Optimization of the Critical Temperature by pharmphresh33

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```									STREAMFLOW SIMULATING USING SRM

by: Jennifer Nemitz
Final Project for:
FOR 520 – Snow Hydrology
Fall 2002

1.0 INTRODUCTION                                          1

1.1 The Snowmelt Runoff Model                        1
1.2 Basin Description                                3

2.0 METHODS                                               5

3.0 RESULTS AND DISCUSSION OF TRIAL VARIATIONS
OF EACH MODEL PARAMETER                             6

3.1 Optimization of the Critical Temperature          7
3.2 Optimization of the Lag Time to Gage              9
3.3 Optimization of the Degree-Day Factor            10
3.4 Optimization of the Runoff Coefficients          12
3.4.1 Snow Runoff Coefficient                 12
3.4.2 Rain Runoff Coefficient                 13
3.5 Optimization of the Rainfall Contributing Area   13

4.0 RESULTS AND DISCUSSION OF THE BEST FIT
MODEL PARAMETERS                                    15

5.0 CONCLUSIONS ABOUT THE USEFULNESS OF SRM               17

6.0 REFERENCES                                            20

APPENDIX                                                  21
1

1.0 INTRODUCTION

The purpose of this exercise is to become familiar with the hydrologic simulation of

streamflow from the seasonal melting of a snowpack for mountainous basins. The

Snowmelt Runoff Model (SRM) was used to simulate streamflow for the Upper Yakima

River Basin for the 1989 melt season. The runoff volume for the entire basin was

calculated using SRM and compared to the measured runoff volume for a 5-month period

running from late February 1989 until early June 1989.    The importance of the critical

mean daily air temperature, the lag time to gage, the degree-day melt factor, the snow

runoff coefficient, the rain runoff coefficient, and the rainfall contributing area was

demonstrated for the simulation of streamflow. Each parameter was varied individually

above and below optimum values. The impact of variation of each parameter was shown

both graphically and by computing correlation coefficients and mean runoff volume

differences between computed and measured runoff.          Based upon the individual

parameter variations, a best-fit set of parameters was chosen to achieve a final runoff

simulation.

1.1 The Snowmelt Runoff Model

Martinec (1975) originally developed the Snowmelt Runoff Model (SRM) for small

European basins. With the advent of satellite remote sensing technology for the aerial

estimation of snow covered area, SRM has been applied to ever-larger basins. The model

has been applied to over 80 basins of varying sized in 25 different countries (De Walle,

2002). The largest basin that SRM has been applied to is the Ganges River Basin at
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917,000 km3 (Seidel et al., 2000). SRM was also tested and accepted by the World

Meteorological Organization (WMO) in relation to runoff simulations (WMO, 1986) and

to partially simulated conditions of real time runoff forecast (WMO, 1992).

SRM computes the daily discharge for a basin by superimposing computed snowmelt

and rainfall water production onto the calculated recession flow according to the

following equation (De Walle, 2002):

A ⋅10000
Q n +1 = [cSn ⋅ a n (Tn + ∆Tn )Sn + c Rn Pn ]            (1 − k n +1 ) + Q n k n +1   (1)
86400

where: Q = average daily discharge
c = runoff coefficient expression the losses as a ratio (runoff/precipitation), with
cs referring to snowmelt and cr to rain
a = degree-day factor
T = number of degree-days
∆T = the adjustment by temperature lapse rate when extrapolating the temperature
from the station to the average hypsometric elevation of the basin or zone
S = ratio of the snow covered area to the total area
P = precipitation contributing to runoff
A = area of the basin or zone
k = recession coefficient indicating the decline of discharge in a period without
snowmelt or rainfall.
n = sequence of days during the discharge computation period.

The number of degree-days, the ratio of snow covered area to total area, and precipitation

need to be measured daily. The temperature lapse rate, snow and rain runoff coefficients,

critical temperature, recession coefficient, and lag times are basin specific and should be

determined for each basin. If the elevation range of a basin is large, elevations zones

should be used and Equation 1 is modified (De Walle, 2002). If elevation zones are used,

the mean hypsometric elevation of each zone must be calculated by using an area-

elevation curve.
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1.2 Basin Description

The basin of interest for this project is the Upper Yakima River Watershed in

Washington (Figure 1, next page). The Upper Yakima Watershed was modeled for the

1989 melt season from late February until early June. The area of the basin is 2130

square miles with a range in elevation from 1200 feet to 6960 feet. The basin was broken

down into 5 elevation zones:

Zone A:   5700-6960 feet
Zone B:   3000-5700 feet
Zone C:   2400-3000 feet
Zone D:   1800-2400 feet
Zone E:   1200-1800 feet.

