Modelling of scattering of seismic waves from a corrugated rough sea surface: a comparison of three methods

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Modelling of scattering of seismic waves from a

corrugated rough sea surface: a comparison of three

methods

Johan O. A. Robertsson, Laws Robert, Chris Chapman, Jean-Pierre Vilotte,

Elise Delavaud

To cite this version:

Johan O. A. Robertsson, Laws Robert, Chris Chapman, Jean-Pierre Vilotte, Elise Delavaud.

Mod-elling of scattering of seismic waves from a corrugated rough sea surface: a comparison of three

methods. Geophysical Journal International, Oxford University Press (OUP), 2006, 167 (1), pp.70-76.

�10.1111/j.1365-246X.2006.03115.x�. �hal-00270819�

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Geophys. J. Int. (2006) 167, 70–76 doi: 10.1111/j.1365-246X.2006.03115.x

GJI

Ma

rine

geoscience

Modelling of scattering of seismic waves from a corrugated rough sea

surface: a comparison of three methods

Johan O. A. Robertsson,

1

Robert Laws,

2

Chris Chapman,

2

Jean-Pierre Vilotte

3

and Elise Delavaud

3

1_{WesternGeco Oslo Technology Centre, Schlumberger House, Solbr˚aveien 23, 1383 Asker, Norway. E-mail: [email protected]} 2_{Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, UK}

3_{Institut de Physique du Globe de Paris, 4, Place Jussieu, 75252 Paris cedex 05, France}

Accepted 2006 June 22. Received 2006 June 21; in original form 2005 December 1

S U M M A R Y

We compare three numerical methods to model the sea surface interaction in a marine seismic reflection experiment (the frequencies considered are in the band 10–100 Hz): the finite-difference method (FDM), the spectral element method (SEM) and the Kirchhoff method (KM). A plane wave is incident at angles of 0◦and 30◦with respect to the vertical on a rough Pierson–Moskowitz surface with 2 m significant wave height and the response is synthesized at 6, 10 and 50 m below the average height of the sea surface. All three methods display an excellent agreement for the main reflected arrival. The FDM and SEM also agree very well all through the scattered coda. The KM shows some discrepancies, particularly in terms of amplitudes.

Key words: finite-difference methods, numerical techniques, rough surface, seismic

mod-elling, seismic wave propagation, synthetic seismograms.

1 I N T R O D U C T I O N

In marine seismic imaging, the reflection response of the rough sea surface can be a significant source of error. In particular, in marine reservoir monitoring applications, such as time-lapse (or 4D), a repeat seismic survey is conducted over a reservoir following a time interval where production of hydrocarbons has occurred. The goal of the repeat survey is to map changes in the reservoir due to the production of oil or gas. The magnitude of these changes can be very weak and in order to resolve them we have to address sources of noise that are as low as 40 dB below the primary reflections from the reservoir (Laws & Kragh 2002). It is in this context that rough sea surface scattering becomes relevant for reflection seismologists. As the primary reflection from the reservoir interacts with the rough sea surface and reflects downwards again (recordings are typically made 5–8 m below the sea surface), the sea surface leaves an imprint that manifests itself in terms of an amplitude perturbation, a phase perturbation as well as coda following the main reflection. All these effects are extensively discussed in Laws & Kragh (2002).

In order to study the effect of rough sea surface perturbations in seismic data, it is necessary to simulate such sea surface shapes and to compute their reflection responses. There are three features that distinguish the reflection seismic rough sea surface problem from the widely studied problem of estimating the average spectral scattering of a wave reflecting at near-grazing incidence and incident from a large distance; see, for example, Thorsos (1988), Stephen (1996) and Hastings et al. (1995, 1997). In our situation, we are dealing with a near-vertical upcoming plane wave, with the receiver close to

the surface and we are interested in the actual shape of the reflection response, not simply its average power spectrum. We are interested in the frequencies from a few Hertz up to about 100 Hz. By invoking the principle of reciprocity, we can calculate equivalently either the response of a receiver near the surface to an upcoming plane wave, or the radiated far field of a source near the surface. The choice is one of convenience for the different methods.

