In contrast with the scalar wave case, the scattering of elastic waves in the long wavelength limit yields data containing a surprising amount of information concerning the nature of the scatterer. We will consider both deterministic and probabilistic versions of the inversion problem pertaining to the above scattering problem. The deterministic version provides theoretical insight into the "blindspots" of an optimal inversion procedure in the hypothetical limit of zero measurement error. The probabilistic version is appropriate for the interpretation of real data containing errors and possible inconsistencies. In the former category our-discussion will start with a review of earlier results obtained by Kohn and Rice, Gubernatis, and the author. Some new results dealing with ellipsoidal inclusions will be discussed.