Algorithm Theoretical Basis Document

GCOM-C Evapotranspiration Index ProductVersion 1.4

Masahiro Tasumi

Department of Forest and Environmental Sciences, University of Miyazaki

March 28, 2021

Abstract

This Algorithm Theoretical Basis Document (ATBD) describes the algorithm forestimating Evapotranspiration Index (ETindex) developed as a research product of
the GCOM-C satellite of the Japan Aerospace Exploration Agency (JAXA). The
ETindex expresses actual evapotranspiration normalized for the weather condition
and is equivalent to the crop coefficient, which has been applied widely for
agricultural and irrigation water management around the globe. The ETindex isconvertible to the actual quantity of evapotranspiration by using a local or global
weather dataset. The ETindex is estimated primarily by GCOM-C land surface
temperature observation, with some additional inputs including those of a Digital
Elevation Model (DEM) and global wind speed data. The final product is a cloud-free
16-day global ETindex map having a spatial resolution of 250 m. Daily actual
evapotranspiration for any weather condition (including cloud-cover days) is further
estimable using a ground-based weather dataset. Additionally, the ETindex isconvertible to volumetric soil water content for locations where information of soil
physical characteristics is available.
Employing two extreme hypothetical surface conditions named “wet surface,”
defined as a surface having zero sensible heat flux, and “dry surface,” defined as asurface having zero ET, the algorithm estimates ETindex by using surface
temperature as an indicator of surface wetness. The algorithm described in this
ATBD has been applied with the MODIS daily land surface temperature map and a
global weather dataset. The estimated result was compared with other independent
global ET datasets and with some other environmental properties. The comparison
implied that the proposed algorithm reasonably estimates ETindex.

Table of Contents

1. Introduction ....................................................................................................................... 1
2. GCOM-C Evapotranspiration Index Estimation Procedure—Overview and
Definitions ........................................................................................................................ 2
2-1. GCOM-C Evapotranspiration Index Map ............................................................2
2-2. Application of GCOM-C ETindex Product to Soil Moisture Estimation ............7
3. Algorithm Descriptions ..................................................................................................... 8
3-1. Basic Strategy for Estimating Ts(wet) and Ts(dry) .............................................8
3-2. Data for Analyses ...................................................................................................93-3. Ts(wet) Estimation ............................................................................................... 11
3-4. Ts(dry) Estimation ...............................................................................................18
3-5. Calibration for the Maximum Limit of ETindex................................................23
3-6. Adjustments and Filtering of ETindex Values on Daily Computation.............24
3-7. ETindex Composite ..............................................................................................27
3-8. Extra Adjustment of ETindex Values by Applying Soil Water Balance Model(Optional - Recommended) .........................................................................................28
3-9. Calculation of Soil Moisture (Optional) ..............................................................29
4. Application to Satellite Imagery and Information about Model Accuracy ................. 30
4.1. Global Application Test using MODIS Land Surface Temperature Product ...30
4.2. Regional Application Test using GCOM-C Land Surface Temperature Product......................................................................................................................................32
4.3. Analysis of ETindex for Model Applicability ......................................................33
4.4. Reality Check of the Estimated Values using Outside Sources ........................34
Acknowledgements ............................................................................................................. 37
References ............................................................................................................................ 38
Appendix: Ts(wet) and Ts(dry) determination model for Shenmu weather
measurement site ............................................................................................................. 11. Introduction
Surface evapotranspiration (ET) and near-surface soil moisture are importantfactors in climate, water balance and circulation, and crop production. The need forsatellite-based, global ET estimation has been increasing with the growth of large-scalewater and environmental issues, such as global warming, increased instability of
weather and climate, water shortages due to population growth, and enhanced humanactivities. Under these circumstances, the estimation algorithm of the global
Evapotranspiration Index (ETindex) has been investigated as one of the GCOM-C
research products. The ETindex is valuable as an index of evapotranspiration efficiency
and soil moisture content. The index is also convertible to the actual amount of ET fromthe land surface by using a global or local near-surface general weather dataset.
Because the ETindex is closely related to soil water availability, the information isfurther convertible to volumetric soil water content (θ), with some additional
information of soil physical characteristics, especially that pertaining to the water
holding characteristic of soil. Although soil moisture estimation is not the main target ofour study for the ETindex product, this Algorithm Theoretical Basis Document (ATBD)
describes the progress toward soil moisture estimation found during the algorithm
development work of the ETindex product.
The first ATBD (version 1.0) was submitted to JAXA in September 2011. The major
update from ATBD ver. 1.0 to ver. 1.1 (submitted in March 2013) was the elimination of
the elevation function in the estimation of wet-surface temperature. The latitude
function was also corrected and modified. The update from ATBD ver. 1.1 to ver. 1.2(submitted in October 2014) included modification of the ETindex regulation method andcorrection of the equations. The update from ATBD ver. 1.2 to ver. 1.3 (submitted in
March 2016) updaters the theoretical background information includes the productaccuracy. The development of the ETindex estimation algorithm is an on-going project.
The detailed procedures for ETindex estimation will continue to be improved, and the
ATBD will be updated as necessary.

12. GCOM-C
Definitions

Evapotranspiration

Index

Estimation Procedure—Overview

and2-1. GCOM-C Evapotranspiration Index Map
Evapotranspiration (ET) is controlled by energy and soil water availability alongwith the near-surface heat and vapor exchange circumstances. The quantity of ET isaffected by several factors including weather related factors such as solar radiation,
surface and air temperatures, humidity, and wind speed and precipitation, as well as
surface topography, land cover, soil type, and status of vegetation. Many of these factors
are strongly dependent on region and time, which makes automated global estimationof ET technically difficult. To develop the information infrastructure of global water
circulation and consumption, by minimizing the technical difficulty caused by thedependency on region and time, we suggest providing global ETindex via GCOM-C earth
observation. The ETindex is ET normalized for the weather condition.

