Shuey (1985) showed that in a 1-D earth, the measured reflection
strength of an event at the surface is approximately linear with the square of
its incidence angle, at angles less than 30 degrees. In a 1-D earth, the NMO
equation gives an approximate relationship between offset and incidence angle.
Claerbout (1995) defines the ``stepout'', *p*, as the spatial
derivative of an event's traveltime curve:

(9) |

(10) |

Figure 12 illustrates the estimation of AVO slope and intercept parameters on a deep reflector in the Green Canyon 3-D data, before and after application of LSJIMP. The reflector, which is well under the multiples in the data, is denoted on the zero offset section with ``O'' symbols. The maximum amplitude in a small time window around the reflection were picked automatically, and make up the input data to the least-squares estimation.

Figure 12

We see that while the parameter estimates contain the same trends before and after LSJIMP, the LSJIMP result is more consistent and less ``noisy'' across midpoint. My implementation of LSJIMP works on a CMP-by-CMP basis, so the results shown in Figure 12 are not smoothed across midpoint. The similarity across midpoint is an expression of the true lithology - lithology which LSJIMP better reveals.

Figure 13 illustrates, as a function of midpoint, the
small time windows taken around the deep reflector shown in Figure
12, before and after LSJIMP. The input data to an AVO
parameter estimation are picked maximum amplitudes within the time window as a
function of . Notice the significant increase in reflector
clarity after LSJIMP. Also recall that the data residuals (e.g., in Figures
8 and 9)
are quite small. Therefore, the cleaner reflection events after LSJIMP in
Figure 13 are not only better for AVO analysis - they
also fit the recorded data in a quantitative fashion. LSJIMP is not an *ad
hoc* post-processing technique.

Figure 13

5/23/2004