Automatic cross‐well tomography by semblance and differential semblance optimization: theory and gradient computation

Plessix, René‐Edouard; Mulder, W.A.; Kroode, A. P. E. ten

doi:10.1046/j.1365-2478.2000.00217.x

Cited by 36 publications

(22 citation statements)

References 16 publications

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“…If the pre-stack migration gives different earth structures, this means that the background slowness used in the migration is erroneous. Several mathematical formulations of this idea have been proposed in the last 20 years, among them Chavent & Jacewitz (1995), Clément et al (2001), Plessix et al (2000) and Symes & Carazzone (1991). In order to compute the gradient of the reformulated cost functions, the authors generally use the adjoint-state formulation because it is the most systematic method, without forgetting that they are mainly mathematicians.…”

Section: S H O T -B a S E D D I F F E R E N T I A L S E M B L A N C Ementioning

confidence: 99%

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

Plessix

2006

Geophysical Journal International

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1,637

890

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S U M M A R YEstimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Fréchet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Fréchet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjointstate variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved.Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established.

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Section: S H O T -B a S E D D I F F E R E N T I A L S E M B L A N C Ementioning

confidence: 99%

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

Plessix

2006

Geophysical Journal International

Self Cite

1,637

890

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“…is a normalized differential semblance coefficient (Symes and Carazzone, 1991;Plessix et al, 2000;Brandsberg-Dahl et al, 2003;Abbad et al, 2010) that should be minimized for the best parameter-fitting. Therefore, the term ð1 − DÞ in equation 1 needs to be maximized.…”

Section: Bootstrapped Differential Semblance (Bds)mentioning

confidence: 99%

High-resolution bootstrapped differential semblance

Abbad

Ursin

2012

GEOPHYSICS

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We formulated two coherency measures, based on the bootstrapped differential semblance (BDS) estimator, that offered higher resolution in parameter tracking than did standard normalized differential semblance. Bootstrapping is a statistical resampling procedure used to infer estimates of standard errors and confidence intervals from data samples for which the statistical properties are unattainable via simple means, or when the probability density function is unkown or difficult to estimate. The first proposed estimator was based on a deterministic sorting of original offset traces by alternating near and far offsets to achieve maximized time shifts between adjacent traces. The near offsets were indexed with odd integers, while the even integers were used to index far offsets that were located at a constant index increment from the previous trace. The second was the product of several BDS terms, with the first term being the deterministic BDS defined above. The other terms were generated by random sorting of traces that alternated near and far offsets in an unpredictible manner. The proposed estimators could be applied in building velocity (and anellipticity) spectra for time-domain velocity analysis, depth-domain residual velocity update, or to any parameter-fitting algorithm involving discrete multichannel data. The gain in resolution provided by the suggested estimators over the differential semblance coefficient was illustrated on a number of synthetic and field data examples.

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“…After the calculation of first arrival traveltime, we trace the wavefront normal directions from the receivers back to the source as the ray path. We also tried the beam method [ Plessix et al , 2000]. A beam is started from a source point and consists of two neighboring rays defined by their take‐off angles.…”

Section: Forward Solver With/without Ray Tracing and Beam Methodsmentioning

confidence: 99%

First arrival stochastic tomography: Automatic background velocity estimation using beam semblances and VFSA

Hu¹,

Stoffa²,

McIntosh³

2008

Geophysical Research Letters

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[1] We present a new tomography method based on the local beam semblance and the very fast simulated annealing (VFSA) global optimization method. The data space is the local beam semblance calculated using local slant stacks for overlapping offset windows, i.e. beam windows, of the original common-shot or common-receiver gathers. On each beam semblance panel, the first coherency peak can be identified with a particular ray parameter, first-arrival traveltime and beam center position. The forward problem can be solved with any ray tracer to find arrivals matching the identified peaks. Our inversion scheme uses VFSA to find the maximum-a-posteriori (MAP) solution and estimates the uncertainty by applying Bayesian analysis of all the sampled models for a specified model parameterization. This integration of automatic local semblance evaluation instead of first-arrival picking and a fast forward modeling method combined with VFSA to determine the optimal model makes our method robust, efficient and accurate. Citation: Hu, C., P. Stoffa, and K. McIntosh (2008), First arrival stochastic tomography: Automatic background velocity estimation using beam semblances and VFSA, Geophys.

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Automatic cross‐well tomography by semblance and differential semblance optimization: theory and gradient computation

Cited by 36 publications

References 16 publications

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

High-resolution bootstrapped differential semblance

First arrival stochastic tomography: Automatic background velocity estimation using beam semblances and VFSA

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