scholarly journals Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. R43-R62 ◽  
Author(s):  
Yunan Yang ◽  
Björn Engquist ◽  
Junzhe Sun ◽  
Brittany F. Hamfeldt
Keyword(s):  
Least Squares ◽  
Optimal Transport ◽  
Waveform Inversion ◽  
Transportation Cost ◽  
Velocity Model ◽  
Seismic Inversion ◽  
Full Waveform Inversion ◽  
Wasserstein Metric ◽  
Full Waveform ◽  
Adjoint State

Conventional full-waveform inversion (FWI) using the least-squares norm as a misfit function is known to suffer from cycle-skipping issues that increase the risk of computing a local rather than the global minimum of the misfit. The quadratic Wasserstein metric has proven to have many ideal properties with regard to convexity and insensitivity to noise. When the observed and predicted seismic data are considered to be two density functions, the quadratic Wasserstein metric corresponds to the optimal cost of rearranging one density into the other, in which the transportation cost is quadratic in distance. Unlike the least-squares norm, the quadratic Wasserstein metric measures not only amplitude differences but also global phase shifts, which helps to avoid cycle-skipping issues. We have developed a new way of using the quadratic Wasserstein metric trace by trace in FWI and compare it with the global quadratic Wasserstein metric via the solution of the Monge-Ampère equation. We incorporate the quadratic Wasserstein metric technique into the framework of the adjoint-state method and apply it to several 2D examples. With the corresponding adjoint source, the velocity model can be updated using a quasi-Newton method. Numerical results indicate the effectiveness of the quadratic Wasserstein metric in alleviating cycle-skipping issues and sensitivity to noise. The mathematical theory and numerical examples demonstrate that the quadratic Wasserstein metric is a good candidate for a misfit function in seismic inversion.

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.

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.

Relaxing the initial model constraint for crustal-scale full-waveform inversion with graph-space optimal transport misfit function

2020 ◽  
Author(s):  
Andrzej Górszczyk ◽  
Ludovic Métivier ◽  
Romain Brossier
Keyword(s):  
Seismic Data ◽  
Optimal Transport ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Wide Angle ◽  
Initial Model ◽  
Full Waveform ◽  
Long Offset ◽  
Obs Data

<p>Investigations of the deep lithosphere aiming at the reconstruction of the geological models remain one of the key sources of the knowledge about the processes shaping the outer shell of our planet. Among different methods, the active seismic Ocean-Bottom Seismometer (OBS) experiments conducted in wide-angle configuration are routinely employed to better understand these processes. Indeed, long-offset seismic data, combined with computationally efficient travetime tomographic methods, have a great potential to constrain the macro-scale subsurface velocity models at large depths. </p><p>On the other hand, decades of development of acquisition systems, more and more efficient algorithms and high-performance computing resources make it now feasible to move beyond the regional raytracing-based traveltime tomography. In particular, the waveform inversion methods, such as Full-Waveform Inversion (FWI), are able to exhaustively exploit the rich information collected along the long-offset diving and refraction wavepaths, additionally enriched with the wide-angle reflection arrivals. So far however, only a few attempts have been conducted in the academic community to combine wide-angle seismic data with FWI for high-resolution crustal-scale velocity model reconstruction. This is partially due to the non-convexity of FWI misfit function, which increases with the complexity of the geological setting reflected by the seismograms. </p><p>In its classical form FWI is a nonlinear least-squares problem, which is solved through the local optimization techniques. This imposes the strong constraint on the accuracy of the starting FWI model. To avoid cycle-skipping problem the initial model must predict synthetic data within the maximum error of half-period time-shift with respect to the observed data. The criterion is difficult to fulfil when facing the crustal-scale FWI, because the long-offset acquisition translates to the long time of wavefront propagation and therefore accumulation of the traveltime error along the wavepath simulated in the initial model. This in turns increases the possibility of the cycle-skipping taking into account large number of propagated wavelengths.</p><p>Searching to mitigate this difficulty, here we investigate FWI with a Graph-Space Optimal Transport (GSOT) misfit function. Comparing to the classical least-squares norm, GSOT is convex with respect to the patterns in the waveform which can be shifted in time for more than half-period. Therefore, with proper data selection strategy GSOT misfit-function has potential to reduce the risk of cycle-skipping. We demonstrate the robustness of this novel approach using 2D wide-angle OBS data-set generated in a GO_3D_OBS synthetic model of subduction zone (30 km x 175 km). We show that using GSOT cost-function combined with the multiscale FWI strategy, we reconstruct in details the highly complex geological structure starting from a simple 1D velocity model. We believe that further developments of OT-based misfit functions can significantly reduce the constraints on the starting model accuracy and reduce the overall risk of cycle-skipping during FWI of wide-angle OBS data.</p>

