filters and the first deghosted output trace may be generated by calculating weighted sums of respective temporal frequency components of the set of temporal frequency domain traces. The weighted sum for a first temporal frequency component may be calculated by multiplying coefficients from the filter corresponding to the first temporal frequency components with the temporal frequency components of the temporal frequency domain traces within an aperture of the respective filter, and summing the products thereof. The plurality of deghosting operators may be implicit finite difference filters.

[0019] In some embodiments, the plurality of deghosting operators may be determined independently from the seismic data, and are a function of velocity, reflectivity, and depth. Each of the plurality of deghosting operators may correspond with one reference depth and may include a plurality of filters, each filter corresponding to a respective temporal frequency component of the set of temporal frequency domain traces. In some embodiments the operators may be localized differential operators. Also, the operators may be applied to the data in a spatial domain, and/or the operators may be applied to the data in a temporal frequency domain. Also, in some embodiments, each of the plurality of deghosting operators may include a filter whose Fourier transform a roximates the Fourier transform of the wavenumber domain deghosting operator G wherein r is a reflectivity of a water-air boundary, z is a

depth, Δχ is a sample interval in a lateral dimension, v is a velocity of an acoustic medium, ω is a frequency, and k is a normalized wavenumber. [0020] Another example method of deghosting seismic data includes the acts of defining a plurality of reference depths spanning a range of actual depths over which a plurality of seismic receivers acquire respective seismic input traces, defining an intermediate depth in the range of actual depths, determining a plurality of spatial deghosting operators for the plurality of reference depths, transforming the seismic input traces into a plurality of plane wave components, applying a spectral deghosting operator to each of the plurality of plane wave components for the intermediate depth to obtain a set of reference deghosted plane wave components, transforming the set of reference deghosted plane wave components to a plurality of intermediate traces, each intermediate trace decomposed into a plurality of temporal frequency components, and applying a first of the plurality of spatial deghosting operators to the temporal frequency components of the intermediate traces to generate a first deghosted set of temporal frequency components.

[0021] In some embodiments, the method may further include transforming the first deghosted set of temporal frequency components to generate a first deghosted output trace. The plurality of spatial deghosting operators may be residual deghosting operators. Also, the intermediate depth may be a first intermediate depth, the spectral deghosting operator may be a first spectral deghosting operator, and the set of reference deghosted plane wave components may be a first set of reference deghosted plane wave components, and the method may further include defining a second intermediate depth and applying a second spectral deghosting operator to the plurality of plane wave components for the second intermediate depth to obtain a second set of reference deghosted plane wave components. In some embodiments, the intermediate depth may be an average depth over which the plurality of seismic receivers acquired the respective seismic input traces.

BRIEF DESCRIPTION OF THE DRAWINGS

[0022] FIG. 1 is a block diagram of a seismic surveying system. [0023] FIG. 2A and 2B illustrate side views of exemplary embodiments of a marine towed seismic streamer.

[0024] FIGS. 3A and 3B illustrate side views of embodiments of a non-towed marine seismic cable.

[0025] FIG. 4 is a flow chart that illustrates one embodiment of a method of deghosting variable depth seismic data. [0026] FIG. 5 is a flow chart that illustrates another embodiment of a method of deghosting variable depth seismic data.

[0027] FIG. 6 is a flow chart that illustrates another embodiment of a method of deghosting variable depth seismic data. [0028] FIG. 7 is a flow chart that illustrates another embodiment of a method of deghosting variable depth seismic data.

[0029] FIG. 8 illustrates an embodiment of a computer system used in a seismic surveying system that is capable of storing and/or processing seismic data, such as the operations illustrated in FIGS. 2 through 7. DETAILED DESCRIPTION

[0030] FIG. 1 illustrates one embodiment of a seismic surveying system 100. The seismic surveying system 100 includes one or more seismic sources 102, one or more seismic receivers 103, a data storage 106, and a data processing apparatus 108. The seismic surveying system 100 may be adapted for acquiring seismic data in any of a number of different geological settings. For example, the seismic surveying system 100 may be adapted for seismic acquisition in a marine-based setting in some embodiments.

[0031] The seismic source(s) 102 may be anything that emits seismic energy. For example the sources 102 may include one or more air guns (e.g., for use in a marine towed-streamer acquisition), one or more vibrators, and so forth. In some examples, the seismic sources 102 may be naturally occurring, such as a geological disturbance, background seismic noise, or seismic activity induced by hydraulic fracturing. As illustrated in FIG. 1, in some examples, the seismic sources may provide seismic source data to a data storage device 106. The seismic source data may include, for example, amplitudes, times, positions, and so forth of seismic source activity that can later be correlated with the received seismic traces from the receivers 103. [0032] Seismic energy emitted by the seismic sources may be detected by one or more seismic receiver(s) 103. Each seismic receiver includes one or more sensors that detect a disturbance of a medium at one or more points in time. For example, a seismic receiver 103 may include a pressure sensor such as a hydrophone in some embodiments. A hydrophone detects amplitudes of a pressure wavefield over time. Another example of a seismic receiver 103 may include a particle motion sensor, which detects the motion of particles over time, which, in turn, can be related to the rate of change of a pressure wavefield over time. The particle motion sensor may detect particle motion in one, two, or three directional components. The particle motion sensor may be, for example, a geophone or a triaxial accelerometer. A motion sensor detects the motion of particles or of an elastic medium over time. A motion sensor may detect velocity, acceleration, or displacement, or some combination of these, and may do so in one, two, or three directional components. A seismic receiver may alternatively or additionally include other types of sensors.

