scholarly journals A robust waveform inversion using a global comparison of modeled and observed data

2019 ◽  
Vol 38 (3) ◽  
pp. 185-192 ◽  
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah
Keyword(s):  
Optimal Transport ◽  
Waveform Inversion ◽  
Alternative Formulation ◽  
Initial Model ◽  
Data Set ◽  
Dirac Delta ◽  
Global Comparison ◽  
L2 Norm ◽  
Highly Nonlinear ◽  
Matching Filter

A high-resolution model of the subsurface is the product of a successful full-waveform inversion (FWI) application. However, this optimization problem is highly nonlinear, and thus, we iteratively update the subsurface model by minimizing a misfit function that measures the difference between observed and modeled data. The L2-norm misfit function provides a simple, sample-by-sample comparison between the observed and modeled data. However, it is susceptible to local minima in the objective function if the low-wavenumber components of the initial model are not accurate enough. We review an alternative formulation of FWI based on a more global comparison. A combination of Radon transform and utilizing a matching filter allows for comparisons beyond sample to sample. We combine two recent developments to suggest the following algorithm for optimal inversion: (1) we compute the matching filter between the observed and modeled data in the Radon domain, which helps reduce the crosstalk introduced in the deconvolution step of computing the matching filter, and (2) we use Wasserstein distance to measure the distance between the resulting matching filter in the Radon domain and a representation of the Dirac delta function, which provides us with the optimal transport between the two distribution functions. We use a modified Marmousi model to show how this Radon-domain optimal-transport-based matching-filter approach can mitigate cycle skipping. Starting from a rather simplified v(z) media as the initial model, the proposed method can invert for the Marmousi model with considerable accuracy, while standard L2-norm formulation is trapped in a local minimum. Application of the proposed method to an offshore data set further demonstrates its robustness and effectiveness.

scholarly journals The application of an optimal transport to a preconditioned data matching function for robust waveform inversion

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R923-R945 ◽  
Author(s):  
Bingbing Sun ◽  
Tariq Alkhalifah
Keyword(s):  
High Resolution ◽  
Probability Distribution ◽  
Transport Theory ◽  
Optimal Transport ◽  
Waveform Inversion ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Matching Filter ◽  
Optimal Transport Theory

Full-waveform inversion (FWI) promises a high-resolution model of the earth. It is, however, a highly nonlinear inverse problem; thus, we iteratively update the subsurface model by minimizing a misfit function that measures the difference between the measured and the predicted data. The conventional [Formula: see text]-norm misfit function is widely used because it provides a simple, high-resolution misfit function with a sample-by-sample comparison. However, it is susceptible to local minima if the low-wavenumber components of the initial model are not accurate. Deconvolution of the predicted and measured data offers an extended space comparison, which is more global. The matching filter calculated from the deconvolution has energy focused at zero lag, like an approximated Dirac delta function, when the predicted data matches the measured one. We have introduced a framework for designing misfit functions by measuring the distance between the matching filter and a representation of the Dirac delta function using optimal transport theory. We have used the Wasserstein [Formula: see text] distance, which provides us with the optimal transport between two probability distribution functions. Unlike data, the matching filter can be easily transformed to a probability distribution satisfying the requirement of the optimal transport theory. Though in one form, it admits the conventional normalized penalty applied to the nonzero-lag energy in the matching filter, the proposed misfit function is metric and extracts its form from solid mathematical foundations based on optimal transport theory. Explicitly, we can derive the adaptive waveform inversion (AWI) misfit function based on our framework, and the critical “normalization” for AWI occurs naturally per the requirement of a probability distribution. We use a modified Marmousi model and the BP salt model to verify the features of the proposed method in avoiding cycle skipping. We use the Chevron 2014 FWI benchmark data set to further highlight the effectiveness of the proposed approach.

scholarly journals Application of Laplace Domain Waveform Inversion to Cross-Hole Radar Data

