Imaging in Random Media

Abstract

We give a self-contained presentation of coherent array imaging in random media, which are mathematical models of media with uncertain small-scale features (inhomogeneities). We describe the challenges of imaging in random media and discuss the coherent interferometric (CINT) imaging approach. It is designed to image with partially coherent waves, so it works at distances that do not exceed a transport mean-free path. The waves are incoherent when they travel longer distances, due to strong cumulative scattering by the inhomogeneities, and coherent imaging becomes impossible. In this article we base the presentation of coherent imaging on a simple geometrical optics model of wave propagation with randomly perturbed travel time. The model captures the canonical form of the second statistical moments of the wave field, which describe the loss of coherence and decorrelation of the waves due to scattering in random media. We use it to give an explicit resolution analysis of CINT which includes the assessment of statistical stability of the images.

Appendices

Appendix 1: Second Moments of the Random Travel Time

We obtain by direct calculation from (75) and (84) that

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [\nu (\vec{{\mathbf{x}}},\vec{\mathbf{y}})\nu (\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}^{{\prime}}) = \frac{\mathcal{L}} {\sqrt{2\pi }\ell}\int _{0}^{1}dt\int _{ 0}^{1}dt^{{\prime}}\right ] \\ & & \mathbb{E}\left [\mu \left (\frac{(1 - t)\vec{\mathbf{y}}} {\ell} + \frac{t\vec{{\mathbf{x}}}} {\ell} \right )\mu \left (\frac{(1 - t^{{\prime}})\vec{\mathbf{y}}^{{\prime}}} {\ell} + \frac{t^{{\prime}}\vec{{\mathbf{x}}}} {\ell} \right )\right ] \\ & & = \frac{\mathcal{L}} {\sqrt{2\pi }\ell}\int _{0}^{1}dt\int _{ 0}^{1}dt^{{\prime}}\,\mathcal{R}\left [\frac{(t^{{\prime}}- t)(\vec{{\mathbf{x}}} -\vec{\mathbf{y}})} {\ell} + \frac{(1 - t^{{\prime}})(\vec{\mathbf{y}}^{{\prime}}-\vec{\mathbf{y}}) + t^{{\prime}}(\vec{{\mathbf{x}}}^{{\prime}}-\vec{{\mathbf{x}}})} {\ell} \right ] \\ & & = \frac{1} {\sqrt{2\pi }}\int _{0}^{1}dt^{{\prime}}\int _{ -\mathcal{L}/\ell}^{\mathcal{L}/\ell}d\tilde{t}\,\mathcal{R}\left [\tilde{t}\,\vec{\mathbf{n}} + \frac{(1 - t^{{\prime}})(\vec{\mathbf{y}}^{{\prime}}-\vec{\mathbf{y}}) + t^{{\prime}}(\vec{{\mathbf{x}}}^{{\prime}}-\vec{{\mathbf{x}}})} {\ell} \right ]\,, {}\end{array}$$

(174)

where we made the change of variables $(t,t^{{\prime}}) \leadsto (\tilde{t},t^{{\prime}})$ with

$$\displaystyle{t^{{\prime}}- t = \frac{\ell} {\mathcal{L}}\tilde{t}.}$$

We can rewrite the result as

$$\displaystyle\begin{array}{rcl} \mathbb{E}\left [\nu (\vec{{\mathbf{x}}},\vec{\mathbf{y}})\nu (\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}^{{\prime}})\right ]& =& \int _{ 0}^{1}dt^{{\prime}}\int _{ -\infty }^{\infty } \frac{d\tilde{t}} {\sqrt{2\pi }}\,1_{[-\mathcal{L}/\ell,\mathcal{L}/\ell]}(\tilde{t})\,\mathcal{R} \\ & &\left [\tilde{t}\,\vec{\mathbf{n}} + \frac{(1 - t^{{\prime}})(\vec{\mathbf{y}}^{{\prime}}-\vec{\mathbf{y}}) + t^{{\prime}}(\vec{{\mathbf{x}}}^{{\prime}}-\vec{{\mathbf{x}}})} {\ell} \right ]\,,{}\end{array}$$

