On the computation of the Fréchet derivatives for seismic waveform inversion in 3D general anisotropic, heterogeneous media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB153-WB163 ◽  
Author(s):  
Bing Zhou ◽  
Stewart Greenhalgh
Keyword(s):  
Anisotropic Medium ◽  
Heterogeneous Media ◽  
Jacobian Matrix ◽  
Displacement Vector ◽  
Waveform Inversion ◽  
Model Parameters ◽  
Seismic Waveform ◽  
Model Parameterization ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

We present a perturbation method and a matrix method for formulating the explicit Fréchet derivatives for seismic body-wave waveform inversion in 3D general anisotropic, heterogeneous media. Theoretically, the two methods yield the same explicit formula valid for any class of anisotropy and are completely equivalent if the model parameterization in the inversion is the same as that used in the discretization scheme (unstructured or structured mesh) for forward modeling. Explicit formulas allow various model parameterization schemes that try to match the resolution capability of the data and possibly reduce the dimensions of the Jacobian matrix. Based on the general expressions, relevant formulas for isotropic and 2.5D and 3D tilted transversely isotropic (TTI) media are derived. Two computational schemes, constant-point and constant-block parameterization, offer effective and efficient means of forming the Jacobian matrix from the explicit Fréchet derivatives. The sensitivity patterns of the displacement vector to the independent model parameters in a weakly anisotropic medium clearly convey the imaging capability possible with seismic waveform inversion in such an anisotropic medium.

Analysis of reflection traveltimes in 3-D transversely isotropic heterogeneous media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1884-1895 ◽  
Author(s):  
Vladimir Y. Grechka ◽  
George A. McMechan
Keyword(s):  
Ray Tracing ◽  
Heterogeneous Media ◽  
Chebyshev Approximation ◽  
Transversely Isotropic ◽  
Singular Value ◽  
Common Source ◽  
Model Parameters ◽  
Value Analysis ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

A two‐point ray‐tracing technique for rays reflected from irregular, but smooth, interfaces in 3-D transversely isotropic heterogeneous media is developed. The method is based on Chebyshev parameterization of curved segments of the reflected rays, of the reflectors, and of the velocity and anisotropy distributions in the model. Chebyshev approximation also can describe the reflection traveltime surfaces to compress traveltime data by replacing them with coefficients of the corresponding Chebyshev series. The advantage of the proposed parameterization is that it gives traveltime as an explicit function of the model parameters. This explicitly provides the Frechét derivatives of the traveltime with respect to the model parameters. The Frechét derivatives are used in two ways. First, a two‐term Taylor series is constructed to relate variations in the model parameters to the corresponding perturbations in the traveltimes. This makes it possible, based on the results of a single ray tracing in a relatively simple model, to predict traveltimes for a range of more complicated models, without any additional ray tracing. Second, singular‐value decomposition of the Frechét matrix determines the influence of various model parameters on common‐source and common‐midpoint traveltimes. The singular‐value analysis shows that common‐source traveltimes depend mainly on the reflector position and shape. The common‐midpoint traveltimes also contain additional information about lateral velocity heterogeneity and anisotropy. However, both of these parameters affect the traveltimes in similar ways and so usually cannot be determined separately.

Optimized Model Parameterization using Compact Full Waveform Inversion

2020 ◽  
Author(s):  
Linan Xu ◽  
Edgar Manukyan ◽  
Hansruedi Maurer
Keyword(s):  
Large Scale ◽  
Waveform Inversion ◽  
Gradient Methods ◽  
Significant Loss ◽  
Inverse Model ◽  
Eigenvalue Decomposition ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Model Parameterization

Summary Seismic Full Waveform Inversion (FWI) has the potential to produce high-resolution subsurface images, but the computational resources required for realistically sized problems can be prohibitively large. In terms of computational costs, Gauss-Newton algorithms are more attractive than the commonly employed conjugate gradient methods, because the former have favorable convergence properties. However, efficient implementations of Gauss-Newton algorithms require an excessive amount of computer memory for larger problems. To address this issue, we introduce Compact Full Waveform Inversion (CFWI). Here, a suitable inverse model parameterization is sought that allows representing all subsurface features, potentially resolvable by a particular source-receiver deployment, but using only a minimum number of model parameters. In principle, an inverse model parameterization, based on the Eigenvalue decomposition, would be optimal, but this is computationally not feasible for realistic problems. Instead, we present two alternative parameter transformations, namely the Haar and the Hartley transformations, with which similarly good results can be obtained. By means of a suite of numerical experiments, we demonstrate that these transformations allow the number of model parameters to be reduced to only a few percent of the original parameterization without any significant loss of spatial resolution. This facilitates efficient solutions of large-scale FWI problems with explicit Gauss-Newton algorithms.

