  Topological data analysis (TDA) is a tool from data science and mathematics
that is beginning to make waves in environmental science. In this work, we seek
to provide an intuitive and understandable introduction to a tool from TDA that
is particularly useful for the analysis of imagery, namely persistent homology.
We briefly discuss the theoretical background but focus primarily on
understanding the output of this tool and discussing what information it can
glean. To this end, we frame our discussion around a guiding example of
classifying satellite images from the Sugar, Fish, Flower, and Gravel Dataset
produced for the study of mesocale organization of clouds by Rasp et. al. in
2020 (arXiv:1906:01906). We demonstrate how persistent homology and its
vectorization, persistence landscapes, can be used in a workflow with a simple
machine learning algorithm to obtain good results, and explore in detail how we
can explain this behavior in terms of image-level features. One of the core
strengths of persistent homology is how interpretable it can be, so throughout
this paper we discuss not just the patterns we find, but why those results are
to be expected given what we know about the theory of persistent homology. Our
goal is that a reader of this paper will leave with a better understanding of
TDA and persistent homology, be able to identify problems and datasets of their
own for which persistent homology could be helpful, and gain an understanding
of results they obtain from applying the included GitHub example code.
