Estimation of daily averagover the Baiyangdian wate

Di Long a,b,*, Yanchun Gao a, Vijaya Institute of Geographic Sciences and Natural ResourcesbDepartment of Biological and Agricultural Engineering,cDepartment of Civil & Environmental Engineering, Tex

a r t i c l e i n f o

Article history:

1. Introduction

Compared with instantaneous net radiation, 24-h integrated netradiation or daily average net radiation (DANR) consisting ofdaily average net shortwave radiation (DANSR) and daily average

net longwave radiation (DANLR) has more applications foratmospheric and hydrologic modeling and water resources man-agement, and especially for quantifying land surface evapotran spi-ration (ET) from satellite imagery (Allen et al., 2006; Bastiaanssen,2000; Bisht et al., 2005; Fortin et al., 2008; Gao and Long, 2008;Gao et al., 2008; Samani et al., 2007). DANR is a critical variablelinking estimates of instantaneous latent heat ux (typically atsatellite overpass time) from energy balance-based models(Bastiaanssen et al., 1998; Su, 2002) and daily estimates of ET

* Corresponding author at: Department of Biological and Agricultural Engineer-ing, Texas A&M University, College Station, TX 77843, United States.

Journal of Hydrology 388 (2010) 217233

Contents lists availab

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.e lsE-mail address: descartes66@tamu.edu (D. Long).estimates from the proposed four observation-based method and that from an existing one observation-based method, against that from the Penmen equation solely using routine meteorological data indicatesthat the proposed method is capable of more accurately simulating DANLR than is the one observation-based method. Using the Penman equation as a reference, results show that overall the proposed methodhas a bias of 2.7 Wm2 and a root mean square error (RMSE) of 12.8 Wm2, whereas the one observa-tion-based method has a bias of 33.3 Wm2 and a RMSE of 39.6 Wm2 across 18 weather stations forsix tested days. In general, simulated DANR is shown to be reasonable over the entire study watershed forthe six clear sky days as a result of the improvement in the parameterization schemes of DANSR andDANLR. The resulting DANR would serve well as a critical variable linking instantaneous evaporative frac-tion to the estimates of daily ET primarily from remotely sensed data.

2010 Elsevier B.V. All rights reserved.Received 17 June 2009Received in revised form 26 January 2010Accepted 27 April 2010

This manuscript was handled byKonstantine P. Georgakakos, Editor-in-Chief,with the assistance of ChristaD. Peters-Lidard, Associate Editor

Keywords:Daily average net radiation (DANR)Daily average net shortwave radiation(DANSR)Daily average net longwave radiation(DANLR)MODISSloping land surfaces0022-1694/$ - see front matter 2010 Elsevier B.V. Adoi:10.1016/j.jhydrol.2010.04.042e net radiation from MODIS data and DEMrshed in North China for clear sky days

P. Singh b,c

Research, Chinese Academy of Sciences, Beijing, 100101, ChinaTexas A&M University, College Station, TX 77843, United Statesas A&M University, College Station, TX 77843, United States

s u m m a r y

Daily average net radiation (DANR) is a critical variable for estimation of daily evapotranspiration (ET)from remote sensing techniques at watershed or regional scales, and in turn for hydrological modelingand water resources management. This study attempts to comprehensively analyze physical mechanismsgoverning the variation of each component of DANR during a day, with the objective to improve param-eterization schemes for daily average net shortwave radiation (DANSR) and daily average net longwaveradiation (DANLR) using MODIS (MODerate Resolution Imaging Spectroradiometer) data products, DEM,and minimum meteorological data in order to map spatially consistent and reasonably distributed DANRat watershed scales for clear sky days. First, a geometric model for simulating daily average direct solarradiation by accounting for the effects of terrain factors (slope, azimuth and elevation) on the availabilityof direct solar radiation for sloping land surfaces is adopted. Specically, the magnitudes of sunrise andsunset angles, the frequencies of a sloping surface being illuminated as well as the potential sunshineduration for a given sloping surface are computed on a daily basis. The geometric model is applied tothe Baiyangdian watershed in North China, with showing the capability to distinctly characterize the spa-tial pattern of daily average direct solar radiation for sloping land surfaces. DANSR can then be success-fully derived from simulated daily average direct solar radiation by means of the geometric model andthe characteristics of nearly invariant diffuse solar radiation during daytime in conjunction withMCD43A1 albedo products. Second, four observations of Terra-MODIS and Aqua-MODIS land surfacetemperature (LST) and surface emissivities in band 31 and band 32 from MOD11A1, MYD11A1 andMOD11_L2 data products for six clear sky days from April to September in the year 2007, are utilizedto simulate daily average LST to improve the accuracy of estimates of DANLR. Comparisons of the DANLRJournal of

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ydro(in units of mm d1) on the basis of an assumption that retrievedevaporative fraction (dened as the ratio between latent heat uxto available energy) remains fairly constant during cloudless daysfor which advection occurs occasionally (Brutsaert and Sugita,1992; Crago, 1996; Kustas et al., 1994a; Shuttleworth et al.,1989). Daily ET can be subsequently obtained using the evapora-tive fraction to partition DANR (Ahmad et al., 2006; Bastiaanssenet al., 1998; Jiang and Islam, 2001; Norman et al., 2003; Su,2002). The evaporative fraction can signicantly affect the spatialrepresentation of estimates of ET across large heterogeneous areas,reecting the combined effects of the soil moisture, the availabilityof radiative energy, vegetation type and its state, and meteorolog-ical conditions on the latent heat ux (Batra et al., 2006; Nishidaet al., 2003). DANR, to a large extent, determines the magnitudeof estimates of ET for a given evaporative fraction from a pixelstandpoint. It is thus believed that a considerable effort shouldbe made to substantially improve the accuracy of both evaporativefraction and DANR, with the objective to make reliable predictionsof ET. On the other hand, although comparisons of satellite-basedlatent heat ux with point-scale ground observations or moreregionally with aircraft for several retrievals have been performed,it seems that to date there have not been universally acceptableapproaches to effectively assessing the accuracy of the extrapo-lated daily ET from evaporative fraction. Discretization ofground-based measurements, number of measurements, differ-ence between satellite-based pixel scales (e.g., 1000 m) and mea-surement scales (e.g., 100 m), and errors associated with suchmeasurements should be taken into account (Bisht et al., 2005;Kempf and Tyler, 2006; McCabe and Wood, 2006). It appears thatalthough evaluation of the accuracy of spatially distributed esti-mates of ET from remote sensing-based energy balance modelscannot be readily performed, improvements to the daily estimatesof ET could be potentially achieved by increasing the number and/or frequency of ground-based observations of each component ofDANR or signicantly enhancing the capability of the parameteri-zation scheme of DANR to represent reality.

In some cases, DANR can be obtained directly from eld mea-surements or weather stations (Bastiaanssen, 2000; Jegede, 1997;Kempf and Tyler, 2006; McCabe and Wood, 2006). Nevertheless,limited eld measurements and weather stations that can provideground-based measurements of DANR or the components of DANR(e.g., DANSR and DANLR) often inhibit practical applications of re-mote sensing-based surface ux models to large heterogeneousareas (Irmak et al., 2003; Samani et al., 2007; Su et al., 2005), espe-cially mountainous areas having exceedingly sparse measurementsor stations. Furthermore, weather stations are generally located inat areas so that the observations of DANR tend to represent lim-ited surrounding areas, which may exclude sloping land surfaces ormountainous areas (Allen et al., 2006; Thornton et al., 2000). Thenmaps of DANR and associated components are derived throughsome type of parameterization scheme incorporating meteorolog-ical and/or remotely sensed data for practical ET estimation overlarge domains (Allen et al., 1998; Bisht et al., 2005; Bois et al.,2008; Choudhury, 1997; Fortin et al., 2008; Hurtado and Sobrino,2001; Jacobs et al., 2000; Kim and Hogue, 2008; Lagouarde andBrunet, 1993; Mahmood and Hubbard, 2002; Samani et al., 2007;Thornton and Running, 1999).

