Description:

"GEOPHYSICS, VOL. 67, NO. 4 (JULY-AUGUST 2002); P. 1304-1325, 10 FIGS., 1 TABLE. 10.1190/1.1500393 In honor of the new millennium (Y2K), the SEG Research Committee is inviting a series of review articles and tutorials that summarize the state-of-the-art in various areas of exploration geophysics. Further invited contributions will appear during the next year or two - Sven Treitel, Chairman, SEG Research Subcommittee on Y2K Tutorials and Review Articles. Y2K Review Article: Seismic Modeling Jose M. Carcione*, Gerard C. Herman^, and A. P. E. ten Kroode** ABSTRACT: Seismic modeling is one of the cornerstone techniques in geophysical data processing. We give an overview of the most common modeling methods in use today: direct methods, integral-equation methods, and asymptotic methods. We also discuss numerical implementation aspects and present a few representative modeling examples for the different methods. INTRODUCTION: Seismic numerical modeling is a technique for simulating wave propagation in the earth. The objective is to predict the seismogram that a set of sensors would record, given an assumed subsurface structure. This technique is a valuable tool for seismic interpretation and an essential part of seismic inversion algorithms. Another important application of seismic modeling is the evaluation and design of seismic surveys. There are many approaches to seismic modeling. We classify them into three main categories: direct methods, integral-equation methods, and ray-tracing methods. To solve the wave equation by direct methods, the geological model is approximated by a numerical mesh, that is, the model is discretized in a finite number of points. These techniques are also called grid methods and full-wave equation methods, the latter since the solution implicitly gives the full wavefield. Direct methods do not have restrictions on material variability and can be very accurate when a sufficiently fine grid is used. Furthermore, the technique can handle the implementation of different rheologies and is well suited for the generation of snapshots which can be an important aid in the interpretation of results. A disadvantage of these general methods, however, is that they can be more expensive than analytical and ray methods in terms of computer time. Integral-equation methods are based on integral representations of the wavefield in terms of waves originating from point sources. These methods are based on Huygens' principle, formulated by Huygens in 1690 in a rather heuristic way (see Figure 1). When examining Huygens' work closer, we can see that he states that the wavefield can, in some cases, be considered as a superposition of wavefields due to volume point sources and, in other cases, as a superposition of waves due to point sources located on a boundary. Both forms of Huygens' principle are still in use today and we have both volume integral equations and boundary integral equations, each with their own applications. We briefly review both methods. These methods are somewhat more restrictive in their application than the above direct methods. However, for specific geometries such as bounded objects in a homogeneous embedding, boreholes, or geometries containing many small-scale cracks or inclusions, integral-equation methods have shown to be very efficient and give accurate solutions. Due to their somewhat more analytic character, they have also been useful in the derivation of imaging methods based on the Born approximation. For example, see Cohen et al. (1986) and Bleistein et al. (2001) for a description of Born inversion methods. Asymptotic methods or ray-tracing methods are very frequently used in seismic modeling and imaging. These methods are approximate since they do not take the complete wave field into account. On the other hand, they are perhaps the most efficient of the methods discussed in this review. Especially for large, three-dimensional models, the speedup in computer time can be significant. In these methods, the wavefield is considered as an ensemble of certain events, each arriving at a certain time (traveltime) and having a certain amplitude. We discuss some of these methods as well as some of their properties. Asymptotic methods, due to their efficiency, have played a very important role in seismic imaging based on the Born approximation for heterogeneous reference velocity models. Another important application of these methods is the modeling and identification of specific events on seismic records." Manuscript received by the Editor January 16, 2001; revised manuscript received January 22, 2002. *OGS, Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, Italy. E-mail: jcarcione@ogs.trieste.it. ^Delft University of Technology, Department of Applied Analysis, Mekelweg 4, 2628 CD Delft, The Netherlands. E-mail: g.c.herman@math.tudelft.nl. "Shell Research SIEP, Post Office Box 60, 2280 AB Rijswijk, The Netherlands. (c) 2002 Society of Exploration Geophysicists. All rights reserved. Seismic Modeling In this overview, we discuss the above methods in some detail and give the appropriate references. In order not to get lost in tedious notations, we discuss only the acoustic formulation so we can concentrate on the differences between methods. Pressure formulation: The pressure formulation for heterogeneous media can be written as (Akian and Richards, 1980, 775) All methods discussed here have been generalized to the elastic case and, where appropriate, references are given. We also present a few examples illustrating the