Journal of Hydrology 468–469 (2012) 35–46
Contents lists available at SciVerse ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Upscaling latent heat flux for thermal remote sensing studies: Comparison
of alternative approaches and correction of bias
Thomas G. Van Niel a,b,⇑, Tim R. McVicar c, Michael L. Roderick b,d,e, Albert I.J.M. van Dijk c,f, Jason Beringer g,
Lindsay B. Hutley h, Eva van Gorsel i
a
CSIRO Land and Water, Private Bag No. 5, Wembley, WA 6913, Australia
Research School of Biology, Australian National University, Canberra, ACT 0200, Australia
CSIRO Land and Water, G.P.O. Box 1666, Canberra, ACT 2601, Australia
d
Research School of Earth Sciences, Australian National University, Canberra, ACT 0200, Australia
e
Australian Research Council Centre of Excellence for Climate System Science, Sydney, NSW 2052, Australia
f
Fenner School of Environment & Society, Australian National University, Canberra, ACT 0200, Australia
g
School of Geography and Environmental Science, Monash University, Clayton, VIC 3800, Australia
h
Research Institute for the Environment and Livelihoods, Charles Darwin University, Darwin, NT 0909, Australia
i
CSIRO Marine and Atmospheric Research, G.P.O. Box 3023, Canberra, ACT 2601, Australia
b
c
a r t i c l e
i n f o
Article history:
Received 22 May 2012
Received in revised form 2 August 2012
Accepted 7 August 2012
Available online 16 August 2012
This manuscript was handled by Geoff
Syme, Editor-in-Chief
Keywords:
Evaporation
Evapotranspiration
Evaporative energy
Instantaneous
One-time-of-day
s u m m a r y
For instantaneous latent heat flux (kE) estimates from thermal remote sensing data to be useful in the
hydrologic sciences, they require integration over longer time frames (e.g., months to years). This is
not trivial because thermal remote sensing data acquired under cloud-free daytime conditions require
upscaling to a monthly energy amount that is both relevant over cloudy periods and considers daytime
and nighttime. Previous work has compared upscaling approaches, but as yet there is no authoritative
comparison that does so under conditions relevant for thermal remote sensing. In this paper we describe,
under the conditions relevant for thermal remote sensing, a generic framework for comparing any
upscaling approach that assumes self-preservation. Then we use eddy-flux data from two sites in contrasting climates to systematically evaluate the accuracy of different upscaling proposals within the
framework. We assumed that the instantaneous estimate of the latent heat flux measured by the
eddy-flux technique would have been measured by a satellite sensor. We then scaled this estimate to
a monthly period using four approaches and compared the result with the observed monthly integral.
This design enabled us to isolate the accuracy of each upscaling method. The four methods upscaled
kE by: (i) observed solar irradiance (S); (ii) modelled solar irradiance from a sine function (SSIN); (iii)
modelled top-of-atmosphere solar irradiance (STOA); and (iv) observed available energy (AE). We showed
that upscaling kE using observed data (S, AE) resulted in underestimation of monthly evaporative energy,
while the use of modelled data (SSIN, STOA) led to overestimation, primarily due to the relationship
between error and both the season (day-of-year) and cloud fraction. Of the two observed fluxes, upscaling
with S resulted in lower overall errors than when using AE (S bias: 1.11 M J m2 d1 or 16%; AE bias:
2.15 M J m2 d1 or 34%). Of the two modelled fluxes, upscaling with STOA had lower errors than the
widely used SSIN method (SSIN bias: 1.03 M J m2 d1 or 14%; STOA bias: 0.91 M J m2 d1 or 13%). We
subsequently developed a simple procedure to minimise bias from all four upscaling approaches, and
concluded that modelled data (STOA) can be used to upscale kE to longer timescales for thermal remote
sensing applications. This study developed the theory to minimise upscaling bias at two sites with
contrasting climates, further work is needed to extend the approach to all global terrestrial climates.
Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction
Thermal remote sensing estimates of latent heat flux (kE,
W m2) are based on data acquired at a specific time-of-day under
⇑ Corresponding author at: CSIRO Land and Water, Private Bag No. 5, Wembley,
WA 6913, Australia. Tel.: +61 8 9333 6705; fax: +61 8 9333 6499.
E-mail address: Tom.VanNiel@csiro.au (T.G. Van Niel).
0022-1694/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jhydrol.2012.08.005
clear-sky conditions. For remote sensing estimates of kE to be useful to hydrology, they must be temporally ‘upscaled’ to longer time
frames over which clouds will influence the surface energy balance
(e.g., monthly time steps over years). Temporal upscaling is usually
performed by assuming that the ratio of total evaporative energy
over specific time-of-daytime kE is equal to that of some other
more readily observed or modelled surface energy flux (i.e., that
the evaporative fraction is constant). This assumption is generally
36
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
known as self-preservation of fluxes (Brutsaert and Sugita, 1992;
Crago and Brutsaert, 1996). The most common fluxes used for temporal upscaling are available energy (AE, W m2) (e.g., Crago, 1996;
Sugita and Brutsaert, 1991) and solar irradiance (S, W m2) (e.g.,
Hatfield et al., 1983; Jackson et al., 1983; Zhang and Lemeur, 1995).
Temporal upscaling kE by assuming self-preservation with AE is
known as the evaporative fraction (EF) method; when self-preservation is assumed with S, it is herein referred to as the solar irradiance (SI) method. Determining the relative merits of using either
the EF or the SI method for upscaling kE has been problematic because in previous studies the scaling fluxes have been inconsistent,
e.g.: (i) either observed or modelled; (ii) integrated over 24-h or
daytime-only; and (iii) observed under clear-sky, all-sky or (rarely)
under the mixed clear and all-sky conditions relevant for hydrological remote sensing. For example, assessment of the SI method’s
performance was done using observed 24-h S by Chávez et al.
(2008), but was done using modelled daytime-only S by Zhang
and Lemeur (1995). Theoretical development and assessment of
the EF method was from point-based observed daytime-only AE
by Crago and Brutsaert (1996) and Crago (1996). When upscaling
instantaneous remote sensing derived kE, modelled 24-h net radiation was used by Su (2005) and Sobrino et al. (2005), even though
ignoring the ground heat flux also introduces bias (Cammalleri
et al., 2012).
A further fundamental assumption involved in upscaling kE by
self-preservation is that all times within the integration period
are cloud-free (Crago, 1996; Jackson et al., 1983). While self-preservation is assumed for upscaling remote sensing estimates of kE,
in practical application the restriction that the entire daytime must
be clear-sky is not enforced (e.g., Anderson et al., 1997, 2007a,
2007b; McVicar and Bierwirth, 2001; McVicar and Jupp, 2002).
Rather, the specific time-of-day observation must be clear-sky,
but the rest of the day can be clear or cloudy. Previous assessments
have largely evaluated upscaling error under either clear-sky conditions (Colaizzi et al., 2006; Zhang and Lemeur, 1995) or all-sky
conditions (Brutsaert and Sugita, 1992; Cammalleri et al., 2012;
Kustas et al., 1994; Ryu et al., 2012; Van Niel et al., 2011). Few
studies have assessed error under the mixed conditions pertinent
to remote sensing by upscaling clear-sky specific time-of-daytime
kE to all-sky evaporative energy (for a notable exception, see Liu
and Hiyama, 2007). Discrepancies between using modelled or observed fluxes, 24-h or daytime-only integration, and different sky
conditions confound the interpretation of results between upscaling studies and therefore a rigorous assessment under clearly defined and standardised conditions is warranted.