Allen et al. (1998) proposed a framework, known as FAO56method and widely used to estimate DANR from routinely ob-served meteorological data for calculating reference ET and quan-tifying crop water requirement. The parameterization scheme ofDANSR seems to be applicable to at areas because of the exclusionof the effects of terrain factors (e.g., slope and azimuth) on solar

218 D. Long et al. / Journal of Hradiation. Regarding the parameterization scheme of DANLR, itshould be noted that the FAO56 equation involving the terms forcorrecting StefanBoltzmann Law using air humidity and cloudi-ness is a site-specic method, not applicable to large heteroge-neous areas. Therefore, to calculate DANR the FAO56 methodshould be used with caution for satellite-based ET estimationacross large heterogeneous areas.

Jacobs et al. (2000) utilized Geostationary Operational Environ-mental Satellite (GOES) data to detect cloud cover throughout aday and subsequently derived instantaneous direct solar radiationand net radiation on a 15-min basis and DANR for ET estimationover wetlands in the Paynes Prairie Preserve, North Central Florida.However, the relatively low spatial resolution of GOES data wouldnot be adequate to estimate DANR and ET over mountainous areas.Bisht et al. (2005) developed a sinusoidal model similar to Lagou-arde and Brunets (1993) methodology to estimate DANR based onestimates of instantaneous net radiation from MODIS data prod-ucts. One of the strengths of this model is direct simulation ofDANR from retrieved instantaneous net radiation for clear skydays, without the requirement of parameterization schemes foreach component of DANR. In addition, the model accounts forthe effects of differences in sunrise and sunset angles on DANR,with specication of varied values of sunrise and sunset anglesfor different a Day of Year (DOY) but same values for the entirestudy region for the same DOY due to domination of at areas overthe Southern Great Plains, United States. It is, however, noted thatthe model would not be suitable for implementation in mountain-ous areas because of existence of a wide range of sunrise and sun-set angles for sloping land surfaces. Moreover, integrating allcomponents of DANR into a simple sinusoidal model would intro-duce certain errors to estimates of DANR resulting from the differ-ence in the temporal phase of shortwave radiation and longwaveradiation throughout a day, showing that the longwave radiationis not negligible but the shortwave radiation is non-existent duringnighttime.

This study focuses on two issues associated with DANR estima-tion. First, in application of DANR and ET estimation across largeheterogeneous domains for clear sky days, the effect of terrain fac-tors on solar radiation should be quantied instead of simplifyingterrain through an assumption of uniform extensive slopes and azi-muths. For instance, some specic sloping land surfaces with steepslopes facing north in middle or high latitude areas may receive so-lar radiation only during a very short period or even may not re-ceive at all. Some sloping land surfaces, by contrast, may beilluminated by direct solar radiation twice a day, meaning thatthere exist two sets of sunrise and sunset angles. These extremeexamples are rare but may be important in some applications tomountainous areas (Allen et al., 2006; Gao et al., 2008). Second,DANLR also serves as a critical component in DANR. In many appli-cations, DANLR has been obtained using one observation of landsurface temperature (LST) from satellite imagery acquired at nearmidday as a surrogate of daily average LST in combination withmeteorological data to calculate daily upwelling and downwellinglongwave radiation (Hurtado and Sobrino, 2001; Kustas et al.,1994b; Lagouarde and Brunet, 1993; Roerink et al., 1997). In addi-tion, DANLR can also be estimated by making use of the FAO56method and incorporating a wealth of meteorological data. How-ever, for satellite observation-based ET estimation, the utility ofexisting methods to parameterize DANLR needs to be furtherexamined in that the difference in representativeness of daily aver-age LST induced by the difference in overpass time from differentsatellite platforms may result in the difference in retrievals ofDANLR. Furthermore, an associated problem has to be investigatedif one observation of LST can appropriately represent daily averageLST for calculating upwelling longwave radiation.

Two MODIS (MODerate Resolution Imaging Spectroradiometer)

logy 388 (2010) 217233sensors, onboard the Earth Observing System EOS-AM (Terra) andEOS-PM (Aqua), have remarkable advantages over other sensors forproviding much more spatially distributed land and atmospheric

data products, such as surface albedo, surface emissivity, atmo-spheric prole temperatures at relatively higher spatial and tem-poral scales, particularly the critical variable LST at most fourobservations per day. This opens a new opportunity to more reli-ably parameterize DANR across large heterogeneous areas and toeliminate the need for a large amount of ground-based measure-ments to estimate DANSR, DANLR, and DANR, and related site-spe-cic calibration for operational ET estimation.

The objectives of this study are to: (1) analyze physical mecha-nisms for variations in each component of DANR with time duringa day, and then improve the parameterization scheme of DANSR byquantifying the effect of terrain factors on solar radiation for slop-ing land surfaces; (2) improve DANLR estimation using four obser-vations of LST from MODIS data products, DEM and minimummeteorological data; and (3) examine the relationship betweenDANR and terrain factors.

watershed. In the recent two decades, drought frequency overBaiyangdian Lake has increased rapidly, in particular since year2000, showing that the water level of this lake declined belowthe warning water level of 6.5 m during several months in a yearand surface runoff into Baiyangdian Lake reduced drastically. How-ever, the precipitation of this watershed has not yet shown an evi-dently decreasing trend. Central and local administrations relevantto water resources management have thus conducted several in-ter-basin water transfer projects from reservoirs and rivers withinor adjacent to this watershed to address severe water shortage cri-ses and to sustain drinking water requirement and ecologicalintegrity over Baiyangdian areas. Therefore, it is critical to explorethe reasons for droughts over this area in the context of climatechange and intensifying human activities through reliable estima-tion of ET and water budget. Reliably modeling DANR is a criticalstep to determine ET amount and distribution over this watershedas indicated above.

wave radiation and to examine the difference in retrievals between

D. Long et al. / Journal of Hydrology 388 (2010) 217233 2192. Study site and data description

2.1. Study site

Located in North China, the study site is the Baiyangdian wa-tershed, extending in latitude from around 37.8 to 40.4N and inlongitude from around 113.3 to 116.6E (Fig. 1). Hebei and ShanxiProvinces and Beijing City contribute to 80.4%, 12.3% and 7.3% of itstotal area of 31, 200 km2, respectively. Elevations decrease fromthe northwest of the watershed, Taihang mountainous areas, tothe southeast plain, ranging from around 2784 m to 0 m, withshowing that mountainous areas (elevation above 100 m) occupyapproximately 53% of this watershed. Eight main streams of theDaqing River provide primary water sources for four irrigation dis-tricts, reservoirs, industrial and municipal use in this watershed, -nally converging to Baiyangdian Lake, the largest lake on the NorthChina Plain. In general, woodland and grassland dominate north-west mountainous areas, and cropland is distributed across plainareas, with statistics of a land use map (Fig. 1) derived from Land-sat TM images in the year 2000 showing that dry land, shrub andmoderate grassland account for 33.6%, 12.1% and 12.0% of the wa-tershed, the three largest land covers, respectively. Mean annualtemperature is between 6.8 C and 12.7 C (the daily maximum va-lue is 43.3 C and the daily minimum value is 30.6 C), with amean annual precipitation of 548 mm and a mean annual panevaporation of 15002000 mm according to historical weather re-cords of recent 50 years from Baoding, Shijiazhuang, Wutainshan,Weixian, and Huailai weather stations within or adjacent to theFig. 1. Location (left) and land use (right) of the Baiyangdia2.2. Data description

Meteorological data on a daily basis relevant to the parameter-ization schemes of DANSR, DANLR and DANR, such as daily meantemperature, daily mean vapor pressure, are determined by anaverage of in situ measurements from 18 weather stations withinthe study watershed. Of the 18 weather stations, Baoding and Fup-ing stations can provide the observations of air temperature, vaporpressure and atmospheric pressure at a 1-h interval. The other sta-tions can provide relevant observations at 6-h intervals. Daily ac-tual sunshine duration is available by accumulating in situmeasurements at 1-h intervals during daytime. Terrain factors, likeslope, azimuth and elevation, can be directly extracted from Shut-tle Radar Topography Mission (SRTM) digital elevation models(resampled to 100 m).

