Traveltime and slowness model

In a cross-well experiment, the seismic source is located at various positions in one well and receivers are deployed in the other well. The seismic waves generated by the source propagate through the medium within the cross-well region, and are recorded by the receivers. Each pair of source and receiver generates a seismic trace from which we can pick the traveltime of the first arrival (or first break). Therefore, from a cross-well experiment, one can obtain a set of first arrival traveltimes $\{t_i(r);\ i=1,2,\cdots,N\}$ that are the functions of receiver depth r. Subscript i indicates the source number, and N is the total number of sources used in the experiment.

The medium between two wells is characterized by a slowness model. The slowness model is usually described as a linear superposition of a set of basis functions:

$$m(x,z)=\sum^M_{j=1}m_j\beta_j(x,z),$$ (1)

where $\{\beta_j(x,z);\ j=1,2,\cdots,M\}$ is a set of known basis functions, and $\{m_j;\ j=1,2,\cdots,M\}$ is a set of unknown coefficients that determines the slowness model. The choices of basis functions should depend on the experimental geometry, prior knowledge of the slowness model, and the resolution and accuracy of the reconstruction. The goal of the traveltime tomographic inversion is to reconstruct the slowness model of the medium between the two wells by using the traveltime information picked from the data.