WATER RESOURCES RESEARCH, VOL. 43, W04426, doi:10.1029/2006WR005224, 2007
Analyzing spatial and temporal variability of annual
water-energy balance in nonhumid regions of China
using the Budyko hypothesis
Dawen Yang,1 Fubao Sun,1 Zhiyu Liu,2 Zhentao Cong,1
Guangheng Ni,1 and Zhidong Lei1
Received 2 June 2006; revised 4 January 2007; accepted 16 January 2007; published 25 April 2007.
[1] On the basis of long time series of climate and discharge in 108 nonhumid catchments
in China this study analyzes the spatial and temporal variability of annual water-energy
balance using the Budyko hypothesis. For both long-term means and annual values of
the water balances in the 108 catchments, Fu’s formula derived from the Budyko
hypothesis is confirmed. A high correlation and relatively small systematic error between
the values of parameter v in Fu’s equation optimized from the water balance of individual
year and calibrated from the long-term mean water balance show that Fu’s equation
can be used for predicting the interannual variability of regional water balances. It has
been found that besides the annual climate conditions the regional pattern of annual
water-energy balance is also closely correlated with the relative infiltration capacity (Ks/ir),
relative soil water storage (Smax/E0), and the average slope (tan b). This enables one to
estimate the parameter v from catchment characteristics without calibration from the long
time series of water balances. An empirical formula for the parameter v in terms of the
dimensionless landscape parameters is proposed. Applications of Fu’s equation
together with the parameter v estimated by this empirical formula have shown that Fu’s
equation can predict both long-term mean and annual value of actual evapotranspiration
accurately and predict both long-term mean and interannual variability of runoff
reasonably. This implies that the Fu’s equation can be used for predicting the annual water
balance in ungauged basins.
Citation: Yang, D., F. Sun, Z. Liu, Z. Cong, G. Ni, and Z. Lei (2007), Analyzing spatial and temporal variability of annual waterenergy balance in nonhumid regions of China using the Budyko hypothesis, Water Resour. Res., 43, W04426, doi:10.1029/2006WR005224.
1. Introduction
[2] Because of marked variability of the hydrological
cycle in time and space, one of the most basic issues in
hydrology is to understand the key controlling factors and
predict the spatial and temporal variability of the annual
water balance [Milly, 1994; Wolock and McCabe, 1999].
Evapotranspiration links with both water and energy balances and plays a key role in the climate-soil-vegetation
interactions. The primary controls on the long-term mean
annual evapotranspiration (E) are precipitation (P) and
potential evapotranspiration (E0 ). Budyko [1948, 1974]
proposed a semiempirical expression for the coupled waterenergy balances, here defined as the Budyko hypothesis,
which is a partition of annual water balance as a function of
the relative magnitude of water and energy supply.
1.1. Fu’s Equation: Analytical Solutions to the
Budyko Hypothesis
[3] Bagrov [1953] made the first attempt to derive the
Budyko curve theoretically by imposing an additional
condition on the derivatives dE/dP = 1 (E/E0)u, where
u denotes the effects of catchment characteristics. This
condition was further developed by Mezentsev [1955] as
dE/dP = [1 (E/E0)u]s, from which the integral achieved
is E/P = 1/[1 + (P/E0)u ]1/u when s equals (u + 1)/u. This is
the same as Choudhury’s equation [Choudhury, 1999]
proposed consulting the Turc-Pike equation [Turc, 1954;
Pike, 1964]. Recall that the expression of dE/dP is empirically proposed, rather than from strict derivations. On the
basis of phenomenological considerations, Fu [1981] gave
the differential forms of the Budyko hypothesis as @E
@P =
@E
f (E0 E, P), when E0 = const; and @E
=
f
(P
E,
E0),
0
when P = const. Through dimensional analysis and mathematical reasoning (see Fu [1981] and Zhang et al. [2004] for
more details), Fu [1981] finally achieved the analytical
solutions (called Fu’s equation) to the Budyko hypothesis as
v 1=v
E
E0
E0
¼1þ 1þ
or
P
P
P
v 1=v
E
P
P
¼1þ 1þ
E0
E0
E0
1
State key Laboratory of Hydro-Science and Engineering and Department of Hydraulic Engineering, Tsinghua University, Beijing, China.
2
Bureau of Hydrology, Ministry of Water Resources, Beijing, China.
Copyright 2007 by the American Geophysical Union.
0043-1397/07/2006WR005224
ð1Þ
where v is a constant of integration, and its values range
(1, 1).
