Reservoir Operators React to Uncertainty in Snowmelt Runoff Forecasts

Mountain snowmelt is the primary water supply for almost 2 billion people (Mankin et al. 2015). This crucial water supply varies from year to year and from place to place, which adds uncertainty to decisions about water resource management. Through years of experience, water managers know that forecasts contain errors (Rayner et al. 2005; Whateley et al. 2015). In theory, forecasts add value, but this is realized only if water managers respond to them (Brown et al. 2015; Gong et al. 2010; Krzysztofowicz 1986; Pagano et al. 2001; Rayner et al. 2005; Turner et al. 2020; Whateley et al. 2015). A pressing question arises: How does the uncertainty of forecasts affect water management? We empirically examined how water managers respond to forecast uncertainty by analyzing a historical record of uncertainty in operational water-supply forecasts and the corresponding water management decisions. The goal was to determine whether forecast uncertainty affects water releases. Holding the expected forecast constant, does a more uncertain forecast cause managers to release more, or less? Or does the degree of forecast uncertainty have no effect? We used a fixed-effects panel model in a case study of basins in the Sierra Nevada to determine the association between forecasts of April–July runoff, their uncertainty, and volumes released from the reservoirs.

California’s Sierra Nevada has a historical record of publicly available data to assess the relation between operational forecast skill and water management decisions. The individual basins are diverse, with a mix of geologic, hydrologic, and water management characteristics (Fig. 1; Table 1). Water management decisions matter every year, because the median April–July flows from seasonal snowmelt are similar in magnitude to the total storage capacity of the reservoirs in the Sierra Nevada (Rittger et al. 2016), and California experiences larger interannual fluctuations in precipitation than any other US state (Dettinger et al. 2011). The history of forecasting water supplies originating as Sierra Nevada snow stretches back a century, and these forecasts from Bulletin 120 (B120) (California Cooperative Snow Survey 1930–2021) embody baseline information that is available to water managers for making informed decisions on storage and releases from reservoirs in preparation for the upcoming summer. Pagano et al. (2004) and Harrison and Bales (2016) analyzed their accuracy and showed that forecast skill increases as the season progresses. However, the April 1 B120 forecasts of April–July flows show forecasts exceeding runoff by 50% or more in 1 year in 10 and by 100% or more in 1 year in 50 (Fig. 2).

Reservoir managers in the Sierra Nevada rely on a combination of operational forecasts, predetermined allocations of water, and rules for managing flood risk to guide operations and determine water releases. The B120 forecast influences the decisions to increase or decrease the annual allocations of water within California’s major water systems (CADWR 1963–2021; USBR 1992–2021), regulate environmental releases (State Water Resources Control Board 1978, 1999), and provide guidance for flood protection. Within the constraints of these predetermined allocations of water and winter flood pool storage, water managers have the freedom to make subjective decisions about when to release water and how much to release, creating the possibility that forecast uncertainty could affect water releases.

California has a network of field measurements and a record of all operational forecasts. We assembled a panel data set of $34\mathrm{years}$ (1985–2019) of water data from the 14 high-elevation basins (Stillinger 2020). Fig. 3 illustrates the data in the panel data model for the Tuolumne River in 2011. We analyzed the B120 April–July runoff forecast, which included a confidence range (90% and 10% exceedance probabilities). These forecasts are based on regression analyses from measured snow water equivalent, rain, and streamflow. We used the April 1 forecasts because they are the final forecasts available before the start of the forecasted period, so they represent the information available to water managers at the start of spring runoff. The median (50% probability) forecast is used as the forecast ($F$), and the uncertainty in the forecast ($U$) is calculated as the 10% exceedance forecast minus the 90% exceedance forecast. This measure represents the level of uncertainty confronting water managers at the time of the forecast and varies from year to year, as seen in the individual box plots for each basin’s uncertainty range [Fig. 4(b)]. Basin releases ($R$) are defined as the measured monthly flow below the terminal reservoir for the 14 basins in Fig. 1. Each year there are variable amounts of available space to store runoff in relation to how much runoff is expected in the forecasted period. Because overall storage capacity of the basins about equals median annual full natural flow (FNF), i.e., measured flow adjusted for human actions that alter unimpaired flow, most water that flows out of the basins results from water management decisions. Fig. 4 shows the variability in the relationships between basin releases from April through July and FNF, and between available space to store runoff and the forecast. In the panel data model, all reservoirs in a basin are treated as a single large reservoir because we were interested in the total available reservoir space to store incoming flows above the basin release point, i.e., the basin-level response of the anthropogenic modifications of the watershed (Steinschneider et al. 2013). Measured reservoir storage data from March 31 each year were available for 74 reservoirs within the 14 basins. Available space to store runoff ($S$) is the measured storage volume subtracted from the total basin storage capacity—the summed volume of each basin’s reservoir storage capacity, not adjusted for the dead pool. These values represent the starting condition for water managers when they receive the April 1 forecast and began active management of the spring runoff. Individual reservoir information is in Table S1.

