Prestack migration and/or inversion may be implemented in either the time or the frequency domain. In the frequency domain, it is possible to discretize the frequencies with a much larger sampling interval than that dictated by the sampling theorem and still obtain an imaging result that does not suffer from aliasing (wrap around) in the depth domain. The selection of input frequencies can be reduced when a range of offsets is available; this creates a redundancy of information in the wavenumber coverage of the target. In order to optimize the use of this information, we define a new discretization strategy that depends on the maximum effective offset present in the surface seismic survey: the larger the range of offsets, the fewer frequencies are required. The strategy, exact in a homogeneous 1D earth, selects frequencies by making use of the well-known effect of image stretch in normal-moveout (NMO) correction and in migration (usually considered detrimental for the imaging). The strategy is also useful in more general earth models: we apply it to the 2D Marmousi model and recover a continuous range of wavenumbers using only three input frequencies. The Marmousi inversion result accurately predicts all other data frequencies, demonstrating the redundancy of the data.

Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies

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

The problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory. In its roughest formulation, the inverse problem consists of obtaining the Earth model for which the predicted data best fit the observed data. If an adequate forward model is used, this best model will give the best images of the Earth’s interior. Three parameters are needed for describing a perfectly elastic, isotropic, Earth: the density ρ(x) and the Lame parameters λ(x) and μ(x), or the density ρ(x) and the P-wave and S-wave velocities α(x) and β(x). The choice of parameters is not neutral, in the sense that although theoretically equivalent, if they are not adequately chosen the numerical algorithms in the inversion can be inefficient. In the long (spatial) wavelengths of the model, adequate parameters are the P-wave and S-wave velocities, while in the short (spatial) wavelengths, P-wave impedance, S-wave impedance, and density are adequate. The problem of inversion o...

A strategy for nonlinear elastic inversion of seismic reflection data

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).

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Wave-equation traveltime inversion

Density and bulk modulus variations in an acoustic earth are separately recoverable from standard reflection surveys by utilizing the amplitude-versus-offset information present in the observed wave fields. Both earth structure and a variable background velocity can be accounted for by combining the Born and WKBJ approximations, in a "before stack" migration with two output sections, one for density variations and the other for bulk modulus variations. For the inversion, the medium is considered to be composed of a known low-spatial frequency variation (the background) plus an unknown high-spatial frequency variation in bulk modulus and density (the reflectivity). The division between the background and the reflectivity depends upon the frequency content of the source. For constant background parameters, computations are done in the Fourier domain, where the first part of the algorithm includes a frequency shift identical to that in an F-K migration. The modulus and density variations are then determined by observing in a least-squares sense amplitude versus offset wavenumber. For a spatially variable background, WKBJ Green's operators that model the direct wave in a medium with a smoothly varying background are used. A downward continuation with these operators removes the effects of variable velocity from the problem, and, consequently, the remainder of the inversion essentially proceeds as if the background were constant. If the background is strictly depth dependent, the inversion can be expressed in closed form. The method neglects multiples and surface waves and it is restricted to precritical reflections. Density is distinguishable from bulk modulus only if a sufficient range of precritical incident angles is present in the data.

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Robert H. Stolt

Papers

A Born-WKBJ inversion method for acoustic reflection data