The technicaldifficulty of global ET estimation is mitigated effectively by this normalization.
The GCOM-C ETindex is equivalent to the crop coefficient, which has been applied
widely for agricultural and irrigation water management around the globe. The ETindex
is defined by the following equation:

ET𝑖𝑛𝑑𝑒𝑥 =

𝐸𝑇𝑎𝑐𝑡
𝐸𝑇0

(1)

where ET0 is reference ET and ETact is actual ET.
ET0 is typically described as the ET from a well-watered, 12-cm tall grassy field. A
detailed definition of ET0 is the ET from a hypothetical grassy reference field with acrop height of 12 cm, surface resistance of 70 s/m, and albedo of 0.23. ET0 is calculatedusing only near-surface weather data (solar radiation, air temperature, humidity, andwind speed). The calculation procedure of ET0 is described in Allen et al. (1998).
Using the ETindex map and a weather dataset, users can back-calculate actual ETas2𝐸𝑇𝑎𝑐𝑡 = ET𝑖𝑛𝑑𝑒𝑥 × 𝐸𝑇0(2)

Framework of the GCOM-C ETindex Algorithm
In order to achieve a global and near-timely estimation of daily ETindex with aspatial resolution of 250 m, the algorithm should be as simple as possible, with aminimum requirement of input data. In this algorithm, the input data that are used for
ETindex computation are limited to GCOM-C observation data, global DEM data, andnear-surface wind speed data.
Our model adopts, from among several types of ET estimation approaches, anestimation approach that produces an index of surface temperature. Kogan (1995)
suggested that a surface temperature index having the form of Eq. 3, named thetemperature condition index (TCI), can adequately express drought-related vegetationstress:𝑇𝐶𝐼 =𝑇𝑠(𝑚𝑎𝑥) − 𝑇𝑠(𝑎𝑐𝑡)′
𝑇𝑠(𝑚𝑎𝑥) − 𝑇𝑠(𝑚𝑖𝑛)(3)where Ts(max) and Ts(min) are the multi-year weekly maximum and minimum surface
temperatures, respectively, and Ts(act)′ is weekly actual surface temperature (oC).
Kimura (2007) and Senay et al. (2007) proposed indices named the modified
temperature/vegetation dryness index (MTVDI) and the ET fraction (ETf), respectively.
These indices estimate surface evapotranspirative efficiency using the basic structure ofa temperature index, as does the TCI, but the timescales of the actual, maximum, andminimum temperatures are instantaneous rather than weekly or multi-annual. In ourmodel, following their findings, the ETindex is estimated as

𝐸𝑇𝑖𝑛𝑑𝑒𝑥 = 𝐶𝑎𝑑𝑗 ×𝑇𝑠(𝑑𝑟𝑦) − 𝑇𝑠(𝑎𝑐𝑡)𝑇𝑠(𝑑𝑟𝑦) − 𝑇𝑠(𝑤𝑒𝑡)(4)where Ts(act) is the instantaneous actual surface temperature from satellite thermal
3observation (oC) and Ts(wet) and Ts(dry) are the hypothetical wet surface temperatureand the dry surface temperature (oC), which are the instantaneous surface
temperatures when the surface is expected to have zero sensible heat flux and when thesurface is expected to have zero latent heat flux, respectively. The constant C adj is anadjustment factor employed in the algorithm.
The constant Cadj in the equation, representing the maximum limit of the ETindex
occurring when the surface is in a very wet condition, is uniquely employed in the
GCOM-C ETindex algorithm. Figure 1 shows an overview of the algorithm for ETindex
estimation.

Land Surface Temperature(Instantaneous meas.)

Core Algorithm ofETi ndexTs(act)Daily ETi ndex

Latitude
Solar Zenith
Ts(wet)Digital Elevation ModelWind speed(e.g. Re-analysis data)

Ts(dry)16-day ETi ndex

ET(act)ETo computed by
global or local
weather dataset

Land Cover/ Land Use
NDVI
Snow coveradjustments (future study)

Figure 1. Flowchart of the GCOM-C ETindex estimation algorithm.
Calculation Time Interval
The ETindex is first processed on a daily basis, for every day, including cloudy days,
using daytime instantaneous land surface temperature (LST) images. The direct
estimation of ETindex is applicable only for the moment when actual surface temperature
is observed by the GCOM-C satellite. For extension of the estimation to a 24-hour4period, we employ an assumption, similar to that of Allen et al. (2007), that the
instantaneous ETindex computed at image time is the same as the 24-hour average
ETindex. GCOM-C will observe the surface every day or two. However, the presence ofclouds critically degrades or disables surface temperature observation and thus ETindex
estimation. A small contamination by clouds in a pixel cell, which may be undetectableby an automated cloud-mask algorithm, can cause significant overestimation of ETindex
due to a reduction in the retrieved Ts(act) value. To automatically prevent the impact ofcloud contamination in global ETindex estimation, daily estimated ETindex maps arecomposited every 16 days, and the minimum value of ETindex over the 16-day period isselected for each pixel as the representative ETindex value of the 16-day period. However,in some extreme climatic conditions, no single clear-sky day is available within a 16-daycomposite period. In such cases, because of the cloud effect, the pixel value of the ETindex
automatically takes the maximum number, which may be a good approximation because
more than 16 days of continuous cloud is an indicator of the rainy season, during which
the surface is expected to be in a wet condition and the ETindex is expected to take a large
number. We accept this assumption for the following reasons:(1) Such a condition is expected only in limited areas/seasons of the world.
(2) Such a condition typically occurs during the rainy season, when the land surface
beneath the clouds is in a wet condition and the actual ETindex value is likely not far
from the maximum.
(3) We want to avoid any data-lacking pixels. Supplying a complete set of 16-day
ETindex maps, with no data-lacking pixels anywhere, is a big advantage. For example,
annual ET cannot be estimated if a period lacking data is included.
(4) In such a cloudy condition with weak solar radiation, ETo is expected to be small.
Thus, the error in ET is expected to be small even if a somewhat larger error ispresent in the ETindex.
Application of the Output Data
The GCOM-C ETindex is applicable to the following purposes:(1) Estimate actual ET from GCOM-C ETindex and weather data.
Users can derive daily, monthly, and annual actual ET maps from the ETindex map
by applying a global or local weather dataset. The required weather data are
typically obtainable from any major global weather dataset source or from public5weather stations within or near the users’ area of interest.
(2) Use the GCOM-C ETindex map for relative analyses.
The ETindex map is a relative indicator of ET. In a specific ETindex image of an area,
pixels having higher ETindex have higher actual ET. Therefore, the map is usable as
is for relative analyses, e.g., identifying areas having higher/lower ET orunderstanding the yearly trend of ET.

6

2-2. Application of GCOM-C ETindex Product to Soil Moisture Estimation
The GCOM-C ETindex essentially indicates the level of soil wetness from zero (verydry) to the maximum (very wet), and it is valuable for estimating soil moisture. The
additional input data required to estimate soil moisture are the water holding
characteristic of the soil (typically estimated by soil type if directly measured
information is not available) and the average root zone depth of the vegetation (ifvegetation is present over the surface).
The volumetric soil water content (θ) for the average depth at which the soil watersignificantly contributes to surface ET can be estimated from the ETindex. The depth isrepresented as the root zone if the surface is covered by vegetation or by the top 10–15
cm if the surface is in a bare-soil condition. The volumetric soil water content (m3/m3) isdefined by the following equation:
θ = (𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑜𝑖𝑙 𝑤𝑎𝑡𝑒𝑟)/(𝑇𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑜𝑖𝑙)(5)

Additional and more-detailed information about the soil water estimation procedureusing GCOM-C ETindex is available in Tasumi and Kimura (2013).

