Which data residual norm for robust elastic frequency-domain full waveform inversion?

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. R37-R46 ◽  
Author(s):  
Romain Brossier ◽  
Stéphane Operto ◽  
Jean Virieux
Keyword(s):  
Frequency Domain ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Sampling Interval ◽  
Data Sets ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Fitting Procedure ◽  
Full Waveform ◽  
Threshold Criterion

Elastic full-waveform inversion is an ill-posed data-fitting procedure that is sensitive to noise, inaccuracies of the starting model, definition of multiparameter classes, and inaccurate modeling of wavefield amplitudes. We have investigated the performance of different minimization functionals as the least-squares norm [Formula: see text], the least-absolute-values norm [Formula: see text], and combinations of both (the Huber and so-called hybrid criteria) with reference to two noisy offshore (Valhall model) and onshore (overthrust model) synthetic data sets. The four minimization functionals were implemented in 2D elastic frequency-domain full-waveform inversion (FWI), where efficient multiscale strategies were designed by successive inversions of a few increasing frequencies. For the offshore and onshore case studies, the [Formula: see text]-norm provided the most reliable models for P- and S-wave velocities ([Formula: see text] and[Formula: see text]), even when strongly decimated data sets that correspond to few frequencies were used in the inversion and when outliers polluted the data. The [Formula: see text]-norm can provide reliable results in the presence of uniform white noise for [Formula: see text] and [Formula: see text] if the data redundancy is increased by refining the frequency sampling interval in the inversion at the expense of computational efficiency. The [Formula: see text]-norm and the Huber and hybrid criteria, unlike the [Formula: see text]-norm, allow for successful imaging of the [Formula: see text] model from noisy data in a soft-seabed environment, where the P-to-S-waves have a small footprint in the data. However, the Huber and hybrid criteria are sensitive to a threshold criterion that controls the transition between the criteria and that requires tedious trial-and-error investigations for reliable estimation. The [Formula: see text]-norm provides a robust alternative to the [Formula: see text]-norm for inverting decimated data sets in the framework of efficient frequency-domain FWI.

2D frequency-domain elastic full-waveform inversion using the block-diagonal pseudo-Hessian approximation

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. R247-R259 ◽  
Author(s):  
Yuwei Wang ◽  
Liangguo Dong ◽  
Yuzhu Liu ◽  
Jizhong Yang
Keyword(s):  
Frequency Domain ◽  
Wave Velocity ◽  
Hessian Matrix ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Trade Off ◽  
Full Waveform ◽  
Trade Offs ◽  
Block Diagonal

Elastic full-waveform inversion (EFWI) of multicomponent seismic data is a powerful tool for estimating the subsurface elastic parameters with high accuracy. However, the trade-offs between multiple parameters increase the nonlinearity of EFWI. Although the conventional diagonal-approximate Hessian matrix describes the illumination and limited bandwidth effects, it ignores the trade-off effects and decreases the convergence rate of EFWI. We have developed a block-diagonal pseudo-Hessian operator for 2D frequency-domain EFWI to take into account the approximate trade-offs among the P-wave (compressional-wave) velocity, S-wave (shear-wave) velocity, and density without extra computational costs on forward simulations. The Hessian matrix tends toward a block-diagonal matrix as the frequency grows to infinity; thus, the proposed block-diagonal pseudo-Hessian matrix is more accurate at higher frequencies. The inverse of the block-diagonal pseudo-Hessian matrix is used as a preconditioner for the nonlinear conjugate-gradient method to simultaneously reconstruct P- and S-wave velocities and density. This approach effectively mitigates the crosstalk artifacts by correcting the gradients from the trade-off effects and produces more rapid inversion convergence, which becomes more significant at higher frequencies. Synthetic experiments on an inclusion model and the elastic Marmousi2 model demonstrate its feasibility and validity in EFWI.

