Scale space topology and natural scales of geological surfaces

S. A. Stewart1, P. L. Rosin2, G. J. Hay3 & T. J. Wynn4

1BP plc, Chertsey Road, Sunbury on Thames, Middlesex TW16 7LN, UK

2Cardiff University, Queen's Buildings, Newport Road, Cardiff CF24 3XF, UK

3Université de Montréal, Montréal, Québec, Canada, H3C 3J7

4TRACS International, Falcon House, Union Grove, Aberdeen AB10 6XU, UK

Geological surfaces such as bedding planes in sedimentary basins can be mapped over thousands of square kilometres at high resolution using 3D seismic data. These surfaces invariably show superimposed structures at different wavelengths, from metres to tens of kilometres. Separating the superimposed structural elements is relevant to academics studying polyphase deformation, and commercial geoscientists building reservoir models.

Most approaches to extracting features of different structural wavelengths from geological surfaces involve iteratively smoothing or sub-sampling the data to progressively remove shorter wavelength features, producing a stack of sections or maps that can be viewed, for example, as a movie. No methods are currently employed to assign relative significance to slices of this map stack or specific structural elements identified using it. Representatives are chosen using a priori knowledge, data handling constraints or aesthetics.

The space represented by a stack of progressively smoothed maps has, however, been extensively studied in the fields of geomorphology and computer vision, where it is known as scale space. Mathematics and tools to analyse scale space have been developed in these disciplines. Published material includes definition of robust approaches for generating scale space and description of generic scale space topologies. Of particular interest to the analysis of geological sections and maps are a number of methods for automatically identifying the most significant scales, known as natural scales, at which to map a given structure based on scale space characteristics.

Schemes for determining natural scales are generally based on maximising some measure of activity (e.g. information content, magnitude of features such as edges, ridges) normalised to take the effect of scale into account. The scheme used here measures the degree of "wiggliness" of a profile. The curve is smoothed by Gaussian blurring, and those scales maximising the number of points of inflection times the Gaussian scale parameter are recorded as natural scales.

Here we derive natural scales of cross sections from a number of structural settings, including salt diapirs, folds and fault blocks. The most prominent natural scales correspond to recognisable structural components, including basin and fold scale curvature, fault blocks, sedimentary architectures and dataset noise. This method identifies these key wavelengths in a way that enables reconstruction of a surface based on spatially varying natural curvature, but we have yet to optimise the method of rebuilding the surface between the inflections.

Applications include automatically identifying spatial variation in superimposed structures from finite state maps or sections, and automated spatially varying smoothing of mapped surfaces.