Fractality has been found in many areas and has been used to describe the internal features of time series. But is it possible to use fractal theory to improve the performance of hydrological models? This study aims at investigating the potential benefits of applying fractal theory in model calibration. A new criterion named the ratio of fractal dimensions (RD) is defined as the ratio of the fractal dimensions of simulated and observed streamflow series. To combine the advantages of fractal theory with
classical criteria based on squared residuals, a multi-objective calibration strategy is designed. The selected classical criterion is the Nash–Sutcliffe efficiency (

Ever since the first hydrological model was developed, appropriate methods of evaluating the performance of such models have been sought by the hydrological community, and a large variety of efficiency criteria have been proposed and used over the years. Most of these criteria are based on squared residuals or absolute errors (Pushpalatha et al., 2012). Krause et al. (2005) compared nine efficiency criteria, including the correlation coefficient (

Chiew and McMahon (1993) classified calibration criteria into statistical parameters and dimensionless coefficients. Statistical parameters include the mean value, standard deviation, coefficient of skewness, coefficient of variance and quantile points. Most dimensionless coefficients (such as the Pearson correlation coefficient (

Another deficiency of existing criteria is the preference for particular parts of the hydrograph. For example, statistical parameters are easily influenced by extreme individuals and have large uncertainties (Westerberg and McMillan, 2015). Coefficients provide a measure of the overall agreement between simulation and observation, but are still significantly influenced by particular parts of the hydrograph. High flows make a significant contribution to the values of

First introduced by Hurst in 1951, the fractality of streamflow series has been studied for decades (Hurst, 1951). There has been spectacular growth in various areas of fractal theory and multifractal theory (Bai et al., 2019; Davis et al., 1994). According to fractal theory, fractality can be described by the Hurst exponent (rescaled range analysis) (Hurst, 1951), the Hausdorff dimension (the box-counting dimension or local dimension) (Karperien et al., 2008; Falconer, 2004) and the correlation dimension (Grassberger and Procaccia, 1983), for example. These dimensions differ in the schemes used to calculate them, but they are numerically related to and theoretically dependent on each other. While the Hurst exponent calculated with rescaled range analysis is more widely used, the Hausdorff dimension can easily be extended to multifractal analysis and has prospective applications in hydrology (Bai et al., 2019; Zhou et al., 2014). The fractality of a time series is generally considered to reflect its self-affinity, periodicity, long-term memory and irregularity (Bai et al., 2019; Hurst, 1951; Mandelbrot, 2004). Self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the

Unlike a typical statistical evaluation of fluctuation (such as the standard deviation and the distribution function), the Hausdorff dimension takes the order of the data into account. Therefore, compared to classical criteria that are used to compare observed and simulated water balances, the Hausdorff dimension can offer useful insights into the mechanisms that control extreme hydrological events (including floods, droughts and low flows) (Radziejewski and Kundzewicz, 1997). Another difference between fractal dimensions and classical criteria is the influence of an individual datum (or a small number of data). While closer simulated individual data and observed data make the coefficients better, it can shift the Hausdorff dimension of the simulated data closer to or farther from that of the observed data. Thus, to reproduce all the characteristics of the observed streamflow, the simulated and observed streamflows should have similar Hausdorff dimensions and other traditional metrics. Taking all of this into account, the Hausdorff dimension is used for hydrological model calibration in the present study.

Since the fractal dimension describes the fractality of a streamflow series and two different series may have the same fractal dimension, the fractal dimension cannot be used to calibrate a hydrological model independently. Multi-objective optimization approaches are widely used by the hydrological community (Harlin, 1991; Yapo et al., 1998; Liu et al., 2017, 2019; Pan et al., 2017; Shafii and Tolson, 2015). This has resulted in the use of some noncomprehensive but effective criteria as targets, such as the aforementioned hydrological signatures (Shafii and Tolson, 2015; Westerberg and McMillan, 2015), statistical targets and fractal criteria. Nonetheless, the strategy of using the Hausdorff dimension to calibrate hydrological models has not been studied.

