Elastic full-waveform inversion for VTI media: Methodology and sensitivity analysis

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. C53-C68 ◽  
Author(s):  
Nishant Kamath ◽  
Ilya Tsvankin
Keyword(s):  
Objective Function ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Model Parameters ◽  
Valuable Insight ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Adjoint State ◽  
P And Sv Waves ◽  
Vti Media

Most existing implementations of full-waveform inversion (FWI) are limited to acoustic approximations. In this paper, we present an algorithm for time-domain elastic FWI in laterally heterogeneous VTI (transversely isotropic with a vertical symmetry axis) media. The adjoint-state method is employed to derive the gradients of the objective function with respect to the stiffness coefficients and then to a chosen set of VTI parameters. To test the algorithm, we introduce Gaussian anomalies in the Thomsen parameters of a homogeneous VTI medium and perform 2D FWI of multicomponent transmission data for two different model parameterizations. To analyze the sensitivity of the objective function to the model parameters, the Fréchet kernel of FWI is obtained by linearizing the elastic wave equation using the Born approximation and employing the asymptotic Green’s function. The amplitude of the kernel (“radiation pattern”) yields the angle-dependent energy scattered by a perturbation in a certain model parameter. Then we convert the general expressions into simple approximations for the radiation patterns of P- and SV-waves in VTI media. These analytic developments provide valuable insight into the potential of multicomponent elastic FWI and help explain the numerical results for models with Gaussian anomalies in the VTI parameters.

Full-waveform inversion with borehole constraints for elastic VTI media

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. R553-R563
Author(s):  
Sagar Singh ◽  
Ilya Tsvankin ◽  
Ehsan Zabihi Naeini
Keyword(s):  
Objective Function ◽  
Reservoir Characterization ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Rock Physics ◽  
Anisotropic Media ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Trade Offs ◽  
Vti Media

The nonlinearity of full-waveform inversion (FWI) and parameter trade-offs can prevent convergence toward the actual model, especially for elastic anisotropic media. The problems with parameter updating become particularly severe if ultra-low-frequency seismic data are unavailable, and the initial model is not sufficiently accurate. We introduce a robust way to constrain the inversion workflow using borehole information obtained from well logs. These constraints are included in the form of rock-physics relationships for different geologic facies (e.g., shale, sand, salt, and limestone). We develop a multiscale FWI algorithm for transversely isotropic media with a vertical symmetry axis (VTI media) that incorporates facies information through a regularization term in the objective function. That term is updated during the inversion by using the models obtained at the previous inversion stage. To account for lateral heterogeneity between sparse borehole locations, we use an image-guided smoothing algorithm. Numerical testing for structurally complex anisotropic media demonstrates that the facies-based constraints may ensure the convergence of the objective function towards the global minimum in the absence of ultra-low-frequency data and for simple (even 1D) initial models. We test the algorithm on clean data and on surface records contaminated by Gaussian noise. The algorithm also produces a high-resolution facies model, which should be instrumental in reservoir characterization.

scholarly journals Full-waveform inversion, Part 3: Optimization

2018 ◽  
Vol 37 (2) ◽  
pp. 142-145 ◽  
Author(s):  
Philipp Witte ◽  
Mathias Louboutin ◽  
Keegan Lensink ◽  
Michael Lange ◽  
Navjot Kukreja ◽  
...  
Keyword(s):  
Objective Function ◽  
Wave Equations ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Acoustic Wave Equation ◽  
Automated Generation ◽  
Domain Specific ◽  
Full Waveform ◽  
Adjoint State ◽  
Adjoint State Method

This tutorial is the third part of a full-waveform inversion (FWI) tutorial series with a step-by-step walkthrough of setting up forward and adjoint wave equations and building a basic FWI inversion framework. For discretizing and solving wave equations, we use Devito ( http://www.opesci.org/devito-public ), a Python-based domain-specific language for automated generation of finite-difference code ( Lange et al., 2016 ). The first two parts of this tutorial ( Louboutin et al., 2017 , 2018 ) demonstrated how to solve the acoustic wave equation for modeling seismic shot records and how to compute the gradient of the FWI objective function using the adjoint-state method. With these two key ingredients, we will now build an inversion framework that can be used to minimize the FWI least-squares objective function.

scholarly journals Time-domain full-waveform inversion of exponentially damped wavefield using the deconvolution-based objective function

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. R77-R88 ◽  
Author(s):  
Yunseok Choi ◽  
Tariq Alkhalifah
Keyword(s):  
Objective Function ◽  
Frequency Band ◽  
Waveform Inversion ◽  
Low Frequency ◽  
Full Waveform Inversion ◽  
Low Frequencies ◽  
Full Waveform ◽  
Adjoint State ◽  
Recorded Data ◽  
Adjoint State Method

Full-waveform inversion (FWI) suffers from the cycle-skipping problem when the available frequency-band of data is not low enough. We have applied an exponential damping to the data to generate artificial low frequencies, which helps FWI to avoid cycle skipping. In this case, the least-squares misfit function does not properly deal with the exponentially damped wavefield in FWI because the amplitude of traces decays almost exponentially with increasing offset in a damped wavefield. Thus, we use a deconvolution-based objective function for FWI of the exponentially damped wavefield. The deconvolution filter includes inherently a normalization between the modeled and observed data; thus, it can address the unbalanced amplitude of a damped wavefield. We specifically normalize the modeled data with the observed data in the frequency-domain to estimate the deconvolution filter and selectively choose a frequency-band for normalization that mainly includes the artificial low frequencies. We calculate the gradient of the objective function using the adjoint-state method. The synthetic and benchmark data examples indicate that our FWI algorithm generates a convergent long-wavelength structure without low-frequency information in the recorded data.

