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© IWA Publishing 2016 Hydrology Research
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in press
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2016
Probabilistic prediction in ungauged basins (PUB) based
on regional parameter estimation and Bayesian model
averaging
Yanlai Zhou, Shenglian Guo, Chong-Yu Xu, Hua Chen, Jiali Guo
and Kairong Lin
ABSTRACT
Predictions in ungauged basins (PUB) are widely considered to be one of the fundamentally
challenging research topics in the hydrological sciences. This paper couples a regional parameter
transfer module with a probabilistic prediction module in order to obtain probabilistic PUB. Steps in
the proposed probabilistic PUB include: (1) Variable infiltration capacity-three layers (VIC-3L) model
description; (2) three regional parameter transfer schemes for ungauged basins, i.e., regression
analysis, spatial proximity, and physical similarity; (3) probabilistic PUB using Bayesian model
averaging (BMA); and (4) performance evaluation for probabilistic PUB. The study is performed on 12
sub-basins in the Hanjiang River basin, China. The results demonstrate that the mean prediction of
BMA is much closer to the observed data compared with its associated individual parameter transfer
scheme (physical similarity approach), and the probabilistic predictions of BMA can effectively
reduce the uncertainty in runoff PUB better than any associated individual parameter transfer
schemes for two ungauged sub-basins.
Key words
| Bayesian model averaging, physical similarity, predictions in ungauged basins, regional
parameter estimation, regression analysis, spatial proximity
Yanlai Zhou (corresponding author)
Changjiang River Scientific Research Institute,
Wuhan 430010,
China
E-mail: zyl23bulls@whu.edu.cn
Yanlai Zhou
Shenglian Guo
Chong-Yu Xu
Hua Chen
State Key Laboratory of Water Resources and
Hydropower Engineering Science,
Wuhan University, Wuhan 430072,
China
Chong-Yu Xu
Department of Geosciences,
University of Oslo,
Norway
Jiali Guo
Three Gorges University, 443002,
China
Kairong Lin
School of Geography and Planning,
Sun Yat-sen University, Guangzhou 510275,
China
INTRODUCTION
Predictions in ungauged basins (PUB) are widely considered
parameter values are transferred to simulate runoff for the
as an important and challenging research topic in the hydro-
target ungauged basin: regression analysis, spatial proximity,
logical sciences (Sivapalan et al. ; Hrachowitz et al.
physical similarity, and a mixture of them.
). One of the primary research objectives of the PUB
Various studies have been performed to determine
initiative was to improve the ability of existing hydrological
which is best approach between the spatial proximity
models to predict in ungauged basins with reducing uncer-
method and the physical similarity approach. Post & Jake-
tainties (Dong et al. a, b; Parkes et al. ).
man () investigated the relationships between the
Regionalization of parameters is diffusely used to simulate
model parameters of a lumped conceptual rainfall-runoff
runoff in PUB, which is regarded as the process of transfer-
model and the basin landscape attributes of similarly sized
ring parameter values from a donor gauged basin to the
basins. Burn & Boorman (), Johansson (), Sefton
target ungauged basin (Xu , 2003; Xie et al. ;
& Howarth () and Kokkonen et al. () derived the
Kizza et al. ; Hailegeorgis & Alfredsen ). Four regio-
relationships between model parameters and physical catch-
nalization approaches have been typically used for choosing
ment descriptor indices using geographical information
the donor gauged basin whose calibrated and optimized
system. Croke et al. () adopted a simple hydrologic
doi: 10.2166/nh.2016.058
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Hydrology Research
Annual
precipitation varies
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approach to simulate runoff adaption to land-use changes in
resources.
ungauged basins. Goswami et al. () developed a pooling
1,100 mm, with 70–80% of the total amount occurring in
700
to
method of regional parameter estimation coupled with soil
the wet season from May to October. The Hanjiang basin
data and SMAR model in order to simulate flow in
plays a critical role in flood control and water supply in cen-
ungauged basins in France. Zhang & Chiew () evalu-
tral China. The Danjiangkou reservoir located in the middle
ated the disadvantages and advantages of different
reach of the Hanjiang River is the source of water for the
regionalization methods using two rainfall runoff models,
middle route of the South-North Water Division Project
Xinanjiang and SIMHYD in 210 Australian basins. The
(SNWDP), and the Jianghan plain in the down basin is
study showed that the best approach between the spatial
one of the most important bases for commodity grain
proximity method and the physical similarity approach is
production.
