Introduction

The process of making manual velocity picks from velocity semblance scans can be an extremely time consuming bottleneck in routine preprocessing of seismic reflection data, and may even represent a powerful deterrent to performing unaliased spatial velocity analyses for 3-D surveys. A robust method of making automatic velocity picks is therefore of practical interest.

Velocity scans (or spectra), as discussed by Taner and Koehler (1985) for example, are computed by stacking or migrating seismic reflection data along hyperbolic trajectories h parameterized by moveout velocity $\bar{v}_s$and zero-offset traveltime $\tau$. The summation procedure over a range of $\bar{v}_s$ and $\tau$ values at a given CMP location produces a single velocity scan. Semblance S, one of many possible coherency criteria, is defined as the ratio of stacked energy to input energy:

 

$$S(\bar{v}_s,\tau) = \frac{ \left( \sum_h {\rm Data} \left[... ...left( {\rm Data} \left[ h(\bar{v}_s,\tau) \right] \right) ^2 }$$ (1)

Figure a (left) is an example of a CMP gather, and Figure b is the corresponding velocity semblance scan. Velocity picking is the process of defining an optimal stacking velocity trajectory, $\bar{v}_s(\tau)$, along semblance peaks in each velocity scan $S(\bar{v}_s,\tau)$. Dix (1955) and Claerbout (1976) showed that stacking velocity is exactly equivalent to rms velocity $\bar{v}_r$ in a 1-D earth at a fixed (not necessarily vertical) propagation angle. For somewhat vertical incident-angle arrivals in a somewhat 1-D earth, the relationship becomes approximate:

 

$$\bar{v}_s^2(\tau) \sim \bar{v}_r^2(\tau) = \frac{1}{\tau} \int_0^{\tau} v_i^2(t)\,dt$$ (2)

where v_i is the interval velocity model, explicitly a function of the vertical traveltime $\tau$, but implicitly a function of vertical depth z.

Conventionally, the stacking velocities $\bar{v}_s$ are picked manually from velocity scans by human interpretation at a workstation. This can be extremely time consuming and tedious, and may require several man-months for a typical 3-D exploration survey. Hence, it is of practical interest to develop a robust and fast automatic velocity picking computer algorithm. The automatic velocity picks can then be subsequently evaluated by an interpreter for quality control and refined as necessary, saving much time in the overall procedure.

I present a method to simultaneously obtain an optimal fit to the velocity semblance scan and a geologically reasonable interval velocity model, using a nonlinear Monte Carlo optimization. There are two key criteria to my approach: (1) the $\bar{v}_s$ pick must maximize the semblance integral along the rms path in the scan, and (2) the $\bar{v}_s$ pick must correspond to a physically realizable and geologically reasonable v_i interval velocity model. In a related work, Zhang (1991) uses constrained linear and nonlinear optimization to perform automatic velocity picking. Zhang's algorithm gives an accurate fit to the semblance peaks, however, the resulting $\bar{v}_s$ picks can yield unreasonable and even unphysical (imaginary) interval velocities after Dix inversion. Toldi (1989) uses a linearized conjugate gradient method with v_i constraints to find an interval velocity model which maximizes semblance. Toldi's results give physical interval velocities, but the fitting algorithm may easily become trapped in a local minimum since it is a linearization about a initial starting model of the full nonlinear optimization problem. Finally, Rothman (1985) proposed a method for generating random v_i(x,z) earth models, calculating the resulting non-hyperbolic moveout at each CMP gather, and performing a nonlinear optimization of the velocity stack power by simulated annealing. My approach is not as ambitious as that of Rothman, but based on criteria (1) and (2), may be viewed as a desirable extension to the methods of Toldi and Zhang.

This paper proceeds as follows. First I discuss a method to find a global parametric $\bar{v}_r$ and v_i fit by a controlled parameter search. Then I discuss the Monte Carlo random walk perturbations to the parametric $\bar{v}_r$ solution, along with issues of convergence and constraints. Finally, I present the results of the application of the algorithm to 300 marine CMP gathers.

 

cmpscan
Figure 1 A representative CMP gather (a) and its corresponding velocity semblance scan (b).