Unsupervised seismic facies from mixture models to highlight channel features Robert Hardisty*1 and Bradley C. Wallet1 1)University of Oklahoma, OK USA

Summary

Unsupervised seismic facies are a convenient and efficient

tool for interpretation. Expanding upon Zhao et al.’s (2016)

study, Gaussian mixture models are used to show how

features can automatically be generated using machine

learning. The conventional expectation-maximization

algorithm is compared to the neighborhood expectation-

maximization algorithm to highlight the effects of spatial

relations in the data in addition to the measurements of

seismic attributes. The survey being used is a 3D seismic

survey from the Canterbury basin, New Zealand called

Waka-3D

Introduction

Visual examination of seismic facies on large 3D seismic

data sets where there is little a priori geologic information

can be tedious and inaccurate. The process can be more

automated and improved using machine learning. By

teaching a computer how to recognize patterns, features can

automatically be picked. This has the obvious benefit of

quicker interpretations, but moreover it can highlight

features that might otherwise go unnoticed. The Gaussian

mixture model (GMM) provides a flexible framework by

which to accomplish this.

Geologic setting

The seismic survey is located on the Canterbury Basin,

offshore New Zealand (Figure 1). The area lies in the

transition zone of the continental rise and continental slope.

The data set contains many Cretaceous and Tertiary age

paleocanyons and turbidite deposits. Sediments were

deposited in a single transgressive-regressive cycle driven

by tectonics (Zhao et al. 2016). A previously identified

channel feature is analyzed using a Gaussian mixture model

technique.

Theory

Gaussian mixture models for seismic attributes

Gaussian mixture models (GMM) are a well-known semi-

parametric density estimation technique using a weighted

sum of normal, or Gaussian, distributions (Figure 2). An

inherent assumption when using this technique is that the

data comes from different Gaussian distributions.

A multivariate Gaussian distribution can be defined as

φ(𝐱|𝛍, 𝐂) =1

(2π)d2|𝐂|

12

e−1

2(𝐱−𝛍)′𝐂−1(𝐱−𝛍)

where µ is the mean, C is the covariance matrix, and d is the

number of dimensions of x and µ. For seismic attributes, x is

a voxel with dimensions equal to the number of attributes. A

GMM can be expressed as

p(𝐱|ψ) = ∑ πjφ(𝐱| 𝛍𝐣, 𝐂𝐣)

k

j=1

where k is the number of different Gaussian distributions or

components, and πj is the weight of the jth component such

that πj > 0 and ∑ πj = 1kj=1 .

Figure 2: Example of a Gaussian mixture model with three

mixture components. The overall density is estimated as the sum

of the components. (Modified from Wallet et al., 2014)

Figure 1: Aerial view of study area (Modified from Zhao et al.,

2016)

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Unsupervised seismic facies from mixture models

The problem is to estimate (or learn) the parameters of the

GMM, {πj, 𝛍j, 𝐂j} for j= (1… k). Common practice is to

learn the parameters of a Gaussian mixture through the

expectation-maximization (EM) algorithm developed by

Dempster et al. (1977). Dynamic component allocation

(DCA) as proposed by Vlassis and Likas (2002) is used to

avoid user-defined initialization and to make the process

more unsupervised. Dynamic component allocation (DCA)

starts with a single component, and then alternates between

optimization using the EM algorithm and allocation of a new

component for the GMM. The first component is initialized

using the population mean and covariance. Convergence of

DCA occurs when the maximum number of components is

reached.

Neighborhood expectation-maximization (NEM) algorithm

Learning of a GMM using the EM algorithm is a purely

statistical construct and doesn’t consider spatial correlations.

In general, facies are expected to be at least laterally

continuous to some extent. To account for spatial

correlations of the latent space the Neighborhood

expectation-maximization (NEM) algorithm is implemented

and compared to the results of the conventional EM. The

conventional EM algorithm can be viewed as a variant of

coordinate descent on a certain objective function,

D(𝐖, ψ) = ∑ ∑ Wji[log {

N

i=1

k

j=1

Wji} − log {πjφ(𝐱| 𝛍𝐣, 𝐂𝐣)}]

where Wji are the elements of the responsibility matrix, W

(Hathaway, 1986). Ambroise et al. (1996) introduced a

regularization term to take into account the spatial

information of the data,

G(𝐖) =1

2∑ ∑ ∑ Wij ∙ Wpj ∙ Vip

N

p=1

N

i=1

k

j=1

where Vip are the elements of a “neighborhood matrix”, V.

The new objective function then becomes

U(𝐖, ψ) = D(𝐖, ψ) + β ∙ G(𝐖)

where β ≥ 0 and determines the weight of the spatial term,

G(W). The “neighborhood matrix”, V, for this application

has been chosen to be

Vip = {1 if xi and xp neighbors

0 else

, and xi and xp are neighbors if they both lie within a user-

defined window. The benefit of the NEM algorithm is that

the responsibilities of neighboring voxels are considered

when deciding which mixture component a voxel belongs to.

Methods

Latent space modeling

Like all statistical classifiers, GMM’s suffer from the curse

of dimensionality. Latent space modeling is a powerful

technique to project high dimensional data into a lower

dimensional space. In this application, a two-dimensional

latent space generated from Zhao et al. (2016) is considered.

