Pi theorem formulation of flood mapping

While physical phenomena are stated in terms of physical laws that are homogeneous in all dimensions, the mechanisms and patterns of the physical phenomena are independent of the form of the units describing the physical process. Accordingly, across different conditions, the similarity of a process may be captured through a dimensionless reformulation of the physical problem with Buckingham $\Pi$ theorem. Here, we apply Buckingham $\Pi$ theorem for creating dimensionless indices for capturing the similarity of the flood process, and in turn, these indices allow machine learning to map the likelihood of pluvial (flash) flooding over a landscape. In particular, we use these dimensionless predictors with a logistic regression machine learning (ML) model for a probabilistic determination of flood risk. The logistic regression derived flood maps compare well to 2D hydraulic model results that are the basis of the Federal Emergency Management Agency (FEMA) maps. As a result, the indices and logistic regression also provide the potential to expand existing FEMA maps to new (unmapped) areas and a wider spectrum of flood flows and precipitation events. Our results demonstrate that the new dimensionless indices capture the similarity of the flood process across different topographies and climate regions. Consequently, these dimensionless indices may expand observations of flooding (e.g., satellite) to the risk of flooding in new areas, as well as provide a basis for the rapid, real-time estimation of flood risk on a worldwide scale.