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.

## INTRODUCTION

Predictions in ungauged basins (PUB) are widely considered as an important and challenging research topic in the hydrological sciences (Sivapalan *et al.* 2003; Hrachowitz *et al.* 2013). One of the primary research objectives of the PUB initiative was to improve the ability of existing hydrological models to predict in ungauged basins with reducing uncertainties (Merz & Bloschl 2004; Young 2006; Heuvelmans *et al.* 2006; Wagener & Wood 2006; Jin *et al.* 2009; Dong *et al.* 2013a, 2013b; Parkes *et al.* 2013). Regionalization of parameters is diffusely used to simulate runoff in PUB, which is regarded as the process of transferring parameter values from a donor gauged basin to the target ungauged basin (Vandewiele & Elias 1995; Xu 1999; Xie *et al.* 2007; Parajka *et al.* 2007; Bastola *et al.* 2008; Kizza *et al.* 2013; Hailegeorgis & Alfredsen 2015). Four regionalization approaches have been typically used for choosing the donor gauged basin whose calibrated and optimized parameter values are transferred to simulate runoff for the target ungauged basin: regression analysis, spatial proximity, physical similarity, and a mixture of them.

Various studies have been performed to determine which is the best approach between the spatial proximity method and the physical similarity approach. Post & Jakeman (1999) investigated the relationships between the model parameters of a lumped conceptual rainfall-runoff model and the basin landscape attributes of similarly sized basins. Burn & Boorman (1993), Johansson (1994), Sefton & Howarth (1998) and Kokkonen *et al.* (2003) derived the relationships between model parameters and physical catchment descriptor indices using geographical information system. Croke *et al.* (2004) adopted a simple hydrologic approach to simulate runoff adaption to land-use changes in ungauged basins. Goswami *et al.* (2007) developed a pooling method of regional parameter estimation (RPE) coupled with soil data and SMAR model in order to simulate flow in ungauged basins in France. Zhang & Chiew (2009) evaluated the disadvantages and advantages of different regionalization methods using two rainfall runoff models, Xinanjiang and SIMHYD in 210 Australian basins. The study showed that the best approach between the spatial proximity method and the physical similarity approach is hard to identify. Most studies suggested that the use of more information (such as remotely sensed (RS) vegetation data, soil data, climate, and land cover) or a mixture of them can improve the accuracy of runoff simulation in ungauged basins (Ao *et al.* 2006; Kay *et al.* 2006; Duan *et al.* 2006; Götzinger & Bárdossy 2007; Oudin *et al.* 2008; Bulygina *et al.* 2009; Li *et al.* 2009; Kling & Gupta 2009; Samaniego *et al.* 2011; Li *et al.* 2014; Ghahraman & Davary 2014).

Previous studies of PUB mostly focused on the comparison of prediction of individual parameter transfer schemes and their weight averages. Our research aims to compare the probabilistic prediction generated by the Bayesian model averaging (BMA) with that of each individual parameter transfer scheme, in order to see if BMA can effectively reduce the uncertainty in runoff PUB and improve the prediction reliability.

The paper is organized as follows: a brief introduction to the study area; then a general description of the main steps and procedures including: (1) variable infiltration capacity-three layers (VIC-3L) model description, (2) regional parameter transfer schemes, (3) probabilistic prediction using BMA, and (4) performance evaluation; comparison between BMA and its three individual parameter transfer schemes is then discussed, finishing with the conclusions drawn from this study.

## STUDY AREA AND DATA

The Hanjiang River is the largest tributary of the Yangtze River and it passes through the provinces of Shannxi and Hubei in China, and merges into the Yangtze River at Wuhan city. The river is of length 1,570 km and area 159,000 km^{2}. The basin has a sub-tropical monsoon climate and has, as a result, a dramatic diversity in its water resources. Annual precipitation varies from 700 to 1,100 mm, with 70–80% of the total amount occurring in the wet season from May to October. The Hanjiang basin plays a critical role in flood control and water supply in central China. The Danjiangkou reservoir located in the middle reach of the Hanjiang River is the source of water for the middle route of the South-North Water Division Project (SNWDP), and the Jianghan plain in the down basin is one of the most important bases for commodity grain production.

