An error estimate on the direct inversion model in shear stiffness imaging

Abstract

This paper considers an inverse problem of shear stiffness imaging in a linear isotropic two-dimensional acoustic media, where the targeted parameter is the shear modulus μ. For given single component displacement data, the mathematical model to recover the shear modulus appears to be a first-order partial differential equation while a much simpler algebraic model, called the direct inversion, can be derived by eliminating the first derivative terms of the shear modulus from the partial differential equation model. The objective of this paper is to establish a theoretical bound on the relative difference between the true value of the modulus and the approximated value reconstructed from the direct inversion. We exhibit a quantitative estimate of the relative error and present reconstruction examples by the direct inversion from simulated data. We demonstrate that the relative error of the numerical solution is well bounded by the theoretical error estimate.