A new perspective in identifying the precursory patterns of eruptions

Abstract

The complexity of the processes responsible for volcanic eruptions makes a theoretical approach to forecasting the evolution of volcanic unrest rather difficult. A feasible strategy for this purpose appears to be the identification of possible repetitive schemes (patterns) in the pre-eruptive unrest of volcanoes. Nevertheless, the limited availability and the heterogeneity of pre-eruptive data, and the objective difficulty in quantitatively recognizing complex pre-eruptive patterns, make this task very difficult. In this work we address this issue by using a pattern recognition approach applied to the seismicity recorded during 217 volcanic episodes of unrest around the world. In particular, we use two non-parametric algorithms that have proven to give satisfactory results in dealing with a small amount of data, even if not normally distributed and/or characterized by discrete or categorical values. The results show evidence of a longer period of instability in the unrest preceding an eruption, compared to isolated unrest. This might indicate, even if not necessarily, a difference in the energy of processes responsible for the two types of unrest. However, if the unrest is followed by an eruption, it seems that the seismic energy released during the unrest (parameterized by the duration of the swarm and the maximum magnitude recorded) is not indicative of the magnitude of the impending eruption. We also found that, in general, unrest followed by the largest explosive eruptions have a longer repose time than those related to moderate eruptions. This evidence supports the fact that the occurrence of a large eruption needs a sufficient amount of time after the last event in order to re-charge the feeding system and to achieve a closed-conduit regime so that a sufficiently large amount of gas can be accumulated.

Appendices

Appendix A: Binary decision tree

This method was developed by Rounds (1980) and, slightly modified, successfully applied to volcanic data by Mulargia et al. (1992). It can be used only in the 2-class problem and it was originally designed for hierarchically ordered datasets, even though tests on synthetic data have shown very good behavior also on different types of datasets.

Once the data have been collected, and objects and classes have been defined, BDT integrates feature selection and binary decision tree according to the following steps:

• 1. The fixing of a level α for the decision rule. This level represents the risk we accept of a wrong attribution at each step. We used α=0.1.

• 2. The computation of the cumulative distribution in both classes for each feature taken one at a time, and the identification of the feature and the relative threshold value for which the statistical difference between the cumulative of the two classes is the largest. This means that the significance level of this statistical difference must be (a) lower than the level α and (b) lower than the significance level of the statistical difference calculated for any other feature. The feature (if any) for which both the (a) and (b) conditions are satisfied is the first-order feature, often called the "root" of the pattern. On the basis of the root feature and its threshold value, each object is assigned to either one of two subsets formed respectively by data with a value of the root feature lower/higher than the threshold.

• 3. The identification of the second-order features and their thresholds for which the statistical difference again satisfies the (a) and (b) conditions. These features, which are at most two (i.e., one for each subset), are found by reanalyzing all the features in the two subsets separately, as in step 2.

• 4. The repeating of step 3 for each second-order feature in order to identify progressively higher orders, as long as it is possible to find a feature for which the cumulatives in the two classes are statistically different at a significance level lower than α. The progressive branching of the tree gives all the possible patterns. The procedure automatically terminates when no further branching is possible at the given level α.

Steps 2–4 are performed by means of non-parametric Kolmogorov-Smirnov two-sample statistics (Hollander and Wolfe 1973). Note that the use of an a priori fixed level α reduces the possibility of obtaining overfitting patterns.

Appendix B: Fisher's discriminant analysis

This method (see e.g., Duda and Hart 1973) is based on the reduction of the n-dimensional space of the objects (where n is the number of variables describing the objects, i.e., the dimension of the vectors) to an L-1 dimensional space (where L is the number of classes). In our 2-class problem (L=2), Fisher's method simply projects the objects onto a line. The basic idea, called Fisher criterion, is to project the objects onto the direction that maximizes the ratio of the dispersion between the classes to the dispersion within the classes. More rigorously, suppose we have N objects x, each represented by a vector consisting of n components x _k (k=1,...n). Of these, N ₁ belong to class 1 and N ₂ to class 2. We linearly combine the components of x, i.e., the x _k (k=1,...n), in order to obtain a one-dimensional vector y=(y):

$$y = w^{T}_{k} x_{k} $$

(2)

where w _k are the elements of an n-dimensional vector that projects x onto y. In this way, we obtain N objects y=(y) spread over the two classes.

The unknown in Eq.(2) is the projector, i.e., the vector w. As mentioned above, we would like to choose the projection for which the ratio of the dispersion between the classes to the dispersion within the classes is maximum. In order to do this, first we need to define some quantities.