A fixed-effects panel regression model was applied to the 14 Sierra Nevada watersheds described previously. This model allowed us to estimate empirically how forecasted flow, uncertainty in forecasted flow, and other variables were associated with releases, while allowing for fixed effects at the basin level that control for unobserved differences across basins that do not change year-to-year, for example, geomorphology or minimum environmental flow requirements (Steinschneider et al. 2013). The model ensured that standard errors were robust to heteroskedasticity and adjusted for 14 basins (Álvarez et al. 2017). The unit of analysis was the basin, with annual time steps from 1985 to 2019; too few reservoirs had available data prior to 1985 to run the model. Some data were missing, so the model was run as an unbalanced panel with $N=416$ and 398 degrees of freedom. The form of the model iswhere $R_{it}$ = release of water from basin $i$ in year $t$, which we take to be the manager’s decision; $F_{it}$ = forecasted flow; $S_{it}$ = available reservoir space to store runoff the day before the forecast arrives; $U_{it}$ = uncertainty in forecast; and $F_{it}{U}_{it}$ = interaction of forecast with its uncertainty. This interaction term allows for the possibility that a manager reacted differently to a forecast depending on both uncertainty and magnitude. The fixed effect ${\mu }_i$ accounts for basin-specific unobservable variables that do not change over time. The error term ${\epsilon }_{it}$ has a mean of zero and no autocorrelation. This setup enables modeling the relationship between water supply management and spring runoff forecasts by evaluating the associations of forecasts with basin releases and forecast uncertainty.

Our main result was that, all else being equal, basins with larger forecast uncertainty on April 1 released less water, and this effect of uncertainty on releases decreased as the forecast volume increased. The coefficients for the panel data model with fixed effects estimated the marginal change in basin releases from a unit change in each of the predictors, controlling for the heterogeneity between groups (Table 2). By analyzing associations at the basin scale, the preferred model captured 70% of the variability in the outcome of April–July basin releases (adjusted $R^2=0.70$, Wald $F$-test $p\ll 0.001$). The April 1 forecast, forecast range, available storage capacity, and the interaction between forecasted volume and uncertainty all were statistically significant predictors ($p<0.05$) of basin releases. The sign of each estimated coefficient was consistent with first principles. Greater forecast uncertainty and greater available storage both were negatively associated with April–July volumes released from basins, whereas forecast volume and the interaction between forecast uncertainty and forecast volume both were positively associated with release volume. The estimated interaction term, ${\gamma }_1$, was positive, which requires interpretation. Its sign implies that the effect of uncertainty on releases also depends on the magnitude of the forecast. The fact that ${\gamma }_1>0$ implies that when higher flows are forecast, the net effect of uncertainty is weakened (the marginal effect of uncertainty on releases is ${\beta }_2+{\gamma }_1{F}_{it}$, where ${\beta }_2<0$). When forecasted flows are high enough, the influence of uncertainty on releases disappears, above about $5{\mathrm{km}}^3$. The Wald $F$-test ($p<0.001$) rejects the null hypothesis that our model coefficients are not statistically significantly different from zero. To test for nonstationary in basinwide water management, a dummy variable split the data into two periods; this variable was insignificant, and was removed from the final model. Table S2 lists the individual fixed effects, ${\mu }_i$ [Eq. (1)], which characterize the unobserved time-invariant variables specific to each basin. The fixed effects account for heterogeneity that does not vary with time, and therefore can be interpreted as baseline releases for each basin. Our fixed-effects panel model does not explain variability not captured by our predictors.