In this paper, we compare the ability of three numerical simulation methods to synthesize the response due to a plane wave (frequency content of 10–100 Hz) incident on a ‘corrugated’ (varying only in one direction) rough sea surface and being recorded at depths close to the sea surface: 6, 10 and 50 m. The plane wave propagation direction lies within the plane in which the sea surface varies; the problem is invariant under translation perpendicular to this plane. The three methods are: the Kirchhoff method (KM), the spectral element method (SEM) and the finite-difference method (FDM). Whereas the former two naturally incorporate surface topography, a general, accurate and efficient technique to model free-surface to-pography in FDMs does not exist (see review by Moczo et al. 2006). Here, we use a relatively robust technique proposed by Robertsson (1996) to model a free surface with topography in a staggered FD grid (Levander 1988).

2 G E N E R AT I N G T H E S E A S U R FA C E P R O F I L E

The sea surface model used in this comparison was generated us-ing the method described by Thorsos (1988). An average surface

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Modelling of rough sea surface scattering 71

Figure 1. Profile of the corrugated sea surface used in this study. The profile was extracted from a Pierson–Moskowitz synthesized sea surface with 2 m SWH

(Pierson & Moskowitz 1964).

power spectrum is used of the form derived by Pierson & Moskowitz (1964). The derivation of this spectrum is also discussed in, for example, Kinsman (1983) and Carter et al. (1986). It is a simple spectrum describing a fully developed wind-driven sea where the fetch (the distance over which the wind has blown) is infinite. This average spectrum is used to generate a typical surface profile by multiplying it by an array of Gaussian random complex numbers. The method can be extended to create 3-D surfaces as was done in Laws & Kragh (2002). However, in the current comparison, we are using a surface that varies only in one dimension. The signifi-cant wave height (SWH) was set to 2 m (a moderately rough sea as marine reflection seismic data typically are acquired in seas up to 4 m SWH) and the resulting realization of the surface is shown in Fig. 1. The SWH is typically defined (for example, by Carter et al. 1986) to be four times the rms wave amplitude. It corresponds to the subjectively assessed peak-to-peak wave height.

3 T H E K I R C H H O F F M E T H O D

The Kirchhoff approximation is described in general terms in, for example, Clay & Medwin (1977). The application to near-vertical scattering of seismic signals from a rough sea surface has been discussed by Laws & Kragh (2002) and we use their eq. (8). This equation is symmetrical with respect to source and receiver. We are dealing with the scattering of a plane wave approaching the surface at close-to-normal incidence, so issues of multiple scattering and shadowing are not expected to be a problem. In our numerical code, the surface is 3-D and the receiver is a point. The corrugated surface is created simply by extending the slice shown in Fig. 1 laterally without modification so that the results of the 3-D KM can be directly compared with the 2-D approach of the other two methods. The source is at infinite distance away and is represented by upcoming plane waves. In terms of Laws & Kragh’s (2002) eq. (8), R2 is allowed to tend to infinity.

The integral is performed using elementary areas that are trian-gles; each pair of triangles forms a square of side 0.5 m. Numerical experiments were performed to check that there would be no ad-vantage in reducing the grid size further. The faceted surface is continuous but its gradient is not. The use of such a fine grid is not strictly necessary when the shortest wavelength is 15 m, but it allows the use of a simple integration scheme for the evaluation of the integral given in Laws & Kragh’s (2002) eq. (8). The ele-mentary surface normal, dS, is well defined for each triangle of the surface. The function to be integrated is calculated at the position of the centroid of the triangle. The time differentiation of the pulse is performed by a simple divided difference operator using a sample interval of 1/16 ms. The integration surface is extended sufficiently far that at no times of interest are affected by edge effects.

The surface is invariant in one direction and there is no component of the upcoming plane wave in that invariant direction. Thus, the whole 3-D problem is invariant with respect to translation in the invariant direction. Therefore, there are no modifications needed to the results so that they can be compared with the 2-D codes. The only question is one of scaling. This is resolved by calculating the direct arrival by reflecting from a planar surface with a reflection coefficient of unity.

The Kirchhoff approximation can be used for studying acoustic reflections from a free surface, the problem considered here. For re-flections from a general interface and for elastic waves, the angular dependence of the reflection coefficients introduces difficulties con-cerning reciprocity and the dependence of results on details of the surface geometry. For acoustic free-surface reflections, the results are reciprocal and robust to surface details. As this does not appear to have been widely discussed in the literature (see Chapman 2004, exercise 10.6), we outline the argument here.

