We perform a prestack seismic impedance inversion on each of the three
synthetic surveys. The basis of the method is outlined here, and given
in more detail in Lumley and Beydoun (1991), and Lumley (1993).
The first step consists of estimating angle-dependent reflectivity
via a true-amplitude least-squares Kirchhoff prestack migration.
This is obtained by an *l _{2}* estimate of the coefficient in
constant offset sections at each subsurface point:

(14) |

and a separate *l _{1}* estimate of the reflection angles directly from the
data in constant offset sections at each subsurface point:

(15) |

where *D* are the recorded seismic data.
To complete the final estimation of , we make a simple
set of mappings from the separate estimates of and .A map of can be obtained directly as:

(16) |

Then, since there is a one-to-one mapping of to any point , and to the same point , there is a unique map of and to such that:

(17) |

which is the desired result. This completes the angle-dependent reflectivity estimation process.

Once we have estimated ,an inverse problem for three isotropic elastic
parameters can be posed. Under the assumption that *relative contrasts*
in material properties are small at reflecting boundaries, and the reflection
angles are well within the pre-critical region (Aki and Richards, 1980),
a linearization of the
Zoeppritz plane wave reflection coefficients can be made at every subsurface
point :

(18) |

where are the relative contrasts in *P* impedance,
*S* impedance and
density at the reflecting boundary, and are known basis
functions which are analytical in . The three basis functions
are plotted in Figure , with *c _{1}* at the top,

(19) |

where is the shear to compressional velocity ratio *V*_{s}/*V*_{p}.
We invert (18) at every subsurface location by a least-squares method which bootstraps with offset and angle.
This yields an output section each of relative P and S impedance
contrasts.

Figure 23

Figure shows the P impedance inversion difference section for the two surveys before and after one time step of waterflood. The waterflood zone at the top of the reservoir shows the correct increase in P impedance at the well location due to injection of water which is of higher P impedance than the initial light oil in place at lower pore pressure. Figure shows the impedance inversion difference section for the two surveys before and after two time steps of waterflood production. Again, the waterflood zone at the top of the reservoir shows the correct increase in P impedance at the well, and the correct lateral spatial extent. We note that these P impedance sections resemble the migrated sections because, to first order, a (migrated) stack approximates the normal incidence P-P reflection coefficient in the absence of anomalous AVO. However, the impedance inversion results are more accurate in terms of relative impedance contrast estimates than a simple prestack imaging algorithm in general.

Figure shows the S impedance inversion difference section for the two surveys before and after one time step of waterflood. The waterflood zone at the top of the reservoir shows the correct decrease in S impedance at the well location due to injection of water which is of lower S impedance than the initial light oil in place at lower pore pressure. However, the S impedance results are much noisier than the P impedance results, which is expected since they are most sensitive to the relatively fewer far offset trace data. Figure shows the S impedance inversion difference section for the two surveys before and after two time steps of waterflood production. Again, the waterflood zone at the top of the reservoir shows the correct decrease in S impedance at the well, and the correct lateral spatial extent, although in a somewhat more noisy manner.

Finally, the impedance inversions detected the correct opposite polarity changes in P and S impedance between the pre-waterflood survey and the post-waterflood surveys. These two parameters, instead of one single (potentially ambiguous) migrated or stacked reflection amplitude parameter, may be more diagnostic of changes in reservoir petrophysical properties over time-lapse monitor surveys. Furthermore, the magnitude of the impedance changes was recovered reasonably well in data having a significant noise level, since the change in S impedance between surveys is estimated from the data to be on the order of 1.5 times the magnitude of the change in the P impedance after waterflood production. Estimates of absolute or relative changes in the petrophysical properties themselves could be a very valuable tool in monitoring reservoir production processes.

Figure 24

Figure 25

Figure 26

Figure 27

11/16/1997