A HYBRID ALGORITHM

To avoid the problems arising from the poor estimation of p in the picked-based method, I devised a hybrid algorithm, in which the events are still defined through the picking process, but instead of estimating horizontal slownesses, this algorithm uses a probability distribution $\phi(x,p)$ which is computed for each event by using beam-stacks.

Beam-stacks represent a mapping of the x-t plane onto the p-x-t cube ( Biondi, 1990). For each point (x,t) in the object plane a sum is performed over a finite path, centered at that point and parametrized by p, with the result corresponding to the point (p,x,t) in the image cube. Different transforms can be defined depending on the aperture of path (number of points in the sum) and the specific choice of parametric curve: slant-stack ("infinite" aperture) and local slant-stack (finite aperture) for a straight-line path, parabolic-transform for an "infinite" aperture parabolic path, and so on.

The concept of beam-stacks can be generalized to include semblance as well as stacking as the basic transform operation. Biondi (1990) compared the semblance spectrum obtained with straight, parabolic, and hyperbolic finite aperture curves. His results indicated that the hyperbolic-stack has a superior resolution power.

In the hybrid scheme, we interpret the hyperbolic semblance spectrum $\phi_j(x,p)$ as the (non normalized) probability that a given point x of event j have a horizontal slowness p. Figure a shows one plane of the three-dimensional beam-stack of the synthetic data, and Figure b shows the two-dimensional beam-stack $\phi_2(x,p)$ corresponding to the second event ($t_0 \approx 0.8$ sec.) of the same data.

 

beamst
Figure 6 (a) A plane from the beam-stack of the data shown in Figure  for p=0.17 msec/meter and (b) a cut in the same beam-stack along the second event of that data.

For a given p, v, and a point x^j of reflection j equation 1 can be used to define the related point x^j-1 in the reflection above (j-1) which share the same horizontal slowness

x^j-1 = f(x^j,p,v).

I define the function $\phi(x^j,p,v)$, as the likelihood that point x^j and x^j-1 share the same horizontal slowness p, in a medium with velocity v, as the sum of their beam-stacks  

$$\Phi(x^j,p,v) = \phi(x^{j},p) + \phi(x^{j-1},p),$$ (4)

where $\phi(x^{j},p)$ is the beam-stack spectrum (whose maximum value is 1.0).

If the velocity is the true velocity, there will be one (and only one) point x^j for each value of p, for which $\Phi$ will assume its maximum value (2.0). If the velocity is not the true velocity, $\Phi$ will not assume its maximum value for all ps. Based on this assumption I define  

$$\Psi(v) = \sum_{p=0}^{p_{\scriptsize max}} \: \max_{x_{\scriptsize min}}^{x_{\scriptsize max}} \: \{\Phi(x^j,p,v)\}.$$ (5)

The estimated velocity for the interval between j-1 and j is the velocity for which $\Psi$ is maximum  

$$v_{\mbox{est}} \;\; \vert \;\;\;\; \Psi(v_{\mbox{est}}) = \max \Psi(v).$$ (6)

Differently from the picked-based method, the hybrid scheme requires the introduction of a scanning through a set of discretized velocities. For each pair of adjacent reflection events, the interval velocity is estimated by the following procedure:

$$\begin{array} {l} \left [ \begin{array} {l} {\tt For \ eac... ...\max \{ {\displaystyle \sum_{i_p}} \psi(i_v,i_p) \}\end{array}$$

The estimated velocity v_j corresponds to the interval velocity between reflectors $\! j-1$ and $\! j$. Instead of repeating this procedure for all pairs of adjacent reflectors, the actual algorithm has an internal loop in $\! j$ that make it more computationally efficient. Also, the beam-stacks are computed only once, before the loop in v. Figure  compares the true model with the results of two inversion algorithms: one (dashed line in the figure) finds the interval velocity model whose Dix's related hyperbolas best fit the picked events; the other (dotted line in the figure) is the result from the hybrid scheme inversion. The first thing we observe in this figure is that the velocities estimated by the hybrid scheme fits better the true model than both the Dix's fitting and the purely picking-based scheme. Most important, the error in the hybrid-method estimation does not change with depth and, actually, is of the same order as the velocity discretization interval (dv) used for the inversion.

 

vel2
Figure 7 The continuous line represents the true velocity model used to generate the synthetic data shown in Figure ; the dashed line represents the velocity model estimated Dix's least squares fitting of the picked events; and the dotted line represents the velocity model obtained with the hybrid method.