[0033] The seismic receivers 103 may be positioned proximate the seismic sources 102 during a seismic survey. During the seismic survey, one or more seismic sources 102 may be fired, and the one or more seismic receivers 103 may measure one or more disturbances and may generate one or more traces with time sequences of measurements over a period of time. In general, each component of each sensor may generate a trace. Each trace may include or may be associated with corresponding positional information, which may be provided by a navigation system (not shown in FIG. 1). The time sequences in the traces may in some examples be transformed into frequency domain traces, where the trace is a set of frequency samples. In general, a trace may be a set of data associated with a certain location (e.g., a certain x, y location), whether that data be temporal samples, frequency samples, etc.

[0034] The seismic traces generated by the seismic receivers 103 may be provided to the data storage 106 in some embodiments. The data storage 106 may be a local data storage 106 near the seismic receiver 103 and may record seismic traces from a single receiver 103 in some examples, or may be a bulk data storage 106 located at a central station and may record seismic traces from a plurality of different receivers 103 in other examples. The data storage 106 may include one or more tangible mediums for storing the seismic traces, such as hard drives, magnetic tapes, solid state storage, volatile and non-volatile memory, and so forth. In some examples, the seismic traces from the seismic receivers 103 may bypass the data storage 106 and be provided directly to the data processing apparatus 108 in order to at least partially process the seismic traces in real- time or substantially real-time (e.g., to provide quality control information).

[0035] The data processing apparatus 108 may be any computing apparatus that is adapted to process and manipulate the seismic traces from the seismic receivers 103, and, in some embodiments, the seismic source data from the seismic sources 102. The data processing apparatus 108 may be a single computing device, or may be distributed among many computing nodes in some examples. In some examples, different computing apparatuses perform different data processing operations. For example, a first may deghost seismic traces, and another may migrate seismic traces to obtain an image of the earth's subsurface. An image of interest may be a spatial indication of discontinuities in acoustic impedance or the elastic reflectivity of the subsurface, and may be displayed on a tangible medium, such as a computer monitor or printed on a piece of paper. While some embodiments of the data processing apparatus 108 may process the seismic traces until a migrated image is obtained, in other examples, the data processing apparatus 108 may only partially process the seismic traces - for example, the data processing apparatus may merely deghost the seismic traces, and provide the processed and deghosted seismic traces to another process flow for further processing.

[0036] FIGS. 2A and 2B illustrate side views of embodiments 200A, 200B of the seismic surveying system 100 illustrated in FIG. 1. In FIG. 2A a vessel 201 is shown towing a source 202 and several receivers 203 on one or more shaped streamers 210 behind the vessel 201. For the sake of discussion, the embodiment depicted in FIG. 2A illustrates the source 202 and receiver 203 being towed by the same vessel 201, and a streamer 210 or streamers being shaped - e.g., having the receivers 203 spread over a plurality of depths. As will be appreciated, other combinations are possible. For example, in some embodiments, the source 202 and receivers 203 may be towed by separate vessels 201. In other embodiments, one of the source 202 or the receivers 203 may be stationary while the other is towed behind the vessel 201.

[0037] The streamer 210 illustrated in FIG. 2A includes a plurality of segments 21 OA, 210B, each of which includes a plurality of receivers 203 positioned therealong. The first segment 21 OA may slant downwards from the vessel 201 at, for example, .1 to 2 degrees from the horizontal. The second segment 210B may slant upwards towards the sea surface 211 at a similar angle, which may or may not be the same angle at which the first segment 21 OA slants downwards. Because the end of each of the segments 210A, 210B that is attached to control devices is towed at a relatively shallow depth (i.e., the end of the first segment 210A that is attached to the boat, and the end of the second segment 210B that is attached to a tail buoy), it may reduce the amount of lead-in or other positioning cable needed, and may reduce the tension on the streamer 210. The ends of the segments 21 OA, 210B may be towed, for example, at a depth of 12-15 meters. However, in addition to having a portion towed at a relatively shallow depth, other portions of the streamer 210 are towed at relatively deeper depths.

[0038] A plurality of receivers 203 (e.g., pressure sensors such as hydrophones and/or particle motion sensors such as geophones or accelerometers) are positioned along the streamer segments 21 OA, 210B at different depths relative to each other, which may allow the receivers 203 to acquire seismic data for a plurality of different, variable depths. The plurality of depths at which the receivers 203 on the streamer 210 acquire data provide a wide diversity of ghosts, and this diversity can be used, as described below, to more effectively remove the effects of the ghost from the recorded seismic data.

[0039] The segments 21 OA, 21 OB may maintain their respective shapes in some embodiments by one or more streamer positioning devices, such as DigiFIN or DigiBIRD devices from ION Geophysical Corporation. Also, a smoothing interconnect may be used to smooth the transition from the downward slope of the first segment 21 OA to the upward slope of the second segment 21 OB. In some examples, a plurality of streamers 210 may be towed by the vessel 201, whereas in other embodiments a single streamer 210 may be used.

[0040] During operation, the source 202 may emit or "fire" seismic energy (e.g., through an air gun), which may reflect off various portions of the Earth and may be received back at the receivers 203. The signal received at the receivers 203 may be a disturbance of the water caused by a seismic wave that reflected off of an oil and/or gas reservoir 205. In FIG. 2A, the initial propagation of the seismic energy from source 202 is indicated by the line 251, with the energy reflected from the subsurface indicated by line 252, and the ghost reflected from the interface of the air and sea surface 211 indicated by line 253.

