A clear-sky longwave radiation model for remote alpine areas

ARCHIV Arch. Met. Geoph. Biokl., Set. B, 27,159-187 (1979) FUR METEOROLOGIE

GEOPHYSIK UND BIOKLIMATOLOGIE �9 by Springer-Verlag 1979

551.521.3

Department of Geography, University of California, Santa Barbara, California, U. S. A.

A Clear-Sky Longwave Radiation Model for Remote Alpine Areas

D. Marks and J. Dozier

With 12 Figures

Received February 13, 1979

Summary

A theoretically based model accurately calculates incoming longwave radiation under clear sky conditions in a remote alpine area. Data requirements are relatively simple and because the model is theoretically based it can be widely applied. The model calculates atmospheric radiation using a form of Brutsaert's method that is adjusted for variations in air pressure and radiation from adjacent slopes. Tests of both point and areal applications indicate that the method gives reliable results. When applied as a segment of an energy balance snowmelt model, the mode/provides information on energy exchange in an alpine environment and improved snowmelt runoff prediction. Radiation from cloud cover and canopy cover can be more easily isolated by using the model to account for clear sky radiation. It is possible that it can also calibrate atmospheric effects in thermal satellite data, thus enhancing the possibilities for using satellite radiometry in alpine areas.

Zusammenfassung

Ein Modell der langwelligen Strahlung far abgelegene alpine Gebiete

Ein theoretisch fundiertes Modell berechnet zuverl~issig die einfallende langwellige Strah- lung bei klarem Himmel in abgelegenen alpinen Gebieten. Die Datenansprtiche sind relativ einfach und auf Grund der theoretischen Fundierung kann das Modell unter vielfachen Bedingungen angewendet werden. Das Modell berechnet die atmosph~irische Rtickstrahlung unter Anwendung einer Art yon Brutsaert-Methode, welche auf Variation im Luftdruck und auf Strahlung yon benachbarten H~ingen abgestimmt wurde. Die An- wendung auf Met~punkte und Gebiete zeigt, daf~ die Methode zuverl~issige Resultate ergibt. Wird das Modell als Tefl eines Energiebilanzmodelles zur Berechnung der Schnee- schmelze angewendet, so kann es Aussagen tiber den Energieaustausch in einer alpinen Umgebung sowie verbesserte Vorhersagen der Schneeschmelze liefern. Strahlung yon bedecktem Himmel oder yon einer Pflanzendecke kann leichter isoliert werden, denn aus Modellrechnungen kann die Strahlung des unbew61kten Himmels bestimmt werden. M6glicherweise kann das Modell zur Kalibrierung thermischer Satellitendaten bentitzt werden und dadurch die Anwendungsm6glichkeiten der Satellitenradiometrie in alpinen Gebieten verbessern.

0066-6424 /79 /0027 /0159 /$ 05.80

160 D. Marks and J. Dozier

Notation

A Amp AO B Cp ea es e r

g H

In L m

Pa

Pm Po Phase R RH ra T., r, Tw T' vf Z

Zm

C

ea es 2/ 0

o

regression coefficient amplitude of a periodic function mean value of a periodic function regression coefficient specific heat of air (1005 J kg -1K -I ) near surface vapor pressure at a point (rob) saturation vapor pressure (rob) corrected vapor pressure (rob) acceleration due to gravity (9.8 m sec -2) mean horizon angle from the zenith about a point (tad) incoming longwave radiation flux (Win -2) net longwave radiation flux (Wm -2) latent heat of vaporization (2.5.106 J kg -1 ) molecular weight of dry air (.0289 kg mol -x ) near surface air pressure at a point (rob) reference air pressure (mb) air pressure at sea level (rob) phase of a periodic function gas constant (8.3143 J tool -1K - 1 )

relative humidity near surface air temperature at a point (K) reference air temperature (K) surface temperature (K) wet bulb temperature (K) corrected air temperature (K) thermal view factor (dimensionless) elevation of a point (m) reference elevation (m) emissivity coefficient effective atmospheric emissivity surface emissivity temperature lapse rate (Km -1) Stefan-Boltzmann constant (5.6697.10 -s J m -2 K --4 s e c 1 )

potential temperature (K)

1. In t roduc t ion

Fo r an accurate assessment o f the surface energy exchange in rugged snow- covered terrain, each aspect of energy inpu t and ou tpu t must be evaluated. At a po in t this can be done by direct measurement or, for a larger area, by an appropr ia te empir ical or theoret ica l model o f the parameters o f interest . Satel l i te de te rmina t ion o f the surface t empera tu re of a snowcover could be used to mon i to r these energy fluxes if the satelli te da ta can be cal ibra ted for thermal radia t ion input f rom the a tmosphere and surrounding terrain.

A Clear Sky Longwave Radiation Model for Remote Alpine Areas 161

In this paper we present a model of incoming longwave radiation which provides an accurate estimate of this variable under clear skies. This work is part of a larger effort to extend an energy balance snow~elt model over a large mountainous region using satellite data as direct or in- direct input whenever possible. The approach has been to simulate each of the components of surface energy exchange over an alpine area during the snow season, resulting in a complete accounting of the energy budget of the snowcover from which melt and subsequent runoff could be calculated. Such a model has been used with success at a point [2] and a surface energy budget has been calculated over an alpine area [9]. In the latter paper, incoming longwave radiation was not modeled in detail. The model presented herein is intended to provide additional detail to enhance further energy budget snowmelt studies and possibly to provide a technique that can be used to assess the utility of thermal satellite radiometry in an alpine area when such data become available.

2. Background

The net longwave radiation at the surface is

In = l i - ( 1 - e s ) Ii-esoTas = es ( I i - e T~s ). (1)