We present a common framework that is useful for assessing
any method that makes the self-preservation assumption, whether
using observed or modelled fluxes and which is explicit regarding
time and sky conditions. The current paper extends the all - sky
daily framework of Van Niel et al. (2011) to consider both modelled
and observed versions of the SI method to assess upscaling error at
a monthly time-step, and evaluates upscaling errors under the
mixed sky conditions relevant to remote sensing applications.
The aims of the current paper are to: (i) compare the error of four
cases in total of the SI and EF methods under conditions relevant to
remote sensing applications; and (ii) evaluate the ability to model
upscaling bias with generic functions suitable for implementation
over large areas (e.g., using variables day-of-year, cloud fraction,
and time-of-daytime, described below).
2. Theory
Upscaling a specific time-of-daytime kE to evaporative energy
by assuming self-preservation with some other flux is based on
the clear-sky assumption, that over the entire integration period,
all times are cloud-free (Crago, 1996; Jackson et al., 1983). The
clear-sky assumption is not pertinent to upscaling of remote sensing kE for large-area hydrological applications because: (i) only the
specific time-of-daytime observation is required to be clear-sky,
while the rest of the day can be clear or cloudy; and (ii) to ensure
long-term mass and energy are balanced, estimation of the all-sky
evaporative energy is required, not just that from clear-sky days.
Estimation of all-sky kE is necessary for accurate estimation of
the mass and energy balances, but particularly problematic for remote sensing applications due to data gaps. Even for polar orbiting
satellites with daily repeat times (e.g., MODIS and AVHRR), the
specific time-of-daytime thermal measurement will often not be
observed due to cloud cover, resulting in a gap for the entire day.
Persistent cloudy conditions in places with a distinct wet season
(e.g., the tropics) can result in few or no kE estimates over a month
(e.g., McVicar and Bierwirth, 2001). Evaluation of upscaling error at
the monthly time step allows for the influence of data gaps to be
intrinsically assessed, and thus is particularly relevant to remote
sensing studies. The theoretical framework of our study is demonstrated next using the specific case of upscaling kE by groundbased observed S. However, the equations are applicable to other
observed or modelled fluxes, as shown further on.
Any specific time-of-daytime kE based on data observed by satellite under clear-sky conditions is upscaled to monthly all-sky
evaporative energy amount by the following approximation based
on the assumption of self-preservation:
Dt kEm ¼ kEðt Þ Dt kEm
Dt Sm
kEðt Þ ;
kEðt Þ
Sðt Þ
ð1Þ
where Dt kEm and Dt kSm are the average evaporative and solar
energy over a given month, respectively (J m2 d1); Dt the number
of seconds in the 24-h day (86,400); t = (t tSR)/Dtd (proportion) is
the normalised time-of-daytime associated with a satellite overpass
calculated as the number of seconds past sunrise (t tSR) divided by
the number of seconds of the daytime (Dtd); kEm and Sm the
monthly mean kE and S under all sky conditions, respectively
(W m2); and kEðt Þ and S(t) specific time-of-daytime kE and S observed under clear sky conditions (W m2). Note, Eq. (1) is valid as
long as S(t) – 0.
Because S is both more readily observed and modelled than kE,
the approximation given by Eq. (1) is potentially useful for largearea modelling of evapotranspiration by remote sensing. The
accuracy of Eq. (1), however, varies due to differences in surface,
planetary boundary layer, and free atmosphere conditions, all
possibly resulting in a lack of self-preservation over days to
months. Following Van Niel et al. (2011), Eq. (1) can be modified
to include a correction factor, which accounts for the error due to
the approximation made by the self-preservation assumption,
yielding:
Dt kEm ¼ kEðt Þ Dt Sm
bs ðt Þ:
Sðt Þ
ð2Þ
When both S and kE are observed, bs(t) can be calculated directly as:
Dt kEm
bs ðt Þ ¼
kEðt Þ
,
,
Dt Sm kEm kEðt Þ
¼
¼ aS;m =aS ðt Þ;
Sðt Þ
Sðt Þ
Sm
ð3Þ
where aS,m is the monthly all-sky evaporative ratio (in this case
with respect to S), and where aS(t) is the clear-sky specific timeof-daytime evaporative ratio with respect to S. Note, Eq. (3) is valid
for all Sm kEðt Þ–0. The assumption for studies implementing the
approach of Eq. (1) is that bS(t) is unity (Van Niel et al., 2011),
therefore deviation of bS(t) from unity serves as a measure of
how good the self-preservation assumption is for any given site,
for any given day, for any given time-of-daytime. The error of
37
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
assuming self-preservation between kE and S (eS(t), J m2 ) is
expressed, then, by (see Supplementary material for derivation):
eS ðt Þ ¼ Dt kEm ½bS ðt Þ1 1 ¼ Dt kEm aS ðt Þ aS;m
:
aS;m
et al., 2007; Guerschman et al., 2009; Leuning et al., 2005; O’Grady
et al., 2000; Schymanski et al., 2007; Van Gorsel et al., 2007). Energy
balance closure is a frequently used diagnostic at eddy covariance
flux sites. The daily averaged hourly energy balance closure is within
10% at Tumbarumba (Leuning et al., 2005), while daily energy balance closure is within 8–15% at Howard Springs (Beringer et al.,
2007). Following Van Niel et al. (2011), the flux tower data were
treated as reference and energy balance was not forced to close.
The sub-daily flux observations were integrated to average
monthly evaporative energy for each month. Time-of-daytime past
sunrise for all valid flux measurements were normalised by their
associated daytime lengths calculated using the US Nautical Almanac Office algorithm (NAO, 1990) and categorised into the closest
of 12 normalised time-of-daytimes (0.00, 0.083, 0.167, . . . , 1.00),
allowing for analysis to be performed between specific time-ofdaytime and monthly integrated fluxes. These comparisons were
only made for a flux on any particular day if both the specific
time-of-daytime observation had an atmospheric transmittance
P0.70 (i.e., was clear-sky), and if > 15 days worth of valid sub-daily
observations were measured over the month.
Observed monthly cloud cover data at Bureau of Meteorology
sites used to correct for upscaling error were from Jovanovic
et al. (2011). That dataset was based on the mean of visual 9 a.m.
and 3 p.m. (local time) observations of total cloud amount with a
precision of one-eighth of the sky fraction (i.e., ‘oktas’). We extracted data from the two closest stations to each flux tower site
described above. These were Darwin Airport, Northern Territory
(Station ID:014015, 12.420°S, 130.890°E, 27 km from Howard
Springs), and Wagga Wagga, New South Wales (Station
ID:072150, 35.160°S, 147.460°E, 83 km from Tumbarumba).
There were no missing cloud data during the flux site period of record for both sites. Mean monthly cloud amount were converted to
monthly cloud fraction (cf).
ð4Þ
Eq. (4) represents the self-preservation error for a specific time-ofdaytime for a given day under conditions relevant for thermal remote sensing applications. That is, when the specific time-of-daytime must be under a clear sky, but when the rest of the time can
be clear or cloudy.
The monthly bias (Jm-2d-1) at normalised time of satellite overpass (t) relevant to remote sensing studies is calculated using all
days of the month (c) where the specific time of satellite overpass
kEðt Þ and S(t) occurred under clear skies, and is given by:
biasS ðt Þ ¼
c
1X
eS ðt Þ:
c 1
ð5Þ
The mean monthly bias (J m2 d1) at normalised time of satellite overpass, t, is calculated by averaging Eq. (5) over all months,
n, that have a near complete record by:
biasS ðt Þ ¼
n
1X
biasS ðt Þ;
n 1
ð6Þ
and the Root Mean Squared Difference (RMSD) for the given time of
satellite overpass over those same months (J m2 d1) is:
RMSDS ðt Þ ¼
n
1X
½biasS ðt Þ2
n 1
!0:5
ð7Þ
:
Some days of the month can have clouds at the specific time of
satellite overpass. When this occurs, c is a subset of the number of
days of the month, and assessment of monthly upscaling error
from Eqs. (5)–(7) includes error associated with daily data gaps
consistent with remote sensing applications.