The robustness of the results to surface details is important as for numerical calculations the surface must be approximated by some basis functions. As mentioned above, we have used triangular planar elements. In its basic form, the Kirchhoff surface integral can C

2006 Schlumberger Cambridge Research, GJI, 167, 70–76

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72 J. O. A. Robertsson et al.

be written as (Clay & Medwin 1977, appendix A10):

U= S U1 ∂U2 ∂n − U2 ∂U1 ∂n d S (1)

(for simplicity, we have followed the notation of Clay & Medwin 1977, where further details can be found). In this expression, U1is taken as the reflected wavefield (Clay & Medwin 1977, eq. A10.2.1):

U1= RUs, (2)

where Usis the incident wavefield and R the reflection coefficient,

and U2 is Green’s function between the receiver and the surface. Details need not concern us here (Laws & Kragh 2002, have given details of the final expression evaluated in their eq. 8), except that in the reflected wavefield the reflection coefficient is R= −1, that is, a constant, independent of the incident wavefield direction and the surface normal. Suppose we have two numerical representations of the surface, SA and SB(for the sake of argument, we can take the

boundaries of the two representations as coincident). Using Green’s theorem, the difference in the Kirchhoff integrals for the two sur-faces can be written as

UA− UB = − SA+SB U1 ∂U2 ∂n − U2 ∂U1 ∂n d S = VAB U1∇2_U 2− U2∇2U1 d V (3)

[VABis the (signed) volume between the surfaces SA and SB; the

sign of the volume integral depends on the relative positions of the two surfaces and the sign convention for the surface normal, n]. The wavefields, U1, Us and U2vary smoothly and if the two surfaces

SAand SBare close together, the volume integral will be small. In

practical terms, we require that the separation of the two surfaces is small compared with the wavelength of interest, but there is no requirement on the detailed shape of the interface. In other words, a smooth interface can be approximated by a staircase or linear facets and provided the elements are small enough, numerically robust re-sults will be obtained. The numerical rere-sults depend on the position of the interface but not its derivatives. This contrasts with the ar-gument when the reflection coefficient depends on the ray direction and surface normal. The wavefield U1and the surface integrals then will depend on derivatives of the surface. Green’s theorem cannot be used to reduce the differences to a small volume integral. The Kirchhoff surface integral will be non-robust, depending on small numerical differences in the surface representation. This, together with the breakdown of reciprocity, are unsolved issues in the use of the Kirchhoff integral method for general reflections. For the subject of this research note, fortunately, they need not concern us.

4 T H E F I N I T E - D I F F E R E N C E M E T H O D The FDM that we use is a staggered stress–velocity formulation of the first-order partial-differential equations describing viscoelas-tic wave propagation in two space dimensions; it is appropriate to model the response from a line source under a corrugated sea sur-face. It is fourth-order accurate in space and second-order accurate in time (Robertsson et al. 1994). In principle, for this study, a much simpler acoustic scheme will be sufficient and will produce equiva-lent results. However, the viscoelastic properties allow for a simple ‘sponge-type’ absorbing boundary condition to be implemented by gradually increasing the attenuation along the edges of the compu-tational domain as described by Robertsson et al. (1994). As will be evident from below, this study confirms that there are no inherent

problems with modelling acoustic wave propagation using an elastic or viscoelastic scheme.

The stress-imaging method for modelling a flat free-surface FDM boundary condition was introduced by Levander (1988). The stress-imaging technique applies explicit boundary conditions to the stress–tensor component(s) located at the grid plane coinciding with the free surface, and uses imaged values of the stress–tensor com-ponents above the free surface, assuming their antisymmetry about the free surface to ensure that the free-surface boundary conditions are satisfied.

To model the rough sea surface reflection, we use the viscoelas-tic technique for rough-surface topography by Robertsson (1996). The technique implicitly discretizes the rough sea surface such that it fits a staircase-shaped surface on the FD grid. The technique is relatively straightforward to implement (at least in two space di-mensions) and is capable of modelling also an irregular acoustic free-surface boundary condition as a special case of the more gen-eral viscoelastic case. The method by Robertsson (1996) can be viewed as a generalization of the stress-imaging method of Levan-der (1988) with one important modification. Instead of updating the particle velocities in the vicinity of the free surface such that the free-surface condition is explicitly satisfied by using second-order accurate difference approximations, the particle velocities are sim-ply set to zero above the free surface. Stresses, on the other hand, are imaged such that tractions perpendicular to the free surface always are zero at the free surface.