Full-waveform inversion of multicomponent data for horizontally layered VTI media

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. WC113-WC121 ◽  
Author(s):  
Nishant Kamath ◽  
Ilya Tsvankin
Keyword(s):  
Joint Inversion ◽  
Hessian Matrix ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Media Analysis ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Full Waveform ◽  
The Time Domain ◽  
Vti Media

Although full-waveform inversion (FWI) has shown significant promise in reconstructing heterogeneous velocity fields, most existing methodologies are limited to acoustic models. We extend FWI to multicomponent (PP and PS) data from anisotropic media, with the current implementation limited to a stack of horizontal, homogeneous VTI (transversely isotropic with a vertical symmetry axis) layers. The algorithm is designed to estimate the interval vertical P- and S-wave velocities ([Formula: see text] and [Formula: see text]) and Thomsen parameters [Formula: see text] and [Formula: see text] from long-spread PP and PSV reflections. The forward-modeling operator is based on the anisotropic reflectivity technique, and the inversion is performed in the time domain using the gradient (Gauss-Newton) method. We employ nonhyperbolic semblance analysis and Dix-type equations to build the initial model. To identify the medium parameters constrained by the data, we perform eigenvalue/eigenvector decomposition of the approximate Hessian matrix for a VTI layer embedded between isotropic media. Analysis of the eigenvectors shows that the parameters [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] (density is assumed to be known) can be resolved not only by joint inversion of PP and PS data, but also with PP reflections alone. Although the inversion becomes more stable with increasing spreadlength-to-depth ([Formula: see text]) ratio, the parameters of the three-layer model are constrained even by PP data acquired on conventional spreads ([Formula: see text]). For multilayered VTI media, the sensitivity of the objective function to the interval parameters decreases with depth. Still, it is possible to resolve [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for the deeper layers using PP-waves, if the ratio [Formula: see text] for the bottom of the layer reaches two. Mode-converted waves provide useful additional constraints for FWI, which become essential for smaller spreads. The insights gained here by examining horizontally layered models should help guide the inversion for heterogeneous TI media.

scholarly journals Land seismic multiparameter full waveform inversion in elastic VTI media by simultaneously interpreting body waves and surface waves with an optimal transport based objective function

2019 ◽  
Vol 219 (3) ◽  
pp. 1970-1988 ◽  
Author(s):  
Weiguang He ◽  
Romain Brossier ◽  
Ludovic Métivier ◽  
René-Édouard Plessix
Keyword(s):  
Objective Function ◽  
Surface Waves ◽  
Least Squares ◽  
Optimal Transport ◽  
Waveform Inversion ◽  
Body Waves ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Initial Model ◽  
Full Waveform

SUMMARY Land seismic multiparameter full waveform inversion in anisotropic media is challenging because of high medium contrasts and surface waves. With a data-residual least-squares objective function, the surface wave energy usually masks the body waves and the gradient of the objective function exhibits high values in the very shallow depths preventing from recovering the deeper part of the earth model parameters. The optimal transport objective function, coupled with a Gaussian time-windowing strategy, allows to overcome this issue by more focusing on phase shifts and by balancing the contributions of the different events in the adjoint-source and the gradients. We first illustrate the advantages of the optimal transport function with respect to the least-squares one, with two realistic examples. We then discuss a vertical transverse isotropic (VTI) example starting from a quasi 1-D isotropic initial model. Despite some cycle-skipping issues in the initial model, the inversion based on the windowed optimal transport approach converges. Both the near-surface complexities and the variations at depth are recovered.

Source-independent time-lapse full-waveform inversion for VTI media

Geophysics ◽  
2021 ◽  
pp. 1-51
Author(s):  
Yanhua Liu ◽  
Ilya Tsvankin
Keyword(s):  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Time Lapse ◽  
Model Complexity ◽  
Double Difference ◽  
Full Waveform Inversion ◽  
Reservoir Properties ◽  
Hydrocarbon Production ◽  
Full Waveform ◽  
Adjoint State

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.

Analysis of optimal transport and related misfit functions in full-waveform inversion

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. A7-A12 ◽  
Author(s):  
Yunan Yang ◽  
Björn Engquist
Keyword(s):  
Optimal Transport ◽  
Waveform Inversion ◽  
Wasserstein Distance ◽  
Model Parameters ◽  
Full Waveform Inversion ◽  
Wasserstein Metric ◽  
Full Waveform ◽  
Adjoint State ◽  
State Method ◽  
Adjoint State Method

Full-waveform inversion has evolved into a powerful computational tool in seismic imaging. New misfit functions for matching simulated and measured data have recently been introduced to avoid the traditional lack of convergence due to cycle skipping. We have introduced the Wasserstein distance from optimal transport for computing the misfit, and several groups are currently further developing this technique. We evaluate three essential observations of this new metric with implication for future development. One is the discovery that trace-by-trace comparison with the quadratic Wasserstein metric works remarkably well together with the adjoint-state method. Another is the close connection between optimal transport-based misfits and integrated techniques with normalization as, for example, the normalized integration method. Finally, we study the convexity with respect to selected model parameters for different normalizations and remark on the effect of normalization on the convergence of the adjoint-state method.

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.

Objective-Function Behavior in Acoustic Full-Waveform Inversion

2013 ◽  
Vol 56 (5) ◽  
pp. 685-703
Author(s):  
DONG Liang-Guo ◽  
CHI Ben-Xin ◽  
TAO Ji-Xia ◽  
LIU Yu-Zhu
Keyword(s):  
Objective Function ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Full Waveform

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.

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