Scaling Theory of Floods for Developing a Physical Basis of Statistical Flood Frequency Relations
Authors: Gupta Vijay 
Published: February 2017

Prediction of floods at locations where no streamflow data exist is a global issue because most of the countries involved don’t have adequate streamflow records. The United States Geological Survey developed the regional flood frequency (RFF) analysis to predict annual peak flow quantiles, for example, the 100-year flood, in ungauged basins. RFF equations are pure statistical characterizations that use historical streamflow records and the concept of “homogeneous regions.” To supplement the accuracy of flood quantile estimates due to limited record lengths, a physical solution is required. It is further reinforced by the need to predict potential impacts of a changing hydro-climate system on flood frequencies. A nonlinear geophysical theory of floods, or a scaling theory for short, focused on river basins and abandoned the “homogeneous regions” concept in order to incorporate flood producing physical processes. Self-similarity in channel networks plays a foundational role in understanding the observed scaling, or power law relations, between peak flows and drainage areas. Scaling theory of floods offers a unified framework to predict floods in rainfall-runoff (RF-RO) events and in annual peak flow quantiles in ungauged basins.

Theoretical research in the course of time clarified several key ideas: (1) to understand scaling in annual peak flow quantiles in terms of physical processes, it was necessary to consider scaling in individual RF-RO events; (2) a unique partitioning of a drainage basin into hillslopes and channel links is necessary; (3) a continuity equation in terms of link storage and discharge was developed for a link-hillslope pair (to complete the mathematical specification, another equation for a channel link involving storage and discharge can be written that gives the continuity equation in terms of discharge); (4) the self-similarity in channel networks plays a pivotal role in solving the continuity equation, which produces scaling in peak flows as drainage area goes to infinity (scaling is an emergent property that was shown to hold for an idealized case study); (5) a theory of hydraulic-geometry in channel networks is summarized; and (6) highlights of a theory of biological diversity in riparian vegetation along a network are given.

The first observational study in the Goodwin Creek Experimental Watershed, Mississippi, discovered that the scaling slopes and intercepts vary from one RF-RO event to the next. Subsequently, diagnostic studies of this variability showed that it is a reflection of variability in the flood-producing mechanisms. It has led to developing a model that links the scaling in RF-RO events with the annual peak flow quantiles featured here.

Rainfall-runoff models in engineering practice use a variety of techniques to calibrate their parameters using observed streamflow hydrographs. In ungagged basins, streamflow data are not available, and in a changing climate, the reliability of historic data becomes questionable, so calibration of parameters is not a viable option. Recent progress on developing a suitable theoretical framework to test RF-RO model parameterizations without calibration is briefly reviewed.

Contributions to generalizing the scaling theory of floods to medium and large river basins spanning different climates are reviewed. Two studies that have focused on understanding floods at the scale of the entire planet Earth are cited.

Finally, two case studies on the innovative applications of the scaling framework to practical hydrologic engineering problems are highlighted. They include real-time flood forecasting and the effect of spatially distributed small dams in a river network on real-time flood forecasting.