7

3. Algorithm Descriptions
Estimation of wet surface temperature (Ts(wet)) and dry surface temperature(Ts(dry)) is key to the estimation of ETindex, as shown in Equation 4 and in Figure 1.
This section describes the computational algorithms of Ts(wet) and Ts(dry).

3-1. Basic Strategy for Estimating Ts(wet) and Ts(dry)The GCOM-C ETindex is estimated by indexing surface temperature, which
requires two hypothetical surface temperatures, the wet surface temperature (Ts(wet))
and the dry surface temperature (Ts(dry)). Ts(wet) is the hypothetical surface
temperature assuming that the surface is in a very wet condition. The “wet” surface isdefined here as the surface having latent heat equivalent to available energy (i.e.,
sensible heat is zero). Ts(dry) is the hypothetical surface temperature assuming that thesurface is in a very dry condition. The “dry” surface is defined here as the surface having
zero latent heat flux (i.e., zero ET). The definitions of the two extreme surfaces areequivalent to the “cold pixel” and the “hot pixel” of the SEBAL ET estimation algorithmof Bastiaanssen et al. (1998). Whereas the cold and hot pixels of SEBAL are specificpixels of an image, the Ts(wet) and Ts(dry) of the GCOM-C ETindex algorithm are nottemperatures of specific pixels but temperatures assigned for every pixel individually.
Theoretically, Ts(wet) and Ts(dry) can be determined by solving surface radiationand heat balance (i.e., energy balance) by employing specific assumptions for wet anddry surfaces. However, we have found operational difficulty in obtaining the theoreticalsolutions of Ts(wet) and Ts(dry). The theoretical approach determines Ts(wet) andTs(dry) by solving energy balance equations to derive long-wave radiation from earth tosky (Lout) and then converts Lout to the hypothetical temperatures. The structuralproblem of this procedure is that all uncertainties and errors in energy balancecomputation are cumulated and passed to Ts(wet) and Ts(dry). Therefore, input data
quality and robust definition of the hypothetical surface characteristics are critical for
successful estimation of the hypothetical temperatures. Because we cannot satisfy theserequirements of input data quality and robust definitions of the surface characteristics,
using the theoretical approach is operationally difficult. Instead, the GCOM-C ETindex
algorithm takes an empirical approach to determine Ts(wet) and Ts(dry).

8

3-2. Data for Analyses
The development of empirical equations for Ts(wet) and Ts(dry) requires
near-surface radiation and heat balance measurements. The current version of thealgorithm relies largely on micrometeorological data, supplied by Dr. Kimura of Tottori
University, measured at Shenmu, China, during 2005 to 2008. The site is located in apart of the Loess Plateau, in a semiarid region of China. Photographs of the weather
station and the surrounding area are shown in Figure 2. The data include airtemperature, humidity, wind speed and direction, four separated radiation components(solar radiation, reflectance, and longwave radiations from earth and from sky), and soil
moisture for 10 different depths. The details of the data measurement are described in
Kimura (2007).
Using the weather data, Ts(wet) and Ts(dry) were computed by an energy balance
model described in the Appendix of this document. An example calculation result for
September 9, 2005, is shown in Figure 3.

Figure 2. Photographs of weather station (left) and surrounding area (right) of theShenmu measurement site (38o47′ N, 110o21′ E; 1224 m a.s.l.).

9

900

40

675

30

450

20

225

Solar Radiation (W/m2)

Surface Temperature (oC)

50

Ts(act)Ts(wet)Ts(dry)

10

0
1

3

5

7

9 11 13 15 17 19 21 23
Hour

Rs

Figure 3. Ts(wet) and Ts(dry) estimated by energy balance computation, plotted withmeasured actual Ts and Rs at Shenmu, for 9/9/2005.

10

3-3. Ts(wet) Estimation
Wet and dry surface temperatures at 10:30 A.M. on clear-sky days tend to have alinear relationship with radiation, such as solar radiation (Rs) or net radiation (Rn)
(Tasumi, 2010, 2011). Thus, Ts(wet) and Ts(dry) are estimated using Rs or Rn.
Considering the theoretical energy and temperature relationship, Rn is a more
appropriate term to use than Rs, because Rs is only one part of the surface radiation
balance. Nevertheless, we suggest adopting Rs to estimate Ts(wet) and Ts(dry) because
using Rn results in some operational problems that are difficult to solve.
Considering the global application of the algorithm with GCOM-C satellite
measurements, estimating Rs is much easier than estimating Rn, because theevaluation of longwave radiation from sky to land surface is difficult. Moreover, thecritical problem of using Rn for Ts(wet) and Ts(dry) estimations is the difference
between Rn over an actual surface and Rn over the hypothetical wet and dry surfaces.
Rn over the hypothetical surfaces is the appropriate term to use to estimate Ts(wet) andTs(dry). However, Rn cannot be determined without knowing the hypothetical surface
temperatures. Thus, we cannot use Rn over the hypothetical surfaces for theestimations of hypothetical temperatures. On the other hand, the solar radiation of the
hypothetical surfaces is the same as the actual solar radiation. Thus, we canoperationally use Rs for hypothetical surfaces without any problem.
A problem related to the application of actual (not hypothetical) Rn to estimate
Ts(wet) and Ts(dry) is visualized in Figure 4. For a certain small area at 10:30 A.M., thespatial difference of net radiation depends on surface temperature and albedo. Hot andbright desert has much lower net radiation than cold forest. If Ts(wet) is estimated byactual Rn using an empirical linear regression equation, the estimated Ts(wet) of desert
will be much lower in desert than in forest, which is probably not realistic. If the hot andbright desert is in a very wet condition, Rn of the surface should be very similar to Rn of
the cold forest, and the Ts(wet) estimation of the area will be much more homogeneous.
This problem is avoided if we use Rs as the independent variable to estimate thehypothetical surface temperatures.

11

Input satellite image
Cold forest

Hot andbright desert

Net Radiation

Solar Radiation

High
Same

Ts(wet) (oC)Ts(wet) (oC)Low

Rn (W/m2)

Ts(wet) estimated by Rn

Rs (W/m2)

Ts(wet) estimated by Rs

High
Same
Low

Figure 4. Expected result of Ts(wet) estimated by net radiation and by solar radiation.

12

Figure 5 shows the relationship between solar radiation and Ts(wet) at Shenmu,
China, at about 10:30 A.M. (solar time) for clear-sky days during 2005–2008. A clear
linear relationship, with large scatter, between Rs and Ts(wet) is confirmed. If Ts(wet) is
estimated by the linear regression equation shown in the figure, the standard error of
estimation is 7.83 oC, which is large.