Spectrally, longwave radiation occurs in wavelengths above 4/~m, and the major sources are the atmosphere and surrounding terrain. The sun also emits longwave radiation, but because of its temperature only abou t 0.9% of its energy is at wavelengths longer than 4/~m. Integration of Planck's equation shows that the radiation emitted by a 5800 K black body at these longer wavelengths exceeds 3.10 s Wm -2 but the earth-sun distance reduces this to about 13 Wm -~ at the top of the earth's atmosphere. Atmospheric absorption reduces this to a negligible amount at the surface, and it is thus reasonable to treat solar and atmospheric radiant fluxes separately. Deter- mining longwave radiation flux from the atmosphere is simpler than more general radiative transfer problems for two reasons: (1) There is no scattering; the atmosphere, as an optically active gas, partially absorbs and partially transmits incident longwave radiation, but it does not appreciably scatter the transmitted radiation [ 10]. (2) For most applications, a parcel of atmosphere can be treated as though it were in thermodynamic equilibirum [20]; thus the emitted radiation from a parcel o f atmosphere can be described by its density, temperature, pressure, and relative concentrations of optically active substances. Moreover, determining net radiation in the longwave region is simpler than for the solar region, because the spectral albedo is much less varied at wavelengths longer than 4/~m. For snow, O'Brien and Munis [14] showed that reflectance is nearly zero above 2.5 #m,


so in the wavelength band where most terrestrial and atmospheric radiation occurs (8 -12 /am) , a snow surface may be assumed to be a near perfect ab- sorber. The atmosphere is a mixed gas made up of nitrogen and oxygen along with several trace gases. A few of these trace gases, water vapor, carbon dioxide, and ozone, are responsible for most atmospheric absorption and subsequent thermal emission [ 11, 15 ]. Of these three gases, water vapor is by far the most important because of its highly variable mixing ratio. The water vapor content of air may vary from 0 to 4% by mass, and is the most important atmospheric constituent for thermodynamic considerations [6]. Carbon dioxide and ozone are important only in the upper atmosphere where water vapor concentrations are very small. In the absence of scattering and reflec- tion we know, from Kirchhoff 's law for thermal radiation, that incoming longwave radiation from the atmosphere must equal radiation transmitted through the atmosphere plus radiation emitted by the atmosphere. However, for considerations of the surface energy balance it is not important to separate transmitted from emitted radiation. The two can be combined in what Paltridge and Platt [15] describe as an emissivity approximating technique. This technique treats the atmosphere as a grey body with an effective atmospheric emissivity such that

Ii = ea o T 2 (2)

The effective atmospheric emissivity ea is a function of the temperature and quantity of the emitting source, mainly water vapor. Variations of this functional relationship are found in many of the atmospheric radiation models in wide use and are typical of calculations based on radiation charts. The most common approach for estimating thermal radiation at the surface has been to approximate atmospheric emissivity, based either upon appropriate measurements near the surface [3, 4, 5, 21 ] or upon temperature and vapor pressure profiles [ 10]. In a remote alpine area atmospheric soundings are generally unavailable, so the emissivity approximation must be based on surface measurements. Approximations by Brunt's method [4] or by similar methods suffer from the necessity to calibrate the appropriate regression coefficients at each location. Brutsaert [5] derived an alternative approach. By substituting exponential approximations for temperature, pressure, and water vapor density, he was able to integrate Schwarzschild's transfer equation to yield an approximation for effective atmospheric emissivity

ea = 1 . 2 4 ( e a / T a ) 1/7 �9 (3)

Under alpine conditions a reliable empirical model is very difficult to develop, whereas Brutsaert's more theoretical model is sensitive to fluctu-


ations in the parameters which most affect longwave radiation input at the surface. To be useful in energy balance studies over large areas, however, such a model should have relatively simple data requirements. The model presented herein is a modification of Brutsaert's and satisfies this constraint. For the test cases, the model is driven by a few discontinuous field measurements collected in higher portions of the southern Sierra Nevada during the 1977 and 1978 snow seasons. The model gives reliable results over large alpine areas of extreme topographic relief and can be driven by measurements of surface meteorological parameters which are relatively easy to obtain.

3. Longwave Radiation Model for an Alpine Area

For a longwave radiation model to have practical application in a remote alpine area it must have several attributes. It must have data requirements which are relatively easy to satisfy either by active field measurements or by unattended remote collection systems. It must account for variations in input parameters caused by the extreme topographic and diurnal variations likely to occur in such a region. It must be able to respond to changes in meteorological and climatological conditions as they occur seasonally. The model described in this paper satisfies these requirements. The model has been developed over a mountainous portion of the southern Sierra Nevada as a major segment of an energy balance snowmelt model. It wiU also be used to calibrate thermal satellite data for remote determination of surface temperatures in the area. At present it is configured to account for incoming longwave radiation from the atmosphere and from surrounding slopes. Radiation from clouds and canopy cover are ignored, but will be accounted for when subsequent data become available. Eqs. (2) and (3) provide a technique for calculating incoming longwave radiation which is based on physical grounds rather than correlation with observational data. Brutsaert points out that the coefficient, 1.24, is relatively insensitive to variations in the typical parameters used to derive it, and that because the right side ofeq. (3) is taken to the 1/7 power it is also insensitive to minor changes in Ta. As it is presented, this approach assumes a standard atmosphere, at the 1000 mb (10 s Pa) level, and a lapse rate of - .0065 Km -1 . In coastal or lowland locations with minimum relief these assumptions are valid, and the approach gave results that compared favorably with field measurements when used to compute incoming longwave radiation near the south coast of France [13 ]. In alpine areas the assumption of a standard atmosphere is not valid. In the southern Sierra Nevada surface air pressure may vary from 850 mb to 550 mb with elevation alone. In order to use eq. (3) to compute effective atmospheric emissivity, it must be adjusted for the actual air pressure of the


point of interest. The adjustment corrects the near surface air temperature, Ta, and vapor pressure, ea, at elevation z, to their sea level equivalents using a standard temperature lapse rate ( - .0065 K m -1 ), and an assumption of constant relative humidity

T ' = Ta + (.0065 z) (4)

R H = ea/es[Ta ] (5)

e' = R H es[T'] (6)

The corrected air temperature, T', and vapor pressure, e', are then used with the surface air pressure at the point to compute its effective emissivity

ea = [1.24(e'/T') 1/7 ] (Pa/1013 ). (7)

Eq. (7) accounts for atmospheric emissivity only. In an alpine area of high relief we must account for radiation from the surrounding terrain as well. The emissivity of these surrounding slopes is assumed to be very close to unity [12, 17, 19]. The portion of the hemisphere which is obscured from view of the atmosphere is determined by comparison with elevations of the surrounding terrain. The unobscured portion is termed the thermal view factor, Vf, and is expressed as a number between 0.0 and 1.0 [9]

Vf= cos 2 (H), (8)

The average horizon angle H includes the slope on which the point falls if that slope has an angle greater than the horizon angle. From eqs. (4) through (8) incoming longwave radiation can be computed for any point for which we know the elevation, near surface air-pressure, vapor pressure, and air temperature, by

Ii -~ (eao T~a) Vf + (esoT4s) (1-Vf). (9)

Because the view factor, Vf, is usually larger than 0.90, it is not necessary to very accurately determine either the surface temperature Ts or the surface emissivity es. Reasonable approximations are adequate. Two other factors, radiation from cloud cover and from forest canopy cover, are not being considered at this time. The isolation of radiation from these sources requires a much more thorough data collection effort than was possible for the present work. However, the results of the work, under clear skies and on open slopes, indicate that it should be possible to model incoming longwave radiation under all other conditions as well.

