## 1. Introduction

Actual evapotranspiration ET_{act} can be estimated in different ways. Among them are (i) in situ measurements using lysimeters or flux towers, (ii) the crop coefficient approach by Allen et al. (Allen et al. 1998), and (iii) remote sensing algorithms. Kalma et al. (Kalma et al. 2008) divide the latter into two main classes: empirical/statistical approaches and energy balance approaches. The first correlates ET_{act} either to air–surface temperature differences (e.g., Caselles et al. 1998) or to vegetation indices (e.g., Neale et al. 1989). The second requires solving the energy balance of the surface (e.g., Bastiaanssen et al. 1998).

The main disadvantage of in situ measurements is that they do not capture the spatial variability of diverse and large terrains. The crop coefficient approach requires knowledge of a land-cover-specific crop coefficient *K*, which is mostly unknown for natural vegetation. The disadvantage of remote sensing algorithms is that sufficient cloud-free data must be available. The advantage, however, is their ability to cover large and diverse terrain and long time series at low costs.

Given the orographic effects of the Andes, an appropriate methodology for estimating ET_{act} in central Bolivia should capture the spatial variability caused by the large differences in altitude and land cover. We implemented an existing satellite-based energy balance approach [Surface Energy Balance Algorithms for Land (SEBAL)] and adapted it for the effects of elevation, slope, aspect, and atmospheric properties, varying strongly throughout the study area. The area consists of the upper part of the Mamoré River basin, extending roughly to 160 000 km^{2} and covering a very diverse terrain with elevations ranging from 250 to 5000 m above mean sea level, including 12 contrasting ecological regions (Ibisch and Mérida 2003) (Figure 1). Among them are southeastern parts of the Amazon, mountainous humid forests (Yungas), dry forests (Chaco), savannahs, and highland vegetation (Puna). The geographical center of the area is formed by the two national parks Amboró and Carrasco, which combined protect about 12 000 km^{2} of mostly the Yungas. Located along the eastern slopes of the Andes, the Yungas form a diverse transition zone between highlands and lowlands (Ibisch and Mérida 2003). These forests contribute to the water supply of the lower agricultural areas and cities, including the 1.5-million-inhabitant city Santa Cruz de la Sierra (Pauquet et al. 2005).

The objectives of this paper are to discuss the adaptations applied to SEBAL and to present spatial- and land-cover-specific estimates of ET_{act} for central Bolivia. The information is useful for the quantification of hydrological services provided by natural ecosystems, including the two national parks Amboró and Carrasco. The following sections present the modifications applied, the input data, results, and a comparison between estimated annual ET_{act} and measured precipitation.

## 2. Methodology

### 2.1. Actual evapotranspiration

_{act}by solving the surface energy balance for the latent heat flux

*λ*ET

_{act}, the energy used for actual evapotranspiration,where

*R*is net radiation (W m

_{n}^{−2}),

*G*

_{0}is the soil heat flux, and

*H*is the sensible heat flux. Net radiation

*R*is the sum of incoming minus the sum of outgoing long- and shortwave radiation. In SEBAL,

_{n}*G*

_{0}is expressed as an empirical function of

*R*, surface albedo, and vegetation cover. Here,

_{n}*H*is estimated from the temperature difference

*δT*between 0.1 and 2.0 m above the surface as follows: First, SEBAL solves the energy balance for a very dry and for a very wet location by assuming that, in the case of the first

_{a}*λ*ET

_{act}and in the case of the latter,

*H*equals zero. Then,

*δT*can be calculated for the dry location and is assumed to equal zero for the wet location. To obtain

_{a}*δT*for all locations, SEBAL expresses

_{a}*δT*as a linear function of altitude-corrected surface temperature

_{a}*T*

_{0dem}[where the subscript dem denotes digital elevation model (DEM)] by interpolating between the values of the dry and the wet location (Figure 2). This interpolation allows for estimation of

*H*and thus

*λ*ET

_{act}for all locations. SEBAL estimates fluxes of Equation (1) for the moment of satellite overpass and clear-sky conditions. (See appendix A for a list of all variables.)