MODIS data products, MOD11_L2 (the level 2 MODIS LST andemissivities for bands 31 and 32 daily data) are retrieved as 1 kmpixel by a generalized split-window LST algorithm (Wan and Doz-ier, 1996). MOD11A1 and MYD11A1 from Terra-MODIS and Aqua-MODIS (the level 3 MODIS LST and emissivities for bands 31 and 32daily data), are generated in a sinusoidally projected tile by map-ping the level 2 LST product on a 1 km grid and are retrieved bythe algorithm taking into account the dependence of retrievedLST on the viewing angle based on the physics-based day/nightLST algorithm (Wan and Li, 1997). MOD11_L2, MOD11A1 andMYD11A1 were utilized to simulate daily surface upwelling long-n watershed in North China with relevant information.

ydrousing one observation of LST from MOD11_L2 and using fourobservations of LST from MOD11A1 and MYD11A1.

MCD43A3, the 16-day composite level 3 gridded albedo prod-ucts in the sinusoidal projection, provides both directional hemi-spherical reectance (black-sky albedo in the extreme case ofcompletely direct illumination) and bihemispherical reectance(white-sky albedo in the extreme case of completely diffuse illumi-nation) at a spatial resolution of 500 m. MOD04 provides daily le-vel 2 aerosol depth at 0.550 lm wavelength product of spatialresolution of 10 km. MCD43A3 and MOD04 were jointly utilizedto simulate land surface albedo in terms of the algorithm devel-oped by Lucht et al. (2000). Six clear sky days in the year 2007 wereselected in terms of the MOD11_L2 LST product with less than 10%cloud cover relative to the entire scene. Table 1 contains DOY, Ter-ra-MODIS overpass time and some variables with respect toweather conditions.

3. Methodology

DANR can be expressed as follows:

Rn;24 1 rSin;24 Sd;24 Ld;24 Lu;24 1where Rn,24 is the DANR (Wm2), r is the land surface albedo ()which is assumed to be similar to the surface albedo during themorning overpass (Bastiaanssen, 2000), Sin,24 is the daily average di-rect solar radiation (Wm2), Sd,24 is the daily average diffuse solarradiation (Wm2), (1 r)(Sin,24 + Sd,24) is also termed daily averagenet shortwave radiation (DANSR), Ld,24 is the daily average down-welling longwave radiation (Wm2), and Lu,24 is the daily averageupwelling longwave radiation (Wm2), (Ld,24 Lu,24) is also termeddaily average net longwave radiation (DANLR).

3.1. Parameterization scheme of DANSR

3.1.1. From S to S for sloping land surfaces

Table 1Day of year (DOY), Terra overpass time and relevant weather conditions for six clearsky days in the year 2007.

Calendar day(DOY)

Terra daytimeoverpass time,UTC

Dailyactualsunshineduration(h)

Dailypotentialduration (h)

Dailycloudiness(%)

25th April (115) 03:15 11.9 13.4 1.19th May (129) 03:25 and 03:30 11.8 13.9 18.615th June (166) 03:45 10.7 14.7 20.919th July (200) 03:35 12.1 14.4 47.213rd August (225) 03:35 and 03:30 12.1 13.6 13.519th September (262) 03:45 10.3 12.1 8.9

220 D. Long et al. / Journal of Hin in,24

Instantaneous direct solar radiation Sin for a given sloping landsurface at a given moment (typically satellite overpass) can be ex-pressed as:

Sin I0d2

cosi sm 2

m Pa101:3 cosi 3

where I0 is the solar constant (around 1367Wm2), d is the EarthSun distance in astronomical units (Appendix A1), s is the atmo-spheric transmissivity (), m is the optical air mass number (), Pais the atmospheric pressure (kPa), which could be assumed to bea function of elevation in this watershed (Pa = 101.3exp(-eleva-tion/8200)), i is the solar zenith angle (rad), and cos(i) is the cosineof solar zenith angle ().A sensitivity analysis of Sin to atmospheric attenuation variableshas been performed in order to quantify the degree to which theyinuence the magnitude of Sin (see Fig. 2). Results show that theatmospheric transmissivity and elevation are positively correlatedwith Sin, with a 10% increase in atmospheric transmissivity and ele-vation resulting in around a 14.6% and 0.061% increase in the mag-nitude of Sin, respectively. On the contrary, a 10% increase in solarzenith angle (SZA) will result in around a 12.5% reduction in Sin.This is because of an increase in the distance of sunlight propagat-ing in the atmosphere and thus an increase in atmospheric atten-uation. It is concluded that the atmospheric transmissivity isquite sensitive to the simulation of Sin. In many cases, instanta-neous atmospheric transmissivity for clear sky days is determinedmerely by elevation (Melesse and Nangia, 2005; Wu et al., 2006).This approximation would lead certain errors to resulting Sin, espe-cially for those days when cloud cover is signicant. However, itshould be pointed out that one of our purposes was to deriveSin,24 rather than Sin. We adopted another method to deal with dailyaverage atmospheric transmissivity in the following text. On theother hand, accurately simulating atmospheric transmissivity atsatellite overpass virtually requires detailed information on atmo-spheric composition with its state from radiosounding data. It is,however, not readily available in most cases. Through a measure-ment of atmospheric transmissivity for clear sky days, Liu and Jor-dan (1960) stated that s is between 0.45 and 0.75. In addition,Gates (1980) stated that under typical clear sky days, s is between0.6 and 0.7. Under extreme clear sky condition, s reaches around0.75. Here, we take s for 0.7 for clear sky days for which good qual-ity MODIS data products were available.

It is worthwhile to note that for at land surfaces, the solar ze-nith angle is simply a function of local standard time, latitude andsolar declination; for sloping land surfaces, it additionally incorpo-rates terrain effects, such as slope and azimuth (Fu, 1983; Morseet al., 2000):

cosi sin dsinu cosa cosu sina cos b cos d cosxcosu cosa sinu sina cos b cos d sina sinb sinx 4

where d is the solar declination (Appendix A2), u is the latitude(rad), a is the slope (rad), b is the azimuth (from due south, clock-wise positive value, counterclockwise negative value, so the rangeof value is [p, p]); x is the solar angle (=p(t 12)/12, and t isthe local standard time).