The latent space was generated using a distance-preserving

SOM (DPSOM) technique with the attribute inputs being

peak spectral frequency, peak spectral magnitude, coherent

energy, and curvedness. The DPSOM algorithm mapped the

4D attribute input to a 2D SOM latent space resulting in 2

seismic attribute volumes, SOM latent axis 1 and SOM latent

axis 2 (Figure 3). Using a GMM as a classifier on these two

axes will produce a single partition volume and a number, k,

of mixture decomposition volumes for unsupervised seismic

facies analysis.

Gaussian mixture models as a classifier

Each component of a GMM attempts to model an underlying

process that generated the data. A GMM is a model based

clustering technique in that that each underlying process is

assumed to be Gaussian in shape. The objective of a

classifier is to find which component is responsible for

producing each voxel.

Figure 3: Horizon slice of (a) seismic amplitude, (b) SOM latent

axis 1, (c) SOM latent axis 2. Purple lines and arrows show a

feature previously interpreted as a muddy channel that cuts throug a

sandy channel (orange arrow).

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Unsupervised seismic facies from mixture models

Usually finding the component responsible for each voxel is

simply done by using the responsibility matrix, W, and

assigning each voxel to the component with the highest

responsibility. However, due to the large size of seismic data

a training set must be used due to memory and time

constraints. The training set is used to learn the parameters

of the GMM. The training set is constructed by uniformly

sampling every 125th voxel (one voxel for every 5th inline,

crossline, and time sample). Once the parameters of a GMM

are learned using a training data set, the responsibility of

each voxel can be calculated individually. For the

conventional EM algorithm, this is simply done by

implementing another E-step that includes the whole

volume. The NEM algorithm is done in a similar manner,

but uses the training data set to approximate the total

population when calculating the penalty term, G(W).

Application

The area of interest has been interpreted as a possible

channel feature by Zhao et al. (2016). The area of interest

consists of 456 crosslines x 576 inlines x 23 time samples.

The SOM latent axis 1 and SOM latent axis 2 are used as

inputs for two different GMM’s; one GMM using the

conventional EM algorithm and another using the NEM

algorithm. The number of components to be found is set to

be four because four prototype vectors were used in the

construction of the latent space axes. For the conventional

EM case, DCA is used to find a GMM with four

components. For the NEM case, the parameters from the EM

case are used for initialization and the spatial weight, β, is

set to 0.1.

Two cross sections are made, A-A’ and B-B’, to show the

channel feature in three dimensions. Previously this was

interpreted by Zhao et al. (2016) as a possible muddy

channel cutting through a sandy channel (Figure 3). In both

the EM and NEM case the sandy channel is dominated by

the 4th component of the mixture model and is colored tan.

Likewise, the muddy channel is dominated by the 2nd and 3rd

components of the mixture model, and are colored red and

green respectively. The NEM algorithm successfully

segments the image into more spatially continuous facies.

However, there are hard right angles similar to how

acquisition footprint looks due to the uniform sampling of

the training set of data.

Cross section A-A’ shows the high amplitude channel being

delineated by the tan colored facies and being surrounded by

the blue colored facies. The NEM algorithm improves the

segmentation by removing the anomalous red facies above

the high amplitude feature. In both EM and NEM the red and

green facies are not within the high amplitude feature.

Figure 4: Horizon slice (a) seismic amplitude for reference (b)

results from EM algorithm, (c) results frm NEM algorithm. Blocky

right angles can be noticed in (c) due to how the training data was

sampled. A-A’ cuts across the flow direction of both channels. B-

B’ goes along the flow direction of the tan-colored channel and

cuts across the blue-green colored channel in the western end of

the profile.

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Unsupervised seismic facies from mixture models

Cross section B-B’ goes more or less along the flow

direction of the tan colored channel (Figure 6). The

combination of red and green colored facies segment the

channel well. The NEM algorithm removes many of the red

colored facies in in the high amplitude areas and replaces

them with tan colored facies.

Conclusions

Gaussian mixture models are a convenient way to

characterize seismic attributes and generate unsupervised

seismic facies to let the data speak for itself. Results may not

correlate to all the geology, but can highlight features that

may be of geological interest.

The NEM algorithm can act like a smoothing operator in the

spatial domain to ensure that facies have some spatial

continuity. Different ways of defining the neighborhood

matrix, along with different values of the spatial weight, β,

should be investigated further. The unsupervised seismic

facies in this paper are using GMM’s as a partitioning

method like k-means; future work using GMM’s as a fuzzy

clustering method may more reveal more complexity in the

data.

Acknowledgements

We would like to thank the New Zealand Petroleum and

Minerals for making the Waka-3D seismic data public. We

would also like to thank the sponsors of the Attribute-

Assisted Seismic Processing and Interpretation (AASPI)

Consortium at the University of Oklahoma. Horizon slices

were generated using Petrel licenses courtesy of

Schlumberger. A special thanks to Tao Zhao for use of his

latent space axes. And thanks to all our colleagues for their

valuable insights.

Figure 5: Profile A-A’ of (a) seismic amplitude, (b) EM algorithm,

and (c) NEM algorithm. Cuts perpindicular to the flow direction of

tan colored channel. The black arrow indicates a high amplitude

feature

Figure 6: Profile B-B’ of (a) seismic amplitude, (b) EM algorithm,

and (c) NEM algorithm. The channel outlined in purple is composed

of all the facies. In the NEM algorithm, C, constrains the red facies

to mostly the channel fill unlike the EM algorithm.

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