comparison of models for forecasting snowmelt runoff volumes

VOL. 16, NO. 5 WATER RESOURCES BULLETIN

AMERICAN WATER RESOURCES ASSOCIATION OCTOBER 1980

COMPARISON OF MODELS FOR FORECASTING SNOWMELT RUNOFF VOLUMES

Mark E. Hawley, Richard H. McCuen, and Albert Rango2

ABSTRACT: Many new models for predicting snowmelt derived runoff have been proposed in the last few years. These new methods are generally more conceptual than the widely used regression models, and many of them require Landsat derived snow covered area data. In order to determine the relative accuracy of these models, a representative sample of the available models was tested on the Kings River watershed. The models chosen for testing are the regression model, the Tangborn model, and the Martinec model. Twenty-four years of data were as- sembled and the data base was split into a number of 12-year subsets. Each model was calibrated with five of these subsets, then tested for accuracy on the remaining data years. Results of the testing program were evaluated by comparing the correlation coefficients between predicted and observed runoff volumes; correlations ranged from less than zero to 0.997. Comparison showed that the Martinec model was most effective for forecast periods of one or two days, while the regression model worked best for forecasts of 60 days or longer. The Tangborn model was less accurate than the Martinec model for periods of ten days or less, and less accurate than the regression models for longer time periods. Accuracy of the Martinec model would probably be better on a smaller watershed, or if a routing method were incorporated. (KEY TERMS: snowmelt runoff; water yield; remote sensing; hydrologic modeling.)

INTRODUCTION Snowmelt runoff is the primary source of water supply in

many mountainous areas of the world, including the western U.S. In order to maximize the benefits of the water supply, forecasts of future volumes of runoff are required. Over the years, a variety of methods of forecasting snowmelt runoff have been developed.

The first models for forecasting snowmelt runoff were strictly empirical; linear regression equations were developed using snow water equivalent and precipitation totals as predictor variables. Recently, a number of conceptual models have been developed that rely less on statistical correlations and more on a theoretically based structure. Additionally, a number of these new models have been designed to utilize new data sources such as measurements of snow covered area(SCA) made from Landsat satellite imagery.

The primary purpose of this study is to test and compare the accuracy of a representative sampling of the available snowmelt models. In order to allow direct comparison of the

results, the models are all tested on the same watershed using the same data base. Because so many models are available, the models were separated into categories reflecting similar characteristics and a model that is characteristic of the cate- gory was included in the test.

Specific analyses are made for the following: 1) to determine the value of Landsat derived SCA data in the various runoff prediction models; 2) to investigate the value of subdividing the watershed into smaller, more homogeneous areas; 3) to determine the effect of delays in data collection, such as that which might occur in the compilation and analysis of Landsat data; 4) to evaluate the accuracy of forecasts based solely on SCA measurements; and 5) to determine the length of record required for model calibration.

MODEL SELECTION

The selection of models was subject to several important constraints. First, the group of models selected should be representative of those currently in use. Second, the models should show significant variation in levels of conceptual development. Third, the set of models should be applicable over a range of forecast periods from one day to the entire snowmelt season, with each model being applicable over a significant range of periods. Fourth, the input requirements of the models should be similar to the data usually available for forecasting. Fifth, a data base that includes the input requirements for all the models must be available for at least one watershed.

Based on these criteria, the following three models were selected: 1) the empirical regression equation; 2) a conceptual water balance model that required calibration using regression equation; and 3) a conceptual model based on an energy bud- get analysis. These three methods cover almost all levels of conceptual development and are applicable for the entire range of forecast periods.

'Paper No. 80061 of the Water Resources Bulletin. Discussions are open until June 1,1981. Respectively, Faculty Research Assistant and Professor, Department of Civil Engineering, University of Maryland, College Park, Maryland 20742; and 2

Senior Research Hydro1 p t , Laboratory for Atmospheric Sciences, Goddard Space Flight Center, Greenbelt, Maryland 20771.

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Hawley, McCuen, and Rango

The Regression Model

The most widely used type of snowmelt model is the linear multiple regression equation. While this model is based on causal relationships, it depends solely on the observed statistical correlations between the predictor variables and the volume of snowmelt runoff. Snow water equivalent, previous season runoff volume, and monthly or seasonal precipitation volumes are the most frequently used predictor variables. Linear regressions are usually developed for making seasonal forecasts but the method may also be used for shorter periods. Because this is the simplest and most widely used type of model, it is included in the set of models selected for testing.