scholarly journals Estimation of the anisotropy parameters from imaging moveout of diving wave in a factorized anisotropic medium

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. C139-C150 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Anisotropic Medium ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Parameters Estimation ◽  
Model Parameters ◽  
Full Waveform ◽  
Vertical Heterogeneity ◽  
Analytical Formulas ◽  
Anisotropy Parameters ◽  
The One

The importance of diving waves is being realized because they provide long-wavelength model information, which can be used to help invert for the reflection information in full-waveform inversion. The factorized model is defined here as a combination of vertical heterogeneity and constant anisotropy, and it admits closed-form description of the traveltime. We have used these resulting analytical formulas to describe the behavior of diving waves in a factorized anisotropic medium, and we used an approximate imaging moveout formulation (residual moveout after imaging) to update the velocity model when the wrong model parameters (isotropic assumption) were used for imaging. We then used these analytical representations of the image moveout to establish a semblance analysis framework to search for the optimal anisotropic parameters. We have also discussed different parameterizations of the factorized medium to find the one that gave the best accuracy in anisotropy parameters estimation.

scholarly journals Electric Potential and Fréchet Derivatives for a Uniform Anisotropic Medium with a Tilted Axis of Symmetry

2009 ◽  
Vol 166 (4) ◽  
pp. 673-699 ◽  
Author(s):  
S. A. Greenhalgh ◽  
L. Marescot ◽  
B. Zhou ◽  
M. Greenhalgh ◽  
T. Wiese
Keyword(s):  
Anisotropic Medium ◽  
Electric Potential ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Axis Of Symmetry

DC resistivity Frechet derivatives for a uniform anisotropic medium with a tilted axis of symmetry

2009 ◽  
Vol 2009 (1) ◽  
pp. 1
Author(s):  
T. Wiese ◽  
S. Greenhalgh ◽  
B. Zhou ◽  
L. Marescot ◽  
M. Greenhalgh
Keyword(s):  
Anisotropic Medium ◽  
Dc Resistivity ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Axis Of Symmetry

Seismic waveform modelling and inversion with velocity-deviatoric stress-isotropic pressure formulation

2021 ◽  
Author(s):  
Yi Zhang ◽  
Luca de Siena ◽  
Alexey Stovas
Keyword(s):  
Waveform Inversion ◽  
Real Data ◽  
Deviatoric Stress ◽  
Model Parameters ◽  
Perfectly Matched Layers ◽  
Current Form ◽  
Anisotropic Models ◽  
Seismic Waveform ◽  
Adjoint State ◽  
Isotropic Pressure

<p>In waveform inversion, most of the existing adjoint-state methods are based on the second-order elastic wave equations subject to displacement. The implementation of the acoustic-elastic coupling problem and free-surface in this formulation is not explicit, especially for arbitrary boundaries. The formulation of velocity-deviatoric stress-isotropic pressure can tackle the above issue. We firstly review the difference between velocity stress equations and velocity-deviatoric stress-isotropic pressure equations. Then the adjoint state of the velocity-stress equations are derived, decomposing stresses into their deviatoric and isotropic parts. To simulate the unbounded wavefield, perfectly matched layers (PML) are embedded into the system of equations. It is modified for cheap computation, which avoids PML-related memory variables by applying complex coordinate stretch to three Cartesian axes in parallel.</p><p>A 3D velocity-deviatoric stress-isotropic stress formulation is implemented with the staggered finite-difference method for several synthetic models (including anisotropic models). And inversions are then performed to reconstruct the model parameters, which is followed by a sensitivity analysis.</p><p>This method has the potential to be used with real data, both for active and passive seismics. However, in its current form, since it does not treat fluid/anisotropic solid interfaces correctly, it is limited to fluid or isotropic solid problems.</p>

3-D seismic full-waveform inversion of the Taurus-Zagros region in Iran and Turkey

2020 ◽  
Author(s):  
Neda Masouminia ◽  
Dirk-Philip van Herwaarden ◽  
Sölvi Thrastarson ◽  
Habib Rahimi ◽  
Lion Krischer ◽  
...  
Keyword(s):  
Upper Mantle ◽  
Waveform Inversion ◽  
Spectral Element ◽  
Earth Model ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Geophysical Research ◽  
Seismic Stations ◽  
Full Waveform ◽  
Seismic Waveform

