A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. R307-R323 ◽  
Author(s):  
Hejun Zhu ◽  
Siwei Li ◽  
Sergey Fomel ◽  
Georg Stadler ◽  
Omar Ghattas
Keyword(s):  
Waveform Inversion ◽  
Computational Cost ◽  
Covariance Operator ◽  
Maximum A Posteriori ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Posterior Distributions ◽  
A Posteriori ◽  
Model Uncertainties ◽  
Full Waveform

Full-waveform inversion (FWI) enables us to obtain high-resolution subsurface images; however, estimating model uncertainties associated with this technique is still a challenging problem. We have used a Bayesian inference framework to estimate model uncertainties associated with FWI. The uncertainties were assessed based on an a posteriori covariance operator, evaluated at the maximum a posteriori model. For the prior distribution, we have used a spatially nonstationary covariance operator based on a plane-wave construction with local dips measured from migrated images. Preconditioned frequency-domain FWI was used to estimate the maximum a posteriori model. Efficient manipulation of the posterior covariance was based on a low-rank approximation of the data misfit Hessian preconditioned by the prior covariance operator. The strong decay of the singular values indicated that data were mostly informative about a low-dimensional subspace of model parameters. To reduce computational cost of the randomized singular value decomposition, we have used a Hessian approximation based on point-spread functions. The 2D numerical examples with synthetic data confirmed that the method can effectively estimate uncertainties for FWI. Visual comparisons of random samples drawn from the prior and posterior distributions have allowed us to evaluate model uncertainties. Furthermore, we found out how statistical quantities, such as means and pointwise standard deviation fields, can be efficiently extracted from the prior and posterior distributions. These fields helped us to objectively assess subsurface images provided by FWI.

Full-waveform inversion for signal enhancement of weak amplitude phases using beamforming and adjoint methods

2020 ◽  
Author(s):  
Maria Koroni ◽  
Andreas Fichtner
Keyword(s):  
Epicentral Distance ◽  
Average Distance ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Signal Enhancement ◽  
Time Shift ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Phase Alignment

<p>In this study, we develop a new adjoint- and full-waveform inversion approach for low-amplitude seismic phases that are typically below noise in individual recordings. The methodology aims at enhancing weak signals from body wave phases, which can be used in full-waveform inversion for inferring structural and boundary parameters in the earth. The new approach is based on the formulation of misfit functionals and corresponding adjoint sources for stacks of suitably time-shifted recordings. </p><p>To tackle this problem, we compute synthetic waveforms using spectral-elements for models with and without topographic variations along mantle discontinuities. We focus on global underside reflections which are reportedly almost always undetectable in real seismograms due to their low amplitudes and are considerably affected by topography. We enforce phase alignment on a chosen reference seismogram recorded at an average distance among the selected stations. A time shift towards the reference is applied to all seismograms according to their epicentral distance calculated by 1-D ray tracing. A set of time shifts is calculated by cross-correlation in time windows around predicted traveltimes of the desired phase. Using this set of time shifts, we sum the waveforms creating the main stack for each model.</p><p>We use the two linear stacks as <em>observed</em> and <em>synthetic</em> (with and without topography, respectively) and develop a least-squares misfit measurement which gives rise to an adjoint source determined by the time shift between stacks. The expectation is that computing the traveltime Fréchet kernel with respect to volumetric and boundary model parameters will show the exact sensitivity of the enhanced signal and save time from computing each station kernel separately. Upon achieving signal enhancement of the desired phases, we can ensure that these can be used for better informing updates of the initial model given the higher quality measurement of the observable.</p><p>This method once fully developed will allow us to leverage information of many recordings by reducing incoherent signal and enhancing weak seismic phases. The computation of sensitivity kernels in our study has a twofold importance. Firstly, it helps us realise whether the stacking technique indeed enhances the desired signal and whether it is ideal for precursor waves. Secondly, the exact sensitivity kernels show us the way of incorporating finite-frequency effects of weak but informative phases and introducing non-linear inversion for improving imaging while reducing some computational cost. </p><p> </p>

scholarly journals Hybrid frequency-domain full-waveform inversion using ray+Born sensitivity kernels

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. R339-R347
Author(s):  
Ramzi Djebbi ◽  
Tariq Alkhalifah
Keyword(s):  
Frequency Domain ◽  
Ray Tracing ◽  
Waveform Inversion ◽  
Hybrid Approach ◽  
Computational Cost ◽  
Equation Modeling ◽  
Inversion Method ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Full Waveform