In the present study, a new criterion – the ratio of fractal dimensions (RD) – is introduced, as well as a new calibration strategy. The criterion and calibration strategy consider the self-affinity, periodicity, long-term memory and irregularity of the hydrograph during model calibration. Three catchments with different climates and geographies are used as case studies. The aim of this study is to examine the applicability of RD as one of the targets of multi-objective calibration and to explore the effects on hydrological model performance when RD is considered. Section 2 describes differences between RD and classical criteria and how RD is used in calibration (the

The box-counting method used to calculate the Hausdorff dimension is based on the idea of separating data into boxes and counting the resulting number of boxes (Mandelbrot, 2004). When adopted to analyse time series, the box-counting method sums adjacent data (places adjacent individuals into boxes) and investigates the effect of changing the resolution (the size of a box) on the results. Figure 1 graphically shows how the box-counting method works with time series. Figure 1a shows how the number of boxes needed to cover all the data (

Flow chart for the use of the box-counting method to calculate the Hausdorff dimension of a time series.

As stated before, the observed and simulated streamflow series should have the same Hausdorff dimension. In this study, a new criterion, the ratio of fractal dimensions (RD), is defined as follows:

Obviously, given that RD is a metric of the deviation of the self-affinity of the simulated streamflow series from that of the observed series, it cannot be used to evaluate the performance of a hydrological model by itself. An immediate thought is to combine RD with another statistical criterion in model calibration. The statistical criterion to be combined
with RD should have three features. Firstly, the statistical criterion should be able to evaluate model
performance in terms of the water balance to some extent. Secondly,
the statistical criterion should evaluate the response of the streamflow to
meteorological forcing. Thirdly, the criterion should calculate model errors over the entire test period. These features ensure that the basic requirements of the strategy
are fulfilled. An additional requirement for the statistical criterion
used in this study is that it is popular within the hydrological community. Therefore, in this study,

The Nash–Sutcliffe efficiency coefficient (

In this work, a set of experiments are performed to illustrate the benefits of using the
proposed

Flow chart of the

The value of the Hausdorff dimension of a particular time series may vary with the resolution applied. This implies that the self-affinity of the time series changes as the resolution changes and, for hydrological processes, the dominant driver of those processes changes. For example, the dominant drivers of the daily and annual temperature cycles are different. The Hausdorff dimension of a joint data series can be used to verify the simulation of the freezing–thawing process that reveals the complex relationship between the hydrological variables (Bai et al., 2019). According to this idea, the Hausdorff dimension determines whether the streamflow components are reasonably simulated. In the present study, the largest temporal resolution is set to 365 d (1 year) in order to eliminate interannual drivers. It is believed that the resulting resolution range is large enough for the Hausdorff dimension to accurately reflect the drivers of hydrological processes.

A small catchment in Tibet named Dong, a medium-sized catchment in southeastern China named Jinhua, and a large catchment located in the middle reach of the Yangtze River and named Xiang are examined in this study.

Dong is a small tributary of the Yarlung Zangbo River, with elevations ranging from 3512 to 5869 m. The area of the Dong catchment is about 43.6 km

Digital elevation models of the study areas.

Jinhua River is a 5536 km

Xiang River is one of the largest tributaries of the Yangtze River, which flows into Dongting Lake, the second largest freshwater lake in mid-China. The area of the Xiang catchment is about 82 400 km

Figure 3 shows the topographies of all the study areas.

The HBV model is a conceptual rainfall-runoff model originally developed by the Swedish Meteorological and Hydrological Institute (SMHI) (Bergström, 1976; Bergström, 1992; Lindström et al., 1997). This model has been successfully used in many studies (Seibert and Vis, 2012; Tian et al., 2015, 2016). The HBV model is composed of precipitation and snow accumulation routines, a soil moisture routine, a quick runoff routine, a baseflow routine and a transform function. The HBV model takes into account the effect of snow melting and accumulation, which is significant in the Dong catchment. The actual evapotranspiration is calculated with a linear function. Two conceptual runoff reservoirs – the upper reservoir and the lower reservoir – are included in the HBV model.

A controlled and elitist genetic algorithm (a variant of NSGA-II) (Deb, 2001) is applied in model calibration. A controlled and elitist GA favours individuals with high fitness values (ranks) as well as individuals that can help increase the diversity of the population, even if they have low fitness values. An important feature of this genetic algorithm is that the individual with the best performance according to any criterion is retained with the lowest rank. This ensures that when using a multi-objective genetic algorithm, the parameter set with the best possible

Since HBV has 14 parameters to calibrate, the number of generations is 2800. Each generation has a population of 600. The crossover fraction is set to 0.8 (average). The Pareto fraction is set to 0.2 (average). The population migrates every 20 generations, and the migration fraction is set to 0.5. These settings ensure that the population is not trapped in a local optimum, which is important because RD has a wider range than traditional criteria. Most of these settings are the default settings applicable to most problems. Only the population of each generation (600) is larger than the default (200) to better represent the Pareto front of the optimization. The meanings of the settings can be found in Deb (2001).

