Uncertainties of Water Budget Components

The development of any water budget involves uncertainty and associated errors. The water-budget residual value is the mathematical difference between all inflow and outflow components, and includes errors associated with those components, as well as unquantified water-budget components such as groundwater fluxes between the lake and bank storage. Uncertainties are expressed as error percentages for components that were measured or estimated, and these uncertainties may refer back to error percentages and methods previously described for Coeur d’Alene Lake (Woods and Beckwith, 1997) and elsewhere (Winter, 1981; Brown, 1987).

Surface-water inflows for gaged tributaries represent most of the total contributing drainage basin of Coeur d’Alene Lake (92 percent); most USGS streamflow records were rated as good, but a few were rated as fair. Records that are rated as excellent, good, or fair are considered within 5, 10, and 15 percent, respectively, of the actual daily streamflow 95 percent of the time (Brennan and others, 2006). A known deficiency exists in the accuracy of the historical flows for the St. Joe River near Chatcolet (S2). Data for spring-time runoff may be underreported (low), possibly by as much as 25 percent, because some flows bypass the site during high-flow events (Molly Wood, U.S. Geological Survey, oral commun., 2008). Therefore, high flows recorded at the St. Joe River near Chatcolet site may be lower than actual flows. Considering all of the sites and reported ratings for each site and water year, an overall rating of fair (15 percent) was assigned to mean annual inflows for all gaged tributaries. Ungaged surface-water inflows were calculated using methods comparable to those used by Woods and Beckwith (1997), who constructed a water budget for water years 1991 and 1992 and assigned a 25-percent error for the ungaged component of that budget. The same percentage of error was adopted for this study.

Surface-water outflows were calculated using the stream-gaging station on the Spokane River at Post Falls Dam (S7), which was rated as good for water years 2000–2005; therefore, the surface-water outflow component of the water budget was assigned a 10 percent error estimate. Similarly, data at the Coeur d’Alene Lake station (L7), which is used to determine changes in lake storage above 2,120 ft elevation, were reported as good for water years 2000–2005; therefore, a 10 percent error was assigned to the net change in lake storage component of the water budgets.

The precipitation component of the water budgets is the quantity of water that fell directly on the lake surface. The NWS weather stations are close to the lake. Errors associated with precipitation data that are averaged over long periods are smaller than those averaged over short periods (Winter, 1981). Precipitation data (table 1) for the mean annual water budget were calculated using precipitation data for the period of record for all weather stations and the total surface area of the lake. A 15 percent error is assigned for the precipitation component of the water budget. Evaporation estimates derived from an energy budget method for longer periods (annual basis) are more accurate than other methods for shorter periods (Winter, 1981). For this study water temperature and wind speed data were available from USGS pelagic stations on Coeur d’Alene Lake (L1 – L6). Also, the evaporation calculations were based on mean monthly, rather than mean daily, time steps as outlined in Allen and Tasumi (2005). Therefore, the evaporation component of the water budgets was assigned a 10 percent error.

The water-use withdrawals and subsequent consumptive use estimates have the least amount of quantifiable information. Little research and data are available for the consumptive use component, and consumptive use estimates are based solely on total withdrawal estimates for which few measured data are available. A 25 percent error was assigned to the consumptive use estimate. Similarly, the seepage losses from Coeur d’Alene Lake and the Spokane River to the SVRP aquifer were difficult to ascertain from historical records. A 15 percent error was assigned to the seepage loss estimates.

The residual value incorporates all of the water-budget errors and unquantified values for aspects of the hydrologic cycle not included in this study. For the study period, the residual value was always negative, and the percentages of the residual value as compared to total inflow or total outflow were within the range of errors for many of the budget components. The standard deviation of the residual value error is equal to the square root of the sum of the standard deviation of individual components’ errors, squared, expressed as:

(8)

The uncertainty in the water budget components, as expressed as the total standard deviation error for the budget components, was 46,400 million ft³, compared to the net residual value term of ‑8,310 million ft³. Uncertainty in the water budget components is five times as large as the net residual value.