### 2.2. Reference evapotranspiration

Reference evapotranspiration ET* _{r}* is the evapotranspiration from a hypothetical surface. In the case of the United Nations Food and Agriculture Organization (FAO) method (Allen et al. 1998), this surface consists of a grass reference crop of 0.12-m height, which is actively growing and adequately watered. It has a surface roughness of 70 (s m

^{−1}) and an albedo of 0.23 (unitless, which is denoted here as -). The latent heat flux of this reference surface is estimated according to the Penman–Monteith equation (appendix B, step 26). As explained in section 2.4, ET

*is used for the purpose of converting instantaneous to daily actual evapotranspiration. Instantaneous ET*

_{r}*is calculated for the instant of satellite overpass under clear-sky conditions. Daily ET*

_{r}*is calculated for all days regardless of the cloud conditions. Here, ET*

_{r}*is estimated on a grid-by-grid basis from daily air temperature; dewpoint temperature; and wind speed and duration of direct sunshine from 58 and 12 meteorological stations, respectively. A DEM is used for the spatial interpolation of meteorological observations as documented in section 2.4.3.*

_{r}### 2.3. Adjustments applied to SEBAL

We adapted SEBAL for the effects of elevation, slope, aspect, and atmospheric properties on incoming solar radiation. We do this by (i) using the angle at which solar radiation reaches mountainous surface and (ii) estimating the attenuation of solar radiation affected by altitude, aerosols, water vapor, and clouds. The complete set of equations of the adapted SEBAL algorithm is given in appendix B. The following paragraphs include only equations that differ from SEBAL.

#### 2.3.1. Instantaneous fluxes used for actual and reference evapotranspiration

##### 2.3.1.1. Solar incidence angle

*θ*(Duffie and Beckman 1991). Here,

*θ*is the angle between a solar beam and a line perpendicular to the surface and ranges from 0 to ½

*π*, where 0 means that the incoming radiation is perpendicular to the surface and ½

*π*means that it is parallel to the surface,where

*δ*is the solar declination (rad),

*ϕ*is the latitude (rad),

*s*is the slope (rad),

*γ*is the surface azimuth angle (rad), and

*ω*is the sun hour angle (rad). The angle

*γ*is equal to the aspect ranging from −

*π*to +

*π*. The aspect is obtained from a digital elevation model. Unfortunately, it is not possible to account for shadows in a simple way.

##### 2.3.1.2. Instantaneous atmospheric transmissivity

*τ*is the ratio of incoming solar radiation at Earth’s surface over incoming solar radiation at the top of the atmosphere and is here used for the estimation of incoming solar radiation. According to the approach of Ineichen and Perez (Ineichen and Perez 2002), clear-sky transmissivity

*τ*

_{1}can be expressed as a function of the Linke turbidity coefficient

*T*, altitude, and solar zenith angle. The coefficient

_{L}*T*gives the number of clean and dry atmospheres necessary to produce the observed extinction of incoming solar radiation. The smallest possible value of

_{L}*T*at sea level is therefore equal to 1. The source and magnitude of

_{L}*T*used in this application are given in section 3 and Table C1 (see appendix C for supporting tables),where

_{L}*τ*

_{1}= clear-sky transmissivity (-);

*a*

_{1}= 5.09 × 10

^{−5}× altitude + 0.868 (m);

*a*

_{2}= 3.92 × 10

^{−5}× altitude + 0.0387 (m);

*f*

_{h}_{1}= exp(−altitude/8000) (-);

*f*

_{h}_{2}= exp(−altitude/1250) (-);

*T*is the Linke turbidity coefficient; and am is the altitude-corrected optical air mass (Kasten and Young 1989) where mr is the relative optical air mass (Rapti 2000),where

_{L}*P*is atmospheric air pressure,

*P*

_{0}is reference atmospheric air pressure at sea level (

*P*

_{0}= 101 325 Pa), and

*ζ*is the solar zenith angle at the time of satellite overpass (rad).

##### 2.3.1.3. Instantaneously incoming solar radiation

*K*

^{↓}is estimated as a function of the solar constant

*S*(W m

^{−2}), the inverse squared relative distance Earth–Sun

*d*(-), slope

_{r}*s*(rad), solar incidence angle

*θ*(rad), and clear-sky transmissivity

*τ*

_{1}(-). According to Morse et al. (Morse et al. 2000),

#### 2.3.2. Daily fluxes used for reference evapotranspiration

##### 2.3.2.1. Daily atmospheric transmissivity

*τ*

_{24}is relevant for the estimation of daily incoming solar radiation. Here,

*τ*

_{24}must account for the presence and movement of clouds throughout the day. This is approximated by combining the spatial information of instantaneous transmissivity of clear and cloudy skies with the temporal information on the duration of direct sunshine as measured by meteorological stations,where