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1.2. Budyko Hypothesis for Regional Variability
[4] The Budyko hypothesis has been widely applied to
investigate the regional variability of annual water balances
in the former USSR [Budyko, 1974], in the United States
[Milly, 1994; Wolock and McCabe, 1999], and in Australia
[Zhang et al., 2004]. However, deviations from the original
Budyko curve were observed [Budyko, 1974; Milly, 1994],
which means that climate alone is insufficient to account for
all the regional variability of mean annual water balance.
Besides the mean climate conditions, the regional variability is also related to climate seasonality [Budyko, 1974;
Dooge, 1992; Milly, 1994; Wolock and McCabe, 1999;
Potter et al., 2005], the spatial average plant available
water-holding capacity of the soil [Milly, 1994; Wolock and
McCabe, 1999; Zhang et al., 2001; Sankarasubramanian
and Vogel, 2002a; Potter et al., 2005], and the variability of
rainfall depths and arrival time [Milly, 1994; Potter et al.,
2005]. Potter et al. [2005] suggested that infiltration-excess
runoff might be an important factor which was ignored in
previous studies. Using six parameters to depict basin terrain and climate, Berger and Entekhabi [2001] developed a
linear regression relation for mean annual water balance,
which was further examined by Sankarasubramanian and
Vogel [2002b]. Characterizing and quantifying the interbasin variability of annual water balance and examining its
dominant controlling factors remain an ongoing and important problem.
[5] Moreover, the regional water balance is determined
by the nonlinear interactions of climate factors, soil properties, and vegetation [Rodriguez-Iturbe and Porporato,
2004; Rodriguez-Iturbe et al., 2006]. Therefore it is desirable to understand the dependence of the Budyko-type
formulae on spatial scale. On the basis of field observations
and water balances in large river basins, Choudhury [1999]
fitted different values of the unique parameter in the empirical Budyko-like formula at field scale (0.07– 1.6 km2)
and regional scale (1.2 – 7.0 M km2), and further argued that
the parameter was mainly dependent on spatial scale.
1.3. Budyko Hypothesis for Interannual Variability
[6] Building on the Budyko hypothesis, simple frameworks [Schaake, 1990; Dooge, 1992; Dooge et al., 1999;
Koster and Suarez, 1999] were suggested for analyzing the
interannual variability of water balance induced from longterm variability of atmospheric forcing variables. The
effectiveness of this approach was confirmed by using an
atmospheric general circulation model [Koster and Suarez,
1999], subsequently examined using observational
data [Milly and Dunne, 2002], and further improved by
introducing an index of soil moisture storage capacity
[Sankarasubramanian and Vogel, 2002a].
[7] If the Budyko hypothesis can be applied for estimating actual evapotranspiration at the annual timescale, this is
important for understanding the changes in water cycle
especially in the context of decreasing pan evaporation
worldwide [Brutsaert and Parlange, 1998; Roderick and
Farquhar, 2002]. Another operational hypothesis, in which
actual and potential evapotranspiration shows complementary in almost diametrical opposition to the Penman proportional hypothesis [Bouchet, 1963; Brutsaert and Stricker,
1979; Parlange and Katul, 1992], has been invoked
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for predicting changes in hydrologic cycle [Brutsaert and
Parlange, 1998; Szilagyi et al., 2001; Kahler and Brutsaert,
2006]. The Bouchet hypothesis has been examined using
annual water balance in a number of catchments [Morton,
1983; Hobbins et al., 2001a, 2001b; Ramirez et al., 2005;
Yang et al., 2006]. Most cases indicated that it is valid
more qualitatively than quantitatively [Sugita et al., 2001;
Brutsaert, 2005]. Regarding the apparent contradiction
between the Bouchet and Penman hypotheses, a recent
research [Yang et al., 2006] suggested that change in actual
evapotranspiration in nonhumid regions is dominated by
change in precipitation rather than in potential evapotranspiration, and the complementary relationship between actual
and potential evapotranspiration comes about because actual
and potential evapotranspiration is correlated via precipitation. In humid regions, change in actual evapotranspiration is
controlled by change in potential evapotranspiration rather
than precipitation, and this is identical to the Penman hypothesis. Fu’s equation can provide a full picture of the
evaporation mechanism at the annual timescale. Therefore
Fu’s equation could be used through top-down analysis for
providing an insight into the dynamic interactions among
climate, soils, and vegetation and their controls on the annual
water balance at the regional scale [Sivapalan, 2003; Farmer
et al., 2003].
