adjoint state
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Geophysics ◽  
2021 ◽  
pp. 1-51
Author(s):  
Yanhua Liu ◽  
Ilya Tsvankin

Time-lapse full-waveform inversion can provide high-resolution information about changes in the reservoir properties during hydrocarbon production and CO2 injection. However, the accuracy of the estimated source wavelet, which is critically important for time-lapse FWI, is often insufficient for field-data applications. The so-called “source-independent” FWI is designed to reduce the influence of the source wavelet on the inversion results. We incorporate the convolution-based source-independent technique into a time-lapse FWI algorithm for VTI (transversely isotropic with a vertical symmetry axis) media. The gradient of the modified FWI objective function is obtained from the adjoint-state method. The algorithm is tested on a model with a graben structure and the modified VTI Marmousi model using three time-lapse strategies (the parallel-difference, sequential-difference, and double-difference methods). The results confirm the ability of the developed methodology to reconstruct the localized time-lapse parameter variations even for a strongly distorted source wavelet. The algorithm remains robust in the presence of moderate noise in the input data but the accuracy of the estimated time-lapse changes depends on the model complexity.


Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Guangchi Xing ◽  
Tieyuan Zhu

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion ( Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. We find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel. After validating the Fréchet kernels using the finite-difference method, we conduct a numerical example to demonstrate the capability of the kernels to characterize both velocity and attenuation anomalies. The kernels of different misfit measurements are presented to investigate their different sensitivities. Our results suggest that rather than the traveltime, the amplitude and the waveform kernels are more suitable to capture attenuation anomalies. These kernels lay the foundation for the multiparameter inversion with the fractional formulation, and the decoupled nature of them promotes our understanding of the significance of different physical processes in the Q-FWI.


2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Samuel Amstutz

PurposeThis paper provides a self-contained introduction to the mathematical aspects of the topological derivative.Design/methodology/approachFull justifications are given on simple model problems following a modern approach based on the averaged adjoint state technique. Extensions are discussed in relation with the literature on the field.FindingsClosed expressions of topological derivatives are obtained and commented.Originality/valueSeveral cases are covered in a unified and didactic presentation. Some elements of proof are novel.


2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Michel C. Delfour

PurposeThe object of the paper is to illustrate how to obtain the topological derivative as a semidifferential in a general and practical mathematical setting for d-dimensional perturbations of a bounded open domain in the n-dimensional Euclidean space.Design/methodology/approachThe underlying methodology uses mathematical notions and powerful tools with ready to check assumptions and ready to use formulas via theorems on the one-sided derivative of parametrized minima and minimax.FindingsThe theory and the examples indicate that the methodology applies to a wide range of problems: (1) compliance and (2) state constrained objective functions where the coupled state/adjoint state equations appear without a posteriori substitution of the adjoint state.Research limitations/implicationsDirect approach that considerably simplifies the analysis and computations.Originality/valueIt was known that the shape derivative was a differential. But the topological derivative is only a semidifferential, that is, a one-sided directional derivative, which is not linear with respect to the direction, and the directions are d-dimensional bounded measures.


2021 ◽  
Vol 5 (3) ◽  
pp. 102
Author(s):  
Fangyuan Wang ◽  
Xiaodi Li ◽  
Zhaojie Zhou

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.


Geophysics ◽  
2021 ◽  
pp. 1-72
Author(s):  
Yu Zhong ◽  
Hanming Gu ◽  
Yangting Liu ◽  
QingHui Mao

Elastic reverse time migration (ERTM) is developed for better characterization of complex structures by imaging multicomponent seismic data. However, conventional ERTM is subject to limitations such as finite recording aperture, limited bandwidth, and imperfect illumination. Elastic least-squares reverse time migration (ELSRTM) can improve imaging accuracy gradually with iterations by minimizing the residuals between observed and calculated multicomponent data. Conventional ELSRTM suffers from crosstalk artifacts caused by coupled elastic wavefields with different wave modes. Decomposing the coupled elastic wavefields into pure P- and S-waves is an effective method to suppress these crosstalk artifacts. Considering the trade-off between calculation accuracy and efficiency, we have developed a new ELSRTM scheme for isotropic media based on decoupled wave equations to suppress these wave mode-related crosstalk artifacts in the images of conventional ELSRTM. Pure wavefields are obtained by solving the decoupled wave equations using the finite-difference (FD) method in our new ELSRTM method. We also derive new decoupled adjoint-state wave equations, which are suitable for the elastic velocity-stress equations in isotropic media. We further propose the gradient equations based on pure wavefields to update the reflectivity images. Synthetic examples demonstrate that our new ELSRTM method can generate images that better represent the subsurface when compared with conventional ERTM and conventional ELSRTM.


Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Carlos A. M. Assis ◽  
Jörg Schleicher

Joint migration inversion (JMI) is a method based on the one-way wave equations that aims at fitting seismic reflection data to estimate an image and a background velocity. The depth-migrated image describes the high spatial-frequency content of the subsurface and, in principle, is true amplitude. The background velocity model accounts mainly for the large spatial-scale kinematic effects of the wave propagation. Looking for a deeper understanding of the method, we briefly review the continuous equations that compose the forward modeling engine of JMI for acoustic media and angle-independent scattering. Then, we use these equations together with the first-order adjoint-state method to arrive at a new formulation of the model gradients. To estimate the image, we combine the second-order adjoint-state method with the truncated-Newton method to obtain the image updates. For the model (velocity) estimation, in comparison to the image update, we reduce the computational cost by simply adopting a diagonal preconditioner for the corresponding gradient in combination with an image-based regularizing function. Based on this formulation, we build our implementation of the JMI algorithm. The proposed image-based regularization of the model estimate allows us to carry over structural information from the estimated image to the jointly estimated background model. As demonstrated by our numerical experiments, this procedure can help to improve the resolution of the estimated model and make it more consistent with the image.


Geophysics ◽  
2021 ◽  
pp. 1-43
Author(s):  
Jiangtao Hu ◽  
Jianliang Qian ◽  
Junxing Cao ◽  
Xingjian Wang ◽  
Huazhong Wang ◽  
...  

First-arrival traveltime tomography is an essential method for obtaining near-surface velocity models. The adjoint-state first-arrival traveltime tomography is appealing due to its straightforward implementation, low computational cost, and low memory consumption. Because solving the point-source isotropic eikonal equation by either ray tracers or eikonal solvers intrinsically corresponds to emanating discrete rays from the source point, the resulting traveltime gradient is singular at the source point, and we denote such a singular pattern the imprint of ray illumination. Because the adjoint-state equation propagates traveltime residuals back to the source point according to the negative traveltime gradient, the resulting adjoint state will inherit such an imprint of ray illumination, leading to singular gradient descent directions when updating the velocity model in the adjoint-state traveltime tomography. To mitigate this imprint, we propose to solve the adjoint-state equation twice but with different boundary conditions: one being taken to be regular data residuals, and the other taken to be ones uniformly, so that we are able to use the latter adjoint state to normalize the regular one and we further use the normalized quantity to serve as the gradient direction to update the velocity model; we call this process the ray-illumination compensation. To overcome the issue of limited aperture, we propose a spatially varying regularization method to stabilize the new gradient direction. A synthetic example demonstrates that the proposed method is able to mitigate the imprint of ray illumination, remove the footprint effect near source points, and provide uniform velocity updates along ray paths. A complex example extracted from the Marmousi2 model and a migration example illustrate that the new method accurately recovers the velocity model, and an offset-dependent inversion strategy can further improve the quality of recovered velocity models.


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