# Imaging and Inversion Algorithms Based on the Source-Type Integral Equation Formulations

Friday, April 21, 2006 - 12:00pm - 1:00pm

Vincent 20

Aria Abubakar (Schlumberger-Doll Research)

Joint work with Tarek M. Habashy (Schlumberger-Doll Research, USA) and Peter van den Berg (Delft University of Technology, The Netherlands).

In this presentation we present a class of inversion algorithms to solve acoustic, electromagnetic and elastic inverse scattering problems of the constitutive material properties of bounded objects embedded in a known background medium. The inversion utilizes measurements of the scattered field due to the illumination of the objects by a set of known wave-fields. By using the source-type integral equation formulations we arrive at two set of equation in terms of the contrast and the contrast sources (the product of the contrast and the fields). The first equation is the integral representation of the measured data (the data equation) and the second equation is the integral equation over the scatterers (the object equation). These two integral equations are solved by recasting the problem as an optimization problem. The main differences of the presented algorithms then other inversion algorithms available in the literature are: (1) We use the object equation itself to constrain the optimization process. (2) We do not solve any full forward problem in each iterative step. We present three inversion algorithms with increasing complexity, namely, the regularized Born inversion (linear), the diagonalized contrast source inversion (semi-linear) and the contrast source inversion (full non-linear). The difference between these three inversion algorithms is in the way we use the object equation in the cost functional to be optimized. Although the inclusion of object equation in the cost functional serves as a physical regularization of the ill-conditioned data equation, the inversion results can be enhanced by introducing an additional regularizer that can help in accounting for any a priori information known about the contrast profile or in imposing any constraints such as limiting the spatial variation of the contrast. We propose to use the multiplicative regularized inversion technique so that there is no necessity to determine the regularization parameter before the optimization is started. This parameter is determined automatically during the optimization process. As numerical examples we present some synthetic and real data inversion from oilfield, biomedical and microwave applications both in two and three-dimensional configurations. We will show that by employing the above approach it is possible to solve full non-linear inverse problem with large number of unknowns using a personal computer with a single processor.

In this presentation we present a class of inversion algorithms to solve acoustic, electromagnetic and elastic inverse scattering problems of the constitutive material properties of bounded objects embedded in a known background medium. The inversion utilizes measurements of the scattered field due to the illumination of the objects by a set of known wave-fields. By using the source-type integral equation formulations we arrive at two set of equation in terms of the contrast and the contrast sources (the product of the contrast and the fields). The first equation is the integral representation of the measured data (the data equation) and the second equation is the integral equation over the scatterers (the object equation). These two integral equations are solved by recasting the problem as an optimization problem. The main differences of the presented algorithms then other inversion algorithms available in the literature are: (1) We use the object equation itself to constrain the optimization process. (2) We do not solve any full forward problem in each iterative step. We present three inversion algorithms with increasing complexity, namely, the regularized Born inversion (linear), the diagonalized contrast source inversion (semi-linear) and the contrast source inversion (full non-linear). The difference between these three inversion algorithms is in the way we use the object equation in the cost functional to be optimized. Although the inclusion of object equation in the cost functional serves as a physical regularization of the ill-conditioned data equation, the inversion results can be enhanced by introducing an additional regularizer that can help in accounting for any a priori information known about the contrast profile or in imposing any constraints such as limiting the spatial variation of the contrast. We propose to use the multiplicative regularized inversion technique so that there is no necessity to determine the regularization parameter before the optimization is started. This parameter is determined automatically during the optimization process. As numerical examples we present some synthetic and real data inversion from oilfield, biomedical and microwave applications both in two and three-dimensional configurations. We will show that by employing the above approach it is possible to solve full non-linear inverse problem with large number of unknowns using a personal computer with a single processor.