4. Methods
3. Study sites and data
The errors associated with four cases of upscaling specific timeof-daytime latent heat flux to monthly evaporative energy were
evaluated. This was the first aim of the current paper. The bias
and RMSD of each of the four upscaling cases was: (i) assessed under the original assumption of self-preservation (i.e., that b is
unity), herein referred to as the ‘null’ model; and (ii) assessed after
adjusting the null model by a function of day-of-year (Yd), cloud
fraction (cf), and normalised time-of-daytime (t), herein referred
to as the ‘adjusted’ or ‘correction’ model. The four cases assessed
temporally upscaling observed latent heat flux by using:
Hourly and half-hourly precipitation (P), net radiation (RN),
ground heat flux (G), S, and kE data were measured at two longterm eddy covariance flux tower sites at Tumbarumba, New South
Wales (35.650°S, 148.151°E), and Howard Springs, Northern Territory, (12.495°S, 131.150°E) Australia. AE was determined as RN less
G. Hourly measurements were made at Tumbarumba over
3965 days between February 2001 and December 2011. Halfhourly measurements were made at Howard Springs over
3365 days between May 2001 and December 2010. The sites are
situated in highly contrasting ecosystems: Tumbarumba is located
in a wet sclerophyll forest with no distinct P seasonality, but where
there is a considerable difference between summer (DJF) and winter (JJA) S and AE (Table 1). Howard Springs is a tropical savanna
site where there is a large seasonal variability in P between summer and winter, but a smaller difference in S and AE (Table 1).
Instrumentation, data quality, and data processing of the flux
tower data have been described in detail previously (Beringer
(i) observed solar irradiance (S) from flux tower measurements;
(ii) modelled solar irradiance based on a sine function (SSIN,
defined below);
(iii) modelled top-of-atmosphere solar irradiance (STOA, defined
below); or
(iv) observed available energy (AE) from flux tower
measurements.
Table 1
Overall and seasonal mean energy amounts are shown for the two sites calculated over their entire period of record. P is expressed as energy equivalent units of MJ m2 d1 and in
mm d1 in parentheses. Energy was averaged over seasons of December, January, and February (DJF); March, April, and May (MAM); June, July, and August (JJA); September,
October, and November (SON); and January – December (Annual).
Time
DJF
MAM
JJA
SON
Annual
Tumbarumba (MJ m2 d1)
Howard Springs (MJ m2 d1)
Dt kE
Dt AE
Dt S
9.8
5.2
2.7
6.5
6.3
15.8
7.7
3.7
12.4
10.4
26.9
16.7
11.4
22.8
20.0
Dt P
7.6
5.8
12.0
9.9
8.8
(3.0)
(2.3)
(4.7)
(3.9)
(3.5)
Dt kE
Dt AE
Dt S
Dt P
11.5
9.9
5.4
6.8
8.2
15.0
13.7
9.8
13.4
12.8
22.2
22.0
20.0
22.6
21.6
33.0
10.5
0.9
7.4
12.4
(12.9)
(4.1)
(0.3)
(2.9)
(4.8)
38
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
Table 2
Description of notation used for selected key variables is provided.
Notation
Description
v
Generic term used to represent the four upscaling cases of S, SSIN, STOA, AE
Dt; Dt d ; t
24-h day length (s); daytime length (s); time after sunrise divided by daytime length (proportion)
kEm ; AE;m ; Sm ; STOAd;m
Monthly average all-sky latent heat flux; available energy; solar irradiance; and daytime top-of-atmosphere solar irradiance (all
in units of W m2)
Dt kEm ; Dt AE;m ; Dt Sm ; Dtd STOAd;m
Monthly average all-sky evaporative energy; available energy; solar energy; and daytime top-of-atmosphere solar energy (all in
units of J m2 d1)
kEðt Þ; AE ðt Þ; Sðt Þ; STOA ðt Þ
Specific time-of-daytime clear-sky latent heat flux; available energy; solar irradiance; and top-of-atmosphere solar irradiance (all
in units of W m2) for a given satellite overpass
av;m
The monthly all-sky evaporative ratio (proportion) with respect to case v (e.g., when v is AE, then aAE;m ¼ kEm =AE;m )
av ðt Þ
The specific time-of-daytime clear-sky evaporative ratio (proportion) on a given day with respect to case v (e.g., when v is S, then
aS ðt Þ ¼ kEðt Þ=Sðt Þ)
bv ðt Þ
Term to correct for error in upscaling case v at specific time-of-daytime on a given day (proportion)
biasv ðt Þ; biasv ðt Þ
The mean bias in upscaling case v for a given month at a specific time-of-daytime; and the mean bias in upscaling case v over a
number of months at a specific time-of-daytime (both in units of J m2 d1)
RMSDv ðt Þ
The root mean squared difference in upscaling case v over a number of months at a specific time-of-daytime (J m2 d1)
Cases (i)–(iii) were instances of the SI method, and case (iv) was
the observed data instance of the EF method. Assessment was performed over the set of near-complete months over seven normalised daytimes (0:25 6 t 6 0:75), herein referred to as the central
part of the daytime. A summary of the notation used here is provided in Table 2 for key variables.
4.1. Null models
4.1.1. S
For case (i), the sub-daily b factors and associated errors were
calculated using Eqs. (3)–(7). For cases (ii)–(iv), modified versions
of these same equations were used, explained below.
4.1.2. SSIN
Ground-based sensor networks of S at the sub-daily time step
are usually not sufficiently dense to allow their use in the SI method over vast areas (McVicar and Jupp, 1999). Upscaling of largearea remote sensing estimates of kE, then, normally require some
model of S. Jackson et al. (1983) used the sine function model of
Monteith (1973) to upscale kE estimated from remote sensing data
to the daily total evaporative energy. This upscaling case is herein
referred to as SSIN (case (ii), above). The convenience of SSIN is in its
algebraic simplification of the integral of the daily solar irradiance.
This means that it does not require numeric integration and is
therefore very simple to implement. SSIN approximates the ratio
of the daily evaporative energy over the specific time-of-daytime
latent heat flux (Jackson et al., 1983), and after adjustment to estimate monthly evaporative energy, it is given by:
Dt kEm Dt Sm
2 Dt d
:
kEðt Þ
Sðt Þ
psinðpt Þ
ð8Þ
The factor required to correct for upscaling with SSIN is defined by
adjusting Eqs. (1)–(3) to account for Eq. (8), yielding:
bSSIN ðt Þ ¼
Dt kEm psinðpt Þ
:
kEðt Þ 2Dtd
ð9Þ
The errors associated with bSSIN ðt Þ were defined by modifying Eqs.
(4)–(7) for use with Eq. (9).
4.1.3. STOA
Case (iii) was based upon top-of-atmosphere solar irradiance
(STOA). Ryu et al. (2012) used STOA to estimate 8-day average kE
associated with MODIS imagery. For consistency, we generalised
the method of Ryu et al. (2012) so that it was not specific to
half-hourly observations of kE, and so it estimated the ratio of
the monthly (rather than daily) evaporative energy. The relevant
approximation of upscaling kE becomes:
Dt kEm Dtd STOAd;m
;
kEðt Þ
STOA ðt Þ
ð10Þ
where STOAd;m (W m2) is the monthly daytime average modelled
STOA. The modelled specific time-of-daytime STOA (STOA(t), W m2)
was defined by Ryu et al. (2012) as:
2pY d
cosð/ðt ÞÞ;
STOA ðt Þ ¼ SSC 1 þ 0:033cos
Y d max
ð11Þ
with SSC being the solar ‘constant’ (ca. 1360 W m2), Yd being the
day-of-year, Y d max the maximum Yd (365 or 366) for the specified
year, and /(t) being the specific time-of-daytime solar zenith angle
(radians). Following the same procedure as above, the factor required to correct for the use of modelled STOA is given by:
bSTOA ðt Þ ¼
Dt kEm STOA ðt Þ
:
Dt d STOAd;m kEðt Þ
ð12Þ
The errors associated with bSTOA ðt Þ were defined by modifying Eqs.
