The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena, such as amplitude-offset variations and shear-wave events, which can only be explained by using the more correct elastic wave equation. Not only are such phenomena ignored by acoustic theory, but they are also treated as undesirable noise when they should be used to provide extra information, such as S-wave velocity, about the subsurface. The problems of using the conventional acoustic wave equation approach can be eliminated via an elastic approach. In this paper, equations have been derived to perform an inversion for P-wave velocity, S-wave velocity, and density as well as the P-wave impedance, S-wave impedance, and density. These are better resolved than the Lame parameters. The inversion is based on nonlinear least squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between the observed data and the modeled data corresponding to these earth parameters. The iterations are based on the preconditioned conjugate gradient algorithm. The fundamental requirement of such a least-squares algorithm is the gradient direction which tells how to update the model parameters. The gradient direction can be derived directly from the wave equation and it may be computed by several wave propagations. Although in principle any scheme could be chosen to perform the wave propagations, the elastic finite-difference method is used because it directly simulates the elastic wave equation and can handle complex, and thus realistic, distributions of elastic parameters. This method of inversion is costly since it is similar to an iterative prestack shot-profile migration. However, it has greater power than any migration since it solves for the P-wave velocity, S-wave velocity, and density and can handle very general situations including transmission problems. Three main weaknesses of this technique are that it requires fairly accurate a priori knowledge of the low-wavenumber velocity model, it assumes Gaussian model statistics, and it is very computer-intensive. All these problems seem surmountable. The low-wavenumber information can be obtained either by a prior tomographic step, by the conventional normal-moveout method, by a priori knowledge and empirical relationships, or by adding an additional inversion step for low wavenumbers to each iteration. The Gaussian statistics can be altered by preconditioning the gradient direction, perhaps to make the solution blocky in appearance like well logs, or by using large model variances in the inversion to reduce the effect of the Gaussian model constraints. Moreover, with some improvements to the algorithm and more parallel computers, it is hoped the technique will soon become routinely feasible.

Nonlinear two-dimensional elastic inversion of multioffset seismic data

A new linearized AVO inversion technique is developed in a Bayesian framework. The objective is to obtain posterior distributions for P‐wave velocity, S‐wave velocity, and density. Distributions for other elastic parameters can also be assessed—for example, acoustic impedance, shear impedance, and P‐wave to S‐wave velocity ratio. The inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the Zoeppritz equation. The solution is represented by a Gaussian posterior distribution with explicit expressions for the posterior expectation and covariance; hence, exact prediction intervals for the inverted parameters can be computed under the specified model. The explicit analytical form of the posterior distribution provides a computationally fast inversion method. Tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero. With realistic noise levels, acoustic impedance was the best determined parameter, ...

Bayesian linearized AVO inversion

Soit Ω un domaine borne dans R n , n≥3 avec une frontiere C 1,1 . On considere l'operateur Lγ(u)=⊇•(γ⊇u) ou γ(x) est une fonction a valeur reelle dans C 1,1 (Ω) avec une borne superieure positive. On definit la forme quadratique Qγ sur H 1/2 (∂Ω) par Qγ(f)=∫ Ω γ(x)|⊇u(x)| 2 dx ou u∈H 1 (Ω) est la solution unique a Lγu=0 dans Ω, u| ∂Ω =f. On etudie la reconstruction de γ a partir de Qγ

Reconstructions from boundary measurements

Part 1 Preliminary statistics: random variables random nunmbers probability probability distribution, distribution function and density function joint and marginal probability distributions mathematical expectation, moments, variances and covariances conditional probability Monte Carlo integration importance sampling stochastic processes Markov chains homogeneous, inhomogeneous, irreducible and aperiodic Markov chains the limiting probability. Part 2 Direct, linear and iterative-linear inverse methods: direct inversion methods model based inversion methods linear/linearized inverse methods iterative linear methods for quasi-linear problems Bayesian formulation solution using probabilistic formulation. Part 3 Monte Carlo methods: enumerative or grid search techniques Monte Carlo inversion hybrid Monte Carlo-linear inversion directed Monte Carlo methods. Part 4 Simulated annealing methods: metropolis algorithm heat bath algorithm simulated annealing without rejected moves fast simulated annealing very fast simulated reannealing mean field annealing using SA in geophysical inversion. Part 5 Genetic algorithms: a classical GA schemata and the fundamental theorem of genetic algorithms problems combining elements of SA into a new GA a mathematical model of a GA multimodal fitness functions, genetic drift, GA with sharing, and repeat (parallel) GA uncertainty estimates evolutionary programming - a variant of GA. Part 6 Geophysical applications of SA and GA: 1-D seismic waveform inversion pre-stack migration velocity estimation inversion of resistivity sounding data for 1-D earth models inversion of resistivity profiling data for 2-D earth models inversion of magnetotelluric sounding data for 1-D earth models stochastic reservoir modelling seismic deconvolution by mean field annealing and Hopfield network. Part 7 Uncertainty estimation: methods of numerical integration simulated annealing - the Gibbs' sampler genetic algorithm - the parallel Gibbs' sampler numerical examples.