The zonal hypsometric mean elevations were previously calculated in an unpublished

thesis and are shown in Figure 2.          All air temperature, precipitation, measured

streamflow, and snow cover data and an initial set of model parameters were given for

the five elevation zones. The recession constant for the basin was also know from the

previously mentioned unpublished thesis.

Figure 2. Zonal mean hypsometric mean elevations for the Upper Yakima Watershed.
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Figure 1. Map of the Upper Yakima Watershed showing locations of the main streams,
rivers and water bodies within the basin.
5

2.0 METHODS

The computed runoff volume for the Upper Yakima Watershed was simulated and

compared to the measured (actual) runoff volume for the watershed.          The critical

temperature, lag time to gage, degree-day factor, snow runoff coefficient, rain runoff

coefficient, and rainfall contributing area were initially varied individually to find an

optimal value and to visually see how the variation affected the simulated runoff.

Parameters were varied both above and below the optimum value. For simplicity and to

avoid over-fitting the model parameters, the parameters were kept constant over time and

over the elevation zones. The volume difference (%) and the coefficient of determination

(R2) were calculated to determine the optimal values.

Based upon the optimization of the individual model parameters, an optimal

combination of all six parameters was determined. Since the parameters may not vary

independently of each other, the final optimal values for each parameter did not

necessarily have to be the same as the optimal values determined for the independent

individual tests. The individual optimal values were merely used as a starting point for

the parameter combination optimization.       As with the individual parameters, the

combined parameters were initially kept constant over time and over the elevation zones.

However, due to an over-prediction of runoff through the second half of March and a

very large under-prediction of runoff in the first half of May, some of the parameters

were varied with time to achieve the R2 values requested as a final product for the

assignment. The model parameters were still held constant over elevation zone.
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3.0 RESULTS AND DISCUSSION OF TRIAL VARIATIONS OF EACH MODEL
PARAMETER

The critical temperature, lag time to gage, degree-day factor, snow runoff coefficient,

rain runoff coefficient, and rainfall contributing area were initially varied individually to

find an optimal value. The starting set of parameters is graphed in Figure 3 and their

values are presented in Table 1. Only one parameter at a time was varied with the rest of

the parameters remaining at their starting values. Each parameter was varied both above

and below the optimal value to determine how perturbations affect each parameter. The

volume difference (%) and the coefficient of determination (R2) were calculated to

determine the optimal values. In addition, a visual assessment on how each perturbation

affected the matching of the timing and magnitude of the flow between the calculated and

measured curves was performed. The snowmelt depletion and accumulative melts curves

for each zone were also visually analyzed and discussed where significant changes

occurred. All additional graphs for non-optimal tries are presented in the Appendix.

Figure 3. Measured (red) vs. computed (green) runoff for initial parameter values.
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Table 1 - Initial Model Parameters

Snow Runoff    Rain Runoff     Rainfall
Lag Time   Degree Day
Tcrit (Tc)                             Coefficient   Coefficient   Contributing
(L)     Factor (AN)
(Cs)           (Cr)       Area (RCA)

Value          1           6         0.100          0.200         0.200            1

3.1 Optimization of the Critical Temperature

The critical temperature (Tc) is the temperature at which precipitation changes from

rain into snow for a given hydrologic basin. The range for critical temperature generally

varies from 0 to 2ºC. In our calculations, the critical temperature was used to determine

if the measured precipitation for our basin fell as rain or snow. If the average daily

temperature was above the critical temperature, the precipitation was assigned to be

rainfall and the precipitation immediately contributes to runoff. If the average daily

temperature was below the critical temperature the precipitation was assigned to be

snowfall and is kept in storage by SRM until it is melted on subsequent warm days.

Computed runoff curves were calculated using critical temperatures of 0, 1, 2, 3, 4,

and 5˚C (Table 2). The optimal runoff curve was calculated using a critical temperature

of 4˚C (Figure 4). Values of the critical temperature below the optimal value shifted the

computed runoff curve upward resulting in an increasingly larger over-prediction of the

computed runoff. Values of the critical temperature above the optimal value shifted the

computed runoff curve downward resulting in an under-prediction of the computed

runoff. As the critical temperature is raised, more of the precipitation is assigned as snow,

which now must be melted before runoff can be assigned and the computed runoff curve
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is shifted downward. As the critical temperature is lowered, more of the precipitation is

assigned as rainfall and immediately contributes to runoff. The computed runoff curve

would shift upwards. No significant changes were observed in the snowmelt depletion

and accumulative melt curves.