The Water Balance Model

Water balance models are structured using a simple conceptual representation of the runoff processes. W l e they reflect causality of the processes, they involve simplifying as- sumptions to eliminate detailed data requirements; these as- sumptions also simplify the evaluation of model coefficients. The water balance is an accounting of all the water entering and leaving the basin The volume of water stored in the snowpack is estimated from precipitation or water equivalent data; allowances are made for losses due to evaporation, transpiration, and ground water storage; the remaining volume is the seasonal snowmelt runoff forecast. The loss rates are estimated conceptually in some water balance models and empirically in others.

Tangborn (1976, 1978) developed and refined a runoff model of the water balance type. The model structure was developed conceptually; the volume of water stored on the basin is estimated using the observed winter runoff, Rw, and the observed winter precipitation, Pw. The predicted runoff, R,, is related to Pw and Rw using equations of the form:

Rs = a t b P w -

in which a and b are constants that are evaluated using regression. The model may be used for any length of forecast from one day to the entire snowmelt season. The most recent form of the Tangborn model includes a test season approach to im- proving model accuracy; the test season is a method of cor- recting for seasonal bias. The Tangborn model is included in the set of models t o be tested because of its basic conceptual development and the ease with which it may be calibrated.

The Martinec Model

Short term runoff forecasts generally require models of greater complexity than those developed for seasonal forecasts. A short term forecast is based on an estimate of the amount of snow that will be melted in a given time period, rather than on an estimate of the total volume of water stored in the snowpack The rate at which the snow r,elts is determined by the amount of energy available for that purpose. Therefore, most short term forecasting techniques are based on estimates of the amount of energy available for melting the snow. Some

techniques require such data as albedo, level of incoming solar radiation, and cloud cover; because these data are not readily available, calibrating and testing these models is difficult.

Martinec (1975) developed one of the less complex conceptual snowmelt models for short term forecasts. The volume of water produced by melting snow on a given day is estab- lished by multiplying the daily temperature index (in degree- days) by the snow covered area and a degree-day factor. The daily temperature index indicates the amount of energy available for melting snow on a given day. It may be calculated using either daily high and low temperatures or hourly values provided by the National Weather Service. Temperatures may be adjusted for any difference in elevation between the snowmelt area and the temperature station by using a constant temperature lapse rate (Barry and Chorley, 1968). Whichever data are wed, all temperatures below freezing are considered to be at the freezing point; then the average number of degrees above freezing for the day is calculated. This figure is the daily temperature index, expressed in degree-days; it must always be either at or above the freezing point. The degree-day factor relates the daily temperature index to the depth of the water column produced by snowmelt on that day; it is expressed as depth of water per degree-day and may be estimated from field measurements or calibrated analytically. When the degree-day factor is multiplied by the temperature index, a depth estimate is obtained; multiplying this depth by the snow covered area gives an estimate of the volume of water produced by snowmelt.

On large mountainous watersheds, the routing time for water traveling from the source (snowpack) to the watershed outlet is often more than one day. Therefore, of the water produced by snowmelt on day n, one portion can be expected to appear as runoff on day n, while the remaining water will appear as runoff on subsequent days. A recession coefficient, K, is included in the Martinec model to account for this process. This coefficient represents the proportion of the water generated on day n that does not leave the watershed on that day; K times the generated volume is an estimate of the amount of water that becomes recession flow.

Other terms of the Martinec model are included to account for precipitation and for losses due to evaporation and ground water storage. If the daily temperature index is above freezing, the precipitation (extrapolated from base station data) is added to the generated melt water for the day; if the daily temperature index is at the below freezing point, the precipitation is assumed to be in the form of snow, which does not contribute to runoff on that day. Losses to evaporation and ground water storage are accounted for by a runoff coefficient, c, which is calibrated empirically.