(4)–(7) for use with Eq. (12).
4.1.4. AE
Case (iv) assessed the ability of AE to upscale kE and is the observed data version of the EF method. The factor required to correct
for upscaling kE with AE is given by (see Van Niel et al. (2011) for
full details):
bAE ðt Þ ¼
kEm AE ðt Þ
;
AE;m kEðt Þ
ð13Þ
where AE;m and AE ðt Þ are the monthly mean all-sky and the specific
time-of-daytime clear-sky AE, respectively (W m2). The errors
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
associated with bAE ðt Þ were defined by modifying Eqs. (4)–(7) for
use with Eq. (13).
4.2. Adjusted models
Our second aim was to evaluate the ability to correct for upscaling bias with generic functions suitable for implementation over
large areas. Upscaling bias can be caused by any number of reasons. A cyclic pattern in bias may occur as the evaporative ratio
varies seasonally and/or if there is a non-uniform distribution of
precipitation throughout the year (Table 1). The presence of clouds
decreases kE less than AE, causing an increase in the evaporative
fraction when compared to that under clear-sky conditions (Crago,
1996; Van Niel et al., 2011). Asymmetry in the diurnal progression
between latent heat flux and the scaling flux would cause a timevarying upscaling bias (e.g., kE might lag or lead S if stomatal conductance changes throughout the day or if there is more moisture
to evaporate in the morning than in the afternoon). The intent of
the current analysis was to define a minimal set of predictor variables that were fundamentally related to the definition of bias from
Eqs. (3)–(5) and that could be implemented over large areas as to
be useful for remote sensing applications.
Since the specific time-of-daytime evaporative ratio is a component of bv(t), and thereby a component of the bias, see Eq. (3), it is
a reasonable assumption that upscaling bias should vary as a function of normalised time-of-daytime (t). Because bv(t) compares
monthly all-sky evaporative ratio to specific time-of-daytime
clear-sky evaporative ratio (Eq. (3)), cf is also likely to be an important predictor of upscaling bias. Finally, as the heat and water balance, and associated evaporative ratios vary seasonally (Table 1),
bv(t) and the bias should demonstrate seasonal variability as a
function of Yd. At the very least, then, the generic correction model
for upscaling bias should consider these three predictor variables:
Yd, cf, and t.
The predictor variable cf has a direct biophysical influence on
upscaling error, and is thus termed a direct variable (Austin,
1980, 2002). Yd and t are indirect variables because they have no
direct biophysical influence on upscaling error (Austin, 1980,
2002), but have an indirect relationship with upscaling error via
multiple other direct variables mentioned above like stomatal conductance, moisture availability, and seasonal energy balance cycles. We assessed the relative influence of all three predictor
variables, but for Yd and t, we do not model the many possible direct underlying causes for their influence. First, the general functions need to be expressed and their relative influence assessed,
which is the focus of the current analysis.
Four sets of functions of the three predictor variables were defined, one for each upscaling case using Boosted Regression Tree
(BRT) analysis with the gbm package version 1.6–3.1 (Ridgeway,
2006) in R version 2.3–1. BRT analysis was used because it allowed
for assessment of the relative influence of all three predictor variables and for their general functions to be defined. As cf can vary
both as a function of t, and Yd, BRT was also chosen because it handles the interactions between predictor variables. For each case,
the Tumbarumba and Howard Springs upscaling biases were
pooled over the central part of the daytime, resulting in 8608 records input into each of the four BRTs. Tree complexity was set
to three, allowing for interactions between the three predictor
variables to be modelled, and following the methods outlined in
Elith et al. (2008), the learning rate (lr) was set so that the BRTs
used at least 1000 trees to fit the models (lr ranged from 0.03 to
0.09) and the ‘bag fraction’ was set to 0.5. Relative influence of
the predictor variables was estimated and curvilinear functions
for each predictor variable was output after accounting for the
average effects of the other two predictor variables in the model.
Functions were expressed after subtracting the mean value, herein
39
referred to as the zero-centred functions (Elith et al., 2008). These
zero-centred functions were defined in order to estimate monthly
upscaling bias.
The observed upscaling cases of S and AE, and the modelled
upscaling cases of SSIN and STOA had very similar zero-centred functions, which allowed for generalisation. The four sets of zero-centred functions were reduced to two sets by averaging the original
two zero-centred functions of the observed upscaling cases (S
and AE), and by averaging the original two zero-centred functions
of the solar irradiance model upscaling cases (SSIN and STOA). We
performed a least squares fit of the two final zero-centred functions to generic equations so the estimate of monthly bias could
be applied over vast areas. The two sets of generic zero-centred
functions were used to estimate monthly bias for each of the four
upscaling cases by:
biasestv ðt Þ ¼ lv þ fv0 ðY d Þ þ g v0 ðcf Þ þ hv0 ðt Þ;
ð14Þ
where biasestv(t) is the estimated monthly bias (J m2 d1) for
upscaling case ‘v’; fv0 ðY d Þ ¼ fv ðY d Þ fv ðY d Þ, which is the function,
f, between upscaling bias and day-of-year after centring it’s mean
to zero; g v0 ðcf Þ ¼ g v ðcf Þ g v ðcf Þ, which is the function, g, between
upscaling bias and cloud fraction after centring it’s mean to zero;
hv0 ðt Þ ¼ hv ðt Þ hv ðt Þ, which is the function, h, between upscaling
bias and normalised
time-of-daytime
after centring it’s mean to
zero; lv ¼ fv ðY d Þ þ g v ðcf Þ þ hv ðt Þ =3, which is the grand mean
of the three different functions; and ‘v’ represents one of the four
defined upscaling cases (i.e., S, SSIN, STOA, AE). As explained above,
there were two sets of fv0(Yd), gv0(cf), and hv0(t); one for the
observed upscaling cases, and one for the modelled upscaling cases.
Estimated bias for each upscaling case was then used to adjust the
basic null model of monthly evaporative energy by:
Dt kEm ðt Þadjv ¼ Dt kEm ðt Þnull v biasestv ðt Þ;
ð15Þ
where Dt kEm ðt Þadjv is the adjusted estimate of monthly all-sky
evaporative energy from observations made at time, t (J m2 d1)
for upscaling case ‘v’, and Dt kEm ðt Þnull v is the null model estimate
of monthly all-sky evaporative energy from observations made at
time, t (J m2 d1) for upscaling case ‘v’, which for the specific
example of upscaling with S, is given by:
Dt kEm ðt Þnull S ¼
c
1X
Dt Sm
kEðt Þ Sðt Þ
c 1
ð16Þ
5. Results
Monthly bias calculated from Eq. (5) averaged over all seven
central daytimes is shown for the four upscaling cases’ null models
in Fig. 1. Positive bias indicated that the estimated mean monthly
evaporative energy was higher than observed mean monthly evaporative energy; negative bias indicated that the estimated mean
monthly evaporative energy was lower than observed mean
monthly evaporative energy. Time series of null model bias in
Fig. 1 revealed that: (i) using S to upscale kE tended to result in a
bias that was closer to zero than when using AE, especially at Tumbarumba (Fig. 1a); (ii) using S or AE to upscale kE resulted in an
underestimation of the monthly evaporative energy (i.e., S and AE
bias was negative); (iii) using either of the modelled solar irradiances (SSIN or STOA) to upscale kE resulted in a similar or sometimes
lower (i.e., closer to zero) monthly bias than when using either of
the observed data cases (S or AE) for the central daytime period,
especially at Tumbarumba (Fig. 1a); (iv) using SSIN or STOA to upscale
kE resulted in an overestimation of the mean monthly evaporative
energy (i.e., SSIN and STOA bias was positive); and (v) bias of the
observed upscaling cases (S or AE) was complementary in nature
to the bias resulting from the modelled upscaling cases (SSIN or STOA).
