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

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.

💡 Research Summary

The manuscript addresses the challenging inverse problem of passive imaging, where one seeks to recover an unknown medium by exploiting ambient noise rather than controlled sources. The authors focus on the situation in which the recorded signals at sensor locations are processed through cross‑correlation (or equivalently auto‑covariance) to produce a data set that is quadratic in the underlying wavefield. This quadratic transformation dramatically increases the dimensionality of the data and introduces a strong non‑linearity into the forward operator, making standard linear‑measurement based inversion techniques inefficient and theoretically fragile.

The paper first formulates the forward problem in a unified abstract setting. The unknown parameter (\theta) belongs to a real Banach space (X) and enters a family of linear elliptic operators (D(\theta):U\to W^{*}) in an affine way, i.e. (D(\theta)=K+B(\cdot)\theta). For a given source (f) the state (u) solves (D(\theta)u=f), which defines the parameter‑to‑state map (S_f(\theta)=D(\theta)^{-1}f). In classical inverse problems the measurement operator (M:U\to Y) is linear, and the forward map is simply (F=M\circ S_f). In passive imaging, however, the source is a random field with known covariance, and the observable is the sample covariance of the state, \