The complete Martinec model has the form

Qn = c(d T SCA + P)A(l-K) + K Qn-l

in which Qn is the predicted runoff for day n, Qn-l is the runoff on the previous day, c is the runoff coefficient, d is the degree-day factor, T is the temperature index, SCA is the snow

91 5

Comparison of Models for Forecasting Snowmelt Runoff Volumes

covered area (in percent), P is the precipitation, A is the total watershed area, and K is the recession coefficient. The first part of the equation represents the amount of runoff that is generated on day n and leaves the watershed the same day; the second term represents the amount of recession flow on day n. The Martinec model was chosen for this study because (1) it may be used in the specific investigations outlined previously, (2) it is applicable to a range of forecast periods, and (3) it requires less data than most of the other conceptual models.

THE KINGS RIVER BASIN: A TEST WATERSHED

Data requirements for each model are different. While regression models can use almost any hydrologic variable as a predictor, the variables considered in this study were snow water equivalent, the total winter precipitation, snow covered area, and the product of the snow covered area and the snow water equivalent, which is a conceptual estimate of the volume of water in the snowpack. The Tangborn model requires only precipitation and runoff volumes; the required frequency of measurement depends on the length of both the forecast and the test seasons. The Martinec model requires daily values of temperature, precipitation, runoff, and snow covered area. A suitable data base for calibrating and testing all three models consists of daily values of snow covered area, temperature, precipitation, and runoff and also snow water equivalent data during the spring. A record length of at least 20 years is desirable,

A data base that includes almost all of the necessary data is available for the Kings River watershed in the Sierra Nevada mountains of California. The exception is that snow covered area data are not available on a daily basis. In the years 1952- 1973, SCA measurements were made by the U.S. Army Corps of Engineers using low altitude aircraft mapping. Since 1973, SCA data has been obtained using Landsat imagery. In both methods, only four to eight observations were made per year. The required daily SCA values were generated from the available data by interpolation.

The Kings River basin is a large watershed (1545 square miles) with elevations ranging from less than 1,000 to nearly 13,000 feet (Rango, et al., 1979). There are precipitation and temperature stations on the watershed, as well as many snow courses where snow water equivalent is measured during the spring and winter. Though there are a number of reservoirs on the watershed, the Kings River Water Association computes the daily unimpaired runoff that would result if no reservoirs or other hydrologic structures existed. The data base was as- sembled from a number of sources (Table 1).

CALIBRATION OF THE MODELS

All of the models require calibration before they can be tested for accuracy. Calibration involves determining the values for the various model parameters that result in the most accurate predictors for the calibration data.

TABLE 1. Data Sources.

Data Source

National Weather Service Daily precipitation Daily temperature National Weather Service Daily unimpeded runoff Snow covered area NASA

Snow water equivalent

Kings River Water Association

U.S. Army Corps of Engineers California Department of Water Resources

Split Sample Analysis

In order to assess the accuracy of the models for real-time prediction, the data set must be split into two subsets so that the models can be tested using data that were not used in calibration. One data subset is used in calibrating the models, while the rest of the data are reserved for testing. The data base in this study consists of 24 years of record, with con- siderable range in the runoff volumes. Because of the small sample size, the estimated accuracy of the models is likely to depend on the way in which the sample is divided.

Accuracy is a measure of the fit between the observed and predicted values of the criterion variable (McCuen, 1979). Model calibration is an attempt to eliminate model bias so that the random component of the error variance can be used to assess the reliability of a model. However, for small samples, calibration may not lead to an unbiased model. For example, if the sample is separated in such a way that the high runoff years are used for calibration and the low runoff years are used for testing, the resulting accuracy is expected to be less (i.e., the bias will be more significant) than if the sample is split by ranking the data years in order of decreasing runoff, then selecting the odd ranked years for calibration and the even ranked years for testing. In the high vs. low case, the test data lie outside the range of the calibration data, whereas in the odd vs. even case, the data subsets are much more alike; the bias would be expected to be greater for the former than for the latter case. Therefore the data set was split in five ways, and the models were tested to assess both the bias and precision components of accuracy. Each of the models was calibrated with each of the five calibration sets and tested with the corresponding test data sets. The data set was sequen- tially organized by ranking the years in order of decreasing runoff for three periods: April 1-September 30, April 1- June 30, and May 1 May 31. Then each of the three lists was divided into high vs. low and even vs. odd subsets, as shown in Table 2. Only five divisions resulted, rather than six, because splitting the April ]-September 30 list into high vs. low resulted in the same division as splitting the April 1 -June 30 list with the high vs. low arrangement (see Table 2).