(175)

and note that the integrand converges to $\mathcal{R}\left [\tilde{t}\,\vec{\mathbf{n}} + \frac{(1-t^{{\prime}})(\vec{\mathbf{y}}^{{\prime}}-\vec{\mathbf{y}})+t^{{\prime}}(\vec{{\mathbf{x}}}^{{\prime}}-\vec{{\mathbf{x}}})} {\ell} \right ]$ pointwise, from below, as $\mathcal{L}/\ell \rightarrow \infty$. Since $\mathcal{R}$ is integrable by assumption, we obtain from the Lebesgue dominated convergence theorem that

$$\displaystyle\begin{array}{rcl} \mathbb{E}\left [\nu (\vec{{\mathbf{x}}},\vec{\mathbf{y}})\nu (\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}^{{\prime}})\right ]& \approx & \lim _{ \mathcal{L}/\ell\rightarrow \infty }\int _{0}^{1}dt^{{\prime}}\int _{ -\infty }^{\infty } \frac{d\tilde{t}} {\sqrt{2\pi }}\,1_{[-\mathcal{L}/\ell,\mathcal{L}/\ell]}(\tilde{t}) \\ & & \mathcal{R}\left [\tilde{t}\,\vec{\mathbf{n}} + \frac{(1 - t^{{\prime}})(\vec{\mathbf{y}}^{{\prime}}-\vec{\mathbf{y}}) + t^{{\prime}}(\vec{{\mathbf{x}}}^{{\prime}}-\vec{{\mathbf{x}}})} {\ell} \right ] \\ & =& \int _{-\infty }^{\infty } \frac{d\tilde{t}} {\sqrt{2\pi }}\,\mathcal{R}\left [\tilde{t}\,\vec{\mathbf{n}} + \frac{(1 - t^{{\prime}})(\vec{\mathbf{y}}^{{\prime}}-\vec{\mathbf{y}}) + t^{{\prime}}(\vec{{\mathbf{x}}}^{{\prime}}-\vec{{\mathbf{x}}})} {\ell} \right ]\,,{}\end{array}$$

(176)

as stated in (86).

Appendix 2: Second Moments of the Local Cross-Correlations

Using the Gaussian windows in (147), we obtain

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [\left \vert \mathcal{C}(\overline{\tau }_{o}(\vec{{\mathbf{x}}},\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}),\tilde{\tau }_{ o}(\vec{{\mathbf{x}}},\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}});\vec{{\mathbf{x}}},\vec{{\mathbf{x}}}^{{\prime}})\right \vert ^{2}\right ] = \frac{2\pi } {(32\pi ^{3}L^{2})^{2}B^{4}}\iint _{-\infty }^{\infty } \\ & & d\omega _{1}d\omega _{2}\,e^{-\frac{(\omega _{1}-\omega _{o})^{2}} {B^{2}} -\frac{(\omega _{2}-\omega _{o})^{2}} {B^{2}} }\quad \times \iint _{-\infty }^{\infty } \\ & & d\tilde{\omega }_{1}d\tilde{\omega }_{2}e^{-\frac{(\tilde{\omega }_{1}^{2}+\tilde{\omega }_{2}^{2})} {2} \left ( \frac{1} {\Omega _{\Phi }}^{2} + \frac{1} {2B^{2}} \right ) } \\ & & \mathbb{E}\left [e^{i\big(\omega _{1}-\omega _{2}+\frac{\tilde{\omega }_{1}-\tilde{\omega }_{2}} {2} \big)\nu _{\tau }(\vec{{\mathbf{x}}},\vec{\mathbf{y}})-i\big(\omega _{1}-\omega _{2}-\frac{\tilde{\omega }_{1}-\tilde{\omega }_{2}} {2} \big)\nu _{\tau }(\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}) }\right ]\,, {}\end{array}$$

(177)

and it remains to evaluate the expectation. Because ν _τ is approximately Gaussian, we have

$$\displaystyle\begin{array}{rcl} \mathbb{E}\left [e^{i\big(\Delta \omega +\frac{\Delta \tilde{\omega }} {2} \big)\nu _{\tau }(\vec{{\mathbf{x}}},\vec{\mathbf{y}})-i\big(\Delta \omega -\frac{\Delta \tilde{\omega }} {2} \big)\nu _{\tau }(\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}) }\right ] \approx e^{-\frac{1} {2} \mathbb{E}\left [\big(\Delta \omega +\frac{\Delta \tilde{\omega }} {2} \big)\nu _{\tau }(\vec{{\mathbf{x}}},\vec{\mathbf{y}})-\big(\Delta \omega -\frac{\Delta \tilde{\omega }} {2} \big)\nu _{\tau }(\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}})\right ]^{2} }\,,& & {}\\ \end{array}$$

