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

Probabilistic formulation of inverse problems leads to the definition of a probability distribution in the model space. This probability distribution combines a priori information with new information obtained by measuring some observable parameters (data). As, in the general case, the theory linking data with model parameters is nonlinear, the a posteriori probability in the model space may not be easy to describe (it may be multimodal, some moments may not be defined, etc.). When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sucient, as we normally also wish to have infor

/pdf/monte-carlo-sampling-of-solutions-to-inverse-problems-vkxbgptqug.pdf

Monte Carlo sampling of solutions to inverse problems

SUMMARY Monte Carlo direct search methods, such as genetic algorithms, simulated annealing etc., areoften used to explore a ﬁnite dimensional parameter space. They require th e solving of theforward problem many times, that is, making predictions of observables from an earth model.The resulting ensemble of earth models represents all ‘information’ collected in the searchprocess. Search techniques have been the subject of much study in geophysics; less attention isgiven to the appraisal of the ensemble. Often inferences are based on only a small subset of theensemble, and sometimes a single member.This paper presents a new approach to the appraisal problem. To our knowledge this is the ﬁrsttime the general case has been addressed, that is, how to infer information from a completeensemble, previously generated by any search method. The essence of the new approach is touse the informationin the available ensembleto guidea resamplingofthe parameterspace. Thisrequires no further solving of the forward problem, but from the new ‘resampled’ ensemblewe are able to obtain measures of resolution and trade-off in the model parameters, or anycombinations of them.The new ensemble inference algorithm is illustrated on a highly non-linear waveform inversionproblem.It is shownhow the computationtime and memoryrequirements scale with the dimen-sion of the parameter space and size of the ensemble. The method is highly parallel, and mayeasily be distributed across several computers. Since little is assumed about the initial ensembleof earth models, the technique is applicable to a wide variety of situations. For example, it maybe applied to perform ‘error analysis’ using the ensemble generated by a genetic algorithm, orany other direct search method.Key words: numerical techniques, receiver functions, waveform inversion.

http://rses.anu.edu.au/~malcolm/papers/pdf/na2_rep_col.pdf

Geophysical inversion with a neighbourhood algorithm—II. Appraising the ensemble

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

[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

The aim of inverting seismic waveforms is to obtain the “best” earth model. The best model is defined as the one producing seismograms that best match (usually under a least‐squares criterion) those recorded. Our approach is nonlinear in the sense that we synthesize seismograms without using any linearization of the elastic wave equation. Since we use rather complete data sets without any spatial aliasing, we do not have the problem of secondary minima (Tarantola, 1986). Nevertheless, our gradient methods fail to converge if the starting earth model is far from the true earth (Mora, 1987; Kolb et al., 1986; Pica et al., 1989).

Wavelengths of Earth structures that can be resolved from seismic reflection data

We present a new subsurface angle-domain seismic imaging systemforgeneratingandextractinghigh-resolutioninformation about subsurface angle-dependent reflectivity. The system enables geophysicists to use all recorded seismic data in a continuousfashiondirectlyinthesubsurfacelocalangledomainLAD, resulting in two complementary, full-azimuth, common-imageangle gather systems: directional and reflection. The complete setofinformationfrombothtypesofanglegathersleadstoaccurate, high-resolution, reliable velocity model determination and reservoir characterization. The directional angle decomposition enables the implementation of specular and diffraction imaging in real 3D isotropic/anisotropic geological models, leading to simultaneous emphasis on continuous structural surfaces and discontinuous objects such as faults and small-scale fractures. Structural attributes at each subsurface point, e.g., dip, azimuth andcontinuity,canbederiveddirectlyfromthedirectionalangle gathers. The reflection-angle gathers display reflectivity as a function of the opening angle and opening azimuth. These gathers are most meaningful in the vicinity of actual local reflecting surfaces,wherethereflectionanglesaremeasuredwithrespectto the derived background specular direction. The reflection-angle gathers are used for automatic picking of full-azimuth angle-domainresidualmoveoutsRMOwhich,togetherwiththederived background orientations of the subsurface reflection horizons, provide a complete set of input data to isotropic/anisotropic tomography. The full-azimuth, angle-dependent amplitude variations are used for reliable and accurate amplitude versus angle andazimuthAVAZanalysisandreservoircharacterization.The proposed system is most effective for imaging and analysis below complex structures, such as subsalt and subbasalt, high-velocity carbonate rocks, shallow low-velocity gas pockets, and others. In addition, it enables accurate azimuthal anisotropic imaging and analysis, providing optimal solutions for fracture detectionandreservoircharacterization.

Full-azimuth subsurface angle domain wavefield decomposition and imaging Part I: Directional and reflection image gathers

Embodiments of the invention provide a system and method for converting coordinate systems for representing image data such as for example seismic data, including accepting a first set of seismic data, mapping the first set of seismic data to a second set of seismic data, where the dimensionality of the second set of seismic data is less than the dimensionality of the first set of seismic data, and generating image data by processing the second set of seismic data.

System and method for full azimuth angle domain imaging in reduced dimensional coordinate systems

The complete solution to an inverse problem, including information on accuracy and resolution, is given by the a posteriori probability density in the model space. By running a modified simulated annealing, samples from the model space can be drawn in such a way that their frequencies of occurrence approximate their a posteriori likelihoods. Using this method, maximum likelihood estimation and uncertainty analysis of seismic background velocity models are performed on multioffset seismic data. The misfit between observed and synthetic waveforms within the time windows along computed multioffset travel times, is used as an objective function for the simulated annealing approach. The real medium is modeled as a series of layers separated by curved interfaces. Lateral velocity variations within the layers are determined by interpolation from specified values at a number of sampling points. The input data consists of multioffset seismic data. Additionally, zero-offset times are used to migrate the reflectors in time to the depth domain. The multioffset times are calculated by an efficient ray-tracing algorithm which allows inversion of a large number of seismograms. The a posteriori probability density for this problem is highly multidimensional and highly multimodal. Therefore, the information contained in this distribution cannot be adequately represented by standard deviations and covariances. However, by sequentially displaying a large number of images, computed from the a posteriori background velocity samples and the data, it is possible to convey to the spectator a better understanding of what information we really have on the subsurface.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/91JB02278%4010.1002/%28ISSN%292169-9356.MATHGEO1

Monte Carlo estimation and resolution analysis of seismic background velocities

We have developed an angle-based system, which defines the interaction between incident and reflected waves at a specific image point. We provide details about the technique used for angle-domain decomposition and imaging; in particular, the creation of two types of 3D subsurface angle gathers, as discussed in Part 1. The system, referred to as local angle domain (LAD) consists of two subsystems: directional and reflection. In the directional subsystem, the orientation of the ray-pair normal is represented by dip and azimuth. The reflection angle system is defined by the opening angle between the phase velocities of the incident and reflected rays, and by the opening azimuth, which describes the orientation of the ray-pair incidence plane measured in the ray-pair reflection plane. The directional angles are defined in a rotated local frame, and the opening azimuth is defined in the ray-pair reflection plane. The vertical axis of the local frame is collinear with a preset background normal to the physical ...

Zvi Koren

Papers

Wavelengths of Earth structures that can be resolved from seismic reflection data

Full-azimuth subsurface angle domain wavefield decomposition and imaging Part I: Directional and reflection image gathers

System and method for full azimuth angle domain imaging in reduced dimensional coordinate systems

Monte Carlo estimation and resolution analysis of seismic background velocities

Full-azimuth subsurface angle domain wavefield decomposition and imaging: Part 2 — Local angle domain