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Quantitative passive imaging by iterative holography: The example of helioseismic holography

Björn Müller, Thorsten Hohage, Damien Fournier, Laurent Gizon

Abstract

In passive imaging, one attempts to reconstruct some coefficients in a wave equation from correlations of observed randomly excited solutions to this wave equation. Many methods proposed for this class of inverse problem so far are only qualitative, e.g., trying to identify the support of a perturbation. Major challenges are the increase in dimensionality when computing correlations from primary data in a preprocessing step, and often very poor pointwise signal-to-noise ratios. In this paper, we propose an approach that addresses both of these challenges: It works only on the primary data while implicitly using the full information contained in the correlation data, and it provides quantitative estimates and convergence by iteration. Our work is motivated by helioseismic holography, a well-established imaging method to map heterogenities and flows in the solar interior. We show that the back-propagation used in classical helioseismic holography can be interpreted as the adjoint of the Fréchet derivative of the operator which maps the properties of the solar interior to the correlation data on the solar surface. The theoretical and numerical framework for passive imaging problems developed in this paper extends helioseismic holography to nonlinear problems and allows for quantitative reconstructions. We present a proof of concept in uniform media.

Quantitative passive imaging by iterative holography: The example of helioseismic holography

Abstract

In passive imaging, one attempts to reconstruct some coefficients in a wave equation from correlations of observed randomly excited solutions to this wave equation. Many methods proposed for this class of inverse problem so far are only qualitative, e.g., trying to identify the support of a perturbation. Major challenges are the increase in dimensionality when computing correlations from primary data in a preprocessing step, and often very poor pointwise signal-to-noise ratios. In this paper, we propose an approach that addresses both of these challenges: It works only on the primary data while implicitly using the full information contained in the correlation data, and it provides quantitative estimates and convergence by iteration. Our work is motivated by helioseismic holography, a well-established imaging method to map heterogenities and flows in the solar interior. We show that the back-propagation used in classical helioseismic holography can be interpreted as the adjoint of the Fréchet derivative of the operator which maps the properties of the solar interior to the correlation data on the solar surface. The theoretical and numerical framework for passive imaging problems developed in this paper extends helioseismic holography to nonlinear problems and allows for quantitative reconstructions. We present a proof of concept in uniform media.
Paper Structure (23 sections, 10 theorems, 81 equations, 8 figures, 1 table)

This paper contains 23 sections, 10 theorems, 81 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. A model problem
  3. Diagonals of operator kernels
  4. Fréchet derivative and adjoint of the forward operator
  5. On the algorithmic realization of iterative regularization methods
  6. Avoiding the computation of $\operatorname{Corr}$
  7. Iterative regularization methods without image space vectors
  8. Noise and likelihood modelling
  9. Forward problems in local helioseismology
  10. Acoustic oscillations in the Sun
  11. Source model in helioseismology
  12. Iterative helioseismic holography
  13. Relations to conventional helioseismic holography and other methods
  14. Kernels and resolution
  15. Inversions
  16. ...and 8 more sections

Key Result

Proposition 1

Under Assumption ass:well_posed the problem eqs:forward is well posed in the sense that for all $s\in H^{-1}_0(\Omega)$ there exists a unique $\psi\in H^1(\Omega)$ satisfying eqs:forward in the weak sense, and $\psi$ depends continuously on $s$ with respect to these norms.

Figures (8)

  • Figure 1: The left panel shows the sound speed and density obtained from the Solar Model S Dalsgaard1996 in the solar core, the convection zone (CZ), and the radiation zone (RZ). The right panel shows the potential close to the surface for $\omega/2 \pi=3~$mHz.
  • Figure 2: The sound speed sensitivity kernel $K(\bi{x},\cdot)$ in the $r-\theta$-plane as defined in \ref{['eq:kernel_scalar']} for a three-dimensional uniform medium with $c_0=200$ km/s and the $l$-range is $0 \leq l<100$ for four different target positions $\bi{x}$. We have averaged the sensitivity kernels over 100 frequencies in the frequency regime $2.75$--$3.25$ mHz and normalized with $K(\bi{x}, \bi{x})$ at the target location $\bi{x}$.
  • Figure 3: The sound speed sensitivity kernel $K(\bi{x},\cdot)$ in the $r-\theta$-plane as defined in \ref{['eq:kernel_scalar']} in a spherically stratified solar-like background medium and spherical harmonics degrees $0 \leq l<100$ for four different target positions. We have averaged the sensitivity kernels over 100 frequencies in the frequency regime $2.75$--$3.25$ mHz and normalized with $K(\bi{x}, \bi{x})$ at the target location $\bi{x}$. For better comparisons, we have multiplied the sensitivity kernels with the sound speed.
  • Figure 4: The sound speed sensitivity kernels in a spherically stratified background medium and the $l$ range is $0 \leq l<100$. In the top panels we present the kernels for a uniform medium with $c_0=200$ km/s (as in Fig. \ref{['fig: sound speed kernel_Uniform']}), and in the second line the kernels for a solar-like medium (as in Fig. \ref{['fig: sound speed kernel_Sun']}). In the first column, we show the kernels in the radial direction, and in the second column the kernels in the angular direction. We have averaged the sensitivity kernels over 100 frequencies in the frequency regime $2.75$--$3.25$ mHz. Furthermore, we compare the width of the sensitivity kernels to the classical resolution limit of $\lambda/2$. For better comparisons, we have multiplied the sensitivity kernels with the sound speed.
  • Figure 5: Matrix-valued sensitivity kernel for joint inversion for sound speed $c$ and damping $\gamma$. The left two panels exhibit the diagonal entries, and the right panel the cross-kernel in a uniform two-dimensional medium in a rectangular box of $[0.2\,R_{\odot}, 0.4\,R_{\odot}]^2$. The target location is indexed by a red cross. The kernels are normalized by the maximal value of the sound-speed kernel.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof : Proof of Proposition \ref{['prop:diagonal']}
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • ...and 10 more