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Topological data analysis (TDA) is a tool from data science and mathematicsthat is beginning to make waves in environmental science. In this work, we seekto provide an intuitive and understandable introduction to a tool from TDA thatis particularly useful for the analysis of imagery, namely persistent homology.We briefly discuss the theoretical background but focus primarily onunderstanding the output of this tool and discussing what information it canglean. To this end, we frame our discussion around a guiding example ofclassifying satellite images from the Sugar, Fish, Flower, and Gravel Datasetproduced for the study of mesocale organization of clouds by Rasp et. al. in2020 (arXiv:1906:01906). We demonstrate how persistent homology and itsvectorization, persistence landscapes, can be used in a workflow with a simplemachine learning algorithm to obtain good results, and explore in detail how wecan explain this behavior in terms of image-level features. One of the corestrengths of persistent homology is how interpretable it can be, so throughoutthis paper we discuss not just the patterns we find, but why those results areto be expected given what we know about the theory of persistent homology. Ourgoal is that a reader of this paper will leave with a better understanding ofTDA and persistent homology, be able to identify problems and datasets of theirown for which persistent homology could be helpful, and gain an understandingof results they obtain from applying the included GitHub example code.

A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science