Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

Grana, Dario; Fjeldstad, T.; Omre, Henning

doi:10.1007/s11004-016-9671-9

Cited by 162 publications

(45 citation statements)

References 41 publications

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“…Following Grana et al . () and Fjeldstad and Omre (), we adopt a linearized and Gaussian likelihood for the forward model for d given

κ

. More specifically, we assume each of

p (m | κ)

and

p (d | m)

to be Gaussian, the conditional mean of d given m to be a linear function of m and the conditional covariance matrix of d given m not to be a function of m .…”

Section: Methodsmentioning

confidence: 99%

“…Different methods have been used for inverting seismic data to elastic properties and LFCs, both deterministic approaches such as optimization‐based methods (Aster, Borchers and Thurber ; Sen and Stoffa ) and probabilistic approaches such as Bayesian inversion (Tarantola ). Using the Bayesian framework, a linearized relation between the data and the elastic properties is commonly used and a Gaussian likelihood function is adopted (see, for example Buland and Omre ; Gunning and Glinsky , and the discussion in Grana, Fjeldstad and Omre ). When inverting to elastic properties, the prior is also often assumed to be Gaussian, in which case the posterior becomes Gaussian with analytically available mean and covariance (see again Buland and Omre ).…”

Section: Introductionmentioning

confidence: 99%

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A Bayesian model for lithology/fluid class prediction using a Markov mesh prior fitted from a training image

Tjelmeland

Luo

Fjeldstad

2019

Geophysical Prospecting

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We consider a Bayesian model for inversion of observed amplitude variation with offset data into lithology/fluid classes, and study in particular how the choice of prior distribution for the lithology/fluid classes influences the inversion results. Two distinct prior distributions are considered, a simple manually specified Markov random field prior with a first‐order neighbourhood and a Markov mesh model with a much larger neighbourhood estimated from a training image. They are chosen to model both horizontal connectivity and vertical thickness distribution of the lithology/fluid classes, and are compared on an offshore clastic oil reservoir in the North Sea. We combine both priors with the same linearized Gaussian likelihood function based on a convolved linearized Zoeppritz relation and estimate properties of the resulting two posterior distributions by simulating from these distributions with the Metropolis–Hastings algorithm. The influence of the prior on the marginal posterior probabilities for the lithology/fluid classes is clearly observable, but modest. The importance of the prior on the connectivity properties in the posterior realizations, however, is much stronger. The larger neighbourhood of the Markov mesh prior enables it to identify and model connectivity and curvature much better than what can be done by the first‐order neighbourhood Markov random field prior. As a result, we conclude that the posterior realizations based on the Markov mesh prior appear with much higher lateral connectivity, which is geologically plausible.

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“…Following Grana et al . () and Fjeldstad and Omre (), we adopt a linearized and Gaussian likelihood for the forward model for d given

κ

. More specifically, we assume each of

p (m | κ)

and

p (d | m)

to be Gaussian, the conditional mean of d given m to be a linear function of m and the conditional covariance matrix of d given m not to be a function of m .…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Bayesian model for lithology/fluid class prediction using a Markov mesh prior fitted from a training image

Tjelmeland

Luo

Fjeldstad

2019

Geophysical Prospecting

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“…As shown in previous works (Grana and Della Rossa ; Grana et al . ; Grana, Fjeldstad, and Omre ), if the rock physics model is linear with associated matrix F , then the posterior conditional distribution m | d is a Gaussian mixture

f_{m | d} (m | d) = \sum_{k = 1}^{N_{F}} λ_{k | d} {scriptN}_{n_{m}} (m, bold-italic μ_{m | d, k}, Σ_{m | d, k}),

with conditional mean

μ_{m | d, k} = μ_{m | k} + Σ_{m | k} F^{T} {((), boldF {boldΣ}_{bold-italicm false| bold-italick} {boldF}^{T} + {boldΣ}_{ε})}^{- 1} (d - F μ_{m | k}),

…”

Section: Methodsmentioning

confidence: 99%

Statistical facies classification from multiple seismic attributes: comparison between Bayesian classification and expectation–maximization method and application in petrophysical inversion

Grana

Lang

2016

Geophysical Prospecting

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We present here a comparison between two statistical methods for facies classifications: Bayesian classification and expectation–maximization method. The classification can be performed using multiple seismic attributes and can be extended from well logs to three‐dimensional volumes. In this work, we propose, for both methods, a sensitivity study to investigate the impact of the choice of seismic attributes used to condition the classification. In the second part, we integrate the facies classification in a Bayesian inversion setting for the estimation of continuous rock properties, such as porosity and lithological fractions, from the same set of seismic attributes. The advantage of the expectation–maximization method is that this algorithm does not require a training dataset, which is instead required in a traditional Bayesian classifier and still provides similar results. We show the application, comparison, and analysis of these methods in a real case study in the North Sea, where eight sedimentological facies have been defined. The facies classification is computed at the well location and compared with the sedimentological profile and then extended to the 3D reservoir model using up to 14 seismic attributes.

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“…In a real-world setting, a Gaussian model may also not be the obvious choice to describe the prior model, but the Gaussian prior model assumption is needed to make use of equations 11 and 12. Recently, Grana et al (2017) propose a method that allows using Gaussian prior models with a non-Gaussian 1D marginal distribution. Sabeti et al (2017) perform direct sequential simulation to allow using non-Gaussian 1D distributions.…”

Section: Bayesian Linearized Avo Inversion -Numerical Examplementioning

confidence: 99%

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

Madsen

Hansen

2018

GEOPHYSICS

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A linearized form of Zoeppritz equations combined with the convolution model is widely used in inversion of amplitude variation with offset (AVO) seismic data. This is shown to introduce a "modeling error," compared with using the full Zoeppritz equations, whose magnitude depends on the degree of subsurface heterogeneity. Then, we evaluate a methodology for quantifying this modeling error through a probability distribution. First, a sample of the unknown probability density describing the modeling error is generated. Then, we determine how this sample can be described by a correlated Gaussian probability distribution. Finally, we develop how such modeling errors affect the linearized AVO inversion results. If not accounted for (which is most often the case), the modeling errors can introduce significant artifacts in the inversion results, if the signal-to-noise ratio is less than 2, as is the case for most AVO data obtained today. However, if accounted for, such artifacts can be avoided. The methodology can easily be adapted and applied to most linear AVO inversion methods, by allowing the use of the inferred modeling error as a correlated Gaussian noise model.

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Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

Cited by 162 publications

References 41 publications

A Bayesian model for lithology/fluid class prediction using a Markov mesh prior fitted from a training image

A Bayesian model for lithology/fluid class prediction using a Markov mesh prior fitted from a training image

Statistical facies classification from multiple seismic attributes: comparison between Bayesian classification and expectation–maximization method and application in petrophysical inversion

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

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