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The extended adjoint state and nonlinearity in correlation-based passive imaging

Tram Thi Ngoc Nguyen

TL;DR

The paper addresses passive imaging from ambient noise by formulating it as a covariance-based inverse problem for linear elliptic PDEs. It introduces the extended adjoint state on the squared domain to enable an efficient backpropagation scheme that reduces PDE solves and applies across various PDE models. A key contribution is the explicit derivation of covariance-gradient formulas via an extended adjoint PDE, yielding practical backpropagators for $a,b,c$-type problems and beyond. The work also analyzes the extreme nonlinearity induced by covariance measurements, establishing a tangential-cone-condition–like structure to pave the way for convergence guarantees in iterative regularization. Together, these results offer a universal, parallelizable framework for covariance-based passive imaging with potential real-world impact in solar physics, geophysics, and related fields.

Abstract

This articles investigates physics-based passive imaging problem, wherein one infers an unknown medium using ambient noise and correlation of the noise signal. We develop a general backpropagation framework via the so-called extended adjoint state, suitable for any elliptic PDE; crucially, this approach reduces by half the number of required PDE solves. Applications to several different PDE models demonstrate the universality of our method. In addition, we analyze the nonlinearity of the correlated model, revealing a surprising tangential cone condition-like structure, thereby advancing the state of the art towards a convergence guarantee for regularized reconstruction in passive imaging.

The extended adjoint state and nonlinearity in correlation-based passive imaging

TL;DR

The paper addresses passive imaging from ambient noise by formulating it as a covariance-based inverse problem for linear elliptic PDEs. It introduces the extended adjoint state on the squared domain to enable an efficient backpropagation scheme that reduces PDE solves and applies across various PDE models. A key contribution is the explicit derivation of covariance-gradient formulas via an extended adjoint PDE, yielding practical backpropagators for -type problems and beyond. The work also analyzes the extreme nonlinearity induced by covariance measurements, establishing a tangential-cone-condition–like structure to pave the way for convergence guarantees in iterative regularization. Together, these results offer a universal, parallelizable framework for covariance-based passive imaging with potential real-world impact in solar physics, geophysics, and related fields.

Abstract

This articles investigates physics-based passive imaging problem, wherein one infers an unknown medium using ambient noise and correlation of the noise signal. We develop a general backpropagation framework via the so-called extended adjoint state, suitable for any elliptic PDE; crucially, this approach reduces by half the number of required PDE solves. Applications to several different PDE models demonstrate the universality of our method. In addition, we analyze the nonlinearity of the correlated model, revealing a surprising tangential cone condition-like structure, thereby advancing the state of the art towards a convergence guarantee for regularized reconstruction in passive imaging.

Paper Structure

This paper contains 15 sections, 11 theorems, 76 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Inverse problems with correlation data
  3. Parameter identification in PDEs
  4. Correlation measurement
  5. Passive imaging with covariance data
  6. Sensitivity and adjoint state
  7. Linear measurements
  8. Covariance measurements
  9. The $a, b, c$-problems and extension
  10. Passive imaging for $a, b, c$ parameter inference
  11. Discussion on extensions
  12. Nonlinearity analysis
  13. Auxiliary results
  14. The TTC-like linearization error
  15. Conclusion and outlooks

Key Result

Lemma 3.1

\newlabellem:linearized-state0 The parameter-to-state map $S$ in S is Fréchet differentiable, with derivative and with the state $u=S(\theta)$; we refer to $\phi$ as the linearized state.

Figures (3)

  • Figure 1: Demonstration. Left: Noise source distributed through medium. Middle: Noise signal recorded at sensors. Right: Correlation between pairs of sensors.
  • Figure 1: Extended adjoint state computation. Left via \ref{['adj-ext-1']}: parallel PDE solution $\psi_i, i=1\ldots I$ -- outer product with $q_i$ -- take $\sum_i^I$. Right via \ref{['adj-extend']}: parallel PDE solution $\Psi(\cdot,x'_n), n=1\ldots N$ -- collect and arrange in column-wise.
  • Figure 1: Tangential cone condition (TCC)

Theorems & Definitions (32)

  • Remark 1
  • Lemma 3.1: Linearized state
  • Proof 1
  • Lemma 3.2: Adjoint state
  • Proof 2
  • Remark 2: Adjoint - linear measurement
  • Definition 3.3
  • Remark 3: Covariance and decomposable structure
  • Lemma 3.4: Extended adjoint state
  • Proof 3
  • ...and 22 more