First-break traveltime tomography with the double-square-root eikonal equation |

The first-break traveltime tomography with DSR eikonal equation (DSR tomography) can be established by following a procedure analogous to the traditional one with the shot-indexed eikonal equation (standard tomography). To further reveal their differences, in this section we will derive both approaches.

For convenience, we use slowness-squared instead of velocity in equations 1, 3 and 4. Based on analysis in Appendix A, the velocity model and prestack cube are Eulerian discretized and arranged as column vectors of size and of size . We denote the observed first-breaks by , and use and whenever necessary to discriminate between computed from shot-indexed eikonal equation and DSR eikonal equation.

The tomographic inversion seeks to minimize the (least-squares) norm of the data residuals. We define
an objective function as follows:

Here and are gradient vector and Hessian matrix, respectively. We may evaluate the gradient by taking partial derivatives of equation 5 with respect to , yielding

where is the Frechét derivative matrix and can be found by further differentiating with respect to .

We start by deriving the Frechét derivative matrix of standard tomography. Denoting

Here we assume that there are in total shots and use for first-breaks of the th shot. Applying to both sides of equation 9, we find

Kinematically, each contains characteristics of the th shot. Because shots are independent of each other, the full Frechét derivative is a concatenation of individual , as follows:

Inserting equation 11 into equation 7, we obtain

where, similar to , stands for the observed first-breaks of the th shot.

Figure 3 illustrates equation 12 schematically, i.e. the gradient produced by standard tomography. The first step on the left depicts the transpose of the th Frechét derivative acting on the corresponding th data residual. It implies a back-projection that takes place in the plane of a fixed position. The second step on the right is simply the summation operation in equation 12.

cartonstd
The gradient produced by standard
tomography. The solid curve indicates a shot-indexed
characteristic.
Figure 3. |
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To derive the Frechét derivative matrix associated with DSR tomography, we first re-write equation
1 with definition 8

We recall that and are of different lengths. Meanwhile in equation 13, both and have the size of . Clearly in equation 14 and must achieve dimensionality enlargement. In fact, according to Figure 1, and can be obtained by spraying such that and . Therefore, and are essentially spraying operators and their adjoints perform stackings along and dimensions, respectively.

In Appendix B, we prove that has the following form:

Note that unlike equation 12, equation 16 can not be expressed as an explicit summation over shots.

Figure 4 shows the gradient of DSR tomography. Similarly to the standard tomography, the gradient produced by equation 16 is a result of two steps. The first step on the left is a back-projection of prestack data residuals according to the adjoint of operator . Because contains DSR characteristics that travel in prestack domain, this back-projection takes place in and is different from that in standard tomography, although the data residuals are the same for both cases. The second step on the right follows the adjoint of operators and and reduces the dimensionality from to . However, compared to standard tomography this step involves summations in not only but also .

cartondsr
The gradient produced by
DSR tomography. The solid curve indicates a DSR
characteristic, which has one end in plane and the
other in plane . Compare with Figure 3.
Figure 4. |
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First-break traveltime tomography with the double-square-root eikonal equation |

2013-10-16