Optimization approach to traveltime inversion for a medium with laterally inhomogeneous curvilinear layers

Abstract

Summary Determination of interval velocities is often performed under the assumption, that Dix’s formula gives us reasonable values. This formula has been derived for a medium with horizontal homogeneous layers. If there are strong lateral changes in the shallow interval velocities, Dix’s formula can cause large errors in the velocity model (Al-Chalabi, 1979). For a medium with curvilinear boundaries, a Dix’s type inversion has been developed (Chernjak, Gritsenko, Hubral, Krey, Goldin). These formulas suggest layerwise determination of interval velocities and assume locally-homogeneous layers. As analytically shown by one of the authors (Blias 1988), not taking into account lateral velocity changes may lead to significant errors in interval velocities. There is another problem with layer-by-layer interval velocity calculation. A small error in the first layer causes a bigger error for the second layer and so on. These two errors, in their turn, cause much a bigger error in the third layer and so no. This process may result in very large errors for the deep layers. The reason for these errors lies in the connection between stacking velocities for deep horizons and second derivatives of the shallow interval velocities (Blias, 1981). It implies that small errors in the second derivatives of interval velocities may lead to big estimation errors for deep layers. These two problems show that for complicated geology (velocity anomalies) other methods should be developed. These two problems show that for complicated geology (velocity anomalies) other methods should be developed.