Location of 12 primary streamflow stations in the Ankang basin (upper reach of Hanjiang basin) is shown in Figure 1. The Ankang basin contains 98 precipitation stations, nine weather stations, 12 streamflow stations of sub-basins (10 sub-basins have observed data, Mumahe and Renhe sub-basins with in the Ankang basin are ungauged). The streamflow information – digital elevation model (DEM), forcing, soil and vegetation data, and so on – is required for VIC-3L model implementation and calibration. The data include the following: (1) daily streamflow and weather data (download from http://www.escience.gov.cn/) from 1980–1986 and 1987–1990 are used for calibration and verification, respectively; (2) DEM data (download from http://www.gscloud.cn/) of 0.009 degree (around 1 × 1 km^{2} cell size) spatial resolution for the Ankang basin are derived and used to delineate the sub-basin boundary and stream network; (3) vegetation type data are taken from the global land cover classification generated by the University of Maryland with a one-kilometre pixel resolution; (4) vegetation parameters are based on the vegetation from the Land Data Assimilation System; (5) the soil parameters are derived from the soil classification information of the global 5 min data provided by the National Atmospheric and Oceanic Administration.

## METHODOLOGY

### Procedures

A procedure coupling a regional parameter transfer module with probabilistic prediction module is developed to obtain probabilistic prediction in ungauged basins. It consists of two modules, namely a regional parameter transfer module and probabilistic prediction. The regional parameter transfer module is aimed at obtaining model parameter estimates from a limited number of calibrated basins and then regionalizing them to uncalibrated basins based on the spatial proximity approach, physical similarity approach (similar characteristics of climate as well as physicality), and multiple regression analysis, which is described in detail in the next section. Probabilistic prediction is designed to infer a prediction by weight averaging over many different regional parameter transfer schemes based on the BMA method, which is described in detail in the section ‘Hydrological probability prediction’. The main steps and procedures include: (1) VIC-3L model description; (2) regional parameter transfer scheme; (3) probabilistic prediction using BMA; and (4) performance evaluation.

### VIC-3L model description

The VIC-3L model has one kind of bare soil and different vegetation types in each grid cell (Liang & Xie 2001; Xie *et al.* 2003). It includes both the saturation and infiltration excess runoff processes in a grid cell with a consideration of the sub-grid scale soil heterogeneity, and the frozen soil processes for cold climate conditions. The one-dimensional Richard equation is used to describe the vertical soil moisture movement and the moisture transfer between soil layers obeys the Darcy law. The ARNO method is used to describe base flow which takes place only in the lowest layer. The routing model represented by the unit hydrograph method for overland flow and the linear Saint-Venant method for channel flow, allows runoff to be predicted (Liang *et al.* 1994).

The VIC-3L model has 10 hydrological parameters that need to be calibrated, as shown in Table 1. A similar description for VIC-3L parameters was made by Xie *et al.* (2007).

Number | Variable | Description | Units |
---|---|---|---|

1 | b | The shape of the variable infiltration capacity curve | / |

2 | D _{m} | The maximum base flow from the lowest soil layer | mm/day |

3 | D _{s} | The fraction of D where non-linear base flow begins _{m} | / |

4 | W _{s} | The fraction of the maximum soil moisture where non-linear base flow occurs | / |

5 | Dep _{1} | The depth of top layer soil | m |

6 | Dep _{2} | The depth of middle layer soil | m |

7 | Dep _{3} | The depth of lower layer soil | m |

8 | x | The regulation capacity of river channel for streamflow | / |

9 | k | The propagation time of steady flow in river channel | h |

10 | ckg | The regulation capacity of slope land for base flow | / |

Number | Variable | Description | Units |
---|---|---|---|

1 | b | The shape of the variable infiltration capacity curve | / |

2 | D _{m} | The maximum base flow from the lowest soil layer | mm/day |

3 | D _{s} | The fraction of D where non-linear base flow begins _{m} | / |

4 | W _{s} | The fraction of the maximum soil moisture where non-linear base flow occurs | / |

5 | Dep _{1} | The depth of top layer soil | m |

6 | Dep _{2} | The depth of middle layer soil | m |

7 | Dep _{3} | The depth of lower layer soil | m |

8 | x | The regulation capacity of river channel for streamflow | / |

9 | k | The propagation time of steady flow in river channel | h |

10 | ckg | The regulation capacity of slope land for base flow | / |

### Regional parameter transfer

Spatial proximity approach – The spatial proximity approach uses the parameter values from the geographically closest gauged catchment hypothesizing that neighboring catchments should behave similarly.

