INTRODUCTION

The process of velocity analysis transforms field data in (x,t) coordinates to velocity coordinates ($v,\tau$) where $\tau$ is zero-offset time. Given any three coordinates, the fourth can be determined from them. Claerbout (SEP-65) suggested making v the computed coordinate, because it alone of the four lacks the a priori sampling established at data acquisition time. The name ``pixel-precise'' comes from that the default data space (x,t) sampling is used. This method has interesting consequences in computation cost, interpolation and truncation artifacts, and inverse transformations (Claerbout, op cit.). In this article I examine the results of the forward transformation of field data.

Since any of the four coordinates can be calculated from the other three, this suggests the four velocity analysis methods of Table 1.

$$\tau,v$$ 3|l|input variables: t,x

1|c|LOOPED vari- 2|c|CALCULATED

1|c|able (cost) 2|c|variable 1|c|results

$$x,v,\tau$$ (1) t pull conventional

$$x,\tau,t$$ (2) v push fewer artifacts

$$\tau$$ (3) x,v,t push similar to conventional

$$v,\tau,t$$ (4) x pull not studied

Line 1 is conventional hyperbola summation. Each point in velocity space generates a hyperbola to be summed along in data space. The term pull refers to taking a sample from data space based on a calculation using velocity space coordinates. Push moves a sample to velocity space based on a calculation using data space coordinates.

The pixel-precise method is the same as calculating v (Line 2). It will be referred to as ``velocity-push'' in this article.

 

Figure 1: Velocity spectra of the same input gather: three attributes of three methods. The count is the samples summed into each output location. The semblance is the mean-squared divided by power. The last column is the semblance weighted by the mean.

The final method studied in this article is the $\tau$-push method to see whether the push aspect confers similar results to velocity-push.

The computation cost of each method is determined by multiplying the length of the three looping coordinates in Table 1. The velocity-push method has the special property that computation cost is independent of number of velocities used. The ``number of velocities'' means rounding the calculated velocity to a bin size, then summing together all the data points in the bin to improve signal-to-noise.