GB2521598A

GB2521598A - Multi-dimensional deconvolution using exact boundary conditions

Info

Publication number: GB2521598A
Application number: GB1321231.1A
Authority: GB
Inventors: Lasse Amundsen; Johan Olof Anders Robertsson
Original assignee: Statoil Petroleum ASA
Current assignee: Equinor Energy AS
Priority date: 2013-12-02
Filing date: 2013-12-02
Publication date: 2015-07-01
Also published as: GB2520979A; GB201321231D0; WO2015082421A1; GB201321517D0; GB2520979A9; WO2015082419A1

Abstract

A method is provided for performing multi-dimensional deconvolution of geophysical data, wherein the method is characterised by using exact boundary conditions 22, 24. The method may comprise using a forward solver of the wave equation in combination with the boundary conditions. The boundary conditions may comprise a recording surface and an emitting surface. Other aspects of the invention relate to methods of redatuming a wavefield by means of finite-difference (FD) injection or injecting the wavefield as source terms in the wave equation. Another relates to a method of redatuming receivers to a different depth using FD injection and a final one relates to a method of performing multiple suppression by redatuming a wavefield using FD-injection and using exact boundary conditions to achieve multiple suppression.

Description

Multi-Dimensional Deconvolution using Exact Boundary Conditions

FIELD OF THE INVENTION

The invention relates to multi-dimensional deconvolution, which can take out the geophysical response of the geology in geophysical data above the receiver datum. It can be used for example for multiple suppression, or in combination with redatuming of data, to remove the response of the geology overburden at the new datum. It can be used for acoustic, elastic, or electromagnetic data.

BACKGROUND OF THE INVENTION

Seismic data are acquired to provide images of the sub surface. To provide detailed answers of the location and properties of sub-surface targets, the seismic acquisition and processing chain can be thought of in terms of four distinct steps: Seismic data acquisition Seismic data processing or data conditioning Imaging/Inversion Interpretation Here we are concerned with steps 2 and 3 above, ie data conditioning and imaging/inversion.

A similar chain from acquisition to interpretation is valid for electromagnetic data, of which one example is marine Controlled Source ElectroMagnetic (m-CSEM) surveying (Eidesmo et al., 2002; Ellingsrud et al., 2002). The methods that we address here are for seismic (acoustic and elastic) data. However, by using elastic-electromagnetic mathematical equivalences, which consist of recasting Maxwell's equations and the constitutive relations for an electromagnetic medium into a form that is mathematically identical with that of the basic equations for elastic waves, the methods we derive here for seismic waves can by persons skiHed in the art be derived for electromagnetic fields. In the following, our focus is mainly on seismic surveying.

During the data condition step, raw seismic data are pre-conditioned to remove noise and to make the data fit assumptions of the imaging/inversion step. One particular problem in data conditioning is that of removing multiples corresponding to seismic energy that bounces multiple times between reflectors. Many imaging algorithms assume that the data are free of multiples as these can create spurious unreal apparent reflectors and noise in the final image. We make distinctions between different types of multiples. Of particular importance are surface-related multiples corresponding to energy that bounces off the surface of the Earth (where sources and receivers typically are located in seismic surveys) one or more times before being recorded. Note that there are also ghosts corresponding to the immediate bounce off the surface of the Earth right when the source is excited (the source ghost) or when the data are recorded (the receiver ghost). Ghosts are a different problem and not considered to be multiples. Another type of multiples is intrabed multiples which correspond to energy that bounces multiple times between subsurface layers before being recorded.

The imaging/inversion step can be done using a wide range of different methods that all make different assumptions with respect to the data. Traditionally, many methods (such as Kirchhoff migration) assume that all data correspond to primary reflections only from sub-surface layers. Clearly, any multiples will be considered as noise when using such methods. Other methods, such as Reverse Time Migration (RTM), can in principle cope with other data than just simply primary reflections. However, in practice, many RTM implementations will assume that at least surface-reflated multiples have been removed from the data.