Calibration of the Regression Models

A stepwise regression program was used to calibrate the regression models. The models were calibrated for forecast periods of 15, 30, 45, 60,90, 120, and 150 days starting both

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on April 1 and on May 1. Four single predictor models were developed; snow water equivalent (SWE), snow covered area (SCA), winter precipitation (P,), and the product of SWE and SCA were used as predictors. Also, four two predictor models were calibrated using the following combinations: SWE and Pw; SWE and SCA; Pw and SCA; and Pw and SCA-SWE.

TABLE 2 . Data Separation for Split Sample Testing.

Data Set No. Runoff Period Calibration Set Test Set

1 May 1-May 31 even ranked years odd ranked years 2 May I-May 31 low ranked years high ranked years 3 April 1-June 30 even ranked years odd ranked years 4 April 1-June 30 low ranked years high ranked years 5 April 1-Sept. 30 even ranked years odd ranked years

Calibration of the Tangborn Model

In addition to finding the optimum values of the parameters of the Tangborn model, it was necessary to determine both the optimum winter starting date and the optimum test season length. Tests showed that October 1 was the best date to use as the first day of the winter season and that a test season of one day provided as accurate estimates as a longer test season. However, the model did not seem to be very sensitive to either of these factors.

The purpose of a test season is to reduce or eliminate the bias that results from small sample calibration. The test season modification to Equation ( 1 ) is described by Tangborn (1978). Basically, a prediction is made of the runoff expected during a short test season; the error of the test season prediction should correlate well with the actual prediction season error because both are caused by the inaccuracy of the estimate of basin storage. Therefore, the test season error is used in the prediction season equation, resulting in the form:

Rs = a + b(Pw + Pt) - (rC, + €$) - cet ( 3 )

in which & is the predicted runoff; Pw and Pt are the winter and test season precipitation totals, respectively; Rw and Rt are the winter and test season runoff totals, respectively, et is the test season error; and a, b, and c are coefficients determined by regression.

The Tangborn model was calibrated for forecast periods of 1, 2 , 3 , 5, 10, and 15 days starting on May 1, May 15, June 1, and June 15 and also for forecast periods of 15, 30, 45, 60, 90, and 120 days starting on April 1 and May 1.

Calibration of the Martinec Model

The basic form of the Martinec model is given in Equation (2 ) . The parameters that require calibration are the runoff coefficient, the degree day factor, and the regression coefficient. Martinec (1975) suggests that the recession coefficient be defined as a function of the previous day runoff:

Q n t l = b Kn = - aQn-1

Qn (4)

The coefficients a and b are calibrated using regression. This method of estimating K results in the Martinec model:

Qn = c(dTSCA + P)A(l-aQn-t) + aQn-l @ + I ) (5)

By using this method of estimating K, a different value of K is calculated for each day of the prediction period; this should result in higher accuracy than using a constant recession coeffi- cien t .

Equation (5) contains four model parameters that must be calibrated. Calibration may be accomplished by any of three methods: analytical numerical, or subjective optimization. With subjective optimization, initial values of the parameters are selected using the tester’s knowledge of the system and then trial-and-error is used to adjust the initial values in order to improve the goodness-of-fit statistics. Analytical optimization is a more precise and reproducible method of calibration. The criterion function is defined to be the sum of squares of the prediction errors. The partial derivatives of the criterion function are evaluated with respect to each of the parameters to be optimized. After setting the partial derivatives equal to zero, the resulting simultaneous equations are solved for the values of the parameters that will minimize the value of the objective function. With numerical optimization, initial estimates of the parameter values are selected and used to make predictions; the sum of squares of the prediction errors is used as a criterion function. The parameter values are adjusted by 5 or 10 percent, one at a time, and the predictions and sum of squares are again calculated. If adjusting the value of the parameter resulted in a lower sum of squares, the initial parameter estimate is replaced with the adjusted value. Iteration continues until the optimum set of values is reached (i.e., until the criterion function is minimized).

The Martinec model was calibrated by all three methods. Results of the analytical and numerical methods were nearly identical, but the numerical method was simpler to use. There- fore, the results of the numerical optimization method were used to judge the accuracy of the Martinec model. Accuracy was assessed for forecast periods of 1 , 2 , 3 , 5 , 10, and 15 days starting on May 1 , May 15, June 1, and June 15.

