From Fault Likelihood to Fault Networks: Stochastic Seismic Interpretation Through a Marked Point Process with Interactions

Abstract

Faults are crucial subsurface features that significantly influence the mechanical behavior and hydraulic properties of rock masses. Interpreting them from seismic data may lead to various scenarios due to uncertainties arising from limited seismic bandwidth, possible imaging errors, and human interpretation noise. Although methods addressing fault uncertainty exist, only a few of them can produce curved and sub-seismic faults simultaneously while quantitatively honoring seismic images and avoiding anchoring to a reference interpretation. This work uses a mathematical framework of marked point processes to approximate fault networks in two dimensions with a set of line segments. The proposed stochastic model, namely the Candy model, incorporates simple pairwise and nearby connections to capture the interactions between fault segments. The novelty of this approach lies in conditioning the stochastic model using input images of fault probabilities generated by a convolutional neural network. The Metropolis–Hastings algorithm is used to generate various scenarios of fault network configurations, thereby exploring the model space associated with the Candy model and reflecting the uncertainty. Probability level sets constructed from these fault segment configurations provide insights on the obtained realizations and on the model parameters. The empty space function produces a ranking of the generated fault networks against an existing interpretation by testing and quantifying their spatial variability. The approach is applied to two-dimensional sections of seismic data, in the Central North Sea.

Appendix A Supplementary Materials

1.1 Appendix A.1 Algorithm for Simulating the Candy Model

In this paper, the MH (algorithm 1) allows us to sample the distribution of the Candy model.

Algorithm 1

Candy model simulation with MH

1.2 Appendix A.2 Comparison Between Candy Model Realizations and the Thinning-Based Skeleton Through the Kolmogorov–Smirnov (KS) Test

In complement to the analysis carried out in Sect. 4.3.2, the Kolmogorov–Smirnov (KS) test is used for comparing the Candy realizations and the thinning-based interpretation. This is a non-parametric statistic, designed for comparing two empirical distributions, by defining the largest absolute difference between the two cumulative distribution functions as a measure of disagreement (Lopes et al. 2008).The null hypothesis states that the samples $\tilde{F}_{\theta _k}(j)$ and ${\widehat{F}}_\textrm{skeleton}(j)$, for $j = 1,\ldots , n$, are drawn from the same distribution. A confidence level of $95 \%$ is chosen, that is, indicating that the null hypothesis will be rejected in favor of the alternative if the p-value is less than 0.05.

Fig. 15

Interpretation on image A. Interpreted fault networks (first column) with the lowest energy value and their distance maps (second column). Parameter set $\theta _{i=1,\ldots ,5}$ are those Table 5. The red arrows and circles indicate isolated features in the skeleton and seldom visited areas in the Candy model realizations, respectively

The analysis for the KS test is carried out using the scipy.stats.ks_2samp function from the SciPy library. This test performs the two-sample KS test of goodness of fit, which compares the underlying distributions of two independent samples from the continuous empty space functions of the Candy realizations and the skeleton. The test is carried out on three images (Fig. 13), each one using five parameters (Table 5), in which $N_\textrm{sim} = 100$ simulations are run. The comparison between the two interpretations is done for each parameter set, through the median $\tilde{F}_{\theta _k}$ curves obtained for each model parameter set $\theta _k$ with the empty space ${\widehat{F}}_\textrm{skeleton}$ of the skeleton.

Figure 14 shows the results of the Kolmogorov–Smirnov statistical test on boxes in either green or red. For the A image, the p-values of all parameter sets are greater than 0.05, which means that the null hypothesis is not rejected. This is consistent with the results of Fig. 14, even though the envelope test seems to be more discriminating. Actually, envelopes show how the realizations of the Candy model from such an image fill the space in a way quite close to the skeleton process. For image B, only the p-value corresponding to the parameter set $\theta _2$ is less than 0.05, indicating that Candy model realizations fill the space in a way quite different from the skeleton for this image (which is also consistent with the two other images). For image C, only the p-value of the parameter set $\theta _3$ is greater than 0.05, which indicates the parameter set induces realizations filling the space in a closer way than the skeleton.