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Seismic tomography is a non-linear problem. A standard technique is to itteratively assume a linear relation between the change in slowness and the change in travel times Biondi (1990); Etgen (1990) and then re-linearize around the new model. In ray-based methods, this amounts to assuming stationary ray paths and reflection locations to construct a back projection operator Stork and Clayton (1991). The change in this back projection operator from non-linear iteration to non-linear iteration can be thought of as an important second order effect.

By formulating our back projection operator in terms of vertical travel-time ($\tau$) rather than depth our reflector locations become more stable Biondi et al. (1997); Clapp and Biondi (1998). We show that the corresponding back projection operator is less sensitive to our initial velocity estimate. Therefore, our back projection operator changes less from non-linear to non-linear iteration, making the estimation less likely to get stuck in local minima.

THEORY Velocity estimation is fundamentally an inverse problem. The correct solution is to do full wave-form inversion Mora (1987); Tarantola (1986), but is generally impractical. Instead we start from the idea that there is a non-linear operator that relates slowness ($\bf s_{}$)and travel time ($\bf t$),

\bf t\approx \bf T_{n} \bf s_{} .\end{displaymath} (1)

We then attempt to approximate $\bf T_{n}$ by doing a two term Taylor expansion around our initial guess at the slowness field (a version of Newton's minimization method):
\bf t&\approx&\bf T_{n} \bf s_{0} + \bf T_{0}' \bf \Delta s\ \n...
 ...\Delta s\nonumber \\ \bf \Delta t&\approx&\bf T_{0}' \bf \Delta s.\end{eqnarray}

$\bf s_{0}$
is our initial guess at slowness,
$\bf T_{0}'$
is a linear operator describing the relationship between, slowness and travel times given the initial slowness model. In ray-based methods we usually use some stationary ray paths based on the initial slowness model,
$\bf \Delta s$
is the change in slowness,
$\bf t_0$
are the modeled travel times applying $\bf T_{0}'$ to $\bf s_{0}$,
$\bf \Delta t$
are the difference between the modeled travel times, $\bf t_0$,and the measured travel times, $\bf t$.

After inverting for $\bf \Delta s$, we have a new estimate for our slowness field:
\bf s_1 = \bf s_{0} + \bf \Delta s.\end{displaymath} (3)
We can then re-linearize around this new model ($\bf s_1$), constructing a new tomography operator $\bf T_{1}'$. We repeat this procedure until $\bf \Delta t$ is negligible. Figure 1 is a graphical representation of the method.

Figure 1
Newton's method applied to ray based tomography.

There are two problems with this approach. First, Newton's method is only guaranteed to converge to a local minima, we hope that by applying regularization Clapp and Biondi (1999) we can avoid this problem. And second, we are only using the first term in our Taylor expansion, which means that when our higher order derivatives are large, are descent direction will be wrong, and we will converge at a much slower rate. When using rays, this problem occurs when the initial guess at ray paths and reflector locations are too far from their correct locations.

We can obtain a measure of how inaccurate our linear approximation is by looking at how much our linearized tomography operator changes from non-linear iteration to non-linear iteration (the difference erence between $\bf T_{0}'$ and $\bf T_{1}'$.) The smaller the difference, the more accurate our linearization, and the less likely our estimate well diverge.

By forming our tomography in ($\tau,x$) rather than (z,x) space, we reduce the change in $\bf T_{1}'$ from $\bf T_{0}'$. The fundamental reason is that our data is in time rather than depth. In depth, reflector positions and layer boundaries change significantly from iteration to iteration, while in tau, they hardly change at all Biondi et al. (1997).

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