Wave-equation traveltime tomography by global optimization |

where and are the source and receiver locations, is the modeled data with velocity , and is the observed data. By setting the first derivative of equation (1) around the velocity to zero, the velocity update can be expressed as follows:

where is the step size, is the source signature, and and are the forward wave propagation operator and its adjoint, respectively.

The objective function wave-equation traveltime tomography can be written as follows:

where is the lag of the maximum cross-correlation between the observed data and the data modeled by a velocity model . Again, the first derivative of equation (3) around the lags is set to zero to get the velocity update, which can be expressed as follows:

where is defined as follows:

By examining equations (2) and (4), it can be shown that (WT) can handle much larger velocity errors than (FWI).

Now, I cast the picking procedure of the lags as a global optimization problem with an objective function as follows:

where and are a sparse representation of the source and receiver locations, is a bicubic spline interpolation operator that maps the sparse coordinates and to the original coordinates and , and evaluate the correlation value at .

The goal of the global optimization is to maximize the function described by equation (6), which is to maximize the stacking power along the interpolated spline surface. The searching procedure is a simulated annealing algorithm, which varies the spline points along the time axis in a stochastic sense until a satisfying solution is reached. In the following section, I show the results of using such global scheme to pick the correlation lags.

velali
The true velocity model used to create the data.
Figure 1. |
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Wave-equation traveltime tomography by global optimization |

2010-05-19