The most important concepts of the method by Robertsson (1996) can be summarized as follows. Each of the equations for the particle velocities consists of two derivatives (one vertical and one horizon-tal) of the stress–tensor components. By first performing all vertical imaging of the stress–tensor components to ensure that the appropri-ate stress components vanish at the free surface and then calculating and adding only the vertical derivatives in the update of the particle velocities, the free-surface condition is satisfied completely in all grid-cells along the boundary. Next, the horizontal imaging of the stress components is carried out followed by an update with the re-maining horizontal derivatives in the equations, again satisfying the free-surface boundary condition. As a result the free-surface condi-tion is satisfied everywhere along the free surface during the update of particle velocities.

Although the technique by Robertsson (1996) offers a robust method to model a rough sea surface, it suffers from lower accuracy compared to the interior of the FD method. Roughly, 15–20 grid-points per minimum wavelength are needed to model wave prop-agation in the vicinity of the free-surface topography compared to 5–8 grid-points per minimum wavelength for the interior of the FD grid (Levander 1988). To avoid having to pay this price in terms of computational cost on the entire grid, Robertsson & Holliger (1997) used a grid-refinement technique that uses a three times finer grid only in the close vicinity of the free surface, which also is used in this study. Careful convergence tests were carried out to ensure that no numerical artefacts were introduced by using the grid-refinement scheme in this context.

The plane waves impinging on the sea surface in this study were introduced inside the FD grid as described in Robertsson et al. (1996). An analytical solution for a plane wave was used. This was tapered towards the edges to avoid edge effects. In addition, the spatial extent of the FD grid was sufficiently large for residual edge diffractions caused by the truncation of the plane wave not to propagate to the recording locations within the time window of in-terest. Since the distance of propagation within the FD grid is small and the propagating wavefield in the water column is significantly

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oversampled in order to ensure an accurate representation of the rough sea surface, errors due to numerical dispersion are negligible (this was confirmed through the convergence tests).

5 T H E S P E C T R A L E L E M E N T M E T H O D SEM is a high-order variational method combining the flexibility of finite-element methods with the accuracy of spectral methods. The theoretical background and application to elastodynamics are detailed in Komatitsch & Vilotte (1998) and Chaljub et al. (2003). Hereafter, the formulation of the SEM for acoustic wave propagation is briefly outlined within the context of rough sea surface reflection.

5.1 Acoustic wave propagation

The SEM formulation in this study is based on a potential formula-tion of acoustic wave propagaformula-tion (Komatitsch et al. 2000; Chaljub

et al. 2003). The irrotational velocity fieldv is expressed as the

gradient of a scalar potentialφ, that is, v = −∇ φ. The governing scalar wave equation yields

1

c2φ =¨ 1

ρ∇ · (∇φ), (4)

together with the initial conditions,

φ(x, 0) = φ0(x); φ(x, 0) = ˙φ˙ 0(x), (5) where c denotes the acoustic wave speed andρ the density. The pressure p can be retrieved as p= ρ ˙φ. A dot over a symbol indicates partial differentiation with respect to time.

On the rough free surface denoted by∂, pressure should be zero and therefore the boundary condition is given by

˙

φ(x, t) = 0 ∀¯x ∈ ∂. (6)

The unbounded physical acoustic domain is mapped to a bounded computational domain, ⊂ 2_{, surrounded, apart from the upper} free surface, by perfectly matched layers (PMLs). Implementation of PMLs within the SEM is detailed in Festa & Vilotte (2005) and Festa et al. (2005). The boundary of the computational domain∂ can therefore be decomposed into∂ = ∂ ∪ ∂pml, where∂pml denotes the external boundary of the PML regions along which homogeneous Dirichlet boundary conditions are assumed.

5.2 Variational formulation

In contrast with FDM, the SEM is based on a variational formula-tion of the scalar wave eq. (4) (see e.g. Chaljub et al. 2003). The variational formulation can be stated as searching for the couple (φ, ¨φ), such that ∀ψ: ρ c2ψ ¨φd = − ρ∇ψ∇φd, (7)

whereψ denotes admissible potential variations, and both φ and

ψ are assumed to be square integrable scalar fields, together with

their first-order spatial derivatives, defined on and zero on the boundary.

