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Gaussian mixture model

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Book Chapter

Author(s)
Gerard Schuster
Series: Geophysical Developments Series
Published: 18 December 2024
DOI: 10.1190/1.9781560804048.ch21
EISBN: 9781560804048
... for equation 21.35 . Figure adapted from Giraud et al. (2017) . This chapter presents the theories of discriminant analysis and the Gaussian mixture model (GMM). 21.3.2 Gaussian discriminant analysis 21.3.1 Linear discriminant analysis 21.1 Introduction 21.3 Linear and Gaussian...
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Journal Article
Journal: Interpretation
Published: 20 January 2022
Interpretation (2022) 10 (1): T181–T193.
... convolutional embedded clustering with adaptive Gaussian mixture model (AGMM-MDCEC) for seismic waveform clustering. Trainable feature extraction and clustering layers in AGMM-MDCEC are implemented using neural networks. We fuse the two independent processes of feature extraction and clustering...
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Journal Article
Journal: Geophysics
Published: 30 December 2021
Geophysics (2022) 87 (2): M59–M72.
... statistical distribution of porosity, the nonlinear relationship between porosity and seismic elastic parameters, and the uncertainty of porosity estimation, we have used a Gaussian mixture model deep neural network (GMM-DNN) to invert porosity from seismic elastic parameters. We use a Gaussian mixture model...
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Journal Article
Journal: Geophysics
Published: 30 August 2021
Geophysics (2021) 86 (5): R685–R699.
... characteristics on inversion results based on prior information to obtain a stable and unique solution. However, it is difficult to find an appropriate regularization to describe the actual subsurface geology. We have developed a new acoustic impedance inversion method via a patch-based Gaussian mixture model...
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Journal Article
Journal: Geophysics
Published: 11 February 2021
Geophysics (2021) 86 (2): R221–R236.
... depth trends. Then, the marginal prior models for the petrophysical properties and elastic attributes are multivariate Gaussian mixture models. The likelihood model is assumed to be Gauss-linear to allow for analytic computation. A recursive algorithm that translates the Gibbs formulation of the Markov...
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Journal Article
Journal: Interpretation
Published: 12 October 2020
Interpretation (2020) 8 (4): SS47–SS62.
.... Quantitative petrophysical signatures for each rock unit are obtained from sample measurements, such as the increasing density of the PK/VK unit with depth and the remanent magnetization of the HK unit, and are represented as a Gaussian mixture model (GMM). This GMM guides the PGI toward generating a 3D quasi...
Journal Article
Journal: Interpretation
Published: 28 May 2019
Interpretation (2019) 7 (3): SE93–SE111.
... probabilistic formulation of Gaussian mixture models (GMMs) allows for the number and shape of clusters to be determined in a more objective manner using a Bayesian framework that considers a model’s likelihood and complexity. Furthermore, the development of alternative expectation-maximization (EM) algorithms...
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(a) Example event associated from the Gaussian Mixture Model Associator (GaMMA), with the predicted P and S phase marked by blue and red bars, respectively. (b,d) The distribution of earthquakes (circles) associated from the (b) Rapid Earthquake Association and Location (REAL) and (d) GaMMA. (c) Event association results from the REAL and GaMMA. The earthquakes were colored by focal depth. The color version of this figure is available only in the electronic edition.
Published: 21 August 2024
Figure 3. (a) Example event associated from the Gaussian Mixture Model Associator (GaMMA), with the predicted P and S phase marked by blue and red bars, respectively. (b,d) The distribution of earthquakes (circles) associated from the (b) Rapid Earthquake Association and Location (REAL
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A six‐cluster Gaussian mixture model (GMM) clustering is applied on the continuous velocity map A in (a). Each pixel is clustered to be a six‐cluster discrete label map X in (b). Pixels with similar velocity information have been assigned the same label. The color version of this figure is available only in the electronic edition.
Published: 25 April 2024
Figure 2. A six‐cluster Gaussian mixture model (GMM) clustering is applied on the continuous velocity map A in (a). Each pixel is clustered to be a six‐cluster discrete label map X in (b). Pixels with similar velocity information have been assigned the same label. The color version
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Joint distribution of the Gaussian mixture model for the velocity and porosity from (a) the Gassmann model prediction, (b) the BR model prediction, and (c) the BR model prediction aided by the Monte Carlo simulation.
Published: 08 May 2023
Figure 11. Joint distribution of the Gaussian mixture model for the velocity and porosity from (a) the Gassmann model prediction, (b) the BR model prediction, and (c) the BR model prediction aided by the Monte Carlo simulation.
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(a) The prior distribution of the Gaussian mixture model estimated from the well-log data with a good initialization, and (b) histogram and (c) bivariate distribution estimated from the Monte Carlo simulated data (2000 samples) for the model parameters.
Published: 08 May 2023
Figure 10. (a) The prior distribution of the Gaussian mixture model estimated from the well-log data with a good initialization, and (b) histogram and (c) bivariate distribution estimated from the Monte Carlo simulated data (2000 samples) for the model parameters.