40

Ts(wet) (oC)

30
20

10
0
-10

y = 0.06 x - 30.34
R² = 0.68

-20
-30
200

400

600

800

1000

1200

Rs (W/m2)
Figure 5. Relationship between measured solar radiation and estimated Ts(wet) at
Shenmu, China, at about 10:30 A.M. (solar time) for clear-sky days during 2005–2008.
The primary reason for the large scatter in Figure 5 is seasonal differences causedby the thermal inertia of the Earth. Figure 6 uses exactly the same data used in Figure
5, but shown as monthly averages. A clear seasonal trend of Rs and Ts(wet) is shown,and the trend can be explained theoretically by the thermal inertia of the Earth. In thisversion of ATBD, the Rs vs Ts(wet) relationship with the seasonal trend was expressedas the following equation:

𝐷𝑜𝑌+𝑆𝑑𝑎𝑦Ts(wet) = 0.06Rs − 30.34 − sin (365× 2𝜋) × 𝑓(𝐿𝑎𝑡) − 𝐶𝑡𝑜𝑝 × (𝑧𝑝 − 𝑧𝑏 )(6)

where Ts(wet) is in oC, Rs is in W/m2, DoY is day of year, Sday is the phase shift of thesine curve calibrated as 37 in the Northern Hemisphere and 220 in the Southern
Hemisphere, z is elevation in meters, and f(Lat) is the amplitude of the sine curve
13determined using latitude (deg.) by f(lat) = -0.0021*Lat2+0.3449*|Lat|-2.9864 (0.0 ≤f(lat) ≤ 10.0), Ctop is a coefficient for the topographic adjustment that is tentatively
calibrated as 0.0098 (oC /m), which is equivalent to dry-adiabatic lapse rate, zp is theelevation of the pixel (m), and zb is the lowest elevation of the surrounding 15 km by 15
km area (m), where the pixel is at the center of the 15 km by 15 km area. The zp and zb

Ts(wet) (C)are intended to be derived using a 250 m resolution digital elevation model (DEM).

35
30
25
20
15
10
5
0
-5
-10
-15

Fall
Springmonthly average
estimation by equation

400

600

800

1000

Rs (W/m2)
Figure 6. Monthly trend of Rs and Ts(wet) at 10:30 a.m. on clear-sky days along withestimated Ts(wet) from Equation 6.
The first component of Equation 6 “0.06Rs – 30.34” is the average linear
relationship between Rs and Ts(wet), as shown in Figure 5. The second component of
the equation is the sine function describing the seasonal trend between Rs and Ts(wet)calibrated by the Shenmu data (Figure 6). The third component f(Lat) expresses theamplitude of the sine curve. At Shenmu (latitude = 38.78o), the amplitude of the sine
curve was calibrated as 7.0. If the seasonal trend of the Rs vs Ts(wet) relationship is dueto the thermal inertia of the Earth, the amplitude of the sine curve should change withlatitude or with the difference of solar radiation that develops the seasons. The seasonal
trend would be minimum at the equator and maximum at the location where thedifference of solar radiation between summer and winter is largest.
Figure 7 shows the yearly trend of clear-sky solar radiation at 10:30 A.M. by latitudein the Northern Hemisphere. The seasonal difference of solar radiation is smallest near14the equator and largest at about 60o latitude. Assuming that the amplitude of the sine
curve depends on the difference of solar radiation between summer and winter andassuming that the best value for amplitude at latitude = 38.78o (Shenmu) is 7.0, theClear-sky Solar Radiation (W/m2)

amplitude of the sine curve was calibrated as shown in Figure 8.

1000

800

0
10

600

20
30

400

40

50

200

60
70

0

0

60

120

180

240

300

360

Day of Year
Figure 7. Yearly trend of clear-sky solar radiation at 10:30 A.M. by latitude from 0o to 70oin the Northern Hemisphere calculated assuming the atmospheric condition of the

Amplitude of sine curve

Shenmu measurement site.

12
y = -0.0021x2 + 0.3449x - 2.9864
R² = 0.9949

10
8
6
4
2
0
0

20

40

60

80

Latitude (deg)15

Figure 8. Calibration of the amplitude of the sine curve according to the difference of
solar radiation between summer and winter. The amplitudes from 0–10o latitude wereset as zero because of no clear winter at these latitudes.
The first version of the ATBD (ver. 1.0) contained an elevation function in Equation
8. However, it was eliminated in the previous ATBD (ver. 1.1) because an analysis
indicated that the elevation function was inappropriate (Tasumi, 2013). This version of
the ATBD temporarily follows the decision made for version 1.1 for impact of elevation.
Future study is required to adequately represent the effect of elevation on Ts(wet)estimation.
The solar radiation used in Equation 6 is estimated using the solar zenith angle
image of the satellite and the DEM. The following equations are based on the clear-sky
solar radiation estimation procedure summarized in Allen et al. (1998), organized for
application to ETindex estimation. Clear-sky solar radiation is calculated by

Rs = τ ∙ Ra

(7)

where τ is atmospheric transmittance and Ra is extraterrestrial solar radiation (W m-2).
Atmospheric transmittance is calculated asτ = 0.75 + 2 × 10−5𝑧(8)where z is the elevation (m) given by the DEM.Ra can be calculated by the following
equation:Ra =𝐺𝑠𝑐 ∙ 𝑐𝑜𝑠𝜃𝑑2

(9)where Gsc is the solar constant of 1367 W m-2, cosθ is the cosine of the solar zenith angle,
where the solar zenith angle is supplied as a part of the satellite image, and d is therelative distance between Earth and Sun calculated as
16

d=√

1
2𝜋
1 + 0.033 ∙ 𝑐𝑜𝑠 (∙ 𝐷𝑜𝑌)365( 10 )

where DoY is day of year.

173-4. Ts(dry) Estimation
Compared to wet surface temperature (Ts(wet)), dry surface temperature (Ts(dry))is much more sensitive to the micrometeorological condition over the surface (Kanda etal., 2010; Tasumi, 2011). Therefore, direct estimation of Ts(dry) without sufficient
information of the micrometeorology over the surface might entail a considerable
uncertainty in estimation accuracy. One operational solution to the problem is toestimate the difference between Ts(dry) and Ts(wet) (i.e., Ts(dry–wet)), instead of direct
estimation of Ts(dry). Ts(dry–wet) is expected to be more stable than Ts(dry) for
changing micrometeorological condition. After estimating Ts(dry–wet), Ts(dry) is
computed using the following equation:

𝑇𝑠(𝑑𝑟𝑦) = Ts(wet) + Ts(dry − wet)

( 11 )The relationship between Rs and Ts(dry–wet) at 10:30 A.M. for clear-sky days at
Shenmu, China, is shown in Figure 9. The figure clearly shows the impact of wind speed
on Ts(dry–wet). Within a similar wind speed condition, Ts(dry–wet) is linearly related to
Rs. Ts(dry–wet) becomes smaller as wind speed becomes higher. No seasonal trend or
effect of thermal inertia was found in the Ts(dry–wet) vs Rs relationship, probably
because the seasonal impacts available in Ts(wet) and Ts(dry) are eliminated when the
difference is taken.
The impact of wind speed on Ts(dry–wet) is explained by Figure 10. This figure
shows the relationships between wind speed and the hypothetical surface temperatures
for same season (May) for days having similar intensity of solar radiation (900–950 Wm-2) at 10:30 A.M. for clear-sky days at Shenmu. The graph indicates that higher windspeed decreases the temperatures for both wet surface and dry surface, because windpromotes ET or sensible heat exchange. However, the decrement of surface temperatureby wind speed is larger for dry surface than for wet surface, and this is the reason whyTs(dry–wet) becomes smaller as wind speed becomes higher.