Table 2 - Statistical Parameters when Tcrit is Varied

Tcrit = 0   Tcrit = 1   Tcrit = 2   Tcrit = 3   Tcrit = 4   Tcrit = 5

Volume
-22.1663    -1.5908     1.4064      2.7005      4.4566      5.8712
Difference (%)

Correlation
-0.502     0.4698      0.5317      0.5588      0.5856      0.5837
Coefficient

Figure 4. Measured (red) vs. computed (green) runoff for the optimal Tcrit = 4˚C.
9

3.2 Optimization of the Lag Time to Gage

Lag times (L) for runoff / snowmelt to travel through the basin and flow through the

gage are typically determined directly from the basin hydrographs from previous years.

Generally, the larger the basin, the larger the lag time will be. However, since we do not

have information concerning previous years’ hydrographs for our basin, an estimate can

be made instead for basin size. Since our basin is greater than 5000 km2, a lag time of

greater than 12 hours can be expected from the WMO intercomparison test (WMO,

1986).

Computed runoff curves were calculated using critical lag times of 0, 6, 12, 18, and 24

hours (Table 3). The optimal runoff curve was calculated using a lag time of 18 hours

(Figure 5). Values of the lag time below the optimal value shifted the computed runoff

curve to the left. Values of the lag time above the optimal value shifted the computed

runoff curve to the right. In both cases, the shifts caused computed and measured runoff

peaks and valleys to fall out of sync. No significant changes were observed in the

snowmelt depletion and accumulative melt curves.

Table 3 – Statistical Parameters When Lag
Time to Gage is Varied

0 hrs     6 hrs    12 hrs    18 hrs    24 hrs

Volume
-1.477   -1.5908   -1.223   -0.2861   -0.7994
Difference (%)

Correlation
0.4491   0.4698    0.4849   0.4952     0.4926
Coefficient
10

Figure 5. Measured (red) vs. computed (green) for the optimal Lag Time = 18 hours.

3.3 Optimization of the Degree Day Factor

The degree-day factor (AN) allows one to calculate how much melt is produced from

a snowpack in a given day for a given hydrologic basin. In our calculation, the degree-

day factor was used to determine the potential melt for each day. If the average daily

temperature was above 0ºC, the potential melt was assigned a value of the degree-day

factor times the average daily temperature. If the average daily temperature was below

0ºC, the potential melt was assigned a value of zero as no melt was assumed to occur.

Computed runoff curves were calculated using degree-day factors of 0.050, 0.070,

0.100, 0.200 and 0.300 (Table 4). The optimal runoff curve was calculated using a

degree-day factor of 0.090 (Figure 6). Values of the degree-day factor below the optimal

value shifted the computed runoff curve downward resulting in an increasingly larger
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under-prediction of the computed runoff due to the decreased melt prediction. Values of

the degree-day factor above the optimal value shifted the computed runoff curve upward

resulting in an increasingly larger over-prediction of the computed runoff due to the

increased melt prediction. Doubling the degree-day factor more than doubled the total

snowmelt depth (from 42 in. to 102 for zone E) on the snow depletion curve and the

accumulated melt depth (from 27 in. to 66 in.) on the accumulated melt depth curve.

Table 4 - Statistical Parameters the Degree-Day
Factor is Varied

0.05      0.07     0.09    0.100      0.200        0.3

Volume
31.6742   18.3873   5.0753   -1.5908   -68.5315   -135.8446
Difference (%)

Correlation
0.0685    0.369    0.4837   0.4698    -2.3915    -10.3928
Coefficient

Figure 6. Measured (red) vs. computed (green) runoff for the optimal Degree-Day
Factor = 0.090.
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3.4 Optimization of the Runoff Coefficients

The runoff coefficient accounts for the difference (loss) between the available water

volume and the outflow volume from a basin. The runoff coefficient is necessary as

water     can    be    lost   to     infiltration,   evaporation,    sublimation,   interception,

evapotranspiration, snowfall gage catch efficiency, and lack of precipitation data in

mountainous regions. Depending on the dominant process for the time of year, the runoff

coefficient will differ. For example, losses would be expected to be largest late in the

snowmelt season as evapotranspiration begins to take hold. There are separate runoff

coefficients for rain (cR) and snow (cS).

