Preconditioning of full‐waveform inversion in viscoacoustic media

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 130-145 ◽  
Author(s):  
Emmanuel Causse ◽  
Rune Mittet ◽  
Bjørn Ursin
Keyword(s):  
Waveform Inversion ◽  
Synthetic Data ◽  
Phase Distortion ◽  
Full Waveform Inversion ◽  
Amplitude Distortion ◽  
Full Waveform ◽  
Hessian Operator ◽  
Gradient Based ◽  
Dispersion And Attenuation ◽  
Phase Spectra

Intrinsic absorption in the earth affects the amplitude and phase spectra of the seismic wavefields and records, and may degrade significantly the results of acoustic full‐waveform inversion. Amplitude distortion affects the strength of the scatterers and decreases the resolution. Phase distortion may result in mislocated interfaces. We show that viscoacoustic gradient‐based inversion algorithms (e.g., steepest descent or conjugate gradients) compensate for the effects of phase distortion, but not for the effects of amplitude distortion. To solve this problem at a reasonable numerical cost, we have designed two new forms of preconditioning derived from an analysis of the inverse Hessian operator. The first type of preconditioning is a frequency‐dependent compensation for dispersion and attenuation, which involves two extra modeling steps with inverse absorption (amplification) at each iteration. The second type only corrects the strength of the recovered scatterers, and requires two extra modeling steps at the first iteration only. The new preconditioning methods have been incorporated into a finite‐difference inversion scheme for viscoacoustic media. Numerical tests on noise‐free synthetic data illustrate and support the theory.

scholarly journals Adding Prior Information in FWI through Relative Entropy

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 599
Author(s):  
Danilo Cruz ◽  
João de Araújo ◽  
Carlos da Costa ◽  
Carlos da Silva
Keyword(s):  
Inverse Problem ◽  
Relative Entropy ◽  
Global Minimum ◽  
Prior Information ◽  
Waveform Inversion ◽  
Petroleum Industry ◽  
Synthetic Data ◽  
Full Waveform Inversion ◽  
Full Waveform

Full waveform inversion is an advantageous technique for obtaining high-resolution subsurface information. In the petroleum industry, mainly in reservoir characterisation, it is common to use information from wells as previous information to decrease the ambiguity of the obtained results. For this, we propose adding a relative entropy term to the formalism of the full waveform inversion. In this context, entropy will be just a nomenclature for regularisation and will have the role of helping the converge to the global minimum. The application of entropy in inverse problems usually involves formulating the problem, so that it is possible to use statistical concepts. To avoid this step, we propose a deterministic application to the full waveform inversion. We will discuss some aspects of relative entropy and show three different ways of using them to add prior information through entropy in the inverse problem. We use a dynamic weighting scheme to add prior information through entropy. The idea is that the prior information can help to find the path of the global minimum at the beginning of the inversion process. In all cases, the prior information can be incorporated very quickly into the full waveform inversion and lead the inversion to the desired solution. When we include the logarithmic weighting that constitutes entropy to the inverse problem, we will suppress the low-intensity ripples and sharpen the point events. Thus, the addition of entropy relative to full waveform inversion can provide a result with better resolution. In regions where salt is present in the BP 2004 model, we obtained a significant improvement by adding prior information through the relative entropy for synthetic data. We will show that the prior information added through entropy in full-waveform inversion formalism will prove to be a way to avoid local minimums.

Tackling cycle skipping in full-waveform inversion with intermediate data

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. R411-R427 ◽  
Author(s):  
Gang Yao ◽  
Nuno V. da Silva ◽  
Michael Warner ◽  
Di Wu ◽  
Chenhao Yang
Keyword(s):  
Objective Function ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Intermediate Data ◽  
Full Waveform ◽  
L2 Norm ◽  
Recorded Data

Full-waveform inversion (FWI) is a promising technique for recovering the earth models for exploration geophysics and global seismology. FWI is generally formulated as the minimization of an objective function, defined as the L2-norm of the data residuals. The nonconvex nature of this objective function is one of the main obstacles for the successful application of FWI. A key manifestation of this nonconvexity is cycle skipping, which happens if the predicted data are more than half a cycle away from the recorded data. We have developed the concept of intermediate data for tackling cycle skipping. This intermediate data set is created to sit between predicted and recorded data, and it is less than half a cycle away from the predicted data. Inverting the intermediate data rather than the cycle-skipped recorded data can then circumvent cycle skipping. We applied this concept to invert cycle-skipped first arrivals. First, we picked up the first breaks of the predicted data and the recorded data. Second, we linearly scaled down the time difference between the two first breaks of each shot into a series of time shifts, the maximum of which was less than half a cycle, for each trace in this shot. Third, we moved the predicted data with the corresponding time shifts to create the intermediate data. Finally, we inverted the intermediate data rather than the recorded data. Because the intermediate data are not cycle-skipped and contain the traveltime information of the recorded data, FWI with intermediate data updates the background velocity model in the correct direction. Thus, it produces a background velocity model accurate enough for carrying out conventional FWI to rebuild the intermediate- and short-wavelength components of the velocity model. Our numerical examples using synthetic data validate the intermediate-data concept for tackling cycle skipping and demonstrate its effectiveness for the application to first arrivals.