hard to identify. Most of studies had to suggest that the
Location of 12 primary streamflow stations in the
use of more information (such as remotely sensed (RS) veg-
Ankang basin (upper reach of Hanjiang basin) is shown in
etation data, soil data, climate, and land cover) or a mixture
Figure 1. The Ankang basin contains 98 precipitation
of them can improve the accuracy of runoff simulation in
stations, 9 weather stations, 12 streamflow stations of sub-
ungauged basins (Duan et al. ; Götzinger & Bárdossy
basins (10 sub-basins have observed data, Mumahe and
; Oudin et al. ; Bulygina et al. ; Li et al. ;
Renhe sub-basins with in the Ankang basin are ungauged).
Kling & Gupta ; Li et al. ).
The streamflow information – DEM, forcing, soil and veg-
Previous studies of PUB mostly focused on the compari-
etation data, and so on – is required for VIC-3L model
son of prediction of individual parameter transfer schemes
implementation and calibration. The data include: (1) daily
and their weight averages. Our research aims to compare
streamflow and weather data (download from http://www.
the probabilistic prediction generated by the Bayesian
escience.gov.cn/) from 1980–1986 and 1987–1990 are used
model averaging (BMA) with that of each individual par-
for calibration and verification, respectively; (2) DEM data
ameter transfer scheme, in order to see if BMA can
(download from http://www.gscloud.cn/) of 0.009 degree
effectively reduce the uncertainty in runoff PUB and
(around 1 × 1 km2 cell size) spatial resolution for the
improve the prediction reliability.
Ankang basin are derived and used to delineate the sub-
The paper is organized as follows: a brief introduction to
basin boundary and stream network; (3) vegetation type
the study area; then a general description of the main steps
data are taken from the global land cover classification gen-
and procedures including: (1) variable infiltration capacity-
erated by the University of Maryland with a one-kilometre
three layers (VIC-3L) model description, (2) regional par-
pixel resolution; (4) vegetation parameters are based on
ameter transfer schemes, (3) probabilistic prediction using
the vegetation from the Land Data Assimilation System;
BMA, and (4) performance evaluation; comparison between
(5) the soil parameters are derived from the soil classifi-
BMA and its three individual parameter transfer schemes is
cation information of the global 5 min data provided by
then discussed and, finishing with the conclusions drawn
the National Atmospheric and Oceanic Administration.
from this study.
METHODOLOGY
STUDY AREA AND DATA
Procedures
The Hanjiang River is the largest tributary of the Yangtze
River and it passes through the provinces of Shannxi and
A procedure coupling a regional parameter transfer module
Hubei in China, and merges into the Yangtze River at
with probabilistic prediction module is developed to obtain
Wuhan city. The river is of length 1,570 km and area
probabilistic prediction in ungauged basins. It consists of
2
159,000 km . The basin has a sub-tropical monsoon climate
two modules, namely a regional parameter transfer
and has, as a result, a dramatic diversity in its water
module and probabilistic prediction. The regional parameter
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Figure 1
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Location of 12 main streamflow stations of the Ankang basin in China.
transfer module is aimed at obtaining model parameter esti-
the Darcy law. The ARNO method is used to describe base
mates from a limited number of calibrated basins and then
flow which takes place only in the lowest layer. The routing
regionalizing them to uncalibrated basins based on the
model represented by the unit hydrograph method for over-
spatial proximity approach, physical similarity approach
land flow and the linear Saint-Venant method for channel
(similar characteristics of climate as well as physicality),
flow, allows runoff to be predicted (Liang et al. ).
and multiple regression analysis, which is described in
The VIC-3L model has ten hydrological parameters that
detail in the next section. Probabilistic prediction is designed
need to be calibrated, as shown in Table 1. A similar descrip-
to infer a prediction by weight averaging over many different
tion for VIC-3L parameters was made by Xie et al. ().
regional parameter transfer schemes based on the BMA
method, which is described in detail in the section named
Regional parameter transfer
‘Hydrological probability prediction’. The main steps and
procedures include: (1) VIC-3L model description; (2)
In the regional parameter transfer study, the parameters of
regional parameter transfer scheme; (3) probabilistic predic-
individual sub-basins with similar climate characteristics
tion using BMA; and (4) performance evaluation.
and underlying surface, as well as the individual sub-basin
are assumed to have the same values. Three regional par-
VIC-3L model description
ameter transfer schemes, i.e., spatial proximity approach,
physical similarity approach, and multiple regression analy-
The VIC-3L model has one kind of bare soil and different veg-
sis are tested in the study.
etation types in each grid cell (Liang & Xie ; Xie et al.
