Robert M. Hirsch

Contamination by nutrients, major ions, and metals poses a major threat to global water sustainability. Understanding how these pollutants vary across time and space requires long-term monitoring and robust statistical approaches. Traditional methods, however, often struggle to account for streamflow variability, seasonality, and nonlinear responses. Introduced in 2010, the Weighted Regressions on Time, Discharge, and Season (WRTDS) method offers a flexible, data-driven framework that generates both observed and flow-normalized estimates of concentration and load. Over the past 15 years, WRTDS has become a state-of-the-art tool for water-quality science and management, with applications spanning a wide range of hydrologic, climatic, and policy contexts─including major watersheds across North America, Europe, Asia, Australia, and the Arctic. In this review of WRTDS, we document the method’s major advancements, examine its expanding geographic and thematic applications, and summarize its relevance to water-quality management programs and policies worldwide. We also discuss its performance relative to other regression and machine-learning approaches. Finally, we identify key priorities for future development to support the continued evolution of WRTDS as a trusted and practical tool for scientists and managers working to protect and sustain water resources.

Koocanusa Reservoir (KOC) is a waterbody that spans the United States (U.S.) and Canadian border. Increasing concentrations of total selenium (Se), nitrate + nitrite (NO₃^-, nitrite is insignificant or not present), and sulfate (SO₄^2-) in KOC and downstream in the Kootenai River (Kootenay River in Canada) are tied to expanding coal mining operations in the Elk River Watershed, Canada. Using a paired watershed approach, trends in flow-normalized concentrations and loads were evaluated for Se, NO₃^-, and SO₄^2- for the two largest tributaries, the Kootenay and Elk Rivers, Canada. Increases in concentration (SO₄^2- 120%, Se 581%, NO₃^- 784%) and load (SO₄^2- 129%, Se 443%, NO₃^- 697%) in the Elk River (1979-2022 for NO₃^-, 1984-2022 for Se and SO₄^2-) are among the largest documented increases in the primary literature, while only a small magnitude increase in SO₄^2- (7.7% concentration) and decreases in Se (-10%) and NO₃^- (-8.5%) were observed in the Kootenay River. Between 2009 and 2019, the Elk River contributed, on average, 29% of the combined flow, 95% of the Se, 76% of the NO₃^-, and 38% of the SO₄^2- entering the reservoir from these two major tributaries. The largest increase in solute concentrations occurred during baseflows, indicating a change in solute transport and delivery dynamics in the Elk River Watershed, which may be attributable to altered landscapes from coal mining operations including altered groundwater flow paths and increased chemical weathering in waste rock dumps. More recently there is evidence of surface water treatment operations providing some reduction in concentrations during low flow times of year; however, these appear to have a limited effect on annual loads entering KOC. These findings imply that current mine water treatment, which is focused on surface waters, may not sufficiently reduce the influence of mine-waste-derived solutes in the Elk River to allow constituent concentrations in KOC to meet U.S. water-quality standards.

Many studies investigate river floods by analyzing annual maximum series that record the largest flow of each year, including many within‐bank events inconsequential for human communities. Fewer focus on larger floods, especially at the continental scale. Using 473 streamgages across the conterminous United States with near‐natural flow from 1966 to 2015, we characterized the seasonality, occurrence, and climatic associations of the 10 largest and 2 largest floods at each site. These are often overbank events that have ecosystem functions and pose risks to humans. We grouped sites into 14 clusters corresponding to climatic and physiographic regions and characterized their flood seasonality at a monthly resolution using a probabilistic method. We then evaluated annual occurrence regionally and nationally, and seasonal occurrence regionally, by complementing a traditional approach to trend analyses with a novel method based on expected occurrence. Relationships between flood occurrence and climate indices were also investigated. Large floods have strong seasonality in some regions, but in areas with numerous flood‐generating mechanisms, seasonality is more complex. There is little evidence nationally that large riverine floods are more or less frequent than expected in recent years and only two regions show significant trends in annual counts; few show seasonal trends. We found some regional relationships between flood counts and climate indices, annually and seasonally; nationally the Pacific North American pattern is related to annual counts of the 2 largest floods. Large‐flood occurrence was generally stable across the United States in the last five decades; this may or may not continue with projected warming.

