Abstract “Modeling of seismic wave propagation is a core component in almost every aspect of exploration seismology, ranging from survey design methods to imaging and inversion algorithms. The last time SEG published a reprint volume on numerical modeling was in 1990. Since then, the last two decades has seen a step change in the application and use of Â"full wave equationÂ" modeling methods enabled by the tremendous increase in available computational power. Full waveform inversion, reverse time migration and 3D elastic finitedifference synthetic data generation are examples of modeling applications that are currently having a fundamental impact on our business. In Numerical Modeling of Seismic Wave Propagation: Gridded Two-way Wave-equation Methods, readers will find many of the wellknown and referenced papers from the exploration seismic community as well as some of the key papers that have impacted other fields of seismology. Because the modeling literature is vast, we have limited the scope of the reprint volume to papers over the last two decades on modeling methods based on the full wave equation. The reprint volume will be of particular interest to researchers and practitioners interested in modeling methods and their applications. The searchable CD includes the 114-page book of abstracts and the full papers.”

Abstract Modeling of seismic wave propagation plays a key role in almost every aspect of exploration seismology. Fundamentally, it provides us with a means of understanding the character of recorded seismic data. Although analytical or semianalytical solutions exist for a number of canonical models, these are often insufficient to explain the full range of phenomena that arise in complex heterogeneous earth models governed by, for example, anisotropic, viscoelastic, or poroelastic rheologies. Typical manifestations of such complex phenomena that are observed widely in surface seismic data include scattering, generation of multiples, or interface waves. Another application area of seismic modeling is in survey evaluation and design, where different acquisition geometries and subsurface model hypotheses are assessed to choose an optimal acquisition and processing strategy. Furthermore, modeling is at the core of many seismic data processing and imaging/inversion algorithms, including noise attenuation (removal of coherent noise by first modeling it), multiple suppression, migration, inversion, etc. Further up the seismic value chain, modeling also is very useful to guide final interpretation of an image, both to evaluate whether key features in the data are real or the result of (for instance) multiples, and to assess lithology or fluid content. Finally, generating synthetic data sets for research has proven extremely valuable for the exploration-seismics community in benchmarking and testing new processing algorithms, e.g., Marmousi (Versteeg, 1994), SMART JV Pluto model (Stoughton et al., 2001), the SEG salt model (Aminzadeh et al., 1995), and the SEAM initiative (Fehler, 2009; Symes et al., 2009). Thus, it should

Abstract This work presents a new constitutive law for linear viscoelastic and anisotropic media, to model rock behaviour and its effects on wave propagation. In areas with high dissipation properties (e.g. hydrocarbon reservoirs), the interpretation of seismic data based on the isotropic and purely elastic assumption might lead to misinterpretations or, even worse, to overlooking useful information. Thus, a proper description of wave propagation requires a rheology which accounts for the anisotropic and anelastic behaviour of rocks. The present model is based on the following mechanical interpretation; each eigenvector (eigenstrain) of the stiffness tensor of an anisotropic solid defines a fundamental deformation state of the medium. The six eigenvalues (eigenstiffnesses) represent the genuine elastic parameters. Since they are independent of the reference system, they have an intrinsic physical content. From this fact and the correspondence principle we infer that in a real medium the rheological properties depend essentially on six relaxation functions, which are the generalization of the eigenstiffnesses to the viscoelastic case. The existence of six or less complex moduli depends on the symmetry class of the medium. We probe the new stress-strain relation with homogeneous viscoelastic plane waves, and give expressions for the slowness, attenuation, phase velocity, energy velocity (wavefront) and quality factor of the different wave modes.

Abstract 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 only marginally more expensive than analogous elastic schemes. The efficient implementation of absorbing boundaries may therefore be a good reason for also using the viscoelastic scheme in purely elastic simulations. We illustrate our method and the importance of accurately modeling anelastic media through 2-D and 3-D examples from shallow marine environments.

Abstract We address the basic theoretical and algorithmic aspects of memory-efficient implementation of realistic attenuation in the staggered-grid finite-difference modeling of seismic-wave propagation in media with material discontinuities. We show that if averaging is applied to viscoelastic moduli in the frequency domain, it is possible to determine anelastic coefficients of the averaged medium representing a material discontinuity. We define (1) the anelastic functions in a new way, as being independent of anelastic coefficients, and (2) a new coarse spatial distribution of the anelastic functions in order to properly account for material discontinuities and, at the same time, to have it memory efficient. Numerical tests demonstrate that our approach enables more accurate viscoelastic modeling than other approaches.

Abstract A numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory is developed in this paper. For a demonstration of the method, the Cole–Cole model of viscoelastic relaxation is adopted here. A formulation of the Cole–Cole model based on internal variables satisfying fractional relaxation equations is applied. In order to avoid integrating and storing of the entire history of the variables, a new method for solving fractional differential equations of arbitrary order based on a set of secondary internal variables is developed. Using the new method, the velocity–stress equations and the fractional relaxation equations are reduced to a system of first-order ordinary differential equations for the velocities, stresses, primary internal variables as well as the secondary internal variables. The horizontal spatial derivatives involved in the governing equations are calculated by the Fourier pseudospectral (PS) method, while the vertical ones are calculated by the Chebychev PS method. The physical boundary conditions and the non-reflecting conditions for the Chebychev PS method are also discussed. The global solution of the first-order system of ordinary differential equations is advanced in time by the Euler predictor–corrector methods. For the demonstration of our method, some numerical results are presented.

Abstract Until recently, the term “elastic” usually implied two-dimensional (2-D) and isotropic. In this limited context, the divergence and curl operators have found wide use as wave separation operators. For example, Mora (1987) used them in his inversion method to allow separate correlation of P and S arrivals, although the separation is buried in the math and not obvious. Clayton (1981) used them explicitly in several modeling and inversion methods. Devaney and Oristaglio (1986) used closely related operators to separate P and S arrivals in elastic VSP data. With the current widespread interest in anisotropy, it seems useful to extend the wave-type separation concept to anisotropic media. We give a simple geometrical explanation of why divergence and curl are wave-type separation operators in the isotropic case and then show how to construct wave-type separation operators for general 2-D anisotropic media. We demonstrate the method on a heterogeneous strongly anisotropic finite-difference example. Extending existing isotropic 2-D algorithms based on wave-type separation to include anisotropy seems to be straightforward.