The null space of a generally anisotropic medium in linearized surface reflection tomography

Surface reflection tomography is an inversion method that attempts to deter mine simultaneously the subsurface elastic parameters and reflector depths from pre-stack seismic data. To date, this technique has been applied to ac oustic and isotropic elastic media. In this paper, we study surface reflect ion tomography in the anisotropic elastic case. Anisotropic media are descr ibed by many more elastic parameters than are needed to describe isotropic media; these additional parameters are useful for obtaining rock properties such as crack density, pore shape and fracture strike that cannot be found with isotropic methods. Similar to isotropic surface reflection tomography , many features of an anisotropic model are poorly resolved due to limited ray path coverage; the vertical smearing of isotropic tomography occurs in each elastic parameter in the anisotropic problem. Unlike the isotropic pro blem, however, there is additional indeterminacy in the solution of the ani sotropic tomography problem because of ambiguity amongst the several elasti c parameters needed to describe anisotropic media. Also unlike the isotropi c problem, the reflector depths participate very strongly in this ambiguity . We investigate the nature of this indeterminacy by studying the null spac e for linearized tomography, that is, the class of model perturbations of a background medium which, to first order, cause no perturbation at all in t he surface reflection traveltimes. Such model perturbations cannot be deter mined from the traveltime perturbations; describing these null space model perturbations gives insight into the indeterminacy in the anisotropic probl em. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formula tion Our first set of results concerns the 'elastic' null space: perturbations o f the elastic parameters (with zero depth perturbation) which cause no pert urbation in the traveltimes. The results here are very similar to previous results for cross-well transmission tomography. As expected, the 'elastic' null space is larger than the isotropic null space due to the ambiguity amo ngst the elastic parameters. We identify three categories of model perturba tions in the 'elastic' null space. The first category consists of model per turbations for which the perturbation in each of the individual elastic par ameters is itself either in the isotropic null space, or in a closely relat ed set of model perturbations which we call the odd isotropic null space. E lements in the second category are anisotropic versions of the most well-kn own isotropic null space elements: perturbations which are polynomials in t he horizontal variable with coefficients which are functions of the depth v ariable satisfying certain linear integral constraints; unlike the isotropi c problem, the integral constraints in the anisotropic problem couple toget her the several elastic parameters. The third category consists of model pe rturbations satisfying zero boundary conditions at the surface and at the r eflector for which a specific linear combination of integrals and derivativ es of the several elastic parameters is in the isotropic null space. In par ticular, there are model perturbations in this third category which represe nt anomalies that are completely contained in the interior of the model and yet are in the 'elastic' null space; this behaviour is different from the isotropic problem. These categories are sufficient to describe the 'elastic ' null space completely. We demonstrate that every model perturbation in th e 'elastic' null space is the sum of an element in the first category (indi cating an indeterminacy of the same nature as in the isotropic problem in e ach of the elastic parameters separately) and an element in the third categ ory (indicating an ambiguity amongst the parameters). The second category g ives a rich family of examples of sums of null space elements in the first and third categories, and thereby gives a sense of just how large the 'elas tic' null space is. Moreover, we show that the traveltime perturbations caused by an elastic pe rturbation determine only a small number of features of the elastic perturb ation which distinguish between the several elastic parameters. We identify these features precisely: they are functions of the horizontal variable re presenting vertical averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the vertical velocity app ear more prominently in these features than those influencing the horizonta l velocity, and these features are closely related to the zero-offset trave ltimes and the normal moveout velocities. Our second set of results concerns the so-called velocity-depth ambiguity, more precisely here the elastic parameter versus the depth parameter ambigu ity. We demonstrate that for any smooth depth perturbation, there is a pert urbation in the elastic parameters for which the combined elastic and depth perturbations cause zero traveltime perturbations; moreover, this can be a ccomplished by an elliptically anisotropic elastic perturbation with a vert ical axis of symmetry, perturbing the vertical velocity alone. This implies that the velocity-depth ambiguity cannot be resolved here as it can be in the isotropic case. We show an example of an anticlinal structure embedded in a homogeneous background whose reflection traveltimes exactly match thos e from a model with elliptically anisotropic elastic perturbations overlyin g a flat reflector. We also draw a number of conclusions from these results for the special case of transversely isotropic media with a vertical symme try axis.