scholarly journals Constrained Full Waveform Inversion for Borehole Multicomponent Seismic Data

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 45
Author(s):  
Marwan Charara ◽  
Christophe Barnes
Keyword(s):  
Inverse Problem ◽  
Spatial Correlation ◽  
Seismic Data ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Model Space ◽  
Elastic Model ◽  
Full Waveform Inversion ◽  
Spatial Correlations ◽  
Full Waveform

Full-waveform inversion for borehole seismic data is an ill-posed problem and constraining the problem is crucial. Constraints can be imposed on the data and model space through covariance matrices. Usually, they are set to a diagonal matrix. For the data space, signal polarization information can be used to evaluate the data uncertainties. The inversion forces the synthetic data to fit the polarization of observed data. A synthetic inversion for a 2D-2C data estimating a 1D elastic model shows a clear improvement, especially at the level of the receivers. For the model space, horizontal and vertical spatial correlations using a Laplace distribution can be used to fill the model space covariance matrix. This approach reduces the degree of freedom of the inverse problem, which can be quantitatively evaluated. Strong horizontal spatial correlation distances favor a tabular geological model whenever it does not contradict the data. The relaxation of the spatial correlation distances from large to small during the iterative inversion process allows the recovery of geological objects of the same size, which regularizes the inverse problem. Synthetic constrained and unconstrained inversions for 2D-2C crosswell data show the clear improvement of the inversion results when constraints are used.

Seismic full-waveform inversion using minimization of virtual scattering sources

Geophysics ◽  
2020 ◽  
Vol 85 (3) ◽  
pp. R299-R311
Author(s):  
Donguk Lee ◽  
Sukjoon Pyun
Keyword(s):  
Newton Method ◽  
Scattering Problem ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Inverse Scattering Problem ◽  
Simulated Data ◽  
Model Space ◽  
Full Waveform Inversion ◽  
Velocity Inversion ◽  
Full Waveform

Full-waveform inversion (FWI) is a powerful tool for imaging underground structures with high resolution; however, this approach commonly suffers from the cycle-skipping issue. Recently, various FWI methods have been suggested to address this problem. Such methods are mainly classified into either data-space manipulation or model-space extension. We developed an alternative FWI method that belongs to the latter class. First, we define the virtual scattering source based on perturbation theory. The virtual scattering source is estimated by minimizing the differences between observed and simulated data with a regularization term penalizing the weighted virtual scattering source. The inverse problem for obtaining the virtual scattering source can be solved by the linear conjugate gradient method. The inverted virtual scattering source is used to update the wavefields; thus, it helps FWI to better approximate the nonlinearity of the inverse scattering problem. As the second step, the virtual scattering source is minimized to invert the velocity model. By assuming that the variation of the reconstructed wavefield is negligible, we can apply an approximated full Newton method to the velocity inversion with reasonable cost comparable to the Gauss-Newton method. From the numerical examples using synthetic data, we confirm that the proposed method performs better and more robust than the simple gradient-based FWI method. In addition, we show that our objective function has fewer local minima, which helps to mitigate the cycle-skipping problem.

Source-domain full waveform inversions

Geophysics ◽  
2020 ◽  
pp. 1-50
Author(s):  
Yulang Wu ◽  
George A. McMechan
Keyword(s):  
Differential Operators ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Velocity Model ◽  
Low Noise ◽  
Full Waveform Inversion ◽  
Virtual Source ◽  
Reflected Waves ◽  
Source Domain ◽  
Full Waveform

Conventional full waveform inversion (FWI) updates a velocity model by minimizing the data residuals between predicted and observed data, at the receiver positions. We propose a new full waveform inversion to update the velocity model by minimizing virtual source artifacts, at the receiver positions, in the source domain (SFWI). Virtual source artifacts are created by replacing the propagating source wavefield by the forward-time observed data at the receiver positions, as a data-residual constraint. Therefore, no matter whether the velocity model is correct or not, the data residuals, at the receiver positions, are always forced to be zero. If the velocity model is correct, this data-residual constraint has no effect on the wavefield, since the predicted data is the same as the observed data. However, if the estimated velocity model is incorrect, the mismatch between the replaced forward-time observed data and the incorrect predicted upgoing waves (e.g., reflected waves) at the receiver positions, will produce downgoing artifact waves. Thus, the data-residual constraint behaves as a virtual source to create artifact wavefields. By minimizing the virtual source artifacts (equivalent to producing the artifact wavefield), the velocity model can be iteratively updated toward the true velocity model. Similar to conventional FWI, SFWI can be implemented in either the frequency or the time domain, which is unlike previous source-domain solutions, which have to be implemented only in the frequency domain, to solve the normal equations. SFWI does more over-fitting of noisy observed data than conventional FWI does, because noise is amplified by the differential operators when calculating the virtual source artifacts. Tests on synthetic data show that the SFWI inverts for the velocity model more accurately than conventional FWI for noise-free or low-noise data.