Over recent years, the trend has been that data conditioning and imaging/inversion techniques increasingly are based on using the full wave equation. Although the increase in computational cost can be very significant compared to traditional techniques based on for instance a ray assumption (such as Kirchhoff migration), the quality of results can be superior, particularly where we face complex imaging challenges as wave equation based techniques better honour the physics of the seismic experiment. Two particular wave equation based techniques that have proven highly effective at removing multiples and imaging seismic data are the Amundsen demultiple method (Amundsen, 2001) and F{TM. So-called Multi-Dimensional Deconvolution (MDD) is a key tool for both of these methods and many other multiple elimination and imaging methods allowing the effects of an overburden to be removed from the data being processed. In particular, for the Amundsen demultiple method, the sea surface can be replaced by an infinite water layer so that all surface-related multiples disappear.

Seismic interferometry has found many applications in acquisition, processing and imaging of seismic data, in particular for Green's function retrieval (i.e., the indirect prediction of the response between a source and a receiver location which has not been explicitly recorded). MDD was investigated and implemented for such applications by Wapenaar, van der Neut and co-workers (van der Neut, 2012; Wapenaar et al., 2011; Wapensar and van der Neut, 2010). Traditional" Green's function retrieval by crosscorrelation may suffer from irregularities in the source distribution, asymmetric illumination, intrinsic losses, etc. MDD may overcome these limitations. Whereas a traditional crosscorrelation method gives a Green's function of which the source is smeared in space and time (quantified by a space-time point-spread function), MDD removes this effect and thus deblurs and deghosts the source of the Green's function obtained by correlation.

Another application of MDD is for redatuming which may or may not be part of an imaging step. It has recently been demonstrated that one can redatum the seismic data from the recording datum to a new datum in the subsurface geology with a process called autofocusing. This process requires no information of the geology between the water layer and the new depth datum. This opens for application of MDD at the new depth datum, and imaging of geology beneath this datum, without any influence of wavefields of the overburden (see e.g. Broggini, 2013).

Whether the application is for redatuming, multiple attenuation, imaging or Green's function retrieval, the underlying principles of MDD remain the same. Given that the receiver measurements on any plane allow wavefield decomposition into total upgoing and total downgoing wavefields, the upgoing and downgoing wavefields form an integral relationship with the seismic response of the geology beneath the receiver plane: the "underburden" response. The resulting seismic response is not affected by the medium above the receiver plane, ndependent of what this geology is. In other words the seismic response is not affected by the "overburden".

Thus, from the separate upgoing and downgoing wavefields, by solving the integral equation the seismic underburden response of the medium can be found. The underburden response can be conventionally imaged to get a clean and accurate image of the underburden geology.

The solution of the integral equations is called Multi-Dimensional Deconvoution (MDD).

The integral equations are Fredholm integral equations of the first kind. Such integral equations are, in general, ill-conditioned, and their accurate solution may be difficult and numerically expensive to obtain.

SUMMARY OF THE INVENTION

The invention provides methods as set out in the accompanying claims.

BRIEF DESCRIPTION OF THE FIGURES

Figure 1 shows the ideal acquisition geometry of a marine seismic experiment, in which sources are located below receivers, in which a method can be implemented to remove free-surface related multiples; Figure 2 shows the geometry ol an EBC simulation for removing the effects of the free surface (ocean surface) by means of MDD; FigureS shows the depth levels used in a method of removing surface related multiples where sources are located above receivers; Figure 4 shows the structure of a model used in a method of imaging using autofocussing and MDD; and Figure 5 shows a computer suitable for carrying out methods described herein.

DESCRIPTION OF PREFERRED EMBODIMENTS

We describe how MDD can be implemented in an alternative way. Although MDD has a large range of applications, which our method is applicable to, we will first focus our description of the method on the application of multiple attenuation methods for acoustic and ocean bottom seismic data by Amundsen (2001) and Amundsen et al. (2001), and the land and borehole seismic multidimensional deconvolution method in Holvik and Amundsen (2005). When seismic data are measured in the water column or on the seabed, MDD will remove multiples, giving cleaner data for imaging of the subsurface.