An expression of Sin,24 can be obtained through an integral of Sinfrom sunrise anglex1 to sunset anglex2. The instantaneous atmo-spheric attenuation sm in Sin should be replaced by the daily atmo-spheric transmissivity (a + bn/N) and then the expression should bedivided by the total length of 1 day 2p (in the unit of solar angle)for calculating the integral mean on the range of [p, p]:

Sin;24 a b nN I0

2pd2

Z x2x1

sin d u cos d cosx m

cos d sinb sina sinxdx

a b nN

I02pd2

u sin dx2 x1 m cos dsinx2 sinx1 sin b sina cos dcosx2 cosx1 5

u sinu cosa cosu sina cosb 6

v cosu cosa sinu sina cos b 7

logy 388 (2010) 217233N 12x2 x1p

8

olar

ydrowhere n is the actual sunshine duration (h), which was obtained byinterpolating 18 point-based observations of actual sunshine dura-tion by an inverse distance square method, N is the potential sun-shine duration (h), which can be expressed by the sunrise andsunset angles, x1 and x2, a is a regression constant characterizingthe fraction of extraterrestrial radiation reaching the Earth on over-cast days (n = 0), a + b is the fraction of extraterrestrial radiationreaching the Earth on clear sky days (n = N). Values of a and bmay vary with geographical locations and climate zones. The FAOmethod (Allen et al., 1998) recommends that a be taken as 0.25and b be taken as 0.50, if no actual solar radiation data are availableand no calibration has been carried out for improved a and b esti-mates. Chen et al. (1995) made a regression analysis using observa-tions of daily net radiation over North China Plain and concludedthat a and b could be specied as 0.17 and 0.54.

It is obvious that for theoretically solving Eq. (5), a parameteri-zation scheme of sunrise and sunset angles, which incorporates theeffects of terrain factors on the duration of sloping land surfacesbeing illuminated, the frequency of being illuminated during aday, and subsequently on the magnitude of the availability of di-rect solar radiation for sloping land surfaces, is virtually critical.

Fig. 2. Sensitivity analysis of instantaneous direct sD. Long et al. / Journal of HThe quantied x1 and x2 are thus input to Eq. (5) to computeSin,24. Apparently the parameterization scheme of sunrise and sun-set angles is only determined by the geometric relationship be-tween the solar incidence and the sloping surface, beingindependent of Eq. (5).

3.1.2. x1 and x2 for sloping land surfacesThe reasons why the critical solar angles (x1 and x2) should be

quantied and the relationship between critical solar angels forsloping land surfaces and for at land surfaces should be exploredare as follows: As we know, critical solar angles for at land sur-faces are symmetrical (Appendix A3), exhibiting the same absolutevalues but with inverse signs. However, the critical solar angles forsloping land surfaces probably show different absolute values,moreover, even the same sign. For instance, for certain sloping landsurfaces facing north and located in middle or high latitudes inwinter, they may receive solar irradiance merely in the morningor in the afternoon. For some particular cases, they may be illumi-nated twice during daytime, namely having two sets of x1 and x2.Consequently, if those sloping land surfaces were assumed to beat surfaces in the calculation of critical solar angles for estimatingdirect solar radiation, it would lead to gross errors in the estimatesof Sin,24 and eventually DANR and ET.The computed critical solar angles for sloping land surfaces canbe directly obtained by allowing Sin in Eq. (4) to be zero, obtainingat most two real roots with opposite or identical signs (AppendixA4 and A5). Only after comparing the computed critical solar an-gles for sloping land surfaces with that for at land surfaces interms of a set of physical and mathematical principles can wedetermine the absolute values with their signs of the actual criticalsolar angles for sloping land surfaces.

Let xs1 and xs2 be the roots of A4, respectively, and xs2 >xs1.Both xs1 and xs2 are a function of not only the latitude and solardeclination determining the macroscopic distribution of direct so-lar radiation from the perspective of the scene, but also the slopeand azimuth causing the microscopic variation in direct solar radi-ation for a specic sloping land surface. Additionally, let the sun-rise and sunset angles for at land surfaces be xH and xH (xHis absolutely positive) which are simply a function of latitudeand solar declination (Appendix A3). All mathematical expressionsof xs1, xs2, xH, and xH can also be found in Fu (1983), Gao et al.(2008) and the Appendix. Here, a comprehensive mathematicalanalysis of the relationship between (xs1, xs2) and (xH, xH)has been made in order to specify x and x for a given sloping

radiation Sin to atmospheric attenuation variables.

logy 388 (2010) 217233 2211 2

surface. The solutions of critical solar angles should inherently sat-isfy a set of physical and mathematical principles simultaneouslyas follows:

A. Only when cos(i)P 0 can the sloping land surface be illumi-nated, otherwise it would produce nonphysical solutions,such as the presence of receiving direct solar radiation whenspecic terrain actually deates sunlight.

B. The sunrise angle for sloping land surfaces,x1, should not beearlier than that for at land surfaces at the same latitude.Similarly, the sunset angle for sloping land surfaces, x2,should not be later than that for at land surfaces.

On the basis of the two basic principles stated above, the sun-rise and sunset angles can be specied by exploring the relation-ship among xs1, xs2, xH and xH.

a. Ifxs1 6x 6xs2, cos(i)P 0 can be satised; thus the sunriseand sunset angles can be specied in terms of the principleB, namely, x1 = max(xs1, xH); x2 = min(xs2, xH).

b. If x xs2, cos(i)P 0 can be satised; thus thesunrise and sunset angels can be specied in terms of prin-ciple B. There are four kinds of possibilities in case b.

(a) If computed |xs1| xH and |xs2| xH con-tradicting principle B, accordingly there is also one setof critical solar angle for this case, namely, x1 =xs2,x2 =xH.

(d) If computed |xs1| xH, both contradict-ing principle B, therefore the specic sloping land sur-face cannot receive direct solar radiation during thewhole day.

The four possibilities discussed in case b are illustrated in asketch in Fig. 3.

c. If real roots forxs1 andxs2 are non-existent, the sunrise andsunset angles can be specied directly in terms of principlesA and B. Therefore, there are two possibilities in case c.(a) If whatever value x is within its domain of denition

[p, p], cos(i)P 0 can be satised, corresponding toFig. 3. A sketch for illustrating determination of sunrise afraction of the daytime to the length of day (24 h) to the magnitudeof Sd

Sd;24 Sd N24 10

3.2. Parameterization scheme of DANLR

Numerous theories and practices have shown that althoughinstantaneous net longwave radiation accounts for only a smallportion of total instantaneous net radiation, daily average net long-wave radiation DANLR is not negligible due to its domination in thenighttime. In general, DANLR contributes a negative quantity toDANR in that the EarthAtmosphere system is able to balancethe accumulated energy available from the shortwave radiationin the daytime by constantly emitting longwave radiation duringthe whole day. Thereby, DANLR is also critical in the calculationof DANR.

DANLR is dened as the difference between the daily averagedownwelling longwave radiation Ld,24 from the atmosphere andthe daily average upwelling longwave radiation Lu,24 from theEarths surface as follows:

L24 Ld;24 Lu;24 earT4a erT4s 11where ea is the daily average atmospheric emissivity () which wascalculated by Brutsaerts (1975) formula shown in Eq. (12), r is theIt can be seen from Eq. (9) that Sd is 0.3 times the difference be-tween Sin not being attenuated by the atmosphere and Sin beingattenuated by the atmosphere, a relatively constant proportion.Pa involved in m and s do not vary greatly for a particular locationon a daily basis. Thereby, it seems that Sd,24 may rely on the solarzenith angle.