[8] Using a comprehensive data set available from nonhumid regions of China, this study is aimed to examine Fu’s
equation from both the long-term water balance in different
catchments and annual water balance of individual catchments, to explore both regional and interannual variability
in annual water balances and their control factors. In
particular, this paper attempts to establish a regional relationship between the parameter (v) in Fu’s equation and a
limited number of dimensionless landscape characteristics
to enable its application to ungauged basins.
2. Data and Methodology
2.1. Study Area and Data Available
[9] Climate and soil control vegetation dynamics, and
vegetation exerts important control on the entire water
balance and feedback to the atmosphere in nonhumid regions [Rodriguez-Iturbe and Porporato, 2004]. Understanding the water balances in nonhumid regions is a valuable
method for understanding the climate-soil-vegetation interactions. In this study, 108 catchments located in the Yellow
River basin, the Haihe River basin and several inland basins
are selected as the study catchments (see Figure 1). All
selected catchments have relatively few human interferences, such as dams and irrigation projects. The land use
changes in the study areas during the last two decades were
reported to be in range of 2 – 5% [Liu and Buheaosier,
2000]. From the Yellow River basin, 9 catchments located
on the Tibetan Plateau and 54 catchments located on the
Loess Plateau are chosen. In addition, 38 catchments are
chosen from the Haihe River basin and 7 catchments from
several inland river basins in Gansu province of western
China. Table 1 summarizes the basic physiographic characteristics of the selected catchments. The drainage areas
range from 272 to 94,800 km2, and the dryness index
(E0 =P) varies in the range 1 7.
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Figure 1. Study area. Triangles represent hydrological gauges, circles represent meteorological gauges,
and the numbers represent the catchment numbers.
[10] Monthly discharge data from 1951 to 2000 for the
108 catchments have been provided by the Hydrological
Bureau of the Ministry of Water Resources of China. The
shortest record length is 6 years, the longest is 46 years,
median record length is 29 years, and 88 catchments have
more than 20 years discharge data (see Table 1). By
ignoring the interannual change of water storage in the
catchments, actual evapotranspiration is calculated from the
annual water balance and used as the ‘‘measured’’ actual
evapotranspiration for analysis. Daily meteorological data
of 238 gauges from 1951 to 2000 are obtained from the
China Administration of Meteorology, which consists of
precipitation, mean, maximum and minimum air temperature, sunshine duration, wind speed, and relative humidity.
Daily precipitation data at hydrological stations are
also available. Additionally, daily data of solar radiation at
47 meteorological stations (among the 238 stations) are
available. The data in the same data period for both
discharge and meteorological observation have been chosen
for this analysis.
[11] The catchment extent is extracted using digital elevation model (DEM) of 1 km resolution, and resampled to
10 km resolution for calculating the areal average values of
hydroclimatic variables. The procedures for calculating
catchment average precipitation and potential evapotranspiration are (1) a 10 km gridded data set covering the study
area is interpolated from the gauge data (see Yang et al.
[2004] for more details), (2) potential evapotranspiration at
the daily timescale is estimated in each grid using the
Penman equation recommended by Shuttleworth [1993],
and (3) the catchment average values are then calculated
for each variable. For estimating net radiation (see
Appendix A of Yang et al. [2006] for the details), solar
radiation is calculated by an empirical equation involving
the sunshine duration, in which the parameters (as, bs of
equation (A3) of Yang et al. [2006]) are calibrated using the
observed solar radiation for each month at the 47 stations. The
values of as and bs for each grid are obtained from the nearest
station. The first-order estimate of net long-wave radiation is
derived from the relative sunshine duration, minimum and
maximum near surface air temperature and vapor pressure, as
per the method recommended by Allen et al. [1998].
2.2. Description of Catchment Characteristics
[12] In addition to climate, topography, soil and vegetation are main factors affecting the partitioning of rainfall
into runoff and evapotranspiration, which are considered in
the parameter v in Fu’s equation. The average slope tan b
of a catchment is used for representing the topography, and
is estimated as the average slope of all hillslopes derived
from the 1 km DEM. The hydraulic conductivity of soil
controls the rainfall infiltration and thus the supply of soil
water for evaporation. Considering the rainfall intensity, the
relative infiltration capacity [Berger and Entekhabi, 2001] is
used for indicating the soil property of infiltration-excess. In
this study, the relative infiltration capacity is defined as the
ratio of saturated hydraulic conductivity Ks (mm hr1)
to mean precipitation intensity ir (mm hr1) in 24 hours
(M. Sivapalan, personal communication, 2006). The saturated hydraulic conductivity is obtained from the Global
Soil Data Task [International Geosphere-Biosphere
Programme, 2000] with a 10 km resolution. The mean precipitation intensity is averaged for rainy days of the study period.