ACCURACY ASSESSMENT USING THE TEST DATA

Each model was tested with each of the five test data sets corresponding to the data for which it had been calibrated. Accuracy of the resulting predictions was measured by the correlation coefficient between predicted and observed values and by the standard error of estimate. In evaluating the relative accuracy of the various models, it was necessary to compare these goodness-of-fit statistics. Because each model was tested on only 12 years of data, tests of significance were used

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to assess the significance of differences in correlation coefficients. The test used in this study is based on the Fisher R-to-Z transformation:

z = %loge (-- 1tR ) 1 -R

in which R is the sample correlation coefficient. The correlation coefficients are compared using the test statistics

(7 1

in which Z1 and Z2 are the Z values corresponding to the correlations for the two samples, respectively, R1 and R2; N1 and N2 are the number of observations in the respective samples; and z is the value of a random variable having a standard normal distribution. The test statistic value is compared with a critical value obtained from a table of the normal distribution function to determine the significance of a difference in correlation.

In order to draw conclusions from the test results, a level of significance must be chosen. A level of significance of 20 percent was chosen as reasonable for use in this study.

Effect of Data Set on Model Accuracy

As noted above, each model was calibrated and tested using five different data sets; the accuracy of the results varied with the data set. In data sets Nos. I , 3 , and 5, the sample was split into even ranked years for Calibration vs. odd ranked years for testing; in data sets Nos. 2 and 4, the split was low ranked years for calibration, high ranked years for testing (see Table 2). Thus, data sets Nos. 2 and 4 were biased compared to sets Nos. 1, 3, and 5 .

Correlation coefficients between predicted and observed values ranged from 0.3 1 1 to 0.992 for the unbiased sets, and from 0.076 to 0.966 for the biased sets in the testing program. The number of the various test situations (combination of model, input variables, date of forecast, and forecast length) is too large to allow inclusion of all the goodness-of-fit statistics in this paper; only a sample tabulation is included (see Table 3). The complete tabulation is available in Hawley (1979). Only the trends are reported here.

Generally, the results of testing with the unbiased data were significantly better than those obtained using the biased data sets. This result was expected because in using data sets Nos. 1 . 3, and 5 the test data generally were within the range of data for which the model had been calibrated. In the regression models, sets Nos. 1 , 3, and 5 were consistently more accurate for prediction periods of 90 days or more, but not for shorter periods. The Tangborn model was more accurate with sets Nos. 1 , 3, and 5 for all lengths of forecast, especially during Jme . The Msrtinec model was more accurate with

sets Nos. 1, 3, and 5 only for forecasts in June; in May, the differences in correlation were not significant.

TABLE 3. Sample Tabulation of Summary Statistics for Testing of Tangborn Models; Forecast Date: May 1.

Data Set No. Length of

Forecast (days) 1 2 3 4 5

15 R* 0.740 0.494 0.689 0.525 0.828 Se** 68.000 82.900 78.200 82.000 58.900 Sy*** 96.400 91.000 102.800 91.800 100.000

30 R Se SY

45 R Se SY

60 R Se SY

90 R Se SY

120 R Se SY

150 R Se SY

0.899 0.770 0.903 0.778 0.943 119.800 132.500 112.600 135.500 90.000 260.600 197.900 249.600 205.900 258.500

0.915 0.838 0.908 0.818 0.979 177.100 175.600 1f9.000 182.800 86.600 417.300 307.100 407.400 303.300 409.000

0.936 0.910 0.955 0.890 0.980 203.400 171.600 169.200 183.200 110.600 551.900 394.600 545.500 383.200 534.500

0.948 0.934 0.976 0.923 0.980 249.800 205.800 169.400 215.900 147.100 749.400 548.000 749.000 536.300 696.100

0.949 0,931 0.978 0.925 0.978 266.200 228.500 178.100 233.300 163.300 807.000 596.200 807.900 584.700 746.900

0.950 0.930 0.978 0.925 0.978 269.400 236.000 181,500 239.700 167.000 819.500 612.000 822.800 600.600 756.800

*R = Correlation coefficient. **Se = Standard error or estimate.