All 600 Pareto-optimized solutions for the last generation are used in the
following analysis. In GA optimization with the

Several tools are utilized to investigate the effects of using RD in hydrological model evaluation.

Pearson's correlation coefficient (

To elucidate how the model is altered when RD is used as one of the objectives, the relationships between the parameters and RD are analysed. The
distance correlation

To study the influences of specific parts of the hydrograph on the simulation when using RD, the fast flow and baseflow are analysed separately. The HBV model is slightly modified to output the simulated fast flow and baseflow at every time step. Observed streamflow series are divided into the fast flow and the baseflow using the Water Engineering Time Series PROcessing tool (WETSPRO tool) introduced by Willems (2009). WETSPRO separates the fast flow from the slow flow using filter theory with several filter parameters, including the recession constant and the the average ratio of the fast flow volume to the total flow volume. The

Figure 4 shows the RD–

A single-objective calibration was performed to support an assumption made in Sect. 3.3: that, in this work, the NSGA II algorithm can find the best

Comparison of the best

The models in the Pareto front with the best RD, best

Typical examples of models: those with the best RD, the best

A precondition for adopting the RD–

Table 2 lists hydrological signatures of typical models selected by the

Hydrological signatures of the typical models in all three cases.

Table 2 shows the hydrological signatures of the observed and simulated flow series in the three cases. Most of the hydrological signatures, including the lag-1 autocorrelation, relative variation and maximum monthly flow, of the simulated series are close. The lag-1 autocorrelations of the simulated series are similar to the autocorrelations of the observed flow series. The lag-1 autocorrelations for the Dong and Xiang flow series are more than 0.9 while those for the Jinhua series are between 0.75 and 0.77. The relative variances of the flow series for the Dong and Xiang catchments are smaller than 1, while those for the Jinhua flow series are more than 1.8. These results show that catchments of different types are well simulated by the HBV model. The maximum and minimum monthly flows of the simulated and observed series are significantly different. In all three cases, the maximum monthly flows of the simulated series are similar to each other and slightly smaller than the maximum monthly flows of the observed flow series. The minimum monthly flow is the only hydrological signature used in this study that identifies the models with the best RD and the best

The parameter set in the Pareto front varies depending on the case considered. The distance correlations (

Ranges of the determinative parameters and distance correlations (

Relationship between

Relationship between

Table 3 lists the determinative parameters for the three cases. The distance correlation (

The explicit relationship between the parameters and the criteria confirms that the
effect of RD is not random. Six parameters (

Figure 7 shows the relationship between

Relationship between KF and RD in the three cases.

Response of fast flow to surface water storage. For each case, the fast
flow responses of the typical models with the best RD and

Figure 9 shows how the fast flow changes with the surface water storage
with different KF and

The baseflow (slow flow) factor (KS) is related to RD in all cases. Figure 10 shows the relationship between KS and RD. The trend in KS is the same in all three cases. However, the range of variation in KS is different for the three cases. The largest value of KS in Dong (0.153) is much larger than that in Jinhua (0.063) and Xiang (0.048). The smallest value of KS in Dong (0.016) is also larger than that in Jinhua (0.005) and in Xiang (0.010). The KS of the best

Relationship between KS and RD in the three cases.

Relationship between percolation and RD in the three cases.

Relationship between degree-day factor and RD in the three cases.

The percolation is significantly related to RD in all cases. The range of percolation in Dong is larger than in the other cases. Figure 11 shows the relationship between percolation and RD in each of the three cases. Percolation increases in Dong and decreases in the other two cases with increasing RD. The KS and percolation determine the way that HBV models the baseflow. The percolation in Dong is larger than in the other cases, which is a reflection of the arid climate of the Dong catchment. The percolation is larger in Jinhua than in Xiang because the slope in the Jinhua catchment is larger.