We examined the variation in Sdwith the solar zenith anglewith-in the range of [0, p/2] in an attempt to determine some kind of ap-proach to simulating Sd,24. The variations in Sin and Sd under theconditions of the atmospheric transmissivity of 0.7 and 0.6 and theelevations of 20 m and 1200 m have been modeled (see Fig. 4),respectively, based on Eq. (9). The results suggest that Sin varies dra-matically with the solar zenith angle during a day whereas Sd re-mains fairly constant during daytime except for short periods aftersunrise and before sunset (solar zenith angle approaches 90). Addi-tionally, Sin at higher elevations is greater than that at lower eleva-tions. On the contrary, Sd at higher elevations is smaller than thatat lower elevations. As for the effect of atmospheric transmissivityon solar radiation, Sin increases with atmospheric transmissivitywhile Sd decreases with an increase in atmospheric transmissivity.

It may be concluded that Sd remains fairly constant during day-nd sunset angles for sloping land surfaces in case b.

StefanBoltzmann constant (5.67 108 Wm2 K4), Ta is the dailyaverage air temperature at screen level (K). Maps of daily average

temperature can be made by multi-variate regression analysisbased on the observations of air temperature from weather stations

Fig. 4. Variations in direct and diffuse solar radiation (Wm2) with solar zenith angle on the range of [0, p/2] under different atmospheric transmissivity and elevationconditions.

D. Long et al. / Journal of Hydrology 388 (2010) 217233 223Fig. 5. Distributions of sunrise and sunset angles (rad) of the Baiyangdian watershed on February 9, May 9, August 13 and November 11.

and their corresponding elevations, longitudes and latitudes. e is theland surface emissivity which can be calculated using a nonlinearformula (Liang, 2004) shown in Eq. (13). Ts is the land surface tem-perature (K) which can be obtained fromMOD11_L2, MOD11A1 andMYD11A1 data products.

ea 1:24ea=Ta1=7 12

e 0:273 1:778e31 1:807e31e32 1:037e32 1:774e232 13where ea is the daily average vapor pressure at screen level (hPa),e31 and e32 are the emissivities in MODIS channels 31, 32, respec-tively, which can also be obtained from MOD11_L2 data product.

It should be pointed out that plenty of research and applica-tions relative to estimation of DANLR have only taken advantageof one near midday observation of LST from some type of remo-tely sensed images. On the one hand, remotely sensed imageswith high spatial but low temporal resolution or fewer bands(e.g., Landsat TM and ASTER) are not capable of offering sufcientinformation on thermal infrared bands for retrieving LST. On theother hand, remotely sensed images with high temporal resolu-tion (e.g., MODIS) have not yet been adequately exploited for esti-mating DANLR. In many applications, estimation of Lu,24 isgenerally based on an assumption that if a satellite overpassesa study site at near midday, the instantaneous LST retrieved from

Table 2Distributions of sunrise and sunset angles (rad) for 2 days in the summer half-year and 2 days in the winter half-year over the Baiyangdian watershed in the year 2007.

Calendar day(DOY)

AveragexH

Rang of x1 Range of x2 The number of pixels notreceiving direct solar radiation

The number of pixels havingtwo sets of sunrise and sunset angles

9th May (129) 1.826 [1.836, 0.405] [0.442, 1.836] 529 112nd August (224) 1.781 [1.789, 0.335] [0.404, 1.789] 215 19th February (40) 1.349 [1.357, 1.337] [1.261, 1.357] 522 01st November (305) 1.351 [1.354, 1.369] [1.287, 1.354] 571 0

224 D. Long et al. / Journal of Hydrology 388 (2010) 217233Fig. 6. Variations in simulated daily average direct solar radiation (Wm2) with slopes for different azimuths across the Baiyangdian watershed on April 25, 2007.

some kind of remotely sensed data can be taken as the daily aver-age LST for calculating Lu,24 (Hurtado and Sobrino, 2001; Kustaset al., 1994b; Lagouarde and Brunet, 1993; Roerink et al., 1997).However, the use of one observation of remotely sensed LST ac-quired at near midday perhaps needs to be further examined, gi-ven that the time of polar orbiting satellites overpass varies muchwith remote sensing systems, which would result in the differ-ence in the magnitude of LST and thus the estimates of Lu,24.Comparisons of estimates of DANLR from the use of one observa-tion of Terra-MODIS LST and that from the weather data-basedPenman equation (Penman, 1948) given as Eq. (14) were per-formed to investigate the usefulness of one observation-basedmethod.

L24 rT4max;d T4min;d

2

!0:56 0:25 eap 0:1 0:9 nN

14

where Tmax,d and Tmin,d are the daily maximum and minimum tem-perature (K), respectively; and ea is the daily average vapor pressure(kPa). Eq. (14) has been proven to be capable of reliably estimatingDANLR across the study site (Yin et al., 2008). It should be notedthat Eq. (14) is a site-specic empirical equation. It was developed

and calibrated to estimate DANLR by circumventing the require-ment of LST and land surface emissivity that had not been readilyavailable for physically-based Eq. (11) through conventional meth-ods, especially at watershed or regional scales. The advent of re-mote sensing techniques provides an opportunity to capturespatially consistent and distributed variables (e.g., LST, and e). Inaddition, it should be emphasized that one of our purposes is to de-velop a method to estimate DANLR directly based on Eq. (11) usingremotely sensed data in conjunction with minimum of meteorolog-ical data. It is expected that the retrieved distributed DANLR has ahigher spatial resolution in comparison with the predictions madefrom the Penman equation. The proposed method would eliminatethe need for calibration required by the Penman equation when ap-plied to other regions.

MODIS data has prominent advantages over other remotelysensed data that it can offer a wealth of thermal infrared informa-tion from both Terra and Aqua satellites. We may utilize at mostfour observations of remotely sensed LST for a given area to trackthe diurnal cycle of LST, greatly helpful in retrieving DANLR overlarge heterogeneous areas. The four observations of MODIS LSTare acquired around 10:30 a.m. and 10:30 p.m. for Terra-MODIS,and 1:30 a.m. and 1:30 p.m. for Aqua-MODIS, respectively. Every

D. Long et al. / Journal of Hydrology 388 (2010) 217233 225Fig. 7. Variations in simulated daily average direct solar radiation (Wm2) with slopes for different azimuths across the Baiyangdian watershed on September 19, 2007.

two snapshots of each satellite are just in the processes of the riseand fall of LST during a day. This particularity is greatly benecialto capturing the diurnal cycle of LST and thus to estimating Lu,24.Since meteorological variables (e.g., air temperature) and land sur-face uxes (e.g., radiation) basically present similar periodic uctu-ations with solar zenith angle for clear sky days, or just havecertain lag phrase, LST which is greatly affected by the variationpatterns of meteorological variables and land surface uxes fol-lows closely the diurnal variation in the air temperature as well.We rst tted a three order polynomial function using the fourobservations of Terra-MODIS and Aqua-MODIS temperatures, andthen calculated the average value of this tting function on[0, 24]. The tting of a three order polynomial function can be ex-pressed as

Ts R 240 a1t3 a2t2 a3t a4dt

24 3456a1 192a2 12a3 a4 15

where ai (i = 1, . . . , 4) are regression coefcients which can be ob-tained by tting a polynomial function using the four observationsof MODIS LST. Ultimately, it seems logical that the averagevalue could serve as the daily average LST in Eq. (11) to calculateDANLR.

4. Results and discussion

4.1. Sunrise and sunset angles

The distributions of sunrise and sunset angles across the studywatershed for 4 days are presented in Fig. 5 and Table 2, in which2 days are in the summer half-year (9 May and 13 August) and2 days are in the winter half-year (9 February and 1 November)for explicitly illustrating their essential spatial patterns.