***Sy = Standard deviation of observed values.

Effect of Forecast Length on Model Accuracy

A comparison of the correlation coefficients for various forecast lengths was performed for each model The regression equations were significantly more accurate for forecasts of 60, 90, and 120 days than for 15 or 30 day periods; the same was true of the Tangborn models, but the Tangborn results for 15 day periods were better than for periods of 1 , 2, or 3 days. The Martinec model was significantly better for 1 and 2 day periods than for longer forecasts.

Differences in Accuracy of the Models

The results of this study indicate that the Martinec model is the best of those tested for forecast periods of 1 , 2, or 3 days. For longer forecast periods, the regression models generally gave the best predictions. The best regression models were formed using winter precipitation and snow water equivalent as predictors; the least accurate regressions were those using only snow covered area as a predictor variable. The Tangborn model was generally less accurate than the regression

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models for long term predictions, and less accurate than the Martinec model for short forecast periods.

Effect of Spatial Sepuration on Model Accuracy

In an effort to determine the effect of subdividing the watershed into smaller, more homogeneous areas, the Martinec model was modified to incorporate these subdivisions. The watershed was divided into six elevation zones, and precipitation, temperature, and snow covered area data were generated for each zone. Temperatures were extrapolated from base station readings using a constant lapse rate; precipitation was assumed constant over the watershed, but if the zonal temperature value was below freezing, the effective precipitation was considered to be zero. The resulting equation has the form

which is the same as Equation ( 5 ) except for the areal subdi- vision.

Testing of this model showed that accuracy of prediction was not significantly better when the area was subdivided. This may be partly due to the lack of a routing mechanism in the model; on a watershed as large as the Kings River basin, routing is an important process. The result may also be due to the difficulties in calibrating the parameters caused by model insensitivity.

Effect of Snow Covered Area Data on Model Accurac-v The inclusion of snow covered area data in the regression

models did not improve the accuracy of the forecasts in this test. This may be because the regression models were used only for forecast dates of April 1 and May 1 ; later in the season, SCA data may be more indicative of basin storage. An- other problem in using SCA was that in many of the data years observations of SCA were not available before May 1, and the April 1 values were generated by extrapolation from the later data.

The Martinec model requires SCA as input, and this was the best short term model tested. Thus, it appears that SCA is useful for short prediction periods, but for periods of more than 15 days, other variables are at least as useful.

The SCA Model. Linear regression seemed to be the most sensible structure for a prediction model using only SCA as input. The models based only on SCA generally did not per- form as well as did the models based on other variables, especially for forecasts made on May 1. The trend was less pronounced for forecasts made on April 1, possibly because of the fact that in some years the April 1 data was generated from later observations, rather than being collected on April 1 .

Importance of Real-Time Data. The Martinec model was used to evaluate the effect of delays in data reporting. The test was performed by assuming that data from the previous days was delayed for periods of one, three, and five days. The

delayed data were the daily SCA, temperature, precipitation, and runoff.

In order to forecast future runoff, the Martinec model requires predictions of daily temperature, precipitation, and SCA for the days included in forecast period. These predicted data are derived from observations of the data made over the previous few days. A delay in reporting the data collected on the previous few days may result in less accurate predictions of the required data, which in turn will result in less accurate runoff forecasts.

The results of the testing program indicate that for forecast periods of one or two days the accuracy when real-time data are available is significantly better than when the data are delayed for three days or more. For forecasts of more than three days, accuracy was not significantly affected by delays in data reporting of five days or less.

Length of Record for Calibration. Each of the models used in this study was calibrated with five different data sets; of these five sets, three were constructed so that the calibration data were representative of the test data, and the other two were constructed so that the calibration data and the test data covered entirely separate ranges of values. The correlations and standard errors of estimate for each model calibration are tabulated in Hawley (1979); a sample tabulation appears in Table 4.

TABLE 4. Sample Tabulation of Summary Statistics for Calibration of Regression Models Predictor With Variables:

Snowpack Index and October-March Precipitation.