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Global Optimization Methods in Geophysical Inversion

This paper presents a new traveltime inversion method based on the wave equation. In this new method, designated as wave-equation traveltime inversion (WT), seismograms are computed by any full-wave forward modeling method (we use a finite-difference method). The velocity model is perturbed until the traveltimes from the synthetic seismograms are best fitted to the observed traveltimes in a least squares sense. A gradient optimization method is used and the formula for the Frechet derivative (perturbation of traveltimes with respect to velocity) is derived directly from the wave equation. No traveltime picking or ray tracing is necessary, and there are no high frequency assumptions about the data. Body wave, diffraction, reflection and head wave traveltimes can be incorporated into the inversion. In the high-frequency limit, WT inversion reduces to ray-based traveltime tomography. It can also be shown that WT inversion is approximately equivalent to full-wave inversion when the starting velocity model is 'close' to the actual model.Numerical simulations show that WT inversion succeeds for models with up to 80 percent velocity contrasts compared to the failure of full-wave inversion for some models with no more than 10 percent velocity contrast. We also show that the WT method succeeds in inverting a layered velocity model where a shooting ray-tracing method fails to compute the correct first arrival times. The disadvantage of the WT method is that it appears to provide less model resolution compared to full-wave inversion, but this problem can be remedied by a hybrid traveltime + full-wave inversion method (Luo and Schuster, 1989).

/pdf/wave-equation-traveltime-inversion-ou1eqd1296.pdf

Wave-equation traveltime inversion

We present a multidimensional multiple‐attenuation method that does not require any subsurface information for either surface or internal multiples. To derive these algorithms, we start with a scattering theory description of seismic data. We then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering. These include (1) the idea that the inversion process can be viewed as a series of steps, each with a specific task; (2) the realization that the inverse‐scattering series provides an opportunity for separating out subseries with specific and useful tasks; (3) the recognition that these task‐specific subseries can have different (and more favorable) data requirements, convergence, and stability conditions than does the original complete inverse series; and, most importantly, (4) the development of the first method for physically interpreting the contribution that individual terms (and pieces of terms) in the inv...

An inverse-scattering series method for attenuating multiples in seismic reflection data

This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task, and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four task-specific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks, i.e., no terms corresponding to coupled tasks are ever computed. This inversion in stages provides a tremendous practical advantage. The achievement of a task is a form of useful information exploited in the redefined and restarted problem; and the latter represents a critically important step in the logic and overall strategy. The individual subseries are analysed and their strengths, limitations and prerequisites exemplified with analytic, numerical and field data examples.

https://iopscience.iop.org/article/10.1088/0266-5611/19/6/R01/pdf

Inverse scattering series and seismic exploration

Seismic migration and inversion describe a class of closely related processes sharing common objectives and underlying physical principles. These processes range in complexity from the simple NMO‐stack to the complex, iterative, multidimensional, prestack, nonlinear inversion used in the elastic seismic case. By making use of amplitudes versus offset, it is, in principle, possible to determine the three elastic parameters from compressional data. NMO‐stack can be modified to solve for these parameters, as can prestack migration. Linearized, wave‐equation inversion does not inordinately increase the complexity of data processing. The principal part of a migration‐inversion algorithm is the migration. Practical difficulties are considerable, including both correctable and intrinsic limitations in data quality, limitations in current algorithms (which we hope are correctable), and correctable (or perhaps intrinsic) limitations in computer power.

Migration and inversion of seismic data

This paper is an overview of the current state of multiple attenuation and developments that we might anticipate in the near future. The basic model in seismic processing assumes that reflection data consist of primaries only. If multiples are not removed, they can be misinterpreted as, or interfere with, primaries. This is a longstanding and only partially solved problem in exploration seismology. Many methods exist to remove multiples, and they are useful when their assumptions and prerequisites are satisfied. However, there are also many instances when these assumptions are violated or where the prerequisites are difficult or impossible to attain; hence, multiples remain a problem. This motivates the search for new demultiple concepts, algorithms, and acquisition techniques to add to, and enhance, our toolbox of methods.

Multiple attenuation; an overview of recent advances and the road ahead (1999)

A multiple attenuation method derived from an inverse scattering series is described. The inversion series approach allows a separation of multiple attenuation subseries from the full series. The surface multiple attenuation subseries was described and illustrated in Carvalho et al. (1991, 1992). The internal multiple attenuation method consists of selecting the parts of the odd terms that are associated with removing only multiply reflected energy. The method, for both types of multiples, is multidimensional and does not rely on periodicity or differential moveout, nor does it require a model of the reflectors generating the multiples. An example with internal and surface multiples will be presented.

Arthur B. Weglein

Papers

An inverse-scattering series method for attenuating multiples in seismic reflection data

Inverse scattering series and seismic exploration

Migration and inversion of seismic data

Multiple attenuation; an overview of recent advances and the road ahead (1999)

Inverse Scattering Series For Multiple Attenuation: An Example With Surface And Internal Multiples