Physical similarity approach – The physical similarity approach transfers the entire set of parameter values from a physically similar catchment whose attributes (climatic and physical) are similar to those of the target ungauged one. The use of more information, such as RS vegetation data and soil data in the physical similarity approach can improve runoff estimates in ungauged basins.

Multiple regression analysis – The multiple regression analysis approach establishes a relationship between VIC parameter values calibrated on gauged catchments and catchment descriptors or attributes (such as climatic, vegetation, and soil data), and then the VIC parameter values for the ungauged catchments are estimated from these attributes and the established relationships. Three regression analysis equations are used to establish relationships between dependent variables (VIC parameters) and independent variables (15 climatic as well as soil characteristic variables), described as follows.

*j*th dependent variable, are independent variables, is fitting error and is assumed as . Fifteen independent variables comprising six climatic characteristic variables and nine soil characteristic variables are used, as shown in Table 2.

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 | % | |

10 | Climatic characteristic variables | T | Annual mean temperature | °C |

11 | P | Annual mean precipitation | mm | |

12 | E | Annual mean evaporation from water surface | mm | |

13 | Cv_T | Coefficient of variation for monthly temperature during 1 year | / | |

14 | Cv_P | Coefficient of variation for monthly rainfall during 1 year | / | |

15 | Cv_E | Coefficient of variation for monthly evaporation from water surface during 1 year | / |

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 | % | |

10 | Climatic characteristic variables | T | Annual mean temperature | °C |

11 | P | Annual mean precipitation | mm | |

12 | E | Annual mean evaporation from water surface | mm | |

13 | Cv_T | Coefficient of variation for monthly temperature during 1 year | / | |

14 | Cv_P | Coefficient of variation for monthly rainfall during 1 year | / | |

15 | Cv_E | Coefficient of variation for monthly evaporation from water surface during 1 year | / |

For a detailed description of methods for weather forcing data, vegetation dataset, and soil dataset based on regionalization and grid in this paper, readers are referred to Xie *et al.* (2007).

### Hydrological probability prediction – Bayesian model averaging

BMA is a statistical technique designed to infer a prediction 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 accounting for between-scheme and within-scheme uncertainty (Ajami *et al.* 2007). Below is a brief description of the basic ideas of this method.

*Q*to be predicted on the basis of input data

*D*= [

*I*,

*O*] (

*I*denotes the input forcing data, and

*O*stands for the observational flow data).

*f*= [

*f*,

_{1}*f*, …,

_{2}*f*] is the ensemble of the

_{k}*K*-member predictions. The probabilistic prediction of BMA is given by where is the posterior probability of the prediction given the input data

*D*and reflects how well the scheme fits

*Y*. Actually is just the BMA weight , and better performing predictions receive higher weights than poorer performing ones, all weights are positive and should add up to 1. is the conditional probability density function (PDF) of the prediction

*Q*conditional on and

*D*. For computation convenience, is always assumed to be a normal PDF and is represented as , where is the variance associated with scheme prediction and observations

*O*. In order to make this assumption valid, some techniques such as Box-Cox transformation are needed to make the data approximately normally distributed and to narrow the data range (Poirier 1978).

### EM algorithm for BMA parameter estimation

To estimate BMA weight and scheme prediction variance , the expectation-maximization (EM) algorithm, which has proved to be an efficient technique for BMA calculation based on the assumption that *K*-member predictions are normally distributed, is described in this section (Duan *et al.* 2007).

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 steps are iterated to convergence when there is no significant change between two consecutive iterative log-likelihood estimations. In the EM algorithm, a latent variable (unobserved quantity) is used as an assistant for estimating BMA weight . For a detailed description of the EM algorithm for a BMA scheme, readers are referred to Dong *et al.* (2013b).

### Estimation of probabilistic prediction

After estimating BMA weight and prediction variance , we use the Monte Carlo method to generate BMA probabilistic prediction for any time *t* (Hammersleym & Handscomb 1975). The procedures are described as follows.