applicability of the different methods. Since the applicability regions of the methods are rather different and, largely, nonoverlapping, the models are different but typical for each technique. Hopefully, they will provide the reader with some idea of the types of problems or geometries to which a particular modeling method is best suited. The modeling methods discussed here have in common that they are applicable to various two- and three-dimensional geometries. This implies that we do not discuss methods especially suited for plane-layered media, despite the fact that these methods are certainly at least as important and often used as those we discuss here. For an excellent overview of these plane wave summation (or slowness) methods, see Ursin (1983). DIRECT METHODS: where V is the gradient operator, p is the pressure, c is the compressional-wave velocity, ρ is the density, and f is the body force. This is a second-order partial differential equation in the time variable. Velocity-stress formulation: Instead of using the wave equation, wave propagation can also be formulated in terms of a system of first-order differential equations in the time and space variables. Consider, for instance, propagation of SH-waves. This is a two-dimensional phenomenon with the particle velocity, say v2, perpendicular to the plane of propagation. Newton's second law and Hooke's law yield the velocity-stress formulation of the SH-wave equation (Akian and Richards, 1980, 780): We consider finite-difference (FD), pseudospectral (PS), and finite-element (FE) methods. These methods require the discretization of the space and time variables. Let us denote them by (x, t) = (jdx, n dt), where dx and dt are the grid spacing and time step, respectively. We first introduce the approximate mathematical formulations of the equation of motion, and then consider the main aspects of the modeling as follows: (1) time integration, (2) calculation of spatial derivatives, (3) source implementation, (4) boundary conditions, and (5) absorbing boundaries. All these aspects are discussed using the acoustic and SH-wave equations. Mathematical formulations: For simplicity, we consider the acoustic and SH-wave equations which describe propagation of compressional and pure shear waves, respectively. and ω denote stress and μ is the shear modulus. The solution to equation (2) subject to the initial condition v(0) = v0 is formally given by v(t) = exp(t H)v0 + f exp(r H) F(t - T)dT, t where exp(t H) is called the evolution operator because application of this operator to the initial condition vector (or to the source vector) yields the solution at time." Ключевые слова: numerical modeling, coefficient, eikonal equation, spectral accuracy, ray tracing, spatial derivative, geophys, object, computational aspect, large variation, ray-tracing method, coordinate transformation, calculation, seismic modeling, problem, plane-wave kernel, original french, differential formulation, approximation, review, satisfied explicitly, function, poisson ratio, gabor wavelet, liu, point, accuracy, acoust soc, velocity function, mapping transformation, initial condition, finite-difference, discus, wave equation, xk, kosloff, constant, nodal pressure, traveltime function, time, parasitic mode, poissons ratio, eigenvalue, finite difference, heterogeneous medium, numerical solution, fourier, grid, evolution operator, space, scheme, nyquist wavenumber, numerical method, bounded extent, solved, boundary, embedding medium, da, large compared, crack, absorbing strip, wavefront, expl, element, vector, field, fd, travel, acoustic, huygens, integral, pressure formulation, seismic wave, born approximation, absorbing boundary, result, interpolation, matrix, take-off angle, modeling, phase velocity, physical, caustic, seismic source, relation, smooth function, relaxation time, seism, eikonal, based, differential equation, wave propagation, chebyshev method, central frequency, ray, formulation, ray coverage, anisotropic medium, case, large, asymptotic, numerical, strain component, velocity, elastic formulation, finiteelement method, direct method, form, finite-element method, spatial, taylor series, location, method, derivative, geophysics, space spanned, ax, frequency, large negative, synthetic seismogram, greens function, discrete delta, explicit, interval, equation, heterogeneous formulation, staggered grid, representation, medium, dt, step, elastic, finite, material property, high, stationary phase, traveltimes, differential, slowness vector, lumiere, directional force, incident wave, wavefront construction, configuration space, math, source, term, seismic, one-sided difference, seismic exploration, queiroz filho, huygens principle, boundary condition, variable, order, numerical mesh, integral equation, solution, operator, regular grid, specific geometry, dx, ha, phase, domain, sponge method, soc, pressure, wave, interface, maximum wavenumber, propagating wavefront, accurate, efficient, dynamic elasticity, appl, discussed, time history, internat, amplitude, wavefield, chebyshev operator, integral-equation, volume, algorithm, velocity-stress formulation, general, taylor expansion, carcione, chebyshev polynomial, robertsson, sampling point, arrival, model, surface, numerical dispersion, application, prosp, lagrangian interpolants, heaviside function, technique, stationarity condition, number, elastic wave, traveltime, chebyshev, reflectedrefracted ray, difference, property, condition, propagation, instance, mathematical formulation, grid point, phys