1. Spatial proximity approach – The spatial proximity
). It includes both the saturation and infiltration excess
approach uses the parameter values from the geographi-
runoff processes in a grid cell with a consideration of the
cally closest gauged catchment hypothesizing that
sub-grid scale soil heterogeneity, and the frozen soil processes
neighboring catchments should behave similarly.
for cold climate conditions. The one-dimensional Richard
2. Physical similarity approach – The physical similarity
equation is used to describe the vertical soil moisture move-
approach transfers the entire set of parameter values from
ment and the moisture transfer between soil layers obeys
a physically similar catchment whose attributes (climatic
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Table 1
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Hydrological parameters in VIC-3L model
Number
Variable
Description
Units
1
b
The shape of the variable infiltration capacity curve
/
2
Dm
The maximum base flow from the lowest soil layer
mm/day
3
Ds
The fraction of Dm where non-linear base flow begins
/
4
Ws
The fraction of the maximum soil moisture where non-linear base flow occurs
/
5
Dep1
The depth of top layer soil
m
6
Dep2
The depth of middle layer soil
m
7
Dep3
The depth of lower layer soil
m
8
x
The regulation capacity of river channel for stream flow
/
9
k
The propagation time of steady flow in river channel
h
10
ckg
The regulation capacity of slope land for base flow
/
and physical) are similar to those of the target ungauged
where
one. The use of more information, such as RS vegetation
X1j , X2j , , Xmj are independent variables, εj is fitting
data and soil data in the physical similarity approach can
error and is assumed as εj ∼ N(0, σ 2 ). Fifteen independent
improve runoff estimates in ungauged basins.
variables comprising six climatic characteristic variables
3. Multiple regression analysis – The multiple regression
analysis approach establishes a relationship between
Yj
denotes
the
jth
dependent
variable,
and nine soil characteristic variables are used, as shown in
Table 2.
VIC parameter values calibrated on gauged catchments
For a detailed description of methods for weather for-
and catchment descriptors or attributes (such as climatic,
cing data, vegetation dataset, and soil dataset based on
vegetation, and soil data), and then the VIC parameter
regionalization and grid in this paper, readers are referred
values for the ungauged catchments are estimated from
to Xie et al. ().
these attributes and the established relationships. Three
regression analysis equations are used to establish
relationships
between
dependent
variables
(VIC
Hydrological probability prediction – Bayesian model
averaging
parameters) and independent variables (fifteen climatic as
well as soil characteristic variables), described as follows.
The regression analysis equations are the linear
regression analysis equation,
Yj ¼ β 0 þ β 1 X1j þ β 2 X2j þ þ β m Xmj þ εj
by weight averaging over many different regional parameter
transfer schemes. This method is not only a pathway for
scheme combination but also a coherent approach for
(1)
accounting for between-scheme and within-scheme uncertainty (Ajami et al. ). Below is a brief description of
the basic ideas of this method.
the square-root regression analysis equation,
qffiffiffiffiffi
qffiffiffiffiffiffiffi
qffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffi
Yj ¼ β 0 þ β 1 X1j þ β 2 X2j þ þ β m Xmj þ εj
BMA is a statistical technique designed to infer a prediction
Let us consider a quantity Q to be predicted on the basis
of input data D ¼ [I, O] (I denotes the input forcing data, and
(2)
O stands for the observational flow data). f ¼ [ f1, f2, …, fk] is
the ensemble of the K-member predictions. The probabilistic
prediction of BMA is given by
and the logarithmic regression analysis equation,
log (Yj ) ¼ β 0 þ β 1 log (X1j ) þ β 2 log (X2j ) þ þ β m log (Xmj ) þ εj
(3)
p(QjD) ¼
K
X
k¼1
p( fk jD) pk (Qj fk , D)
(4)
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Y. Zhou et al.