40
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
Fig. 1. Monthly null model bias calculated from Eq. (5) averaged over the seven time periods of the central part of the daytime for the four upscaling cases is shown for (a)
Tumbarumba; and (b) Howard Springs. The legend in (a) applies to (b).
To partially correct for the unadjusted biases (Fig. 1), we defined
generic adjusted models. As described above, we pooled the observed Tumbarumba and Howard Springs biases and fit the results
of the BRT analysis to equations that were generically applicable.
Because the two observed upscaling cases (S or AE) and the two
modelled cases (SSIN or STOA) had very similar underlying functions,
it allowed for a single generic equation to be defined for each predictor variable for the observed cases, and a single equation to be
defined for each predictor variable for the modelled cases, thereby
reducing the number of equations by half. The physical basis for
combining the two observed cases together and the two modelled
upscaling cases together is what causes the complementarity between the observed and modelled upscaling cases, explained in detail later. The equations of the three BRT zero-centred functions
and the mean biases used in Eq. (14) were:
fv0 ðY d Þ ¼
g v0 ðcf Þ ¼
8
>
< SjAE ;
>
: SSIN jSTOA ; 1:577 sin
(
hv0 ðt Þ ¼
1:151 sin
SjAE ;
9
;
d
;
p Y dYmax
þ 0:994 >
1:115 2:074 cf
SSIN jSTOA ; 0:683 þ 1:583 cf
SjAE ;
d
=
p Y dYmax
0:725 >
;
ð17Þ
ð18Þ
3:220 þ 22:246 t 50:207 t 2 þ 36:424 t3
SSIN jSTOA ; 5:277 þ 33:550 t 68:740 t 2 þ 45:566 t 3
)
;
ð19Þ
8
9
S;
1:097 >
>
>
>
>
>
<
2:118 =
AE ;
lv ¼
> SSIN ;
0:862 >
>
>
>
>
:
;
STOA ;
0:711
ð20Þ
Eqs. (17)–(19) are shown in Fig. 2 compared to the BRT output.
Relative influence of the three predictor variables from the BRT
analysis for the observed upscaling cases was 60% for Yd; 26% for
cf; and 14% for t. Relative influence of the three predictor variables
for the solar irradiance model upscaling cases was 63% for Yd; 22%
for cf; and 15% for t. The y-axis of the functions in Fig. 2 represent
the estimated bias less the mean bias. For both Yd (Fig. 2a and b)
and cf (Fig. 2c and d), the modelled and observed upscaling cases
were nearly opposing, while for t, the zero-centred functions were
nearly the same (Fig. 2e and f). Bias as a function of Yd (Fig. 2a and
b) revealed a rounded shape in winter and a pointed shape in summer. This response was generally seen in the bias time series at
both sites (Fig. 1), but was particularly evident at Howard Springs
(for example, see in Fig. 1b the rounded winter response of bias
around June 2008 and the peakier summer response around January 2009 for SSIN and STOA). The function of Yd defined in the current
study is representative of the southern hemisphere. For the
northern hemisphere, the analogous equation of S or AE would
be: 1:151 sinðp ½1 jY d ðY d max =2Þj=Y d max Þ 0:725; and for SSIN
or STOA, it would be: 1:577 sinðp ½1 jY d ðY d max =2Þj=Y d max Þþ
0:994. Because the function of t is so similar for observed vs. modelled upscaling cases, these two equations could be combined if
further generalisation was desired (Fig. 2e and f and Eq. (19)). If
parsimony were the main objective, it could be argued to omit
the function of t altogether, since it described the least variability
in bias. For this study, we maintain the two separate equations for
the functions describing t and included them for estimating bias.
The mean adjusted model bias and RMSD associated with each
of the upscaling cases are shown as a function of Yd in Figs. 3 and 4
for Tumbarumba and Howard Springs, respectively, as it was the
strongest predictor variable. The null model’s bias and RMSD are
also provided so improvement by using Eq. (15) with the functions
defined in Eqs. (17)–(19) could be assessed. Corresponding summaries by cf and t are provided in Supplementary material Figs. 1–
4. Adjustment by Yd, cf, and t improved both the mean bias and the
RMSD over the central daytime period at both sites for all four
upscaling cases (Figs. 3 and 4). Most improvement was attained
for the observed upscaling cases (Figs. 3a, d and 4a, d). The errors
after implementing the previously published correction for the EF
method by Van Niel et al. (2011) is provided for comparison at both
sites (Figs. 3d and 4d). The current adjustment model resulted in
lower errors than the previous model of Van Niel et al. (2011) at
both sites (Figs. 3d and 4d), especially at Howard Springs (Fig. 4d).
The pooled null and pooled adjusted scatter plots are shown in
Fig. 5 for all upscaling cases, and summarised in Table 3. As noted
previously, implementing the adjustment model had a larger benefit on the observed data cases (S and AE), see Table 3 and Fig. 5. S
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
had the lowest RMSD of all cases before and after adjustment (Table 3). Under the null model, STOA had the lowest overall bias, and
an RMSD that was only 3% higher than that of S (Table 3). When
considering the null models only, then, this indicated that STOA
would likely be preferable to S for the upscaling of remote sensing
kE because of its ease in implementation and its very similar error
values to that of observed S. It is an important distinction though,
that STOA and S bias had opposite signs (i.e., STOA predicted evaporative energy that was too high, while S predicted evaporative energy
that was too low). After adjustment by functions of Yd, cf, and t,
there was a clear advantage to using either of the observed data
upscaling cases over the two modelled upscaling cases (Fig. 5
and Table 3), but this advantage can only be exploited where
observations are made over extensive areas at sub-daily time step.
After correction there was almost no preference between the two
modelled upscaling cases if both RMSD and bias were considered
(Fig. 5 and Table 3).
Although the current correction factors have been applied to
point-based time series, they are amenable to spatio-temporal
41
data. Visual representation of estimated S bias for the nominal
overpass time of MODIS Aqua over Australia (t = 0.67) using Eq.
(14) with the parameters in Eqs. (17)–(20) is provided in the Supplementary material Fig. 5 using spatially interpolated long-term
January and July cloud fraction data.
6. Discussion
At both sites upscaling with S produced a much lower bias (i.e.,
closer to zero) and RMSD than when upscaling using AE (Figs. 1, 3
and 4). This agreed with Liu and Hiyama (2007), who showed the
ratio of kE over S to be more constant than the ratio of kE over
AE. In contrast, Chávez et al. (2008) showed that overall, upscaling
based on AE produced lower RMSD, but larger mean bias than when
upscaling with S, and therefore neither flux was clearly preferred
as a scaling mechanism over the other in that study. Because the
Chávez et al. (2008) study was conducted over one month in
mid-summer, the difference between total daytime-only and total
24-h AE would be minimal because nighttime AE would not tend to
Fig. 2. The equations used to estimate bias in Eq. (14) are shown (lines) against the BRT output (points). The zero-centred functions for the observed upscaling cases of S or AE
are shown in (a, c and d), while the zero-centred functions for modelled cases of SSIN or STOA are shown in (b, d and f). Zero-centred functions: (a and b) Yd for the southern
hemisphere using Eq. (17); (c)–(d) cf using Eq. (18); and (e and f) t using Eq. (19). The legend in (a) applies to (c and e); the legend in (b) applies to (d and f). The y-axis
represents the estimated bias less the mean bias for each predictor variable.
