Predicting "D" with Limited Stream Temperature Data

The model's error is naturally very sensitive to errors in estimating the dampening factor "D", but this is expected to be only a linear relationship. The graph below shows the mean absolute error as a function of the departure from the best "D". All 96 station-years are evaluated for each departure. The departure from linear at low estimates of "D" is because the predicted temperature is limited at 0.0°C. Estimates of "D" that are off by more than about 0.1 will seriously impact the error of the model. However, there is every reason to believe that "D" is easy to estimate, at least with midsummer stream temperature data.

Even with limited water temperature data, "D" can be estimated as the sum of the mean water temperature over the sum of the air temperatures for the same days, with both sets of temperatures demeaned for the mean air temperature. The table below evaluates the expected results of estimating "D" with 1 to 31 days of consecutive mean stream temperature measurements. Only days within the period from July 15 through August 15 were used. All possible combinations of the existing data were evaluated in each case. These results indicate that "D" is easily estimated with limited mean stream temperature measurements. An estimate based on one day has an 87% change of being a good estimate, but could be pretty bad. An estimate based on only 8 consecutive days is guaranteed to be a good estimate.

N is the number of consecutive days used to estimate "D",
SS is the sample size and necessarily decreases with longer time series,
PCT is the percent of the results within 0.1 of the known best value,
MEAN is the mean absolute departure from the known best "D", and
MAX is the maximum absolute departure from the known best "D".

NSSPCTMEANMAX
1 2972 87.5841 0.0492 0.7107
2 2876 93.5327 0.0385 0.3463
3 2780 97.0504 0.0308 0.2035
4 2684 98.8823 0.0259 0.1698
5 2588 99.5750 0.0230 0.1339
6 2492 99.9197 0.0211 0.1085
7 2396 99.9583 0.0204 0.1036
8 2300 100.0000 0.0199 0.0919
9 2204 100.0000 0.0191 0.0795
10 2108 100.0000 0.0187 0.0776
11 2012 100.0000 0.0182 0.0787
12 1916 100.0000 0.0176 0.0707
13 1820 100.0000 0.0171 0.0669
14 1724 100.0000 0.0168 0.0591
15 1628 100.0000 0.0162 0.0512
16 1532 100.0000 0.0154 0.0460
17 1436 100.0000 0.0145 0.0430
18 1340 100.0000 0.0139 0.0376
19 1244 100.0000 0.0135 0.0376
20 1148 100.0000 0.0130 0.0377
21 1052 100.0000 0.0124 0.0357
22 956 100.0000 0.0119 0.0328
23 860 100.0000 0.0117 0.0316
24 764 100.0000 0.0113 0.0326
25 668 100.0000 0.0110 0.0329
26 572 100.0000 0.0106 0.0315
27 476 100.0000 0.0097 0.0318
28 380 100.0000 0.0088 0.0305
29 284 100.0000 0.0081 0.0287
30 188 100.0000 0.0071 0.0273
31 94 100.0000 0.0062 0.0245

01 NOV 2001, updated on 05 NOV 2001
dlg@rapid.msu.montana.edu