A backscattering model based on corrector theory of homogenization for the random Helmholtz equation. (English) Zbl 1419.35244
Summary: This work concerns the analysis of wave propagation in random media. Our medium of interest is sea ice, which is a composite of a pure ice background and randomly located inclusions of brine and air. From a pulse emitted by a source above the sea ice layer, the main objective of this work is to derive a model for the backscattered signal measured at the source/detector location. The problem is difficult in that, in the practical configuration we consider, the wave impinges on the layer with a non-normal incidence. Since the sea ice is seen by the pulse as an effective (homogenized) medium, the energy is specularly reflected and the backscattered signal vanishes in a first order approximation. What is measured at the detector consists therefore of corrections to leading order terms, and we focus in this work on the homogenization corrector. We describe the propagation by a random Helmholtz equation, and derive an expression of the corrector in this layered framework. We moreover obtain a transport model for quadratic quantities in the random wavefield in a high frequency limit.
MSC:
35R30 | Inverse problems for PDEs |
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
35Q60 | PDEs in connection with optics and electromagnetic theory |
86A05 | Hydrology, hydrography, oceanography |
86A40 | Glaciology |
35Q35 | PDEs in connection with fluid mechanics |
35Q85 | PDEs in connection with astronomy and astrophysics |
35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |