A three-term (3T) amplitude-variation-with-offset projection is a weighted sum of three elastic reflectivities. Parameterization of the weighting coefficients requires two angle parameters, which we denote by the pair [Formula: see text]. Visualization of this pair is accomplished using a globe-like cartographic representation, in which longitude is [Formula: see text], and latitude is [Formula: see text]. Although the formal extension of existing two-term (2T) projection methods to 3T methods is trivial, practical implementation requires a more comprehensive inversion framework than is required in 2T projections. We distinguish between projections of true elastic reflectivities computed from well logs and reflectivities estimated from seismic data. When elastic reflectivities are computed from well logs, their projection relationships are straightforward, and they are given in a form that depends only on elastic properties. In contrast, projection relationships between reflectivities estimated from seismic may also depend on the maximum angle of incidence and the specific reflectivity inversion method used. Such complications related to projections of seismic-estimated elastic reflectivities are systematized in a 3T projection framework by choosing an unbiased reflectivity triplet as the projection basis. Other biased inversion estimates are then given exactly as 3T projections of the unbiased basis. The 3T projections of elastic reflectivities are connected to Bayesian inversion of other subsurface properties through the statistical notion of Bayesian sufficiency. The triplet of basis reflectivities is computed so that it is Bayes sufficient for all rock properties in the hierarchical seismic rock-physics model; that is, the projection basis contains all information about rock properties that is contained in the original seismic.

A Bayesian linearized inversion (BLI) framework is used to analyze how uncertainty in the low-frequency model (LFM) affects the solution in prestack seismic inversion. Two related effects are considered: sensitivity addresses how a change in the LFM changes the inversion solution, whereas uncertainty addresses how uncertainty in the LFM is propagated into the inversion results. The posterior covariance matrix in the BLI equations does not depend on the LFM, and it can be concluded mistakenly that uncertainty in the inversion result is independent of uncertainty in the LFM. Instead, a broader characterization of uncertainty can be made by using a hierarchical BLI framework that includes uncertainties in the LFM. The standard BLI equations are adequate to describe uncertainty in relative rock-property inversion which can be obtained by subtracting the logarithm of the LFM from the logarithm of the inversion solution. Uncertainty in relative rock-property inversion is independent of both the particular LFM used in the solution and the uncertainty associated with the LFM. On the other hand, uncertainty in absolute rock-property inversion is independent of the particular LFM used but does depend on the uncertainty of that LFM. The uncertainty of absolute rock-property inversion is always greater than the uncertainty of the associated relative rock-property inversion.

Quantitative relationships between true elastic properties are not always applicable to inverted elastic properties. We use the important example of two-term AVO reflectivity inversion to demonstrate this. AVO literature includes many examples of two-term AVO equations, and we provide a general relationship to convert between inverted reflectivities obtained when using these various equations. Conversion between inverted reflectivities should be based on this equation, rather than standard elastic-property relationships. This is of immediate importance when applying the method of chi-angle projections to estimate elastic reflectivities.

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