This section describes the inversion of a single Rpp and offset gather by SVD to produce a single depth trace each of P-impedance (Ip), S-impedance (Is), and density (). Recall that these parameters are of the form , where is the change in a given property value, and is the average property value, i.e. -- relative changes in elastic material properties.
Figure shows the three elastic parameter traces Ip(z;x), Is(z;x) and (as a function of depth z, for a fixed surface position x). The first depth trace is the P-impedance trace Ip. The correct model values (in %) from Table are plotted in parentheses, adjacent to the m/i estimated peak values. The Ip estimates are seen to be reasonably accurate. Again the first event at 1960 m is weak due to the mute pattern in the shot records affecting the Rpp estimation, and the events below the gas (2960 m) are weak due to attenuation through the low Q gas layer. The Is estimates are also seen to be quite reasonable, although perhaps less robust than Ip. Also, although the Is relative change of 0% has been correctly estimated at the gas sand, there are wavelet sidelobe artifacts which would make it difficult to resolve in field data situations. A similar situation applies to the impedance change at 3100 m where two closely spaced reflections overlap into one apparent reflection. This is analogous to a thin bed resolution problem and linearized amplitude tuning effects. Although the Is estimate is correct at 3100 m (somewhat fortuitously), the overlap of two distinct wavelets erroneously suggests a single impedance change of a large negative polarity which is clearly incorrect. Lastly, the density variation is completely unreliable, except for perhaps the 1960 m event which has the correct polarity and roughly approximate amplitude due to its large specular angle coverage (). This will be explained further in the next section under the topic of choice of elastic parameterization for the inversion.
In summary, with accurate synthetic data and a realistic recording geometry and background model, we are able to recover very reasonable Ip estimates, somewhat less robust Is estimates, and very little (if any) information about density variations.