3. 1 Model Sensitivity

Eqs. (7) and (9) were evaluated across a wide range of values of air temperature and vapor pressure likely to occur in an alpine area. Figs. 1 and 2 show the sensitivity of these equations to variations in their input parameters.


An elevation 0f 3000 m and a view factor of 1.0 were used for the test. Fig. 1 illustrates that computed emissivity is not very sensitive to reasonable variations in either air temperature or vapor pressure. At 280 K, doubling the vapor pressure from 5 to 10 mb increases emissivity from 0.59 to 0.64, or only about 8%, and, under saturated conditions, an increase in temperature from 270 K to 280 K results in an increase in emissivity from 0.60 to

I

8

E r ~

>~

EMISSIVITY

TEMPERATURE (~

Fig. 1. Sensitivity of emissivity, eq. (7), to variations in air temperature and vapor pressure, at an elevation of 3000 m. Shaded area indicates above-saturation vapor pressure

0.65, also only about 8%. This is particularly helpful in an areal estimate of effective atmospheric emissivity because it shows that even if air temperature and vapor pressure are roughly approximated the model will give useful results. To apply the model an estimate of air temperature and vapor pressure must be made for all times and elevations in a remote alpine area where meteorological data collection is sporadic at best. Under these conditions, relatively simple functions of the diurnal and altitudinal variations are used. Fig. 2 shows that computed incoming longwave radiation is more sensitive to minor variations in air temperature than vapor pressure. This sensitivity results because air temperature Ta in eq. (9) is raised to the fourth power. However, air temperature has a smoother temporal and spatial distribution than does vapor pressure, and it is therefore easier to approximate its variations. In addition, accurate temperature data are easier to acquire.


Vapor pressure is diff icult to measure under any circumstances, par t icular ly at t empera tures near the freezing point .

The evaluat ion of model sensitivity shows that it responds smooth ly to variat ions in input parameters . I t does no t require an es t imate o f these parameters beyond an approx ima t ion of their diurnal and al t i tudinal variations. Therefore , extensive data col lect ion is no t necessary to drive the model in a r emote alpine area.

~, INCOMING LONGWAVE RADIATION (Wm -2)

LU n"

TEMPERATURE (OK)

Fig. 2. Sensitivity of incoming longwave radiation, eq. (9), to variations in air temperature and vapor pressure. Similar to Fig. 1

4. Point Simulation Results

4.1 The Study Area and Test Sites

The mode l was tes ted during the snow seasons o f 1977 and 1978 in higher por t ions of the southern Sierra Nevada. This area is well adapted to such a mode l because a large por t ion o f the snow ca tchment area is above t imber- line. Five test areas were selected represent ing a variety of exposures and elevations. Each of the test areas is approx ima te ly five minutes of la t i tude and longi tude , though the topographic maps (Fig. 3) include surrounding terrain which is necessary for calculat ion of the view factor . Each test area is accessible on skis for field data col lect ion during the snow season.


TEST AREAS FOR LONGWAVE RADIATION MODEL W118~'14'00 " WI IEP(~CO" WI I8~8 '0C P WII8~ '

�9

w " ~ i

WII O(~'

N ~ o ~ Mammoth Lakes N3.~o~ Bullfroq l a k e - Kearsarge Pass Area Valentine Reserve Area Mammoth Mountain Area

Fig. 3. Topographic maps of five test areas located in the southern Sierra Nevada. Maps are drawn from NCIC Digital Terrain data which has been resampled to a grid spacing of 100 meters with a contour interval of 100 meters. Measurement s i t e s are shown as �9

The test areas were represented as digit ized grids o f elevations taken f rom the NCIC Digital Terrain Tapes. The grid spacing was resampled f rom 63.5 meters to 100 meters to reduce compu ta t ion t ime, and the digit ized grids were used. ini t ial ly to locate measurement poin ts and to compu te view factors for each o f these. When the s imulat ion is run over the entire test area, incoming longwave radia t ion is c o m p u t e d for each grid point .

168 D. Marks and J, Dozier

There has been some concern over the accuracy and reliability of the NCIC Digital Terrain Tape data [18]. However, a comparison of the elevations of 30 points on 15 minute topographic maps with their elevations on the Terrain Tapes indicated that the mean positional error was acceptable for this application. Table 1 indicated that the errors were only slightly larger

Table 1. Summary of Digital Terrain Tape Positional Errors

RMS latitudinal error: 50 m RMS longitudinal error: 58 m RMS vertical error: 56 m Combined positional error: 95 m

than half the grid spacing. Moreover, they were not systematic. The only time significant positional deviation was observed was when trying to locate a prominent feature such as peak. This was not an important problem when cons!dering simulation over the entire area. Topographic information was used to determine the air pressure at a point and to determine the thermal view factor, Vf, as described by eq. (8). The view factor is determined using a technique developed by Dozier and Outcalt [9] that computationally is straight-forward, but which can require extensive computer time. The mean horizon angle, H, requires a comparison with every other grid point. This can be expensive for a grid of ten thousand or more points. For this reason the view factor calculations were done on a smaller computer (a PDP 11/45) dedicated to research projects and later transferred to a larger time-sharing machine (an Itel AS-6). The problem is eased by the fact that view factors need only be computed once for each test area.

4.2 Data Collection Procedures

Three sources of data were used to drive and test the model: digital terrain data, weather station data from the Bishop, California weather station (Bishop WSO), and field data collected during the snow season. Weather station data from Bishop WSO were used only to calculate air pressure at each point in the test areas. Air temperature and vapor pressure information were not used because Bishop WSO lies a thousand meters or so below the snow line in the bottom of the Owens Valley, and experiences frequent low level inversions. Because of this condition an average daily air pressure and temperature were all that could be used to correlate the Bishop data to conditions in the study area, Pressure at a given elevation is derived from the hydrostatic equation and the equation of state

10, P~ t K T L JJ


A temperature lapse rate of -0 .006 K m -a was used because it best approximated average conditions. Field data collected consisted of wet and dry bulb temperature, measured using a sling psychrometer, and a measurement of incident longwave radiation using an Eppley Pyrgeometer equipped with a silicon-coated dome. These instruments are portable enough to permit carrying them into remote field areas by hiking or skiing. During the 1977 and 1978 snow seasons we made a number of data collection trips. Data from six of these trips were used to test and calibrate the model. Data was collected during all types of weather conditions from severe storms to clear, cloudless weather. Data from the 1977 snow season represent a drought year, while data from 1978 represent a very heavy snow year. Measurements were made at numerous times and elevations during each trip. For each measurement point, effective atmospheric emissivity and incoming longwave radiation were computed using eqs. (7) and (9). These values were then compared to the measured value from the pyrgeometer. The view factor and elevation for each point were determined from the digital terrain data. Air pressure was determined from eq. (10). Air temperature was measured directly and vapor pressure was calculated using measured wet bulb temperature and the psychrometric equation

~ Cpl'o 1 ea = es-[.622LJ-- (Ta-Tw) . (11)

The temperature of the surrounding slopes was approximated by an estimate of the mean daily air temperature, which was derived by averaging field measurements. Differences between north and south facing slopes were assumed to cancel each other and were ignored. A surface emissivity of 0.95 was used in all cases.

