Passive Imaging with Ambient Noise Under Wave Speed Mismatch: Mathematical Analysis and Wave Speed Estimation

Abstract

It is known that waves generated by ambient noise sources and recorded by passive receivers can be used to image the reflectivities of an unknown medium. However, reconstructing the reflectivity of the medium from partial boundary measurements remains a challenging problem, particularly when the background wave speed is unknown. In this paper, we investigate passive correlation-based imaging in the daylight configuration, where uncontrolled noise sources illuminate the medium and only ambient fields are recorded by a sensor array. We first analyze daylight migration for a point reflector embedded in a homogeneous background. By introducing a searching wave speed into the migration functional, we derive an explicit characterization of the deterministic shift and defocusing effects induced by wave-speed mismatch. We show that the maximum of the envelope of the resulting functional provides a reliable estimator of the true wave speed. We then extend the analysis to a random medium with correlation length smaller than the wavelength. Leveraging the shift formula obtained in the homogeneous case, we introduce a virtual guide star that remains fixed under migration with different searching speeds. This property enables an effective wave-speed estimation strategy based on spatial averaging around the virtual guide star. For both homogeneous and random media, we establish resolution analyses for the proposed wave-speed estimators. Numerical experiments are conducted to validate the theoretical result.

Figures (9)

• Figure 1: Daylight configuration for passive imaging of two different types of medium.
• Figure 2: The sensor array and three reference frames $(\widehat{{\itbf e}}_1,\widehat{{\itbf e}}_3)$, $(\widehat{\bf}_1,\widehat{\bf}_3)$, and $(\widehat{{\itbf g}}_1,\widehat{{\itbf g}}_3)$. The unit vector $\widehat{{\itbf e}}_2=\widehat{\bf}_2=\widehat{{\itbf g}}_2$ are orthogonal to the plane of the figure.
• Figure 3: We pick the example that ${{\itbf z}}_r = (-5,0,50)$, $v=1.1$ and calculate $\alpha_r$ and $\alpha_f$ according to (\ref{['eq: alpha_def']}) with ${{\itbf z}}_f = \phi_v({{\itbf z}}_r)$. We plot the function ${\mathcal{G}}_{\alpha_r,\alpha_f}(\xi_1,0,\xi_3,0)$ showing its real part, imaginary part, and modulus from left to right.
• Figure 4: We pick the example that $a_0 = 20$, $\omega_0=4$, $c_0=1$, and ${{\itbf z}}_r = (10,0,20\sqrt{2})$. We plot the profile of $\left|{\mathcal{G}}_{\alpha_r,\alpha_f}\left(\frac{-a_0\omega_0}{c_0|{{\itbf z}}_r|}\eta_1, 0,\frac{-a_0^2\omega_0}{c_0|{{\itbf z}}_r|^2}\eta_3, \frac{-a_0^2\omega_0}{c_0|{{\itbf z}}_r|}\left(\left(\frac{c_0}{c_s}\right)^2-1\right)\right)\right|^2$ for three different choices of $c_s$: $c_s = 0.8$ (left), $c_s$ (middle) and $c_s = 1.2$ (right). The peaks of the three plots takes value $0.87$, $1$ and $0.96$ respectively.
• Figure 5: Illustration for the effective wave speed estimation. For any fixed $c_s$, we pick the searching point at ${{\itbf z}}_s$, and the virtual guide star is positioned at ${{\itbf z}}_g= \phi_v^{-1}({{\itbf z}}_s)$ with $v = c_s/c_0$.

Theorems & Definitions (21)

• Proposition 2.1
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• proof
• Theorem 4.1
• proof
• Remark 4.2