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.

Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Guangchi Xing ◽  
Tieyuan Zhu
Keyword(s):  
Wave Equation ◽  
Fractional Laplacian ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Physical Processes ◽  
Full Waveform ◽  
Adjoint State ◽  
The Finite Difference Method ◽  
Adjoint State Method ◽  
Laplacian Operators

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.

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.

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.

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

3D elastic full-waveform inversion of surface waves in the presence of irregular topography using an envelope-based misfit function

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. R1-R11 ◽  
Author(s):  
Dmitry Borisov ◽  
Ryan Modrak ◽  
Fuchun Gao ◽  
Jeroen Tromp
Keyword(s):  
Surface Wave ◽  
Objective Function ◽  
Surface Waves ◽  
Large Scale ◽  
Body Wave ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Near Surface ◽  
Full Waveform

Full-waveform inversion (FWI) is a powerful method for estimating the earth’s material properties. We demonstrate that surface-wave-driven FWI is well-suited to recovering near-surface structures and effective at providing S-wave speed starting models for use in conventional body-wave FWI. Using a synthetic example based on the SEG Advanced Modeling phase II foothills model, we started with an envelope-based objective function to invert for shallow large-scale heterogeneities. Then we used a waveform-difference objective function to obtain a higher-resolution model. To accurately model surface waves in the presence of complex tomography, we used a spectral-element wave-propagation solver. Envelope misfit functions are found to be effective at minimizing cycle-skipping issues in surface-wave inversions, and surface waves themselves are found to be useful for constraining complex near-surface features.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document