From Fig. 5 and Table 2, it appears that the adopted geometricmodel is capable of capturing the spatial variability in sunriseand sunset angles under complex terrain conditions over the entirescenes. First, sunrise angles are earlier for the summer half-yearthan those for the winter half-year and sunset angles for the sum-mer half-year are later compared with those for the winter half-year. In other words, the direct solar radiation availability for slop-ing surfaces in the summer half-year is larger than that in the win-ter-half year. Second, there are marked differences between thesunrise angles for sloping surfaces and those for at surfaces acrossthis watershed for a specic day. For instance, the difference be-tween the earliest sunrise angle and the latest sunrise angle onNovember 1 incredibly reaches the order of 10.4 h (1.369 +1.354 = 2.723 rad), and similarly, the difference between the earli-est sunset angle and the latest sunset angle also spans a long per-iod of 10.1 h (1.354 + 1.287 = 2.641 rad). This highlights the

226 D. Long et al. / Journal of Hydrology 388 (2010) 217233Fig. 8. Frequency distributions of the difference between simulated Sin,24 (Wm2) from(Wm2) from the adopted geometric model for sloping land surfaces with slopes larger tspecied).assuming the land surface to be horizontal across the study watershed and Sin,24han 25 and 45 on April 25 and September 19, respectively (a bin size of 5 Wm2 is

heterogeneity in the terrain of the study watershed. For some spe-cic sloping land surfaces in the study watershed, they can only beilluminated by direct sunlight late in the morning or early in theafternoon that are quite shorter compared with the at surface atthe same latitude. Third, the number of pixels that could not re-ceive direct sunlight or have two sets of sunrise and sunset anglesis relevant to their specic slopes, azimuths and DOY. There wasapproximately an area of 5.71 km2 not receiving direct solar radi-ation on November 1. In addition, there was only one pixel onMay 9 and August 13 being illuminated twice a day. It is noted thatthe accuracy of solar critical angles depends largely on the resolu-tion and accuracy of DEM used. This means the ner DEM, themore accurate magnitudes of sunrise and sunset angles.

In summary, if the sunrise and sunset angles for the whole wa-tershed were simulated irrespective of terrain factors, it wouldintroduce gross errors to the simulation of sunshine duration andSin,24 for sloping surfaces. What we have done is restoring realisticillumination conditions in simulation of Sin,24 by adequately takingaccount the effects of terrain factors on deriving critical solar an-gles, which would greatly improve the spatial representation ofestimates of DANR and ET.

4.2. Daily average direct solar radiation

Daily average direct solar radiation can be derived throughinputting quantitatively determined sunrise and sunset angles toEq. (5) and observations of actual solar duration and other terrain

Sin,24 for sloping land surfaces will decrease with an increase inslopes up to lower than Sin,24 for the at land surface at certainlarge slopes. Namely, the sloping land surfaces facing south havethe potential to receive more direct solar radiation compared withthe at land surfaces (slope = 0). In addition, the hottest slope andmodeled Sin,24 for April 25 were generally smaller than that for Sep-tember 19 (25, 39.5 and 39.5 for the sloping surface facing south,southwest and southeast on 25 April; 29.8, 45.5 and 38.6 on 19September). This could be ascribed to different solar declinationvalues, April 25 having a smaller solar declination value than Sep-tember 19 (12590 for 25 April, 0350 for 19 September that is quiteclose to the autumnal equinox in 2007). By contrast, the slopingland surface facing north (north, northwest and northeast) couldnot present similar variation trends as the sloping land surface fac-ing south, exhibiting that Sin,24 for the surface facing north declinedrapidly with an increase in slopes from the at surface (slope = 0).The declining rate for September 19 was more rapid than that forApril 25. As for the variation trends for different azimuths, Sin,24for the land surface facing due south and due north vary withslopes more dramatically than that facing southwest or southeastand northwest or northeast. The southwest and southeast azi-muths tend to receive much more direct solar radiation comparedwith other azimuths of this study watershed on the two testeddays.

Frequency distributions of the difference between the simu-lated Sin,24 from assuming the entire land surface to be at and thatfrom the geometric model for sloping land surfaces larger than 25

D. Long et al. / Journal of Hydrology 388 (2010) 217233 227parameters. For examining the characteristics of the variation indirect solar radiation with terrain factors and evaluating the utilityof the adopted geometric model to estimate Sin,24, the variationtrends in Sin,24 with slopes for given azimuths on April 25 and Sep-tember 19 have been, respectively, examined (see Figs. 6 and 7).

Both days explicitly show that there is a kind of hottest slopefor the sloping land surface facing south (due south, southwest,and southeast), presenting that from 0 to the hottest slope, Sin,24increases with slopes. However, if slopes exceed the turning point,Fig. 9. Simulated DANSR (Wm2) over the Baiyangdianand 45 on April 25 and September 19, respectively, are shown inFig. 8. The results indicate that the largest differences between twomethods would incredibly reach the order of 199.4 Wm2 and204.6 Wm2 on April 25 and September 19, respectively. In gen-eral, the simple way would underestimate Sin,24 by 10.6 Wm2

and 17.0 Wm2 when the slope is larger than 25 for the twotested day. Furthermore, with an increase in slope from larger than25 to larger than 45, the difference between the two methodswould increase, showing that the standard deviation increasedwatershed for six clear sky days in the year 2007.

signicantly from 39.6 Wm2 to 60.1 Wm2 on April 25 and from57.9 Wm2 to 70.3 Wm2 on September 19, respectively.

To sum up, it was found that the proposed geometric model hasthe capability to capture the characteristics of the variation in di-rect solar radiation with not only the latitude and solar declination,but also terrain factors (slope, azimuth, elevation) across the entirewatershed. Without accounting for terrain effects, the resultswould not present the detailed spatial heterogeneity and temporalvariation trends in modeled Sin,24, and would result in gross errorsunder some specic terrain conditions.

4.3. Daily average net shortwave radiation

After Sin,24 and Sd,24 were simulated on the basis of the geomet-ric model and the characteristic of diffuse solar radiation, DANSRcan be ultimately calculated in combination with the surface albe-do from MODIS black-sky and white-sky albedo products(MCD43A3, 500 m) and MODIS aerosol optical depth product(MOD04, 1000 m). It is noted here that the resolution of Sin,24and Sd,24 (100 m) differs from that of MODIS data products. Sin,24and Sd,24 had to be thus resampled to 500 m by bilinear interpola-tion so as to be consistent with the resolution of MODIS albedoproducts.

Fig. 9 and Table 3 show the essential characteristics of the spa-tial distribution of DANSR from the proposed method over theBaiyangdian watershed. It can be inferred that terrain factors areprimarily responsible for the spatial variability in DANSR underthe condition of the uniform distribution of actual sunshine dura-tion in Eq. (5). The retrieved DANSR on April 25, May 9, August 13,and September 19 distinctly show that DANSR over northwestmountainous areas is larger than that over southeast plain areas,resulting primarily from the difference in atmospheric attenuationbetween mountainous and plain areas. Higher elevation areas usu-

Table 3Statistics about DANSR (W m2) over Baiyangdian watershed for six clear sky days inthe year 2007.

Calendar day (DOY) Maximum Minimum Mean Standarddeviation

25th April (115) 347.1 94.8 292.8 18.59th May (129) 349.4 132.5 301.4 15.415th June (166) 366.1 171.0 304.4 15.019th July (200) 382.0 184.1 335.6 11.113rd August (225) 354.8 104.4 300.0 17.719th September (262) 342.1 21.9 249.8 27.9

Fig. 10. Comparisons of estimates of DANLR (Wm2) from the one observation-based m18 weather stations over the Baiyangdian watershed for six clear sky days in the year 2

from.m.,.