scholarly journals On cycle-skipping and misfit functions modification for full-wave inversion: comparison of five recent approaches

Geophysics ◽  
2021 ◽  
pp. 1-85
Author(s):  
Arnaud Pladys ◽  
Romain Brossier ◽  
Yubing Li ◽  
Ludovic Métivier
Keyword(s):  
Least Squares ◽  
Field Data ◽  
Optimal Transport ◽  
Waveform Inversion ◽  
Imaging Method ◽  
Full Waveform Inversion ◽  
Wave Inversion ◽  
Full Waveform ◽  
Full Wave ◽  
Definition Of

Full waveform inversion, a high-resolution seismic imaging method, is known to require sufficiently accurate initial models to converge toward meaningful estimations of the subsurface mechanical properties. This limitation is due to the non-convexity of the least-squares distance with respect to kinematic mismatch. We propose a comparison of five misfit functions promoted recently to mitigate this issue: adaptive waveform inversion, instantaneous envelope, normalized integration, and two methods based on optimal transport. We explain which principles these methods are based on and illustrate how they are designed to better handle kinematic mismatch than a least-squares misfit function. By doing so, we can exhibit specific limitations of these methods in canonical cases. We further assess the interest of these five approaches for application to field data based on a synthetic Marmousi case study. We illustrate how adaptive waveform inversion and the two methods based on optimal transport possess interesting properties, making them appealing strategies applicable to field data. Another outcome is the definition of generic tools to compare misfit functions for full-waveform inversion.

Source-independent time-domain vector-acoustic full-waveform inversion

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. R489-R505 ◽  
Author(s):  
Yu Zhong ◽  
Yangting Liu
Keyword(s):  
Particle Velocity ◽  
Time Domain ◽  
Seismic Data ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Adjoint State ◽  
Dual Sensor ◽  
Seismic Acquisition

Dual-sensor seismic acquisition systems that record the pressure and particle velocity allow the recording of the full-vector-acoustic (VA) wavefields. Most previous studies have focused on data-domain processing methods based on VA seismic data; whereas, few studies focused on using full-VA seismic data in full-waveform inversion (FWI). Conventional acoustic FWI only takes advantage of the pressure recordings to estimate the medium’s velocity model. Some artifact events will appear in the adjoint-state wavefields based on the conventional acoustic FWI method. These artifact events further reduce the accuracy of acoustic FWI. To simultaneously use pressure and vertical particle velocity recordings, we introduced a new time-domain VA FWI method. The VA FWI method can take advantage of directivity information contained in the VA seismic data. Thus, the adjoint-state wavefields based on VA FWI are more accurate than those from the conventional acoustic FWI method. In addition, we applied a convolution-based objective function to eliminate the effects of the source wavelet and implement a time-domain multiscale strategy in VA FWI. Synthetic examples are presented to demonstrate that VA FWI can improve the accuracy of acoustic FWI in the presence and absence of a free surface in the acoustic case. In addition, VA FWI does not significantly increase the computation and memory costs, but it has better convergence when compared with conventional acoustic FWI.

scholarly journals Full-Waveform Inversion for Imaging Faulted Structures: A Case Study from the Japan Trench Forearc Slope