4. 3 Model Accuracy

The results of the point tests of the model showed that it gives reliable results under clear skies independent of the time of day or the elevation of the measurement point (Table 2). The results were divided into four cate- gories: clear sky and cloudy sky measurements made during the snow season, measurements taken during 1977 at a forested ~ite after most of the snow had melted, and a set of clear sky measurements taken during the 1978 snow season. The last two groups of measurements were included to determine the magnitude of the effect of canopy cover on the results, and to see if model accuracy would vary significantly from one snow season to the next. An RMS error was computed for each group shown in Table 2. Under clear skies during both 1977 and 1978 the results are within 10-15% of the measured mean, which is an acceptable tolerance for modeling the

1'2 Arch. Met. Geoph. Biokl. B. Bd. 27, H. 2 ~ 3


Table 2. Measurements Taken During the 1977 Snow Season

* All cloud cover conditions, 49 measurement points, mean value = 254 Wm -2 , RMSE = 64 Wm -2

* Clear sky conditions, 19 measurement points, mean value = 235 Wm 2, RMSE = 31 Wm -2

* Cloudy sky conditions, 30 measurement points, mean value = 266 Wm -2 , RMSE = 78 Wm -2

Measurements Under a Forest Canopy at the Valentine Reserve

* Clear sky conditions, 28 measurement points, mean value = 31-8 Wm 2, RMSE = 71 Wm -2

Measurements Taken During the 1978 Snow Season

* Clear sky conditions, 7 measurement points, mean value = 211 Wm -2 , RMSE = 26 Wm 2

surface energy balance over a large area. It is regrettable that more 1978 data are no t available due to a failure of the pyrgeometer early in the season. The data shown represent the only clear sky measurements taken during 1978 unti l well into the melt season. The errors described above did not have a normal distr ibution about zero. In all cases the computed incoming longwave radiation was less than the measured. This was expected, as the model in its present form accounts for clear sky radiat ion only, ignoring addit ional sources of thermal radiat ion input . Some measurement errors can be a t t r ibuted to poor field techniques. At times field assistants inadvertent ly included themselves and other warm objects in the pyrgeometer 's field of view. This was particularly evident in readings taken during storms or at night when the ins t rument was only reluctant ly taken out of the tent. Such measurements would produce a positive bias causing model errors to appear larger than their true values. Errors also occurred in the measurement of air temperature and vapor pressure, but as was stated earlier, these were probably less critical. While it is difficult to quant i fy this positive bias caused by measurement errors, the total magnitude of this error is probably less than 5%. This problem will be solved in future measurement seasons by una t tended measurement systems and by improved field techniques.

4.4 Distribution o f Errors Among Model Parameters

The dis tr ibut ion of errors among model parameters was examined to see if the model had any systematic errors or particular weak points. Separate tests were made of the data collected during the 1977 snow season, the 1978 snow season, and the data collected in the forested site after the snow had melted. For the 1977 snow season data, Fig. 4 shows the relationship


between calculated incoming longwave radiation and incoming longwave radiation as measured by the pyrgeometer. It is clear from the figure that the clear sky relationship has a systematic bias, but is otherwise strong. Cloudy sky data show no definable relationship. As discussed earlier, statistical analysis of the model errors is difficult because the errors have a positive bias and are not normally distributed about zero. For clear sky

525 Zl

_ ~ ~ % 0

:D 2 2 5

o h i ~ O

175

t '5%0 ,75 2~o 'zz5 z;o ' ~ o ' 3 ; o 2 7 ~

CALCULATED I (Wrrf 2)

Fig. 4. Plot of calculated vs measured incoming longwave radiation for measurements taken during the 1977 snow season under clear and cloudy sky conditions. Circles, �9 indicate clear skies and triangels, A, indicate cloudy skies

values, the model was tested by a simple regression analysis of observed vs calculated incoming longwave radiation. The systematic error is treated as a constant and the deviations from this are assumed to be normally distributed with mean zero. The results of the analysis (Table 3) show a strong correlation coeffient, that the slope of the clear sky values in Fig. 4 is not

Table 3. Regression Analysis of Clear Sky Measured vs. Calculated Incoming Longwave Radiation

Correlation Coefficient: 0.88 Regression Coefficient (Slope): 0.97 Standard error of slope: 0.13 Upper limit: 1.25 Lower limit: 0.70 Intercept of Regression: 33.02

12"

172 D. Marks and J. Dozier: A Clear Sky Longwave Radiation Model

significantly different from 1.0 at the 95% confidence level, and that the intercept of the regression equation is about the same as the RMS error shown in Table 2. To further isolate the errors from the effects of the magnitude of the output, we converted them to a difference between a measured and calculated actual emissivity, where the actual emissivity is

e = I i / ( a T4a). (12)

