Interferometric imaging condition for wave-equation migration

Wave propagation and scale regimes

Acoustic waves characterized by pressure $p \left (x,y,z,t \right)$ propagate according to the second order acoustic wave-equation for constant density

$$\frac{\partial^2 p}{\partial t^2} = v^2 \nabla^2 p + F_{\lambda}(t) \;,$$ (21)

where $F_{\lambda}\left (t \right)$ is a wavelet of characteristic wavelength $\lambda$.

Given the parameters $l$ (size of inhomogeneities), $\lambda$ (wavelength size), $L$ (propagation distance) and $\sigma$ (noise strength), we can define several propagation regimes.

The weak fluctuations regime characterized by waves with wavelength of size comparable to that of typical inhomogeneities propagating over a medium with small fluctuations to a distance of many wavelengths. This regime is characterized by negligible back scattering, and the randomness impacts the propagating waves through forward multipathing. The relevant length parameters are related by

$$l \sim \lambda \ll L \;,$$ (22)

and the noise strength is assumed small

$$\sigma \ll 1 \;.$$ (23)

The diffusion approximation regime characterized by waves with wavelength much larger than that of typical inhomogeneities propagate over a medium with strong fluctuations to a distance of many wavelengths. This regime is characterized by traveling waves that are statistically stable but diffuse with time. Back propagation of such waves in a medium without random fluctuations results in loss of resolution. The relevant length parameters are related by

$$l \ll \lambda \ll L \;,$$ (24)

and the noise strength is not assumed small

$$\sigma \sim 1 \;.$$ (25)

Interferometric imaging condition for wave-equation migration