*a*is the transmissivity of the cloudy sky

_{s}*τ*

_{2},

*b*is the difference in transmissivity between clear and cloudy sky (

_{s}*τ*

_{1}−

*τ*

_{2}),

*N*is the maximum possible sunshine (h day

^{−1}), and

*n*is the actual duration of sunshine (h day

^{−1}). The transmissivity of a cloudy sky

*τ*

_{2}is obtained from the sum of the optical depth of a clear sky

*d*

_{1}and the optical depth of clouds

*d*

_{2}. According to the relation between transmissivity and optical depth given by Jacob [Jacob 1999, Equation (7.31)],where

*d*

_{3}is the optical depth of a cloudy sky (-),

*d*

_{1}is the optical depth of a clear sky (-), and

*d*

_{2}is the optical depth of clouds (-). Source and magnitude of

*d*

_{2}are given in section 3 and Table C2;

*d*

_{2}enters the algorithm as a single number, presenting the areal average of

*d*

_{2}of the study area.

##### 2.3.2.2. Daily incoming solar radiation

*ω*by (2

*π*/48) from (−

*π*/2) to (+

*π*/2). Note that

*τ*

_{24}is used here, not

*τ*

_{1}, assuming

*τ*

_{24}being constant throughout the day. Thus,where cos(

*θ*) is cos(

_{i}*θ*) with varying

*ω*.

##### 2.3.2.3. Net longwave radiation of the reference surface

*τ*

_{24}/

*τ*

_{1}. Both ratios are equal because all other factors cancel out. Thus,where

*L*

_{n}_{24accr}is accumulated 24-h net longwave radiation (J m

^{−2}day

^{−1}) used for ET

*,*

_{r}*σ*is Stefan–Boltzmann constant (

*σ*= 5.67 × 10

^{−8}W m

^{−2}K

^{−4}), and

*e*is actual vapor pressure (kPa). Instantaneous net longwave radiation of the reference surface is only relevant for clear-sky conditions, because only for those cases will SEBAL be used to determine the surface energy balance terms. For clear skies, the ratio of actual to clear-sky radiation equals 1, leading towhere

_{a}*L*

_{nr}is instantaneous net longwave radiation (W m

^{−2}). Note that Equations (12) and (13) are applied only to the reference surface. Outgoing longwave radiation for real surfaces are estimated according to appendix B, step 30.

#### 2.3.3. Further modifications

Further adjustments of SEBAL include a different choice on the estimation of atmospheric emissivity *ε _{a}* and soil heat flux

*G*

_{0}. SEBAL expresses atmospheric emissivity

*ε*as a function of transmissivity

_{a}*τ*

_{1}(Bastiaanssen et al. 1998, computation step 14). We prefer to express

*ε*as a function of actual vapor pressure (Brunt 1932; appendix B, step 28). The reason for this preference is that

_{a}*ε*and

_{a}*τ*

_{1}have different sensitivities for different factors. Here,

*ε*is most sensitive to the abundance of water vapor within a few hundred meters above the surface and less sensitive to factors that may determine

_{a}*τ*

_{1}such as the abundance of aerosols from biomass burning. With respect to

*G*

_{0}, we choose the approach from Su (Su 2002) (appendix B, step 33).

### 2.4. Interpolations

#### 2.4.1. Converting instantaneous to daily fluxes

_{act}from remote sensing data implies that the estimated fluxes represent the fluxes occurring at the moment of satellite overpass. For most applications, however, it is much more interesting to obtain daily or yearly fluxes. This may be done by multiplying the evaporative fraction Λ =

*λ*ET/(

*R*−

_{n}*G*

_{0}), with 24-h averaged net radiation, assuming Λ to be constant throughout the day (e.g., Bastiaanssen et al. 2005). Alternatively, daily fluxes may be obtained by (i) estimating a crop coefficient

*K,*the ratio of instantaneous actual to instantaneous reference latent heat flux, and (ii) multiplying

*K*with daily reference evapotranspiration (e.g., Bashir et al. 2008). Thus,where ET

_{24}is 24-h accumulated actual evapotranspiration (mm day

^{−1}),

*K*is the crop coefficient (-), and ET

_{r}_{24}is 24-h accumulated reference evapotranspiration (mm day

^{−1}). This method relies on the assumption that

*K*is constant throughout the day, which is verified in section 5.

