Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

/pdf/an-overview-of-full-waveform-inversion-in-exploration-59sei7fzvh.pdf

An overview of full-waveform inversion in exploration geophysics

SUMMARY By specifying a discrete matrix formulation for the frequency^space modelling problem for linear partial diierential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non-linear inverse methods, such as the gradient (steepest descent) method, the Gauss^Newton method and the full Newton method We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency-domain inversion (FDI)The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general The FDI methods only require that the original partial diierential equations can be expressed as a discrete boundary-value problem (that is as a matrix problem) Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the mis¢t function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a ¢nal computation to compute a step length) This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss^Newton method, and the full Hessian matrix used in the full Newton method In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss^Newton method, can be e⁄ciently computed using a backpropagation approach similar to that used to compute the gradient vector The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of ¢rst-order multiples (such as free-surface multiples in seismic data) Another interpretation is that this term predicts changes in the gradient vector due to second-order non-linear eiects In a numerical test, the Gauss^Newton and full Newton methods prove eiective in helping to solve the di⁄cult non-linear problem of extracting a smooth background velocity model from surface seismic-re£ection data

Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion

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.

/pdf/global-optimization-methods-in-geophysical-inversion-3xaeou1fz2.pdf

Global Optimization Methods in Geophysical Inversion

A least-squares migration algorithm is presented that reduces the migration artifacts (i.e., recording footprint noise) arising from incomplete data. Instead of migrating data with the adjoint of the forward modeling operator, the normal equations are inverted by using a preconditioned linear conjugate gradient scheme that employs regularization. The modeling operator is constructed from an asymptotic acoustic integral equation, and its adjoint is the Kirchhoff migration operator. We tested the performance of the least-squares migration on synthetic and field data in the cases of limited recording aperture, coarse sampling, and acquisition gaps in the data. Numerical results show that the least-squares migrated sections are typically more focused than are the corresponding Kirchhoff migrated sections and their reflectivity frequency distributions are closer to those of the true model frequency distribution. Regularization helps attenuate migration artifacts and provides a sharper, better frequency distribution of estimated reflectivity. The least-squares migrated sections can be used to predict the missing data traces and interpolate and extrapolate them according to the governing modeling equations. Several field data examples are presented. A ground-penetrating radar data example demonstrates the suppression of the recording footprint noise due to a limited aperture, a large gap, and an undersampled receiver line. In addition, better fault resolution was achieved after applying least-squares migration to a poststack marine data set. And a reverse vertical seismic profiling example shows that the recording footprint noise due to a coarse receiver interval can be suppressed by least-squares migration.

Least-squares migration of incomplete reflection data

[1] Monte Carlo inversion techniques were first used by Earth scientists more than 30 years ago. Since that time they have been applied to a wide range of problems, from the inversion of free oscillation data for whole Earth seismic structure to studies at the meter-scale lengths encountered in exploration seismology. This paper traces the development and application of Monte Carlo methods for inverse problems in the Earth sciences and in particular geophysics. The major developments in theory and application are traced from the earliest work of the Russian school and the pioneering studies in the west by Press [1968] to modern importance sampling and ensemble inference methods. The paper is divided into two parts. The first is a literature review, and the second is a summary of Monte Carlo techniques that are currently popular in geophysics. These include simulated annealing, genetic algorithms, and other importance sampling approaches. The objective is to act as both an introduction for newcomers to the field and a comprehensive reference source for researchers already familiar with Monte Carlo inversion. It is our hope that the paper will serve as a timely summary of an expanding and versatile methodology and also encourage applications to new areas of the Earth sciences.

/pdf/monte-carlo-methods-in-geophysical-inverse-problems-59l6iaiob5.pdf

Monte carlo methods in geophysical inverse problems

We propose an iterative method for the linearized prestack inversion of seismic profiles based on the asymptotic theory of wave propagation. For this purpose, we designed a very efficient technique for the downward continuation of an acoustic wavefield by ray methods. The different ray quantities required for the computation of the asymptotic inverse operator are estimated at each diffracting point where we want to recover the earth image. In the linearized inversion, we use the background velocity model obtained by velocity analysis. We determine the short wavelength components of the impedance distribution by linearized inversion of the seismograms observed at the surface of the model. Because the inverse operator is not exact, and because the source and station distribution is limited, the first iteration of our asymptotic inversion technique is not exact. We improve the images by an iterative procedure. Since the background velocity does not change between iterations. There is no need to retrace rays,...