5.3 Diffracted field decomposition

For the scattering of an incident plane wave, the potential field may be decomposed into an incident and a diffracted field, for example,

φ = φi_{+ φ}d _where_φi_{denotes the incident potential field, known}

analytically, andφd_{the diffracted potential field that must be} com-puted. The problem can be readily rewritten in terms of the diffracted potential field as ρ c2ψ ¨φ d_{d = −} ρ∇ψ∇φ d_d, ₍₈₎

where now ˙φd_{= − ˙φ}i_on_{∂ and φ}d_{= 0 on ∂pml. This allows} effi-cient computation of diffracted fields for various angles of incidence of the incoming plane wave, using the PML formulation.

5.4 SEM discretization

Conforming SEM approximation is detailed in Komatitsch & Vilotte (1998). First, the geometrical approximation is based on the decom-position of the domain into non-overlapping elements e, quad-rangles in two space dimensions (Komatitsch & Vilotte 1998). Each of these elements is individually mapped to a reference element K= [−1, 1]2_{, using a smooth invertible mapping Fe} _{: K}_{→ e.} Secondly, inside the reference element K, the solution is expanded on to a discrete basis constructed as the tensor-product of the Lagrange polynomial basis associated with 1-D Gauss–Lobatto– Legendre (GLL) quadrature nodes. The discrete inner products are constructed using numerical integration based on the tensor-product of the GLL 1-D quadrature in the reference element K. At the end, the variational form (8) leads to the following algebraic system of ordinary differential equations:

M ¨φd_{= −F}int₍_φd₎ _in ˙

φd _{= − ˙φ}i _on _∂

φd _{= 0 on ∂pml}_, ₍₉₎

where M is a diagonal mass matrix and Fint₍_φd_{) denotes the vector} containing the internal forces at the global nodes.

The time evolution of the system is then discretized using a second-order explicit scheme, implemented in a staggered way with respect to the pressure (Festa & Vilotte 2005):

M ˙φd n+1 − M ˙φdn= −tFint φd n+1 2 φd n+1 2 − φ d n−1 2 = t ˙φ d n ˙ φd n+1 = − ˙φni+1 on (10) 5.5 Numerical simulations

The computational domain is meshed into quadrangles defined with respect to the reference element by a local quadratic geometrical mapping. The surface is therefore approximated by a piecewise C0 second-order polynomial interpolation.

To ensure an accurate approximation of the rough surface, the elements have a characteristic size of 2.5 m. A polynomial order of 4 has been used in all the numerical simulations leading to at least five points per minimal wavelength and a CFL condition (Courant

et al. 1928) of 0.2, corresponding to a time discretizationt =

58.5μs.

5.6 Numerical tests

Synthetic responses were generated with all three methods, using a corrugated Pierson–Moskowitz surface of 2 m SWH as described above. The response due to a plane wave consisting of a Ricker C

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Figure 2. Rough sea surface (2 m SWH) reflected response for a plane wave incident vertically from below and received by a hydrophone placed at 6 m depth

(top row), 10 m depth (middle row) and 50 m depth (bottom row). The receiver is directly below the 0 m lateral position shown in Fig. 1. The right-hand column is an enlargement of the scattered coda in the plots in the left-hand column. The black lines: FDM response. The grey lines: KM response. The dashed lines: SEM response.

wavelet (second-derivative of a Gaussian pulse) with 50 Hz centre frequency (frequency band of roughly 10–100 Hz) incident under different angles on the rough sea surface was simulated.

First, we carried out careful convergence tests to ensure that de-tails of the reflected and scattered response were not contaminated by numerical artefacts in the respective method. The FDM deter-mined the spatial discretization of the sea surface used for all three methods, since it required the densest sampled sea surface.

Fig. 2 shows the reflected response for a plane wave incident vertically from below and recorded at 6, 10 and 50 m average depth below the rough sea surface. To the left-hand side, we have plotted

the response from the three methods on top of each other on a scale such that the main peak of the reflected arrival remains unclipped. The main reflected event as simulated by the three methods is very similar between the three methods: the error energy between any of the methods compared to the total recorded signal ranges from 50 ppm (parts per million) for SEM and FDM at 10 m depth to 1250 ppm for KM and SEM at 50 m depth. On this scale, we see that the amplitude perturbations and coda following the main reflection accumulate with depth as the sea surface acts as concave/convex mirrors with focal depths of the order of the radius of curvature of the sea surface. The rough sea surface also introduces a time shift

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Figure 3. Rough sea surface (2 m SWH) reflected response for a plane wave incident at an angle of 30◦to the vertical and received by a hydrophone placed at 6 m depth. The plot on the right-hand side is an enlargement of the scattered coda in the plot on the left-hand side. The black lines: FDM response. The grey lines: KM response. The dashed lines: SEM response. The traces are shorter than those of Fig. 2, because scattering from the edge of the sea surface limits the maximum time that can be computed.

of the main reflection which is greatest close to the sea surface and gradually reduces with depth below it.