The MDD method can also be applied to m-CSEM data as shown by Nordskag et al. (2008). They showed that MDD eliminates from the recorded multi-component source, multi-component receiver marine electromagnetic measurements the effect of the physical source radiation pattern and the scattering response of the water-layer.

Having introduced the new method for multiple elimination we will descrbe a second application related to imaging.

We first describe a method for surface-related multiple elimination, where sources are located below receivers.

The concept and method of Exact Boundary Conditions (EBC) has been introduced by van Manen et al. (2007a, 2007b) for performing exact wave field simulations for finite-volume scattering problems. Recently, a rigorous theoretical foundation for EBC's with generalizations to higher dimensions has been introduced by Vasmel et al. (2013).

Briefly, the EBC method works as follows. The objective is to connect a numerical simulaton to Green's functions from a different state (physical or simulated) so that the numerical simulation will include the effects of that state outside the numerical simulaton domain. By state we refer to an environment (real or simulated) characterized by parameters of the medium or sources affecting wave propagation where a geophysical experiment takes place. Two surfaces are of particular importance. First, the emitting surface S' where a wavefield is emitted on the edge of the computational domain. Second, the recording surface where waves are recorded in the numerical simulation. The medium between the recording surface and the emitting surface must be essentially identical in the numerical simulation and the (physical or numerical) state where the Green's functions that will be used for extrapolation were generated, but away from this sub-volume they can be arbitrarily different. The algorithm for connecting the numerical simulation with the physical experiment follows an algorithm for coupling numerical finite-difference computations (van Manen et al., 2007a, 2007b) where we use equation (4) in van Manen et al. (2007a), which is a time-discrete version of Green's second identity: {(i"" .1-mi".O)x (1) where the caret denotes time sampled quantities, is the sampled pressure at time-step " and location, O(rt,1_m:r,O) is the Green's function at time step -between and r, is a location on the integration surface r with normal, and is a spatial gradient operator normal to the integration surface. Note that the usual time-integral in Green's second identity is implicit within the recursion in equation (1).

Green's functions for the numerical simulation connecting ç( and S" are either pre-computed using a wave propagation simulation technique or as in our case they are measured data from a seismic survey. Acoustic waves are recorded along at discrete time steps. These data are extrapolated to S by means of equation (1) using the pre-computed Green's functions. The extrapolated data comprise a discrete time series that is added to a stored buffer jr" (fTMt.Lm) containing future values to be emitted along. At each time-step, equation (1) is thus evaluated as many times as the number of samples in the discrete Green's functions. At time-step /+ldata from the stored buffer are emitted on r In this fashion we are able to link a numerical simulaton to Green's functions from a different state (physical or simulated) so that the numerical simulation will include the effects of that state outside the numerical simulaton domain.

We describe an alternative way of implementing MDD using a forward solver for partial-differential equations in combination with EBO's. In a preferred embodiment the MDD implementation is based around a space-time domain finite-difference solver in combination with EBC. Note that similar results can also be obtained using a pseudo-spectral method for instance corresponding to a wavenumber-time solver.

Figure 1 shows the ideal acquisition geometry of a marine seismic experiment where the new method can be implemented directly to remove free-surface related multiples.

The ideal acquisition geometry consist of: A carpet" 2 of receivers 4, recording pressure and vertical component of particle velocity, located at a constant depth hi in the water column. The carpet" 2 is essentially evenly spaced in the lateral directions essentially satisfying the Nyquist sampling criterion.

A "carpet" 6 of explosive type (monopole) sources 8 located at a constant depth h2>hl n the water column. The "carpet" 6 is essentially evenly spaced in the lateral directions essentially satisfying the Nyqust sampling criterion.

