Johan O. A. Robertsson

Viscoelastic finite‐difference modeling

Real earth media disperse and attenuate propagating mechanical waves. This anelastic behavior can be described well by a viscoelastic model. We have developed a finite‐difference simulator to model wave propagation in viscoelastic media. The finite‐difference method was chosen in favor of other methods for several reasons. Finite‐difference codes are more portable than, for example, pseudospectral codes. Moreover, finite‐difference schemes provide a convenient environment in which to define complicated boundaries. A staggered scheme of second‐order accuracy in time and fourth‐order accuracy in space appears to be optimally efficient. Because of intrinsic dispersion, no fixed grid points per wavelength rule can be given; instead, we present tables, which enable a choice of grid parameters for a given level of accuracy. Since the scheme models energy absorption, natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary. The viscoelastic scheme is ...

Modeling of a constant Q: Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique

Linear anelastic phenomena in wave propagation problems can be well modeled through a viscoelastic mechanical model consisting of standard linear solids. In this paper we present a method for modeling of constant Q as a function of frequency based on an explicit closed formula for calculation of the parameter fields. Several standard linear solids connected in parallel can be tuned through a single parameter to yield an excellent constant Q approximation. The proposed method enables substantial savings in computations and memory requirements. Experiments show that the new method also yields higher accuracy in the modeling of Q than, e.g., the Pade approximant method.

A numerical free-surface condition for elastic/viscoelastic finite-difference modeling in the presence of topography

An accurate free-surface boundary condition is important for solving a wide variety of seismic modeling problems. In particular, for earthquake site studies or shallow environmental investigations the surface of the earth may have a significant impact on the outcome of simulations. Computations based on several elastic/viscoelastic flat horizontal free-surface conditions are compared and benchmarked against an analytical solution. An accurate and simple condition is found and then generalized to allow for irregular free surfaces. This new method is simple to implement in conventional staggered finite-difference schemes, is computationally efficient and enables modeling of highly irregular topography. The accuracy of the method is investigated and criteria for sampling of the wavefield are derived.

On the use of multicomponent streamer recordings for reconstruction of pressure wavefields in the crossline direction

Three-component measurements of particle motion would bring significant benefits to towed-marine seismic data if processed in conjunction with the pressure data. We show that particle velocity measurements can increase the effective Nyquist wavenumber by a factor of two or three, depending on how they are used. A true multicomponent streamer would enable accurate data reconstruction in the crossline direction with cable separations for which pressure-only data would be irrecoverably aliased. We also show that conventional workflows aimed at reducing these aliasing effects, such as moveout correction applied before interpolation, are compatible with multicomponent measurements. Some benefits of velocity measurements for deghosting data are well known. We outline how the new measurements might be used to address some long-standing deghosting challenges of particular interest. Specifically, we propose methods for recovering de-ghosted data between streamers and for 3D deghosting of seismic data at the stream...

Interferometric modeling of wave propagation in inhomogeneous elastic media using time reversal and reciprocity

Time reversal of arbitrary, elastodynamic wavefields in partially open media can be achieved by measuring the wavefield on a surface surrounding the medium and applying the time reverse of those measurements as a boundary condition. We use a representation theorem to derive an expression for the time-reversed wavefield at arbitrary points in the interior. When this expression is used to compute, in a second point, the time-reversed wavefield originating from a point source, the time-reversed Green’s function between the two points is observed. By invoking reciprocity, we obtain an expression that is suitable for modeling of wave propagation through the medium. From this we develop an efficient and flexible two-stage modeling scheme. In the initial phase, the model is illuminated systematically from a surface surrounding the medium using a sequence of conventional forward-modeling runs. Full waveforms are stored for as many points in the interior as possible. In the second phase, Green’s functions between ...