#### 2.4.2. Obtaining air temperature for time of satellite overpass

*L*

^{↓}; (ii) net longwave radiation of the reference surface

*L*

_{nr}; (iii) virtual air temperature

*T*; and (iv) temperature-related constants, including the slope of saturation vapor pressure curve Δ, latent heat of vaporization

_{υ}*λ*, and saturation vapor pressure

_{r}*e*(Table C3). The available air temperature measurements from meteorological stations consist of two values per day, namely, a minimum value and a maximum value valid for 24 h. However, the algorithm requires an estimation of the air temperature valid for the moment of satellite overpass. To obtain this value, we interpolate between minimum and maximum values using a general sine function. The function was calibrated and validated with hourly data from one month for two stations: one located in the highlands and one located in the lowlands. Unfortunately, more hourly station data were not available. The function is then used to obtain air temperature values valid for the moment of satellite overpass from observed minimum and maximum values,wherewhere

_{s}*T*

_{min/max}is the minimum and maximum air temperature (°C) and

*t*

_{min/max}is the time of minimum and maximum air temperature (

*t*

_{min}= 6.5 h and

*t*

_{max}= 15 h). The mean difference between interpolated and measured air temperature at the time of satellite overpass equals 1°C, with a standard deviation of 1°C.

#### 2.4.3. Spatial interpolations

We spatially interpolate meteorological measurements of air temperature, dewpoint temperature, wind speed, and duration of direct sunshine. Dewpoint temperature and duration of direct sunshine are directly spatially interpolated by applying geostatistics (kriging). The spatial interpolation of air temperature is done by (i) converting absolute temperature to potential air temperature, (ii) spatially interpolating the obtained values, and (iii) converting them back to absolute air temperature. Wind speed is spatially interpolated by (i) estimating wind speed at 200-m height above the weather stations assuming a logarithmic wind profile, (ii) spatially interpolating the obtained values, and (iii) bringing them back to the surface, again assuming a logarithmic wind profile (appendix B, step 21). For the last step, a proper estimation of the surface roughness *z*_{0m} is necessary. We develop a surface roughness map by assigning *z*_{0m} values to land-cover types according to the Davenport–Wieringa classification (Stull and Aherns 2000) (Table C4). The spatial interpolation of wind speed therefore requires a land-cover map, which may be obtained from classified satellite data such as Moderate Resolution Imaging Spectroradiometer (MODIS) land-cover type (MOD12).

### 2.5. Choice of wet and dry pixel

SEBAL estimates the near surface to air temperature difference *δT _{a}* between 0.1 and 2.0 m above the surface from the linear interpolation between the surface temperature for a dry and wet location. For such locations, we choose the hottest and coldest pixel from the altitude corrected surface temperature

*T*

_{0dem}. Varying slightly among the different days, the hottest pixel is located in the very dry highlands (−18.9°S, −66.3°W), whereas the coldest pixel is located at a lake in the lowlands (−15.8°S, −64.3°W).

## 3. Data

The required input data consist of meteorological data and satellite data. We obtain the latter from the MODIS sensor (carried by the polar orbiting satellite *Terra*) and use the optical depth of clouds (MOD06), land surface temperature and surface emissivity (MOD11), land-cover type (MOD12), enhanced vegetation index (MOD13), and albedo (MOD43) (King et al. 2003). The MODIS data have a grid size of 1 km with respect to MOD06, MOD11, and MOD12; 250 m in the case of MOD13; and 500 m in the case of MOD43. This application considers the hydrological year of October 2003–September 2004. Because of year-round cloud cover, we obtain only one image per month for MOD11, MOD13, and MOD43. We use 16 MOD06 images and a single MOD12 image. The chosen dates of each image are documented in Table C5. Dates differ among MOD11, MOD13, and MOD14, because the latter two present temporal mosaics of 8, 8, and 16 days respectively. The required meteorological data consist of daily air temperature, dewpoint temperature, wind speed, and duration of direct sunshine. The first three are obtained from 58 meteorological stations and the latter are from 12 meteorological stations provided by the National Climatic Data Center (NCDC) and Freemeteo (see online at http://freemeteo.com), respectively. Next to satellite data and meteorological observations, we use the Linke turbidity coefficient provided by Solar Radiation Data (SODA; see online at http://www.soda-is.com/eng/index.html) (Table C1). This Linke turbidity coefficient is based on (i) global information from satellites measuring global clear-sky radiation [National Aeronautics and Space Administration (NASA) Surface Radiation Budget Project (SRB)], precipitable water vapor [NASA Water Vapor Project (NVAP)], and aerosol optical depth (Pathfinder) and (ii) ground information about turbidity from radiation or aerosol measurements [Aerosol Robotic Network (AERONET)]. Also, this adaptation of SEBAL requires the use of a digital elevation model.

