Abstract
Characterizing buried sedimentary structures through the use of seismic data is part of many geoscientific projects. The evolution of seismic acquisition and processing capabilities has made it possible to acquire evergrowing amounts of data, increasing the image resolution so that sedimentary objects (geobodies) can be imaged with greater precision within sedimentary layers. However, exploring and interpreting them in large datasets can be tedious work. Recent practice has shown the potential of automated methods to assist interpreters in this task. In this paper, a new semisupervised methodology is presented for identifying multifacies geobodies in threedimensional seismic data, while preserving their internal facies variability and keeping track of the input uncertainty. The approach couples a nonlinear datadriven method with a novel supervised learning method. It requires a prior delineation of the geobodies on a few seismic images, along with a priori confidence in that delineation. The methodology relies on a learning of an appropriate data representation, and propagates the prior confidence to posterior probabilities attached to the final delineation. The proposed methodology was applied to threedimensional real data, showing consistently effective retrieval of the targeted multifacies geobodies masstransport deposits in the present case.
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The authors are grateful to the CGG Houston office for the provision of and permission to publish data, and to Karine Labat for proofreading the article.
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Appendix A: Definitions and Proofs
Appendix A: Definitions and Proofs
A.1 Definition of the GLCM Features and the EnvelopeBased Feature Used for Initial Data Representation (Sect. 2.1.1)
The twodimensional GLCM contrast, correlation, energy and homogeneity features are defined below
where \(M_{ij}\) is the (i, j) component of the GLCM matrix M computed for a specific vector; sums are done over all pairs of pixels in the image; and n is the number of gray levels used (\(n^2\) is the number of elements in M).
The envelopebased feature \(f_\mathrm{{env}}\) is computed from the envelopeseismic image \(S_\mathrm{{env}}\) where every trace has been replaced by its envelope. Denoting \(G^{\sigma = 3}_{2D}\) as the twodimensional Gaussian filtering with standard deviation \(\sigma = 3\) pixels, and \(q_{95}\) as the 0.95 quantile of a set of values, the computation of the envelopebased feature can be resumed to
All features are in a range close to [0 1].
A.2 Proof for Posterior Probability Assignment
This section gives a proof for the method presented in Sect. 2.2. Let us recall that here the goal is to minimize the error \(E_i\) associated with each cluster \(C_i\), thus obtaining \(y_i^\mathrm{{opt}}\), the optimized posterior probability associated with \(C_i\). With \(N_i\) as the number of points in \(C_i\) and \(p_{k,i} = p(s_{k}C_{i})\) as the proportion of points in \(C_{i}\) whose prior probability value is \(s_{k}\), one can write
Let us assume i is fixed, write E in the following for \(\frac{E_{i}}{N_i}\), and omit the i index in the rest of this section. One may now study the variations of E with respect to y in order to find \(y^\mathrm{{opt}}\) which minimizes E.
A.2.1 Variations of E Outside [0, 1]
If \(y < s_1 = 0\), then: \( E(y) = \sum _{k \in 1:n}(s_{k}  y)\ p_{k}\), and every term of this sum decreases when y increases up to \(s_1\); in other words: E is a strictly decreasing function of y for \(y < s_1\).
If \(y > s_n = 1\), then: \( E(y) = \alpha \sum _{k \in 1:n}(y  s_{k})\ p_{k}\), and every term of this sum decreases when y decreases down to \(s_n\); in other words: E is a strictly increasing function of y for \(y > s_n\).
Consequently (as expected), \(y^\mathrm{{opt}} \in [s_{1}, s_{n}] = [0, 1]\).
A.2.2 Variations of E Inside [0, 1]
Let \(k \in 1:n1\). Let \(y, d > 0\), such that\(s_{k} \le y < y + d \le s_{k+1}\). Then one can note that
As \(\sum _{j \in 1:n}p_{j} = 1\) (sum of proportions), one can then write (\(d > 0\)) that
Thus, the variation of E does not depend on y on \([s_k, s_{k+1}]\): E is monotonous on this interval. More precisely
Different Cases According to the Distribution of p
If \(p_{1} > \frac{1}{\alpha + 1}\), then E is strictly increasing on \([s_{1}, s_{2}]\), so also on \([s_{1}, s_{n}]\). E being strictly decreasing for \(y < s_1\) (see above), E reaches its minimum on \(y^\mathrm{{opt}} = s_1\) (i.e. 0).
If \(p_{1} \le \frac{1}{\alpha + 1}\), then one can define q as follows
There are now two cases:

Case 1: \(\sum _{j \in 1:q}p_{j} = \frac{1}{\alpha + 1}\) Here, the subcase \(q = n\) corresponds to the case when \(\alpha = 0\). The optimal value will then be \(y^\mathrm{{opt}} = 1\) ; note that this value will be for any cluster considered: taking \(\alpha = 0\) is not interesting. In the subcase \(q < n\): E is constant on \([s_{q}, s_{q+1}]\), and \(y^\mathrm{{opt}}\) can take any value in \([s_{q}, s_{q+1}]\). Here it is taken as \(y^\mathrm{{opt}} = s_{q}\) (for consistency with the subcase \(q = n\)).

Case 2: \(\sum _{j \in 1:q}p_{j} < \frac{1}{\alpha + 1}\) Here, the subcase \(q = n\) corresponds to a function E strictly decreasing on \([s_{1}, s_{n}]\) ; E being strictly increasing for \(y > s_{n}\) (see above), E is minimum for \(y^\mathrm{{opt}} = s_{n} = 1\). In the subcase \(q < n\): E is strictly decreasing on \([s_{q}, s_{q+1}]\) and strictly increasing on \([s_{q+1}, s_{q+2}]\) (from q’s definition), so E is minimum for \(y^\mathrm{{opt}} = s_{q+1}\).
A.3 Results of the Dimension Reduction
Figure 10 shows the results of the FS dimension reduction and its relationship with the principal components space, as explained in Sect. 3.2.
The alignment of features of a same cluster along a line crossing the origin of the graph confirms that the clusters were formed according to the positive and negative correlations of features. It also ensures that at least one feature was selected among each group of correlated features.
For scale 1, there were four possible orientations; one feature per orientation was selected for all datasets except the faroffset dataset. For scale 2, there were eight possible orientations. For the fullstack and nearoffset datasets, one feature per typical orientation was selected, except for orientations [0, 2] and [2, 0]; similarly for the midoffset dataset, orientations [0, 2] and [2, 1] were not selected. For the faroffset dataset, two features of the same orientation were selected, so that three orientations were not represented in the selected feature set: [2, 0], [2, 2] and \([1,2]\).
A.4 Detailed Results on EM
Results presented in Table 3 are detailed support for Fig. 6c.
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Le Bouteiller, P., Charléty, J. SemiSupervised MultiFacies Object Retrieval in Seismic Data. Math Geosci 52, 817–846 (2020). https://doi.org/10.1007/s11004019098228
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Keywords
 Seismic interpretation
 Object recognition
 Semisupervised analysis
 Multifacies geobody