Xian-Yun Wu

One-way and one-return approximations (de wolf approximation) for fast elastic wave modeling in complex media

The De Wolf approximation has been introduced to overcome the limitation of the Born and Rytov approximations in long range forward propagation and backscattering calculations. The De Wolf approximation is a multiple-forescattering-single-backscattering (MFSB) approximation, which can be implemented by using an iterative marching algorithm with a single backscattering calculation for each marching step (a thin-slab). Therefore, it is also called a one-return approximation. The marching algorithm not only updates the incident field step-by-step, in the forward direction, but also the Greens function when propagating the backscattered waves to the receivers. This distinguishes it from the first order approximation of the asymptotic multiple scattering series, such as the generalized Bremmer series, where the Greens function is approximated by an asymptotic solution. The De Wolf approximation neglects the reverberations (internal multiples) inside thin-slabs, but can model all the forward scattering phenomena, such as focusing/defocusing, diffraction, refraction, interference, as well as the primary reflections. In this chapter, renormalized MFSB (multiple-forescattering–single-backscattering) equations and the dual-domain expressions for scalar, acoustic and elastic waves are derived by using a unified approach. Two versions of the one-return method (using MFSB approximation) are given: one is the wide-angle, dual-domain formulation (thin-slab approximation) (compared to the screen approximation, no small-angle approximation is made in the derivation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation for the wave-medium interaction, it can be clearly seen that the forward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or reflected waves, are mainly controlled by impedance perturbations. Later in this chapter the validity of the thin-slab and screen methods, and the wide-angle capability of the dual-domain implementation are demonstrated by numerical examples. Reflection coefficients of a plane interface, derived from numerical simulations by the wide-angle method, are shown to match the theoretical curves well up to critical angles. The methods are applied to the fast calculation of synthetic seismograms. The results are compared with finite difference (FD) calculations for the elastic French model. For weak heterogeneities ( ± 15 % perturbation), good agreement between the two methods verifies the validity of the one-return approach. However, the one-return approach is about 2–3 orders of magnitude faster than the elastic FD algorithm. The other example of application is the modeling of amplitude variation with angle (AVA) responses for a complex reservoir with heterogeneous overburdens. In addition to its fast computation speed, the one return method (thin-slab and complex-screen propagators) has some special advantages when applied to the thin-bed and random layer responses.

Wavefield and AVO modeling using elastic thin-slab method

We propose a dual-domain, one-way, elastic thin-slab method for fast and accurate amplitude variation with offset AVOmodeling.Inthismethod,thewavefieldpropagatesin the wavenumber domain and interacts with heterogeneity in the space domain. The approach requires much less memory andistwotothreeordersofmagnitudefasterthanafull-wave methodusingfinitedifferenceorfiniteelement.Thethin-bed AVO and AVOs with lateral parameter variations have been conducted using the thin-slab method and compared with reflectivity and finite-difference methods, respectively. It is shown that the thin-slab method can be used to accurately modelreflectionsformostsedimentaryrocksthathaveintermediateparameterperturbations20%forP-wavevelocity and 40% for S-wave velocity. The combined effects of overburdenstructureandthescatteringassociatedwithheterogeneitiesonAVOhavebeeninvestigatedusingthethin-slab method. Properties of the target zone and overburden structure control the AVO trends at overall offsets. Scattering associated with heterogeneities increases local variance in the reflected amplitudes and becomes significant for the sedimentarymodelswithweakreflections.InterpretationofAVO observations based on homogeneous elastic models would therefore bias the estimated properties of the target. Furthermore, these effects can produce different apparent AVO trendsindifferentoffsetranges.

Simulation of High-Frequency Wave Propagation in Complex Crustal Waveguides Using Generalized Screen Propagators

In the crustal waveguide environment, the major part of wave energy is carried by forward propagating waves, including forward scattered waves. Therefore, neglecting backscattered waves in numerical modeling will not modify the main features of regional waves in most cases. By neglecting backscattering in the theory, the wave modeling becomes a forward marching problem in which the next step of propagation depends only on the present values of the wavefield in a transverse cross-section and the heterogeneities between the present cross-section and the next one (wavefield extrapolation interval). The saving of computation time and computer memory is enormous. A half-space screen propagator (generalized screen propagator) has been developed to accommodate the free-surface boundary condition for modeling SH wave propagation in complex crustal waveguides. The SH screen propagator has also been extended to handle irregular surface topography using conformal or non-conformal topographic transforms. The screen propagator for modeling regional SH waves has been calibrated extensively against some full-wave methods, such as the wavenumber integration, finite-difference and boundary element methods, for different crustal models. Excellent agreement with these full-wave methods demonstrated the validity and accuracy of the new one-way propagator method. For medium size problems, the screen-propagator method is 2–3 orders of magnitude faster than finite-difference methods. It has been used for the simulation of Lg propagation in crustal models with random heterogeneities and the related energy partition, attenuation and blockage. It is found that the leakage attenuation of Lg waves caused by large-angle forward scattering by random heterogeneities, which scatters the guided waves out of the trapped modes and leaking into the mantle, may contribute significantly to Lg attenuation and blockage in some regions. In the case of P-SV elastic screen propagators, plane wave reflection calculations have been incorporated into the elastic screen method to handle the free surface. Body waves, including the reflected and converted waves, can be calculated by real wavenumber integration; while surface waves (Rayleigh waves) can be obtained with imaginary wavenumber integration. Numerical tests proved the validity of the theory and methods.

AVO Modeling Using One-return Approximation

In this paper we apply elastic thin-slab propagator, which is based on one-return approximation and scattering theory, to AVO forward modeling. For medium perturbation of about 20%, such as shale/oil sand and shale/gas sand cases, the re ection coe cients are accurate up to 450 incident angle for P waves and 200 incident angle for S wave, which can meet the requirement in most AVO analyses. Numerical results show that thin-slab propagator can easily handle thin-bed e ects as well as lateral variations in lithology.

Modeling Porous Layers In Elastic Media With a Hybrid Method

A hybrid method which combines one-way approximation method and the propagator matrix method ( the full-wave solution ) is introduced and extended to including twophase media based on the equivalent propagator matrix of two-phase media derived in this paper. It can handle more complicated structure than one-way approximation method and the propagator matrix method used alone. It can handle strong heterogeneous and/or two-phase media layers embedded in arbitrarily weak heterogeneous media. Numerical examples show good agreement between one-way approximation method and the matrix method for weak heterogeneous media. By means of the hybrid method, the reflection seismograms in elastic solid and the transmission fields through two-phase media with different porosi ve and permeability, are modeled, respectively.