42
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
Fig. 3. Mean monthly bias and RMSD are shown by Yd for the null (i.e., original) and adjusted (adj.) models at Tumbarumba for (a) S; (b) SSIN; (c) STOA; and (d) AE. The legend in
(a) applies to (b–d). The null model errors were calculated using Eqs. (6) and (7) for all t in the central daytime period, while the adjusted model errors were calculated after
Eqs. (6) and (7) with consideration of Eq. (15). The errors of correcting the null model for available energy proposed by Van Niel et al. (2011) are labelled by VN2011.
Fig. 4. Mean monthly bias and RMSD are shown by Yd for the null (i.e., original) and adjusted (adj.) models at Howard Springs for (a) S; (b) SSIN; (c) STOA; and (d) AE. The legend
in (a) applies to (b–d). The null model errors were calculated using Eqs. (6) and (7) for all t in the central daytime period, while the adjusted model errors were calculated
after Eqs. (6) and (7) with consideration of Eq. (15). The errors of correcting the null model for available energy proposed by Van Niel et al. (2011) are labelled by VN2011.
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
43
Fig. 5. Scatter plots of pooled data from both sites for all seven time periods of the central daytime period are shown for the uncorrected monthly null models in (a, c, e and g)
whereas the adjusted models are shown in (b, d, f, and h) for upscaling cases using S, SSIN, STOA, and AE, respectively. Number of observations is 1167 months for each figure
part. The 1:1 line is the solid line and the line of best-fit is the dash line.
be strongly negative in the mid-west of the USA in mid-summer,
but would rather tend towards zero. This likely explains why the
study of Chávez et al. (2008) did not show a similarly large bias
to our study when using AE to upscale kE.
Kustas and Norman (1996) state that use of S to upscale kE
would only be suitable for clear sky conditions, while the use of
AE should be more suited to the influence of clouds. In our current
study, we compared the two fluxes across the mixed sky conditions
44
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
Table 3
Error statistics of the pooled data associated with Fig. 5 are provided across the seven central daytime periods over both sites for the four upscaling cases original, or ‘Null’ models
compared with that after adjusting with Y d , cf , and t using Eqs. (14) and (15) with the polynomials defined in Eqs. (17) and (19). Combined number of observations is
1
1167 months. v stands for the specific upscaling case used in the equations. biasv ð0:25 0:75Þ ðMJ m2 d Þ was determined using Eq. (6) over the seven times of the central
1
daytime period; RMSDv ð0:25 0:75Þ ðMJ m2 d Þ was determined using Eq. (7) over the seven times of the central daytime period.
v
S
SSIN
STOA
AE
Null model
Adjusted model
RMSDv ð0:25 0:75Þ
biasv ð0:25 0:75Þ
r2
RMSDv ð0:25 0:75Þ
biasv ð0:25 0:75Þ
r2
1.55
1.71
1.59
2.41
1.11
1.03
0.91
2.15
0.87
0.88
0.89
0.87
0.88
1.18
1.13
0.99
0.16
0.10
0.12
0.18
0.92
0.87
0.88
0.90
relevant to remote sensing studies, and found the solar irradiance
(both observed and modelled) to be a better scalar than AE, contradicting the expectation of Kustas and Norman (1996). Kustas
and Norman (1996), however, used total daytime-only AE to upscale kE. Without the influences of nighttime AE considered, as discussed above, there may be little difference between scaling with S
or AE. Our findings support the recommendations of Zhang and
Lemeur (1995) and of Liu and Hiyama (2007), that modelling solar
irradiance is more suited to large-area remote sensing applications
because of its relatively simpler implementation and comparable
or higher accuracies when compared to AE.
Anderson et al. (2007a) assumed self-preservation between kE
and the Priestley–Taylor potential evapotranspiration in order to
gap-fill missing data due to clouds, while accounting for soil moisture depletion via a two-source resistance energy balance model at
an hourly time step. Gap-filling in this manner allows for upscaling
by numerical integration (Anderson et al., 2007a). Anderson et al.
(2007a) did not compare the performance of other modelled or observed fluxes for upscaling kE, but found that error increased from
10% when the model was run using clear-sky land surface temperature to 19% during cloudy periods when self-preservation was assumed with potential evaporation. To avoid confusion due to this
extrapolation error, Anderson et al. (2007b) subsequently used
clear-sky average model output for long-term hydrological
analysis over the continental US. Using composites of clear-sky
estimates of kE was deemed beneficial for detecting moisture
stress (Anderson et al., 2007b), but it would not be beneficial for
closing long-term mass and energy balances. We showed that
upscaling bias was driven by the difference between clear-sky
specific time-of-daytime evaporative ratio and the all-sky monthly
evaporative ratio (av(t) av,m, Eq. (4)), thus ignoring this interaction causes bias and is detrimental for long-term hydrological
studies.
Although we found upscaling kE with S resulted in the lowest
overall error, sub-daily S is not commonly observed over a dense
enough network (McVicar and Jupp, 1999) to make it a practical
solution for upscaling large-area kE. Therefore, large-area upscaling
of kE would be most readily done by modelling solar irradiance.
We found that the null versions of both SSIN and STOA had lower bias
and RMSD than the null version of AE; null STOA had lower bias and
comparable RMSD to null S when used to upscale kE to monthly
evaporative energy (Table 3, Figs. 3 and 4). Previous studies have
also shown that using modelled solar irradiance to upscale kE
can compare favourably with observed data. The study of Colaizzi
et al. (2006) showed that overall, using SSIN to upscale kE resulted
in lower RMSD and mean bias than when using observed 24-h
RN. Similarly, Zhang and Lemeur (1995) found little difference in
correlation coefficients between using observed total daytime-only
AE (r = 0.96) when compared to SSIN (r = 0.94). Our results reinforce
those findings, while providing the direct comparison between observed and modelled solar irradiance (rather than between RN or AE
and modelled solar irradiance as was the case for the studies of
Colaizzi et al. (2006) and Zhang and Lemeur (1995)).
Brutsaert and Sugita (1992) and Ryu et al. (2012) used STOA to
upscale kE, but neither compared STOA to the already established
SSIN method of Jackson et al. (1983). We found that upscaling kE
by either STOA or SSIN resulted in similar errors, but that STOA slightly
outperformed SSIN when considering the null models (Table 3). Besides the current study, Brutsaert and Sugita (1992) also inspected
the ability of cloud amount to correct for upscaling with STOA. They
calibrated STOA to S using a linear regression and found that cloud
corrected STOA upscaled kE to daytime values (r = 0.90) to about
the same as S did (r = 0.91) (Brutsaert and Sugita, 1992). Brutsaert
and Sugita (1992) did not compare the cloud corrected STOA to the
null version of STOA, so it is unknown whether their application of
cloud data improved its ability to upscale kE. They also did not attempt to correct observed fluxes by cloud cover (Brutsaert and
Sugita, 1992), whereas we showed that observed fluxes had the
higher capacity for improvement when corrected by cloud data,
along with day-of-year and time-of-daytime (Table 3). Our study
also showed that while cloud data could explain 25% of upscaling
error, it was secondary to the relative influence of the seasonal variation in energy balance expressed through day-of-year, which explained 60%.
We showed that adjustment by Yd, cf, and t allowed substantial
reduction in bias and RMSD for all four upscaling cases, and in particular, for the AE case (i.e., the EF method) and the S upscaling case
(i.e., the observed variant of the SI method; Table 3). The correction
proposed here resulted in greater improvements than the correction to the EF method of Van Niel et al. (2011) (Figs. 3d and 4d).
