Improving adaptive waveform inversion by local 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.

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Adaptive traveltime inversion

Geophysics ◽

10.1190/geo2018-0595.1 ◽

2019 ◽

Vol 84 (4) ◽

pp. U13-U29 ◽

Cited By ~ 9

Author(s):

Bingbing Sun ◽

Tariq Alkhalifah

Keyword(s):

Least Squares ◽

Penalty Function ◽

Waveform Inversion ◽

Velocity Model ◽

Second Order ◽

P Wave ◽

Order Moment ◽

Traveltime Inversion ◽

First Order ◽

Matching Filter

We have developed a method to obtain a misfit function for robust waveform inversion. In this method, called adaptive traveltime inversion (ATI), a matching filter that matches predicted data to measured data is computed. If the velocity model is relatively accurate, the resulting matching filter is close to a Dirac delta function. Its traveltime shift, which characterizes the defocusing of the matching filter, is computed by minimization of the crosscorrelation between a penalty function such as [Formula: see text] and the matching filter. ATI is constructed by minimization of the least-squares errors of the calculated traveltime shift. Further analysis indicates that the resulting traveltime shift corresponds to a first-order moment, the mean value of the resulting matching filter distribution. We extend ATI to a more general misfit function formula by computing different order moment of the resulting matching filter distribution. Choosing the penalty function in adaptive waveform inversion (AWI) as [Formula: see text], the misfit function of AWI is the second-order moment, the variance of the resulting matching filter distribution with zero mean. Because our ATI method is based on a global comparison using deconvolution, such as AWI, it can resolve the “cycle skipping” issue. We evaluate our ATI misfit function and compare it with state-of-the-art options such as least-squares inversion (L2 norm), wave-equation traveltime inversion, and AWI using schematic examples before moving to more complex examples, such as the Marmousi model. For the Marmousi model, starting with a 1D [Formula: see text] model, with data without low frequencies (no energy below 3 Hz), a meaningful estimation of the P-wave velocity model is recovered. Our ATI misfit function (first-order moment) indicates comparable performance with the AWI misfit function (the second-order moment). We also include a real data example from the Gulf of Mexico to demonstrate the effectiveness of our method.

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

Geophysics ◽

10.1190/geo2018-0413.1 ◽

2019 ◽

Vol 84 (6) ◽

pp. R923-R945 ◽

Cited By ~ 6

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.

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Applications of adaptive-matching filter in full-waveform inversion

10.1190/segam2016-13821369.1 ◽

2016 ◽

Author(s):

Hejun Zhu ◽

Sergey Fomel

Keyword(s):

Waveform Inversion ◽

Full Waveform Inversion ◽

Full Waveform ◽

Matching Filter

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Robust full-waveform inversion with Radon-domain matching filter

Geophysics ◽

10.1190/geo2018-0347.1 ◽

2019 ◽

Vol 84 (5) ◽

pp. R707-R724 ◽

Cited By ~ 5

Author(s):

Bingbing Sun ◽

Tariq Alkhalifah

Keyword(s):

Radon Transform ◽

Waveform Inversion ◽

Delta Function ◽

Measured Data ◽

Dirac Delta Function ◽

Full Waveform Inversion ◽

Dirac Delta ◽

Full Waveform ◽

Matching Filter ◽

Filter Approach

Cycle skipping is a severe issue in full-waveform inversion. One option to overcome it is to extend the search space to allow for data comparisons beyond the “point-to-point” subtraction. A matching filter can be computed by deconvolving the measured data from the predicted ones. If the model is correct, the resulting matching filter would be a Dirac delta function in which the energy is focused at zero lag. An optimization problem can be formulated by penalizing this matching filter departure from a Dirac delta function. Because the matching filter replaces the local sample-by-sample comparison with a global one using deconvolution, it can reduce the cycle-skipping problem. Because the matching filter is computed using the whole trace of the measured and predicted data, it is prone to unwanted crosstalk of different events. We perform the deconvolution in the Radon domain to reduce crosstalk and improve the inversion. We first transform the measured and the predicted data into the [Formula: see text] domain using the local Radon transform. We then perform deconvolution for the trace indexed by the same slope value. The main objective of the proposal is to use the slope information embedded in the Radon-transform representation to separate the events and reduce the crosstalk in the deconvolution step. As a result, the objective function tends to be more convex and stabilizes the inversion process. The result obtained for the modified Marmousi model demonstrates the proposed Radon-domain matching-filter approach can converge to a meaningful model given data without the low frequencies of less than 3 Hz and a [Formula: see text] initial model. Compared to the conventional time-space matching-filter approach, the Radon-domain approach indicates fewer artifacts in the model and better fitting of the measured data. The result corresponding to the Chevron 2014 benchmark data set also indicates the good performance of the proposed approach.

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