scholarly journals Accelerating full-waveform inversion with attenuation compensation

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. A13-A20 ◽  
Author(s):  
Zhiguang Xue ◽  
Junzhe Sun ◽  
Sergey Fomel ◽  
Tieyuan Zhu
Keyword(s):  
Wave Equation ◽  
Convergence Rate ◽  
Waveform Inversion ◽  
Adjoint Operator ◽  
Synthetic Data ◽  
Low Rank ◽  
Full Waveform Inversion ◽  
Time Step ◽  
Velocity Models ◽  
Full Waveform

The calculation of the gradient in full-waveform inversion (FWI) usually involves crosscorrelating the forward-propagated source wavefield and the back-propagated data residual wavefield at each time step. In the real earth, propagating waves are typically attenuated due to the viscoelasticity, which results in an attenuated gradient for FWI. Replacing the attenuated true gradient with a [Formula: see text]-compensated gradient can accelerate the convergence rate of the inversion process. We have used a phase-dispersion and an amplitude-loss decoupled constant-[Formula: see text] wave equation to formulate a viscoacoustic FWI. We used this wave equation to generate a [Formula: see text]-compensated gradient, which recovers amplitudes while preserving the correct kinematics. We construct an exact adjoint operator in a discretized form using the low-rank wave extrapolation technique, and we implement the gradient compensation by reversing the sign of the amplitude-loss term in the forward and adjoint operators. This leads to a [Formula: see text]-dependent gradient preconditioning method. Using numerical tests with synthetic data, we demonstrate that the proposed viscoacoustic FWI using a constant-[Formula: see text] wave equation is capable of producing high-quality velocity models, and our [Formula: see text]-compensated gradient accelerates its convergence rate.

From classical reflectivity-to-velocity inversion to full-waveform inversion using phase-modified and deconvolved reverse time migration images

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. S31-S49 ◽  
Author(s):  
Chen Tang ◽  
George A. McMechan
Keyword(s):  
Waveform Inversion ◽  
Time Integration ◽  
Multiscale Approach ◽  
Full Waveform Inversion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Velocity Inversion ◽  
Full Waveform ◽  
Time Migration ◽  
Gradient Based

To obtain a physical understanding of gradient-based descent methods in full-waveform inversion (FWI), we find a connection between the FWI gradient and the image provided by reverse time migration (RTM). The gradient uses the residual data as a virtual source, and RTM uses the observed data directly as the boundary condition, so the FWI gradient is similar to a time integration of the RTM image using the residual data, which physically converts the phase of the reflectivity to the phase of the velocity. Therefore, gradient-based FWI can be connected to the classical reflectivity-to-velocity/impedance inversion (RVI). We have developed a new FWI scheme that provides a self-contained and physically intuitive derivation, which naturally establishes a connection among the amplitude-preserved RTM, the Zoeppritz equations (amplitude variation with angle inversion), and RVI, and combines them into a single framework to produce a preconditioned inversion formula. In this scheme, the relative velocity update is a phase-modified and deconvolved RTM image obtained with the residual data. Consistent with the deconvolution, the multiscale approach applies a gradually widening low-pass frequency filter to the deconvolved wavelet at early iterations, and then it uses the unfiltered deconvolved wavelet for the final iterations. Our numerical testing determined that the new method makes a significant improvement to the quality of the inversion result.

scholarly journals Skeletonized wave-equation inversion in vertical symmetry axis media without too much math

2017 ◽  
Vol 5 (3) ◽  
pp. SO21-SO30 ◽  
Author(s):  
Shihang Feng ◽  
Gerard T. Schuster
Keyword(s):  
Wave Equation ◽  
Symmetry Axis ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Traveltime Inversion ◽  
Anisotropic Models ◽  
Full Waveform ◽  
Starting Model

We have developed a tutorial for skeletonized inversion of pseudo-acoustic anisotropic vertical symmetry axis (VTI) data. We first invert for the anisotropic models using wave-equation traveltime inversion. Here, the skeletonized data are the traveltimes of transmitted and/or reflected arrivals that lead to simpler misfit functions and more robust convergence compared with full-waveform inversion. This provides a good starting model for waveform inversion. The effectiveness of this procedure is illustrated with synthetic data examples and a marine data set recorded in the Gulf of Mexico.

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