18

30

Ts(dry-wet) (oC)

25

20
15

0-1 m/s
1-2 m/s
2-3 m/s
3-4 m/s
4-5 m/s
5-6 m/s
6-7 m/s> 7 m/s

10
5

0
300

500

700

900

1100

Rs (W m-2)
Figure 9. Relationship between measured solar radiation and estimated Ts(dry–wet) foreight wind speed groups at Shenmu, China, at about 10:30 A.M. (solar time) on clear-sky
days during 2005–2008.

60

Surface temperature (oC)Ts(wet)Ts(dry)

50
40
30
20
10
0

2

4

6

8

Wind speed (m/s)
Figure 10. Relationships between wind speed and the hypothetical surface
temperatures.

19

Considering the relations among Ts(dry–wet), Rs, and wind speed, we propose theTs(dry–wet) estimation method summarized in Figure 11. Specifically, Ts(dry–wet) is
proportional to Rs and the proportional factor changes with wind speed. The
proportional factor was calibrated as shown in Figure 12, which uses the data shown in
Figure 9. The slopes of the linear regression lines for the wind speed groups in Figure 9
were plotted with the average wind speed of each wind speed group. The resultant
equation for Ts(dry–wet) estimation is the following:𝑇𝑠(𝑑𝑟𝑦 − 𝑤𝑒𝑡) = (−0.0023𝑢 + 0.0301)𝑅𝑠 (Ts(dry − wet) ≥ 0)

( 12 )

where Ts(dry–wet) is in oC, u is wind speed at the height of 2 m in m s-1, and Rs is solar
radiation in W m-2.

Ts(dry-wet) (oC)Weaker wind

Stronger wind00

Rs (W m-2)

Figure 11. Proposed Ts(dry–wet) estimation method.

20

Proportional Factor

0.03

0.02

0.01
y = -0.00230 x + 0.03010
R² = 0.97836
0
0.0

1.5

3.0

4.5

6.0

7.5

9.0

Wind Speed (m s-1)
Figure 12. Calibration of the proportional factor.
Equation 12 applies when using wind speed at the height of 2 m. In the case inwhich the input wind speed data are for a different height, the wind speed data must be
converted to wind speed at 2 m using the following equation:

𝑢2 = 𝑢ℎ

𝑙𝑛 (𝑧

2

𝑜𝑚

𝑙𝑛 (𝑧

ℎ

𝑜𝑚

)

( 13 ))

where h is the height of the input wind speed data, uh is wind speed at height h (m), andzom is surface roughness for momentum transport (m).
The zom is determined using the land cover/land use map. Recommended values ofzom for different land uses, shown in Table 1, were determined by considering therepresentative values summarized by Kondo (2000).

21

Table 1. Recommended zom by land use type.

Land use
Metroporitan
ForestTown
Agriculture
Rangeland
Water/Snowzom (m)
2
0.6
0.3
0.05
0.05
10 -3

22

3-5. Calibration for the Maximum Limit of ETindex
By definition, ETindex = 1 means that the surface has an ET value similar to the one
from the hypothetical grassy reference field. The condition of such a surface is different
from the “wet” surface determined for Ts(wet). Usually, the reference field has positive
sensible heat during the satellite overpass time, and therefore the ETindex of the wet
surface exceeds 1.0. Namely, the adjustment factor, Cadj, of Equation 4 should exceed
1.0.
The Cadj of Equation 4 was calibrated using Shenmu data. Figure 13 shows acomparison of the latent heat flux from the “reference” surface defined by FAO and fromthe “wet” surface defined for our Ts(wet) estimation. The result shows that the ET fromthe wet surface is 1.23 times greater than the ET from the reference surface if therelationship is assumed to be linear without intercept; however, a non-linear trend wasfound. The non-linearity indicates that Cadj depends on the intensity of the latent heat
flux at the reference or the wet surface, which cannot be determined in this algorithm
before determination of Cadj. Therefore, assuming a non-linear relationship, orassuming a linear relationship with intercept in Equation 13, is operationally difficult.
In the current version of ATBD, we assume the Cadj of Equation 4 to be 1.23 by adopting
a proportional assumption between the reference and the wet surface.

LE for H=0 surface (W m-2)

800

600

400

200

y = 1.23 x
R² = 0.76

0
0

200

400

600

800

LE for reference surface (W m-2)
Figure 13. Comparison of latent heat flux from reference surface and from the “wet”
surface at about 10:30 A.M. (solar time) for clear-sky days, Shenmu, China.

23

3-6. Adjustments and Filtering of ETindex Values on Daily Computation
ETindex expresses the efficiency of ET. It is also strongly related to the soil watercontent. ETindex = 0 means that there is no soil water to evapotranspirate and that ET
is thus zero. ETindex = 1.23 (i.e., maximum limit) means that sensible heat from thesurface is zero at the satellite overpass time (e.g., 10:30 A.M.), which is the wettest
condition that the satellite can identify via surface temperature. Thus, the mostsuccessful result would be that the calculated ETindex is between zero and 1.23, without
any extra regulation.
However, there is no guarantee that the empirical equations employed in the
GCOM-C ETindex algorithm are applicable to all regions and conditions. To improve theoperational accuracy of ETindex estimation, we suggest additional regulations thatmight be applicable for the ETindex. Future studies for model improvement will be
conducted to minimize this type of regulation by improving the ETindex estimation
equations themselves.
Minimum and Maximum Values of ETindex
By the definition of dry surface, it is important to limit the minimum value of theETindex to zero. The physical meaning of a negative ETindex is condensation of vapor,which is difficult at a dry surface during the satellite overpass time on clear-sky days.
Therefore, a negative value of ETindex is most likely evidence of underestimation ofETindex. Additionally, it is effective to limit the maximum value of ETindex to 1.23,
which represents the wet surface condition (H is zero). Theoretically, the actual ETindex
can exceed the ETindex for the defined wet surface condition in some situations (e.g., in
the case of the surface having negative sensible heat at 10:30 A.M. due to, for example,
advection of sensible heat). However, setting a maximum number by accepting someunderestimation in such rare conditions is a realistic strategy in this type of global
application. A global application must target a very wide range of surface, subsurface,and climatic conditions. Also, the data processing should be fully automated. The
empirical equations in the algorithm might work poorly in some specific conditions.
Thus, setting these minimum and maximum limits of ETindex can help avoid theoccurrence of large error. In this algorithm, ETindex is limited as24

0 ≤ 𝐸𝑇 𝐼𝑛𝑑𝑒𝑥 ≤ 1.23 .

