Recurrence Lacunarity for the Analysis of Spatial Data

Quantifying the structure and heterogeneity of complex spatial patterns is a key challenge in the analysis of spatial data across many scientific disciplines, including geoscience and geomorphology. The complexity of spatial patterns can be analysed by considering the recurrence of specific properties. While recurrence plot based methods are well established for analysing dynamical systems, their application to spatial patterns has received less attention. Here, we propose a novel approach that combines spatial recurrence analysis with a measure from fractal geometry, lacunarity, which originally quantifies homogeneity in spatial patterns. Applied to recurrence plots, it is referred to as recurrence lacunarity (RL) and quantifies the homogeneity of recurrences. Although recurrence plots can be generated from higher-dimensional data, RL has not yet been calculated for higher-dimensional (spatial) recurrence plots. To address this gap, we evaluate the RL of spatial data by validating the method using synthetic test patterns and then applying it to analyse hillslope gradients of river catchments near the Mendocino Triple Junction. The results demonstrate that the RL effectively detects and quantifies differences in the spatial structure of the catchments, which can be related to local uplift rates and the geological setting. RL provides a robust measure for comparing diverse spatial data sets and for quantifying how their spatial structure relates to external parameters, and may also be used as features in machine-learning models, complementing existing descriptors of spatial structure.