We define m _i as the average vector of class i:

$${\mathbf{m}}_{i} = \frac{1}{{N_{i} }}{\sum\limits_{{\mathbf{x}} \in {\text{class }}i} {\mathbf{x}} }$$

(3)

We also define m as the average of all the x:

$${\mathbf{m}} = \frac{1}{N}{\sum\limits_{\mathbf{x}} {\mathbf{x}} }$$

(4)

Thus, the dispersion matrix within the class i is given by:

$$S_{i} = {\sum\limits_{{\mathbf{x}} \in {\text{class }}i} {({\mathbf{x}} - {\mathbf{m}}_{i} )} }({\mathbf{x}} - {\mathbf{m}}_{i} )^{T} $$

(5)

and the dispersion within all of the classes is

$$S_{w} = {\sum\limits_{i = 1}^2 {S_{i} } }$$

(6)

The total dispersion matrix is given by

$$S_{T} = {\sum\limits_{\mathbf{x}} {({\mathbf{x}} - {\mathbf{m}})} }({\mathbf{x}} - {\mathbf{m}})^{T} $$

(7)

It follows that

$$ \begin{array}{*{20}l} {{S_{T} } \hfill} & {{ = {\sum\limits_{i = 1}^2 {{\sum\limits_{{\mathbf{x}} \in {\text{class }}i} {({\mathbf{x}} - {\mathbf{m}}_{i} + {\mathbf{m}}_{i} - {\mathbf{m}})({\mathbf{x}} - {\mathbf{m}}_{i} + {\mathbf{m}}_{i} - {\mathbf{m}})} }^{T} } }} \hfill} \\ {{} \hfill} & {{ = {\sum\limits_{i = 1}^2 {{\sum\limits_{{\mathbf{x}} \in {\text{class }}i} {({\mathbf{x}} - {\mathbf{m}}_{i} )({\mathbf{x}} - {\mathbf{m}}_{i} )} }^{T} + {\sum\limits_{i = 1}^2 {{\sum\limits_{{\mathbf{x}} \in {\text{class }}i} {({\mathbf{m}}_{i} - {\mathbf{m}})({\mathbf{m}}_{i} - {\mathbf{m}})} }^{T} } }} }} \hfill} \\ {{} \hfill} & {{ = S_{w} + {\sum\limits_{i = 1}^2 {N_{i} } }({\mathbf{m}}_{i} - {\mathbf{m}})({\mathbf{m}}_{i} - {\mathbf{m}})^{T} } \hfill} \\ \end{array} $$

(8)

The second addendum of the right side term of Eq. (8) is a dispersion matrix S _b that gives an idea of the dispersion between the partial means m _i over the different classes and the total mean m:

$$S_{b} = {\sum\limits_{i = 1}^2 {N_{i} } }({\mathbf{m}}_{i} - {\mathbf{m}})({\mathbf{m}}_{i} - {\mathbf{m}})^{T} $$

(9)

In order to achieve the vector w ^* that maximizes the ratio of the S _b to the S _w , we need to project these matrixes onto the y space and compute the w ^* such that:

$$\frac{{|w^{{*T}} S_{b} w^{*} |}}{{|w^{{*T}} S_{w} w^{*} |}} = {\mathop {\max }\limits_w }\frac{{|w^{T} S_{b} w|}}{{|w^{T} S_{w} w|}}$$

(10)

Once the maximization has been carried out, Fisher's analysis projects the x vectors onto the y space, which is a line. Then, each object y is assigned to the class i whose mean m _i , projected onto the same line, is closest to y.

Appendix C: Branch-and-bound technique

This technique (see e.g., Fukunaga 1990) allows us to select the subset of relevant features among those available. In fact, given n features for each object, apart from few statistical PR algorithms (e.g., BDT) that automatically provide the subset of features by which the classification is carried out (named optimal subset), most of the statistical PR algorithms just perform the pattern recognition and the classification of the objects, but do not explicitly provide the optimal subset. The basic concept in the selection of the optimal subset of features is to find, among all possible subset of the n features, the one leading to the lowest classification error and consisting of the smallest number of features. In such a situation, we are confident that we are considering all of the important variables (otherwise the classification error would not be the lowest) and we are excluding the irrelevant ones (otherwise the number of features in the optimal subset would not be the smallest).

A simple, but very time consuming way to find such an optimal subset consists of exploring the performance of the chosen statistical PR algorithm on all the possible subsets of the n features. This becomes prohibitive as n increases, since we have to explore ${\sum\nolimits_{k = 1}^n {{\left( {\begin{array}{*{20}c} {n} \\ {k} \\ \end{array} } \right)}} }$ subsets. In order to avoid the application of the chosen statistical PR algorithm to all the possible subsets of features, the branch-and-bound technique was developed. This technique is applied iteratively n times; at each iteration k (k=1,...n), it allows for the identification of the suboptimal subset consisting of k features by applying the statistical PR algorithm only to the "most promising" subsets of k features. The suboptimal subset is then the one consisting of k features and leading to the lowest classification error.

The branch-and-bound method relies on a basic assumption; i.e., it assumes that the noise introduced by irrelevant features does not deteriorate the signal given by the relevant features. In a previous study we have tested the validity of this assumption for algorithms BDT and FIS. Based on this assumption, when a certain subset of k features does not produce a good discrimination rule, the branch-and-bound method assumes that any other subset of k+l (l=1,...n-k) features containing those k features will not be the optimal one. In this way, a considerable portion of all the possible subsets is discarded a priori thus saving computation time and effort.