By formulating our back projection operator in terms of vertical travel-time () 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 ()and travel time (),
We then attempt to approximate by doing a two term Taylor expansion around our initial guess at the slowness field (a version of Newton's minimization method):
After inverting for , we have a new estimate for our slowness field:
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 and .) The smaller the difference, the more accurate our linearization, and the less likely our estimate well diverge.
By forming our tomography in () rather than (z,x) space, we reduce the change in from . 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).