( 14 )

Additional Regulation for minimum value of ETindex by NDVI (Optional)
Figure 14 shows the relationship between NDVI and ETindex in an agricultural
region of Urmia, Iran, for 6 Landsat satellite images during May to September, 2016.
The selected six dates were “dry” images, where had no or little precipitation during theprevious periods. ETindex dataset used in the analysis has been verified by Tasumi(2019). The spatial resolution was degraded to 270m for simulate GCOM-C, where,
GCOM-C has 250m at-nadir resolution and typical operational resolution might be in
the range of 250 to 300m (Tasumi et al., 2019). Because vegetation needs water to live,
and because ET occurs when water is available, ETindex value is difficult to be in thegray-colored triangle region of the figure.

1.4

ETindex (270m res.)

1.2
1

5/22

7/9

7/25

8/10

8/26

9/11

0.8
0.6
0.4
0.2
0
0.2

0.4

0.6

0.8

NDVI (270m res.)

Figure 14. Relationship between NDVI and ETindex in an agricultural region of the
Urmia, Iran, for 6 Landsat images during May to September, 2016.
The result of the analysis indicates that the following additional regulation may be
applicable for improve the ETindex estimation accuracy;

𝐸𝑇 𝐼𝑛𝑑𝑒𝑥(𝑚𝑖𝑛) = 1.70 × 𝑁𝐷𝑉𝐼 − 0.55

( 15 )

25

Regulation of ETindex for Snow/Ice-covered Pixels and for No-sunlight Pixels
On snow/ice surfaces, most of the available energy is expected to be used for
melting (i.e., consumed as latent heat of fusion or melting), and thus only a small
portion of the available energy is spent on evaporation. Because detection of this verysmall latent heat is difficult with this type of empirical approach, an operationallyreasonable method would be to assign zero ETindex for snow/ice surfaces. This
limitation would not induce a large error, especially in the absolute value of ET, because
the available energy itself is generally small on snow/ice surfaces, with their highsurface reflectance and small gradient of absolute vapor pressure due to the cold airtemperature.
Additionally, some locations on the Earth, typically within the north or south polar
circles, may have no solar radiation at 10:30 A.M. This condition violates the applicationof some of the ETindex estimation equations. For such a surface, ETindex should be
assigned as zero. Such a condition occurs in the winter season at high latitude, and thesurface is likely to be covered by snow. Again, this limitation would not induce a large
error, especially in the absolute value of ET, because the available energy itself isgenerally small on snow/ice surfaces, with their high surface reflectance and small
gradient of absolute vapor pressure due to the cold air temperature.

26

3-7. ETindex Composite
16-day composite ETindex map is computed, simply by adopting the minimum
ETindex value within each 16-day period (i.e. apply Minimum Value Composite). Taking
the minimum value is to reject cloud contaminated pixels. Satellite-derived surface
temperature, which is the primary input for the ETindex estimation, is very sensitive tocloud availability. Even tiny contamination of clouds can under-estimate surface
temperature by 5 oC, and it can underestimates ETindex significantly.

27

3-8. Extra Adjustment of ETindex Values by Applying Soil Water Balance Model(Optional - Recommended)
The minimum value composite procedure described in the previous section isnecessarytorejectcloud-contaminatedpixels.

However,theprocesscausesunder-estimation of ETindex in some conditions. Applying soil water balance model iseffective to improve estimation accuracy of ETindex. The soil water balance model isapplied after ETindex is converted to ET. Following procedure is an example formonthly ET and ETindex estimation:(1) Compute ETindex for satellite observation dates.
(2) Compute 16-day composite images of ETindex.
(3) Compute daily ET (mm/day) using equation 16 with ETo computed by weather data.
ETindex in the equation is the representative number (i.e. minimum value) of the
16-day, and ETo is daily values including cloudy days.

𝐸𝑇𝑎𝑐𝑡 = ET𝑖𝑛𝑑𝑒𝑥 × 𝐸𝑇0( 16 )(4) Compute monthly ET (mm/month) by integrating daily ET derived by equation 16.
(5) Compute soil evaporation of the month, using weather data and standard soil
condition, adopting soil water balance model suggested by FAO (Allen et al., 1998).
(6) If monthly ET computed by step (4) is less than evaporation derived by soil waterbalance (step (5)), over-write the ET value with the result of soil water balance
model. The result is the monthly ET (mm/month) adjusted by soil water balance
model.
(7) Compute monthly ETindex using equation 1, with the output of step (6) andmonthly ETo.

28

3-9. Calculation of Soil Moisture (Optional)
Estimating soil moisture requires information about soil water holding capacity.
With this information, one can convert ETindex to soil moisture by adopting the SWEST
method summarized in Tasumi and Kimura (2013).
The soil moisture estimation procedure has been tested for a natural grassland
condition under semiarid climate using the weather and soil moisture observation dataof Shenmu, China. The result indicated that soil moisture was sufficiently estimated at
this location (Figure 15). The estimation error of the mean soil water content from zero
to 25 cm depth was about 0.029 (i.e., estimation error of soil water was 2.9% of total
volume of soil in 1 σ) on average for three years (Tasumi and Kimura, 2013).

1:1

22

θ Estimated (%)

18

14
2004

10

2005
6

2006

2
2

6

10

14

18

22

θ Measured (%)
Figure 15. Performance of GCOM-C SM estimation model applied to Shenmu, China,
during 2004–2006 (average soil moisture for root zone, 0–25 cm ).

29

4. Application to Satellite Imagery and Information about Model Accuracy
Application tests and accuracy analyses have been made through the past years.
Some of the results listed below were derived using the older version of the algorithm.

4.1. Global Application Test using MODIS Land Surface Temperature Product
The ETindex was estimated for the entire globe using the MODIS daily surface
temperature map for the years 2001–2007 and daily global near-surface wind speed
data for the corresponding years supplied by Dr. Mabuchi of Chiba University, Japan.
The result of the annual average value for 2001 is shown in Figure 16. Using the daily
information, daily ETindex maps were calculated. The daily maps contained effects ofclouds and those had to be eliminated. In this application, the daily maps were grouped
into series of 16 days, and the minimum pixel values of the 16-day series were picked up
by assuming that the minimum values are the cloud-free, representative ETindex of the
16 days. For the pixels having no data during the 16 days, the ETindex value of 1.23 (i.e.,the maximum value) was assigned by assuming that such pixels should be wet pixels(16 days of continuous clouds would likely bring some rainfall). No regulation using
NDVI or land cover was applied in this application.