Figure 2 shows how EBC's can be used to remove the effects of the free surface through finite-difference simulations. The result will substantially be equivalent to so-called Amundsen demultiple (Amundsen, 2001). A finite-difference (FD) model 10 is defined which comprises a homogenous medium 12 with the water column properties.

All edges (14, 16, 18) of the FD simulation except for the bottom edge 20 have absorbing boundary conditions applied to them so that all energy that is incident on them is attenuated. The remaining bottom edge 20 has the EBC applied to it.

An EBC consists of two surfaces: a recording surface 22 and an emitting surface 24.

The emitting surface 24 coincides with the bottom edge 20 of the FD simulation. The recording surface 22 can be located at some distance above the emitting surface 24 in the FD model. It is important that the emitting surface 24 coincides with the depth level where the carpet 6 of sources 8 was located during the acquisition of the seismic data and that the recording surface 22 coincides with the depth level where the carpet 2 of receivers 4 was located. The acquired seismic data will now represent the Green's functions in the EBC's so that we can generate the response between any source-receiver configurations above the depth hi where the receivers 4 were located. The use of EBC's naturally "stitches together" real acquired Green's functions with synthetic Green's functions generated implicitly using the FD simulation on a grid with an infinite half space of water (the effect of absorbing boundaries (14, 16, 18) is to mimic an infinite medium). The EBC's themselves serve two purposes: 1. To attenuate any waves that impinge on the emitting surface 24 (it acts as an absorbing boundary condition).

2. To emit waves that would be reverberated back inside the finite-difference model corresponding to reflections from the sub surface below the emitting surface 24.

The resulting generated data will be free of the effects of the free surface (ie the ocean surface).

We note a couple of important points: In principle there is no need for the FD model to be homogenous. The only requirement is that the medium 12 is substantially identical between the real acquired data and the FD model in the region between hi and h2 (between the recording surface 22 and the emitting surface 24).

We do not need to use FD to simulate the data. EBC's can be used with any numerical method. In particular, if the medium consists of a homogenous half space, we can use analytical methods to implement the EBC's.

We next describe a method for surface-related multiple elimination, where sources are located above receivers.

Unfortunately, most seismic data are not acquired as shown in Figure 1. Instead, sources are typically located at a depth level that is shallower than the receivers (ie.

sources located at hi and receivers located at h2). To apply the method to such a source/receiver configuration we need to pre-process the data by redatuming the receivers 4 to a depth level shallower than the sources 8. Redatuming can be achieved in several ways, for instance, building on the well-known integral representation of the wave equation. A particularly attractive alternative means by which we can do this is to use the method of FD-injection (Robertsson and Chapman, 1999, 2000) for wavefield separation. Redatuming is carried out on the up-and down-

going wavefield separately.

If the medium is known, FD-injection can be used to forward propagate the up-going wavefield data through the model to a new datum level. By recording the data in a different location (ie a higher receiver location, corresponding with receivers at a shallower depth) we can retrieve the up-going part of the recorded data that would have been recorded at that new location due to the same source location. Referring to FigureS, the redatuming can be carried out as follows: * Define a homogeneous medium with water layer properties. Apply absorbing boundary conditions on all sides (or alternatively use a free surface at the top with a modified recording configuration compared to the description in the next couple of steps).

* Define a FD-injection surface coinciding with the carpet" of receivers in the acquired data.

* Inject the recorded data so that the up-going data radiates up-wards. The down-going data will automatically be radiated downwards and will be attenuated as it hits the absorbing boundary below the injection surface. Note that an FD-injection surface should ideally be closed. However, since we are forward propagating a wavefield in a homogenous medium all contributions will come from the one boundary corresponding to the receiver locations.

* Record the wavefield at four different depth levels: 1. The depth level h0 to which we wish to redatum the wavefield.

2. The depth level -h0 which corresponds to the mirror position of the desired depth level across where the sea surface would have been located.

3. The depth level -h2 -a which corresponds to the mirror position of a location immediately below the actual recording depth level across where the sea surface would have been located.

4. The depth level h2 + A which corresponds to a location immediately below the actual recording depth level but just above the absorbing boundary.