2021 ◽  
Author(s):  
Ehsan Jamali Hondori ◽  
Chen Guo ◽  
Hitoshi Mikada ◽  
Jin-Oh Park
Keyword(s):  
Seismic Data ◽  
Initial Velocity ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Control Measures ◽  
Japan Trench ◽  
Full Waveform Inversion ◽  
Full Waveform ◽  
Time Migration ◽  
Shallow Sediments

AbstractFull-waveform inversion (FWI) of limited-offset marine seismic data is a challenging task due to the lack of refracted energy and diving waves from the shallow sediments, which are fundamentally required to update the long-wavelength background velocity model in a tomographic fashion. When these events are absent, a reliable initial velocity model is necessary to ensure that the observed and simulated waveforms kinematically fit within an error of less than half a wavelength to protect the FWI iterative local optimization scheme from cycle skipping. We use a migration-based velocity analysis (MVA) method, including a combination of the layer-stripping approach and iterations of Kirchhoff prestack depth migration (KPSDM), to build an accurate initial velocity model for the FWI application on 2D seismic data with a maximum offset of 5.8 km. The data are acquired in the Japan Trench subduction zone, and we focus on the area where the shallow sediments overlying a highly reflective basement on top of the Cretaceous erosional unconformity are severely faulted and deformed. Despite the limited offsets available in the seismic data, our carefully designed workflow for data preconditioning, initial model building, and waveform inversion provides a velocity model that could improve the depth images down to almost 3.5 km. We present several quality control measures to assess the reliability of the resulting FWI model, including ray path illuminations, sensitivity kernels, reverse time migration (RTM) images, and KPSDM common image gathers. A direct comparison between the FWI and MVA velocity profiles reveals a sharp boundary at the Cretaceous basement interface, a feature that could not be observed in the MVA velocity model. The normal faults caused by the basal erosion of the upper plate in the study area reach the seafloor with evident subsidence of the shallow strata, implying that the faults are active.

Full-waveform inversion for full-wavefield imaging: Decades in the making

2021 ◽  
Vol 40 (5) ◽  
pp. 324-334
Author(s):  
Rongxin Huang ◽  
Zhigang Zhang ◽  
Zedong Wu ◽  
Zhiyuan Wei ◽  
Jiawei Mei ◽  
...  
Keyword(s):  
Model Building ◽  
Seismic Imaging ◽  
Structural Information ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Data Sets ◽  
Full Waveform Inversion ◽  
Data Types ◽  
Velocity Models ◽  
Full Waveform

Seismic imaging using full-wavefield data that includes primary reflections, transmitted waves, and their multiples has been the holy grail for generations of geophysicists. To be able to use the full-wavefield data effectively requires a forward-modeling process to generate full-wavefield data, an inversion scheme to minimize the difference between modeled and recorded data, and, more importantly, an accurate velocity model to correctly propagate and collapse energy of different wave modes. All of these elements have been embedded in the framework of full-waveform inversion (FWI) since it was proposed three decades ago. However, for a long time, the application of FWI did not find its way into the domain of full-wavefield imaging, mostly owing to the lack of data sets with good constraints to ensure the convergence of inversion, the required compute power to handle large data sets and extend the inversion frequency to the bandwidth needed for imaging, and, most significantly, stable FWI algorithms that could work with different data types in different geologic settings. Recently, with the advancement of high-performance computing and progress in FWI algorithms at tackling issues such as cycle skipping and amplitude mismatch, FWI has found success using different data types in a variety of geologic settings, providing some of the most accurate velocity models for generating significantly improved migration images. Here, we take a step further to modify the FWI workflow to output the subsurface image or reflectivity directly, potentially eliminating the need to go through the time-consuming conventional seismic imaging process that involves preprocessing, velocity model building, and migration. Compared with a conventional migration image, the reflectivity image directly output from FWI often provides additional structural information with better illumination and higher signal-to-noise ratio naturally as a result of many iterations of least-squares fitting of the full-wavefield 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.

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