Application of a new 2D time-domain full-waveform inversion scheme to crosshole radar data

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. J53-J64 ◽  
Author(s):  
Jacques R. Ernst ◽  
Alan G. Green ◽  
Hansruedi Maurer ◽  
Klaus Holliger
Keyword(s):  
Time Domain ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Radar Data ◽  
Data Sets ◽  
Full Waveform Inversion ◽  
Arrival Times ◽  
Full Waveform ◽  
First Arrival ◽  
Radar Wave

Crosshole radar tomography is a useful tool in diverse investigations in geology, hydrogeology, and engineering. Conventional tomograms provided by standard ray-based techniques have limited resolution, primarily because only a fraction of the information contained in the radar data (i.e., the first-arrival times and maximum first-cycle amplitudes) is included in the inversion. To increase the resolution of radar tomograms, we have developed a versatile full-waveform inversion scheme that is based on a finite-difference time-domain solution of Maxwell’s equations. This scheme largely accounts for the 3D nature of radar-wave propagation and includes an efficient method for extracting the source wavelet from the radar data. After demonstrating the potential of the new scheme on two realistic synthetic data sets, we apply it to two crosshole field data sets acquired in very different geologic/hydrogeologic environments. These are the first applications of full-waveform tomography to observed crosshole radar data. The resolution of all full-waveform tomograms is shown to be markedly superior to that of the associated ray tomograms. Small subsurface features a fraction of the dominant radar wavelength and boundaries between distinct geological/hydrological units are sharply imaged in the full-waveform tomograms.

scholarly journals Elastic envelope inversion using multicomponent seismic data without low frequency

2014 ◽  
Vol 1 (2) ◽  
pp. 1757-1802
Author(s):  
C. Huang ◽  
L. Dong ◽  
Y. Liu ◽  
B. Chi
Keyword(s):  
Seismic Data ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Low Frequency ◽  
Velocity Model ◽  
Inversion Method ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Full Waveform ◽  
Multicomponent Data

Abstract. Low frequency is a key issue to reduce the nonlinearity of elastic full waveform inversion. Hence, the lack of low frequency in recorded seismic data is one of the most challenging problems in elastic full waveform inversion. Theoretical derivations and numerical analysis are presented in this paper to show that envelope operator can retrieve strong low frequency modulation signal demodulated in multicomponent data, no matter what the frequency bands of the data is. With the benefit of such low frequency information, we use elastic envelope of multicomponent data to construct the objective function and present an elastic envelope inversion method to recover the long-wavelength components of the subsurface model, especially for the S-wave velocity model. Numerical tests using synthetic data for the Marmousi-II model prove the effectiveness of the proposed elastic envelope inversion method, especially when low frequency is missing in multicomponent data and when initial model is far from the true model. The elastic envelope can reduce the nonlinearity of inversion and can provide an excellent starting model.

Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCC105-WCC118 ◽  
Author(s):  
Romain Brossier ◽  
Stéphane Operto ◽  
Jean Virieux
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Frequency Domain ◽  
Wave Velocity ◽  
Waveform Inversion ◽  
Surface Effects ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Velocity Models ◽  
Full Waveform

Quantitative imaging of the elastic properties of the subsurface at depth is essential for civil engineering applications and oil- and gas-reservoir characterization. A realistic synthetic example provides for an assessment of the potential and limits of 2D elastic full-waveform inversion (FWI) of wide-aperture seismic data for recovering high-resolution P- and S-wave velocity models of complex onshore structures. FWI of land data is challenging because of the increased nonlinearity introduced by free-surface effects such as the propagation of surface waves in the heterogeneous near-surface. Moreover, the short wavelengths of the shear wavefield require an accurate S-wave velocity starting model if low frequencies are unavailable in the data. We evaluated different multiscale strategies with the aim of mitigating the nonlinearities. Massively parallel full-waveform inversion was implemented in the frequency domain. The numerical optimization relies on a limited-memory quasi-Newton algorithm thatoutperforms the more classic preconditioned conjugate-gradient algorithm. The forward problem is based upon a discontinuous Galerkin (DG) method on triangular mesh, which allows accurate modeling of free-surface effects. Sequential inversions of increasing frequencies define the most natural level of hierarchy in multiscale imaging. In the case of land data involving surface waves, the regularization introduced by hierarchical frequency inversions is not enough for adequate convergence of the inversion. A second level of hierarchy implemented with complex-valued frequencies is necessary and provides convergence of the inversion toward acceptable P- and S-wave velocity models. Among the possible strategies for sampling frequencies in the inversion, successive inversions of slightly overlapping frequency groups is the most reliable when compared to the more standard sequential inversion of single frequencies. This suggests that simultaneous inversion of multiple frequencies is critical when considering complex wave phenomena.