For representing the vegetation and soil effect on the annual
water balance, the plant extractable water capacity recommended by Dunne and Willmott [1996] is employed, which is
given as
Smax ¼ qf qw droot
ð2Þ
where droot = min(drmax, dTop), drmax is the maximum depth
of the root for each type of vegetation, while the vegetation
type is obtained from the USGS Global Land Cover
Characteristics Data Base version 2.0 (http://edcdaac.usgu.
gov/glcc/globe_int.html) with a 1 km resolution; dTop
denotes the depth of the topsoil for each soil type, which
is derived from a 5 min resolution data set [Food and
Agricultural Organization (FAO), 2003]; the moisture
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YANG ET AL.: VARIABILITY IN ANNUAL WATER BALANCE
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Table 1. Basic Characteristics of the 108 Study Catchments
Mean Values, mm/yr
Number
2
E0
Ks
ir
Smax
E0
tan b
v
966
1026
1036
1011
886
947
965
10.30
4.85
4.72
6.72
14.24
5.36
4.40
0.077
0.088
0.067
0.076
0.075
0.062
0.071
0.069
0.080
0.045
0.035
0.123
0.055
0.037
1.3
1.7
1.7
1.4
1.7
1.8
1.6
Tibetan Plateau
310
275
443
367
588
404
384
322
511
423
519
424
500
403
642
444
320
294
902
822
849
916
987
899
896
827
901
8.72
7.66
8.87
7.21
5.26
4.68
3.48
8.25
7.01
0.081
0.077
0.071
0.063
0.079
0.079
0.071
0.074
0.073
0.021
0.028
0.039
0.038
0.029
0.052
0.045
0.035
0.035
2.3
2.4
2.0
2.2
2.3
2.4
2.3
2.1
2.5
Loess Plateau
431
415
398
386
392
381
395
345
413
400
403
339
433
425
445
395
368
312
425
356
436
372
480
431
488
427
347
315
377
341
413
381
451
406
464
421
486
444
500
463
511
473
453
412
472
439
523
504
497
474
565
530
529
486
395
366
451
406
574
547
532
511
570
525
462
424
487
467
650
565
682
606
472
450
453
421
461
418
600
554
505
453
524
485
371
356
442
421
428
399
522
497
488
449
659
536
655
561
570
493
905
896
868
975
908
975
905
933
1024
993
990
979
963
1013
997
906
942
881
901
920
876
881
870
905
911
921
928
878
858
871
855
877
766
953
952
848
950
940
962
853
891
886
1004
938
873
884
979
943
957
959
3.12
4.22
2.93
4.36
3.21
2.83
3.11
3.68
3.66
2.51
2.51
2.39
3.85
3.74
3.03
2.46
2.42
1.72
1.58
4.43
1.54
1.88
1.71
1.61
1.68
2.02
4.28
2.91
3.10
1.71
1.84
2.02
2.70
3.89
3.48
2.76
3.62
4.18
5.97
1.57
3.44
2.25
3.69
2.98
1.89
1.65
1.66
2.34
2.47
3.17
0.066
0.085
0.087
0.086
0.099
0.065
0.123
0.155
0.093
0.070
0.074
0.061
0.113
0.087
0.089
0.080
0.111
0.064
0.069
0.115
0.118
0.075
0.131
0.170
0.117
0.154
0.115
0.068
0.061
0.171
0.146
0.160
0.097
0.108
0.131
0.165
0.066
0.086
0.113
0.118
0.121
0.110
0.080
0.068
0.085
0.138
0.190
0.136
0.135
0.126
0.023
0.023
0.028
0.013
0.023
0.016
0.027
0.039
0.011
0.019
0.013
0.018
0.034
0.006
0.008
0.016
0.010
0.013
0.014
0.027
0.016
0.016
0.016
0.014
0.018
0.021
0.023
0.017
0.018
0.018
0.018
0.025
0.021
0.042
0.051
0.041
0.030
0.026
0.030
0.018
0.032
0.023
0.025
0.022
0.017
0.021
0.024
0.029
0.020
0.026
3.7
3.7
3.8
2.3
3.7
2.2
4.4
2.6
2.1
2.2
2.3
2.7
2.6
2.4
2.5
2.9
2.7
2.9
3.1
3.3
3.5
2.9
3.3
4.3
3.8
3.8
3.3
2.9
2.9
4.6
4.5
3.7
3.4
3.8
3.1
4.1
3.5
3.0
2.8
4.1
3.0
3.5
3.1
3.4
3.1
4.1
3.0
2.7
3.0
2.8
Area, km