[0041] As also illustrated in FIG. 2A, one or more of the receivers 203 may detect both primary and ghost reflections. For example, in addition to the energy that initially propagates along line 251, seismic energy may also propagate along line 254 in FIG. 2A from the source 202, and this energy may reflect off of the subsurface reservoir 205 and propagate along line 255 as a primary reflection towards the streamer 210. The primary reflection 255 also reflects off of the sea surface 211, and propagates back towards the streamer 210 as a ghost 256. Both the ghost reflection 256 as well as the primary reflection 252 may be incident on a single receiver 203, and the ghost reflection 256 and the primary reflection 252 may be overlapping in some cases, which can cause the reduction in seismic energy or notches as described above. In general, ghost reflections 253, 256 may be detected at similar or different receivers 203 as the primary reflections 252, 255. In practice, of course, entire wavefields propagate through the medium, but for simplicity, the wave propagation is shown using two sets of single lines 251, 252, 253 and 254, 255, 256 in FIG. 2A.

[0042] As mentioned above, each receiver 203 generates one or more traces. Each trace may include one or more samples of the disturbance of the water caused by a passing pressure wavefield. In some embodiments, the traces generated by the respective receivers 203 may be transmitted to a storage medium on the vessel 201. The received and stored signals may, in some embodiments, be partially or fully processed by computers or servers on-board the vessel 201 in real-time, near real-time, or in some cases may not be processed at all on-board but simply recorded for processing at a later time.

[0043] Referring now to FIG. 2B, another embodiment 200A is shown, with a shaped streamer 210. The shaped streamer 210 in FIG. 2B is similar to the shaped streamer 210 illustrated in FIG. 2A, except that the shaped streamer 210 in FIG. 2B includes four segments 21 OA, 210B, 2 IOC, 210D, instead of just two segments. In general, any number of segments may be used in a given streamer 210 - including streamers 210 with only a single sloped segment - to achieve a distribution of receivers 202 at various depths. The segments 21 OA, 210B in FIG. 2A, or the segments 210A, 210B, 210C, 210D in FIG. 2B may define one or more V-shaped portions of the streamers 210. In FIG. 2B, the segments may define a W-shaped portion. Also, in FIGS. 2 A and 2B, the V-shaped portion is concave towards the water surface 211, but in other embodiments, the V-shaped portion may be convex. Also, the segments 21 OA, 210B, 2 IOC, 210D need not necessarily be geometrically symmetrical - for example, one segment 21 OA may have a first slope, and another segment 210B may have a second slope, and so forth. [0044] Referring now to FIGS. 3 A and 3B, embodiments 300A, 300B of a non-towed marine seismic cable 310 are shown. In FIG. 3A, the cable 310 is anchored to the sea floor 309 by one or more anchors 314, and includes two segments 310A, 310B, which are similar to the segments 21 OA, 210B, 2 IOC, 210D in FIGS. 2A and 2B. In FIG. 3B, the cable 310 is attached to a buoy 316, and includes two segments 310A, 310B. The cables 310 in FIGS. 3 A and 3B each have a plurality of seismic receivers positioned therealong, which acquire seismic data at a plurality of different depths. In some examples, the cables 310 may include pressure sensors but may not necessarily include particle motion sensors typically found in ocean-bottom cable systems. The cables 310 may thus be lighter and less expensive to manufacture and deploy.

[0045] Turning now to FIG. 4, a flow chart illustrating one embodiment of a method 400 of deghosting variable depth seismic data is shown. In operation 402, a plurality of reference depth values {z_l, z_2, ... z_n} is defined. The reference depth values span a range of actual depths over which the seismic receivers 203 acquire seismic data - for example between 12 and 200 meters. The reference depths may be defined in a manner such that a maximum phase difference of deghosting operators (described in more detail below) for adjacent reference depths is less than a predetermined value - for example less than 0.2 radians.

[0046] In operation 404, the seismic data acquired by the seismic receivers 203 is transformed into a plurality of plane wave components. In some examples the seismic data that is transformed may be from a single shot record. The transformation into plane wave components may be accomplished by, for example, decomposing the seismic data using a 3D Fourier transform over time and two spatial dimensions (x, y), or decomposing the seismic data using a tau-p transform followed by a temporal Fourier transform. The transformation into plane wave components may be computed under the assumption that the data is sparse in the transformed domain, i.e. the assumption that many or most of the plane wave components are of zero or near zero amplitude. Taking the 3D Fourier transform as an example, the seismic data from a single shot record may be decomposed into a plurality of plane wave components, each wave component identified by (k_x, k_y, co). Because the streamers on which the receivers are located are shaped, the horizontal wavenumbers in the transformed data will correspond to the plane that the streamers are in, rather than a plane parallel to the sea surface; this difference, however, may be negligible.

[0047] Referring still to operation 404, before the seismic data is transformed into the plurality of plane wave components, in some embodiments, the seismic data may be interpolated onto a dense x, y grid of virtual receivers (e.g., may be interpolated in the cross line direction for data acquired in a towed streamer survey). The interpolation to the dense x, y grid may be accomplished using a sparseness assumption in some examples. When the data is interpolated, residual information from the seismic traces that is not represented by the interpolated data may be stored and used in subsequent processing (e.g., in migration, etc.) In some embodiments, however, the data is not interpolated before being transformed into the plurality of plane wave components (e.g., if it is assumed that the wavenumber component in the cross line direction is substantially zero), or the data may be interpolated after it is transformed into the plane wave components.

[0048] Referring still to operation 404, before the seismic data is transformed into the plurality of plane wave components, the raw data from acquired traces may need to be corrected using receiver motion correction. The receiver motion correction may, for example, apply differential normal move out to compensate for time offsets, and/or a coordinate transform process (e.g., from vessel stationary coordinates to earth stationary coordinates).

