Introduction

From the estimation of local dips, Lomask (2003a) showed that vertical shifts (time or depth) can be estimated to flatten seismic data in 2-D or 3-D. The basic idea is to integrate local dips or step outs estimated from the data. This integration gives for every point in the data volume a relative vertical (time or depth) delay to one event present on a reference trace. This delay can be used for flattening, where each sample is shifted according to the delay value, or for picking, where one event (or many events) can be followed from the reference trace to everywhere in the data volume by simply stepping up or down according to the delay value. In addition, time/depth shift estimation can be used for many geophysical applications. For instance, Wolf et al. (2004) illustrate how RMS velocities can be estimated without picking. Similarly, Guitton et al. (2004) solve a tomographic problem by inverting the time delays.

Lomask (2003b) identified a non-linear relationship between the local dips and the relative delays. In his approach, however, this property was first ignored by solving simpler linear problems. The goal of this paper is to solve the non-linear problem exactly with a quasi-Newton approach called L-BFGS Guitton and Symes (2003). Solving the non-linear problem allows us to estimate relative time/depth shifts when the local dips are not constant with time or depth, a central assumption in the linear approach of Lomask (2003a).

This paper starts with a presentation of the theory of time/depth delay estimation in 2-D and 3-D. The quasi-Newton method is briefly introduced. Then, the proposed algorithm is tested on 2-D and 3-D data examples. They illustrate the accuracy of the method to compute relative time/depth delays and to perform event picking.