2D frequency-domain elastic full-waveform inversion using time-domain modeling and a multistep-length gradient approach

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. R41-R53 ◽  
Author(s):  
Kun Xu ◽  
George A. McMechan
Keyword(s):  
Fourier Transform ◽  
Frequency Domain ◽  
Time Domain ◽  
Grid Point ◽  
Waveform Inversion ◽  
Synthetic Data ◽  
Full Waveform Inversion ◽  
Domain Modeling ◽  
Full Waveform ◽  
Gradient Approach

To decouple the parameters in elastic full-waveform inversion (FWI), we evaluated a new multistep-length gradient approach to assign individual weights separately for each parameter gradient and search for an optimal step length along the composite gradient direction. To perform wavefield extrapolations for the inversion, we used parallelized high-precision finite-element (FE) modeling in the time domain. The inversion was implemented in the frequency domain; the data were obtained at every subsurface grid point using the discrete Fourier transform at each time-domain extrapolation step. We also used frequency selection to reduce cycle skipping, time windowing to remove the artifacts associated with different source spatial patterns between the test and predicted data, and source wavelet estimation at the receivers over the full frequency spectrum by using a fast Fourier transform. In the inversion, the velocity and density reconstructions behaved differently; as a low-wavenumber tomography (for velocities) and as a high-wavenumber migration (for density). Because velocities and density were coupled to some extent, variations were usually underestimated (smoothed) for [Formula: see text] and [Formula: see text] and correspondingly overestimated (sharpened) for [Formula: see text]. The impedances [Formula: see text] and [Formula: see text] from the products of the velocity and density results compensated for the under- or overestimations of their variations, so the recovered impedances were closer to the correct ones than [Formula: see text], [Formula: see text], and [Formula: see text] were separately. Simultaneous reconstruction of [Formula: see text], [Formula: see text], and [Formula: see text] was robust on the FE and finite-difference synthetic data (without surface waves) from the elastic Marmousi-2 model; satisfactory results are obtained for [Formula: see text], [Formula: see text], [Formula: see text], and the recovered [Formula: see text] and [Formula: see text] from their products. Convergence is fast, needing only a few tens of iterations, rather than a few hundreds of iterations that are typical in most other elastic FWI algorithms.

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.

scholarly journals Multi-scale time-frequency domain full waveform inversion with a weighted local correlation-phase misfit function

2019 ◽  
Vol 16 (6) ◽  
pp. 1017-1031 ◽  
Author(s):  
Yong Hu ◽  
Liguo Han ◽  
Rushan Wu ◽  
Yongzhong Xu
Keyword(s):  
Frequency Domain ◽  
Seismic Data ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Full Waveform Inversion ◽  
Local Correlation ◽  
Time Frequency ◽  
Model Dependence ◽  
Full Waveform ◽  
Frequency Components

Abstract Full Waveform Inversion (FWI) is based on the least squares algorithm to minimize the difference between the synthetic and observed data, which is a promising technique for high-resolution velocity inversion. However, the FWI method is characterized by strong model dependence, because the ultra-low-frequency components in the field seismic data are usually not available. In this work, to reduce the model dependence of the FWI method, we introduce a Weighted Local Correlation-phase based FWI method (WLCFWI), which emphasizes the correlation phase between the synthetic and observed data in the time-frequency domain. The local correlation-phase misfit function combines the advantages of phase and normalized correlation function, and has an enormous potential for reducing the model dependence and improving FWI results. Besides, in the correlation-phase misfit function, the amplitude information is treated as a weighting factor, which emphasizes the phase similarity between synthetic and observed data. Numerical examples and the analysis of the misfit function show that the WLCFWI method has a strong ability to reduce model dependence, even if the seismic data are devoid of low-frequency components and contain strong Gaussian noise.

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