Finite-difference modeling of electromagnetic wave propagation in dispersive and attenuating media

Realistic modeling of electromagnetic wave propagation in the radar frequency band requires a full solution of Maxwell’s equations as well as an adequate description of the material properties. We present a finite‐difference time‐domain (FDTD) solution of Maxwell’s equations that allows accounting for the frequency dependence of the dielectric permittivity and electrical conductivity typical of many near‐surface materials. This algorithm is second‐order accurate in time and fourth‐order accurate in space, conditionally stable, and computationally only marginally more expensive than its standard equivalent without frequency‐dependent material properties. Empirical rules on spatial wavefield sampling are derived through systematic investigations of the influence of various parameter combinations on the numerical dispersion curves. Since this algorithm intrinsically models energy absorption, efficient absorbing boundaries are implemented by surrounding the computational domain by a thin (⩽2 dominant waveleng...

Finite-difference modeling of wave propagation in a fluid-solid configuration

Finite-difference (FD) techniques are widely used to model wave propagation through complex structures. Two main sources of error can be identified: (1) from numerical dispersion and numerical anisotropy and (2) by modeling the response of internal grid boundaries. Conventional discretization criteria to reduce the effects of numerical dispersion and numerical anisotropy have long been established (5-8 gridpoints per wavelength for a fourth-order accurate FD scheme). We analyze the second source of errors, comparing different staggered-grid FD solutions to the Cagniard-de Hoop solution in models with fluid-solid contacts. Our results confirm that it is sufficient to rely on conventional discretization criteria if the fluid-solid interface is aligned with the grid. If accurate modeling of the Scholte wave is required, then a new imaging method we propose should be used to allow for conventional sampling of the wavefield to minimize numerical dispersion. However, for an interface not aligned with the grid (irregular interfaces), a spatial sampling of at least 15 gridpoints per minimum wavelength is required to obtain acceptable results, particularly in seismic seabed applications where Scholte waves may need to be modeled more accurately.

Interferometric surface-wave isolation and removal

Theremovalofsurfacewavesgroundrollfromlandseismicdataiscriticalinseismicprocessingbecausethesewaves tend to mask informative body-wave arrivals. Removal becomes difficult when surface waves are scattered, and data qualityisoftenimpaired.Weapplyamethodofseismicinterferometry, using both sources and receivers at the surface, to estimatethesurface-wavecomponentoftheGreen’sfunction betweenanytwopoints.Theseestimatesaresubtractedadaptively from seismic survey data, providing a new method of ground-roll removal that is not limited to nonscattering regions.

An efficient method for calculating finite-difference seismograms after model alterations

Seismic modeling, processing, and inversion often require the calculation of the seismic response resulting from a suite of closely related seismic models. Even though changes to the model may be restricted to a small subvolume, we need to perform simulations for the full model. We present a new finite‐difference method that circumvents the need to resimulate the complete model for local changes. By requiring only calculations in the subvolume and its neighborhood, our method makes possible significant reductions in computational cost and memory requirements. In general, each source/receiver location requires one full simulation on the complete model. Following these pre‐computations, recalculation of the altered wavefield can be limited to the region around the subvolume and its neighborhood. We apply our method to a 2-D time‐lapse seismic problem, thereby achieving a factor of 15 reduction in computational cost. Potential savings for 3-D are far greater.

Rough-sea deghosting using a single streamer and a pressure gradient approximation

We present a method for receiver ghost correction of towed streamer data that accounts for the rough sea surface. The method explicitly uses the fact that the pressure is zero at the free (sea) surface to estimate the vertical pressure gradient. Continuous elevation measurements of the wave height directly above the hydrophones are required—a measurement which is currently unavailable. The new deghosting method is fundamentally limited to frequencies below the first ghost notch. The lowest‐order implementation requires that the streamer is towed no deeper than approximately 6 m and a receiver spatial sampling interval of about 3 m or less.Using the lowest‐order and simplest implementation of the new method, the rough‐sea error is reduced from 1.5–2.5 dB to about 1–1.5 dB in amplitude and from 20° to 10° in phase, at 50 Hz in a 4‐m significant wave height sea. Higher‐order terms in the approximation promise to further reduce the error.