Table 2
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Soil and climatic characteristic variables
Number
Type
Variable
Description
Units
1
Soil characteristic variables
Sat_h
Saturated hydraulic conductivity
cm/h
2
Vsat
Variability of saturated hydraulic conductivity
(cm/h)^2
3
Bub
Bubble pressure
Pa
4
Qua
Quartz content
%
5
Sat_m
Saturated moisture content
%
6
Per_c
Percentage of critical moisture content
%
7
Per_w
Percentage of wilting moisture content
%
8
Res
Residual moisture content
%
9
Per_v
Percentage of valid moisture content
%
T
Annual mean temperature
W
P
Annual mean precipitation
mm
12
E
Annual mean evaporation from water surface
mm
13
Cv_T
Coefficient of variation for monthly temperature during one year
/
14
Cv_P
Coefficient of variation for monthly rainfall during one year
/
15
Cv_E
Coefficient of variation for monthly evaporation from water surface during
one year
/
10
Climatic characteristic
variables
11
where p( fk jD) is the posterior probability of the prediction fk
C
EM algorithm for BMA parameter estimation
given the input data D and reflects how well the scheme fits
Y. Actually p( fk jD) is just the BMA weight ωk , and better per-
To estimate BMA weight ωk and scheme prediction variance
forming predictions receive higher weights than poorer
σ 2k , the Expectation-Maximization (EM) algorithm, which
performing ones, all weights are positive and should add up
has proved to be an efficient technique for BMA calculation
to 1. pk (Qj fk , D) is the conditional probability density function
based on the assumption that K-member predictions are nor-
(PDF) of the prediction Q conditional on fk and D. For compu-
mally distributed, is described in this section (Duan et al.
tation convenience, pk (Qj fk , D) is always assumed to be a
).
normal PDF and is represented as g(Qj fk , σ 2k ) ∼ N( fk , σ 2k ),
Firstly, if we denote the set of BMA parameters to be
estimated by θ ¼ wk , σ 2k , k ¼ 1, 2, . . . , K , the log form of
where
σ 2k
is the variance associated with scheme prediction
fk and observations O. In order to make this assumption
the likelihood function can be represented as
valid, some techniques such as Box-Cox transformation are
needed to make the data approximately normally distributed
and to narrow the data range (Poirier ).
l(θ) ¼ log (p(QjD)) ¼ log
K
X
ωk g Qj fk , σ 2k
!
(6)
k¼1
The BMA mean prediction is a weight average of the
individual scheme’s predictions, with their posterior probabilities
being
the
weights.
In
the
case
that
the
observations and individual scheme predictions are all normally distributed, the BMA mean prediction can be
It is difficult to maximize the function (6) by analytical
methods. The EM algorithm is an effective method for finding the maximum likelihood by alternating between two
steps, the expectation step and maximization step. The two
expressed as
steps are iterated to convergence when there is no signifiE½QjD ¼
K
X
k¼1
K
X
pð fk jDÞ E g Qj fk , σ 2k ¼
ωk fk
k¼1
cant
(5)
change
between
two
consecutive
iterative
log-likelihood estimations. In the EM algorithm, a latent
variable (unobserved quantity) ztk is used as an assistant
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for estimating BMA weight ωk . For a detailed description of
the EM algorithm for a BMA scheme, readers are referred to
Dong et al. (b).
Estimation of probabilistic prediction
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definition of NS is expressed in following equation.
3
T 2
P
Qtobs Qtsim 7
6
7
6
NS ¼ 61:0 t¼1
7 × 100%
T 2 5
4
P
t
Qobs Qobs
2
(7)
t¼1
After estimating BMA weight ωk and prediction variance σ 2k ,
we use the Monte Carlo method to generate BMA probabilistic prediction for any time t (Hammersleym & Handscomb
). The procedures are described as follows.
where Qtobs and Qtsim are observed and simulated data at
time t, Qobs is the average of observed data, T is the
length of the data series.
2. Daily root mean square error (DRMS):
1. Generate an integer value of k from [1, 2, …, K ] with
probability [ω1 , ω2 , . . . , ωk ]. A specific procedure is
described as follows.
1(a). Set the cumulative weight ω00 ¼ 0 and compute ω0K ¼
DRMS ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uT 2
uP t
t
u
tt¼1 Qobs Qsim
T
(8)
ω0k1 þ ωk for k ¼ 1, 2, …, K.
1(b). Generate a random number u between 0 and 1.
1(c). If ω0k1 u ω0k , this indicates that we choose the
kth member of the ensemble predictions.
All notations have the same meaning as in Equation (7).