Data Length, years
P
1
2
3
4
5
6
7
10961
800
14325
11388
2240
877
2053
26
21
27
18
37
15
41
Inland River
179
195
152
247
275
301
247
8
9
10
11
12
13
14
15
16
20930
45019
98414
715
3083
9022
12573
5043
4007
31
8
17
30
30
32
8
18
31
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
990
4853
10647
2831
1562
1263
2939
650
3829
8645
1121
283
4102
15325
29662
2415
327
913
3468
3992
5891
3208
719
1121
1662
2169
436
3440
774
17180
4715
2266
600
4788
37006
46827
2484
9805
1019
282
14124
40281
4640
10603
2988
19019
928
9713
829
7273
42
42
44
27
10
27
6
7
29
37
26
24
26
45
22
28
24
27
27
28
30
25
22
20
20
25
22
15
31
46
37
28
19
13
44
26
25
29
10
32
29
35
26
29
13
29
6
12
19
11
E
Basins
100
162
127
155
199
235
182
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YANG ET AL.: VARIABILITY IN ANNUAL WATER BALANCE
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Table 1. (continued)
Mean Values, mm/yr
Number
Area, km
Data Length, years
P
E
E0
Ks
ir
Smax
E0
tan b
v
67
68
69
70
12880
3149
8264
426
25
28
21
14
599
582
702
617
561
514
621
557
975
895
1014
1015
1.85
2.81
1.72
3.30
0.165
0.144
0.144
0.160
0.005
0.024
0.006
0.007
3.8
3.1
3.3
3.2
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
2300
3800
5060
19050
20100
17100
1378
1025
1166
1227
13000
1615
2404
2220
1661
2822
372
5060
2950
5120
1927
4700
3674
2890
271.9
2360
25533
15078
1760
2950
4990
4061
4970
8550
14070
5387
6420
23900
38
33
42
46
25
45
39
39
38
39
11
32
39
38
24
28
37
39
38
7
38
46
44
36
15
44
36
36
46
38
38
38
20
25
38
38
38
25
917
938
883
916
920
940
954
912
876
891
925
881
923
899
859
866
862
889
967
993
950
956
992
984
919
934
974
997
996
993
1008
1031
1052
1051
985
1015
1061
1073
3.33
4.29
4.41
3.65
3.31
3.36
4.67
3.52
2.98
5.89
3.37
3.70
4.91
4.36
3.38
2.77
3.78
3.11
3.09
1.19
5.34
5.15
2.97
4.91
6.66
5.67
3.92
3.57
2.49
3.05
3.59
5.46
5.93
4.07
3.35
4.00
4.26
3.03
0.123
0.128
0.139
0.133
0.134
0.094
0.166
0.090
0.064
0.134
0.086
0.159
0.142
0.157
0.141
0.147
0.167
0.154
0.113
0.160
0.139
0.146
0.071
0.097
0.089
0.106
0.097
0.087
0.145
0.134
0.137
0.131
0.121
0.141
0.128
0.124
0.125
0.132
0.026
0.043
0.046
0.030
0.028
0.013
0.031
0.034
0.043
0.041
0.011
0.037
0.032
0.036
0.033
0.030
0.026
0.024
0.035
0.006
0.045
0.041
0.015
0.026
0.041
0.040
0.026
0.023
0.041
0.044
0.057
0.062
0.047
0.013
0.043
0.030
0.036
0.022
4.3
3.1
2.8
3.1
3.5
2.7
2.6
2.6
3.4
2.8
2.9
2.9
2.9
2.5
2.9
3.2
2.7
2.9
2.7
3.3
2.7
2.7
2.8
2.9
2.5
2.9
3.7
2.8
2.7
3.1
3.4
2.9
2.7
3.0
3.1
2.8
2.4
3.6
2
Haihe River Basin
544
522
558
499
549
471
544
487
556
515
406
369
471
412
643
516
756
624
436
393
397
370
575
493
457
417
532
442
646
529
718
592
530
452
721
578
615
520
665
589
485
432
505
443
375
351
418
390
414
366
425
393
461
445
385
360
690
565
635
559
635
576
591
523
544
483
586
527
526
481
547
486
523
447
584
552
Figure 2. Long-term mean values of annual actual evapotranspiration, precipitation, and potential
evapotranspiration for the 108 catchments, plotted in two different but equivalent Budyko-type forms
(plotted in scattering points) together with Fu’s curves with the regional average values of parameter v.
