Inverse scattering for a random potential

Abstract

In this paper we consider an inverse problem for the n-dimensional random Schrödinger equation (∆ − q + k 2 )u = 0. We study the scattering of plane waves in the presence of a potential q which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential q, we uniquely determine the principal symbol of the covariance operator of q. Especially, for n = 3 this result is obtained for the full non-linear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance. 35 4.4. Non-zero-mean potentials 37 Appendix A. Scaling regimes 40 A.1. Two-scale model of a non-smooth scatterer 40 A.2. Inverse scattering with the two-scale model 43 A.3. Effects from the higher order scattering 44 Appendix B. Random variables with Gaussian probability laws 47 References

Inverse random source scattering for the Helmholtz equation in inhomogeneous media

“…, the random function is a distribution in f ∈ W s− d 2 −ǫ,p for any ǫ > 0 and p ∈ (1, ∞) (cf. Lemma 2.6), which is rougher than those considered in [9,15,20,21]; (2) if σ = 0 and s ∈ [ d 2 , d 2 + 1), the results obtained in this paper coincides with the ones given in [20]. The paper is organized as follows.…”

“…The white noise model can also be found in [6] and [5] for the one-dimensional problem and the stochastic elastic wave equation, respectively. Recently, the model of a generalized Gaussian field was developed to handle random processes [9,15]. The random function is said to be microlocally isotropic of order 2s if the covariance operator is a pseudo-differential operator with principal symbol given by µ(x)|ξ| −2s , where µ ≥ 0 is a smooth and compactly support function and is called the micro-correlation strength of the random function.…”

Inverse Elastic Scattering for a Random Source

“…Recently, a different model is proposed in [18,32] to describe random functions. The random function is considered to be a generalized Gaussian random function whose covariance is represented by a classical pseudo-differential operator.…”

“…The authors studied an inverse problem for the two-dimensional random Schrödinger equation where the potential function was random. It is shown that the principle symbol of the covariance operator can be uniquely determined by the backscattered far field [18] or backscattered field [32], generated by a single realization of the random potential and plane waves [18] or a point source [32] as the incident field. A related work can be found in [25] where the authors considered an inverse scattering problem in a half-space with an impedance boundary condition where the impedance function was random.…”

Determining a Random Schrödinger Equation with Unknown Source and Potential

“…However, our study indicates that a single realization of the far-field measurement can be used to uniquely recover the variance function and the potential in certain scenarios. A crucial assumption to make the single-realization recovery possible is that the randomness is independent of the wave number k. Indeed, there are assorted applications in which the randomness changes slowly or is independent of time [5,20], and by Fourier transforming into the frequency domain, they actually correspond to the aforementioned situation. The single-realization recovery has been studied in the literature; see, e.g., [5,19,20].…”