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The expectation of the Gaussian mixture model estimated by the expectation-maximization algorithm for (a) porosity, (b) water saturation, (c) P-wave velocity, and (d) density, from different data types. The number labeled in the lateral axis denotes the different types of the joint (petrophysical and elastic) data samples for the estimation, which correspond to the data from (1) well log, (2) Gassmann model, (3) BR model predictions, and BR model predictions aided by the Monte Carlo simulation with (4) 500 samples and (5) 2000 samples. Each estimation runs for 20 independent computations. The red and blue color circles (qualified by the boxplots) denote the computation results for the reservoir and nonreservoir sections, respectively. The dashed and dotted lines denote the true value obtained from the interpreted log profiles for the reservoir and nonreservoir sections, respectively.
Published: 08 May 2023
Figure 9. The expectation of the Gaussian mixture model estimated by the expectation-maximization algorithm for (a) porosity, (b) water saturation, (c) P-wave velocity, and (d) density, from different data types. The number labeled in the lateral axis denotes the different types of the joint
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Gaussian mixture model of bivariate data with two clusters.
Published: 01 February 2022
Figure 1. Gaussian mixture model of bivariate data with two clusters.
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Celerity prior Gaussian mixture model distributions varying with location, time, and azimuth with colors relating to the number of ray arrivals in the corresponding location–month–azimuth–range bin. All plots correspond to arrivals within 550–600 km from the source. (a) The prior PDFs are added to their latitude value for all node locations between longitudes of 110° W and 100° W for the month of January and at an azimuth 90° from the node. (b) Prior PDFs from the location in New Mexico from Figure 5 (node 28) and keeping the azimuth at 90° varies the month analyzed. PDFs in the plot are scaled by 10% for better visibility and added to the number of the corresponding month (e.g., a PDF value of 0 in January would be at 1). (c) We again use the New Mexico location and January and vary the azimuths from the source.
Published: 01 October 2014
Figure 6. Celerity prior Gaussian mixture model distributions varying with location, time, and azimuth with colors relating to the number of ray arrivals in the corresponding location–month–azimuth–range bin. All plots correspond to arrivals within 550–600 km from the source. (a) The prior PDF s
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Gaussian mixture model in the porosity-velocity domain. In this example, the mixture is made of two components that correspond to hydrocarbon-bearing sand (the right peak) and shale (the left peak). As in the single-Gaussian case, the Gaussian mixture conditional distribution can be analytically derived.
Published: 01 January 2011
Figure 3. Gaussian mixture model in the porosity-velocity domain. In this example, the mixture is made of two components that correspond to hydrocarbon-bearing sand (the right peak) and shale (the left peak). As in the single-Gaussian case, the Gaussian mixture conditional distribution can
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BIC values for the Gaussian mixture models at each stripe (red rectangles indicate minima).
Published: 01 February 2022
Figure 8. BIC values for the Gaussian mixture models at each stripe (red rectangles indicate minima).
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Published: 01 February 2022
Table 2. Chosen Gaussian mixture models at each stripe S a ( T 1 ) (g) n k BIC Mixture proportions 0.2 1 −1381.7 [1.00] 0.4 1 −1157.0 [1.00] 0.6 3 −655.4 [0.33, 0.60, 0.07] 0.8 4 −499.5 [0.47, 0.08, 0.29
Journal Article
Journal: Geophysics
Published: 16 April 2019
Geophysics (2019) 84 (3): R463–R476.
... posterior distribution of the properties of interest. The prior distribution is a Gaussian mixture model, and each component is associated to a potential configuration of the facies sequence along the seismic trace. The low frequency is incorporated by using facies-dependent depositional trend models...
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Estimated Barnett Shale formation properties compared to core data and commercial spectroscopy measurements. Track description from left to right track 1: logging depth; track 2: gamma ray; tracks 3 and 4: apparent resistivity logs and reconstructed resistivity logs by a 2D likelihood function (red curve) and multiple Gaussian mixture model (blue); track 5: simulated density and neutron (limestone) porosity; track 6: simulated and log measurement PEF; tracks 7 and 8: estimated water saturation and uncertainty by a 2D likelihood function (red) and multiple Gaussian mixture model (blue); tracks 9 and 10: estimated porosity and uncertainty a by 2D likelihood function (red curve) and multiple Gaussian mixture model (blue); track 11: mineralogy-weighted concentration from spectroscopy measurements; and track 12: weight concentrations of minerals obtained with stochastic inversion by a 2D likelihood function; the green dots in tracks 7, 8, 9, and 10 identify core data measurements.
Published: 21 November 2014
function (red curve) and multiple Gaussian mixture model (blue); track 5: simulated density and neutron (limestone) porosity; track 6: simulated and log measurement PEF; tracks 7 and 8: estimated water saturation and uncertainty by a 2D likelihood function (red) and multiple Gaussian mixture model (blue
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Fig. 6.
Published: 06 September 2018
Fig. 6. A comparison of the stratigraphic Intervals A, B, and C, which are defined by the positive organic carbon isotope excursion, to the result of a Gaussian Mixture Model that sorted the same data set in δ 13 C org – δ 34 S pyr space. ( a ) A Gaussian Mixture Model was applied to the 69