CHARACTERIZATION OF THE NULL SPACE OF A GENERALLY ANISOTROPIC MEDIUM IN LINEARIZED CROSS-WELL TOMOGRAPHY

Anisotropic traveltime tomography can potentially determine many usefu l rock properties, such as crack density or pore shape, that cannot be found from isotropic methods. Similar to isotropic cross-well transmi ssion tomography, many features of an anisotropic model are poorly res olved owing to limited ray-path coverage; the lateral smearing of isot ropic tomography occurs in each elastic parameter in the anisotropic p roblem. Unlike the isotropic problem, however, there is additional ind eterminacy in the solution of the anisotropic tomography problem becau se of ambiguity amongst the several elastic parameters needed to descr ibe anisotropic media. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is the cla ss of model perturbations of a background medium which, to first order , cause no perturbation at all in the cross-well transmission travelti mes. Such model perturbations cannot be determined from the traveltime perturbations; describing these null-space model perturbations gives insight into the indeterminacy in the anisotropic problem. Complementa ry to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation. As expected, the anisotropic null space is larger than the isotropic null space owing to the ambiguity amongst the elastic parameters. We i dentify three categories of model perturbations in the anisotropic nul l space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itsel f in the isotropic null space. Elements in the second category are ani sotropic versions of the most well-known isotropic null-space elements : perturbations which are polynomials in the depth variable with coeff icients which are functions of the horizontal variable satisfying cert ain linear integral constraints; unlike the isotropic problem, the int egral constraints in the anisotropic problem couple together the sever al elastic parameters. The third category consists of model perturbati ons satisfying zero boundary conditions in the wells for which a speci fic linear combination of integrals and derivatives of the several ela stic parameters is in the isotropic null space. In particular, there a re model perturbations in this third category which represent anomalie s that are completely contained in the interior of the model and yet a re in the null space; this behaviour is in marked contrast to the isot ropic problem. These categories are sufficient to describe the anisotr opic null, space completely. We demonstrate that every model perturbat ion in the null space is the sum of an element in the first category ( indicating an indeterminacy of the same nature as in the isotropic pro blem in each of the elastic parameters separately) and an element in t he third category (indicating an ambiguity amongst the parameters). Th e second category gives a rich family of examples of sums of null-spac e elements in the first and third categories, and thereby gives a sens e of just how large the anisotropic null space is. Moreover, we show t hat the traveltime perturbations determine only a small number of feat ures of an elastic perturbation which distinguish between the several elastic parameters. We identify these features precisely; they are fun ctions of depth representing horizontal averages of combinations of th e elastic parameters and their derivatives. Elastic parameters influen cing the horizontal velocity appear more prominently in these features than those influencing the vertical velocity. All other features of t he anisotropic model are ambiguous amongst the elastic parameters: exc ept for these features, it is completely impossible from the traveltim e perturbations alone to determine which elastic parameter (or which c ombination of the elastic parameters) gives rise to given traveltime p erturbations.