The correction by Van Niel et al. (2011) had two free parameters
that were different at each of the study sites, whereas an advantage
of the current correction method is that it has no free parameters,
and therefore can be implemented generically. The current correction functions were amenable to implementation at the continental level (e.g., Supplementary material Fig. 1), and were also
defined for all four upscaling cases (not just AE as per Van Niel
et al. (2011)). As Howard Springs and Tumbarumba have very different seasonal variations in surface energy balance (Table 2), the
value of defining generic functions at both sites is considerable,
although further evaluation of proposed functions at other sites
is advisable.
The underlying functions describing upscaling error have seldom been explored. Error statistics are usually quantified for a single upscaling approach over numerous sites (Ryu et al., 2012), or
error statistics of various upscaling cases are quantified over a
few sites or even a single site (Chávez et al., 2008; Colaizzi et al.,
2006). Perhaps the most important finding of the current study
was that the error resulting from upscaling kE to monthly evaporative energy when using observed data was complementary in nature to that when kE was upscaled to monthly evaporative energy
by modelled fluxes (Fig. 1). We found that the generic functions
developed to predict bias from Yd, and cf were nearly opposing
for observed vs. modelled upscaling cases, while the generic
function of t was nearly the same for observed and modelled
cases (Fig. 2). The reason for this is clear from inspection of
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
Eq. (4). Upscaling bias is dependent upon the relative error
([av(t) av,m]/av,m). So, an overestimation would result when
the clear-sky specific time-of-daytime evaporative ratio is greater
than the all-sky monthly evaporative ratio (av(t) > av,m), and an
underestimation would occur when the clear-sky specific timeof-daytime evaporative ratio is less than all-sky monthly evaporative ratio (av(t) < av,m).
Explanation of the physical cause of complementary bias between observed and modelled upscaling cases is most direct for
bias and cf. For the observed upscaling cases, presence of clouds
causes a smaller decrease in kE when compared to S or AE (Crago,
1996; Van Niel et al., 2011), causing an increase in as,m or aAE,m
Since av(t) is the clear-sky specific time-of-daytime evaporative
ratio, it is unaffected by cloud cover for all four upscaling cases
(i.e., in the presence of clouds, this variable is not measured). Thus,
the presence of cloud cover causes av,m to become larger with respect to av(t), and therefore results in both an underestimation of
monthly evaporative energy (Fig. 1), and causes a negative relationship between observed data upscaling bias and cf (Fig. 2c).
For the modelled data upscaling cases, increasing cf does not influence the denominator of either av(t) or av,m because those fluxes
are modelled, not observed. Therefore when considering the modelled upscaling cases, cf only influences the numerator of av,m.
Compared to the unaffected av(t), clouds cause kEm (the numerator of av,m) to decrease, lowering av,m, and subsequently resulting
in both an overestimation of monthly evaporative energy (Fig. 1)
and a positive relationship between modelled upscaling bias and
cf (Fig. 2d). The effect of Yd or t causes variation between the
clear-sky specific time-of-daytime and all-sky monthly evaporative ratio as well. For the observed upscaling cases, av(t) av,m
was larger in the winter then in the summer, resulting in bias that
was greater than the mean in winter and less than the mean in
summer (Fig. 2a). Alternately, av(t) av,m was higher in summer
than winter for the modelled upscaling cases, resulting in a relationship between bias and Yd that was complementary to that from
the observed upscaling cases (Fig. 2b). Evaluation of upscaling error and its underlying functional forms in this way provides the
next steps to advancing the understanding of the processes causing
error, rather than only quantifying the error amount.
7. Conclusion
Removing bias in the upscaling procedure is an important goal
otherwise the long-term mass and energy balance estimates may
be detrimentally prejudiced. We showed that generic correction
functions based on day-of-year, cloud fraction, and time-ofdaytime were able to reduce both RMSD and mean bias of all
four upscaling cases throughout the central daytime period. In
our study S had the lowest RMSD both before and after correction, and had low bias before correction and lowest bias after
correction, so was overall the preferable scaling mechanism of
those tested. However, upscaling large-area kE by observed S
would require a network of observation stations across the study
site, all measuring S at a sub-daily time step. Such networks are
not currently commonplace to support large-area hydrological
applications. Of the two upscaling cases that could be easily applied over large areas, STOA had lower error than SSIN before
adjustment by day-of-year, cloud fraction, and time-of-daytime.
Prior to correction STOA had second best RMSD and overall best
bias. After adjustment, there was no preferred scaling flux
between STOA and SSIN. We conclude that using modelled data
to upscale kE seems viable for large-area hydrological remote
sensing applications, but that correction to minimise the overestimation bias from using either STOA or SSIN is essential to allow
better mass and energy balance closure.
45
Synthesising previous comparisons of the utility of different approaches to upscale kE has been confounded by: (i) differences in
specification of daytime-only or continuous 24-h integration of
scaling fluxes; (ii) sometimes using observed data and other times
modelled data; and (iii) assessing error under different sky conditions. We developed and implemented a framework to evaluate
upscaling approaches that make the self-preservation assumption,
clearly highlighting these issues. We laid out the combination of
integration period and sky conditions to assess upscaling error relevant to thermal remote sensing studies suitable for large-area
continuous hydrologic application (i.e., over temporal and spatial
scales over which mass and energy must balance). Previous studies
that assess upscaling error from either (i) specific time-of-daytime
fluxes observed under all-sky rather than only under clear-sky conditions, or (ii) only assess energy integrated over those days that
are completely clear of clouds, do not evaluate the bias relevant
to hydrological remote sensing applications. We recommend that
the framework defined here be used for assessing upscaling error
of remotely sensed kE because the interaction between clear-sky
specific time-of-daytime flux observed by satellite to the integrated monthly all-sky energy required for hydrology can cause
different error characteristics than if these particular conditions
were not specifically addressed. Under the conditions relevant to
remote sensing the bias from upscaling by observed data was
shown to be complementary in nature to the bias when upscaling
by modelled data.
Acknowledgements
Thanks to Dr. Ray Leuning, CSIRO Marine and Atmospheric Research, for providing the Tumbarumba flux tower dataset. MLR
acknowledges support from the Australian Research Council Centre of Excellence for Climate System Science (CE11E0098). This
work was supported by the Water Information Research and
Development Alliance (WIRaDA) between the Bureau of Meteorology and CSIRO’s Water for a Healthy Country Flagship. Thanks to
Dr. Marta Yebra and Dr. YongQiang Zhang, CSIRO Land and Water,
and two anonymous reviewers for their helpful comments that improved this paper.
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.jhydrol.2012.
08.005.
References
Anderson, M.C., Norman, J.M., Diak, G.R., Kustas, W.P., Mecikalski, J.R., 1997. A twosource time-integrated model for estimating surface fluxes using thermal
infrared remote sensing. Remote Sens. Environ. 60 (2), 195–216.
Anderson, M.C., Norman, J.M., Mecikalski, J.R., Otkin, J.A., Kustas, W.P., 2007a. A
climatological study of evapotranspiration and moisture stress across the
continental United States based on thermal remote sensing: 1. Model
formulation. J. Geophys. Res. 112, D10117. http://dx.doi.org/10.1029/
2006JD007506.
Anderson, M.C., Norman, J.M., Mecikalski, J.R., Otkin, J.A., Kustas, W.P., 2007b. A
climatological study of evapotranspiration and moisture stress across the
continental United States based on thermal remote sensing: 2. Surface moisture
climatology. J. Geophys. Res. 112, D11112. http://dx.doi.org/10.1029/
2006JD007507.