/pdf/iterative-asymptotic-inversion-in-the-acoustic-approximation-8umcw109nr.pdf

Iterative asymptotic inversion in the acoustic approximation

SUMMARY

An asymptotic linearized iterative elastic inversion method is proposed to invert 2-D Earth parameters from multicomponent data and is tested numerically. The forward problem is solved by a combination of the Born approximation and ray theoretical methods. We express the perturbed seismogram in terms of perturbations of P- and S-wave impedances and density. The inversion method is based on generalized least squares. We introduce a special form of the ρ2 norm with a weighting function that corrects for geometrical spreading and obliquity of the reflectors. The Hessian for this norm could be estimated in a closed form that is asymptotically valid at high frequencies. We propose a quasi-Newtonian iterative method for the solution of the inverse problem. The first iteration of this inversion method resembles the operator proposed by Beylkin (1985) and Beylkin & Burridge (1990) for the asymptotic inversion of seismic data. Our method is more general than theirs because it can handle arbitrary discrete distributions of sources and receivers. Elastic inversion is generally ill-posed because the problem is overdetermined but undersampled. We study the resolution of the asymptotic inversion method for general sets of sources and receivers. We show that simultaneous inversion for both P- and S-wave impedance is generally ill-conditioned if data for a single scattering mode are available. In particular, it seems that only one parameter can be reliably resolved from marine data. Simultaneous inversion for a finite set of parameters can be resolved only for multicomponent elastic data containing both P-wave and S-wave information. Inversion tests using synthetic data calculated by finite-differences demonstrates that it is possible to invert simultaneously for P and S impedances.

/pdf/two-dimensional-asymptotic-iterative-elastic-inversion-55aax6ea6r.pdf

Two-dimensional asymptotic iterative elastic inversion

Seismic reflection data contain information on small-scale impedance variations and a smooth reference velocity model. Given a reference velocity model, the reflectors can be obtained by linearized migration-inversion. If the reference velocity is incorrect, the reflectors obtained by inverting different subsets of the data will be incoherent. We propose to use the coherency of these images to invert for the background velocity distribution. We have developed a two-step iterative inversion method in which we separate the retrieval of small-scale variations of the seismic velocity from the longer-period reference velocity model. Given an initial background velocity model, we use a waveform misfit-functional for the inversion of small-scale velocity variations. For this linear step we use the linearized migration-inversion method based on ray theory that we have recently developed with Lambare and Virieux. The reference velocity model is then updated by a Monte Carlo inversion method. For the nonlinear inversion of the velocity background, we introduce an objective functional that measures the coherency of the short wavelength components obtained by inverting different common shot gathers at the same locations. The nonlinear functional is calculated directly in migrated data space to avoid expensive numerical forward modeling by finite differences or ray theory. Our method is somewhat similar to an iterative migration velocity analysis, but we do an automatic search for relatively large-scale 1-D reference velocity models. We apply the nonlinear inversion method to a marine data set from the North Sea and also show that nonlinear inversion can be applied to realistic scale data sets to obtain a laterally heterogeneous velocity model with a reasonable amount of computer time.

Nonlinear velocity inversion by a two-step Monte Carlo method.

Two-Dimensional Asymptotic Iterative Elastic Inversion

An efficient and robust non-linear inversion method for velocity optimization combining a global random search followed by a simplex technique is presented. The background velocity field is estimated at different spatial scales by analysing image gathers after iterative prestack depth migrations. First, the global random search is used to determine the main features/trends of the velocity model (large-scale component). Then, the simplex technique improves the resolution of the velocity field by estimating smaller-scale features. A measure of the quality of the velocity model (objective function) is based on flattening off set events in depth-migrated image gathers. To help constrain the solution, the algorithm can incorporate a priori information about the model and a smoothness condition. This 2D velocity estimation offers the benefit of being semi-automatic (requiring minimal human intervention) as well as providing a global and objective solution (which is a useful approach to an interpretation-derived velocity-estimation technique). The method is applied to a real data set where AVO analysis is carried out after prestack depth migration, as structural effects are non-negligible. It is demonstrated that the method can successfully estimate a laterally inhomogeneous velocity model at a computational cost modest compared with an interpretation-based iterative prestack depth velocity-analysis technique.

2D multiscale non‐linear velocity inversion

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