Generate an integer value of k from [1, 2, …,

*K*] with probability . A specific procedure is described as follows.1(a). Set the cumulative weight and compute for

*k*= 1, 2, …,*K*.1(b). Generate a random number

*u*between 0 and 1.1(c). If , this indicates that we choose the

*k*th member of the ensemble predictions.

Generate a value of from the PDF of . Here, represents the normal distribution with mean and variance .

Repeat the above steps (1) and (2) for M times. M is the probabilistic ensemble size. In this paper, we set M = 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.

For each individual scheme in the BMA model, the prediction uncertainty interval can also be constructed, with the Monte Carlo sampling method still being used to approximate the assumed PDF of .

### Performance evaluation indices

#### Performance evaluation indices for mean prediction

There are three indices for evaluating the mean prediction (Dong *et al.* 2013b) presented as follows.

- The Nash-Sutcliffe coefficient of efficiency (NS) – NS is not only an objective function but also a widely used performance criterion. It ranges from minus infinity to 1.0, with higher values indicating better agreement. The definition of NS is expressed in following equation. where and are observed and simulated data at time
*t*, is the average of observed data,*T*is the length of the data series.

#### Performance evaluation indices for probabilistic prediction

Xiong *et al.* (2009) and Dong *et al.* (2013b) presented a set of indices for assessing the probabilistic prediction generated by the uncertainty analysis methods. Three main indices are selected here to assess the probabilistic prediction produced by the BMA model as well as from each individual parameter transfer scheme.

- Containing ratio (CR) – The containing ratio is used for assessing the goodness of the uncertainty interval. It is defined as the percentage of observed data points that are covered in the prediction bounds. where and denote as upper and lower prediction bounds at time
*t*, and is the number of observed data points that are covered in the prediction bounds.

## RESULTS AND DISCUSSION

### Calibration results

Daily streamflow and weather data from 1980 to 1986 are used for calibration. The gauged sub-basins are selected as the primary basins to implement VIC-3L model calibration, which is achieved by matching the total annual streamflow 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 calibration. In the calibration study, the parameters of individual sub-basins with similar climate characteristics and underlying surface are assumed to have the same values. Ten hydrological parameters in the VIC-3L model have been calibrated for the 10 primary sub-basins. Table 3 shows the calibrated parameter values in 10 primary sub-basins in Ankang basin. Their typical ranges and the effect of each parameter on results of simulated streamflow are described below. (1) *b* typically ranges from 0 to 0.50. It describes the total of available infiltration capacity as a function of the relative saturated grid cell area and controls the quantity of runoff generation directly and the water balance. A lower value of *b* gives lower infiltration and yields higher surface runoff (the value of *b* in this paper is the inverse value of *b* in Xie *et al.* 2007). The highest value of *b* in sub-basins 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) *D _{m}* typically ranges from 0 to 6 mm day

^{−1}. (3)

*D*typically ranges from 0 to 1. With a higher value of

_{s}*D*, the base flow will be higher at lower water content in the lowest soil layer. (4)

_{s}*W*typically ranges from 0 to 1. (5)

_{s}*Dep*,

_{1}*Dep*, and

_{2}*Dep*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)

_{3}*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.

Gauge stations | b | D_{s} | D_{m} (mm/day) | W_{s} | Dep_{1} (m) | Dep_{2} (m) | Dep_{3} (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 |

Gauge stations | b | D_{s} | D_{m} (mm/day) | W_{s} | Dep_{1} (m) | Dep_{2} (m) | Dep_{3} (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 |

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 sub-basins. In the next section, we will demonstrate that these calibrated hydrological parameters can be transferred to the ungauged sub-basins with reasonably good results.