Austin, M.P., 1980. Searching for a model for use in vegetation analysis. Plant Ecol.
42 (1), 11–21.
Austin, M.P., 2002. Spatial prediction of species distribution: an interface between
ecological theory and statistical modelling. Ecol. Model. 157 (2–3), 101–118.
Beringer, J., Hutley, L.B., Tapper, N.J., Cernusak, L.A., 2007. Savanna fires and their
impact on net ecosystem productivity in North Australia. Glob. Change Biol. 13
(5), 990–1004.
Brutsaert, W., Sugita, M., 1992. Application of self-preservation in the diurnal
evolution of the surface energy budget to determine daily evaporation. J.
Geophys. Res. 97 (D17), 18377–18382.
46
T.G. Van Niel et al. / Journal of Hydrology 468–469 (2012) 35–46
Cammalleri, C., Ciraolo, G., La Loggia, G., Maltese, A., 2012. Daily evapotranspiration
assessment by means of residual surface energy balance modeling: a critical
analysis under a wide range of water availability. J. Hydrol. 452–453, 119–129.
Chávez, J.L., Neale, C.M.U., Prueger, J.H., Kustas, W.P., 2008. Daily evapotranspiration
estimates from extrapolating instantaneous airborne remote sensing ET values.
Irrig. Sci. 27 (1), 67–81.
Colaizzi, P.D., Evett, S.R., Howell, T.A., Tolk, J.A., 2006. Comparison of five models to
scale daily evapotranspiration from one-time-of-day measurements. Trans.
ASABE 49 (5), 1409–1417.
Crago, R.D., 1996. Conservation and variability of the evaporative fraction during
the daytime. J. Hydrol. 180 (1–4), 173–194.
Crago, R., Brutsaert, W., 1996. Daytime evaporation and the self-preservation of the
evaporative fraction and the Bowen ratio. J. Hydrol. 178 (1–4), 241–255.
Elith, J., Leathwick, J.R., Hastie, T., 2008. A working guide to boosted regression trees.
J. Anim. Ecol. 77 (4), 802–813.
Guerschman, J.P., Van Dijk, A.I.J.M., Mattersdorf, G., Beringer, J., Hutley, L.B., Leuning,
R., Pipunic, R.C., Sherman, B.S., 2009. Scaling of potential evapotranspiration
with MODIS data reproduces flux observations and catchment water balance
observations across Australia. J. Hydrol. 369 (1–2), 107–119.
Hatfield, J.L., Perrier, A., Jackson, R.D., 1983. Estimation of evapotranspiration at one
time-of-day using remotely sensed surface temperatures. Agric. Water Manag.
7 (1–3), 341–350.
Jackson, R.D., Hatfield, J.L., Reginato, R.J., Idso, S.B., Pinter Jr., P.J., 1983. Estimation of
daily evapotranspiration from one time-of-day measurements. Agric. Water
Manag. 7 (1–3), 351–362.
Jovanovic, B., Collins, D., Braganza, K., Jakob, D., Jones, D., 2011. A high-quality
monthly total cloud amount dataset for Australia. Climatic Change 108 (3),
485–517.
Kustas, W.P., Norman, J.M., 1996. Use of remote sensing for evapotranspiration
monitoring over land surfaces. Hydrol. Sci. J. 41 (4), 495–516.
Kustas, W.P., Perry, E.M., Doraiswamy, P.C., Moran, M.S., 1994. Using satellite
remote sensing to extrapolate evapotranspiration estimates in time and space
over a semiarid rangeland basin. Remote Sens. Environ. 49, 275–286.
Leuning, R., Cleugh, H.A., Zegelin, S.J., Hughes, D., 2005. Carbon and water fluxes
over a temperate Eucalyptus forest and a tropical wet/dry savanna in Australia:
measurements and comparison with MODIS remote sensing estimates. Agric.
For. Meteorol. 129 (3–4), 151–173.
Liu, Y., Hiyama, T., 2007. Detectability of day-to-day variability in the evaporative
flux ratio: a field examination in the Loess Plateau of China. Water Resour. Res.
43, W08503. http://dx.doi.org/10.1029/2006WR005726.
McVicar, T.R., Bierwirth, P.N., 2001. Rapidly assessing the 1997 drought in Papua
New Guinea using composite AVHRR imager. Int. J. Remote Sens. 22 (11), 2109–
2128.
McVicar, T.R., Jupp, D.L.B., 1999. Estimating one-time-of-day meteorological data
from standard daily data as inputs to thermal remote sensing based energy
balance models. Agric. For. Meteorol. 96 (4), 219–238.
McVicar, T.R., Jupp, D.L.B., 2002. Using covariates to spatially interpolate moisture
availability in the Murray-Darling Basin A novel use of remotely sensed data.
Remote Sens. Environ. 79 (2–3), 199–212.
Monteith, J.L., 1973. Principles of Environmental Physics. Edward Arnold, London,
241 pp.
NAO, 1990. Almanac for Computers. Nautical Almanac Office, United States Naval
Observatory, Washington, DC (USA).
O’Grady, A.P., Chen, X., Eamus, D., Hutley, L.B., 2000. Composition, leaf area index
and standing biomass of eucalypt open forests near Darwin in the Northern
Territory, Australia. Aust. J. Bot. 48 (5), 629–638.
Ridgeway, G., 2006. Generalized Boosted Regression Models. Documentation on the
R Package ‘gbm’, Version 1.6–3.1. <http://rgm2.lab.nig.ac.jp/RGM2/
func.php?rd_id=gbm:gbm-package> (accessed May 2012).
Ryu, Y., Baldocchi, D.D., Black, T.A., Detto, M., Law, B.E., Leuning, R., Miyata, A.,
Reichstein, M., Vargas, R., Ammann, C., Beringer, J., Flanagan, L.B., Gu, L., Hutley,
L.B., Kim, J., McCaughey, H., Moors, E.J., Rambal, S., Vesala, T., 2012. On the
temporal upscaling of evapotranspiration from instantaneous remote sensing
measurements to 8-day mean daily-sums. Agric. For. Meteorol. 152, 212–222.
Schymanski, S.J., Roderick, M.L., Sivapalan, M., Hutley, L.B., Beringer, J., 2007. A test
of the optimality approach to modelling canopy properties and CO2 uptake by
natural vegetation. Plant, Cell Environ. 30 (12), 1586–1598.
Sobrino, J.A., Gomez, M., Jimenez-Munoz, J.C., Olioso, A., Chehbouni, G., 2005. A
simple algorithm to estimate evapotranspiration from DAIS data: application to
the DAISEX campaigns. J. Hydrol. 315 (1–4), 117–125.
Su, Z., 2005. Estimation of the Surface Energy Balance. Encyclopedia of Hydrological
Sciences. John Wiley & Sons, Ltd., 22 pp.
Sugita, M., Brutsaert, W., 1991. Daily evaporation over a region from lower
boundary layer profiles measured with radiosondes. Water Resour. Res. 27 (5),
747–752.
Van Gorsel, E., Leuning, R., Cleugh, H.A., Keith, H., Suni, T., 2007. Nocturnal carbon
efflux: reconciliation of eddy covariance and chamber measurements using an
alternative to the u-threshold filtering technique. Tellus B 59 (3), 397–403.
Van Niel, T.G., McVicar, T.R., Roderick, M.L., van Dijk, A.I.J.M., Renzullo, L.J., van
Gorsel, E., 2011. Correcting for systematic error in satellite-derived latent heat
flux due to assumptions in temporal scaling: assessment from flux tower
observations. J. Hydrol. 409 (1–2), 140–148.
Zhang, L., Lemeur, R., 1995. Evaluation of daily evapotranspiration estimates from
instantaneous measurements. Agric. For. Meteorol. 74 (1–2), 139–154.