The difference or error between the actual emissivity from measured and computed incoming longwave radiation is mapped against time, elevation, air temperature, vapor pressure, cloud cover, and view factor for each point. Fig. 5 shows this for clear and cloudy sky measurements made during 1977 over a snow cover. There is no significant pattern to the error distribution with time, elevation, or view factor for either clear or cloudy sky data. The range of elevations for cloudy sky data is limited, but there is no real trend to the error distribution. Nor is there any pattern for clear sky error distribution in relation to temperature. However, there is a definite increase in the magnitude of error at lower temperatures under cloudy skies. This is not too surprising when we consider that cloud cover, storms, and colder temperatures usually occur together. Fig. 5 shows the same general pattern for vapor pressure as for temperature. The same explanation holds because during stormy cloudy weather colder temperatures force lower vapor pressures. Fig. 5 illustrates that as cloud cover increases the magnitude of the error also increases. While there are not really enough data to establish a functional relationship, the data indicate that the increase in error is fairly regular. This substantiates earlier findings by Doronin [8] and Cox [7] that a practical model of cloud radiation should be based on the assumption that for a given region and time of year cloud characteristics and subsequent emissivity are predictable. Data from the 1978 snow season are compared with 1977 data in Fig. 6. The 1978 data were collected during January of a record snowfall year when air temperatures were particularly low. These data were also tested by regression analysis to see if the 1978 data differ significantly from the 1977 data, and to see if the results shown in Table 3 change with the inclusion of the 1978 measurements in the analysis. Table 4 shows the results of the combined analysis, and an analysis of the 1978 data alone. For the combined analysis, the correlation coefficient is unchanged, and while the slope is slightly different, it is still not significantly different from 1.0 at the 95% confidence level. The 1978 data represent only seven measurements so the significance of the regression is questionable. However, the correlation coefficient is still good, and while the slope is much smaller, it is still not significantly different from 1.0. It has been suggested by Aase and Idso [1] that a model based on Butsaert's techniques will fail at subfreezing temperatures. Fig. 7 compares the emissiv-

1.0 7

.~i

.8 ~

~ .7" Q~

~ .5'

2 0

.I 0 0

4

"!t' ul

>- .5 l-- y, ~.4 ~ .3 hi

2

.I O0

0 . 85

1.0

.9

.8

W .6 )- F- .5 -~ .& O3

~ .3'

0 ' 2

.&

.&A^ ZX. A

A A A & O o ~ a ~o o 9 ? 0 , O, 0 8 12 16 2 0

TIME (hrs)

1.0

.9

.8

IJJ .6 >- I-- .5

A ~ .4 (,0

lJa

i

.1' O

zooo

1.0

.9

.8

.6 >- I-" .5

A ~ /'- '"A & & A & �9 [ ]

~ t ''B .2

| , O , i i

VIEW FACTOR |.O

. 9

B'

&A& & t= A

o0

, t ~) 4 6 s ,o

VAPOR PRESSURE (rob)

n~

A

0 ~ 0 ,00

ELEVATION (m)

& A

A'" t, A . ~ A A & . A A

a ~ 1 7 6 ~ 00080 o 0 0 0 8 ,

TEMPERATURE (OK);

0

t 12

i

.7' rY ILl .6" >- I.- .5.

.4- U)

.:5. W

z~ A &

zx A & ! zx t~

~/,, ~A, ~, .1' ! A A

, I , i

CLOUD COVER

Fig. 5. Difference between computed emissivity and actual emissivity over the parameters time, elevation, air temperature, vapor pressure, cloud cover, and view factor, for measurements taken during the snow season under clear and cloudy sky conditions. Symbols as given in Fig. 4


350

325

300

r-~ 275 h i n~ ;2) 250 U~ <I LIJ 225

2OO

175

o

%0 o o

.c b o o~%~'- o

ooo

150

CALCULATED I

Fig. 6. Plot of calculated vs measured incoming longwave radiation for measurements

taken under clear skies during the 1977 and 1978 snow seasons. Circles, (3, indicate 1977 snow season measurements, and asterisks, *, indicate 1978 snow season measurements

1,03

,9C

,8C

0 .70 n~ I1: lad ,60

--_. ,5O

AO

LIJ ,20

,10

0 260

* o o o o

~o 0 o o ~ o

~ 1 7 6 1 7 6 8 , * 9 8 , I i

265 270 275 280 285

T E M P E R A T U R E

Fig. 7. Plot of the difference between computed and actual emissivity in respect to air temperature for measurements taken during the 1977 and 1978 snow seasons. Symbols as given in Fig. 6


Table 4. Regression Analysis of Combined 19 77 and 19 78 Clear Sky Measured vs. Calculated Incoming Longwave Radiation

Correlation Coefficient: 0.88 Regression Coefficient (Slope): 0.86 Standard Error of Slope: 0.10 Upper limit: 1.06 Lower limit: 0.67

Regression Analysis of 1978 Clear Sky Measured vs. Calculated Incoming Longswave Radiation Correlation Coefficient: 0.82 Regression Coefficient (Slope): 0~56 Upper limit: 1.01 Lower limit: 0.12

ity error with temperatures for both the 1977 and 1978 clear sky data. While errors do appear slightly larger at colder temperatures, the trend is not well established by these data. Because temperatures in the Sierra Nevada are typically much warmer than in more continental mountain ranges such as the Rocky Mountains, further tests will be made using data collected during much colder temperatures. Data from the forested site at the Valentine Reserve are compared to clear sky data in Fig. 8. The measurements made in the forest follow the same general trend as do the clear sky measurements but the magnitude of the error for the forested site is larger. All but five measurements in the Valentine data set were taken at the same location. These five were taken in more open areas away from forest cover, and are not significantly different from the other clear sky results. Fig. 8 thus shows the magnitude of the effect of canopy cover. Though none of the other measurements sites involved as dense a forest cover as did the Valentine site, many of them were in some proximity to sparser forest cover. It is clear that this could account forsome of the error associated with the other measurements. Fig. 9 illustrates this further. Measurements taken at the forested site are well separated from the others. The five marked measurements taken away from the forest are similar to other clear sky measurement points. With more careful data collection in a variety of forested sites it should be possible to determine a forest canopy view factor, similar to the thermal view factor described in eq. (8). Reifsnyder and Lull [ 17] point out that radiation from a forest is a function of the density of the cover. Price and Dunne [16] indicate that factors such as the snow depth are also important. If functions describing these effects can be developed, the longwave radiation model presented in this paper can be calibrated for radiation from forest cover. It is noteworthy that the Valentine data set was collected in late spring of


450

40o

35o

c~ 3oo w rm

25O w

r o o

% o ~ o

o% zoo e

'5~ 2~o z~o 36o ~ CALCULATED l ( W m "2)

i

45O

Fig. 8. Plot of calculated vs measured incoming longwave radiation for measurements taken during the 1977 snow season under clear skies and after ablation at the Valentine Reserve. Circles, o, indicate 1977 snow season measurements, and diarnonds,~, indicate Valentine Reserve measurements. Specially marked points,'~., indicate Valentine Reserve measurements not taken under a heavy forest canopy

1.0!

.8 QZ '

.6" >- I--- .5.

