Measure of similarity between geological sections accounting for subjective criteria

We propose a set of criteria which are based upon spatial relations between geological structures, in order to develop a measure of similarity between geological sections. By giving different weights to the criteria, the technique can be tuned to different geological problems. The weights are determined automatically when a user ranks sets of test images, without the need to assign numerical ‘a priori’ values. This is a step towards the development of a tool for applications such as numerical inversion, geological image retrieval, geological feature extraction, and image data set browsing. Introduction In previous work (Boschetti and Moresi 2001, Boschetti et al., 2001), we have explored the possibility of using geological modelling within the framework of numerical inversion, as in traditional geophysical inversion. Here the word inversion is used in its mathematical (Tarantola, 1987) rather than its geological sense. In essence, mathematical inversion, also called optimisation, seeks to answer questions such as “Given this target (a geological section, for example), what are the parameters that generated it?”. The term inversion is used since this is exactly the opposite of the aim of most forward geological models, i.e. “Given these parameters, what target will I obtain?”. The goal is to simplify the understanding of, and hopefully quantify, what and how physical and chemical factors control geological processes. This approach has resulted in techniques to invert simple geomechanical problems (Boschetti and Moresi, 2001) and to visualise the resulting high-dimensional parameter space (Boschetti et al., 2002). One of our main concerns has been to keep the expert geologist (the ‘user’) tied into the process, i.e. not to create a ‘black box’ system. The reason is twofold: first, we are aware that, at present, our mathematical and physical understanding of nature does not allow us to answer complicated geological questions, for which we need to add the empirical knowledge of a geologist; secondly, we believe that the interaction between the expert (geologically knowledgeable) user and the inversion algorithm is beneficial to both. It allows the inversion process to overcome basic mathematical hurdles (non-uniqueness, instabilities etc.), and it allows the user to obtain a wider picture and new insights into the problem. Inversion algorithms need to optimise a measure of similarity between the target and the forward model solution. We have included the expert user in our process by simply asking him or her to provide such a measure. In practice, an expert user sits in front of a computer, ranks a number of geological sections according to their similarity to a target, and the algorithm uses the resulting ranking to generate new models that progressively converge towards the target. The process is iterated until the user is satisfied with the result. This approach has proved particularly successful, but also quite time consuming because of the need to rank each set of forward models. Ideally, we would like the user to supervise the inversion rather than run it. This would entail the user giving rough directions to the algorithm and then intervening only when further geological insight is needed. The algorithm must automatically evaluate a measure of similarity between geological sections. This paper presents some steps in this direction. The problem No formal definition of geological similarity is available in the literature. A small amount of work has been attempted based either solely upon geometric relationships, or upon geological criteria tuned for specific problems. Our approach is also purely geometrical. However, we attempt to capture some basic geological meaning by trying to incorporate the user’s criteria for ‘similarity’ between two geological images. This allows us not only to account for subjectivity (or, more exactly, intuition) but also to adapt to different geological problems. This extremely challenging task has been addressed for different non-geological applications, by, among others, the artificial intelligence, machine learning, and image processing communities. Our work builds upon some recent advances in computer vision and can be broadly subdivided into four steps: recognise objects in a geological section; formalise spatial relationships between objects; build a similarity metric; tune such a metric to both a user’s subjective criteria and to the specific problem at hand. Recognising objects in a geological section Before an object can be recognised, it needs to be found. This is usually achieved by some sort of image segmentation algorithm. This is an open area of research in itself and no standard methods are available. We sidestep this problem and proceed with the more general proof of concept of the overall method. In what follows, we assume that we possess an efficient image segmentation algorithm that recognises the objects in the geological section. This will obviously require future work. Once an object has been found, its location (centroid) and shape (length, height, area, elongation