* With these recorded data we can now generate two data sets: 1. We can generated the redatumed and data at the desired depth level h0 apart from the direct wave contribution: fl = -(p(h0) -p(-h0)) and = -(v(h0) + v(-h0)). This operation accounts for the down-going receiver ghosts generated by the up-going wavefield.

2. We can compute the direct wave contribution only at the recording depth level by evaluating p'(h2 + ) = p(h2 + A) + p(-h2 -) and v?(h2 + ) = v(h2 + ) --). Note that the result of this step may be highly useful in itself for source signature estimation or for removal of the direct wave from seismic data.

* We have now redatumed all data to the desired depth level except far the direct wave contribution. This can simply be computed synthetically for instance by generating the direct wave and its ghost on a finite-difference grid with a homogenous medium (water) and a free surface and absorbing boundaries on all other sides. If the source wavelet is known we can simply record the data at the desired depth level and add it to the redatumed fl and i3 data. Alternatively, we can invert for the optimal source wavelet and its location to match the downgoing wavefield at the actual recording level p'(h2 + a) and vf'(h2 + a).

We can now proceed using the method of EBC's to remove surface-related multiples as described above.

We now describe a method of imaging by using autofocussing and MDD.

Building on the known integral equations for seismic data, that give the relationship between the upgoing and downgoing wavefields and the seismic response of the underburden, the known method of autofocusing (referred to above), we propose the following new workflow for imaging of seismic data that have been

separated into upgoing and downgoing wavefields:

redatum the seismic wavefield to a new datum by any method, preferably autofocusing, apply a forward solver of the wave equation, preferably a space-time domain finite-difference solver in combination with EBC to the redatumed data to find the response of the underburden, image the underburden response by any imaging or inversion method.

The objective of MDD is to replace the overburden above a datum by a homogenous medium so that the data at the datum consists of the response due to point sources on the datum from the medium below the datum only.

The first step is to redatum the surface seismic data to a datum at a desired depth level. For simplicity we will assume that the medium properties are constant along the datum. Note that sources and receivers may be located at different depths. The method can inherently cope with acquisition geometries such as these common to for instance Ocean Bottom Cable (OBC) and towed marine seismic data.

The redatuming step can be done using either a traditional method based on for instance the Kirchhoff integral or the FD-injection method (Robertsson and Chapman, 1999, 2000) described above. However, more recently methods for redatuming that do not rely on a velocity model using the so-called Marchenko equations have been proposed (Broggini, 2013).

We will also assume that we have an acoustic medium only. The method can be generalized to elastic media but will require different wavefield constituents to be available at the datum on both the source and receiver sides. It can also be generalized to electromagnetic media.

We now assume that the data have been redatumed to the new datum at depth and that we have adequately sampled pressure-to-pressure and pressure-to-vertical particle velocity source-receiver pairs (i.e., Green's functions) available.

MDD can now be implemented for instance in the space-time domain using for instance a finite-difference method and a model as illustrated in Figure 4. The model has constant medium properties throughout the grid equivalent to those of the recording datum. Along the bottom edge 40 we use an EBC and along all remaining edges (42, 44, 46) we use absorbing boundaries.

By simply carrying out finite-difference simulations on the model in Figure 4 we can compute the response for any source-receiver locations just above the datum (one shot at a time or reciprocally one receiver at a time). The resulting response will have removed all effects of the overburden from the data. These data are now much better suited to provide a reliable image of the under-burden through any suitable imaging method.

Figure 5 shows a computing device 60, which may for example be a personal computer (PC), on which methods described herein can be carried out. The computng device 60 comprises a display 62 for displaying information, a processor 64, a memory 68 and an input device 70 for allowing information to be input to the computing device. The input device 70 may for example include a connection to other computers or to computer readable media, and may also include a mouse or keyboard for allowing a user to enter information. These elements are connected by a bus 72 via which information is exchanged between the components.

