The hydro-stochastic interpolation method based on traditional block Kriging
has often been used to predict mean annual runoff in river basins. A caveat
in such a method is that the statistic technique provides little physical
insight into relationships between the runoff and its external forcing, such
as the climate and land cover. In this study, the spatial runoff is
decomposed into a deterministic trend and deviations from it caused by
stochastic fluctuations. The former is described by the Budyko method (Fu's
equation) and the latter by stochastic interpolation. This coupled method is
applied to spatially interpolate runoff in the Huaihe River basin of China.
Results show that the coupled method significantly improves the prediction
accuracy of the mean annual runoff. The error of the predicted runoff by the
coupled method is much smaller than that from the Budyko method and the
hydro-stochastic interpolation method alone. The determination coefficient
for cross-validation,
The runoff observed at the outlet of a basin is a crucial element for investigating the hydrological cycle of the basin. Because runoff is influenced by both deterministic and stochastic processes, estimating the spatial patterns of runoff and the associated distribution of water resources in ungauged basins has been one of the key problems in hydrology (Sivapalan et al., 2003), and a thorny issue in water management and planning (Imbach, 2010; Greenwood et al., 2011).
In estimating and predicting runoff and regional water resources availability, we have often used regional or global runoff mapping and geostatistical interpolation methods. In these methods, the value of a regional variable at a given location is often estimated as the weighted average of observed values at neighboring locations. This interpolation of runoff, which is assumed as an auto-correlated generalized stochastic field (Jones, 2009), uses secondary information from more than one variable (Li and Heap, 2008). Spatial autocorrelations of the runoff values are measured by the covariance or semi-variance between the runoffs at pairs of locations as a function of their Euclidian distance (such as in ordinary Kriging). The values obtained by the interpolation methods are the best linear unbiased estimate in the sense that the expected bias is zero and the mean squared error is minimized (Skøien et al., 2006). Ordinary Kriging (OK) estimates the local mean as a constant; corresponding residuals are considered random. Because the spatial mean could also be used as a trend or nonstationary variation in space, OK has been developed into various geostatistical interpolation methods, such as Kriging with a trend by incorporating a local trend within a confined neighborhood as a smoothly varying function of the coordinates. Block Kriging (BK) is another extension of OK for estimating a block value instead of a point value by replacing the point-to-point covariance with point-to-block covariance (Wackernagel, 1995).
Unlike precipitation or evaporation, which we often interpolate to find their values at specific locations, runoff is an integrated spatially continuous process in river basins (Lenton and Rodriguez-Iturbe, 1977; Creutin and Obled, 1982; Tabios and Salas, 1985; Dingman et al., 1988; Barancourt et al., 1992; Blöschl, 2005). Streamflows are naturally organized into basins (Dooge, 1986; Sivapalan, 2005); e.g., rivers flow through sub-basins. The river network constrains the water paths from upstream to downstream in a basin. The hierarchically organized river network requires that the sum of the interpolated discharge from sub-basins equals the observed runoff at the outlet of the entire basin. Previous studies have indicated that runoff interpolation may overestimate the actual runoff without adequate information of the spatial variation of the runoff (Arnell, 1995), e.g., neglecting the river network in connecting sub-basins or processing basin runoff at collective points in space (Villeneuve et al., 1979; Hisdal and Tveito, 1993). In nested basins, Gottschalk (1993a, b) developed a hydro-stochastic method to interpolate runoff. It uses the concept that runoff is an integrated process in the hierarchical structure of a river network. The distance between a pair of basins is measured by geostatistical distance instead of the Euclidian distance. The covariogram among points in the conventional spatial interpolation is replaced by the covariogram between basins. In this concept, runoff is assumed spatially homogeneous in basins; i.e., the expected value of the runoff is constant in space (Sauquet, 2006). The observed patterns of runoff reveal systematic deviations from the homogeneity assumption, however, because of the influences from the heterogeneous climate and underlying surface factors.
An alternate method is to describe the hydrological variables of interest in deterministic forms of functions, curves, or distributions, and construct conceptual and mathematical models to predict hydro-climate variability (Wagener et al., 2007). Qiao (1982), Arnell (1992), and Gao et al. (2017) have used such an approach and derived empirical relationships between runoff and its controlling factors of the climate, land cover, and topography in various basins. However, the deterministic method for describing complex runoff patterns suffers from an inevitable loss of information (Wagener et al., 2007) because of the existence of uncertainty in many hydrological processes and especially in observations. Thus, hydrological variables also contain information of a stochastic nature and should be treated as outcomes from deterministic and stochastic processes. A method that combines both deterministic patterns and stochastic variability is Kriging with an external drift (KED) (Goovaerts, 1997; Li and Heap, 2008; Laaha et al., 2013). It takes the deterministic patterns of spatial variables into account and incorporates them as a local trend of a smoothly varying secondary variable, instead of a function of the spatial coordinates.
