Reflection tomography

Traveltimes are a line integral of the slowness along the ray path expressed as

$$t = \int_s w ds,$$ (23)

where w is the slowness along the ray path, and s is the arc length along the ray path. With the slowness field discretized into cells, the forward problem of the traveltime is expressed as  

$${\bf t} = L {\bf w},$$ (24)

where ${\bf t}$ is a vector of traveltimes, ${\bf w}$ is a vector of slowness, and L is a matrix in which a row contains the path lengths of a ray in each cell. This forward problem is a nonlinear equation because the operator L depends on the slowness vector ${\bf w}$.

This nonlinear problem can be linearized by subtracting a reference problem and expressing the forward problem of the traveltime deviations from the reference model as follows:  

$$\Delta {\bf t} = L_0 \Delta {\bf w},$$ (25)

where $\Delta {\bf t}$ is a vector of traveltime deviations predicted from the reference model, $\Delta {\bf w}$ is a vector of slowness deviations from the reference model, and L₀ is a matrix in which a row contains the path lengths in each cell of the ray traced through the reference model, ${\bf w}_0$.For a given traveltime deviation, $\Delta {\bf t}$, the slowness deviation from the reference model, $\Delta {\bf w}$, is obtained by inverting equation () using the least-squares approach.

In order to solve the original nonlinear problem  (), the above linearized inversion is applied iteratively with the updated back projection operator L_i, which is obtained by ray tracing through the new reference slowness model

$${\bf w}_i = {\bf w}_{i-1}+\Delta {\bf w}_{i-1}$$ (26)

until the traveltime deviations $\Delta {\bf t}$ become small enough; the subscript i represents the iteration number.

Therefore, traveltime tomography can be summarized as an iterative two-step process. First, traveltime deviations are measured by comparing picked traveltimes with expected traveltimes obtained through an assumed velocity model. Then the differences are projected back over the traced ray paths through the assumed velocity model to update the model.

In contrast to transmission traveltime tomography, in which computation of the operator L only requires a slowness model to trace rays, reflection traveltime tomography requires additional information about reflectors, such as the dip and location. Therefore, an image space after prestack migration that can provide both the reflector information and the traveltime deviation is a good choice for picking.