etc.) are evaluated and stored. Objects are classified into three categories: faults, layers, and ‘rock bodies’, depending upon their overall shape. Formalising spatial relationships between objects We build on the work of Chang et al. (1987), to which we refer the reader for details of implementation. In Figure 1a we present a geological section containing two dipping faults, one isolated body, a thick deep layer, and a very thin shallow layer. In Figure 1b a grid with a user-defined cell size of 30 pixels is superimposed. Each object is assigned to the grid cell in which its centroid lies. Each cell can either be empty, or contain one or more objects. Spatial relationships between objects are represented as a set of symbols that store information such as “the left fault is above the thick layer and below the shallow one”, or “the block is to the right of the faults, above the thick layer and below the thin one”. For details see Chang et al. (1987). Building a similarity metric In order to simplify the similarity evaluation task, we reject sections that are ‘too different’, or, equivalently, we assign them zero similarity. First, we ask the user to identify the important features in the target image. For example, in the target image in Figure 2, the user may decide, according to the problem at hand, that a solution must have at least two faults and one layer. Then, as a result of the user’s decision, the sections that do not contain at least two faults and one layer are rejected (assigned zero similarity in Figure 2), while the others are accepted for further analysis. The remaining geological sections are then stored as a set of numbers and symbols as described above, representing the objects in the section, their shapes, and relative positions. Now the problem of measuring similarity can be addressed. Table 1 shows a number of criteria that can be employed. The similarity between two images s t v v , can then be measured as the inverse of the distance ) , ( s t v v d between the images in the abstract space generated by the criteria in Table 1, each criterion weighted by an appropriate factor. ( 1 ) ∑ = i si i s t p v v d γ . ) , ( where for each criterion i in Table 1, i p is the weighting factor and ) , ( s i t i i si v v γ γ = is the measure of similarity between the two images for that specific criterion. When two images contain a different number of objects, the match between objects may not be unique. For example, if image A has one fault (Af1) and image B has two faults (Bf1 and Bf2), the criteria related to the fault similarity (criteria 1 and 2 in Table 1) can be computed either between Af1 and Bf1 or between Af1 and Bf2. When this happens the algorithm calculates the measure for each possible match and stores the smaller value. The presence of a different number of objects between the images is accounted for by criterion 7. As mentioned above, the user can choose which objects in the image are most important in order to accept or reject an image (‘main’ objects in the following). Criteria 5 and 6 measure the relative positions between such main objects in the two images. For example, in Figure 3, both images contain one layer, one block and one fault, but the relative horizontal position of the block and the fault is different. In order to account for the presence of additional objects (i.e. objects not considered as ‘main’), criterion 8 evaluates the position of the additional objects by measuring how many are located between matches of main objects. For example, both Figures 4a and b contain one layer, one block, and one fault, but Figure 4b also contains an extra fault. Criterion 8 accounts for the relative position of the extra object. Tuning the metrics It remains to tune the parameters i p in the distance formula ( 1). This corresponds to answering questions such as “is it more important to match the inclination of the faults or to have an equal number of layers?” or “how important is it to have the same number of objects in the sections?” This accounts not only for the specific problem at hand, but also for the subjective interpretation of the user. We address the problem by generating a set of questions aimed at obtaining this information implicitly from the user. Rather than asking the questions directly, we turn them into their equivalent geological form, by asking the user to compare and evaluate geological scenarios. Appendix A contains an example questionnaire. Once the user has answered these questions, our problem reduces to finding a set of i p that respects the user’s rankings. Mathematically, this is an optimisation problem (another ‘inverse’ problem) and, as always in geoscience, a difficult one. Referring to the distance calculation ( 1), we call s γ the vector containing the similarity measure for an image S for each criterion in Table 1. Let us assume we have m images ranked by the user for similarity to a target in such a way that 1 is the top-ranked image (most similar to the target) and m the bottom-ranked one. In order to respect the user’s ranking we need: ( 2 ) ] 1 , 1 [ . . 1 − ∈ > + m s for p p s s γ γ This can be expressed as ( 3 )