The inclusion of deterministic terms in the geostatistical methods has been shown to increase the interpolation accuracy of basin variables, such as mean annual runoff (Sauquet, 2006), stream temperature (Laaha et al., 2013), and groundwater table (Holman et al., 2009). Those deterministic terms are often described by empirical formulae linking spatial features, e.g., variability of the mean annual runoff in elevation (Sauquet, 2006), and the relationship between the mean annual stream temperature and the altitude of gauges (Laaha et al., 2013). As a semi-empirical approach to model the deterministic process of the runoff, the Budyko framework has been popularly used to analyze the relationship between mean annual runoff and the climatic factors, e.g., aridity index (Milly, 1994; Koster and Suarez, 1999; Zhang et al., 2001; Donohue et al., 2007; Li et al., 2013; Greve et al., 2014). Many efforts have been devoted to improving the Budyko method by, for example, including the effects of other external forcing factors, such as land cover (Donohue et al., 2007, 2012; Li et al., 2013; Han et al., 2011; Yang et al., 2007), soil properties (Porporato et al., 2004), topography (Shao et al., 2012; Xu et al., 2013; Gao et al., 2017), hydro-climatic variations of seasonality (Milly, 1994; Gentine et al., 2012; Berghuijs et al., 2014), and groundwater (Istanbulluoglu et al., 2012). However, it has been found that the use of the deterministic equation in the Budyko method alone still comes with large errors in the prediction of runoff in many basins (e.g., Potter and Zhang, 2009; Jiang et al., 2015).
The aim of this study is to combine the stochastic interpolation with the semi-empirical Budyko method to further improve the spatial interpolation/prediction of the mean annual runoff in the Huaihe River basin (HRB), China. In this study, the spatial runoff from sub-basins in the HRB is separated into a deterministic trend and its residuals, which are estimated by the Budyko method and the interpolation method, respectively. The residuals are calculated as the difference between the observed and estimated runoff from the Budyko method, and are used in the stochastic interpolation as described in Gottschalk (1993a, b, 2006). After that, the runoff of any sub-basin is predicted as the sum of the interpolated residuals and the Budyko estimated value. The improved method is tested in the HRB. In addition, the leave-one-out cross-validation approach is applied to evaluate and compare the performances of the three interpolation methods: the Budyko method, hydro-stochastic interpolation, and our coupled Budyko and stochastic interpolation method.
The Budyko method explains the variability of mean annual water balance on a
regional or global scale. It describes the dependence of actual
evapotranspiration (
The parameter
Gottschalk (1993a) described the hydro-stochastic interpolation method based on the Kriging method to predict spatial runoff. Gottschalk's method redefines a relevant distance between basins, and identifies the river network and supplemental water balance constraints as follows.
As a spatially integrated continuous process, the predicted runoff of a
specific unit of an area
The weights are obtained by solving the following set of equations under the
second-order stationary assumption for hydrologic variables (Ripley, 1976),
The sum of the interpolated runoff for each non-overlapping sub-basin should
be equal to the observed runoff at the river outlet. This constraint can be
written as
The theoretical covariogram, Cov(AB), is derived by averaging the point
process covariance function Cov
The distance
The above stochastic interpolation procedure assumes a stationary stochastic
variation of the runoff among sub-basins or spatial homogeneity in runoff
(Sauquet, 2006), despite variations in river networks. For nonstationary
variations in the runoff resulting from spatial heterogeneity in a river
network, the spatial runoff can be decomposed into a nonstationary
deterministic component and a stochastic component:
In this study,
To validate this prediction procedure, we use the leave-one-out
cross-validation method (Kearns and Ron, 1999). In addition to quantifying the performance of our
coupled Budyko and hydro-stochastic interpolation method, we compare and
contrast its performance with the Budyko and hydro-stochastic interpolation
methods alone. Their performances are evaluated by the following metrics
(Laaha and Blöschl, 2006):
The HRB, the sixth largest river basin in China, is used in evaluation of our
coupled model and in its comparison to the other two methods. The HRB has a
strong precipitation gradient from the humid climate in the east and the
semi-humid climate in the west (Hu, 2008). It is one of the major
agricultural areas in China, with the highest human population density in the
country. About 18 billion m
Our study area is in the upstream of the Bengbu Sluice in the HRB and is
121 000 km
The topography and river network of the study area.
Annual precipitation data used in this study are from 1961 to 2000 and are
obtained from a monthly mean climatological dataset at 0.5
Summary of hydro-meteorological data and predicted runoff of the sub-basins in the HRB.
The sub-basins and hydrological stations in the study area.
Actual evapotranspiration
Using
Using
For comparison, the observed runoff is used in the hydro-stochastic
interpolation following the procedure detailed in Sect. 2.2. In order to
obtain the distance
Empirical covariogram (Cov
The interpolated runoff depth (
Interpolation cross-validation errors between the predicted and observed runoff in the 40 sub-basins in the HRB from the three methods.