Figure 16. Estimated annual average ETindex for 2001 from ATBD ver. 1.2.
The 16-day ETindex was then converted to ET by using additional daily global
meteorological data (air temperature, humidity, solar radiation) supplied by Dr.
Mabuchi of Chiba University. The estimation result is shown in Figure 17. In thisapplication, daily ET was calculated by the 16-day ETindex multiplied by the daily ETo,assuming that the ETindex is constant during the 16-day period. This assumption is30necessary to avoid the problem caused by cloud cover. Daily ET maps were thenaccumulated and the annual ET map shown in Figure 17 was derived.

Figure 17. Estimated annual cumulative ET for 2001 from ATBD ver. 1.2.
Note that the annual averaged ETindex shown in Figure 16 is not a simple averageof the 16-day ETindex maps. ETindex is calculated using annual ET and annual ETo. In
the operational procedure, Figure 16 was derived after Figure 17 was derived. Thisapplication test confirmed that operational computation of the algorithm can be
conducted without problem.

31

4.2. Regional Application Test using GCOM-C Land Surface Temperature ProductRegional application tests were made with GCOM-C Land Surface Temperature
product version 1, for five locations in the world; Miyazaki and Tokyo (Japan), Southern
Idaho (USA), Urmia (Iran), and Bangkok (Thailand). Figure 18 shows the result of ET
estimation, with NDVI information. Estimated monthly ET showed reasonable valuesin Miyazaki, Tokyo, Southern Idaho and Urmia, except some mountainous regions,where further refinements of topographic adjustments are required. A malfunction of
the estimation algorithm was found in Bangkok, Thailand, because the seasonalfunction of the current algorithm does not work for low-latitude area in monthly
timescale. Further improvement of the model is required for the low latitudes.

Jan

Feb

Jan

Feb

Mar

Apr

Mar

Apr

May

Jun

May

Jun

Jul

Aug

Jul

Aug

Sep

Oct

Sep

Oct

Nov

Dec

Nov

Dec

Figure 18. Estimated monthly ET for 2019 (left), and the monthly maximum NDVI
(right), around Magic Valley irrigation area of Idaho, USA.

32

4.3. Analysis of ETindex for Model Applicability
The proposed ETindex estimation algorithm has a structure applicable to the entire
globe. However, some of the most important equations are empirical equations
developed by relying heavily on a location in China. One reasonable method to confirm
the adequacy of the estimation algorithm is to calculate the ETindex value without
applying the regulation suggested by Equation 14 and to evaluate the values in “wet”
areas of the world. In “wet” areas, the ETindex is expected to be close to 1.23. If theestimated ETindex were far above 1.23, or far below 1.23, it would be an obvious
indication that our ETindex estimation has a problem.
Figure 19 shows the seasonal trend of estimated ETindex for four selected “wet”
areas of the world: central Africa, the eastern part of China, central South America, andSoutheast Asia. These four “wet” areas were selected manually according the result of
the global application shown in Figure 16. The ETindex values of each area wereaveraged values of 20 sample pixels. Ideally, if the surface is in a wet-enough condition,
ETindex reaches 1.23 (but does not exceed 1.23). The result shown in Figure 18 seems tosufficiently meet the criterion. This analysis indicates that the ETindex estimation isfairly functional, at least in these four selected regions.

1.4
1.2

ET-Index

1
0.8
0.6
Africa_Center

0.4

China_East

0.2

S-America_Center
SE_Asia0Jan. <------------------------------------> Dec.
Figure 19. Seasonal trend of estimated ETindex for four “wet” regions in the world, 2001.

33

4.4. Reality Check of the Estimated Values using Outside Sources
ET and ETindex are closely related to other types of environmental information,such as climate condition, weather (such as rainfall), vegetation distribution, and so on.
A reality check of the estimated ETindex was conducted using the following outside
sources.
(a) Global forest/non-forest map by JAXA (Figure 20)(b) Distribution of arid regions map by the Ministry of the Environment (Figure 21)(c) Estimated vegetation and precipitation maps of Africa by Sanga and Nonomura(2013) (Figure 22).

Figure 20. PALSAR 10-m global forest/non-forest map, 2009.

34

Figure 21. Distribution of arid regions by Ministry of the Environment, Japan.

Figure 22. Estimated vegetation (left) and precipitation (right) maps of Africa.
These maps were compared with our estimated results shown in Figure 16. Notethat the maps shown in Figures 20–22 are not for the year 2001 but are for different
years or for multi-year general information.
The forest distribution (Figure 20) corresponded well with the high ETindex regionsof Figure 16, and the distribution of arid regions (Figure 21) corresponded well with the
low ETindex regions of Figure 16. The distribution of vegetation on the African continent(Figure 22, left) also corresponded well with the ETindex pattern in Figure 16.
Additionally, the monthly precipitation distribution corresponded well with the 16-day
estimated ETindex (Figure 23). The results of the comparison indirectly support theadequacy of the ETindex estimation algorithm.
35

Figure 23. Precipitation distribution in Jan., Apr., Jul., and Oct. compared withcorresponding 16-day ETindex maps.

36

Acknowledgements
This research was conducted under the Research Announcements for the Global Change
Observation Mission (GCOM), funded by the Earth Observation Research Center,
Japan Aerospace Exploration Agency (JAXA/EORC).

37

References
Allen, R. G., Pereira, L. S., Raes, D., and Smith, M. (1998). “Crop Evapotranspiration.”
FAO Irrigation and Drainage Paper 56, Food and Agricultural Organization of
the United Nations, Rome.
Allen, R. G., Tasumi, M., Trezza, R., (2007). Satellite-based energy balance for mapping
evapotranspiration with internalized calibration (METRIC)—model. J. Irrig.
Drain. Eng. ASCE. 133.
Bastiaanssen, W. G. M., M. Menenti, R. A. Feddes, A. A. M. Holtslag, (1998). A remotesensing surface energy balance algorithm for land (SEBAL): 1. Formulation. J.
of Hydr. 212-213:198-212.
Junsei K, (2000). Atmospheric Science near the Ground Surface. University of Tokyo
Press. (in Japanese)
Kanda, E., Tasumi, M., Ito, F., Ozaki, T., Kimura, R., (2010). 全球蒸発散量推定のための
地表面温度指標についての基礎的研究. 2010 年度農業農村工学会講演要旨集.
Kimura, R., (2007). Estimation of moisture availability over the Liudaogou River Basinof the Loess Plateau using new indices with surface temperature. Journal of
Arid Environments, 70:237–252.
Kogan, F. N. (1995). Application of vegetation index and brightness temperature fordrought detection. Adv. Space Res., 15, 91-100.
Mabuchi, K. (2011). A numerical investigation of changes in energy and carbon cycle
balance under vegetation transition due to deforestation in the Asian tropicalregion. Journal of the Meteorological Society of Japan, Vol. 89, No.1, pp.47-65.
Mu, Q., Zhao, M., Running, S. W. 2011. Improvements to a MODIS global terrestrial
evapotranspiration

algorithm.