Consistent with but distinct from previous literature, we showed that when available reservoir storage capacity is similar to annual flow volumes, releases are sensitive to uncertainty in runoff forecasts and that uncertainty plays the largest role when forecast volume is low. Holding the forecast volume fixed, water managers act adversely to risk: facing a more uncertain forecast, they release less water, hedging against the possibility of lower than expected inflows. When the forecast is more precise, water managers have higher confidence that additional inflow will refill reservoirs, so they release more water earlier in the year. By including an interaction in the analysis between forecast volume and forecast uncertainty, we found that the effect of uncertainty on releases is mediated as the forecast volume increases. Hedging one’s bets becomes less important, but only rarely do flow volumes eliminate the influence of uncertainty on reservoir releases.

Models predict that forecasts return the most value to water managers when reservoir storage capacity is between 25% and 100% of the mean annual flow of the river, and that managers will not respond to improved forecasts if storage capacity is significantly greater or dramatically less than the mean annual flow (Barnard 1989; Ødegård et al. 2019; Zhao et al. 2012). Our results empirically confirm this water manger behavior. We presented empirical evidence of the behavioral responses to uncertainty across basins with diverse operations—hydropower, water supply for cities and agriculture, flood control, and environmental flows—extending prior estimates that were limited to forecast skill (Anghileri et al. 2016) or hydropower operations (Rheinheimer et al. 2016).

There is room to improve communication of forecasts and their uncertainty to water managers (Ramos et al. 2010). A good place to start would be clear communication distinguishing uncertainty in the measurement of accumulated precipitation from uncertainty in expected future precipitation. Knowing where the certainty lies within the forecast may give water mangers more confidence in making a decision that would be too risky if the uncertainty were distributed differently (Shukla and Lettenmaier 2011). In watersheds dominated by a wet winter and a dry melt season, streamflow forecasts become a measurement challenge (Pagano et al. 2004). Accurate precipitation measurement in the mountains is difficult; we can measure the snow water equivalent, thereby characterizing precipitation that already has fallen, but in rain-dominated basins, precipitation uncertainty is harder to constrain (Lundquist et al. 2019). In a warmer future, we would expect that some precipitation that now falls as snow will fall as rain instead. New available measurements of the spatial and temporal variability of hydrological conditions could reduce measurement uncertainty in forecasts. Examples include distributed sensor networks (Zhang et al. 2017) and remotely sensed data (Painter et al. 2016). The contribution of rain versus snow to the hydrology in each basin will play a role in the effectiveness of additional observations in reducing forecast uncertainty. The least-expensive uncertainty reduction in forecasts probably is integration of remotely sensed data in operational forecasts of snow-dominated basins.

Results suggest that, all else being equal, basins with larger forecast uncertainty release less water from April through July, the historical period of snowmelt runoff. Narrowing the uncertainty of the forecast, independent of improving the forecast itself, could impact water operations. The diversity of water users and hydrology among the basins in this study lends confidence to the applicability of the results to other gauged basins that are important to water supply. Reducing uncertainty in runoff forecasts is a tractable objective that can drive forecast improvement, would not require reducing forecast error, and could lead to changes in amount and timing of water released from reservoirs.

All data used in our analysis are available with no restrictions from the California Data Exchange Center (CDEC) (https://cdec.water.ca.gov/), and the forecast data are in the monthly issues of B120 (California Cooperative Snow Survey 1930–2021). Because the data in our analysis were scattered throughout those sources, we consolidated the data sets needed to reproduce our reported findings in a publicly accessible repository (Stillinger 2020).

This work was supported by the University of California award LFR-18-548316.