We use Fu's equation, Eq. (2), to evaluate the deterministic trend or
the external drift function,
The empirical residual covariogram of
Cross-validation of the predicted runoff vs. the observation by
The predicted runoff using this procedure is given in Table 1, with the MAE at 71 mm and RMSE at 93 mm over the 40 sub-basins. The largest absolute error is in the sub-basin QL (220 mm), and the smallest in ZM (4 mm) (Table 2). The largest relative error is 47.2 % of the observed runoff in XZ, and the smallest is 1 % of the observed runoff in BLY. They represent the absolute errors of 52 and 8 mm, respectively.
Comparing the results in Table 2, we find that our coupled method of the
deterministic and stochastic processes substantially reduces the runoff
prediction error in the HRB. The MAE and RMSE of the runoff
from our coupled method are much smaller than those from the Budyko or the
hydro-stochastic interpolation method. In cross-validation (Table 2), our
coupled method has
Our correlation analysis between the predicted and observed
The spatial distributions of the runoff in the HRB calculated from the three methods are shown in Fig. 7. They again show significant differences. Compared to the result from our coupled method (Fig. 7c), the Budyko method overestimates the runoff in most of the northern sub-basins (Fig. 7a), where the climate is relatively dry and runoff is small (ranging from 140 to 280 mm). The hydro-stochastic interpolation method underestimates the runoff in some southern sub-basins (Fig. 7b), where the wet climate has fostered extremely high runoff (800–1100 mm), such as in the sub-basins HWH, BLY, and ZC (Table 1). The results from our coupled method are closest to the observed distribution of the runoff among the three methods (Fig. 7d). Compared to the errors in the predicted runoff by the Budyko method and the hydro-stochastic interpolation (Fig. 7 and Table 1), our coupled method reduces the error in 70 % of all the sub-basins (28 of the 40 sub-basins).
Spatial distribution of the mean annual runoff estimated from
In this study, we use the Budyko deterministic method to describe the mean annual runoff, which is an integrated spatially continuous process and determined by both the hydro-climatic elements and the hierarchical river network. A deviation from the Budyko estimated runoff is used by the stochastic interpolation that assumes spatially auto-correlated error. The deterministic aspects of the runoff described in the Budyko method are reflected in the trends at locations (sub-basins), and deviations from the trends caused by the stochastic processes are described by the weights as a function of the autocorrelation and distance. Information from both the Budyko method and the stochastic interpolation are integrated into our coupled method to predict the runoff.
Different from the universal Kriging method, in which the trend is
represented as a linear function of coordinate variables and determined
solely through spatial data calibration (i.e., semi-variogram analysis), the
Budyko method couples water and energy balance and could directly predict
streamflow in ungauged basins. This physically based method relies on using
the spatial trend of runoff and, in our study, it yields the deterministic
coefficient of cross-validation,
Incorporating secondary information into the geostatistical methods improves
the estimate of a predictive variable, e.g., the estimate of groundwater
level by incorporating topography into the collocated co-Kriging (Boezio et
al., 2006), or the estimate of mean annual stream temperature by
incorporating a nonlinear relationship between the mean annual stream
temperature and altitude of the stream gauge into the top Kriging (Laaha et
al., 2013). By incorporating such secondary information and the relationship
between the mean runoff and the climate conditions (the aridity index) into
the Budyko method through coupling with the hydro-stochastic interpolation,
we develop our new coupled Budyko–hydro-stochastic interpolation method. It
can substantially improve the prediction of streamflow in ungauged basins.
This improvement is shown by the higher
While substantial progress has been made by our coupled method, its results show room for improvement to further increase the accuracy of runoff prediction. For example, runoff prediction errors remain large from our coupled method in some sub-basins in the HRB. In the sub-basins MS, QL, HWH, and HNZ, the absolute error of predicted runoff is larger than 150 mm and the relative error of predicted runoff is larger than 20 % of the observed runoff. In the sub-basins BGS and XZ, the relative error of the predicted runoff is larger than 40 % of the observed runoff. These errors are largely attributable to large prediction errors intrinsic to the Budyko method (e.g., MS, QL, HWH, and XZ in Table 1). Possible causes of the errors could be from additional external factors influencing the runoff, such as land cover, soil properties, hydro-climatic variations, and the groundwater. Including some or all of these effects to improve the Budyko method or incorporating these effects as secondary information (e.g., multi-collocated co-Kriging) into our coupled model would help aid our understanding of the deterministic processes and increase the runoff prediction accuracy.
The precipitation dataset can be accessed at
The authors declare that they have no conflict of interest.
We thank the editor Erwin Zehe and the reviewers Mirko Mälicke and Jon Olav Skøien for their valuable comments and suggestions that helped improve this paper substantially. The research was supported by the National Natural Science Foundation of China (nos. 51190091 and 41571130071). Qi Hu's contribution was supported by USDA Cooperative Project NEB-38-088. Edited by: Erwin Zehe Reviewed by: Mirko Mälicke and Jon Olav Skøien