[0049] In operation 406, a deghosting operator is applied to each of the plurality of plane wave components for each of the plurality of reference depths to obtain a plurality of sets of reference deghosted plane wave components, one set of reference deghosted plane wave components for each of the plurality of reference depths. The deghosting operator may be substantially of the form

^{G 1} = . ^X-₂i_Ak \ (equation 1 )

i + r e where r is a reflectivity of the water-air boundary, z is a depth (e.g., the respective reference depth), and k_z is a vertical wavenumber. If the vertical wavenumber k_z is estimated using a far field dispersion relation, the deghosting operator may be

(equation 2)

for 3D transformed Fourier plane waves, where r is a reflectivity of a water-air boundary, z is a reference depth, k_x is a horizontal wave number in a first lateral dimension, k_y is a horizontal wave number in a second lateral dimension, v is a velocity of an acoustic medium, and ω is a frequency.

[0050] For tau-p transformed plane waves, the deghosting operator may be

where r is a reflectivity of a water-air boundary, z is a reference depth, p_x is a ray parameter in a first dimension, p_y is a ray parameter in a second dimension, v is a velocity of an acoustic medium, and ω is a frequency.

[0051] With reference to equations 2 and 3, the deghosting operator may be a spectral operator and may be a complex number "weight" that is applied to its respective plane wave component. Applying relatively straightforward respective deghosting operators to each respective plane wave component in this manner simplifies the complicated problem of applying a deghosting operator to an entire wavefield. However, in some examples, some of the parameters in the deghosting operator may not be readily ascertainable - such as the reflectivity of the air- water boundary (r) or the velocity of the water (v). It may also be difficult to ascertain the actual depth at which a seismic receiver was towed. However, one or more of these parameters may be estimated by minimizing noise in the deghosted output trace after the deghosting operator is applied using proposed values for the parameters. This estimation and minimization may be done using a comprehensive scan of realistic values in some examples, or any suitable optimization or minimization method (e.g., gradient decent, nonlinear conjugate gradient, etc.) may be used. Furthermore, the one or more parameters can be estimated for one or more subsets of data, in time and/or space. [0052] In operation 408, respective sets of reference deghosted plane wave components may be transformed to respective sets of reference deghosted traces. Each respective set of reference deghosted traces may correspond to one of the plurality of reference depths, and may include a respective reference trace for each respective one of the plurality of seismic receivers. The transformation may be to either a time-space domain (x, y, t), or to a temporal frequency domain (x, y, co) in some embodiments. If the sets of reference deghosted plane wave components are merely spatially transformed into reference deghosted frequency domain traces (x, y, co), the sets of deghosted output traces may still be decomposed into pluralities of temporal frequency components ω - and, in this case, a temporal transform may be needed to transform the deghosted frequency output traces into the time-space domain. On the other hand, the transformation from the reference deghosted plane wave components may be both a spatial and temporal transform back to the time-space domain (x, y, t) in some examples, thereby producing deghosted output traces with a time-series of pressure measurements in the time-space domain.

[0053] Referring now to operation 410, after the reference deghosted plane wave components have been transformed to respective sets of reference deghosted traces (e.g., with one set of reference deghosted traces corresponding to each of the plurality of reference depths), one or more deghosted output traces may be generated from the sets of reference deghosted traces. For example, a first deghosted output trace corresponding to one of the seismic receivers at a first actual depth may be generated by interpolating from the sets of reference deghosted traces, where the first actual depth corresponds to a depth at which the first seismic receiver acquired a portion of the seismic data. Interpolating from the sets of reference deghosted traces may include simply selecting one of the sets of reference deghosted traces whose corresponding reference depth is closest to the first actual depth, or, interpolating from the sets of reference deghosted traces may include performing a piecewise linear interpolation from among a plurality of the sets of reference deghosted traces. Alternatively, more complicated interpolation methods may be used.

[0054] As mentioned above with reference to operation 408, the transformation of the sets of reference deghosted plane waves to the sets of reference deghosted traces may include either a spatial transform or a spatial and temporal transform. In embodiments where the transformation in operation 408 is merely a spatial transform, the interpolation among frequency values of the sets of reference deghosted traces may require interpolating a complex number (which can be separated into phase and amplitude interpolations, or can be separated into real and imaginary interpolations). In either case, the deghosted output trace generated in operation 410 may provide a representation of the data corresponding to an actual x, y receiver location substantially without the effects of the ghost interference - whether that representation of the data be a temporal or frequency representation.

[0055] Just as the operations 406 and 408 may be performed for each of the plurality of reference depths, the operation 410 may be performed for any number of seismic output traces - for example, a deghosted output trace may be generated for each receiver x, y location in a survey.

[0056] Referring back to operation 406, because the deghosting operators in equations 1 , 2, and 3 are complex numbers with both phase and amplitude components, they may be applied to the plane wave components in operation 406 in one of several ways. In one embodiment, the entire deghosting operator (e.g., both phase and amplitude components) may be applied to the plane wave components for each reference depth. In another embodiment, and with reference now to FIG. 5, another method 500 involves applying the phase component of the deghosting operator separately from the amplitude component.

[0057] In operation 502 in FIG. 5, a first deghosted output trace may be generated for a first seismic receiver associated with a first offset in a first shot record, similar to operation 410 in

FIG. 4, except that the deghosting operator applied to the plurality of plane wave components is a phase-only deghosting operator (e.g., only the phase component of the deghosting operators in equations 1 , 2, or 3). In operation 504, a second deghosted output trace may be generated for a second seismic receiver associated with a second offset in a second shot record, similar to operation 410 in FIG. 4, except that the deghosting operator used is a phase-only deghosting operator. The second deghosted output trace may be generated for a second actual depth corresponding to a depth at which the second seismic receiver acquired a portion of the seismic data.