RemoteSensingofEnvironment.

115:1781-1800.
Sanga-Ngoie, K., Nonomura, A. 2013. Devising a framework for assessing renewableenergy resources in the African biosphere using GIS and remote sensing
techniques. 平成 24 年度日本リモートセンシング学会九州支部研究発表会論文要旨集 pp42-44.
Senay, G. B., Budde, M., Verdin, J. P., Melesse, A. M. (2007). A coupled remote sensing
andsimplified

surfaceenergybalanceapproachtoestimate

actualevapotranspiration from irrigated fields. Sensors, 7:979-1000.
Senay, G. B., Budde, M., Verdin, J. P. (2011). Enhancing the Simplified Surface Energy
Balance (SSEB) approach for estimating landscape ET: Validation with the
METRIC model. Agricultural Water Management, 98:606-618.
38

Tasumi, M. and Kimura, R. (2013). Estimation of volumetric soil water content over the
Liudaogou River Basin of the Loess Plateau using the SWEST method withspatial and temporal variability. Agricultural Water Management. 118:22-28.
Tasumi, M. (2019). Estimating evapotranspiration using METRIC model and Landsat
data for better understandings of regional hydrology in the western Urmia Lake
Basin. Agricultural Water Management. 226:105805.
Tasumi, M., Moriyama, M., and Shinohara, Y. (2019). Application of GCOM‑C SGLI for
agricultural water management via field evapotranspiration. Paddy and Water
Environment. 17:75-82.

39

Appendix: Ts(wet) and Ts(dry) determination model for Shenmu weather measurement
siteThis model estimates surface temperatures for hypothetical wet and dry surfaces
surrounded by a given weather condition. The wet surface is defined as the surface
having no sensible heat, and the dry surface is defined as the surface having no ET.
Hypothetical temperatures for the wet and the dry surfaces are named the wet surface
temperature (Ts(wet)) and the dry surface temperature (Ts(dry)). The model estimatesTs(wet) and Ts(dry) using surface radiation and energy balance equations with observed
ground-based weather data as input data.
Determination of Ts(dry) was made possible by using the energy balance model
described below. However, direct determination of Ts(wet) was difficult because of thenumerical instability of the iterative computation. To avoid this numerical instability,
the surface temperature for a reference surface, Ts(ref), was employed as an initialintermediate parameter prior to estimating Ts(wet).
Calculation step 1 – determinations of Ts(ref) and Ts(dry)This model derives Ts(ref) and Ts(dry) from surface energy balance computation. Theenergy balance equation is expressed as follows:

Rn = E + G + H ,where Rn is net radiation (W m-2), ℓE is latent heat flux (W m-2), G is soil heat flux (Wm-2), and H is sensible heat flux (W m-2).
Net radiation (Rn)
Rn is calculated from the radiation balance equation:

Rn = (1 − )Rs + L down *  − L up ,where Rs is measured solar radiation (W m-2), α is measured albedo, Ldown is measured
incoming longwave radiation (W m-2), Lup is outgoing longwave radiation (W m-2) from
App. 1the hypothetical wet or dry surface, and ε is surface emissivity (= 0.98). Lup depends on
Ts(ref) and Ts(dry).
Latent heat flux (ℓE) and soil heat flux (G)ℓE for the wet surface is equivalent to ETo by definition. ETo is calculated byfollowing the FAO56 equation for hourly data, using surface temperature instead of air
temperature:

0.408(Rn − G ) 
ETo =

3600
37
+
u 2 (e0 − ea )6
Ts ( wet ) + 273
10, +  (1 + 0.34u 2 )where Rn is net radiation (W m-2), G is soil heat flux (W m-2), Ts(ref) is temperature ofreference surface (C), Δ is saturation slope vapor pressure curve at T (kPa C-1), γ is thepsychrometric constant (kPa C-1), e0 is saturation vapor pressure at temperature T(kPa), ea is actual vapor pressure (kPa), u2 is wind speed (m s-1), and the number
3600/106 is for the conversion from W m-2 to MJ m-2 hr-1.
ETo (mm hr-1) is converted to latent heat flux (W m-2) as

E = ETo *  *

106
,
3600

where ℓ is latent heat of vaporization (J kg-1) calculated as

 = 2.501− 0.002361* Ts (wet ) .
ℓE for the dry surface is zero by definition. G is calculated by the FAO equation as

G = 0.1Rn
G = 0.5Rn(for daytime)(for nighttime) .

In this model, measured albedo is applied when calculating Rn, which slightly
violates the definition of ETo (albedo = 0.23). The ETo used in this model is a valueApp. 2

adjusted by actual albedo of the site. Also, using measured vapor pressure in EToestimation might overestimate ETo in the “wet condition” because the vapor pressure
over the hypothetical wet surface might be higher than the measured value. This is alimitation of the model.
Sensible heat flux
Sensible heat flux is calculated by the bulk equation (Kondo, p. 142, Eq. 5.7–5.8):

H = Cp    C H  u 2  (Ts − Ta )where Cp is the heat capacity of air (J kg-1 K-1), ρ is the density of air (kg m-3), Ts is the
hypothetical surface temperature (i.e., Ts(ref) or Ts(dry)) (C), Ta is the measured airtemperature (C), and CH is the balk coefficient, which is calculated as

CH =

k2
ln(z / z 0 ) ln(z / z T )where k is von Karman’s constant (0.41), z is the reference height (2 m), and z0 and zT
are the surface roughness for momentum and heat transfer (m).
During the sensible heat flux computation, an air stability correction was applied
via iteration using Monin–Obhukov similarity theory. In this model, measured airtemperature was used to estimate H, which would generate an error on H estimation of
the hypothetical wet and dry surfaces, as the air temperature over the hypotheticalsurfaces would be different from the measured air temperature. This is a limitation of
the model.
Determination of Ts(wet) and Ts(dry)The optimal solutions for Ts(ref) and Ts(dry) are then found by the numericaliteration model. The optimal solutions are the values at which the following equations
are valid.

(1 −  )Rs + Ldown *  − Lup = E + G + H(for reference surface)App. 3

(1 −  )Rs + Ldown *  − Lup = G + H(for dry surface)Calculation step 2 – determination of Ts(wet)Ts(wet) is calculated by adjusting Ts(ref) in proportion to the sensible heat:

Ts (wet ) = Ts (ref ) −Ts (dry) − Ts (ref ) H (ref )H (dry) − H (ref )Surface temperature (oC)The equation is visualized in Figure App.1.

Ts(dry)Ts(ref)Ts(wet)00H(ref)

H(dry)

Sensible heat flux (w/m2)
Figure App.1. Ts(wet) determination.

App. 4