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

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## DSR tomography

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:

 (5)

where the superscript represents transpose. A Newton method of inversion can be established by considering an expansion of the misfit function 5 in a Taylor series and retaining terms up to the quadratic order (Bertsekas, 1982):
 (6)

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
 (7)

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

 (8)

as the partial derivative operator in the 's direction, equation 4 can be re-written as
 (9)

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
 (10)

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:
 (11)

Inserting equation 11 into equation 7, we obtain
 (12)

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
Figure 3.
The gradient produced by standard tomography. The solid curve indicates a shot-indexed characteristic.

To derive the Frechét derivative matrix associated with DSR tomography, we first re-write equation 1 with definition 8

 (13)

where and are at sub-surface source and receiver locations, respectively. Note that in equation 13 appears twice. Thus a differentiation of with respect to must be carried out through the chain-rule:
 (14)

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:

 (15)

Combining equations 7 and 15 results in
 (16)

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
Figure 4.
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.

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

Next: Numerical Implementation Up: Theory Previous: DSR eikonal equation

2013-10-16