An analytic generalized inverse for common‐depth‐point and vertical seismic profile traveltime equations

Abstract

This paper presents an analytic formula for the generalized inverse matrix associated with vertical seismic profile (VSP) and common‐depth‐point (CDP) traveltime equations. Its functional form explicitly depends upon the source‐receiver geometry and its validity is restricted to horizontally layered earth models where there is negligible ray bending. The importance of the generalized inverse is that it can be used to derive closed‐form expressions for the covariance matrix, condition number, resolution matrix, and average inverse eigenvalues for the traveltime equations; these formulas are useful in designing source‐receiver geometries which optimize velocity reconstruction by traveltime inversion. These formulas also provide a fundamental understanding of some resolution limitations associated with earth tomography experiments. Previously, resolution characteristics of tomographic inversions were usually estimated by empirical computer tests on specialized earth models. Some specific conclusions drawn from these formulas include the following: (1) The elements of the slowness covariance matrix for uncorrelated traveltime errors depend only upon the length and number of ray segments which terminate in a layer, not on the number of rays which intersect a layer. (2) Better velocity resolution in a layer is achieved by increasing the length (larger source offset) and number (more sources) of rays which terminate in that layer. (3) Decreasing the layer thickness will result in a poorly conditioned set of equations; in fact, the [Formula: see text] condition number of the covariance matrix will increase faster than [Formula: see text] as the number of layers N increases. (4) A tradeoff exists between better depth resolution and an increase in the variance of the slowness estimate. (5) The discrete generalized inverse is a back‐projection operator followed by a directional second‐order difference operator. The latter operator is a local and numerically tractable operator, while the equivalent filtering operation for the classical inverse Radon transform is a global Hilbert transform preceded by a first‐order derivative.