2019 ◽  
Vol 11 (16) ◽  
pp. 1839
Author(s):  
Xu Meng ◽  
Sixin Liu ◽  
Yi Xu ◽  
Lei Fu
Keyword(s):  
Time Domain ◽  
Waveform Inversion ◽  
Radar Data ◽  
Dominant Frequency ◽  
Data Set ◽  
Laplace Domain ◽  
Full Waveform ◽  
Highly Nonlinear ◽  
The Time Domain ◽  
Ground Penetrating

Full waveform inversion (FWI) can yield high resolution images and has been applied in Ground Penetrating Radar (GPR) for around 20 years. However, appropriate selection of the initial models is important in FWI because such an inversion is highly nonlinear. The conventional way to obtain the initial models for GPR FWI is ray-based tomogram inversion which suffers from several inherent shortcomings. In this paper, we develop a Laplace domain waveform inversion to obtain initial models for the time domain FWI. The gradient expression of the Laplace domain waveform inversion is deduced via the derivation of a logarithmic object function. Permittivity and conductivity are updated by using the conjugate gradient method. Using synthetic examples, we found that the value of the damping constant in the inversion cannot be too large or too small compared to the dominant frequency of the radar data. The synthetic examples demonstrate that the Laplace domain waveform inversion provide slightly better initial models for the time domain FWI than the ray-based inversion. Finally, we successfully applied the algorithm to one field data set, and the inverted results of the Laplace-based FWI show more details than that of the ray-based FWI.

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.

Random objective waveform inversion of surface waves

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. EN49-EN61
Author(s):  
Yudi Pan ◽  
Lingli Gao
Keyword(s):  
Objective Function ◽  
Surface Waves ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Initial Model ◽  
S Wave ◽  
Data Set ◽  
Near Surface ◽  
Synthetic Test ◽  
Ground Penetrating

Full-waveform inversion (FWI) of surface waves is becoming increasingly popular among shallow-seismic methods. Due to a huge amount of data and the high nonlinearity of the objective function, FWI usually requires heavy computational costs and may converge toward a local minimum. To mitigate these problems, we have reformulated FWI under a multiobjective framework and adopted a random objective waveform inversion (ROWI) method for surface-wave characterization. Three different measure functions were used, whereas the combination of one measure function with one shot independently provided one of the [Formula: see text] objective functions ([Formula: see text] is the total number of shots). We have randomly chose and optimized one objective function at each iteration. We performed a synthetic test to compare the performance of the ROWI and conventional FWI approaches, which showed that the convergence of ROWI is faster and more robust compared with conventional FWI approaches. We also applied ROWI to a field data set acquired in Rheinstetten, Germany. ROWI successfully reconstructed the main geologic feature, a refilled trench, in the final result. The comparison between the ROWI result and a migrated ground-penetrating radar profile further proved the effectiveness of ROWI in reconstructing the near-surface S-wave velocity model. We also ran the same field example by using a poor initial model. In this case, conventional FWI failed whereas ROWI still reconstructed the subsurface model to a fairly good level, which highlighted the relatively low dependency of ROWI on the initial model.

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.

Assessment of seismic tomographic models of the contiguous United States using intermediate-period 3D wavefield simulation

2021 ◽  
Author(s):  
Tong Zhou ◽  
Ziyi Xi ◽  
Min Chen ◽  
Jiaqi Li
Keyword(s):  
United States ◽  
North America ◽  
Upper Mantle ◽  
Waveform Inversion ◽  
Selection Procedure ◽  
3D Models ◽  
Initial Model ◽  
Crust And Upper Mantle ◽  
Data Set ◽  
Wavefield Simulation

Summary The contiguous United States has been well instrumented with broadband seismic stations due to the development of the EarthScope Transportable Array. Previous studies have provided various 3D seismic wave speed models for the crust and upper mantle with improved resolution. However, discrepancies exist among these models due to differences in both data sets and tomographic methods, which introduce uncertainties on the imaged lithospheic structure beneath North America. A further model refinement using the best data coverage and advanced tomographic methods such as full-waveform inversion (FWI) is expected to provide better seismological constraints. Initial models have significant impacts on the convergence of FWIs. However, how to select an optimal initial model is not well investigated. Here, we present a data-driven initial model selection procedure for the contiguous US and surrounding regions by assessing waveform fitting and misfit functions between the observations and synthetics from candidate models. We use a data set of waveforms from 30 earthquakes recorded by 5,820 stations across North America. The results suggest that the tested 3D models capture well long-period waveforms while showing discrepancies in short-periods especially on tangential components. This observation indicates that the smaller-scale heterogeneities and radial anisotropy in the crust and upper mantle are not well constrained. Based on our test results, a hybrid initial model combining S40RTS or S362ANI in the mantle and US.2016 for Vsv and CRUST1.0 for Vsh in the crust is compatible for future FWIs to refine the lithospheric structure of North America.