As NS appears to have a negative value frequently and
DRMS is sensitive to the differences between the obser-
2. Generate a value of Qt from the PDF of g(Qt =fkt , σ 2k ).
vations and simulations, DRMS is also selected as a
Here, g(Qt =fkt , σ 2k ) represents the normal distribution
performance evaluation index. The smaller the DRMS
with mean fkt and variance σ 2k .
value is, the better the prediction performance.
3. Repeat the above steps (1) and (2) for M times. M is the
probabilistic ensemble size. In this paper, we set M ¼ 100.
After generating the BMA probabilistic ensemble prediction, results are sorted in ascending order. From this, the
90% uncertainty intervals can be derived within the range
of the 5% and 95% quantities.
1. Relative error of total runoff (RE):
0
T
P
B
B
RE ¼ Bt¼1
@
Qtobs
T
P
t¼1
T
P
t¼1
1
Qtsim C
Qtobs
C
C × 100%
A
(9)
For each individual scheme in the BMA model, the prediction uncertainty interval can also be constructed, with
This reflects the relative bias in the simulation of the
the Monte Carlo sampling method still being used to
total runoff amount. A value of RE closes to zero indicates
approximate the assumed PDF of g(Qt =fkt , σ 2k ).
better agreement of total surface runoff.
Performance evaluation indices
Performance evaluation indices for probabilistic
prediction
Performance evaluation indices for mean prediction
Xiong et al. () and Dong et al. (b) presented a set of
indices for assessing the probabilistic prediction generated
There are three indices for evaluating the mean prediction
by the uncertainty analysis methods. Three main indices
(Dong et al. b) presented as follows.
are selected here to assess the probabilistic prediction pro-
1. The Nash-Sutcliffe coefficient of efficiency (NS) – NS is
duced by the BMA model as well as from each individual
not only an objective function but also a widely used per-
parameter transfer scheme.
formance criterion. It ranges from minus infinity to 1.0,
1. Containing ratio (CR) – The containing ratio is used for
with higher values indicating better agreement. The
assessing the goodness of the uncertainty interval. It is
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Table 3
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Calibrated parameters for the 10 primary sub-basins
Gauge stations
b
Ds
Dm (mm/day)
Ws
Dep1 (m)
Dep2 (m)
Dep3 (m)
x
k
ckg
Baohe
0.291
0.706
5.761
0.246
0.005
0.400
0.224
0.235
0.694
0.903
Hanzhong
0.250
0.590
0.518
0.420
0.074
0.204
0.018
0.258
1.878
0.542
Xushuihe
0.257
0.228
0.580
0.859
0.073
0.100
0.010
0.150
0.595
0.695
Youshuihe
0.257
0.428
2.580
0.559
0.053
0.250
0.010
0.050
0.555
0.695
Ziwuhe
0.257
0.428
3.540
0.559
0.053
0.270
0.008
0.258
0.750
0.750
Shiquan
0.257
0.828
5.540
0.359
0.083
0.150
0.010
0.258
0.700
0.950
Chihe
0.258
0.421
2.569
0.701
0.026
0.300
0.010
0.247
0.721
0.588
Zhehe
0.157
0.428
1.040
0.559
0.073
0.110
0.010
0.208
0.995
0.550
Lanhe
0.450
0.631
4.680
0.489
0.085
0.005
0.010
0.250
0.905
0.697
Ankang
0.257
0.828
4.040
0.159
0.140
0.240
0.100
0.100
0.660
0.950
defined as the percentage of observed data points that are
RESULTS AND DISCUSSION
covered in the prediction bounds.
Calibration results
T
N (qtl Qtsim qtu )
CR ¼ t¼1
× 100%
T
(10)
used for calibration. The gauged sub-basins are selected as
where qtu and qtl denote as upper and lower prediction
T
bounds at time t, and N is the number of observed
t¼1
data points that are covered in the prediction bounds.
2. Average band-width (B). Consider
B¼
T
1X
(qt qtl )
T t¼1 u
Daily streamflow and weather data from 1980 to 1986 are
the primary basins to implement VIC-3L model calibration,
which is achieved by matching the total annual stream flow
volume and the shape of the mean daily hydrograph to the
corresponding observations in the Ankang River basin.
