Multi-scale uncertainty quantification in geostatistical seismic inversion

Geostatistical seismic inversion is commonly used to infer the spatial distribution of the subsurface petro-elastic properties by perturbing the model parameter space through iterative stochastic sequential simulations/co-simulations. The spatial uncertainty of the inferred petro-elastic properties is represented with the updated a posteriori variance from an ensemble of the simulated realizations. Within this setting, the large-scale geological (metaparameters) used to generate the petro-elastic realizations, such as the spatial correlation model and the global a priori distribution of the properties of interest, are assumed to be known and stationary for the entire inversion domain. This assumption leads to underestimation of the uncertainty associated with the inverted models. We propose a practical framework to quantify uncertainty of the large-scale geological parameters in seismic inversion. The framework couples geostatistical seismic inversion with a stochastic adaptive sampling and Bayesian inference of the metaparameters to provide a more accurate and realistic prediction of uncertainty not restricted by heavy assumptions on large-scale geological parameters. The proposed framework is illustrated with both synthetic and real case studies. The results show the ability retrieve more reliable acoustic impedance models with a more adequate uncertainty spread when compared with conventional geostatistical seismic inversion techniques. The proposed approach separately account for geological uncertainty at large-scale (metaparameters) and local scale (trace-by-trace inversion).

💡 Research Summary

The paper addresses a fundamental limitation of conventional geostatistical seismic inversion (GSI): the assumption that large‑scale geological “metaparameters” – such as the variogram model describing spatial continuity and the global prior distribution of petrophysical properties – are known and stationary across the entire inversion domain. By fixing these parameters, traditional GSI underestimates the true uncertainty of the inverted elastic models, especially in early‑stage reservoirs where well data are sparse and geological heterogeneity is poorly constrained.

To overcome this, the authors propose a two‑level uncertainty quantification framework that couples standard GSI with adaptive stochastic sampling and Bayesian inference of the metaparameters. The methodology proceeds as follows:

1. Metaparameter Bayesian Updating – The large‑scale parameters (variogram range, anisotropy, facies proportions, and the parameters of the prior probability density functions) are treated as random variables. An adaptive sampling scheme based on the Neighborhood Algorithm (NA) – referred to as NA‑Bayes – explores the high‑dimensional metaparameter space. At each sampled point a likelihood is computed from the misfit between observed seismic data and synthetic data generated from a GSI realization. Gibbs sampling is then used to approximate the normalising integral in Bayes’ theorem, yielding posterior probability density (PPD) functions for each metaparameter. This approach concentrates samples in high‑likelihood regions, dramatically reducing the number of forward simulations compared with brute‑force Markov Chain Monte Carlo.

2. Local‑Scale Stochastic Inversion – With the updated metaparameter posterior, a new set of prior models is generated using sequential Gaussian simulation (or Direct Sequential Simulation) and co‑simulation. The standard GSI loop (generation of impedance realizations, forward seismic modelling, correlation‑based selection of best traces, and subsequent co‑simulation) is then executed. Because the variogram and prior distributions now reflect the posterior uncertainty, each iteration propagates large‑scale geological uncertainty into the ensemble of local‑scale impedance models.

The framework thus separates (i) large‑scale geological uncertainty, captured by the metaparameter posterior, from (ii) small‑scale stochastic uncertainty, inherent to the random paths of the sequential simulation algorithm. The combined posterior ensemble provides a more realistic spread of acoustic impedance (AI) models, avoiding the over‑confident, narrow variance typical of conventional GSI.

The authors validate the approach on both synthetic and real field data. In the synthetic case, the “true” reservoir model is known; conventional GSI, using fixed variogram and prior, fails to recover the true variance and yields biased AI means. The proposed method accurately recovers the true variogram range and facies proportions, and the resulting AI ensemble matches the true model’s mean and standard deviation.

In the field case, only a limited set of wells is available for conditioning. Traditional GSI, which directly adopts the well‑derived prior (biased toward sand‑rich intervals), produces AI models that correlate poorly with blind‑validation wells. After Bayesian updating of the metaparameters, the AI models achieve higher correlation with the blind wells and exhibit a broader, more geologically plausible uncertainty envelope.

Key contributions include:

• Demonstrating that metaparameter uncertainty can be efficiently quantified using adaptive stochastic sampling combined with Gibbs sampling, avoiding the prohibitive cost of full MCMC.
• Providing a practical workflow that integrates seamlessly with existing GSI software, requiring only the addition of a metaparameter sampling loop.
• Showing, through rigorous synthetic and field tests, that the resulting AI ensembles are both more accurate (higher data fit) and more honest (wider, realistic uncertainty).

The paper concludes that incorporating Bayesian metaparameter inference into geostatistical seismic inversion yields a robust, multiscale uncertainty quantification framework. Future work may explore non‑linear forward models, higher‑dimensional facies representations, and parallel implementations to further accelerate the sampling process for large 3‑D surveys.