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 Full-model wavenumber inversion: An emphasis on the appropriate wavenumber continuation

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. R89-R98 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Waveform Inversion ◽  
Velocity Model ◽  
Initial Model ◽  
Blind Test ◽  
Data Set ◽  
Scattering Angle ◽  
Full Waveform ◽  
The Earth ◽  
Least Squares Fit ◽  
Recorded Data

A model of the earth can be described using a Fourier basis represented by its wavenumber content. In full-waveform inversion (FWI), the wavenumber description of the model is natural because our Born-approximation-based velocity updates are made up of wavefields. Our objective in FWI is to access all the model wavenumbers available in our limited aperture and bandwidth recorded data that are not yet accurately present in the initial velocity model. To invert for those model wavenumbers, we need to locate their imprint in the data. Thus, I review the relation between the model wavenumber buildup and the inversion process. Specifically, I emphasize a focus on the model wavenumber components and identified their individual influence on the data. Missing the energy for a single vertical low-model wavenumber from the residual between the true Marmousi model and some initial linearly increasing velocity model produced a worse least-squares fit to the data than the initial model itself, in which all the residual model wavenumbers were missing. This stern realization validated the importance of wavenumber continuation, specifically starting from the low-model wavenumbers, to higher (resolution) wavenumbers, especially those attained in an order dictated by the scattering angle filter. A numerical Marmousi example determined the important role that the scattering angle filter played in managing the wavenumber continuation from low to high. An application on the SEG2014 blind test data set with frequencies lower than 7 Hz muted out further validated the versatility of the scattering angle filtering.

An optimal transport distance for full-waveform inversion: Application to the 2014 Chevron benchmark data set

2016 ◽  
Author(s):  
Ludovic Metivier ◽  
Romain Brossier ◽  
Edouard Oudet ◽  
Quentin Mérigot ◽  
Jean Virieux
Keyword(s):  
Optimal Transport ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Data Set ◽  
Benchmark Data ◽  
Transport Distance ◽  
Full Waveform

Tunnel detection at Yuma Proving Ground, Arizona, USA — Part 2: 3D full-waveform inversion experiments

Geophysics ◽  
2019 ◽  
Vol 84 (1) ◽  
pp. B107-B120 ◽  
Author(s):  
James A. Smith ◽  
Dmitry Borisov ◽  
Harley Cudney ◽  
Richard D. Miller ◽  
Ryan Modrak ◽  
...  
Keyword(s):  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
3D Image ◽  
Initial Model ◽  
3D Inversion ◽  
Data Set ◽  
Tunnel Detection ◽  
Full Waveform ◽  
Wave Methods ◽  
Reduced Data

We have applied time-domain 3D elastic full-waveform inversion (FWI) to a known tunnel constructed 10 m below the surface with no distinguishing surface expressions. Multicomponent inversion experiments that use an initial model estimated from surface wave methods suggest that the vertical sources and the combination of vertical and longitudinal receivers result in the clearest image of the tunnel. We obtain an approximate 3D image of the tunnel using 24 vertical sources and 720 vertical and 720 longitudinal receivers. We find that increasing the number of vertical sources to 216 does not significantly improve the details of the tunnel. Further experimentation indicates that we can detect the tunnel using a reduced data set of 6–10 vertical sources and 216 vertical sensors. In addition, calculating the [Formula: see text] ratio helps remove artifacts from the inverted model and highlights the location of the tunnel. We compare the 3D inversion results with the 2D FWI results for the same tunnel that we previously evaluated. The variety of successful inversion experiments suggest that FWI is capable of imaging shallow tunnels in desert geologies.

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