[0058] In operation 506, the first and second deghosted output traces may be stacked together, creating a stacked trace corresponding to a common gather. Because the first and second deghosted traces correspond to different offsets (and therefore likely correspond to different acquisition depths), and because they have individually had the phase portion of the ghost reflection corrected, they may be constructively summed together to recover a substantially ghost- free stacked trace, without amplifying noise associated with the ghost notches in the original data. [0059] In operation 508, following the stacking of the two (or more) deghosted traces, an amplitude correction may be applied to the stack, which may be either a deterministic or a statistical correction.

[0060] For a deterministic amplitude correction, a list may be created with amplitudes associated with each of the phase-only deghosting operators that would provide a full deghosted spectrum in the first and/or second deghosted output traces if the respective amplitudes were applied contemporaneously with the respective phase-one deghosting operators. The list of amplitudes may be summed during the stacking process of operation 506, and the sum may subsequently be used to deterministically apply the amplitude portion of the deghosting operator post-stack.

[0061] For a statistical amplitude correction, a spectral whitening or blind deconvolution may be applied that fills in spectral notches of the stacked trace. In some embodiments, a

deterministic amplitude correction is performed first, and then a statistical amplitude correction is performed to flatten the spectrum residually. [0062] It will be appreciated that the split deghosting illustrated in FIG. 5 and described herein has broad application. For example, referring to operations 502 and 504, in some embodiments, rather than applying a fully phase-only deghosting operator, the phase component and a portion of the amplitude component of the deghosting operator may be applied to traces pre-stack, and the remaining amplitude portion can be applied post-stack. As another example, the split or partial split deghosting can be applied to any partial stack or to sub-stacks. As still another example, a phase-only deghosting operator, or a partial-split deghosting operator, may be applied to the traces, and the amplitude portion or remainder thereof may be applied post-migration using a statistical, spectral whitening to fill in spectral notches of the migrated seismic image. As still another example, the statistical post-stack or post-migration amplitude correction can be applied to any deghosted data set - even if the full deghosting operator was applied to the plane wave components. Lastly, as still one additional variation, the amplitude of the reflectivity (r) of the deghosting operators in equations 1 , 2, and 3 may be partially or fully applied post-stack or post- migration, rather than pre-stack.

[0063] Turning now to FIG. 6, another embodiment of a method 600 for deghosting variable depth seismic data is shown. The method 600 illustrated in FIG. 6 may in some aspects be similar to the method 400 illustrated in FIG. 4, except that the deghosting operators in the method 600 are spatial deghosting operators that are applied in, for example, the frequency-and-space (f- x) domain rather than the spectral deghosting operators of method 400 that may be applied in the frequency-and-wave number (f-k) domain.

[0064] In operation 602, a plurality of reference depths may be defined that span a range of actual depths over which a plurality of seismic receivers acquire respective seismic input traces, which may each include a plurality of pressure measurements in the time-space domain.

[0065] In operation 604, a plurality of deghosting operators may be determined for the plurality of reference depths - e.g., one deghosting operator may be determined for each of the plurality of reference depths. The deghosting operators may in some embodiments be Hale- McClellan filters, any suitable explicit finite difference filter, any suitable implicit finite difference filter, any suitable localized differential operator, etc. As one other example, each of the plurality of deghosting operators may be a filter whose Fourier transform approximates the Fourier transform of the wavenumber domain deghosting operator

(equation 4)

wherein r is a reflectivity of a water-air boundary, z is a reference depth, Δχ is a sample interval in a lateral dimension along a streamer, v is a velocity of an acoustic medium, ω is a frequency, and k is a normalized wavenumber

[0066] The deghosting operators may be determined (in operation 604) independently from the seismic data and may be a function of velocity (v), reflectivity of the air-water boundary (r), and reference depth (z). Because the deghosting operators are independent from the seismic data, they may be computed in advance at a relatively fine scale.

[0067] In some embodiments, each deghosting operator may correspond with one reference depth and may include a plurality of filters, with each filter corresponding to a respective temporal frequency component of temporal frequency domain traces. Each of the plurality of filters may be defined by a plurality of coefficients, and the number and position of the coefficients may determine the aperture of the respective filter. The coefficients for each of the filters may be calculated as a function of velocity (v), reflectivity (r), and reference depth (z), and may be stored in a table for subsequent reference during operation 608

[0068] In operation 606, a plurality of the seismic input traces may be transformed into a set of temporal frequency domain traces. The seismic input traces that are transformed may include the seismic input traces within the aperture of the first of the plurality of deghosting operators used in operation 608. The aperture of the deghosting operator may encompass respective apertures of each of the filters of the deghosting operator for different frequencies. In some examples, all seismic input traces from a shot record may be transformed into the temporal frequency domain, but in other examples, only those seismic input traces within the apertures of the filters of the deghosting operator(s) to be used are thus transformed. The seismic input traces may come from a single shot record.

[0069] In operation 608, a first of the determined plurality of deghosting operators may be applied to the set of temporal frequency domain traces to generate a first deghosted output trace. The deghosting operator may be applied to the temporal frequency domain traces in the spatial and temporal frequency domains in some examples. The first of the plurality of deghosting operators may be selected from the plurality of deghosting operators by selecting a reference depth associated with the first of the plurality of deghosting operators closest to the actual depth for which a deghosted output trace is to be generated in some examples. In other examples, the operator is selected from the plurality of deghosting operators by interpolating between deghosting operators associated with two or more of the plurality of reference depths.

