Elastic impedance

as partial offset volumes to exploit the AVO information in the data. However, there has been significant asymmetry in the way these volumes could be calibrated and inverted. The amplitudes of near-offset, or intercept, stacks relate to changes in acoustic impedance and can be tied to well logs using synthetics based on acoustic impedance (AI) or inverted, to some extent, back to AI using poststack inversion algorithms. However, there have been no simple analogous processes for far-offset stacks. The symmetry can be largely restored using a function I call elastic impedance (EI). This is a generalization of acoustic impedance for variable incidence angle. EI provides a consistent and absolute framework to calibrate and invert nonzero-offset seismic data just as AI does for zero-offset data. EI, an approximation derived from a linearization of the Zoeppritz equations (Appendix, part 1), is accurate enough for widespread application. As might be expected, EI is a function of P-wave velocity, S-wave velocity, density, and incidence angle. To relate EI to seismic, the stacked data must be some form of angle stack rather than a constant range of offsets. There are several ways of constructing suitable data sets by either careful mute design or by linear combination of intercept and gradient functions. (Part 2 of the Appendix reviews these methods.) EI was initially developed by BP in the early 1990s to help exploration and development in the Atlantic Margins province, west of the Shetlands, where Tertiary reservoirs are typified by class II and class III AVO responses. Figure 1 shows a suite of logs from the Foinaven discovery well drilled in 1992. The 30° elastic-impedance log, EI(30), is broadly similar in appearance to the acoustic-impedance log although the absolute numbers are lower; it is a property of EI that the level decreases with increasing angle. At this well, the sands are predominantly class III and so have slightly higher amplitudes at 30° than at normal incidence. This can be more clearly seen in Figure 2 in which the EI log has been scaled to have approximately the same shale baseline as the AI log. When the sands are class II, a more dramatic difference is evident between the AI and EI logs. The seismic data around Foinaven suffer from very strong peg-leg multiples. Even after demultiple, the signal-to-noise ratio of the near-trace data is often poor, especially from the class II events, whereas the far-offset data are generally of good quality. EI allows the well data to be tied directly to the high-angle seismic which can then be calibrated and inverted without reference to the near offsets. Figure 3 shows part of the EI(30) log from another Foinaven well overlain on an inverted 30° angle stack. The data were inverted using a constrained sparse spike algorithm for which the EI log provided the basis for the constraints and was used to QC the result. An EI log provides an absolute frame of reference and so can also calibrate the inverted data to any desired rock property with which it correlates. In the case of Foinaven, a strong correlation was found between EI(30) and hydrocarbon pore volume, and this relationship was used to estimate the in-place volumes for the field from the inverted 30° seismic volume. Figure 4 shows a section from the inverted 30° volume used to design the trajectory of the first high-angle development well. The oil sands correlated closely with the areas of low elastic impedance. The EI volume was used to design the trajectories of all subsequent development wells.