The two criteria, i.e., NS and RE are used for model
(11)
Table 4
|
Calibration statistics for the 10 primary sub-basins
where the average band-width B is also an index for
Calibration
measuring the performance of estimated uncertainty
Gauge stations
NS (%)
DRMS (m3/s)
RE (%)
Baohe
90.79
30.67
14.00
amplitude is an index to quantify the average deflection
Hanzhong
91.81
23.31
3.90
of the curve of the middle points of the prediction
Xushuihe
85.11
29.70
5.08
bounds from the observed stream flow hydrograph. It is
Youshuihe
88.29
32.14
7.00
defined as
Ziwuhe
86.87
29.82
6.36
Shiquan
89.27
31.51
11.00
Chihe
84.38
40.17
2.71
interval.
3. Average deviation amplitude (D) – The average deviation
T 1X
1 t
t
t D¼
(q þ ql ) Qobs T t¼1 2 u
with notations as defined previously.
(12)
Zhehe
86.02
31.90
13.00
Lanhe
85.65
32.89
4.83
Ankang
87.94
29.32
12.00
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Table 5
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The transferred parameter values of three regionalization approaches
Gauge stations
Schemes
b
Ds
Dm (mm day
Mumahe
A (Youshuijie)
B (Chihe)
C
0.257
0.258
0.258
0.428
0.421
0.121
Renhe
A (Zhehe)
B (Lanhe)
C
0.157
0.450
0.250
0.428
0.631
0.491
1
)
Ws
Dep1(m)
Dep2(m)
Dep3(m)
x
k
ckg
2.580
2.569
1.569
0.559
0.701
0.801
0.053
0.026
0.056
0.250
0.300
0.150
0.010
0.010
0.010
0.050
0.247
0.247
0.555
0.721
0.921
0.695
0.588
0.588
1.040
4.680
0.580
0.559
0.489
0.589
0.073
0.085
0.085
0.110
0.005
0.005
0.010
0.010
0.010
0.208
0.250
0.250
0.995
0.905
0.955
0.550
0.697
0.697
Note: Scheme A denotes spatial proximity approach, scheme B denotes physical similarity approach, and scheme C denotes multiple regression analysis.
calibration. In the calibration study, the parameters of indi-
calibrated hydrological parameters can be transferred to
vidual sub-basins with similar climate characteristics and
the ungauged sub-basins with reasonably good results.
underlying surface are assumed to have the same values.
Ten hydrological parameters in the VIC-3L model have
Testing of parameter transfer
been calibrated for the 10 primary sub-basins. Table 3
shows the calibrated parameter values in 10 primary sub-
Parameter transfer schemes
basins in Ankang basin. Their typical ranges and the effect
of each parameter on results of simulated stream flow are
Three regionalization approaches, i.e., spatial proximity
described below: (1) b typically ranges from 0 to 0.50. It
approach, physical similarity approach, and multiple
describes the total of available infiltration capacity as a func-
regression analysis are used to choose the 10 donor
tion of the relative saturated grid cell area and controls the
gauged sub-basins whose optimized parameter values are
quantity of runoff generation directly and the water balance.
used to model daily runoff for the two ungauged sub-
A lower value of b gives lower infiltration and yields higher
basins (Mumahe and Renhe). Table 5 lists the transferred
surface runoff (the value of b in this paper is the inverse
parameter values of the three regionalization approaches.
value of b in Xie et al. ). The highest value of b in subbasins is only 0.450 for Lanhe sub-basin and the lowest
value is 0.157 for Zhehe sub-basin. The rest of the values
of b in sub-basins are very close to 0.257, because Ankang
basin is in a humid region; (2) Dm typically ranges from 0
to 6 mm day1; (3) Ds typically ranges from 0 to 1. With a
higher value of Ds, the base flow will be higher at lower
water content in the lowest soil layer; (4) Ws typically
ranges from 0 to 1; (5) Dep1, Dep2, and Dep3 range from 0
to 0.40 m. In general, thicker soil depths slow down
seasonal peak flows and increase the loss due to evapotranspiration; (6) x ranges from 0.05 to 0.30; (7) k ranges
typically from 0.50 to 2.0; (8) ckg ranges typically from 0.5
to 1.0.
Table 4 lists the statistical results (NS, DRMS, and RE)
for the 10 primary sub-basins using calibrated hydrological
parameters. In terms of NS, DRMS and RE, the model in
the calibration provides good simulation results for all subbasins. In the next section, we will demonstrate that these
Figure 2
|
The distribution map of vegetation characteristics for the VIC-3L model with a
5 × 5 km grid in the Ankang basin. (The legend numbers denote different
vegetation types).