R2

0.60.30.00.10.3

228 D. Long et al. / Journal of Hydrology 388 (2010) 217233Table 4Bias, root mean square error (RMSE), and coefcient of determination (R2) for DANLRobservations of MOD11A1 and MYD11A1 LST acquired on 10:30 a.m. and 10:30 pmeteorological data across 18 weather stations for six clear sky days in the year 2007

Calendar day (DOY) Using one observation of MODIS LST

Bias (Wm2) RMSE (Wm2)

25th April (115) 43.3 45.89th May (129) 48.7 54.815th June (166) 34.4 41.519th July (200) 27.9 32.613rd August (225) 34.2 35.7

19th September (262) 12.6 16.3 0.4Total 33.3 39.6 0.7ethod and the four observations-based method against the Penman equation across007.

using one observation of MOD11_L2 LST acquired around 10:30 a.m. and using fourand 1:30 a.m. and 1:30 p.m., respectively, relative to the Penman equation using

Using four observations of MODIS LST

(Wm2) Bias (Wm2) RMSE (Wm2) R2 (Wm2)

1 14.0 16.7 0.447 2.5 7.4 0.681 0.3 13.9 0.014 5.5 8.6 0.556 8.9 11.4 0.68

6 8.0 12.1 0.440 2.7 12.8 0.81

ally correspond to lower atmospheric attenuation and thereforelarger shortwave radiation. Furthermore, surface albedo is foundto be an important factor affecting the spatial variation in DANSR.It is obvious that DANSR of Baiyangdian Lake is larger than that ofsurrounding dry land for six tested days. This is probably becauseof the surface albedo of water body and ambient humid environ-ment being smaller than that of dry land, therefore showing a rel-atively larger DADSR. DANSR on June 15 and July 19 did not showsimilar spatial distributions as the other four clear sky days, whichcould be attributed to heterogeneous distribution of observed sun-shine duration across the entire study watershed for the 2 days.

This tampered the spatial pattern of DANSR that should exhibitfor clear sky days. With respect to the variation in DANSR withDOY, the mean of estimates of DANSR increased with dates, peakedon July 19 with a value of 335.6 Wm2, and then decreased withdates. This trend could be closely related to the variation in solardeclination.

4.4. Daily average net longwave radiation

DANLR from four observations of LST from MOD11A1 andMYD11A1 data products (termed four observations-based methodhereafter), and one observation of LST from MODIS11_L2 (termedone observation-based method hereafter) were produced. Theirutility and accuracy were examined in detail on the basis of thePenman equation over 18 sites for six tested clear sky days.

Results (Fig. 10 and Table 4) suggest that a noticeable discrep-ancy between the one observation-based method and the Penmanequation exists, showing the maximum bias and RMSE on the or-der of 48.7 Wm2 and 54.8 Wm2 on May 9, respectively, andthe minimum bias and RMSE on the order of 12.6 Wm2 and16.3 Wm2 on September 19, respectively. The four observa-tions-based method could however dramatically improve theaccuracy of estimates of DANLR, exhibiting good agreement withthe Penman equation in terms of the maximum bias and RMSE ofonly 14.0 Wm2 and 16.7 Wm2 on April 25, respectively. In addi-tion, the results also clearly demonstrate a relatively strong corre-lation between the four observation-based method and thePenman equation in terms of a higher R2 in comparison with the

Fig. 11. Comparisons of estimates of DANLR (Wm2) from the one observation-based method and the four observations-based method against the Penmanequation across 18 sites over the Baiyangdian watershed for the whole studyperiod.

D. Long et al. / Journal of Hydrology 388 (2010) 217233 229Fig. 12. Simulated DANR (Wm2) over the Baiyangdianone observation-based method. Overall, the one observation-basedmethod systematically underestimates DANLR due primarily torelatively higher LST values acquired around 10:30 a.m. comparedwith relatively lower daily LST from the four observations-basedmethod.

Fig. 11 shows a comparison for the two methods against thePenman equation across 18 weather stations for all tested clearwatershed for six clear sky days in the year 2007.

sky days. It is clear that the one observation-based method showeda relatively larger bias and a RMSE on the order of 33.3 Wm2and 39.6 Wm2, respectively. However, the four observations-based method agrees reasonably well with the Penman equation,

indicating a bias, RMSE and R2 of 2.7 Wm2, 12.8 Wm2 and0.81, respectively. Accordingly, it can be concluded that the pro-posed method that combines four observations of MODIS LST withminimum meteorological data (Ta and ea) is of the capability toreliably derive DANLR against the Penman equation that has beenproven to be applicable to the study site based on extensive groundmeasurements.

4.5. Daily average net radiation

After obtaining each component of Eq. (1), DANR can be ulti-mately calculated over the study watershed (see Fig. 12, Table 5).It can be seen from the results that rst, the mean of estimates ofDANR increased from 25 April, peaked on 19 July with a maximumof 229.4 Wm2, and then showed decreased, reaching a minimumof 131.1 Wm2 on September 19. The temporal variation in DANR

Table 5Estimated DANR (Wm2) over the Baiyangdian watershed for six clear sky days in theyear 2007.

Calendar day (DOY) Maximum Minimum Mean Standarddeviation

25th April (115) 347.4 25.1 140.7 21.69th May (129) 285.6 57.8 142.6 24.115th June (166) 402.9 86.7 183.1 27.019th July (200) 409.6 135.0 229.4 23.013rd August (225) 356.8 34.9 178.2 20.119th September (262) 277.0 0.0 131.1 20.6

230 D. Long et al. / Journal of Hydrology 388 (2010) 217233Fig. 13. Relationships between DANR and elevation across the study watershed for six clear sky days in the year 2007.

We acknowledge that the proposed method does not involve acomplex radiative transfer model. Nevertheless, it incorporates a

ydrocan probably be ascribed to the solar declination which profoundlyimpacts DANSR. The closer to the summer solstice (June 22 in2007), the larger radiation energy the land surface could receive un-der the clear sky condition. Second, although the spatial distribu-tion of DANLR attenuates to some degree the effect of terrainfactors on DANR for sloping land surfaces due to DANLR havingno strong relationship with specic slopes and azimuths, an appre-ciable difference in DANR between mountainous and plains areaswas still observed on April 25, May 9, August 13, and September19, respectively. Estimates of DANR across the southeast plain wereapparently larger than that in the northwest mountainous areas onJune 15. This rests in the fact that the distribution of actual sunshineduration was not homogeneous on June 15, with the observationsfrom the southeast sites showing larger values than that from thenorthwest. Owing to clouds obstructing some portions of LSTimages acquired on July 19, the Penman equation was thus usedto produce DANLR to make up those portions, thereby showing rel-atively larger values compared with the entire scene.

The relationship between retrieved DANR and elevation wasinvestigated to show how elevation inuences macroscopic distri-bution of DANR throughout the entire study watershed. In general,atmospheric composites, temperature, humidity and LST may varyto different degrees with elevation, which could induce the varia-tions in direct and diffuse solar radiation, atmospheric downwel-ling and surface upwelling longwave radiation, and eventuallyresult in the variations in DANR and ET from plains to mountainousareas.