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cation as FAO [2003]. The values of Ks, qf, qw, and dTop for
the dominant soil type and the dominant type of vegetation
are firstly transformed into 10 km gridded data sets, the
same resolution as the gridded climatic data set, and then
Smax is calculated for each 10 km pixel, and finally Smax and
Ks are averaged on each catchment from the 10 km gridded
data sets. In this study, Smax is scaled by mean annual
potential evapotranspiration in a dimensionless form, i.e.,
Smax/E0 . The three dimensionless variables for all 108 catchments are given in Table 1.
3. Results and Discussions
Figure 3. Comparison of the parameter v optimized from
annual water balance with the calibrated one from long-term
mean water balance. Each circle denotes one catchment.
contents at field capacity (qf) and wilting point (qw) for each
soil type are estimated at matric pressures of 33 and
1500 kPa respectively from the soil water retention as
recommended by Dunne and Willmott [1996] using van
Genuchten’s formula [van Genuchten, 1980]. Other soil
water properties are taken from International GeosphereBiosphere Programme [2000] using the same soil classifi-
3.1. Fu’s Curve for Mean Annual Water Balance
[13] On the basis of the long-term mean of annual water
balance and climate, Figure 2 presents the water-energy
balance for the 108 catchments in two different but equivalent forms of Fu’s curves, i.e., E/P vs. E0/P and E/E0 vs.
P/E0. Fu’s curves with the average values of the parameter
v for each region (the Tibetan Plateau, Loess Plateau,
Haihe River basin, and inland river basins) are also shown
in Figure 2. In Figure 2 (left) the different regional features
of Fu’s curves stand out more, whereas in Figure 2 (right),
these regional differences tend to be hidden. This tends to
suggest that actual evapotranspiration is governed more by
precipitation rather than potential evapotranspiration in
nonhumid regions [Yang et al., 2006].
[14] This study calibrates the parameter v in Fu’s equation for each catchment in two different methods. From
long-term mean water balance and average climate, the
unique parameter v can be calculated directly using Fu’s
Figure 4. Relationships between (a) v and Ks /ir , (b) v and Smax/E0 , and (c) v and tan b for the
108 catchments. The significance level of 99% is used in the F test for p > 0.99 and n = 108, |r| 0.21.
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Figure 5. Observed values of annual actual evapotranspiration, precipitation, and potential
evapotranspiration for one typical catchment (catchment 76 in the Haihe River basin) plotted in two
different but equivalent Budyko-type forms in scattering points and Fu’s curve with the calibrated
parameter v from long-term mean.
equation (see Table 1). It can also be optimized by minimizing the mean absolute error (MAE) [Legates and
McCabe, 1999] of the estimated annual evapotranspiration.
Figure 3 illustrates the calibrated values versus the optimized values of parameter v for the 108 catchments. A
high correlation (r 2 = 0.975) and relatively small systematic
error (the slope b = 1.07) of the two v values implies that
Fu’s equation can be used for predicting the interannual
variability of regional water balances.
3.2. Spatial Variability of Annual Water-Energy
Balance
[15] The regional variability of annual water-energy balance has already been noted in Figure 2. This variability
Figure 6. Statistical comparisons of the distributions of criteria for evaluating the estimated annual
actual evapotranspiration using Fu’s equation with the calibrated parameter from long-term mean of water
balance and with the estimated parameter from the empirical equation. The criteria includes the mean
absolute error (MAE), the square root of the mean square error (RMSE), the coefficient of determination
r2, and the Nash-Sutcliffe coefficient of efficiency (NSE).
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Table 2. Stepwise Regression Results for Determining the Coefficients a1, b1, c1, and d1 in
Equation (5) by Increasing Numbers of Variables
Model Coefficients
Variables
F
Fa=0.001
r2
b1
c1
d1
ln a1
Ks /ir
Ks /ir , Smax/E0
Ks /ir , Smax/E0 , tan b
66.3
46.8
33.3
11.50
7.41
5.86
0.385
0.471
0.490
0.559
0.476
0.368
—
0.403
0.436
—
—
4.464
1.282
2.087
2.158
may be mainly caused by the differences in annual precipitation and potential evapotranspiration, however, the fact of
parameter v differing from catchment to catchment calls
more detailed research on examining the regional change in
the parameter v of Fu’s equation. Figures 4a, 4b, and 4c
show the correlations between v and the relative infiltration
capacity (Ks/ir), between v and relative soil water storage
(Smax/E0), and between v and the average slope (tan b) of
the 108 catchments, respectively. The correlation coefficients (r) are 0.60, 0.45, and 0.39, respectively. Applying F test, v was shown to be closely correlated with the
three dimensionless factors (at the p > 0.99 significant
level).
