In this post I will present a Python implementation of a new technique for fractal interpolation derived from a paper by Manousopoulos, Drakopoulos, and Theoharis. You may find my code on here on GitHub. Fractal interpolation is useful for data sets that exhibit self similarity at multiple scales, which are difficult to interpolate with polynomials.
In this post I will present a technique for generating a one dimensional (quasi) fractal data set using a modified Matérn point process, perform a simple box-couting procedure, and then calculate the lacunarity and fractal dimension using linear regression. Lacunarity is actually a pretty large topic, and we will only cover one accepted interpretation here. This material was motivated by an interesting paper on the fractal modelling of fractures in tight gas reservoirs. Tight gas reservoirs refer to reservoirs with very low permeability. To provide a sense of perspective, oil reservoirs typically have a permebility of ten to a hundred millidarcies, whereas shale gas reservoirs are usually less than 0.1 microdarcies, which is about the same permeability as a granite countertop.