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The spatial proximity sub-basins for Mumahe and Renhe are
are Chihe and Lanhe, respectively. Figure 2 shows the distri-
Youshuihe and Zhehe, respectively, as shown in Figure 1.
bution map of vegetation characteristics for the VIC-3L
The physical similarity sub-basins for Mumahe and Renhe
model with 5 × 5 km grid in the Ankang basin. Figure 3
Figure 3
|
The distribution map of climatic factors with a 5 × 5 km grid in the Ankang basin.
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Table 6
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Multiple regression analysis results for 10 donor gauged sub-basins
Number
Variable
b
Ds
1
Sat_h
2
Vsat
3
Bub
4
Qua
5
Sat_m
6
Per_c
7
Per_w
√
8
Res
√
9
Per_v
10
T
11
P
12
E
13
Cv_T
√
√
14
Cv_P
√
√
15
Cv_E
Dm
Ws
Dep1
Dep2
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
√
Number of regression
variable
√
√
√
√
5
4
6
5
√
√
√
6
5
F statistics
5.77
18.25
6.62
10.57
3.81
7.21
Regression equation
Square root
Square root
Square root
Linear
Linear
Square root
R2 (%)
66.73
84.91
75.80
69.74
76.63
62.84
shows the distribution map of climatic factors with 5 × 5 km
equation is tested by the Fα (m, n m 1) statistic (m is
grid in the Ankang basin. The vegetation similarity sub-
the number of regression variables, n is the number of
basins for Mumahe and Renhe are Chihe and Lanhe,
sub-basins; Pope & Webster ). The structures of
respectively, in Figure 2. The annual mean temperatures
regression analysis equations for six hydrological par-
W
W
(T ) for Mumahe and Chihe are 14.6 C and 14.8 C, respect-
ameters b, Dm, Ds, Ws, Dep1, and Dep2 are square root,
ively. The annual mean temperatures (T ) for Renhe and
square root, square root, linear, linear and square root,
W
W
Lanhe are 15.2 C and 15.5 C, respectively. The annual
respectively. However, the remaining four hydrological par-
mean precipitation (P) for Mumahe and Chihe are
ameters, Dep3, x, k, and ckg, have no remarkable
1,070 mm and 997 mm, respectively. The annual mean pre-
regression analysis equations because these hydrological
cipitation (P) for Renhe and Lanhe are 1,021 mm and
parameters are affected by many model variables and
1,068 mm, respectively. The annual mean evaporation
catchment descriptors or attributes. In terms of NS, the
from water surface (E) for Mumahe and Chihe are
regression analysis equations can provide reasonably
564 mm and 514 mm, respectively. The annual mean evap-
good fitting results between parameter values calibrated
oration from water surface (E) for Renhe and Lanhe are
on gauged catchments and climatic as well as vegetation
246 mm and 268 mm, respectively. The climatic similarity
variables. The reasonably good fitting results can be
sub-basins for Mumahe and Renhe are Chihe and Lanhe,
demonstrated by the fitting curves between calibrated
respectively, as in Figure 3.
results and regression analysis results of hydrological par-
Table 6 shows multiple regression analysis results for
ameters in the VIC-3L model, as shown in Figure 4. The
10 donor gauged sub-basins. The regression analysis
distribution map of six hydrological parameters in the
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sub-basins can achieve 91.96 and 88.06% in the calibration period as well as 81.72 and 78.31% in the
validation period, which is better than the best associated
individual
parameter
transfer
scheme
prediction
(Scheme B, physical similarity approach). However, in
terms of RE, the mean prediction of BMA (3) performs
worse than its best individual parameter transfer scheme
prediction.
Three indices illustrated in the section named
‘Performance evaluation indices for probabilistic prediction’ are used for assessing the probabilistic prediction
of both BMA (3) and individual parameter transfer
schemes. The results of two ungauged sub-basins for the
whole flow series are also presented in Figure 7. It is
clear that probabilistic prediction of BMA (3) has the largest values of CR and B, and almost the smallest D, in
both calibration and validation periods. In other words,
probabilistic prediction of BMA (3) has better properties
than probabilistic prediction of any individual parameter
transfer schemes in terms of CR and D, but worse in
terms of B. We then compared the differences between
BMA (3) and its individual parameter transfer schemes
Figure 4
|
The fitting curves between calibrated results and regression analysis results of
hydrological parameters in the VIC-3L model.
in probabilistic prediction by graphs. For illustrative purposes, Figure 8 shows the mean prediction and 90%
confidence interval of both BMA (3) and three individual
VIC-3L model with a 5 × 5 km grid in the Ankang basin is
parameter transfer schemes of maximum one month
shown in Figure 5 (calibrated hydrological parameters for
hydrograph for Baohe sub-basin in 1983 during the cali-
10 primary sub-basins, as well as regression analysis results
bration period, respectively. The observations of 1983
of hydrological parameters for Mumahe and Renhe
are presented by dots, and the mean predictions of
ungauged sub-basins).