Salinity in the Colorado River Basin causes an estimated $300 to $400 million per year in economic damages in the United States. To inform and improve salinity‐control efforts, this study quantifies long‐term trends in salinity (dissolved solids) across the Upper Colorado River Basin (UCRB), including time periods prior to the construction of large dams and preceding the implementation of salinity‐control projects. Weighted Regressions on Time, Discharge, and Season was used with data sets of dissolved‐solids and specific‐conductance measurements, collected as early as 1929, to evaluate long‐term trends in dissolved‐solids loads and concentrations in streams from 1929 to 2019 (n = 14). Results indicate that large, widespread, and sustained downward trends in dissolved‐solids concentrations and loads occurred over the last 50–90 years. For 12 of the 14 stream sites with significant downward change, median declines of −38% (range of −14% to −57%) and −40% (range of −9 to −65%) were observed for flow‐normalized concentration and load, respectively. Steepest rates of decline occurred from 1980 to 2000, coincident with the initiation of salinity‐control efforts in the 1980s. However, there was a consistent slowing or reversing of downward trends after 2000 even though salinity‐control efforts continued. Significant decreases in salinity occurred as early as the 1940s at some streams, indicating that, in addition to salinity‐control projects, changes in land cover, land use, and/or climate substantially affect salinity transport in the UCRB. Observed dissolved‐solids trends are likely the result of changes to watershed‐related processes, not due to changes in the streamflow regime.

Spatial and temporal patterns in low streamflows were investigated for 183 streamgages located in the Chesapeake Bay Watershed for the period 1939–2013. Metrics that represent different aspects of the frequency and magnitude of low streamflows were examined for trends: (1) the annual time series of seven‐day average minimum streamflow, (2) the scaled average deficit at or below the 2% mean daily streamflow value relative to a base period between 1939 and 1970, and (3) the annual number of days below the 2% threshold. Trends in these statistics showed spatial cohesion, with increasing low streamflow volume at streamgages located in the northern uplands of the Chesapeake Bay Watershed and decreasing low streamflow volume at streamgages in the southern part of the watershed. For a small subset of streamgages (12%), conflicting trend patterns were observed between the seven‐day average minimum streamflow and the below‐threshold time series and these appear to be related to upstream diversions or the influence of reservoir‐influenced streamflows in their contributing watersheds. Using multivariate classification techniques, mean annual precipitation and fraction of precipitation falling as snow appear to be broad controls of increasing and decreasing low‐flow trends. Further investigation of seasonal precipitation patterns shows summer rainfall patterns, driven by the Atlantic Multidecadal Oscillation, as the main driver of low streamflows in the Chesapeake Bay Watershed.

This text began as a collection of class notes for a course on applied statistical methods for hydrologists taught at the U.S. Geological Survey (USGS) National Training Center. Course material was formalized and organized into a textbook, first published in 1992 by Elsevier as part of their Studies in Environmental Science series. In 2002, the work was made available online as a USGS report.The text has now been updated as a USGS Techniques and Methods Report. It is intended to be a text in applied statistics for hydrology, environmental science, environmental engineering, geology, or biology that addresses distinctive features of environmental data. For example, water resources data tend to have many variables with a lower bound of zero, tend to be more skewed than data from many other disciplines, commonly contain censored data (less than values), and assumptions that the data are normally distributed are not appropriate. Computer-intensive methods (bootstrapping and permutation tests) now improve upon and replace the dependence on t-intervals, t-tests, and analysis of variance. A new chapter on sampling design addresses questions such as “How many observations do I need?” The chapter also presents distribution-free methods to help plan sampling efforts. The trends chapter has been updated to include the WRTDS (Weighted Regressions on Time, Discharge, and Season) method for analysis of water-quality data. This new version contains updated graphics and updated guidance on the use of statistical techniques. The text utilizes R, a programming language and open-source software environment, for all exercises and most graphics, and the R code used to generate figures and examples is provided for download.

Accurate quantification of riverine water‐quality concentration and flux is challenging because monitoring programs typically collect concentration data at lower frequencies than discharge data. Statistical methods are often used to estimate concentration and flux on days without observations. One recently developed approach is the Weighted Regressions on Time, Discharge, and Season (WRTDS), which has been shown to provide among the most accurate estimates compared to other common methods. The main objective of this work was to improve WRTDS estimation by accounting for the autocorrelation structure of model residuals using the first‐order autoregressive model (AR1). This modified approach, called WRTDS‐Kalman Filter (WRTDS‐K), was compared with WRTDS for six constituents including nitrate‐plus‐nitrite (NO_x), total phosphorus, total Kjeldahl nitrogen, soluble reactive phosphorus, suspended sediment, and chloride. Near‐daily concentration records at nine sites were used to generate subsets through Monte Carlo sampling for five different sampling scenarios. Results show that WRTDS‐K provided generally better daily estimates of concentration and flux than WRTDS under these sampling scenarios for all constituents, especially NO_x. The degree of improvement is strongly affected by the underlying sampling scenario, with WRTDS‐K gaining more advantage when more samples are available, and hence more residuals can be exploited. The performance of WRTDS‐K depends on the AR1 coefficient (ρ) and that relationship varies with constituents and sampling scenarios. These results provided recommendations on the optimal ρ for each constituent and sampling scenario. Overall, WRTDS‐K has the potential for broad applications to monitoring records elsewhere, as demonstrated by a pilot application to Chesapeake Bay tributaries.