Fig. 13 shows the relationship between DANR and elevation forsix clear sky days. Overall, DANR decreased slightly with anincrease in elevation, showing a maximum amplitude of 20Wm2 km1 on August 13 and a minimum amplitude of 1.6Wm2 km1 on April 25, respectively, with the exception of Sep-tember 19 showing a small amplitude of 2.5 Wm2 km1 but aninverse variation trend. In addition, the coefcients of determina-tion R2 are generally low, implying that there is no direct relation-ship between DANR and elevation. This means that DANR basicallyremains invariant with an increase in elevation. The variation trendin DANR with elevation appears to be determined by the combinedeffects of each component of DANSR and DANLR. First, with an in-crease in elevation, the atmosphere tends to be rare, and the airdensity, dust, impurity and water content in atmosphere tend toreduce, therefore resulting in an increase in atmospheric transmis-sivity and direct solar radiation. However, diffuse solar radiationwill reduce in terms of the ndings from Fig. 4, thus restrainingthe increase in total shortwave radiation. Second, as to the varia-tion in DANLR with elevation, the air temperature at screen leveland LST can decrease with an increase in elevation, resulting inreductions in atmospheric downwelling and surface upwellinglongwave radiation simultaneously. Eventually, DANR remains rel-atively invariant. However, it shows very scattered points stem-ming from large differences in land cover types, terrain factors,and actual sunshine duration for a specic location.

It should be emphasized that our purpose is to retrieve DANRfrom MODIS data products and DEM in combination with mini-mum meteorological data at regional or watershed scales. Result-ing estimates of DANR could not be directly compared withground-based measurements for the dearth of radiation observa-tions over this area. We acknowledge that the proposed methodwarrants further validation about the nal estimates of DANR.However, the parameterization schemes for each component ofEq. (1) do provide more reasonable estimates compared with exist-ing parameterization schemes that do not account for the effect ofterrain factors, particularly restoring realistic spatial variability in

D. Long et al. / Journal of HDANR across mountainous areas. The proposed scheme for esti-mating DANSR was validated against the Penman equation. Onthe other hand, although a handful of ground-based DANR wouldcomplex framework to quantify solar radiation for sloping surfaces.In addition, it adequately utilizes four observations of MODIS LSTto improve the accuracy of retrievals of DANLR. One can be con-dent that the improvement to DANSR and DANLR would signi-obtain, there are no universally acceptable methods to comparespatially distributed estimates with point-based ground observa-tions due to the discontinuity, scale issues, and limited numberof observations (Bisht et al., 2005; McCabe and Wood, 2006).

5. Conclusions

DANR is a critical variable linking instantaneous latent heat uxto daily ET, but in many applications of estimation of DANR and ETfrom remote sensing techniques. However, existing parameteriza-tion scheme of DANR appears to be less than suitable for capturingits substantial distribution pattern across large heterogeneousareas, particularly mountainous areas because: (1) the parameter-ization scheme of DANSR does not involve a kind of physical mech-anism to characterize the heterogeneity in Sin,24 for sloping landsurfaces, and (2) applicability of the parameterization scheme ofDANLR only using one observation of remotely sensed LST mayvary with satellite platform systems, leading uncertainties to DAN-LR estimation due to different capabilities of near midday LST torepresent the daily average LST.

In this present study, Sin,24 is parameterized through taking intoaccount the effect of terrain factors, such as slope, azimuth and ele-vation on direct solar radiation. Specically, the sunrise and sunsetangles for a given sloping surface is quantied. Besides, the physi-cal mechanism governing the variation in diffuse solar radiationwith solar zenith angle is investigated in order to involve Sd,24 inthe calculation of DANSR. Results indicate that the geometric mod-el has the capability to characterize the variability in Sin,24 over theentire study watershed, explicitly showing that the southwest andsoutheast azimuths have relatively larger magnitudes of direct so-lar radiation compared with other azimuths on April 25 and Sep-tember 19, respectively, and there exist the hottest slopes forsurfaces facing south. Improvements in the spatial representationof Sin,24 would signicantly improve the distributions and magni-tudes of DANSR and DANR. Without incorporating terrain factorsinto the parameterization schemes, the difference in Sin,24 wouldbe as large as 199.4 Wm2 and 204.6 Wm2 on April 25 and Sep-tember 19, respectively. Furthermore, the steeper the slope, thelarger difference would occur, implying the robustness of theparameterization schemes. DANSR can be ultimately derived frommodeled Sin,24, Sd,24 and MODIS albedo products.

An approach to simulating DANLR is proposed through incorpo-rating four observations of MODIS LST and surface emissivitiesfrom MOD11_L2, MOD11A1 and MYD11A1 in conjunction withminimum meteorological data, aiming at circumventing the de-ciency in the use of only one observation of remotely sensed LST.The retrieval accuracy is evaluated on the basis of the Penmanequation that has been shown to be able to provide reliable esti-mates of DANLR across the study site but needs relatively moremeteorological data. Comparisons of retrievals from the four obser-vation-based method and one observation-based method againstthe Penman equation are performed, showing a bias of2.7 Wm2 and a RMSE of 12.8 Wm2 for the proposed methodand a bias of 33.3 Wm2 and a RMSE of 39.6 Wm2 for theone observation-based method across 18 weather stations for sixtested clear sky days.

logy 388 (2010) 217233 231cantly improve the accuracy of DANR estimates, particularly itsspatial distribution across large heterogeneous areas, contributingto operational regional ET estimation for water resources planning

and ood monitoring from remote sensing. It should be noted that

ydroA.1. The EarthSun distance can be expressed as

d 1 0:0167 sin 2pDOY 93:5365

A1

where d is the EarthSun distance, DOY is the day of year.

A.2. The solar declination can be expressed as

d 0:409 sin 2p365

DOY 1:39

A2

A.3. The sunset angle for at land surface given as

xH cos1 tanu tan d A3where u is the latitude (rad). The magnitude of sunrise angle for atland surface is the same as that of sunset angle but with negativesign, namely xH.

A.4. The roots for solving cos(i) = 0, thus the computed critical solarangles can be expressed as

x cos1uv tan d sinb sina

1 u21 tan2 d

q1 u2

0@

1A A4

and the signs of A4 can be specied by Eq. (A5):

x sin1u sin b sina tan d v

1 u21 tan2 d

q1 u2

0@

1A A5

Referencesfor cloudy days, DANSR can also be estimated from the proposedmethod with successive MODIS albedo products and observationsof actual sunshine duration for calculating daily average atmo-spheric transmissivity (a + bn/N). Because of lack of remotelysensed LST in cloudy days, DANLR would be estimated directlywith the Penman equation using meteorological data. Atmosphericradiative transfer models or more sophisticated methods withproles of atmospheric states can also be used to address dailyaverage atmospheric transmissivity and DANLR in cloudy days(Forman and Margulis, 2009).

Acknowledgments

This work was jointly supported by the National Natural Sci-ence Foundation of China (Grant numbers 40871198 and40301007) and the Innovation Project of the Chinese Academy ofSciences (CAS) (Kzcx2-yw-126-04). We would like to very muchthank two anonymous reviewers, editor-in-chief and associate edi-tor for their valuable comments and helpful suggestions.

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D. Long et al. / Journal of Hydrology 388 (2010) 217233 233

Estimation of daily average net radiation from MODIS data and DEM over the Baiyangdian watershed in North China for clear sky daysIntroductionStudy site and data descriptionStudy siteData description

MethodologyParameterization scheme of DANSRFrom Sin to Sin,24 for sloping land surfaces1 and 2 for sloping land surfacesFrom Sd to Sd,24 for sloping land surfaces

Parameterization scheme of DANLR

Results and discussionSunrise and sunset anglesDaily average direct solar radiationDaily average net shortwave radiationDaily average net longwave radiationDaily average net radiation

ConclusionsAcknowledgmentsAppendix AThe EarthSun distance can be expressed asThe solar declination can be expressed asThe sunset angle for flat land surface given asThe roots for solving cos(i)=0, thus the computed critical solar angles can be expressed as

References