Least-Squares Dix Equation

The Dix equation is a nonlinear relationship between RMS velocity and interval velocity. It is, however, linear in the square of the interval velocity. This linearized formulation of the Dix equation was solved by Clapp et al. (1998) by using a preconditioned least-squares optimization with spatial smoothness constraints. In this approach or data fitting goal is to minimize the residual of  

$$\bf W(Cu-d) \approx 0$$ (1)

where $\bf u$ is a vector whose components range over vertical traveltime depth $\tau$ and whose values are the interval velocity squared v²_int. $\bf d$ is the data vector which has the same range as $\bf u$, but whose values are the scaled RMS velocity squared $\tau v^2_{RMS}/\Delta\tau$ where $\tau/\Delta\tau$ is the index on the time axis. $\bf C$ is the casual integration operator. And $\bf W$ is a weight matrix which is proportional to our confidence in RMS velocities.

Since the fitting goal, equation 1, is unstable when there are high frequency variations in RMS velocity, a regularization term is added to penalize this erratic behavior. As done by Valenciano et al. (2003), first order derivatives are used. This system of equations is

$$\begin{eqnarray} \bf W ( Cu - d ) \approx 0 \nonumber\ \epsilon_{\tau} { \bf D... ...prox 0 } \nonumber \ \epsilon_{x} { \bf D_x u} { \bf \approx 0 }\end{eqnarray}$$ (2)

where ${ \bf D_{\tau} }$ and ${ \bf D_x }$ are first-order finite differences derivatives in traveltime and midpoint, respectively. $\epsilon_{\tau}$ and $\epsilon_{x}$ balance the relative importance of the two model residuals, effectively controlling the smoothness.

The approach taken towards the regularization terms will determine whether a smooth or discontinuous model is found. $\ell_2$regularization will produce a smooth result. If a discontinuous velocity is geologically expected, such as for carbonates, $\ell_1$regularization can be used to produce a blocky model Valenciano et al. (2003).