Full-waveform inversion (FWI) using the scattering integral (SI) approach is an explicit formulation of the inversion optimization problem. The inversion procedure is straightforward, and the dependence of the data residuals on the model parameters is clear. However, the biggest limitation associated with this approach is the huge computational cost in conventional exploration seismology applications. Modeling from each of the source and receiver locations is required to compute the update at every iteration, and that is prohibitively expensive, especially for 3D problems. To deal with this issue, we have developed a hybrid implementation of frequency-domain FWI, in which forward modeling is combined with ray tracing to compute the update. We use the sensitivity kernels computed from dynamic ray tracing to build the gradient. The data residual is still computed using finite-difference wavefield modeling. With ray theory, the Green’s function can be approximated using a coarser grid compared to wave-equation modeling. Therefore, the memory requirements, as well as the computational cost, are reduced significantly. Considering that in transmission FWI long-to-intermediate wavelengths are updated during the early iterations, we obtain accurate inverted models. The inversion scheme captured the anomaly embedded in the homogeneous background medium. For more complex models, the hybrid inversion method helps in improving the initial model with little cost compared with conventional SI inversion approaches. The accuracy of the inversion results shows the effectiveness of the hybrid approach for 3D realistic problems.

scholarly journals Full-waveform inversion in acoustic orthorhombic media and application to a North Sea data set

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. C179-C193 ◽  
Author(s):  
Nabil Masmoudi ◽  
Tariq Alkhalifah
Keyword(s):  
North Sea ◽  
Waveform Inversion ◽  
Computational Cost ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Model Inversion ◽  
Multistage Model ◽  
Full Waveform ◽  
Trade Offs

Full-waveform inversion (FWI) in anisotropic media is challenging, mainly because of the large computational cost, especially in 3D, and the potential trade-offs between the model parameters needed to describe such media. By analyzing the trade-offs and understanding the resolution limits of the inversion, we can constrain FWI to focus on the main parameters the data are sensitive to and push the inversion toward more reliable models of the subsurface. Orthorhombic anisotropy is one of the most practical approximations of the earth subsurface that takes into account the natural horizontal layering and the vertical fracture network. We investigate the feasibility of a multiparameter FWI for an acoustic orthorhombic model described by six parameters. We rely on a suitable parameterization based on the horizontal velocity and five dimensionless anisotropy parameters. This particular parameterization allows a multistage model inversion strategy in which the isotropic, then, the vertical transverse isotropic, and finally the orthorhombic model can be successively updated. We applied our acoustic orthorhombic inversion on the SEG-EAGE overthrust synthetic model. The observed data used in the inversion are obtained from an elastic variable density version of the model. The quality of the inverted model suggests that we may recover only four parameters, with different resolution scales depending on the scattering potential of these parameters. Therefore, these results give useful insights on the expected resolution of the inverted parameters and the potential constraints that could be applied to an orthorhombic model inversion. We determine the efficiency of the inversion approach on real data from the North Sea. The inverted model is in agreement with the geologic structures and well-log information.

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.

FULL-WAVEFORM INVERSION WITH SOURCE AND RECEIVER COUPLING EFFECTS CORRECTION

Geophysics ◽  
2021 ◽  
pp. 1-37
Author(s):  
Xinhai Hu ◽  
Wei Guoqi ◽  
Jianyong Song ◽  
Zhifang Yang ◽  
Minghui Lu ◽  
...  
Keyword(s):  
Waveform Inversion ◽  
Nonlinear Least Squares ◽  
Real Data ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Linear Inversion ◽  
Near Surface ◽  
Model Parameter ◽  
Full Waveform ◽  
Coupling Factors

Coupling factors of sources and receivers vary dramatically due to the strong heterogeneity of near surface, which are as important as the model parameters for the inversion success. We propose a full waveform inversion (FWI) scheme that corrects for variable coupling factors while updating the model parameter. A linear inversion is embedded into the scheme to estimate the source and receiver factors and compute the amplitude weights according to the acquisition geometry. After the weights are introduced in the objective function, the inversion falls into the category of separable nonlinear least-squares problems. Hence, we could use the variable projection technique widely used in source estimation problem to invert the model parameter without the knowledge of source and receiver factors. The efficacy of the inversion scheme is demonstrated with two synthetic examples and one real data test.

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.

Full-waveform inversion of salt models using shape optimization and simulated annealing

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. R793-R804 ◽  
Author(s):  
Debanjan Datta ◽  
Mrinal K. Sen ◽  
Faqi Liu ◽  
Scott Morton
Keyword(s):  
Simulated Annealing ◽  
Least Squares ◽  
Level Set ◽  
Waveform Inversion ◽  
Optimization Techniques ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Velocity Models ◽  
Full Waveform ◽  
Starting Model