The column to the right-hand side in Fig. 2 is a zoom-in on the plots in the column on the left-hand side. Note that the vertical scale on these plots is of the order of 10–100 times smaller compared to the scale in the left-hand column in order to be able to resolve the coda. We note that there is a remarkable fit between the FDM and the SEM methods all the way through the coda (the error energy between SEM and FDM compared to the total recorded signal is 2 ppm at 6 m, 7 ppm at 10 m and 87 ppm at 50 m). The shape and arrival time of the events in the coda also agree very well. However, the amplitudes of the individual events in the coda are lower for the KM (the error energy between KM and FDM compared to the total recorded signal is 32 ppm at 6 m, 64 ppm at 10 m and 476 ppm at 50 m).

Fig. 3 shows the reflected response for a plane wave incident with 30◦angle with respect to the vertical recorded at 6 m average depth below the rough sea surface. Again, the plot on the right-hand side is a zoom-in on the coda in the plot on the left-hand side. There is a very close match between the FDM and SEM also for the plane wave incident with 30◦angle with respect to the vertical (the error energy between SEM and FDM compared to the total recorded signal is 3 ppm). The results for the KM match those of the two other methods very well for the main reflected event but are more dissimilar to those of the two other methods in the coda than what was the case of the vertically incident plane wave (the error energy between KM and both FDM and SEM compared to the total recorded signal is about 100 ppm).

6 D I S C U S S I O N

A number of studies in the ocean acoustic literature have previously assessed the accuracy of different FDM and KM for modelling scat-tering from a rough sea surface (e.g. Hastings et al. 1995; Stephen 1996; Hastings et al. 1997; Schneider et al. 1998; Hastings et al. 2001). Mostly, waves incident under low grazing angles have been studied (Stephen 1996; Hastings et al. 1997, 2001). Whereas this situation is of great interest for a large range of ocean acoustic prob-lems, it is less relevant to the rough sea surface scattering problem as encountered in reflection seismic experiments where waves are mostly incident close to the vertical. Although Stephen (1996) and

Hastings et al. (1995, 1997) demonstrated a good agreement be-tween their FD solutions and a KM in terms of average power spec-tra, their conclusions do not translate directly to reflection seismic applications with incident angles close to the vertical, recordings near the sea surface, broad-band waves and the detailed response of amplitude and phase for both the direct reflection and the coda must be analysed separately. These aspects are critical to understand the impact of a rough sea surface on, for instance, time-lapse seismic recordings (Laws & Kragh 2002).

7 C O N C L U S I O N S

In this paper, we have assessed the ability of three different meth-ods to synthesize the effects that a rough sea surface will have on recorded reflection seismic data (in the frequency band 10–100 Hz). The three methods are fundamentally dissimilar from each other. The FDM which is popular in the exploration seismics community for modelling wave propagation in complex earth models is based on an explicit discretization in space and time of the wave equation. The SEM has become established as one of the most efficient and accurate methods to model complex structures on a global scale in, for instance, earthquake applications and is based on a variational formulation of the wave equation. Finally, the KM, which is widely used to model surface scattering (see, for example, Clay & Medwin, 1977), is derived by considering plane wave reflection coefficients and ignores many phenomena which inherently are accounted for in the FDM and SEM methods (multiple scattering, shadowing, etc.). However, the computational efficiency of the KM is superior to that of the FDM and SEM methods and it can be used with 3-D surfaces without excessive computing power requirements.

We conclude that the FDM and SEM produce very similar results for various configurations (different incident angles of the plane wave with respect to the sea surface and recording depths). We were very pleased to see an extremely good match between the two methods also all the way through the coda which consisted of amplitudes that were as low as 500 times smaller compared to the main reflection. This suggests that these methods are giving the correct answer.

The KM produced fairly similar results, particularly for inci-dent/scattering angles close to the vertical. However, there are more differences between KM and the other two than between the other C

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two. The difference of the KM is mainly in amplitude rather than in phase.

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