For comparison purposes, we use daily precipitation data from 27 stations provided by Bolivia’s national service of meteorology and hydrology [Servicio Nacional de Meteorología e Hidrología (SENAMHI)]. To validate the constancy of *K* we use flux tower measurements collected in the context of the Large-Scale Biosphere-Atmosphere Experiment in the Brazilian Amazon (LBA; Miller et al. 2009). These data include half-hourly measurements from 3.5 years (June 2000–November 2003) of latent heat flux, incoming shortwave radiation, water vapor concentration, air temperature, air pressure, and wind speed. The tower is located outside the project area in the Tapajos National Forest in Brazil.

## 4. Results

The results consist mainly of 12 monthly average reference ET maps (mm day^{−1}), 12 monthly crop coefficient *K* maps (-), and the resulting yearly actual evapotranspiration map (mm yr^{−1}). Figure 3 summarizes the range and distribution of yearly ET_{act} for the upper Mamoré River basin. A third of the basin shows a yearly ET_{act} of 750–1000 mm yr^{−1}. Only few areas (1%) show ET_{act} > 1250 mm yr^{−1}. Such areas are identified as Yungas. Figure 4 shows the spatial distribution of *K* (top left) and ET_{act} (top right), with highest values in the sub-Andes, high values in the northern and humid part of the lowlands, lower values in the drier southern part of the lowlands, and lowest values in the much drier highlands. The bottom-left panel of Figure 4 elucidates the spatial detail of the data with higher *K* values for forest and lower *K* values for savannah and deforestation. Two high-resolution Landsat satellite images of the same location are given for comparison. The bottom-right panel of Figure 4 gives yearly ET_{act}/precipitation ratios, as calculated for meteorological stations. Here, we see the lowest ratios in the humid lowlands and the highest ratios in the drier highlands.

Table 1 lists the mean annual ET_{act} and crop coefficient *K* for 14 different subecoregions and land covers. In addition, the average rainfall of each subecoregion is given. Average station data are included for the subecoregions where precipitation was measured from October 2003 to September 2004. Magnitude and spatial distribution of the estimated ET_{act} are realistic: ET_{act} increases with increasing precipitation but remains below measured precipitation. The highest ET_{act} values are found for subecoregions where water availability and transpiration rates are likely to be high, namely, Yungas and flooded Amazon forests. The lowest values are found for subecoregions where water availability and transpiration are low, namely, inter-Andean dry forests, montane Chaco, and semihumid Puna of the Andean highlands. The spatial variability of ET_{act} is higher for drier ecoregions and lower for more humid ecoregions. Deforestation of pre-Andean Amazon forests reduces ET_{act} by 22%.

Annual-mean ET_{act} and *K* for different subecoregions and land cover.

To illustrate the change of ET_{act} throughout different ecoregions, a transect was drawn from the lowlands to highlands (points A and B in Figure 1, respectively), resulting in the profile of Figure 5. The gray area shows yearly ET_{act}, whereas the black line presents the surface elevation. The profile shows the significant difference of ET_{act} between lowlands (point a) and highlands (point b), the decline of ET_{act} for deforested areas (point c), and the extreme high ET_{act} values for Yungas (point d). The latter indicates the important role that Yungas play in the regulation of the hydrological cycle.

## 5. Comparison

Results are compared to rainfall measurements at the locations of meteorological stations. Usually, annual actual evapotranspiration is less than annual precipitation, and the difference between both presents the runoff, neglecting changes in groundwater storage. The bottom-right panel of Figure 4 gives the location of the meteorological stations used. In 23 cases, precipitation exceeds ET_{act}. In four cases, however, ET_{act} exceeds rainfall by 32, 55, 110, and 120 mm yr^{−1}, corresponding to 9%, 13%, 27%, and 42% of annual measured precipitation, respectively (Table C6). Such negative water balances may be due to (i) an overestimation of ET_{act} for these areas, (ii) provision of runoff from surrounding terrain, (iii) changes in groundwater storage, or (iv) lacking representativeness of a point measurement for a pixel value. To remain conservative, we conclude that SEBAL gives reasonable ET_{act} estimations in the lowlands and Andean slopes and possibly overestimates ET_{act} in some parts of the Andes with altitudes above 2700 m.