[0070] Once the appropriate deghosting operator is selected, each filter of the selected deghosting operator (different filters corresponding to different temporal frequency components) may be applied to the respective temporal frequency components of the temporal frequency domain traces within the respective apertures of the respective filters. For an explicit finite difference deghosting operator, for example, the deghosted output trace may be generated by calculating weighted sums (weighted by the coefficient values of the respective filters) of respective temporal frequency components of the set of temporal frequency domain traces. More specifically, in one example, the weighted sum for a first temporal frequency component may be calculated by multiplying coefficients from the filter corresponding to the first temporal frequency component with the temporal frequency components of the temporal frequency domain traces within the aperture of the respective filter, and summing the products thereof. This process may be repeated for each of the temporal frequency components in order to generate the first deghosted output trace (at a first x, y location). The first deghosted trace may be a frequency domain output trace - e.g., may be a set of temporal frequency components corresponding to an x, y location. This process may also be repeated for each output trace location - e.g., each x, y location where a deghosted output trace is desired. [0071] Referring now to operation 610 in FIG. 6, the first deghosted output trace may optionally be transformed to the time-space domain because the first deghosted output trace generated in operation 608 may be in a temporal frequency domain. In other embodiments, however, the deghosted output traces generated in operation 608 may be directly input to subsequent data processing methods - e.g., the temporal frequency domain deghosted output traces may be directly input to subsequent data processing methods.

[0072] With reference now to FIG. 7, another embodiment of a method 700 for deghosting variable depth seismic data is shown. The method 700 illustrated in FIG. 7 may in some aspects be considered to be a hybrid method, or a Fourier finite difference method, and may combine aspects of the method 400 illustrated in FIG. 4 together with aspects of the method 600 illustrated in FIG. 6. In the method 700 illustrated in FIG. 7, one or more spectral deghosting operators may be applied (similar to method 400 in FIG. 4) for one or more intermediate depths, and one or more spatial, residual deghosting operator(s) may be applied to the intermediate results.

[0073] More specifically in one example, in operation 702, a plurality of reference depths may be defined that span a range of actual depths over which a plurality of seismic receivers acquire respective seismic input traces. In operation 704, an intermediate depth is defined, with the intermediate depth being in the range of actual depths over which the seismic receivers acquired the input traces. In operation 706, a plurality of spatial deghosting operators for the plurality of reference depths are determined, with one spatial deghosting operator for each reference depth. These spatial deghosting operators may be space-frequency residual deghosting operators, as described below.

[0074] In operation 708, the seismic input traces are transformed to a plurality of plane wave components, and in operation 710, a spectral deghosting operator is applied to each of the plurality of plane wave components for the intermediate depth to obtain a set of reference deghosted plane wave components, similar to operation 406 in FIG. 4.

[0075] In operation 712, the set of reference deghosted plane wave components are transformed to a plurality of intermediate traces, with each intermediate trace being decomposed into a plurality of temporal frequency components (x, y, co).

[0076] In operation 714, one of the residual, spatial deghosting operators is applied to the temporal frequency components of the intermediate traces to generate a first deghosted set of temporal frequency components, similar to operation 608 in FIG. 6. Operation 714 may be repeated for each of a plurality of output trace locations, each output location using the appropriate spatial deghosting operator depending on the depth of the respective seismic receiver.

[0077] In operation 716, the first deghosted set of temporal frequency components are optionally transformed, to the time-space domain for example, to generate a first deghosted output trace. As with operation 714, operation 716 may be repeated for each of a plurality of output trace locations.

[0078] Referring now back to operations 710 and 714, the first, spectral deghosting operator applied in operation 710 may be an intermediate or average deghosting operator, and the spatial deghosting operators applied in operations 714 may provide the difference between that average and the actual deghosting needed for each given trace based on its distance away from the intermediate depth. In some examples, the intermediate depth may be an average depth over which the plurality of seismic receivers acquired the respective seismic input traces. To that end, if the deghosting operator for operation 710 is given by

^Go = . . I

-2iz₀ w v -k ,_x.-k ,_. (equation 5)

i + r e ^v where z₀ is the intermediate depth, then the residual, spatial deghosting operators needed for operation 714 may be given by (equation 6) where zo is the intermediate depth and z is the respective reference depth (defined in operation 702), so that, when the two are combined, the original, full deghosting operator is applied to the seismic data to obtain the deghosted output trace.

[0079] Referring still to the method 700 in FIG. 7, in some examples, two or more

intermediate depths may be defined, and the seismic data may be transformed into two or more respective sets of plane wave components, to which two or more respective spectral deghosting operators may be applied to obtain two or more respective sets of reference deghosted plane wave components. The two or more respective sets of reference deghosted plane wave components may independently be transformed as per operation 712, and then respective spatial deghosting operators may be applied independently to the two or more transformed sets. [0080] FIG. 8 illustrates an embodiment of a computer system 835 capable of processing seismic data, including for example, a system capable of executing the operations in FIGS. 4 through 7. The computer system 835 illustrated in FIG. 8 may be used as the data processing apparatus 108 in FIG. 1 in some examples. [0081] In some embodiments, the computer system 835 may be a personal computer and/or a handheld electronic device. In other embodiments, the computer system 835 may be an implementation of enterprise level computers, such as one or more blade-type servers within an enterprise. In still other embodiments, the computer system 835 may be any type of server. The computer system 835 may be onboard a vessel (such as vessel 201 shown in FIG. 2), may be on a remotely controlled drone boat, may be on land in a vehicle, may be in land in a facility, or any other place.

