Derivation of analytic generalised inverses for transmission+reflection tomography

A methodology is presented for deriving analytic generalised inverses for arbitrary combinations of transmission and reflection traveltime data. These inverses are valid for negligible ray bending and one-dimensional velocity structure. The following conclusions are drawn from this analysis. (1) The slowness variance of a layer decreases with an increase in the number and length of rays which terminate in it. (2) The slowness variance associated with offset transmission VSP data will always be less than that associated with VSP reflection data. (3) The slowness covariance matrix for VSP transmission data is tridiagonal, i.e. the slowness errors in the ith layer only couple to those from the (i+1)th and (i-1)th layers. On the other hand, reflection VSP data will introduce a mutual coupling term between the slowness errors in the reflecting layer and the surface layer. (4) The slowness variance associated with transmission+reflection VSP data will generally be less than that from reflection or transmission VSP data. Moreover, the slowness covariance matrix for transmission+reflection data will be fully populated, meaning that there will be global coupling of slowness errors. However, if the segment lengths of reflecting rays are somewhat similar then the covariance matrix will be essentially tridiagonal. It is concluded from this analysis that the combination of reflection data with transmission data should most often improve the accuracy of slowness reconstruction. This assumes that the reflection traveltime errors are no more than twice that of the transmission traveltime errors.