Additionally, we validate the applied methodology of temporal interpolation of instantaneous ET_{act}. For this purpose, we assess the constancy of *K* throughout the day for measurements from a flux tower located in Tapajos National Forest in Brazil, which is representative for the Amazon forest. Actual latent heat flux is directly measured, whereas reference latent heat flux is calculated from measured incoming shortwave radiation, water vapor concentration, absolute air temperature, air pressure, and wind speed. The analysis only considers daytime measurements [0700–1800 local time (LT)] and excludes extreme values (i.e., 0 > *K* > 10) likely to result from measurement errors. The summary of this analysis is given in Figure 6 showing a box plot diagram with median, quartiles, minimum, maximum, extreme, and outliers of *K* during the daytime. The diagram indicates that *K* is constant in the afternoon but not in the morning.

The mean daily standard deviation of *K* from 3.5 years equals 0.51, with a mean of all *K* values equal to 1.06. In our case, we estimate *K* for 1100 LT and assume this *K* to be valid for the entire day. Considering Figure 6, it is likely that a *K* estimated at 1100 LT underestimates the real overall *K*.

## 6. Conclusions and discussion

We adapted SEBAL for the effects of elevation, slope, aspect, and atmospheric properties on incoming solar radiation and implement it for central Bolivia. We conclude that the spatial distribution of estimated ET_{act} appears realistic when compared to the topography of the study area: ET_{act} is high in wet areas such as flooded Amazon forests and low in dry areas such as Puna in the Andean highlands. The results show (i) ET_{act} and *K* values for 14 different subecoregions and land covers; (ii) the far above average ET_{act} for Yungas, indicating their key role in regulating the hydrological cycle; and (iii) a reduction of ET_{act} by 22% over deforested land.

The implementation of SEBAL for the upper part of the Mamoré River basin provides information useful for the quantification of the basin’s water balance, hydrological services, impacts of deforestation, and water-use efficiency of crops and agricultural managing techniques. Future investigation should improve the estimation of incoming shortwave radiation by using data from geostationary satellites (Schüttemeyer et al. 2007). We also recommend a sensitivity analysis, which shows the importance of input variables for the determination of the fluxes. This should include a sensitivity analysis with respect to the location of the wet and dry pixel.

## Acknowledgments

The authors thank for the cooperation of the International Center for Tropical Agriculture (CIAT) and Fundación Amigos de la Naturaleza (FAN-Bolivia). We are grateful for the support from Prof. Dr. Pavel Kabat from the Wageningen University and Research Centre (WUR) and Prof. Dr. Wim Bastiaanssen from the Delft University of Technology (TU Delft). We thank the two anonymous reviewers for their important suggestions. We are grateful for the free access to the LBA data and for the support from the NeWater project.

## Appendix A

### Definitions

Variables listed in the paper are given below.

*α* Surface broadband albedo (-). The albedo is given by the MODIS product MOD43.

*α _{r}* Surface albedo of hypothetical grass reference crop [

*α*= 0.23 (-)] (Allen et al. 1998)

_{r}*γ* Surface azimuth angle (rad), equal to the aspect with the range from −180° to +180°

*γ*_{1} Psychrometric constant (kPa °C^{−1})

Γ* _{c}* Ratio of soil heat flux to net radiation for full vegetation canopy [Γ

*= 0.05 (-); Monteith 1973]*

_{c}Γ* _{s}* Ratio of soil heat flux to net radiation for bare soil [Γ

*= 0.315 (-); Kustas and Daughtry 1989]*

_{s}Γ_{sa} Standard atmospheric lapse rate (Γ_{sa} = 6.5 K km^{−1})

*δ* Solar declination angle (rad)

*δT _{a}* Near-surface vertical air temperature difference (K)

*δT*_{ad} Near-surface vertical air temperature difference valid for a dry pixel (K)

Δ Slope of saturation vapor pressure curve (kPa °C^{−1})

Δ_{24} 24-h-mean slope of saturation vapor pressure curve (kPa °C^{−1})

*ε* Ratio of gas constants for dry air and water vapor [*ε* = 0.622 (-)]

*ε*_{0} Surface emissivity (-); *ε*_{0} is given by the MODIS product MOD11

*ε _{a}* Atmospheric emissivity (-)