The degree-day factor is significantly related to RD in Xiang. However, the relationship between the degree-day factor and RD is weak in Dong and Jinhua. Figure 12 shows the relationship between the degree-day factor and RD in each catchment. The degree-day factor is smaller than 0.05 for most of the models selected for Dong, indicating that these models barely have any snowmelt runoff. When

Correlation coefficients and Nash–Sutcliffe efficiency coefficients for fast flow and baseflow, respectively, in last-generation models for the three cases.

The degree-day factors in all the models selected for Jinhua are large, but the
temperature in Jinhua is too high for snow to accumulate. The distance
correlation between the degree-day factor and RD is weak for Dong and Jinhua. The range of the degree-day factor for most of the models for Xiang is 2.8–3.4. This range is small, meaning that the difference in snowmelt runoff between the selected models is too. Upon checking the temperature series for the Xiang catchment, we found that there were 61 d when the average temperature was below 0

As illustrated in Figs. 7, 8, and 10, the three runoff-generation-routine parameters, namely the baseflow factor (KS), the fast flow factor (KF) and the fast flow exponent (

Parameters of WETSPRO.

Fast flow in typical models and observations (representative 3 year hydrographs are shown).

Baseflow in typical models and observations (representative 3 year hydrographs are shown).

Separating both the simulated and the observed streamflow series further reveals how RD influences the model calibration results. The simulated total flow was separated with the WETSPRO tool to ensure that the same separation principle was used for the simulations and observations. Table 4 lists the parameters of WETSPRO for the three cases. The recession constants for the three cases are similar to each other. However, the

Figures 14 and 15 show the separated streamflows of typical models and the separated observed streamflow,
allowing a visual comparison of models based on

The fast flow response of the best-RD model in Dong matches well with the
observed fast flow. The recession of the fast flow in the best-RD model for Dong is too fast and the stable value is nearly zero, which contrasts with observations. The fast flow response of the best-

In all three cases, the best-RD models simulate the baseflow well. RD-selected models accurately simulate the seasonal flow variations in the three catchments. The amplitude of the baseflow fluctuation is close to that seen in the observed baseflow separated by WETSPRO. The discharge also fits with the observed baseflow well. However, in all three cases, the best-

There are two reasons for the unsatisfactory simulation of the fast flow in Dong. The first is that the HBV model is not capable of accurately simulating a mountainous catchment with snowpack, and gauge data for the Dong catchment are scarce. The second is that WETSPRO may have failed to correctly separate the short streamflow series of the Dong catchment. This needs to be further verified.

Further visual demonstrations of the advantage of using RD are provided by Figs. 14 and 15. Fast-flow generation based on RD is more immediate, whereas baseflow generation based on RD is smoother. Both of these are clearly better than the flows generated when RD is not taken into account.

The above results reveal the benefits of using RD and a slight decrease in

This study aimed to examine the possibility of using fractal theory to
improve the performance of hydrological models. The ratio of fractal dimensions (RD) was defined and proposed as a fractal criterion (rather than traditional statistical criteria). A scheme that used a combination of RD and the Nash–Sutcliffe efficiency coefficient (

The main conclusions of this study are as follows:

The trends in the runoff generation routine parameters (namely the fast flow factor, fast flow exponent and baseflow factor) were similar in all three cases studied.

Several parameters were found to be related to RD. For instance, the

The

The HBV model can be found at

The observed hydrometeorological data are available upon request from the corresponding author (yuepingxu@zju.edu.cn).

ZB and YPX designed this study. YW and DM conducted the partial modeling work. All coauthors contributed to the interpretation of the results. ZB conducted the analysis and preparation of the manuscript with support from all coauthors.

The authors declare that they have no conflict of interest.

The authors are indebted to PowerChina Huadong Engineering Corporation Limited, who provided meteorological and hydrological data for the Dong catchment. The National Climate Center of China Meteorological Administration is greatly acknowledged for providing meteorological data on the Jinhua and Xiang catchments, and Zhejiang Hydrological Bureau is acknowledged for providing hydrological data on the Jinhua River. Also, we are very grateful to the editor and two anonymous reviewers for their insightful and constructive comments aimed at improving the quality of our manuscript.

This research has been supported by Zhejiang Key Research and Development Program (2021C03017), the Natural Science Foundation of Zhejiang Province (LZ20E090001), and the Fundamental Research Funds for the Zhejiang Provincial Universities (2021XZZX015).

This paper was edited by Fabrizio Fenicia and reviewed by Charles Onyutha and one anonymous referee.