BMA (3) and its individual parameter transfer schemes
are shown as solid curve. It is shown that the probabilistic prediction of BMA (3) is much broader than that of
Evaluation for mean prediction and probabilistic
any of its individuals. It can be found from Figure 9
prediction
that the results of validation are similar to that of
the calibration period. In a word, the probabilistic
Figure 6 displays the weight estimates of individual par-
prediction of BMA (3) has better performance than
ameter transfer schemes in BMA (3). We check the
its individual parameter transfer schemes for the flow
mean prediction of BMA (3) using three criteria illus-
series.
trated in the section named ‘Performance evaluation
indices for mean prediction’. Results of BMA (3) and its
three individual parameter transfer schemes in the mean
CONCLUSIONS
prediction of two ungauged sub-basins for the whole
flow series are shown in Figure 7. In terms of NS, the
In this paper, the BMA method is used to predict a new
mean prediction of BMA (3) for Mumahe and Renhe
measurement value associated with a combination of
Uncorrected Proof
12
Figure 5
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The distribution map of six hydrological parameters in the VIC-3L model with a 5 × 5 km grid in the Ankang basin (calibrated hydrological parameters for 10 primary sub-basins,
as well as regression analysis results of hydrological parameters for Mumahe and Renhe ungauged sub-basins).
probabilistic prediction in ungauged basins based on three
between BMA (3) and its three individual parameter
individual parameter transfer schemes. The comparison
transfer schemes is made in terms of both mean
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prediction and probabilistic prediction in this study. The
main conclusions are summarized as follows: (1) The
mean prediction of BMA (3) is much closer to the
observed data as compared with its best individual parameter transfer scheme (physical similarity approach)
for two ungauged sub-basins; (2) The probabilistic predictions of BMA (3) have larger containing ratio, larger
average band-width, and smaller average deviation amplitude than any of its individual parameter transfer schemes
for the two ungauged sub-basins. It is worth mentioning
that further works will focus on transferring parameter
approaches based on data mining and machine learning
techniques, such as artificial neural networks and support
vector machine, as well as choosing other hydrological
Figure 6
|
Histogram for the weighted of individual schemes in BMA (3). Scheme A
denotes the spatial proximity approach, Scheme B denotes the physical
similarity approach, and Scheme C denotes multiple regression analysis.
Figure 7
|
models, then it is anticipated that the advantages of
BMA can be generalized.
The simulation results with three parameter transfer schemes in Mumahe and Renhe ungauged sub-basins. Scheme A denotes the spatial proximity approach, Scheme B
denotes the physical similarity approach, and Scheme C denotes multiple regression analysis.
Uncorrected Proof
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Figure 8
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The mean prediction and 90% confidence interval of both BMA (3) and three individual parameter transfer schemes for Mumahe sub-basin in 1983 during the calibration period.
Scheme A denotes the spatial proximity approach, Scheme B denotes the physical similarity approach, and Scheme C denotes multiple regression analysis.
Uncorrected Proof
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Figure 9
Y. Zhou et al.
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The mean prediction and 90% confidence interval of both BMA (3) and three individual parameter transfer schemes for Mumahe sub-basin in 1987 during the validation period.
Scheme A denotes the spatial proximity approach, Scheme B denotes the physical similarity approach, and Scheme C denotes multiple regression analysis.
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ACKNOWLEDGEMENTS
This study is financially supported by the International
Cooperation in Science and Technology Special Project
of=China
(2014DFA71910),
National
Natural
Science
Foundation of China (51509008, 51509141 and 51379223),
Natural Science Foundation of Hubei Province (2015CFB217)
and Open Foundation of State Key Laboratory of Water
Resources and Hydropower Engineering Science in Wuhan
University (2014SWG02).
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