[16] Comparing the v values with the catchment areas, it
is known that v values for the 108 catchments have no
relation (r = 0.025) with catchment sizes (ranging 272 –
94m800 km2). The spatial scale i.e., catchment size, is not
the cause of the change in the v value for the interested
catchment size.
3.3. Interannual Variability of Water-Energy Balance
[17] The Budyko hypothesis is also examined in each
catchment using annual data series of water balance and
average climate. Similar to Figure 2, Figure 5 plots the
annual water balance in the Budyko type curve for one
typical catchment located in the Haihe River basin (the
catchment number is 76). The results show that the Budyko
hypothesis are also valid for interannual variability of waterenergy balance in these nonhumid catchments.
[18] The predictability of Fu’s equation with the calibrated v values is further examined. Figure 6 shows the
cumulative distribution functions of the mean absolute error
(MAE), the square root of the mean square error (RMSE),
the coefficient of determination (r2) [Legates and McCabe,
1999], and Nash-Sutcliffe coefficient of efficiency (NSE)
[Nash and Sutcliffe, 1970] for the predicted annual evapotranspiration using Fu’s equation with the calibrated v
values in each catchment. The ranges of MAE, RMSE, r2,
and NSE for the 108 catchments are 2.84– 67.00 mm, 3.32–
84.25 mm, 0.62– 1.00, and 0.57– 1.00, respectively. In more
than 90% of the catchments, the values of MAE and RMSE
are less than 39.62 mm and 55.35 mm; and the values of r2
and NSE are larger than 0.82 and 0.76, respectively. In conclusion, Fu’s equation can predict the interannual changes in
water-energy balance well in the study areas.
3.4. Empirical Formula of Parameter v in
Fu’s Equation
[19] As shown in Figure 4, the parameter v in Fu’s
equation is closely correlated with the three dimensionless
landscape characteristics. It is possible to estimate the
parameter v from catchment characteristics without measured discharge data for applying to ungauged basins.
Building on previous investigations, this study selects the
above three dimensionless variables as the key descriptors
Figure 7. Predicted values of mean annual evapotranspiration using Fu’s equation with the estimated
parameter w from the empirical formula plotted versus observed values and predicted mean annual runoff
plotted versus observed values. The 1:1 line is plotted for comparison; b denotes the slope of linear
regression, and a denotes the intercept.
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Figure 8. Comparisons of annual values of actual evapotranspiration between the estimation by Fu’s
equation with the empirical formula of parameter v (solid line) and the estimation by the water balance
(circles) in four typical catchments.
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Figure 9. Comparison of observed interannual variability
of runoff sQ (as defined by Koster and Suarez [1999]) with
the predicted values by Fu’s equation with the empirical
formula of parameter v.
for climate and geomorphology of a catchment to determine
v as follows,
v ¼ 1 þ f1
Ks
Smax
f2
f3 ðtan b Þ
ir
E0
ð3Þ
where f1, f2, and f3 are functions to be determined. The
overbar denotes the long-term mean values.
[20] On the basis of phenomenological considerations the
boundary conditions are achieved
f1 Ks =ir ! 0; i:e:; v ! 1; when ir ! 0;
f2 Smax =E0 ! 0; i:e:; v ! 1; when Smax ! 0;
ð4Þ
f3 ðtan b Þ ! 1; when tan b ! 0;
f3 ðtan b Þ ! 0; i:e:; v ! 1; when tan b ! 1:
b1 Ks
Smax c1
expðd1 tan b Þ
ir
E0
ð5Þ
where the values of coefficients a1 and c1 should be larger
than zero, b1 and d1 should be less than zero. Taking
logarithm of equation (5), the values of the coefficients a1,
b1, c1 and d1 can be estimated by stepwise regression analysis (see Table 2). The final form of equation (5) becomes
v ¼ 1 þ 8:652 Ks =ir
0:368
Smax =E0
0:436
predicted v values of the 108 catchments by equation (6)
are 0.491 and 33.3, respectively. Considering the advantage
of model interpretation, a linear formula is also derived in
this study by stepwise regression analysis, i.e., v = 2.947 0.155(Ks/ir ) + 5.882(Smax/E0 ) 2.096 tan b. The coefficient of determination and the statistic for F test of the
predicted v values are 0.436 and 26.8, respectively, which
is very close to equation (6). It should be also noted that the
linear model cannot satisfy the above boundary conditions,
and it is mainly for the convenience of model interpretation.
