Uncertainty in inverted elastic properties resulting from uncertainty in the low-frequency model

Abstract

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.

Estimation and accounting for the modeling error in probabilistic linearized amplitude variation with offset inversion

“…A variety of sources for modeling errors exist in seismic data, such as using a 1D convolutional model to reflect a 3D physical system, the use of the acoustic-wave equation as opposed to the anisotropic visco-elastic wave equation, imperfections in data processing, general anisotropy considerations, the effects of processing the raw data, the coupling of data to physics within the forward model, uncertain wavelet estimates, and uncertainty on the low-frequency model. (see also Ball et al, 2015;Li et al, 2015;Thore, 2015). For example, we expect higher magnitude modeling errors related to using the Zoeppritz equations as opposed to using the full wave equation.…”

Stuck between a rock and a reflection: A tutorial on low-frequency models for seismic inversion

“…The concern about absolute impedances is due to the difficulty of providing reasonable lowfrequency models and the potential that they can negatively bias the absolute estimates and produce incorrect interpretations. Ball et al (2015) show that with certain assumptions, no low-frequency model is required for inversion to relative impedances when they are appropriately defined (Ball et al, 2014). As such, there is an argument for excluding low-frequency models and relying only on the relative properties for interpretation.…”

Seis2Rock: A Data‐Driven Approach to Direct Petrophysical Inversion of Pre‐Stack Seismic Data

“…Bayesian methods, on the other hand, formulate an inverse problem in a probabilistic manner by means of Bayes theorem, and take advantage of the prior distribution to include user‐defined knowledge of the parameters of interest. Under the assumption of Gaussian priors and noise and linear models (such as post‐stack and pre‐stack linearized modeling operators), the posterior solution is also Gaussian and both the mean and covariance matrix can be analytically derived (Ball et al., 2015; Buland & Omre, 2003; Kemper & Gunning, 2014). When the modeling operator is nonlinear, Markov chain Monte Carlo methods (Hansen et al., 2012; Sen & Stoffa, 2013) are instead preferred to approximate the posterior distribution through repeated sampling coupled with by acceptance‐rejection rules.…”