Gauge stations | Calibration | ||
---|---|---|---|

NS (%) | DRMS (m^{3}/s) | RE (%) | |

Baohe | 90.79 | 30.67 | −14.00 |

Hanzhong | 91.81 | 23.31 | −3.90 |

Xushuihe | 85.11 | 29.70 | −5.08 |

Youshuihe | 88.29 | 32.14 | −7.00 |

Ziwuhe | 86.87 | 29.82 | −6.36 |

Shiquan | 89.27 | 31.51 | 11.00 |

Chihe | 84.38 | 40.17 | 2.71 |

Zhehe | 86.02 | 31.90 | 13.00 |

Lanhe | 85.65 | 32.89 | 4.83 |

Ankang | 87.94 | 29.32 | 12.00 |

Gauge stations | Calibration | ||
---|---|---|---|

NS (%) | DRMS (m^{3}/s) | RE (%) | |

Baohe | 90.79 | 30.67 | −14.00 |

Hanzhong | 91.81 | 23.31 | −3.90 |

Xushuihe | 85.11 | 29.70 | −5.08 |

Youshuihe | 88.29 | 32.14 | −7.00 |

Ziwuhe | 86.87 | 29.82 | −6.36 |

Shiquan | 89.27 | 31.51 | 11.00 |

Chihe | 84.38 | 40.17 | 2.71 |

Zhehe | 86.02 | 31.90 | 13.00 |

Lanhe | 85.65 | 32.89 | 4.83 |

Ankang | 87.94 | 29.32 | 12.00 |

### Testing of parameter transfer

#### Parameter transfer schemes

Three regionalization approaches, i.e., spatial proximity approach, physical similarity approach, and multiple regression analysis are used to choose the 10 donor gauged sub-basins whose optimized parameter values are used to model daily runoff for the two ungauged sub-basins (Mumahe and Renhe). Table 5 lists the transferred parameter values of the three regionalization approaches. The spatial proximity sub-basins for Mumahe and Renhe are Youshuihe and Zhehe, respectively, as shown in Figure 1. The physical similarity sub-basins for Mumahe and Renhe are Chihe and Lanhe, respectively. Figure 2 shows the distribution map of vegetation characteristics for the VIC-3L model with 5 × 5 km grid in the Ankang basin. Figure 3 shows the distribution map of climatic factors with 5 × 5 km grid in the Ankang basin. The vegetation similarity sub-basins for Mumahe and Renhe are Chihe and Lanhe, respectively, in Figure 2. The annual mean temperatures (*T*) for Mumahe and Chihe are 14.6 °C and 14.8 °C, respectively. The annual mean temperatures (*T*) for Renhe and Lanhe are 15.2 °C and 15.5 °C, respectively. The annual mean precipitation (*P*) for Mumahe and Chihe are 1,070 mm and 997 mm, respectively. The annual mean precipitation (*P*) for Renhe and Lanhe are 1,021 mm and 1,068 mm, respectively. The annual mean evaporation from water surface (*E*) for Mumahe and Chihe are 564 mm and 514 mm, respectively. The annual mean evaporation from water surface (*E*) for Renhe and Lanhe are 246 mm and 268 mm, respectively. The climatic similarity sub-basins for Mumahe and Renhe are Chihe and Lanhe, respectively, as in Figure 3.

Gauge stations | Schemes | b | D_{s} | D_{m} (mm day^{−1}) | W_{s} | Dep_{1} (m) | Dep_{2} (m) | Dep_{3} (m) | x | k | ckg |
---|---|---|---|---|---|---|---|---|---|---|---|

Mumahe | A (Youshuijie) | 0.257 | 0.428 | 2.580 | 0.559 | 0.053 | 0.250 | 0.010 | 0.050 | 0.555 | 0.695 |

B (Chihe) | 0.258 | 0.421 | 2.569 | 0.701 | 0.026 | 0.300 | 0.010 | 0.247 | 0.721 | 0.588 | |

C | 0.258 | 0.121 | 1.569 | 0.801 | 0.056 | 0.150 | 0.010 | 0.247 | 0.921 | 0.588 | |

Renhe | A (Zhehe) | 0.157 | 0.428 | 1.040 | 0.559 | 0.073 | 0.110 | 0.010 | 0.208 | 0.995 | 0.550 |

B (Lanhe) | 0.450 | 0.631 | 4.680 | 0.489 | 0.085 | 0.005 | 0.010 | 0.250 | 0.905 | 0.697 | |

C | 0.250 | 0.491 | 0.580 | 0.589 | 0.085 | 0.005 | 0.010 | 0.250 | 0.955 | 0.697 |

Gauge stations | Schemes | b | D_{s} | D_{m} (mm day^{−1}) | W_{s} | Dep_{1} (m) | Dep_{2} (m) | Dep_{3} (m) | x | k | ckg |
---|---|---|---|---|---|---|---|---|---|---|---|