It should be appreciated that any of the methods described herein may also include the step of acquiring seismic or electromagnetic data, which may then be processed in accordance with the method.

References Amundsen, L., 1993, Wavenumber-based filtering of marine point-source data: Geophysics, 58, 1335-1 348.

Amundsen, L., 2001, Elimination of free-surface related multiples without need of the source wavelet: Geophysics, 66, 327-341.

Amundsen, L., L. 1. Ikelle, and L. E. Berg, 2001, Multidimensional signature deconvolution and free-surface multiple elimination of ocean bottom seismic data: Geophysics, 66, 1594-1604.

Broggini, F., 2013, Wave field autofocusing and applications to multidimensional deconvolution and imaging with internal multiples: PhD thesis, Colorado School of Mines.

Eidesmo, 1., S. Ellingsrud, L. M. MacGregor, S. Constable, M. C. Sinha, S. Johansen, F. N. Kong, and H.Westerdahl, 2002, Sea bed logging (SBL), a new method for remote and direct identification of hydrocarbon filled layers in deepwater areas: First Break, 20, 144-1 52.

Ellingsrud, S., T. Eidesmo, M. C. Sinha, L. M. MacGregor, and S. Constable, 2002, Remote sensing of hydrocarbon layers by seabed logging (SBL): Results from a cruise offshore Angola: The Leading Edge, 21, 972-982.

Holvik, E., and Amundsen, L., 2005, Elimination of the overburden response from multicomponent source and receiver seismic data, with source designature and decomposition into PP-, PS-, SP-, and 58-wave responses: Geophysics, 70, 843-859.

Nordskag, J. I., L. Amundsen, L. 0. Lgseth, and E. Holvik, 2009, Elimination of the water-layer response from multicomponent source and receiver marine controlled electromagnetic data: Geophysical Prospecting 57, 897-914.

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Robertsson, J. 0. A., and C. H. Chapman, 1999, Method of determining the seismic response caused by model alterations in seismic simulations, Norwegian patent No. 330994, GB Patent No. 2,329,043 and US patent No. 6,125,330.

van der Neut, J., 2012, Interferometric redatuming by multidimensional deconvolution: PhD thesis, T U Delft.

van Manen, D. J., J. 0. A. Robertsson and A. Curtis, 2007a, Exact wave field simulaton for finite-volume scattering problems: Journal of Acoustical Society of America, 122(4), EL115-EL121.

van Manen, D. J., J. 0. A. Robertsson and A. Curtis, 2007b, Method of evaluating the interaction between a wavefield and a solid body: US Patent No. 771 5985B2.

Vasmel, M., Robertsson, J. 0. A., van Manen, D. J., and Curtis, A., 2013, Immersive experimentation in a wave propagation laboratory: J. Acoust. Soc. Am. (Express Letters), 134, EL492-EL498.

Wapenaar, C. P. A., J. van der Neut, E. Ruigrok, D. Draganov, J. Hunziker, and E. Slob, 2011, Seismic interferometry by crosscorrelation and by multidimensional deconvolution: A systematic comparison: Geophysical Journal International, 185, 1335-1364.

Wapenaar, C. P. A., J. van der Neut, 2010, A representation for Green's function retrieval by multidimensional deconvolution: Journal of the Acoustical Society of America, 128, EL366-EL371.