3.4.1 Optimization of the Snow Runoff Coefficient

Computed runoff curves were calculated using snow runoff coefficients 0.100, 0.200,

and 0.300 (Table 5). The optimal runoff curve was calculated using a snow runoff

coefficient of 0.200 (Figure 3). Values of the snow runoff coefficient below the optimal

value shifted the computed runoff curve downward.                   Values of the snow runoff

coefficient above the optimal value shifted the computed runoff curve upward. No

significant changes were observed in the snowmelt depletion and accumulative melt

curves.

Table 5 - Statistical Parameters when cS is Varied

0.100          0.200            0.300

Volume Difference (%)         31.7762         -1.5908         -35.0464

Correlation Coefficient        0.0512          0.4698          -0.3357
13

3.4.2 Optimization of the Rain Runoff Coefficient

Computed runoff curves were calculated using rain runoff coefficients 0.100, 0.200,

and 0.300 (Table 6). The optimal runoff curve was calculated using a rain runoff

coefficient of 0.200 (Figure 3). Values of the rain runoff coefficient below the optimal

value shifted the computed runoff curve downward. Values of the rain runoff coefficient

above the optimal value shifted the computed runoff curve upward. No significant

changes were observed in the snowmelt depletion and accumulative melt curves.

Table 6 - Statistical Parameters when cR is Varied

0.100         0.200          0.300

Volume Difference (%)      15.963         -1.5908       -19.4683

Correlation Coefficient    0.4052         0.4698        0.1035

3.5 Optimization of the Rainfall Contributing Area

The rainfall contributing area (RCA) can be varied as only one of two cases. For the

first case where RCA=0, the snowpack is not assumed to be ripe (snow temperature is

below 0ºC). Rain that falls on the snowpack is assumed to be utilized to warm the

snowpack and is not included as melt. In this case, only the rain that falls on snow-free

areas is considered in the calculation. For the second case where RCA=1, the snowpack

is assumed to be ripe (snow temperature equals 0ºC). The second case assumes that an

amount of water equal to the rainfall input leaves the snowpack. Therefore, rain falling

anywhere in the basin contributes to the calculation of runoff.
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Computed runoff curves were calculated using rainfall contributing values of 0, 1, and

a changeover partway through the year (Table 7).          The optimal runoff curve was

calculated using the mixed rainfall contributing area value with the value changing from

0 to 1 on 4/5/89 (Figure 7). Inspection of the temperature values across all 5 zones

indicated that the majority of the daily temperatures were above freezing meaning that

the snowpack would probably be ripe after 4/5/89. No significant changes were observed

in the snowmelt depletion and accumulative melt curves.

Table 7 - Statistical Parameters when RCA is Varied

0                 1          Mix

Volume Difference (%)      12.4559         -1.5908        5.493

Correlation Coefficient     0.5602         0.4698         0.6230

Figure 7. Measured (red) vs. computed (green) runoff for the optimal Mixed RCA.
15

4.0 RESULTS AND DISCUSSION OF BEST-FIT MODEL PARAMETERS

Based upon the optimization of the individual model parameters, an optimal

combination of all six parameters was determined. Since the parameters may not vary

independently of each other, the final optimal values for each parameter did not

necessarily have to be the same as the optimal values determined for the independent

individual tests. The individual optimal values were merely used as a starting point for

the parameter combination optimization. The process proceeded on a trial and error basis

from the starting point of the individual parameter tests and proceeded until the suggested

R2 values in the range of 0.7-0.8 were achieved. As with the individual parameters, the

combined parameters were initially kept constant over time and over the elevation zones.

However, due to an over-prediction of runoff through the second half of March and a

very large under-prediction of runoff in the first half of May, some of the parameters

were varied with time. The model parameters were always held constant over elevation

zone.

The final optimal combined computed runoff curve was calculated by first starting

with the optimal RCA and an R2 value of 0.6230. The optimal RCA was selected as a

starting point because it was the individual parameter that had the largest influence on

raising the R2 value in the individual trials.    Adding the individual optimal critical

temperature increased the R2 value to 0.6314. However, the optimal degree-day factor

lowered the R2 value and the initial value of AN = 0.100 turned out to produce the best

R2 value when in conjunction with the other parameters. The optimal lag time lowered

the R2 value and a new combined optimal value of L = 12 hours was used. The R2 value
16

for the new optimal lag rose to 0.6327. The snow runoff coefficient was held at the

initial and optimal value. The majority of the temperatures in the basin were above the

freezing temperature after 4/5/89, so the snow runoff coefficient shouldn’t be as

important and the rainfall coefficient. Variation of the snow runoff coefficient before