Data Set No. Length of

Forecast (days) 1 2 3 4 5

15 R 0.835 0.670 0.731 Se 25.400 24.100 26.200 SY 41.800 30.900 34.800

30 R 0.855 0.623 0.834 Se 55.500 44.500 44.000 Sy 96.800 54.200 75.900

45 R 0.916 0.669 0.923 Se 79.900 72.200 62.100 Sy 180.200 88.000 145.400

60 R 0.956 0.841 0.970 Se 85.200 72.600 72.200 SY 263.600 121.400 269.400

90 R 0.962 0.896 0.972 Se 133.600 98.400 120.000 SY 442.600 200.500 460.300

120 R 0.959 0.896 0.966 Se 161.400 111.600 155.800 SY 516.900 227.400 544.700

150 R 0.956 0.899 0.963 Se 174.600 113.000 170.700 Sy 539.200 233.200 572.300

0.730 0.630 23.400 29.400 31.000 34.400

0.633 0.681 44.700 52.700 54.000 65.100

0.681 0.755 71.400 99.100 88.100 136.700

0.853 0.947 73.700 85.800

127.700 241.300

0.941 0.947 69.800 159.600

186.400 450.800

0.941 0.931 79.200 240.600

212.100 596.300

0.943 0.929 80.100 261.200

218.100 636.100

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Comparison of the correlations achieved by calibrating with the various data sets shows that the higher correlations resulted from using sets Nos. 1, 3, and 5, rather than sets Nos. 2 and 4. Correlation coefficients for the first group of data sets ranged from 0.342 to 0.987, while for the second group, the range was from 0.091 to 0.956. The results imply that the selection of calibration data years may be as important as the number of years. In order to further this investigation, each model was also calibrated using all 24 data years; generally, the results of these calibrations were not significantly better than the results obtained using data sets Nos. 1, 3, and 5. Therefore, 12 years of data seems to be sufficient for calibrating each model provided that the 12 years are representative of the entire range of data.

CONCLUSIONS

The following conclusions result from the comparisons

1. Accuracy of the Tangborn model and the regression models is greater if the test data fall within the range of calibration data than if the test data lie outside the range of calibration data.

2 . The regression models are significantly more accurate for forecasts of 60 days or more than for shorter prediction periods.

3. The Tangborn model is more accurate for forecasts of 90 days or more than for shorter prediction periods.

4. The Martinec model is more accurate for forecasts of one or two days than for periods of 3 , 5 , 10, or 15 days.

5. Accuracy of the long-term models seems to be indepen- dent of forecast date; exceptions are the snowpack index regression model and the snow covered area regression model, both of which are more accurate for April 1 forecasts than for May 1 forecasts.

6. The short-term models are least accurate for forecast periods in late May and early June; this coincides with the period of peak flow for most years.

7. The Martinec model is the best of those tested for one and two day forecasts.

8. With the exception of the snow covered area model, the regression models are all roughly equal in accuracy; these regressions are the most accurate of all models tested for forecasts of 60 days or more.

9. Spatial separation of the watershed by elevation zones does not improve the accuracy of the Martinec model on the Kings River basin.

10. Delays in data collection of more than one day may significantly lessen the accuracy of the Martinec model; real-time data is desirable.

11. The regression model using only snow covered area as input data is not as accurate as the other regression models for May 1 forecasts.

12. The sufficiency of a calibration data base is a function of both the number of years of record and the accuracy with which the calibration years represent the total population of

discussed above:

data years. Twelve years appears to be a sufficient length of record for each of the models considered here, as long as the twelve years are representative of the population.

ACKNOWLEDGMENTS

This study was performed under Grant No. NGR 21-002-399 from NASA, Goddard Space Flight Center, Laboratory for Atmospheric Sciences. The authors would like to thank Dr. Jack Hannaford of Sierra Hydrotech, Ms. Kay Krouse of the National Weather Service, and the Kings River Water Association for their help in assembling the data base.

LITERATURE CITED

Barry, R. G. and R. J. Chorley, 1968. Atmosphere, Weather, and Cli- mate. Methuen & Co., Ltd., London, England, pp. 95-96.

Hawley, M. E., 1979. A Comparative Evaluation of Snowmelt Models. Thesis in partial fulfillment of the requirements for the Master of Science degree, Department of Civil Engineering, University of Maryland, College Park, Maryland.

Martinec, J., 1975. Snowmelt - Runoff Model for Stream Flow Fore- casts. Nordic Hydrology 6:145-154.

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comparison of models for forecasting snowmelt runoff volumes

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