[0082] A keyboard 840 and mouse 841 may be coupled to the computer system 835 via a system bus 848. The keyboard 840 and the mouse 841 , in one example, may introduce user input to the computer system 835 and communicate that user input to a processor 843. Other suitable input devices may be used in addition to, or in place of, the mouse 841 and the keyboard 840. An input/output unit 849 (I/O) coupled to the system bus 848 represents such VO elements as a printer, audio/video (A/V) I/O, etc.

[0083] Computer 835 also may include a video memory 844, a main memory 845 and a mass storage 842, all coupled to the system bus 848 along with the keyboard 840, the mouse 841 and the processor 843. The mass storage 842 may include both fixed and removable media, such as magnetic, optical or magnetic optical storage systems and any other available mass storage technology. The bus 848 may contain, for example, address lines for addressing the video memory 844 or the main memory 845.

[0084] The system bus 848 also may include a data bus for transferring data between and among the components, such as the processor 843, the main memory 845, the video memory 844 and the mass storage 842. The video memory 844 may be a dual -ported video random access memory. One port of the video memory 844, in one example, is coupled to a video amplifier 846, which is used to drive one or more monitor(s) 847. The monitor(s) 847 may be any type of monitor suitable for displaying graphic images, such as a cathode ray tube monitor (CRT), flat panel, or liquid crystal display (LCD) monitor or any other suitable data presentation device.

[0085] The computer system includes a processor unit 843, which may be any suitable microprocessor or microcomputer. The computer system 835 also may include a communication interface 850 coupled to the bus 848. The communication interface 850 provides a two-way data communication coupling via a network link. For example, the communication interface 850 may be a satellite link, a local area network (LAN) card, a cable modem, and/or wireless interface. In any such implementation, the communication interface 850 sends and receives electrical, electromagnetic or optical signals that carry digital data representing various types of information.

[0086] Code received by the computer system 835 may be executed by the processor 843 as the code is received, and/or stored in the mass storage 842, or other non- volatile storage for later execution. In this manner, the computer system 835 may obtain program code in a variety of forms. Program code may be embodied in any form of computer program product such as a medium configured to store or transport computer readable code or data, or in which computer readable code or data may be embedded. Examples of computer program products include CD- ROM discs, ROM cards, floppy disks, magnetic tapes, computer hard drives, servers on a network, and solid state memory devices. Regardless of the actual implementation of the computer system 835, the data processing system may execute operations that allow for processing seismic data, including for example the operations illustrated in FIGS. 4 through 7 and otherwise as described herein.

[0087] The apparatuses and associated methods in accordance with the present disclosure have been described with reference to particular embodiments thereof in order to illustrate the principles of operation. The above description is thus by way of illustration and not by way of limitation. Various modifications and alterations to the described embodiments will be apparent to those skilled in the art in view of the teachings herein. Those skilled in the art may, for example, be able to devise numerous systems, arrangements and methods which, although not explicitly shown or described herein, embody the principles described and are thus within the spirit and scope of this disclosure. Accordingly, it is intended that all such alterations, variations, and modifications of the disclosed embodiments are within the scope of this disclosure.

[0088] In methodologies directly or indirectly set forth herein, various steps and operations are described in one possible order of operation, but those skilled in the art will recognize that the steps and operations may be rearranged, replaced, or eliminated without necessarily departing from the spirit and scope of the disclosed embodiments.

[0089] All relative and directional references (including: upper, lower, upward, downward, upgoing, downgoing, left, right, top, bottom, side, above, below, front, middle, back, vertical, horizontal, and so forth) are given by way of example to aid the reader's understanding of the particular embodiments described herein. They should not be read to be requirements or limitations, particularly as to the position, orientation, or use of the invention unless specifically set forth in the claims. Connection references (e.g., attached, coupled, connected, joined, and the like) are to be construed broadly and may include intermediate members between a connection of elements and relative movement between elements. As such, connection references do not necessarily infer that two elements are directly connected and in fixed relation to each other, unless specifically set forth in the claims. Furthermore, the included figures may not be drawn to scale - for example, FIGS. 2A through 3B in particular are merely illustrative and not to scale.

Claims

What is claimed is:

1. A method of deghosting seismic data, comprising:

defining a plurality of reference depths spanning a range of actual depths over which a plurality of seismic receivers acquire seismic data;

transforming the seismic data into a plurality of plane wave components;

applying a deghosting operator to each of the plurality of plane wave components for each of the plurality of reference depths to obtain a plurality of sets of reference deghosted plane wave components, one set of reference deghosted plane wave components for each of the plurality of reference depths;

transforming respective sets of reference deghosted plane wave components to respective sets of reference deghosted traces; and

generating a first deghosted output trace corresponding to one of the seismic receivers at a first actual depth by interpolating from the sets of reference deghosted traces.

2. The method of claim 1, wherein each set of reference deghosted traces corresponds to one of the plurality of reference depths and includes a respective reference trace corresponding to each respective one of the plurality of seismic receivers. 3. The method of claim 1 , wherein the seismic data is decomposed into the plurality of plane wave components using a 3D Fourier transform over time and two spatial dimensions.

4. The method of claim 1, wherein the seismic data is decomposed into the plurality of plane wave components using a tau-p transform followed by a temporal Fourier transform.

5. The method of claim 1, wherein the plurality of reference depths is defined such that a maximum phase difference of deghosting operators for adjacent reference depths is less than a predetermined value. 6. The method of claim 1, wherein the first seismic receiver is associated with a first offset in a first shot record, and further comprising generating a second deghosted output trace corresponding to a second seismic receiver associated with a second offset in a second shot record.