where we let $\Delta \omega =\omega _{1} -\omega _{2}$ and $\Delta \tilde{\omega } =\tilde{\omega } _{1} -\tilde{\omega }_{2}$. The exponent follows from (84) and (91)

$$\displaystyle\begin{array}{rcl} \mathbb{E}\left [\Big(\Delta \omega + \frac{\Delta \tilde{\omega }} {2} \Big)\nu _{\tau }(\vec{{\mathbf{x}}},\vec{\mathbf{y}}) -\Big (\Delta \omega -\frac{\Delta \tilde{\omega }} {2} \Big)\nu _{\tau }(\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}})\right ]^{2}& \approx & \frac{2\big[(\Delta \omega )^{2} + \frac{(\Delta \tilde{\omega })^{2}} {4} \big]} {\Omega ^{2}} {}\\ & & \left [1 -\int _{0}^{1}dt\,e^{-\frac{t^{2}\vert {\mathbf{x}}-{\mathbf{x}}^{{\prime}}\vert ^{2}} {2\ell^{2}} }\right ]\,. {}\\ \end{array}$$

Substituting in (177), we obtain

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}\left [\left \vert \mathcal{C}(\overline{\tau }_{o}(\vec{{\mathbf{x}}},\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}}),\tilde{\tau }_{ o}(\vec{{\mathbf{x}}},\vec{{\mathbf{x}}}^{{\prime}},\vec{\mathbf{y}});\vec{{\mathbf{x}}},\vec{{\mathbf{x}}}^{{\prime}})\right \vert ^{2}\right ] \\ & & \quad \approx \frac{2\pi } {(32\pi ^{3}L^{2})^{2}B^{4}}\iint _{-\infty }^{\infty }d\omega _{ 1}d\omega _{2}\,e^{-\frac{(\omega _{1}-\omega _{o})^{2}} {B^{2}} -\frac{(\omega _{2}-\omega _{o})^{2}} {B^{2}} } \\ & & \quad \times \iint _{-\infty }^{\infty }d\tilde{\omega }_{ 1}d\tilde{\omega }_{2}e^{-\frac{(\tilde{\omega }_{1}^{2}+\tilde{\omega }_{2}^{2})} {2} \left ( \frac{1} {\Omega _{\Phi }}^{2} + \frac{1} {2B^{2}} \right )-\big[\frac{(\omega _{1}-\omega _{2})^{2}} {\Omega ^{2}} +\frac{(\tilde{\omega }_{1}-\tilde{\omega }_{2})^{2}} {4\Omega ^{2}} \big]\big[1-\int _{0}^{1}dt\,e^{-\frac{t^{2}\vert {\mathbf{x}}-{\mathbf{x}}^{{\prime}}\vert ^{2}} {2\ell^{2}} }\big] }\,,{}\end{array}$$

(178)

and after evaluating the Gaussian integrals, we obtain (149).

Conclusion

This article reviews basic results on coherent array imaging in random media. The random model is motivated by the uncertainty of the small-scale fluctuations of the wave speed in complex media with numerous inhomogeneities. We consider a simple model of wave propagation in random media that captures nevertheless canonical scattering effects on the coherent part of the waves, and consider two imaging methods: migration imaging and CINT imaging. They are both related to an approximation of the solution of the least squares data fit formulation of the inverse problem. Migration imaging is superficially connected to the time reversal process, in the sense that it involves the back-propagation to the imaging region of the time reversed waves measured at the receivers in the array. However, the back-propagation is in a surrogate medium, not in the real one as in the time reversal process, because the medium is not known in imaging. We know only its smooth part, but not its inhomogeneities, which is why we model it as random. This subtle difference between imaging and time reversal has profound effects in random media. We give an explicit and self-contained study of these effects and show that migration imaging is not useful when the waves propagate longer than a scattering mean-free path in random media. The CINT method images by back-propagating to the imaging region local cross-correlations of the measurements at the array. We analyze in detail these local cross-correlations in order to explain why they are useful in imaging. Moreover, we give an explicit resolution analysis of CINT, which includes an assessment of its statistical stability, and illustrate the results with numerical simulations.

Cross-References

• Inverse Scattering

• Large-Scale Inverse Problems in Imaging

• Synthetic Aperture Radar Imaging

• Wave Phenomena