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When data driven reduced order modeling meets full waveform inversion

Liliana Borcea, Josselin Garnier, Alexander V. Mamonov, Jörn Zimmerling

Abstract

Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the non-convexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is non-iterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion. In this arxiv version two important typos are corrected when compared to the published version. The typo was in the second equation in Theorem 3 and carried over into Corollary 1. The proofs are correct.

When data driven reduced order modeling meets full waveform inversion

Abstract

Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the non-convexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is non-iterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion. In this arxiv version two important typos are corrected when compared to the published version. The typo was in the second equation in Theorem 3 and carried over into Corollary 1. The proofs are correct.
Paper Structure (23 sections, 7 theorems, 91 equations, 9 figures, 2 algorithms)

This paper contains 23 sections, 7 theorems, 91 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction to waveform inversion. Paper motivation and outline
  2. The data driven ROMs
  3. Data transformation
  4. The snapshots and the propagator
  5. Data driven Galerkin approximations
  6. The data driven ROMs
  7. The ROM propagator
  8. The ROM of the wave operator
  9. Comparison of the ROMs
  10. Regularization of the ROM computation
  11. Waveform inversion
  12. The estimated internal wave
  13. Inversion with the estimated internal wave
  14. Inversion with the ROM wave operator
  15. The inversion algorithms. Pros and cons
  16. ...and 8 more sections

Key Result

Lemma 1

\newlabellem.1 Define the new data matrix ${\itbf D}(t)$ by the mapping for $t > 0$. Its entries have the integral (inner) product expression where is the solution of the homogeneous wave equation with initial state

Figures (9)

  • Figure 1.1: Illustration of a two-parameter search applied to the true piecewise constant model $c({{\itbf x}})$ (left plot), in a closed rectangular cavity. The sources and receivers are co-located and shown as black $\times$. The probing signal contains frequencies in the interval $[2,10]$Hz. The search parameters are the depth of the slanted fast layer (interface position), that varies over the range indicated by the black arrows, and the ratio of the wave speed inside and above the layer (contrast). The FWI objective function (\ref{['eq:FWI']}) is shown in the right plot. The true parameters are indicated by $\textcolor{magenta}{\bigcirc}$.
  • Figure 1.2: Illustration of three data acquisition setups: With an active array (left), where the sources and receivers are co-located; with a towed-streamer (middle); with a passive array of receivers (blue triangles) and uncontrolled random sources (yellow) dispersed throughout the medium (right).
  • Figure 3.1: Illustration of the internal waves at ${{\itbf x}}$ indicated with red $\times$ in two media containing a ring shaped inclusion (top plots). The active array is linear and placed at the top of the plots, with sensors indicated by green triangles. The wave speed is shown in the colorbar in units of m/s. The axes are cross-range (measured along the array) and range (measured orthogonal to the array), in units of the central wavelength. The bottom plots show the internal waves as functions of the sensor index $s$ (abscissa) and time in units of $\tau$ (ordinate). From left to right: $u^{(s)}(j\tau,{{\itbf x}};w=\bar{c})$ calculated with the constant wave speed $\bar{c} = 1$km/s; the true wave $u^{(s)}(j\tau,{{\itbf x}})$ in the first medium, the estimated $u^{(s),{\rm est}}(j\tau,{{\itbf x}};w= \bar{c})$ in the first medium; the true wave $u^{(s)}(j\tau,{{\itbf x}})$ in the second medium and the estimated $u^{(s),{\rm est}}(j\tau,{{\itbf x}};w= \bar{c})$ in the second medium.
  • Figure 3.2: Display of the log of the objective functions for the setup in Fig. \ref{['fig:FWI_topo']}. The axes are the same as in the right plot there. The true model is indicated by $\textcolor{magenta}{\bigcirc}$.
  • Figure 3.3: Left plot: Illustration of the search speed \ref{['eq:CamembertSearch']} for $9$ choices of $\alpha$ and $\beta$. Right plots: Log of objective functions parametrized by $\beta$ (abscissa) and $\alpha$ (ordinate). The true model is indicated by $\textcolor{magenta}{\bigcirc}$.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Lemma 1
  • theorem 1
  • theorem 2
  • Remark 2.1
  • theorem 3
  • Corollary 1
  • Proposition 4.1
  • theorem 4
  • Remark 4.2