An unbiased estimator of coefficient of variation of streamflow

Abstract

Given increasing demand for high frequency streamflow series (HFSS) at daily and subdaily time scales there is increasing need for reliable metrics of relative variability for such series. HFSS can exhibit enormous relative variability especially in comparison with low frequency streamflow series formed by aggregation of HFSS. The product moment estimator of the coefficient of variation C, defined as the ratio of sample standard deviation to sample mean, as well as ten other common estimators of C, are shown to provide severely downward biased and highly variable estimates of C for very long records of highly skewed and periodic HFSS particularly for rivers which exhibit zeros. Resorting to the theory of compound distributions, we introduce an estimator of C corresponding to a mixture of monthly zero-inflated lognormal distributions denoted as a delta lognormal monthly mixture ΔLN3MM model. Through monthly stratification, our ΔLN3MM model accounts for the seasonality, skewness, multimodality, and the possible intermittency of HFSS. In comparisons among estimators, our ΔLN3MM based C estimator is shown to yield much more reliable and approximately unbiased estimates of C not only for small samples but also for very large samples (tens of thousands of observations). We document values of C in the range of [0.18, 42,000] with a median of 1.9 and an interquartile range of [1.34, 3.75] for 6807 daily streamflow series across the U.S. from GAGES-II dataset, with the highest values of C occurring in arid and semiarid regions. A multivariate analysis and national contour map reveal that extremely large values of C, never previously documented, tend to occur in arid watersheds with low runoff ratios, which tend to also exhibit a considerable number of zero streamflows.