7. The method of claim 6, further comprising stacking the first and second deghosted output traces to form a stacked trace.

8. The method of claim 7, wherein the deghosting operator applied to the plurality of plane wave components for the plurality of reference depths is a phase-only deghosting operator, further comprising applying an amplitude correction to the stacked trace.

9. The method of claim 8, further comprising:

storing a list of amplitudes associated with each of the phase-only deghosting operators that would provide a full deghosted spectrum in the first and/or second deghosted output traces if applied contemporaneously with respective phase-only deghosting operators; and

deterministically applying the amplitude correction to the stacked trace using the stored list of amplitudes as a function of time and frequency.

10. The method of claim 6, wherein the deghosting operator applied to the plurality of plane wave components is a phase-only deghosting operator, further comprising:

migrating the first and second deghosted traces to form a seismic image; and

statistically applying an amplitude correction to the migrated seismic image to fill in spectral notches of the seismic image.

11. The method of claim 1 , wherein the deghosting operator is substantially of the form

, —Hz \kA

+ r e wherein r is a reflectivity of a water-air boundary, z is a respective reference depth, and k_z is a vertical wavenumber.

12. The method of claim 11, wherein the vertical wave number k_z is estimated using a far field dispersion approximation, such that the deghosting operator is substantially of the form

1

G ^l {k_x,k_y )- l + r e

wherein r is a reflectivity of a water-air boundary, z is a respective reference depth, k_x is a horizontal wave number in a first lateral dimension, k_y is a horizontal wave number in a second lateral dimension, v is a velocity of an acoustic medium, and ω is a frequency, or is substantially of the form

wherein r is a reflectivity of a water-air boundary, z is a respective reference depth, p_x is a ray parameter in a first dimension, p_y is a ray parameter in a second dimension, v is a velocity of an acoustic medium, and ω is a frequency.

13. The method of claim 1 , further comprising interpolating the seismic data to a dense receiver x-y grid prior to decomposing the seismic data into the plurality of plane wave components.

14. The method of claim 1, wherein the plurality of seismic receivers are positioned on a plurality of variable depth streamers towed behind a vessel together with one or more seismic sources.

15. The method of claim 1, wherein the plurality of seismic receivers are positioned on a non-towed, variable depth cable in a water column.

16. The method of claim 1, wherein the respective sets of reference deghosted plane wave components are spatially transformed to the respective sets of reference deghosted traces, the respective sets of reference deghosted traces still decomposed into pluralities of temporal frequency components when generating the first deghosted output trace, wherein the method further comprises transforming the first deghosted output trace to a time domain to obtain a time- series of pressure measurements in the time-space domain.

17. The method of claim 1, wherein the respective sets of reference deghosted plane wave components are spatially and temporally transformed to the respective sets of reference deghosted traces, the respective sets of deghosted traces including time-series of pressure measurements in the time-space domain.

18. A method of deghosting seismic data, comprising:

applying a deghosting operator to a plurality of plane wave components of seismic data for a plurality of defined reference depths to obtain a plurality of sets of reference deghosted plane wave components;

transforming respective sets of reference deghosted plane wave components to respective sets of reference deghosted traces; and

generating a first deghosted output trace corresponding to one of the seismic receivers at a first actual depth from the sets of reference deghosted traces.

19. A method of deghosting seismic data, comprising:

defining a plurality of reference depths spanning a range of actual depths over which a plurality of seismic receivers acquire respective seismic input traces;

determining a plurality of deghosting operators for the plurality of reference depths; transforming a plurality of the seismic input traces into a set of temporal frequency domain traces; and

applying a first of the plurality of deghosting operators to the set of temporal frequency domain traces to generate a first deghosted output trace.

20. The method of claim 19, wherein the plurality of seismic input traces transformed into the set of temporal frequency domain traces includes seismic input traces within an aperture of the first of the plurality of deghosting operators.

21. The method of claim 19, wherein the first deghosted output trace is in a temporal frequency domain, further comprising transforming the first deghosted output trace to the time- space domain.

22. The method of claim 19, wherein each of the plurality of deghosting operators includes a plurality of filters for a plurality of temporal frequency components of the temporal frequency domain traces, and each filter is applied to the temporal frequency components of the temporal frequency domain traces within the aperture of the respective filter.

23. The method of claim 22, wherein each of the plurality of filters is defined by a respective set of coefficients.

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reflectivity of a water-air boundary, z is a depth, Δχ is a sample interval in a lateral dimension, v is a velocity of an acoustic medium, ω is a frequency, and k is a normalized wavenumber.

28. A method of deghosting seismic data, comprising:

defining a plurality of reference depths spanning a range of actual depths over which a plurality of seismic receivers acquire respective seismic input traces;

defining an intermediate depth within the range of actual depths;

determining a plurality of spatial deghosting operators for each of the plurality of reference depths;

transforming the seismic input traces into a plurality of plane wave components;

applying a spectral deghosting operator to each of the plurality of plane wave components for the intermediate depth to obtain a set of reference deghosted plane wave components;

transforming the set of reference deghosted plane wave components to a plurality of intermediate traces, each intermediate trace decomposed into a plurality of temporal frequency components; and applying a first of the plurality of spatial deghosting operators to the temporal frequency components of the intermediate traces to generate a first deghosted set of temporal frequency components.

29. The method of claim 28, wherein the plurality of spatial deghosting operators are residual deghosting operators.

30. The method of claim 28, wherein the intermediate depth is an average depth over which the plurality of seismic receivers acquired the respective seismic input traces.