Mumahe | A (Youshuijie) | 0.257 | 0.428 | 2.580 | 0.559 | 0.053 | 0.250 | 0.010 | 0.050 | 0.555 | 0.695 |

B (Chihe) | 0.258 | 0.421 | 2.569 | 0.701 | 0.026 | 0.300 | 0.010 | 0.247 | 0.721 | 0.588 | |

C | 0.258 | 0.121 | 1.569 | 0.801 | 0.056 | 0.150 | 0.010 | 0.247 | 0.921 | 0.588 | |

Renhe | A (Zhehe) | 0.157 | 0.428 | 1.040 | 0.559 | 0.073 | 0.110 | 0.010 | 0.208 | 0.995 | 0.550 |

B (Lanhe) | 0.450 | 0.631 | 4.680 | 0.489 | 0.085 | 0.005 | 0.010 | 0.250 | 0.905 | 0.697 | |

C | 0.250 | 0.491 | 0.580 | 0.589 | 0.085 | 0.005 | 0.010 | 0.250 | 0.955 | 0.697 |

*Note:* Scheme A denotes spatial proximity approach, Scheme B denotes physical similarity approach, and Scheme C denotes multiple regression analysis.

Table 6 shows multiple regression analysis results for 10 donor gauged sub-basins. The regression analysis equation is tested by the statistic (*m* is the number of regression variables, *n* is the number of sub-basins; Pope & Webster 1972). The structures of regression analysis equations for six hydrological parameters *b*, *D _{m}*,

*D*,

_{s}*W*,

_{s}*Dep*, and

_{1}*Dep*are square root, square root, square root, linear, linear and square root, respectively. However, the remaining four hydrological parameters,

_{2}*Dep*,

_{3}*x*,

*k*, and

*ckg*, have no remarkable regression analysis equations because these hydrological parameters are affected by many model variables and catchment descriptors or attributes. In terms of

*NS*, the regression analysis equations can provide reasonably good fitting results between parameter values calibrated on gauged catchments and climatic as well as vegetation variables. The reasonably good fitting results can be demonstrated by the fitting curves between calibrated results and regression analysis results of hydrological parameters in the VIC-3L model, as shown in Figure 4. The distribution map of six hydrological parameters in the VIC-3L model with a 5 × 5 km grid in the Ankang basin is shown in Figure 5 (calibrated hydrological parameters for 10 primary sub-basins, as well as regression analysis results of hydrological parameters for Mumahe and Renhe ungauged sub-basins).

Number | Variable | b | D_{s} | D_{m} | W_{s} | Dep_{1} | Dep_{2} |
---|---|---|---|---|---|---|---|

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 | √ | √ | √ | |||

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 | |

R^{2} (%) | 66.73 | 84.91 | 75.80 | 69.74 | 76.63 | 62.84 |

Number | Variable | b | D_{s} | D_{m} | W_{s} | Dep_{1} | Dep_{2} |
---|---|---|---|---|---|---|---|

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 | √ | √ | √ | |||

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 | |

R^{2} (%) | 66.73 | 84.91 | 75.80 | 69.74 | 76.63 | 62.84 |

### Evaluation for mean prediction and probabilistic prediction

Figure 6 displays the weight estimates of individual parameter transfer schemes in BMA (3). We check the mean prediction of BMA (3) using three criteria illustrated in the section ‘Performance evaluation indices for mean prediction’. Results of BMA (3) and its three individual parameter transfer schemes in the mean prediction of two ungauged sub-basins for the whole flow series are shown in Figure 7. In terms of *NS*, the mean prediction of BMA (3) for Mumahe and Renhe 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 ‘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 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 parameter transfer schemes of maximum one month hydrograph for Baohe sub-basin in 1983 during the calibration period, respectively. The observations of 1983 are presented by dots, and the mean predictions of 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 any of its individuals. It can be found from Figure 9 that the results of validation are similar to that of the calibration period. In a word, the probabilistic prediction of BMA (3) has better performance than its individual parameter transfer schemes for the flow series.

## CONCLUSIONS

In this paper, the BMA method is used to predict a new measurement value associated with a combination of probabilistic prediction in ungauged basins based on three individual parameter transfer schemes. The comparison between BMA (3) and its three individual parameter transfer schemes is made in terms of both mean 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; and (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 models, then it is anticipated that the advantages of BMA can be generalized.

## 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).