<p><span>We present an interpretation of a 3-D velocity model resulting from a regional analysis of earthquake waveforms. This model contains 3-D structure of the crust and upper mantle beneath the Arabian-Eurasian collision zone in eastern Turkey and Iran. We use full-waveform inversion (FWI) of three-component recordings from permanent networks. FWI can exploit all parts of a seismogram, including body and multi-mode surface waves in a broad range of frequencies. This allows us to constrain seismic structure of both the crust and the upper mantle.</span></p><p><span>In our method we simulate 3-D visco-elastic wavefields using a spectral-element method (Fichtner <em>et al,</em>2018). Our numerical mesh honors topography of the surface. We compare observed and synthetic waveforms using time-frequency phase misfits. Using adjoint techniques, we then compute sensitivity kernels with respect to the model parameters, which are V<sub>SV</sub>, V<sub>SH</sub>, V<sub>PV</sub>, and V<sub>PH</sub>. Finally, the kernels enable the iterative solution of the nonlinear inverse problem with the help of the L-BFGS algorithm and without a need for crustal corrections.</span></p><p><span>For this study we obtained seismic waveform data of 59 earthquakes within the magnitude range of Mw 4.5 to 6.3 that occurred in the region between 2012 and 2016. These events were recorded by 398 broadband seismic stations belonging to the two national Iranian networks and freely available seismic stations of the Turkish Network, made available by IRIS.</span></p><p> <span>Starting from the first generation of the Collaborative Seismic Earth Model (Afanasiev <em>et al</em>.2019), we first constrained longer-wavelength structure. To this end, we considered 3-component recordings from a subset of 37 events in the period range from 50 to 80 s. This band was successively broadened by reducing the shorter period from 50 s to 40 s, and finally to 20 s. For each period band, the number and the length of measurement windows are increased; the number of events is also increased to 59 to use the complete dataset. After 46 iterations our model can explain recordings of events, which were not used in the inversion. The results provide to discuss about high-velocity anomaly beneath the Zagros and the shallow low velocities beneath Central Iran using cross-sections to investigate lateral variation of seismic velocity in the lithosphere.</span></p><p><span>REFERENCES</span></p><p><span>Afanasiev, M., Boehm, C., van Driel, M., Krischer, L., Rietmann, M., May, D. A., Knepley, M. G., Fichtner, A., 2019. Modular and flexible spectral-element waveform modelling in two and three dimensions. Geophysical Journal International 216, 1675-1692, doi: 10.1093/gji/ggy469.</span></p><p><span>Fichtner, A., van Herwaarden, D.-P., Afanasiev, M., Simute, S., Krischer, L., Cubuk-Sabuncu, Y., Taymaz, T., Colli, L., Saygin, E., Villasenor, A., Trampert, J., Cupillard, P., Bunge, H.-P., Igel, H., 2018. The Collaborative Seismic Earth Model: Generation I. Geophysical Research Letters 45, doi: 10.1029/2018GL077338.</span></p>

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

A new parameter set for anisotropic multiparameter full-waveform inversion and application to a North Sea data set

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. U25-U38 ◽  
Author(s):  
Nuno V. da Silva ◽  
Andrew Ratcliffe ◽  
Vetle Vinje ◽  
Graham Conroy
Keyword(s):  
North Sea ◽  
Waveform Inversion ◽  
Real Data ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Radiation Patterns ◽  
Data Set ◽  
Full Waveform ◽  
Seismic Image ◽  
Central North Sea

Parameterization lies at the center of anisotropic full-waveform inversion (FWI) with multiparameter updates. This is because FWI aims to update the long and short wavelengths of the perturbations. Thus, it is important that the parameterization accommodates this. Recently, there has been an intensive effort to determine the optimal parameterization, centering the fundamental discussion mainly on the analysis of radiation patterns for each one of these parameterizations, and aiming to determine which is best suited for multiparameter inversion. We have developed a new parameterization in the scope of FWI, based on the concept of kinematically equivalent media, as originally proposed in other areas of seismic data analysis. Our analysis is also based on radiation patterns, as well as the relation between the perturbation of this set of parameters and perturbation in traveltime. The radiation pattern reveals that this parameterization combines some of the characteristics of parameterizations with one velocity and two Thomsen’s parameters and parameterizations using two velocities and one Thomsen’s parameter. The study of perturbation of traveltime with perturbation of model parameters shows that the new parameterization is less ambiguous when relating these quantities in comparison with other more commonly used parameterizations. We have concluded that our new parameterization is well-suited for inverting diving waves, which are of paramount importance to carry out practical FWI successfully. We have demonstrated that the new parameterization produces good inversion results with synthetic and real data examples. In the latter case of the real data example from the Central North Sea, the inverted models show good agreement with the geologic structures, leading to an improvement of the seismic image and flatness of the common image gathers.

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