4/5/89 did not increase the R2 value. The rain runoff coefficient turned out to be the

parameter with the biggest influence on the R2 value in the combined parameter trials. A

very large runoff spike occurred between 5/4/89 and 5/17/89 that was severely

underestimated.     For the remainder of the simulation, the runoff was mildly

underestimated. The only parameter that whose variation affected this spike was the rain

runoff coefficient. During the period from the beginning of the simulation until 5/3/89,

the rain runoff coefficient was assigned a value of 0.200. From 5/4/89 to 5/17/89, the

rain runoff coefficient was assigned a value of 0.600. From 5/18/89 until the end of the

simulation, the rain runoff coefficient was assigned a value of 0.250. The dramatic

increase necessary in the rain runoff coefficient does occur late in the melt season and

perhaps could be the result of direct channel flow within the basin and opposed to

generalized overland flow (De Walle, 2002).             A summary of the best-fit model

parameters is presented in Table 8. The best-fit computed runoff curve is presented in

Figure 8.

Table 8 – Best Fit Model Parameters

Snow Runoff                             Rainfall
Lag Time   Degree Day                      Rain Runoff
Tcrit (Tc)                             Coefficient                          Contributing
(L)     Factor (AN)                    Coefficient (Cr)
(Cs)                               Area (RCA)

Value          4          12         0.100          0.200       Varied (0.200-0.600)       Mix
17

Figure 8. Measured (red) vs. computed (green) runoff for the best-fit model parameters.

5.0 CONCLUSIONS ABOUT THE USEFULNESS OF SRM

SRM is a semi-distributed, deterministic model in which snowcover is broken down

by elevation zone. Therefore, SRM predicts a single streamflow value from a given set

of parameters. In general, distributed models can better predict the timing of events and

better predict the results of unusual snowmelt events as compared to statistical models.

SRM is at a disadvantage as it is one of the simpler distributed models as uses the degree-
18

day approach as opposed to the energy budget approach and does not account for such

parameters as forest cover, cold content, slope/aspect effects, liquid routing, or

interception.   However, the simplicity of SRM may be an advantage as the other

parameters are often very hard to quantify on a basin wide scale. In fact, the SRM

outperforms many other more complex models both in terms of the R2 value and in the

volume difference.

On a more personal note stemming from my experiences with this project, the model

output for SRM, as with any model, can be highly dependent upon the user. Since some

of the parameters may be interconnected, a choice to start with the optimization of one

parameter versus another may change the end result. For our example, if I chose to

optimize the snow runoff coefficient before the rainfall contributing area, my end R2

value would be about the same, but I would have corrected the over-calculation in late

March at the expense of other points on the curve. I chose to optimize the rainfall

contributing area first simply because I could understand how that parameter worked.

Common sense would indicate that the snow would become ripe if the average daily

temperature was consistently well above freezing. The snow runoff coefficient is a

parameter derived from previous knowledge of basin behavior. I only knew that I was

changing a parameter to achieve a best fit, without understanding the other ramifications.

The same could be said for the under-calculation in May and the rain runoff coefficient.

Though my solution looks good and achieves the prescribed R2 values, I would not stake

my career on it (not yet anyway, maybe in the future).
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In summary, anyone could endlessly change parameters to achieve a best-fit curve

without any knowledge of the concepts behind the fit such a snowmelt hydrology or the

basin characteristics. A nice looking solution does not have to equate with a correct,

applicable solution. However, if a user understands the concepts behind the model and

can apply those concepts to the basin of interest, a model can become a very powerful

tool. Any model is only as good as what goes into it, whether data or the knowledge and

interpretation of the user.
20

6.0 REFERENCES

De Walle, D. (2002) Class Notes.

Martinec, J. (1975) Snowmelt-Runoff Model for stream flow forecasts. Nordic
Hydrology 6(3), 145-154.

Seidel, K., Martinec, J., and Baumgartner, M. F., (2000) Modeling runoff and impact of
climate change in large Himalayan basins. Proceedings of International
Conference on Integrated Water Resources Management for Sustainable
Development, New Delhi, India, 1020-1028.

WMO (1986) Intercomparison of Models of Snowmelt Runoff. Operational Hydrology
Report 23, WMO, Geneva, Switzerland.

WMO (1992) Simulated real-time intercomparison of hydrological models. Operational
Hydrology Report 38, WMO, Geneva, Switzerland.

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