[23] This study applies Fu’s equation together with the
parameter v estimated by equation (6) to predict the mean
annual values of actual evapotranspiration and runoff for the
108 catchments. Indicated by the results in Figure 7, the
predicted mean annual evapotranspiration in the 108 catchments can explain 95.2% (94.7% for a regression through
the origin [Snedecor and Cochran, 1980]) of the variance of
the observed values, and the predicted mean annual runoff
can explain 62% (57.2% for a regression through the origin)
of the observed values (where the runoff coefficient is
0.112 ± 0.057).
[24] It also applies to predict the annual values of actual
evapotranspiration for the 108 catchments at the same data
period. Figure 8 displays the time series of the predicted annual
values of actual evapotranspiration compared with the measured values in four typical catchments from the four regions,
respectively. Generally, the accuracy declined slightly
comparing with the predictions using the calibrated v values
(also see Figure 6), whereas only in 5 of the 108 catchments, the
values of NSE are less than 0.6. Figure 9 compares the observed
interannual variability of runoff sQ (defined as the standard
deviations of annual values of runoff, following Koster and
Suarez [1999]) with the values predicted by Fu’s equation
using the estimated v values by equation (6). The correlation
coefficient r is 0.824, the gradient b is 0.72, and the
intercept a is 9.0 mm. All results show that the parameter
v can be estimated from regional characteristics by an
empirical formula i.e., equation (6) without calibration, and
Fu’s equation is reliable and robust for predicting annual
evapotranspiration in different regions.
4. Conclusions
[21] Considering the correlations shown in Figure 4 and
the above boundary conditions, the functional forms of f1,
f2, and f3 are selected and thus equation (3) is obtained as
v ¼ 1 þ a1
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expð4:464 tan bÞ
ð6Þ
[22] Comparing with the calibrated values of v, the coefficient of determination r2 and the statistic for F test of the
[25] Through analyzing annual water balances in 108 nonhumid catchments of China, the spatial and temporal variability in annual water-energy balances have been examined.
For both long-term water balances in the 108 catchments and
annual water balances of individual catchments, Fu’s formula
derived from the Budyko hypothesis is confirmed. The
Budyko-type curves plotted for the 108 catchments have
shown significant regional patterns. By examining the spatial
changes in the calibrated parameter v in Fu’s equation from
the long-term water balance in the 108 catchments, it is
understood that in addition to the mean climate conditions,
the regional feature of water-energy balance is also closely
correlated to the relative infiltration capacity (Ks/ir ), relative
soil water storage (Smax/E0), and the average slope (tan b),
but has nearly no correlation (r = 0.025) with the spatial
scale (i.e., the catchment area). It is found that the optimized
values of Fu’s parameter v from annual values of water
balance in each of the 108 study catchments have a high
correlation and relatively small systematic error comparing
with the calibrated v values from long-term mean annual
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water balance in the same study periods. This implies that
Fu’s equation can be used for predicting the interannual
variability of regional water balances. Furthermore, indicated
by the results, Fu’s equation can predict accurately both longterm mean and interannual actual evapotranspiration.
Through a stepwise regression analysis, an empirical formula
of the parameter v has been derived, and proved to be able to
predict annual actual evapotranspiration accurately, as well as
predict the mean annual and interannual variability of runoff
reasonably. This is especially useful for predictions in ungauged basins.
[26] Acknowledgments. This research was partially sponsored by the
Core Research for Evolutional Science and Technology (CREST) program
of the Japan Science and Technology Agency (JST) and was partially
supported by the National 973 Project of China (2006CB403405) and the
National Natural Science Foundation of China (50679029). The authors
would like to express their appreciation to Murugesu Sivapalan, Marc
Parlange, Michiaki Sugita, Steve Melching, and two anonymous reviewers,
whose comments and suggestions led to significant improvements in the
submitted manuscript as well as raising our interest in the spatial and
temporal variability of annual water balances. We would also like to thank
Baopu Fu for his insightful suggestions.
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