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Given boundary conditions 8, equation 6 describes the traveltime
of a velocity model with a plane-wave source at the surface. For a given
, equation
7 is a first-order linear PDE on
and thus computation of
is
straightforward. Our idea is inspired by a natural logic: if the resulting
does not satisfy equation
5, we need to modify
in a way such that the misfit decreases, and repeat the
process until convergence.
Mathematically, we define a cost function
based on equation 5:
![\begin{displaymath}
f (z,x) = \nabla x_0 \cdot \nabla x_0 - v_d^2 w\;,
\end{displaymath}](img51.png) |
(9) |
where for convenience we use slowness-squared
instead of
. Note that
is
dimensionless. The discretized form of equation 9 reads
![\begin{displaymath}
\mathbf{f} = (\nabla \mathbf{x_0} \cdot \nabla) \mathbf{x_0...
...
- \mbox{diag}(\mathbf{v_d} \star \mathbf{v_d}) \mathbf{w}\;.
\end{displaymath}](img54.png) |
(10) |
In equation 10,
and
are all column vectors after
discretizing the computational domain
. For example,
is the discretized column vector of
. The vector
may require interpolation because it is in
while
the discretization is in
. The interpolation can be done after forward mapping from
to
at current velocity model. We denote an operator which is a matrix
. The other operator
expands a vector
into a diagonal matrix. Finally, the symbol
stands for an element-wise vector-vector multiplication.
As is common in many optimization problems, we seek to minimize the least-squares norm of
:
![\begin{displaymath}
E [\mathbf{w}] = \frac{1}{2} \Vert\mathbf{f}\Vert^2 = \frac{1}{2} \mathbf{f}^T \mathbf{f}\;,
\end{displaymath}](img63.png) |
(11) |
where the superscript
stands for transpose. The Gauss-Newton method in optimization
requires linearizing the cost function in equation 10:
![\begin{displaymath}
\frac{\partial f}{\partial w} =
2 (\nabla x_0 \cdot \nabla...
...ial w} -
2 v_d w \frac{\partial v_d}{\partial w} - v_d^2\;.
\end{displaymath}](img65.png) |
(12) |
The Fréchet derivative matrix
required by inversion is the discretized
form of equation 12, i.e.,
. In Appendix
B we find that
is a cascade and summation of several parts. An update
at current
is found by solving the following normal equation arising from
the Gauss-Newton approach (Björck, 1996):
![\begin{displaymath}
\delta \mathbf{w} = \left[ \mathbf{J}^T \mathbf{J} \right]^{-1} \mathbf{J}^T (- \mathbf{f})\;.
\end{displaymath}](img69.png) |
(13) |
Equations 11 and 13 together suggest a nonlinear inversion
procedure for solving the original system of PDEs 5, 6 and 7. The inversion
is analogous to traveltime tomography but with more complexity. The cost 9 can be interpreted as
difference between modeled and observed geometrical spreadings. However, both of them depend on the model
,
while in traveltime tomography the observed arrival times are independent of
. The forward modeling in our case
involves two steps, which construct a curvilinear coordinate system that is sensitive to lateral velocity
variations. On the other hand, the forward modeling in traveltime tomography consists of only one step. Last but not least, unlike
traveltime tomography, we have observations everywhere in the computational domain, except for areas excluded due to instabilities of the numerical implementation, as we will
discuss later.
Before introducing a numerical implementation, we would like to point out several important facts
and assumptions that make a successful time-to-depth conversion possible by the proposed method:
- Caustics must be excluded from the computational domain. In regions where caustics develop, the gradient
goes to infinity and the cost function is not well-defined. For all numerical examples in this
paper, we do not encounter this issue. In the Discussion section, we provide a possible strategy to cope with
this limitation.
- According to derivations in Appendix B, the calculation of
depends on
values of
and
.
Thus the input
should be differentiable. This requirement can be satisfied during
estimation by
using regularization (Fomel, 2003).
- Similarly to all nonlinear inversions, the proposed method requires a
prior model that is sufficiently close to desired model at the global minimum
. Meanwhile, to
guarantee stability and a smooth output, some form of regularization should be imposed during
inversion (Zhdanov, 2002; Engl et al., 1996).
- Our formulation does not change the ill-posed nature of the original problem. One
assumption is that condition 8 describes all in-flow domain boundaries of
and
. In other words, the image rays are only allowed to be either parallel to or exiting (out-flow)
all other boundaries of the computational domain except the surface.
For the prior model, we adopt the Dix-inverted model. In other words, we seek to refine the interval
velocity given by equation 2 by taking the geometrical spreading of image rays into consideration
according to equation 3.
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Up: Theory
Previous: Connection between time- and
2015-03-25