*ζ* Solar zenith angle (rad)

*θ* Solar incidence angle (rad)

*λ* Latent heat of vaporization (*λ* = 2.45 × 10^{6} J kg^{−1})

*λ*_{24r} 24-h-mean latent heat of vaporization used for ET* _{r}* (J kg

^{−1})

*λ*ET_{act} Latent heat flux (W m^{−2})

*λE _{r}* Instantaneous Penman–Monteith latent heat flux (W m

^{−2})

*λE*_{r24} Daily accumulated Penman–Monteith latent heat flux (J m^{−2} day^{−1})

*λ _{r}* Latent heat of vaporization used for ET

*(J kg*

_{r}^{−1})

*ρ* Water density (*ρ* = 1000 kg m^{−3})

*ρ*_{air} Atmospheric air density (kg m^{−3})

*ρ*_{air24} 24-h-mean atmospheric air density (kg m^{−3})

*σ* Stefan–Boltzmann constant (*σ* = 5.67 × 10^{−8}) (W m^{−2} K^{−4})

*τ*_{1} Clear-sky transmissivity (-)

*τ*_{2} Cloudy-sky transmissivity (-)

*τ*_{24} Daily-mean transmissivity (-)

*ϕ* Latitude (rad)

Ψ* _{h}* Surface layer stability correction term for heat (-)

Ψ_{m(200m)} Surface layer stability correction term for momentum (-)

*ω* Sun hour angle (rad)

*a*_{1} 5.09 × 10^{−5} × altitude + 0.868 (m)

*a*_{2} 3.92 × 10^{−5} × altitude + 0.0387 (m)

am Altitude-corrected air mass (Kasten and Young 1989)

*a _{s}* Calibration constant (

*a*

_{s}= τ_{2})

*b _{s}* Calibration constant (

*b*=

_{s}*τ*

_{1}−

*τ*

_{2})

*c*_{1} 0.66 (hPa^{1/2})

*c*_{2} 0.044 (hPa^{1/2})

*c _{p}* Air specific heat at constant pressure, constant (

*c*= 1004 J kg

_{p}^{−1}K

^{−1})

*d*_{1} Optical depth of a clear sky (-)

*d*_{2} Optical depth of clouds (-)

*d*_{3} Optical depth of a cloudy sky (-)

*d _{n}* Day number of the year (1 January = 1)

*d _{r}* Inverse squared relative distance Earth–Sun (-)

*e*_{0} 0.611 kPa

*e _{a}* Actual vapor pressure (kPa)

*e*_{a24} 24-h-mean vapor pressure (kPa)

*e _{s}* Saturation vapor pressure (kPa)

*e _{s}*

_{24}24-h-mean saturation vapor pressure (kPa)

ET_{24} 24-h actual evapotranspiration (mm day^{−1})

ET_{act} Instantaneous actual evapotranspiration (mm s^{−1})

ET_{md} Multiple day actual evapotranspiration (mm per period)

ET_{r24} Daily accumulated Penman–Monteith reference evapotranspiration (mm day^{−1})

EVI Enhanced vegetation index given by MOD13

EVI_{max} EVI for full vegetation cover

EVI_{min} EVI for bare soil

*f*_{h1} Exp(−altitude/8000) (-)

*f*_{h2} Exp(−altitude/1250) (-)

*g* Acceleration due to gravity (*g* = −9.807 m s^{−2})

*G*_{0} Instantaneous soil heat flux (W m^{−2})

*G*_{0w} Soil heat flux density for water (W m^{−2})

*G _{r}* Instantaneous soil heat flux density (W m

^{−2}) (

*G*= 0.1

_{r}*R*) (Allen et al. 1998)

_{n}*G*_{r24} 24-h reference soil heat flux density (*G _{r}*

_{24}= 0 J m

^{−2}day

^{−1})

*H* Sensible heat flux (W m^{−2})

*H*_{1} First approximation of the sensible heat flux valid for all pixels (W m^{−2})

*H _{p}* Scale height for pressure (

*H*= 7290 m)

_{p}*K* Crop factor (-)

*k* von Kármán constant [*k* = 0.4 (-)]

*K*^{↓} Incoming shortwave radiation expressed in horizontal equivalent (W m^{−2})

^{−2} day^{−1})

*K*_{md} Crop factor valid for the period of interest (-)

*L* Monin–Obukhov length parameter (m)

