WRIR 01-4195:
Ground-Water Discharge Determined from Estimates of Evapotranspiration,
Death Valley Regional Flow System, Nevada and California
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By Gaius J. Roemer, GeoTrans
The objective of this analysis is to quantify the uncertainty associated with estimates of annual ground-water discharge from the nine discharge areas addressed in the main body of this report (table 2). Because discharge estimates are expected to be used as targets for calibrating ground-water flow models, it was considered beneficial to quantify, at least in a general sense, the uncertainty associated with these estimates. The results of this evaluation can be used to better understand the uncertainty associated with each discharge estimate and to appropriately weight estimates to best constrain model results. For this effort, uncertainty is evaluated through Monte Carlo simulations performed using Crystal Ball (Decisioneering, 1996, Crystal Ball Version 4.0), a Microsoft Excel add-in. The input parameters required for Monte Carlo analysis include an estimate of the annual precipitation rate for each discharge area, and estimates of the acreage and ET rate for each of the ET units within a discharge area. Although 10 ET units are identified throughout the study area (table 1), not all are present within each discharge area. In this analysis, a total of 141 input parameters were used to evaluate the uncertainty in estimates of discharge: 61 acreages, 61 ET rates, and 9 precipitation rates (tables 5 and 7).
For this basic analysis, each input parameter is assumed to be characterized by a normal distribution centered about a mean value. The mean of each input parameter is the value of the parameter as estimated in tables 5 and 7. The spread about the mean is described by the coefficient of variability (CV), which is defined as the standard deviation divided by the mean. The CV used for the acreage of each ET unit listed in table 7 is assumed to be 10 percent. A CV value of 10 percent is considered reasonable based on accuracy assessments of about 90 percent for ET-unit classifications of the Ash Meadows and Oasis Valley dischare areas (Laczniak and others, 1998; and Reiner and others, 2002, respectively). The CV for each ET rate was determined from ranges listed in table 4, and for each precipitation rate from measurements given in tables 8 and 9. CV values for the ET rate and precipitation parameters are computed on the assumption that ranges represent ±2 standard deviations of a normal population (95 percent of the measurements are contained in the range; table 11).
The general procedure used to quantify the uncertainty in estimates of ground-water discharge consists of four basic steps:
This process of randomly selecting values for each input parameter and calculating the annual discharge from each discharge area is termed a realization. A test performed to determine the number of realizations needed to produce stable estimates of the standard deviation generated frequency distributions and statistics of annual discharge from sample sizes of 500; 1,000; 2,000; and 3,000 realizations. Test results indicated that a sample size of 1,000 realizations was sufficient.
To exemplify this general procedure, Monte Carlo results of the Ash Meadows discharge area (table 12; figs. 17 and 18A) are discussed in some detail. Table 12 gives the mean acreage, ET rate and precipitation rate calculated from 1,000 realizations. The table also gives the adjusted ET rate (calculated by subtracting the mean precipitation rate from the mean ET rate), annual ET, and annual ground-water discharge computed for each ET unit and the totals. Summary tables for the eight other discharge areas are given in tables 13, 14, 15, 16, 17, 18, 19, and 20.
The eight input parameters to which annual ground-water discharge in Ash Meadows is most sensitive are shown in figure 17. The sensitivity of each parameter is measured by rank correlation (correlation based on ranks rather than on values). Because ET units 6 and 5, respectively, are the largest contributors to ground-water discharge at Ash Meadows, their parameters have the greatest effect on the estimate. The sensitivity of the precipitation rate always is negative because it is subtracted from the ET rate to calculate ground-water discharge.
The five input parameters from each of the nine discharge areas having the greatest effect on annual ground-water discharge are shown in figure 17. Because the precipitation rate is an essential component in calculating ground-water discharge from every ET unit, it always is one of the more sensitive input parameters. Typically the precipitation rate and the ET rate associated with the largest ET unit are the two most sensitive parameters. The lone exception is Oasis Valley, where ET units 4 and 6 have the largest acreage but the ET rate associated with ET unit 8 is the most sensitive parameter. This anomaly can be explained in part by (1) the low CV of the ET rate for ET unit 4 (0.07) relative to that of ET units 6 and 8 (0.29 and 0.28, respectively; table 11), and (2) the high ET rate of ET unit 8 relative to ET unit 6 (1.92 and 1.19 ft/yr, respectively, table 16).
