Identifying Scale-dependent Spatial and Temporal Patterns in Earth Science Data: Lacunarity-based Techniques

Spatial and temporal datasets that comprise distributions of events along a transect/timeline together with their magnitudes can display scale-dependent changes in persistence or anti-persistence that may contain signatures of underlying physical processes. Lacunarity is a technique that was originally developed for multiscale analysis of data and characterizes the distribution of spaces or gaps in a pattern as a function of scale. In this study, we demonstrate how lacunarity may be modified in order to reveal scale-dependent changes in 1-dimensional data related to fractures, sedimentary layering and rainfall. In order to address whether fractures found along a 1-dimensional transect (scanline) occur in clusters, we compare the lacunarity of a given fracture-spacing data to a theoretical random lacunarity curve. Further, we introduce the concept of 1st derivative of log-transformed lacunarity and demonstrate that this function can find the inter-cluster spacing and possible fractal behaviour over certain scales. It will be demonstrated how this same technique may be applied to a time-series, e.g., rainfall data, to see whether such events occur in clusters over certain time-scales. Next, the “event magnitudes” (e.g., fracture aperture) were added to each event data point (e.g., fracture) thus, yielding a 1-dimensional non-binary dataset and it was tested whether the dataset shows scale-dependent changes in terms of anti-persistence and persistence. The concept of lacunarity ratio, LR, is introduced, which is the lacunarity of a given dataset normalized to the lacunarity of its random counterpart. This randomization however, is different from the one used in the previous technique. In case of our fracture dataset for example, the random sequence is generated by leaving the locations of fractures unaltered and randomly reallocating the magnitudes along the dataset. It was demonstrated that LR can successfully delineate scale-dependent changes in terms of anti-persistence and persistence. In addition to the fracture data already mentioned here (spacing and apertures from NE Mexico), the one used for developing this technique, it was applied to two different types of data: a set of varved sediments from Marca Shale and, a hundred-year rainfall record from Knoxville, TN, USA. While the fracture data showed anti-persistence at small scales (within cluster) and random behavior at large scales, the rainfall data and varved sediments both appear to be persistent at small scales becoming random at larger scales. It was no surprise to find such striking similarity between the spatial “sedimentary” data and the time-dependent rainfall data because in rock records, the former is often considered to be a proxy for the latter. In general, such differences in behavior with respect to scale-dependent changes in anti-persistence to random, persistence to random, or otherwise, maybe be related to differences in the physicochemical properties and processes contributing to multiscale datasets.