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![]() | First-break traveltime tomography with the double-square-root eikonal equation | ![]() |
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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:
We start by deriving the Frechét derivative matrix of standard tomography. Denoting
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.
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cartonstd
Figure 3. The gradient produced by standard tomography. The solid curve indicates a shot-indexed characteristic. |
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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:
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
.
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cartondsr
Figure 4. The gradient produced by DSR tomography. The solid curve indicates a DSR characteristic, which has one end in plane ![]() ![]() |
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![]() | First-break traveltime tomography with the double-square-root eikonal equation | ![]() |
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