Descriptive statistics and a frequency chart generated from 1,000 realizations of annual ground-water discharge for Ash Meadows are listed in table 21 and shown in figure 18A. The mean, standard deviation, and the coefficient of variability (CV) of the simulated ground-water discharge for all nine discharge areas are compared in table 22. Assuming that CV is a reasonable estimator of the relative uncertainty (larger values represent a greater uncertainty), the discharge estimates for Oasis Valley and the Tecopa/California Valley area are most certain (0.12 and 0.11, respectively) and those for Stewart Valley and Sarcobatus Flat are least certain (0.42 and 0.48, respectively). The largest CV values (those greater than 0.20) are associated with discharge areas made up of five or fewer ET units and/or with discharge areas dominated by open playa (ET unit 10). Sarcobatus Flat, which has only four ET units and 10,771 acres of open playa, has a CV of 0.48. Whereas, Oasis Valley, which has nine ET units and only one acre of open playa, has a CV of 0.12. In general, the smaller the number of ET units the greater the uncertainty. Where fewer units are present, the greater uncertainty can be explained by the fact that deviations from the mean are less likely to compensate each other. In discharge areas dominated by open playa, the CV of the discharge estimate is high (table 22) because (1) the CV of the unit itself is high (0.38, table 11), and (2) the ET rate is low and usually not much greater than the estimated precipitation rate. Because ET and precipitation rates are similar in magnitude, the CV of their difference is larger than the CV of either parameter. Figure 18B-I presents descriptive statistics and frequency charts generated from 1,000 realizations of ground-water discharge in the eight other discharge areas.
An analysis of the Ash Meadows discharge area was used to examine the uncertainty associated with the classification procedure. In this analysis, classified ET units similar in terms of their spectral response (fig. 6) were correlated using a correlation coefficient of -1.0. Differences in the standard deviations generated from correlating different combinations of three similar ET units in Ash Meadows are listed in table 22. The three correlated ET units are ET unit 3 (dense wetland vegetation), ET unit 4 (dense meadow and forested vegetation), and ET unit 5 (dense to moderately dense grassland vegetation). The small differences (less than 2 percent, table 22) in the standard deviation of the first three combinations are attributed to (1) the decreased likelihood that the sampled area will be much greater than or less than their respective means, and (2) similarities in their ET rates. The program does not allow correlation of ET units 3, 4, and 5 by a coefficient of -1.0. Instead, the program determines the maximum allowable correlation coefficient at -0.5. Because effects of correlation are shown by this example to be minimal, any further efforts to correlate ET units were considered unnecessary.
An additional analysis was performed to evaluate uncertainty related to the assumption of a 10 percent CV for ET-unit acreage. Five Monte Carlo simulations of 1,000 realizations using CV values for acreage of 10, 20, 30, 40, and 50 percent resulted in similar mean values having standard deviations that varied nearly proportionally with changes in the input CV. These results indicate that the predicted uncertainty in the estimate is nearly proportional to the CV of the acreage.
The accuracy of the uncertainty and sensitivity analyses can be improved by better quantifying the errors associated with the parameters used to compute ground-water discharge. The errors most significant to the procedure used are those errors inherent in the methods used to measure ET-unit acreage and to calculate ET rates. Of particular significance are errors associated with using spectral data from Thematic Mapper imagery to classify ET units and the Bowen ratio solution to compute ET rates. Having a more thorough and rigorous analysis of these errors would result in more realistic formulations of the distributions describing the acreage and ET rate of the individual ET units. Although these more accurate error estimates would improve estimates of uncertainty, they probably would not have a substantial effect on estimates of annual ground-water discharge.
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