Unbiased Estimates and Confidence Intervals for Riverine Loads

Abstract

An accurate estimation method for seasonal, annual, or decadal riverine constituent loads, which are the masses that pass through a river cross-section into a receiving water body during target periods, has been investigated on the premise of high-frequency discharge and infrequent water quality (WQ) observations (e.g., Philips et al., 1999; Webb et al., 1997) due to the high cost and labor of WQ monitoring and analysis (Horowitz, 2013; Verma et al., 2012). Load estimation studies have targeted the WQ parameters related to water pollution or erosion, such as nutrients or suspended sediments (SS), from catchments or watersheds with various land uses. Among the load estimation methods (LEMs), the rating curve (RC) method (RCM) is very common, whereas the river load estimates of nutrients and SS by this method can contain large biases and variances (e.g., Lee et al., 2016; Walling & Webb, 1981). Load estimation studies have also retrospectively evaluated the uncertainty in the load estimates by several LEMs, including the RCM using the Monte Carlo method based on high-frequency WQ data sets, daily data sets (

Unbiased Estimates and Confidence Intervals of Riverine Loads for Low‐Frequency Water Quality Monitoring Strategies

“…The proposed methods also improve the load estimates obtained with the RCM using IS (Tada & Tanakamaru, 2021) because the simulated sampling probabilities used in the Monte Carlo method described in Section 3.2.2 can eliminate the maximum sample size limitation. In the RCM using IS, the maximum sample size for PPS samples was introduced to avoid duplicate sample elements, which lowers the coverage of the CIs of estimates because PPS sampling is a form of UPSWR sampling.…”

“…Notably, sophisticated RCs require large data sets that span multiple years, such as 60 sample elements per 10 years in load estimation for the WRTDS model (Hirsch & De Cicco, 2015). The small sample size targeted in this study, ranging from 12 to 20 sample elements for annual load estimation, would inflate the uncertainty of partial coefficients of multiple explanatory variables and, in turn, increase the uncertainty of load estimates by such RCs (Tada & Tanakamaru, 2021). Sophisticated RCs are superior to the simple RC in Equation when data sets with longer periods, such as decadal data, are used in load estimation.…”

“…In addition, the load estimator, as a stratum‐size weighted average of the stratum means with stratified sampling based on SRS (Thomas & Lewis, 1995; Verhoff et al., 1980), is also an HT estimator, although the ratio estimator (Beale, 1962) with stratified sampling employed in Great Lake studies (International Reference Group on Great Lakes Pollution from Land Use Activities & Whitt, 1977) was biased. The load estimates by the RCM become an HT estimator when p j in UPSWOR is proportional to the estimated loading rates by an RC (Tada & Tanakamaru, 2021). As shown here, for unbiased load estimation, the relationship between an LCM and sampling probability defined by the sampling strategy must satisfy Equation ; otherwise, load estimates become biased.…”

“…This discussion of QMLE also holds for the smearing estimator and MVUE. To keep load estimates obtained with the RCM unbiased, samples acquired with sampling probabilities proportional to the expected loading rates by an RC must be used (Tada & Tanakamaru, 2021; Thomas, 1985).…”

“…The SALT method is a realization of PPS sampling at WQ monitoring sites using a cumulative-total (e.g., Arnab, 2017) or inverse function method. Tada and Tanakamaru (2021) explained that PPS sampling is a type of discretized importance sampling (IS, e.g., Hammersley & Handscomb [1964]) and proposed the rating curve method (RCM) using IS as a practical method for obtaining unbiased load estimates and the corresponding CIs. However, PPS sampling is not feasible at WQ monitoring sites because it requires real-time observation of the auxiliary variables related to loading rates and a programmable autosampler.…”