Use of Regression Models


To demonstrate their use as a tool for estimating the rate at which water moves through the Delta, the regression models were used to calculate total and partial replacement rates at two different flow and wind conditions that are roughly representative of spring and summer conditions (figs. 16 and 17). Spring conditions were represented by a Williamson River inflow of 60 m³/s and strong northwesterly winds as measured on May 22, 2008. Summer conditions were represented by a Williamson River inflow of 13 m³/s and weak winds as measured on August 10, 2008. The average May lake elevation for the period of record from water year 1975 to 2010 (station 1150700, U.S. Geological Survey, 2011) ranged from 4,141.2 to 4,143.2 ft. Over that range, the total replacement rate of Tulana water under typical spring conditions could be expected to range from 82 to 105 m³/s, and the partial replacement rate of Tulana water with Williamson River water from 14 to 27 m³/s (fig. 16). Thus, between approximately 17 and 26 percent of the water that passes through Tulana comes from the Williamson River, the rest coming from Upper Klamath and Agency Lakes through (and over, at the highest elevations) the levees on the north and south side of Tulana. Similarly, the total replacement rate of Goose Bay water ranges from 32 to 48 m³/s, and the partial replacement rate of Goose Bay water with Williamson River water ranges from 25 to 38 m³/s, meaning that approximately 78 percent of the water that passes through Goose Bay comes from the Williamson River under typical spring conditions, the rest coming from Upper Klamath Lake over the levees on the south side of Goose Bay, and also from Tulana across the Williamson River channel. Of the water that leaves the Williamson River channel before it enters Upper Klamath Lake, between 64 percent (at a lake elevation of 4,141.2 ft) and 58 percent (at a lake elevation of 4,143.2 ft) can be expected to flow through the Goose Bay side of the Delta under conditions representative of spring. 


During spring, when larval suckers drift down the Williamson River, it is of interest to understand whether they are transported completely passively, where they are likely to go, and how long they are likely to spend on either side of the Delta. Because of the inverse relation between volume replacement rate and the theoretical replacement time (figs. 6 and 7), the time it takes for a parcel of water from the Williamson River to pass through Goose Bay is likely to be much less than it takes for a parcel of water from the Williamson River to pass through Tulana; the theoretical replacement time in Goose Bay under typical spring conditions ranges from 1.7 to 2.2 days between a lake elevation of 4,141.2 and 4,143.2 ft (fig. 16C). 


During summer, management questions are more likely to concern water quality. For some period of time, the restored lands of the Delta can be expected to be a source of nutrients to the water column. As vegetation matures, that relation may reverse and the wetlands might remove nutrients from the water column. In either case, the rate at which water flows over the landscape, in combination with the magnitude of the nutrient fluxes to or from Delta soils, determines the concentration of nutrients in the water as it leaves the Delta and enters Upper Klamath Lake. A first-order approximation to the concentration of a nutrient of interest that would be added to water flowing over the Delta could be obtained by making the simplifying assumption that either side of the Delta acts as a continuously stirred tank reactor, in which case the steady-state contribution of the Delta soils to the water-column concentration would be K / RR, where RR is the total replacement rate, and K is the rate at which the mass of nutrient is added, summed over the entire surface area involved. For example, Kuwabara and others (2010) measured summer benthic fluxes of orthophosphate from Tulana soils in July 2009, and estimated the load to the water column at 66 kg/d. At that rate, and under the assumed summer conditions, the steady-state contribution of Delta soils to the concentration of orthophosphate in the waters of Tulana is estimated to range from 0.088 mg/L at an elevation of 4,140.5 ft to 0.023 mg/L at an elevation of 4,142.4 ft, the upper end of the range in average August lake elevation for the period of record (fig. 17C). The calculated concentration of orthophosphate is higher in Goose Bay than Tulana because the total replacement rate is lower (fig. 17A), and ranges from 0.79 mg/L at an elevation of 4,140.6 ft to 0.044 mg/L at an elevation of 4,142.4 ft. The regression equations provide estimates of concentration at lower lake elevations, but those estimates become imprecise at the low replacement rates associated with the lowest lake elevations (fig. 17C), indicating that the range of validity of the regression model is more limited than shown in table 3 under the assumed conditions. Additionally, the regressions for partial replacement in Goose Bay and Tulana produce negative values below a lake elevation of about 4,140.8 ft, further showing that the range of validity in the regression models can be more limited than shown in table 3 under some combinations of assumed conditions.


The limitations of this work are substantial, and because the regressions cannot be validated with observations, the accuracy cannot be definitively assessed; the results should be interpreted accordingly. First, the regressions are based on 1-day simulations of numerical tracers. The 1-day timeframe provided a good correspondence between the results, as measured by total or partial replacement of water in the Delta, and the observed winds, which, while having generalized seasonal characteristics of being stronger in the spring and autumn and weaker in the summer, vary from day to day. Therefore, calculating replacement over a longer time frame integrates the results of consecutive days when conditions might have been quite different. Extrapolating these results beyond a day is difficult, and could only be done in a probabilistic sense by using a distribution of wind characteristics. Although this is possible, at some point the advantage of using a simple estimation technique is lost and it is wiser to move to using a spatially explicit model to get a more accurate result. Second, the range of validity, particularly in lake elevation, is limited and more exploration could be done to determine at what lake elevation the exchange with Goose Bay becomes severely curtailed, which occurs between an elevation of 4,140.5 and 4,141.5 ft. Third, the range of wind conditions considered was limited to winds with a westerly component. While these are the most common winds over Upper Klamath Lake, winds do occasionally reverse and come from the south to southeast. These winds are likely to move water from the Williamson River preferentially into Tulana instead of Goose Bay, and the regression equations have not been tested and would likely fail under these conditions. Finally, as the Delta vegetation matures, one can expect significant changes in how water moves through the Delta, based on the depth of water and the type of vegetation that thrives in that depth. 


Nonetheless, the results presented here are useful in that they provide insight into how movement of water through the Delta responds to hydrology as defined by the lake elevation and Williamson River inflow, and to wind forcing conditions at the lake surface. The regression equations provide a quick and easy-to-use means of making rough estimates of how fast water moves through both sides of the Delta and how the water entering the Delta at the Williamson River might be partitioned between Goose Bay and Tulana under varying conditions. These estimates can inform management decisions aimed at restoring fisheries, as they influence the extent to which the Delta areas are utilized by fish larvae in the spring. As demonstrated, the regression equations can also be used to estimate the steady-state water column concentrations that would result for assumed, uniform flux rates of water quality constituents in the Delta. 


First posted March 29, 2012