Three-term amplitude-variation-with-offset projections

Abstract

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.

Sandstone detection using three‐term amplitude variation with offset projection: A case study

“…The scalar noise variance shows a sinusoidal path with low noise levels, as demonstrated by Ball et al. (2018). Noise in A , B and C is therefore suppressed by choosing the projection angle along the sinusoidal path.…”

“…The incident angle range was 12˚, 22˚, 29˚, 36˚, 42˚and 47˚, based on synthetic angle stacks used in the next section. The scalar noise variance shows a sinusoidal path with low noise levels, as demonstrated by Ball et al (2018). Noise in A, B and C is therefore suppressed by choosing the projection angle along the sinusoidal path.…”

“…The three‐term (3T) amplitude variation with offset (AVO) projection proposed by Ball et al. (2018) is an extension of the traditional two‐term (2T) AVO projection. A , B and C are blended using two projection angles controlling their relative contributions: $$$$\begin{eqnarray}{R_{\chi ,\psi }} &=& {\rm{\;}}A\cos \chi \cos \psi + B\sin \chi \cos \psi + C\sin \psi {\rm{\;}}\nonumber\\ &=& \left( {A\cos \chi + B\sin \chi } \right){\rm{\;}}\cos \psi + C\sin \psi {\rm{\;,}}\end{eqnarray}$$$$where the parameter $$ - 180^\circ \le \chi \le 180^\circ $$ controls the relative contribution of A and B , and the parameter $$ - 90^\circ \le \psi \le 90^\circ $$ controls the relative contribution of C .…”

“…Note that more sophisticated frequency domain approach (Ball et al., 2018) can be adopted for step 3, which allows frequency dependent noise and angle‐dependent wavelets to be accounted for; this is outside the scope of this paper. Figure 3 shows a comparison between noise‐free synthetics and synthetics containing correlated noise.…”

“…The three-term (3T) amplitude variation with offset (AVO) projection proposed by Ball et al (2018) is an extension of the traditional two-term (2T) AVO projection. A, B and C are blended using two projection angles controlling their relative contributions:…”