Statistical estimation of total discharge volumes

Calculating the volumes of water discharged by streams is becoming increasingly important in water accounting and deciding how much water to allocate to competing uses. Water accounting is particularly important in Australia, as the driest inhabited continent and also in the face of potential impacts of a changing climate. Stream networks all over the world are littered with gauging stations, which take regular measurements of steam flow in order to help natural resource managers make decisions regarding water allocation. Estimating total discharge volumes is also of utmost importance when estimating pollutant loads from catchments. In order to calculate the total discharge volume, one must integrate the hydrograph (the graph of stream flow with time) over the period of interest. The simplest method to perform the integration is a trapezoidal scheme, however this fails to account for a number of sources of uncertainty inherent in the hydrograph, namely: (i) what happens between the discrete observations; (ii) gauging stations measure water height and flow is estimated using a rating curve between height and flow; and (iii) there are measurement errors associated with the height data recorded at gauging stations. We present a Monte Carlo method that employs: (i) nonparametric stochastic differential equations (SDEs) to bridge the gaps between discrete observations; and (ii) the Weighted Nadaraya-Watson estimator to estimate the conditional distribution of log-flow given water height. The output of the method is an ensemble of hydrographs that are faithful to the observed data, but incorporating these uncertainties/errors. Integrating the members of this ensemble gives rise to a distribution for the total discharge volumes and properly accounts for the imperfect information available. We demonstrate the methods using hydrographic data from Obi Obi Creek in the Mary River catchment, Queensland, Australia and examine the uncertainty inherent in total discharges when integrating over a single month and over an entire year. We also introduce an artificial gap of 375 days into the hydrograph and demonstrate how well our simulated diffusions replicate the dynamics of stream flow. Whilst our Monte Carlo method is useful for estimating total discharge volumes, the nonparametric SDEs used also appear to have good potential as stochastic rainfall-runoff models in their own right.