EnKF-levelset method for an acoustics inverse medium scattering problem

This paper demonstrates a numerical method for reconstructing penetrable obstacles in a homogeneous background medium. The ensemble Kalman method is proposed as an inversion solver, and the level set technique is discussed as a flexible way to tracking the boundary. Through some numerical examples, we show that the proposed method is effective and flexible even without the priori number knowledge of the scatterers.


Introduction
In this paper, we consider a special kind of inverse medium scattering problem for determining the shape of permeable obstacles in a homogeneous background. It is assumed that the permeable obstacle to be piecewise constant. This assumption can arises in many situations such as layered medium. Unlike the direct problem, the inverse problem admits great challenges due to its nonlinearity and illposedness. Over the past decades, many researchers have been devoted to this problem and put forward different numerical methods, including Gauss-Newton method, linear sampling method, extended sampling method, etc. For more details one can refer to [1][2][3].
In this paper, we consider the ensemble Kalman filter (EnKF) as the inversion method to reconstruct the shape of the scatterer. The EnKF is a widely used methodology [3][4][5], which can be deduced from the perspective of both Bayesian statistics and optimization. For the inverse problem, the EnKF uses an ensemble of particles, through a sequential process, the states of these particles can be improved by combing the dynamic model and the data. We will show that in the whole inversion process, we do not need to evaluate the Fréchet derivative of the forward operator and its adjoint [6]. Compared with the traditional Newton-type algorithm, it is very attractive in many cases when the Fréchet derivative of the forward operator and its adjoint are hard to achieve. To be flexibly deal with the topological changes of the scatterer boundary, we introduce the level set technique [7] into the EnKF algorithm. To the best of our knowledge, this combination is rarely reported in the field of inverse scattering [8]. Through some numerical experiments, we show that our algorithm is effective and flexible, and it can not only deal with one scatterer, but also can recover the shape of multiple scatterers.
The organization of the paper is as follows. In Section 2, we briefly discuss the inverse medium scattering problem and the level set parameterization. The EnKF algorithm is proposed in Section 3. In Section 4, we do some numerical experiments to show the effectiveness and flexibility of our algorithm.

The inverse medium problem
Denote by 0 k  is the wave number and d is incident direction. The propagation of the time-harmonic plane wave ( ) i u x scattered by an inhomogeneous medium can be modeled by the following Helmholtz equation 2 2 (1 ( )) 0, in , where ( ) u x is the total field, which can be regarded as the superposition of the incident field ( ) i u x and the scattering field ( ) (3) is the Sommerfeld radiation condition. The properties of the inhomogeneous medium is described by ( ) b x . In this paper we The inverse medium scattering problem (IMSP) considered in this paper is stated as follows: x are measurement points that collected on a closed surface  outside D .

Level-set parameterization
For the sake of simplicity, in this section we first transform the problem into the form of operator equations. According to (1)   the observation operator and then taken the measurement noise into consideration, the problem IMSP can be written as ( )( ) , where z is the measurement data and  is the noise. In this paper we always assume  to be Gaussian with mean zero and covariance C . Next, we consider the level-set parameterization of the problem (5 From (6) we can see that the geometric field of interest can be obtained from the level set field through the relationship of the map 2 : ( ) S L     . Together with (5) and (6), we can finally transform the problem IMSP as: for given z find  such that ( ) ,

Ensemble Kalman filter
Based on the previous discussion of the level set parameterization of the inverse problem, in this section we propose the ensemble Kalman method to reconstruct the geometry of the scatterer. Let : . The basic idea of the EnKF method is to use an ensemble of particles, through an update process by combining model (8) and (9) to get an improved state from the current one. The improved state can be obtained through the 3DVar technique by minimizing The initial ensemble of particles in the EnKF approach can be generated from the prior distribution. In this paper, we assume the prior is Gaussian of the Whittle-Martérn type. The covariance function of the prior is given by where ( )   is the Gamma function,  is the smooth parameter, l is the length scale and K  is the th order second kind modified Bessel function. After the initial ensemble is generated, the update procedure of the EnKF method is stated as