Analytical Propagation of Runoff Uncertainty Into Discharge Uncertainty Through a Large River Network

The transport of freshwater from continents to oceans through rivers has traditionally been estimated by routing runoff from land surface models within river models to obtain discharge. This paradigm imposes that errors are transferred from runoff to discharge, yet the analytical propagation of uncertainty from runoff to discharge has never been derived. Here we apply statistics to the continuity equation within a river network to derive two equations that propagate the mean and variance/covariance of runoff errors independently. We validate these equations in a case study of the rivers in the western United States and, for the first time, invert observed discharge errors for spatially distributed runoff errors. Our results suggest that the largest discharge error source is the joint variability of runoff errors across space, not the mean or amplitude of individual errors. Our findings significantly advance the science of error quantification in model‐based estimates of river discharge. Plain Language Summary Computer programs are routinely used to simulate the movement of water on Earth, including the flow of freshwater in rivers. A common approach for calculating river flow is to run a first computer program that estimates how much of the rain is partitioned into runoff, evaporation, or storage, then a second that transports the runoff through rivers. These computer programs are based on a simplified representation of nature and are imperfect. Until now, it has been unclear how the imperfections are transferred from one program to the next, and therefore challenging to understand the sources of errors. In this study, we present a methodology that can be used to backtrack errors in runoff, an otherwise unknown quantity, from observed errors in river flow. Our results reveal a surprising error source and can help diagnose limitations in simulations of the Earth's water cycle. Key Points We present the first analytical equations for propagating runoff uncertainty into discharge uncertainty through a river network Our methodology allows the inverse inference of runoff errors from observed discharge errors at continental scales The largest discharge error source is the joint variability of runoff errors across space, not the mean or amplitude of individual errors