A good starting model is imperative in full-waveform inversion (FWI) because it solves a least-squares inversion problem using a local gradient-based optimization method. A suboptimal starting model can result in cycle skipping leading to poor convergence and incorrect estimation of subsurface properties. This problem is especially crucial for salt models because the strong velocity contrasts create substantial time shifts in the modeled seismogram. Incorrect estimation of salt bodies leads to velocity inaccuracies in the sediments because the least-squares gradient aims to reduce traveltime differences without considering the sharp velocity jump between sediments and salt. We have developed a technique to estimate velocity models containing salt bodies using a combination of global and local optimization techniques. To stabilize the global optimization algorithm and keep it computationally tractable, we reduce the number of model parameters by using sparse parameterization formulations. The sparse formulation represents sediments using a set of interfaces and velocities across them, whereas a set of ellipses represents the salt body. We use very fast simulated annealing (VFSA) to minimize the misfit between the observed and synthetic data and estimate an optimal model in the sparsely parameterized space. The VFSA inverted model is then used as a starting model in FWI in which the sediments and salt body are updated in the least-squares sense. We partition model updates into sediment and salt updates in which the sediments are updated like conventional FWI, whereas the shape of the salt is updated by taking the zero crossing of an evolving level set surface. Our algorithm is tested on two 2D synthetic salt models, namely, the Sigsbee 2A model and a modified SEG Advanced Modeling Program (SEAM) Phase I model while fixing the top of the salt. We determine the efficiency of the VFSA inversion and imaging improvements from the level set FWI approach and evaluate a few sources of uncertainty in the estimation of salt shapes.

Full waveform inversion with sparse structure constrained regularization

2018 ◽  
Vol 26 (2) ◽  
pp. 243-257 ◽  
Author(s):  
Zichao Yan ◽  
Yanfei Wang
Keyword(s):  
Large Scale ◽  
Regularization Parameter ◽  
Waveform Inversion ◽  
Difference Operators ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Structure Information ◽  
Full Waveform ◽  
Ill Posed ◽  
Scale Optimization

AbstractFull waveform inversion is a large-scale nonlinear and ill-posed problem. We consider applying the regularization technique for full waveform inversion with structure constraints. The structure information was extracted with difference operators with respect to model parameters. And then we establish an {l_{p}}-{l_{q}}-norm constrained minimization model for different choices of parameters p and q. To solve this large-scale optimization problem, a fast gradient method with projection onto convex set and a multiscale inversion strategy are addressed. The regularization parameter is estimated adaptively with respect to the frequency range of the data. Numerical experiments on a layered model and a benchmark SEG/EAGE overthrust model are performed to testify the validity of this proposed regularization scheme.

Image-guided sparse-model full waveform inversion

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. R189-R198 ◽  
Author(s):  
Yong Ma ◽  
Dave Hale ◽  
Bin Gong ◽  
Zhaobo (Joe) Meng
Keyword(s):  
Waveform Inversion ◽  
Model Space ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Widespread Application ◽  
Low Frequencies ◽  
Sparse Model ◽  
Image Guided ◽  
Full Waveform ◽  
Recorded Data

Multiple problems, including high computational cost, spurious local minima, and solutions with no geologic sense, have prevented widespread application of full waveform inversion (FWI), especially FWI of seismic reflections. These problems are fundamentally related to a large number of model parameters and to the absence of low frequencies in recorded seismograms. Instead of inverting for all the parameters in a dense model, image-guided full waveform inversion inverts for a sparse model space that contains far fewer parameters. We represent a model with a sparse set of values, and from these values, we use image-guided interpolation (IGI) and its adjoint operator to compute finely and uniformly sampled models that can fit recorded data in FWI. Because of this sparse representation, image-guided FWI updates more blocky models, and this blockiness in the model space mitigates the absence of low frequencies in recorded data. Moreover, IGI honors imaged structures, so image-guided FWI built in this way yields models that are geologically sensible.

Simultaneous estimation of velocity and density in acoustic multiparameter full-waveform inversion using an improved scattering-integral approach

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. R399-R415 ◽  
Author(s):  
Jizhong Yang ◽  
Yuzhu Liu ◽  
Liangguo Dong
Keyword(s):  
Newton Method ◽  
Waveform Inversion ◽  
Integral Approach ◽  
Normal Equation ◽  
Simultaneous Estimation ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Trade Off ◽  
Full Waveform ◽  
Diffraction Patterns

Density is known to be difficult to reconstruct in multiparameter full-waveform inversion (FWI). This difficulty results from the similarity of the diffraction patterns of velocity and density at small scattering angles. In addition, the sensitivities of seismic data with respect to velocity and density have different orders of magnitude which make the inversion ill-conditioned. The inverse Hessian has been shown to mitigate the coupling effects and rescale the magnitudes of different parameters, such that reliable updates for all parameters are available. We have investigated the possibility of simultaneous estimations of velocity and density in acoustic media using the truncated Gauss-Newton method. The model updates are calculated using a matrix-free conjugate gradient solution of the Gauss-Newton normal equation. The gradients of the misfit function with respect to the model parameters and the Hessian-vector products are computed using an improved scattering-integral approach. To give some insights into the trade-off effects between velocity and density, and the imaging resolution in FWI, the sensitivity kernels of both parameters are numerically calculated in homogeneous background models, and their spatial distributions and characteristics are analyzed. The synthetic experiments on a canonical inclusion model and the 2004 BP model confirm that, in cases in which the Gauss-Newton approximate Hessian, especially its off-diagonal blocks, is accurately taken into account, the truncated Gauss-Newton method can effectively mitigate the trade-off effects between velocity and density and provide accelerated convergence rate. Hence, well-resolved velocity and density models are expected.

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