. 2

�9 O 0

i .88

j O o ~ , ~

o $ a 8~

VIEW FACTOR

Fig. 9. Plot of the difference between computed emissivity and actual emissivity over the view factor parameter for measurements taken during the snow season under clear skies and after ablation at the Valentine Reserve. Symbols as given in Fig. 8


a low snow year, after most of the snow had melted. The meteorological characteristics of the area during that period were much different than they had been during the snow season and air temperature and vapor pressures were significantly higher. The model produced reliable results in spite of these differences showing an increase in the calculated incoming longwave radiation as we would expect. For measurements made in sparse forest or clearings, the results are indistinguishable from values computed during the snow season. This is a good indication that the model will work adequately regardless of seasonal climatic variation.

5. Application of the Model Over an Area

The previous section describes how the model works at a point for which we have a measurement of air temperature, vapor pressure, and elevation. For an application of the model, it must be extended over a large area where we do not have measurements at every point.

5.1 Areal Estimate o f the Input Parameters

To compute clear sky incoming longwave radiation at a point we must know the elevation of the point, its near surface air temperature and vapor pressure, and the mean air pressure and air temperature at a reference location. With an estimate of these parameters for every point we can calculate the incoming longwave radiation over the area. The test areas were represented as digitized grids of elevations. A view factor was calculated for each point and stored as part of the topographic informa~ tion. Air pressure was computed for each point by eq. (10) using mean values of Bishop WSO data. Of the parameters necessary, only air temperature, Ta, and vapor pressure, ea, must be estimated for every point. Air temperature and vapor pressure vary with both time of day and elevation (air pressure). Variation with time is straight forward and can be represented as a periodic function. Variation with elevation is more complex and not likely to be a linear function. Both air temperature and vapor pressure were corrected to simplify variation with elevation. Air temperature was corrected to its sea level equivalent or potential temperature [6]:

6) = Ta( I OOO/pa)R/mCp. (13)

This correction removes variation in temperature produced solely by adia- batic effects of pressure changes with elevation. Under most conditions the potential temperature lapse rate over snow should be small or close to zero. Vapor pressure was assumed to have an exponential lapse rate.


A multiple regression technique based on field measurements was used to predict potential temperature and vapor pressure at all times (in radians) and elevations

Q,log ea =AO+ A. cos(Time)+ B. sin(Time)+C. Elev. (14)

The periodic diurnal variation is expressed by the second and third terms in eq. (14), and the assumption of linear variation with elevation is expressed by the last term. The first term is the mean value of the function. Though the results of this represent only an approximation of the true air temperature and vapor pressure, the values are well within the sensitivity tolerances shown in Figs. 1 and 2. For data from the five test sites collected on different dates and under a variety of weather conditions, the analysis of the regression (Table 5) shows that the differences between observed and predicted temperatures or vapor pressures were never more than a few degrees or millibar. In all cases the potential temperature change with elevation was not significantly different from zero at the 95% confidence level. For four of the five data sets the same was true for vapor pressure. There was a very small negative vapor pressure lapse rate for the Valentine Reserve data set probably caused by the lack of snow cover and difference between forested and unforested sites. The large amount of variance left unexplained for some of the data sets is caused by the small size and incomplete temporal sample of these data sets. Asterisks indicate values which could not be computed because the measured and predicted values were equal giving a perfect regression. The last two columns of Table 5 show that under all conditions studied in the southern Sierra Nevada, potential temperature and vapor pressure lapse rates are not different from zero, for zero lies within the confidence interval placed on all regression slopes. Thus, an assumption of neutral stability is adequate for this model. By converting air temperature to potential temperature, all systematic variation of temperature with elevation can be accounted for. The table also shows that variation in vapor pressure with elevation is so slight that it appears almost constant at all elevations. While this is probably incorrect, it is adequate for this model. The model is not sensitive to minor variation in vapor pressure, as shown by Figs. 1 and 2. Variation of vapor pressure with change in elevation is an order of magnitude or so less than diurnal variation, probably because a snow cover is a free evaporating surface. The decision to ignore a near surface vapor pressure lapse rate reduces the model complexity and data requirements and should not introduce a significant error in the results. When more data are available, a more accurate vapor pressure lapse rate will be substituted. A mean reference elevation was selected to establish the mean daily values for potential temperature and vapor pressure. For the test sites this was


Table 5. Analysis o f Regression on Air Temperature and Vapor Pressure Using Eq. (14)

% Explained Mean Computed Lapse Lower Upper Variance Residual Rate per meter Limit Limit

Mammoth Mountain Area 4/13/77 (7 measurements)

0 12% 6 K -0.004 K -0.077 0.069 ea 46% 1.25 mb -0.001 mb -0.003 0.002

Mount Langley Area 4/24/77 (20 measurements)

0 89* 2 K 0.003 K -0.001 0.005 ea *** 1.01 mb -0.000 mb *** ***

Kearsarge Pass Area 4/29/77 (8 measurements)

O 35% 1 K -0.020 K -0.082 0.041 ea 82% 1.06 mb -0.003 mb -0.008 0.002

Horsehoe Maedow Area 5/13/77 (25 measurements)

@ 52% 3 K 0.000 K -0.006 0.006 ea *** 1.13 mb -0.000 mb *** ***

Valentine Reserve Area 5/30/77 (28 measurements)

(3 85% 3 K -0.006 K -0.016 0.004 ea 83% 1.19 mb. -0.001 mb -0.002 0.001

Horsehoe Meadow Area 1/26/78 (7 measurements)

(3 66% 4 K 0.003 -0.005 0.011 ea 75% 0.22 mb 0.001 -0.004 0.006

(Note: alpha = 0.05)

taken as the average elevat ion o f all measurement poin ts used to compu te the regression coefficients. I f variat ion with elevation is zero, then poten t ia l t empera tu re or vapor pressure at a given t ime can be c o m p u t e d f rom the ampl i tude and phase o f the diurnal funct ion , relative to the mean diurnal value

@, log (ea) = A m p �9 cos ( 2 7r - (T irne-Phase) /24 } + A O. (15)

Potent ia l t empera tu re , O, and log vapor pressure, log (ea), can then be conver ted to Ta and ea by

T a = O/(lO00/Pa)R/mCp

ea = exp {log (ea) ) .