Claims

CLAIMS: 1. A method of performing multi-dimensional deconvolution of geophysical data, wherein the method is characterised by using exact boundary conditions.
2. A method as claimed in claim 1, which further comprises using a forward solver of the wave equation in combination with said exact boundary conditions.
3. A method as claimed in claim 1 or 2, wherein said geophysical data is seismic data.
4. A method as claimed in claim 3, when also dependent on claim 2, wherein said wave equation is an elastic wave equation.
5. A method as claimed in claim 3, when also dependent on claim 2, wherein said wave equation is an acoustic wave equation.
6. A method as claimed in any one of claims 1 to 2 claim, wherein said geophysical data is electromagnetic data.
7. A method as claimed in claim 6, when also dependent on claim 2, wherein said wave equation is an electromagnetic wave equation or diffusion equation.
8. A method as claimed in any preceding claim, which further comprises using a finite-difference solver in combination with said exact boundary conditions.
9. A method as claimed in any preceding claim, which further comprises using an analytical solution in combination with said exact boundary conditions
10. A method as claimed in any preceding claim, wherein said exact boundary conditions comprise a recording surface and an emitting surface.
11. A method as claimed in claim 10, which comprises performing a numerical simulaton involving a medium between said recording surface and said emitting surface, and wherein said medium between the recording surface and the emitting surface used in said numerical simulation is substantially the same as the real medium in the corresponding part of the earth where said geophysical data were acquired.
12. A method as claimed in any preceding claim, which further comprises using said method to reduce multiples corresponding to waves that bounce between at least two reflectors.
13. A method as claimed in claim 12, wherein said multiples include surface-related multiples corresponding to energy that bounces off the surface of the earth one or more times before being recorded.
14. A method as claimed in claim 12 or 13, wherein said multiples include intrabed multiples corresponding to energy that bounces multiple times between subsurface layers before being recorded.
15. A method as claimed in any preceding claim, which further comprises redatuming.
16. A method as claimed in claim 15, which further comprises imaging.
17. A method as claimed in claim 16, wherein said imaging is performed using reverse time migration.
18. A method as claimed in claim 15, which further comprises inversion.
19. A method as claimed in claim 18, wherein said inversion is performed using full waveform inversion.
20. A method as claimed in any one of claims 15 to 19, wherein said redatuming is carried out by performing auto-focussing.
21. A method as claimed in any preceding claim, which further comprises using said multi-dimensional deconvolution to perform interferometric Green's functions retrieval.
22. A method as claimed in any preceding claim, wherein said geophysical data are processed in common shot gathers.
23. A method as claimed in any one of claims 1 to 21, wherein said geophysical data are processed in common receiver gathers.
24. A method of redatuming a wavefield in a homogeneous medium by means of FD-injection or injecting the wavefield as source terms in the wave equation
25. A method of redatuming a wavefield in an inhomogeneous medium by means of FD-injection or injecting the wavefield as source terms in the wave equation.
26. A method of redatuming receivers to a different depth level using FD-injection for separation of a wavefield into up-going and down-going wavefields, and carrying out redatuming on the up-going and down-going wavefield separately.
27. A method as claimed in claim 26, which comprises using FD-injection to forward propagate said up-going wavefield to a new datum level, and recording said up-goingwavefield at said new datum level.
28. A method as claimed in claim 26 or 27, which further comprises defining a homogeneous medium with water layer properties, and applying absorbing boundary conditions on all sides of said medium.
29. A method as claimed in any one of claims 26 to 28, which further comprises defining a FD-injection surface coinciding with a layer of receivers in acquired data.
30. A method as claimed in claim 29, which further comprises injecting recorded data so that the up-going data radiates up-wards.
31. A method as claimed in any one of claims 26 to 30, which further comprises recording the wavefield at the following depth levels: a) a depth level h0 to which we wish to redatum the wavefield; and b) a depth level -h0 which corresponds to the mirror position of the desired depth level across where the sea surface would have been located;
32. A method as claimed in claim 31, which further comprises recording thewavefield at the following depth levels:a) a depth level -h2 -a which corresponds to the mirror position of a location immediately below the actual recording depth level across where the sea surface would have been located; and b) a depth level h2 + A which corresponds to a location immediately below the actual recording depth level and above the absorbing boundary.
33. A method of performing multiple suppression comprising the steps of:redatuming a wavefield using FD-injection; andusing exact boundary conditions to achieve multiple suppression.
34. An apparatus comprising at least a processor, a memory, and an input device, wherein said apparatus is programmed to carry out the method of any preceding claim.
35. A computer-readable medium containing computer-readable instructions for performing a method as claimed in any of claims ito 33.