Model misspecification and bias in the least-squares algorithm: Implications for linearized isotropic AVO

Abstract

When inversions use incorrectly specified models, the estimated least-squares model parameters are biased. Their expected values are not the true underlying quantitative parameters being estimated. This means the least-squares model parameters cannot be compared to the equivalent values from forward modeling. In addition, the bias propagates into other quantities, such as elastic reflectivities in amplitude variation with offset (AVO) analysis. I give an outline of the framework to analyze bias, provided by the theory of omitted variable bias (OVB). I use OVB to calculate exactly the bias due to model misspecification in linearized isotropic two-term AVO. The resulting equations can be used to forward model unbiased AVO quantities, using the least-squares fit results, the weights given by OVB analysis, and the omitted variables. I show how uncertainty due to bias propagates into derived quantities, such as the χ-angle and elastic reflectivity expressions. The result can be used to build tables of unique relative rock property relationships for any AVO model, which replace the unbiased, forward-model results.