(16)

(17)

Table 6 shows the coeff ic ients used in eq. (15) for bo th air t empera tu re and vapor pressure at each test site. The results of this regression approach are encouraging because the technique is simple and it indicates tha t mul t ip le data col lect ion sites are no t necessary to drive the model . The rela t ionship

180 D. Marks and J. Dozier: A Clear Sky Longwave Radiation Model

Table 6. Regression Coefficients for Potential Temperature, (3, and log Vapor Pressure

Amplitude Phase Mean Mean Elev (m) Value

Mount Langley (3 8.59 13.52 3266 306 (4/24/77) log (ea) 0.49 13.92 3266 1.76

Kearsarge Pass (3 0.65 15.51 3267 301 (4/29/77) log (ea) 0.18 16.59 3267 1.42

Horsehoe Meadow (3 5.18 12.13 2929 300 (5/13/77) log (ea) 0.34 11.96 2929 1.62

Valentine Reserve (3 8.62 13.76 2875 311 (5/30/77) log (ea) 0.49 13.55 2875 2.38

Horseshoe Meadow (3 6.22 13.44 3025 299 (1/26/78) log (ea) 0.49 13.63 3025 1.28

of temperature and vapor pressure is good in that in all five cases the phase for each is almost the same. Data from the Mammoth Mountain test site were not evaluated because they were collected over only a 3 hour time span. In order to predict a reasonable amplitude and phase, the regression equation must have data over more than 1/4 of the diurnal period. For the other test sites the coefficients shown in Table 6 are a good reflec- tion of the measured values and as such can be used reliably to predict air temperature or vapor pressure at any time or elevation during the diurnal cycle. The very small amplitude for both variables observed for the Kearsarge Pass test site is reflected in the measured data. The measurements were made during an intense storm and the diurnal variations were damped during the period. The fact that the technique responds to variations in local conditions is expected and necessary. The simulation of longwave radiation over the test sites was accomplished by evaluating the model at every point in the digitized grids used to represent the test sites. Elevation and view factor were calculated for each point. Air pressure was estimated by eq. (10). Air temperature was calculated using eqs. (15) and (16) and vapor pressure from eqs. (15) and (17). Surface temperature was estimated using the mean daily air temperature for the elevation of the point. Eq. (7) was then used to calculate effective atmospheric emissivity from the air temperature, vapor pressure, and air pressure computed for each point. Finally for each point incoming longwave radiation was computed from eq. (9) using computed emissivity, air temperature, view factor, and surface temperature. Simulations were run for maximum, minimum, and mean daily values for each of the test sites except Mammoth Mountain where only one simulation was appropriate for reasons discussed above. A sample of the output of

"x k ~ , ...,--_ \\~o eoo l

Minimum Value 0 0 1 0 h r s

W 118 ~ 13~00 " W 118~

L 7 0 ' . 2 3 0

Mean Value 0610/1810 hrs

~o,

~,-&.) L J ~ -\'.,%

1,~,9o."~ ~--,~. Maximum Value 1210hrs

contour interval 10W~ a scale I 0 I 2 3 4 5 km

Fig. 10. Incoming longwave radiation Horseshoe Meadow - Cottonwood Pass Area, May 13, 1977

Minimum Value OI30hm

wlleoL3,oo ,, w l~ ' os 'OO"

,:~... ,~eo,~ o , t \ -~ ,,,.,-,.2o% , ~

~ x

Mean Value 0730 hrs

Maximum Value 1330 hrs

contour interval IOWm -2 scale I O I 2 25 4 5 km

Fig. l 1. Incoming longwave radiation Horseshoe Meadow - Cottonwood Pass Area, January 26, 1978

D. Marks and J. Dozier: A Clear Sky Longwave Radiation Model 183

January 26,1978

W I I 8 ~ ' ' W 1 1 8 ~ "

m y 13, 1977

stole contour intervQI I MJ m _

I 0 I 2 3 4 5km

Fig. 12. Integrated incoming longwave radiation Horseshoe Meadow - Cottonwood Pass Area, 24 hours

these s imulat ions is shown in Figs. 10 and 11. The response o f the simula- t ion to t opography is striking, but no t unexpec ted given the consis tent response o f the various input parameters to the same. Diurnal variat ion is p robab ly more cri t ical in the long run, because o f its higher f requency oscillations. The variat ions appear to range f rom about 30 Wm -2 to as much as 100 Wm-2 over one diurnal cycle. The mode l was also in tegra ted over a 24 hour per iod for both the 1977 and 1978 da ta (Fig. 12). The in tegra ted values are o f a surprisingly large magnitude - from 13 to 20 MJ m -2 for bo th dates - which are o f the same order


of magnitude as the solar input over a 24 hour period. It is interesting that while the range of values are the same for both dates, the distribution of those values is different.

6. Discussion of Model Results

6.1 Validity o f Model Assumptions

The model provides reasonable results at all times and elevations given the assumptions upon which it is based. Under clear skies in areas devoid of forest canopy, the model does very well. It remains, however, that a significant portion of an alpine snow catchment area is likely to be at least sparsely forested. It is also evident from the data collected - more measurements were made during cloud cover and storms than under-clear skies - that cloud cover provides a significant input to incident thermal radiation during the snow season. The model presented in this paper gives only part of the story. The importance of this model toward a better understanding of thermal radiation input in an alpine area is that it isolates input from two of four possible sources. If the source of incoming thermal radiation in an alpine area is limited to the atmosphere, clouds, surrounding slopes, and forest canopy cover, then by isolating inputs from one or two of these, the problem of determining the effect of the others becomes easier. The model as presented in this paper will allow us to calibrate our measurements for the effect of the atmosphere and surrounding terrain, making it easier to isolate cloud radiation or forest canopy radiation.

6.2 Usefulness o f the Model as Part o f an Energy Balance Snowmelt Model

The model as presented can provide information for use as input to an energy balance snowmelt model extended over a large area. The model provides areal information on clear sky incoming longwave radiation at all times and elevations in the study area. It responds well to both seasonal and diurnal variations in critical input parameters. The data needed to drive the model are relatively easy to obtain. Though the terrain and view factor data are complicated to compute initially, this input remains constant and need be done only once for any given area. The computations of air temperature and vapor pressure must be recalibrated from in situ data each time the model is run, but this information is not difficult to simulate either spatially or temporally. With additional data collection, estimates of these parameters could be further simplified. The regression coefficients used in eq. (15) could be replaced with weekly or monthly averages. The technique illustrates that one or two measurement sites run continuously or semi-continuously would


be adequate for a large area. The fact that diurnal variation is more critical than variation with elevation eliminates the need for simultaneous measurements at several elevations, or difficult and expensive field data collection for every simulation. Finally the model is inexpensive to run once it has been calibrated and the terrain files have been constructed. On an Itel AS-6, the model can be run for a specified time over a terrain grid of about 10 minutes of latitude and longitude spaced at 100 meters (28,000 points) in less than 1 minute of CPU time, for under $ 10.00. It can be integrated over a 24 hour period for $ 20.00 to ~; 40.00. Further cost reductions could be achieved by using mean daily values and a longer time step. The model accuracy is within a useful tolerance. The previous analysis showed that the model systematically under-predicts incoming longwave radiation because it does not account for input from forest cover and clouds. For clear skies, the error is not biased toward any type of event; the error has essentially the same distribution over all of the model parameters. The overall magnitude of the clear sky RMS error - about 30 Wm-2 _ is considerably less than we would anticipate for other energy balance parameters such as latent or sensible heat exchanges. Model errors will be reduced in the future with the development of effective canopy and cloud cover models.

