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

Abstract

Surface reflection tomography is an inversion method that attempts to determine simultaneously the subsurface elastic parameters and reflector depths from pre-stack seismic data. To date, this technique has been applied to acoustic and isotropic elastic media. In this paper, we study surface reflection tomography in the anisotropic elastic case. Anisotropic media are described 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 problem, however, there is additional indeterminacy in the solution of the anisotropic tomography problem because of ambiguity amongst the several elastic parameters needed to describe anisotropic media. Also unlike the isotropic problem, the reflector depths participate very strongly in this ambiguity. We investigate the nature of this indeterminacy by studying the null space for linearized tomography, that is, the class of model perturbations of a background medium which, to first order, cause no perturbation at all in the surface reflection traveltimes. 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. Complementary to computational approaches towards identifying the null space for discrete formulations of tomography, we study a continuum formulation. Our first set of results concerns the ‘elastic’ null space: perturbations of the elastic parameters (with zero depth perturbation) which cause no perturbation 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 amongst the elastic parameters. We identify three categories of model perturbations in the ‘elastic’ null space. The first category consists of model perturbations for which the perturbation in each of the individual elastic parameters is itself either in the isotropic null space, or in a closely related set of model perturbations which we call the odd isotropic null space. Elements in the second category are anisotropic versions of the most well-known isotropic null space elements: perturbations which are polynomials in the horizontal variable with coefficients which are functions of the depth variable satisfying certain linear integral constraints; unlike the isotropic problem, the integral constraints in the anisotropic problem couple together the several elastic parameters. The third category consists of model perturbations satisfying zero boundary conditions at the surface and at the reflector for which a specific linear combination of integrals and derivatives of the several elastic parameters is in the isotropic null space. In particular, there are model perturbations in this third category which represent 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 the ‘elastic’ null space is the sum of an element in the first category (indicating an indeterminacy of the same nature as in the isotropic problem in each of the elastic parameters separately) and an element in the third category (indicating an ambiguity amongst the parameters). The second category gives 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 ‘elastic’ null space is. Moreover, we show that the traveltime perturbations caused by an elastic perturbation determine only a small number of features of the elastic perturbation which distinguish between the several elastic parameters. We identify these features pre-cisely: they are functions of the horizontal variable representing vertical averages of combinations of the elastic parameters and their derivatives. Elastic parameters influencing the vertical velocity appear more prominently in these features than those influencing the horizontal velocity, and these features are closely related to the zero-offset traveltimes 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 ambiguity. We demonstrate that for any smooth depth perturbation, there is a perturbation in the elastic parameters for which the combined elastic and depth perturbations cause zero traveltime perturbations; moreover, this can be accomplished by an elliptically anisotropic elastic perturbation with a vertical 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 those from a model with elliptically anisotropic elastic perturbations overlying a flat reflector. We also draw a number of conclusions from these results for the special case of transversely isotropic media with a vertical symmetry axis.