*L*^{↓} Downward emission of IR radiation from the atmosphere (W m^{−2})

*L*^{↑} Upward emission of IR radiation from Earth’s surface (W m^{−2})

*L*_{n24accr} Accumulated 24-h net outgoing longwave radiation (J m^{−2} day^{−1}) used for ET_{r}

*L*_{nr} Instantaneous net outgoing longwave radiation (W m^{−2})

*M*_{1} Wind speed measured at the weather station (m s^{−1})

*M*_{200} Wind speed at 200-m height above the weather stations

*M*_{200}(*x*, *y*) Corrected wind speed at 200-m height for every pixel

*M*_{2r} Wind speed 2 m above the reference surface (m s^{−1})

mr Relative optical air mass

*n* Actual duration of sunshine (h)

*N* Maximum possible sunshine (h)

*P* Atmospheric pressure (Pa, kPa)

*P*_{0} Average sea level pressure (*P*_{0} = 101 325 Pa)

*r* Mixing ratio (kg_{dry air}/kg_{water vapor})

*r*_{24} 24 h-mean mixing ratio (-)

*r*_{ah1} First approximation of aerodynamic resistance to heat transport (s m^{−1})

*r*_{ah2} Second approximation of aerodynamic resistance to heat transport (s m^{−1})

*r*_{ar} Aerodynamic resistance valid for the reference surface (s m^{−1})

*R _{d}* Gas constant for dry air (

*R*= 287.053 Pa K

_{d}^{−1}m

^{−3}kg

^{−1})

*R _{n}* Net radiation (W m

^{−2})

*R*_{n24accr} 24-h accumulated reference net incoming radiation (J m^{−2} day^{−1})

*R*_{nr} Instantaneous net radiation valid for the reference surface (W m^{−2})

*r _{s}* Surface resistance (

*r*= 70 s m

_{s}^{−1})

*R _{υ}* Water-vapor gas constant (

*R*= 461.5 J K

_{υ}^{−1}kg

^{−1})

*s* Slope (rad)

*S* Solar constant (*S* = 1367 W m^{−2})

*t* Local longitude time in decimal hours (h)

*T* Absolute air temperature at satellite overpass (K)

*T*_{0} Land surface temperature (K)

*T*_{0dem} Artificial surface temperature corrected for the altitude (K)

*T*_{24} 24-h-mean absolute air temperature (K, °C)

*T _{c}* 273 K

*T _{d}* Dewpoint temperature (K) from meteorological station

*T _{L}* Linke turbidity coefficient

*T _{υ}* Virtual air temperature (K)

*T*_{υ24} 24-h mean virtual air temperature (K)

*u** Friction velocity (m s^{−1})

^{−1})

VF Vegetation fraction (-)

*x*_{1} *T*_{0dem} of a wet pixel (K)

*x*_{2} *T*_{0dem} of a dry pixel (K)

*y*_{2} Near-surface vertical air temperature difference valid for a dry pixel (K)

*z* Elevation (m)

*z*_{0m} Surface roughness (m)

*z*_{0r} Reference roughness length (*z*_{0r} = 0.014 76 m) (Allen et al. 1998)

*z*_{1} 0.1 m (Bastiaanssen et al. 2005)

*z*_{2} 2.0 m (Bastiaanssen et al. 2005)

*z*_{10} Height above surface of wind speed measurement (*z*_{10} = 10 m)

*z*_{200} Blending height (=200 m)

*z _{i}* Heights above the surface that define the surface to air temperature difference

*δT*(m)

_{a}## Appendix B

### Complete Algorithm

Our application of SEBAL consists of 56 calculation steps. Steps 1–56 need to be run for every cloud-free day for which satellite data are obtained. Climate data generated for the calculation of ET_{act} are also input for ET_{act}. Clear distinctions are made between variables related to instantaneous, 24-h-accumulated, and 24-h-average actual and reference processes. The complete algorithm is shown below.

## Appendix C

### Supporting Tables

Tables C1–C6 provide detailed information about input, output, and validation data, including values, relations, dates, and comparisons.

Clear-sky Linke turbidity coefficient for 17°S, 64°W and zero altitude for 12 months.

Mean optical depth of clouds *d*_{2} in the study area for selected days.

Relation between input and output variables.

Lookup table for surface roughness.

Day numbers of applied satellite data.

Precipitation and actual evapotranspiration at meteorological stations (2003–04).

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