6.3 App#cations of the Model for Satellite Radiometry

The use of satellite remote sensing data to determine the thermal characteristics of an alpine snow cover has been limited even though data are, or will be, available at a spatial and temporal resolution unmatched by ground based techniques - presently from NOAA and HCMM satellites and in the future from Landsat-D. One of the major drawbacks against using remote sensing data has been an inability to calibrate the data for the effects of the atmosphere and topography. The longwave radiation model described in this paper will allow us to calibrate thermal remote sensing data so that it can be used to determine surface temperature characteristics of the snow cover. With a digital terrain grid resampled to a resolution similar to the satellite data, the model can be used to determine atmospheric radiation at each point. This value can then be used as the atmospheric contribution to the thermal brightness of the point recorded by the satellite. The satellite data must be radiometrically and positionally corrected before the results will be meaningful. Thermal brightness, less the atmospheric component, can then be converted to surface temperature. Over a vegetated surface this can be difficult due to the wide variation in surface emissivity. For snow, however, the thermal properties are well known. Their variance is reasonably systematic and can be modeled. Under any conditions, the emissivity of snow is very close to unity. An initial test of this application of the model

13 Arch. ~e t . Oeoph. Biokl. B. Bd. 27, H. 2--3


will be done during a per iod when the snowcover is known to be isothermal so that the surface t empera tu re is known at all points . A compar ison will be made be tween thermal data from NOAA, HCMM, and Landsat-D (unfor tuna te ly , the thermal channel on Landsat-3 has failed) when they become available to evaluate the effects o f d i f ferent resolut ion cells. These efforts have been inhibi ted to date by the lack of available data in digital form. By providing p roper a tmospher ic cal ibrat ion for satelli te thermal data the model provides the o p p o r t u n i t y to use satelli te r ad iomet ry in an alpine area of high topographic variat ion. I f proven successful, this may provide us with a new tool for de termining the surface evergy exchange over an alpine snow cover.

Acknowledgements

This paper is based upon a Master's Thesis submitted to the Department of Geography, Uni,~ersity of California at Santa Barbara. The work was supported by the National Aeronautics and Space Administration, Grants NSG-5155 and NSG-5262, the National Oceanic and Atmospheric Administration, Grant 04-8-MO, and by research funds from the University of California, Santa Barbara.

References

1. Aase, J. K., Idso, S. B.: A Comparison of Two Formula Types for Calculating Long- wave Radiation From the Atmosphere. Water Resour. Res. 14, 623-625 (1978).

2. Anderson, E. A.: A Point Energy and Mass Balance Model of a Snow Cover. Nat. Oceanic Atmos. Admin., Tech. Mem. NWS-HYDRO-19 (1976).

3. Anderson, E. A., Baker, D. R.: Estimating Incident Terrestrial Radiation Under All Atmospheric Conditions. Quart. J. R. met. Soc. 97, 519-536 (1967).

4. Brunt, D.: Notes on Radiation in the Atmosphere. Quart. J. R. met. Soc. 58, 389-420 (1932).

5. Brutsaert, W.: On a Derivable Formula for Long-wave Radiation From Clear Skies. Water Resour. Res. 11,742-744 (1975).

6. Byers, H. R.: General Meteorology, 4th ed. New York: McGraw-Hill 1974. 7. Cox, S. K.: Observations on Cloud Infrared Effective Emissivity. J. Atmos. Sci. 33,

287-289 (1976). 8. Doronin, Yu. P.: Thermal Interaction of the Atmosphere and the Hydrosphere in

the- Arctic. Jerusalem: Keter Publishing House 1970. 9. Dozier, J., Outcalt, S. I.: An Approach Toward Energy Balance Simulation Over

Rugged Terrain. Geog. Analysis 11, 65--85 (1979). 10. Elsasser, W. M., Culbertson, M. F. : Atmospheric Radiation Tables. Boston: American

Meteorological Soc. 1960. 11. Fraser; R. S., Curran, R. J.: Effects of the Atmosphere on Remote Sensing. In:

Remote Sensing of Environment (Lintz, J., Simonett, D. S., eds.), pp. 34-85. Reading: Addison-Wesley 1976.


12. Kondratyev, K. Ya.: Radiation in the Atmosphere. New York: Academic Pres~ 1969. 13. Merimer, M., Seguin, E.: Comment on "On a Derivable Formula for Long-wave

Radiation From Clear Skies" by W. Brutsaert. Water Resour. Res. 12, 1327-1328 (1976).

14. O'Brien, H. W., Munis, A.: Red and Near Infrared Spectral Reflectance of Snow. In: Operational Applications of Satellite Snowcover Observations (Rango, A., ed.), pp. 345-360. NASA SP-391, 1975.

15. Paltridge, G. W., Platt, C. M. R.: Radiative Processes in Meteorology and Climatology. Amsterdam: Elsevier t976.

16. Price, A. G., Dunne, T.: Energy Balance Computations of Snowmelt in a Subarctic Area. Water Resour. Res. 12, 686-694 (1976).

17. Refisnyder, W. E., Lull, H. W.: Radiant Energy in Relation to Forest. U. S. Dept. Agr. Tech. Bull. 1344 (1965).

18. Stow, D. A.: Analysis of Landsat/MSS and Digital Terrain Tape Data in a Geobase Information Systems Context: Ventura County Study Area. M. A. Thesis, Univ. Calif., Santa Barbara, 1978.

19. U.S. Army Corps of Engineers: Snow Hydrology. Portland: U. S. Army Corps of Engineers, North Pacific Division, 1956.

20. Valley, S. L., ed.: Handbook of Geophysics and Space Environments. Bedford: Air Force Cambridge Research Laboratory 1965.

21. Yamamoto, G.: On Nocturnal Radiation. Tohoku Univ. Sci. Rep. 5, 24-43 (1950).

Authors' address: D. Marks and Dr. J. Dozier, Department of Geography, University of California, Santa Barbara, CA 93106, U. S. A.

13"

Documents

A clear-sky longwave radiation model for remote alpine areas