skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Waveform inversion via reduced order modeling
We introduce a novel approach to waveform inversion based on a data-driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system that probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low-frequency information, due to multiple minima of the data fit objective function, we find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least-squares data fit objective function displays multiple local minima.  more » « less
Award ID(s):
2110265
PAR ID:
10412167
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
GEOPHYSICS
Volume:
88
Issue:
2
ISSN:
0016-8033
Page Range / eLocation ID:
R175 to R191
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Communications on Pure and Applied Mathematics)
    Varadhan, S.R.S. (Ed.)
    Full-waveform inversion (FWI) is today a standard process for the inverse problem of seismic imaging. PDE-constrained optimization is used to determine unknown parameters in a wave equation that represent geophysical properties. The objective function measures the misfit between the observed data and the calculated synthetic data, and it has traditionally been the least-squares norm. In a sequence of papers, we introduced the Wasserstein metric from optimal transport as an alternative misfit function for mitigating the so-called cycle skipping, which is the trapping of the optimization process in local minima. In this paper, we first give a sharper theorem regarding the convexity of the Wasserstein metric as the objective function. We then focus on two new issues. One is the necessary normalization of turning seismic signals into probability measures such that the theory of optimal transport applies. The other, which is beyond cycle skipping, is the inversion for parameters below reflecting interfaces. For the first, we propose a class of normalizations and prove several favorable properties for this class. For the latter, we demonstrate that FWI using optimal transport can recover geophysical properties from domains where no seismic waves travel through. We finally illustrate these properties by the realistic application of imaging salt inclusions, which has been a significant challenge in exploration geophysics. 
    more » « less
  2. ; ; ; ; ; (, GEOPHYSICS)
    null (Ed.)
    The quantitative reconstruction of subsurface earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field, but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and it defines the full reciprocity-gap waveform inversion (FRgWI) method. This misfit functional provides additional features compared to the more traditional least-squares approaches, in particular, in that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be chosen arbitrarily. Based on 3D experiments, FRgWI is shown to behave better than full-waveform inversion in the same context. It allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. However, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance, with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to crosstalk than least-squares shot stacking. 
    more » « less
  3. ; (, Physics in Medicine & Biology)
    Abstract Objective. With the ultimate goal of reconstructing 3D elasticity maps from ultrasound particle velocity measurements in a plane, we present in this paper a methodology of inverting for 2D elasticity maps from measurements on a single line.Approach. The inversion approach is based on gradient optimization where the elasticity map is iteratively modified until a good match is obtained between simulated and measured responses. Full-wave simulation is used as the underlying forward model to accurately capture the physics of shear wave propagation and scattering in heterogeneous soft tissue. A key aspect of the proposed inversion approach is a cost functional based on correlation between measured and simulated responses.Main results. We illustrate that the correlation-based functional has better convexity and convergence properties compared to the traditional least-squares functional, and is less sensitive to initial guess, robust against noisy measurements and other errors that are common in ultrasound elastography. Inversion with synthetic data illustrates the effectiveness of the method to characterize homogeneous inclusions as well as elasticity map of the entire region of interest.Significance. The proposed ideas lead to a new framework for shear wave elastography that shows promise in obtaining accurate maps of shear modulus using shear wave elastography data obtained from standard clinical scanners. 
    more » « less
  4. ; (, GEOPHYSICS)
    We have formulated the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we find that the adjoint wave propagator preserves the dispersion and compensates the amplitude, whereas the time-reversed adjoint wave propagator behaves identically to the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit [Formula: see text] parameterization, which avoids the implicit [Formula: see text] in the conventional viscoacoustic/viscoelastic full-waveform inversion ([Formula: see text]-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, whereas 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 the 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 [Formula: see text]-FWI. 
    more » « less
  5. ; ; ; ; ; (, IEEE Transactions on Geoscience and Remote Sensing)
    The mixed Lp-norm, 0 ≤ p ≤ 2, stabilization algorithm is flexible for constructing a suite of subsurface models with either distinct, or a combination of, smooth, sparse, or blocky structures. This general purpose algorithm can be used for the inversion of data from regions with different subsurface characteristics. Model interpretation is improved by simulta- neous inversion of multiple data sets using a joint inversion approach. An effective and general algorithm is presented for the mixed Lp-norm joint inversion of gravity and magnetic data sets. The imposition of the structural cross-gradient enforces similarity between the reconstructed models. For efficiency the implementation relies on three crucial realistic details; (i) the data are assumed to be on a uniform grid providing sensitivity matrices that decompose in block Toeplitz Toeplitz block form for each depth layer of the model domain and yield efficiency in storage and computation via 2D fast Fourier transforms; (ii) matrix-free implementation for calculating derivatives of parameters reduces memory and computational overhead; and (iii) an alternating updating algorithm is employed. Balancing of the data misfit terms is imposed to assure that the gravity and magnetic data sets are fit with respect to their individual noise levels without overfitting of either model. Strategies to find all weighting parameters within the objective function are described. The algorithm is validated on two synthetic but complicated models. It is applied to invert gravity and magnetic data acquired over two kimberlite pipes in Botswana, producing models that are in good agreement with borehole information available in the survey area. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email