A new fractal-theory-based criterion for hydrological model calibration

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 ratio of fractal dimensions (RD) is defined as the ratio of fractal 10 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 Nash-Sutcliffe efficiency (E). The E-RD strategy is tested in three study cases with different climate and geography. The results of experiment reveal that, from most aspects, introducing RD into model calibration makes the simulation of streamflow components 15 more reasonable. Besides, in calibration, only little decrease of E occurs when pursuing better RD. We therefore recommend choosing the best E among the parameter sets whose RD is around 1.


Introduction
Since the first hydrological model was developed, proper methods to evaluate the performance of 20 models have been pursued by hydrological community and a large variety of criteria have been proposed and used over the years. Most of the criteria are based on the squared residuals or absolute errors (Pushpalatha et al., 2012). Krause et al. (2005) compared nine efficiency criteria including correlation coefficient (r 2 ), Nash-Sutcliffe efficiency ( ), index of agreement (d) and their variants, but none of them show overall dominance. Kling-Gupta efficiency was developed by Gupta et al. 25 (2009) and Kling et al. (2012) to provide a diagnostically interesting decomposition of the Nash-删除了: to 删除了: investigate
Since firstly introduced by Hurst in 1951, fractality of streamflow series has been studied for decades (Hurst, 1951). There has been a spectacular growth in fractal theory, which was expended to various areas and to multifractal theory Davis et al., 1994). Following the fractal theory, descriptions of fractality include the Hurst exponent (rescaled range analysis) (Hurst, 1951), Hausdorff 110 dimension (box-counting dimension or local dimension) (Jelinek, 2008;, and correlation dimension  etc. The difference of these dimensions is the calculation scheme of fractal dimensions, and they are numerically related and theoretically dependent. While the Hurst exponent calculated with rescaled range analysis was more widely used, the Hausdorff dimension could be expanded to multifractal analysis easily and has perspective 115 applications in hydrology Zhou et al., 2014). The fractality of time series is generally considered as a reflection of self-affinity, periodicity, long-term memory and irregularity Hurst, 1951;Mandelbrot, 2004). Self-affinity is a feature of a fractal whose pieces are scaled by different amounts in the x-and y-directions, and fractal dimensions represent the self-affinity of time series. The self-affinity of time series is the similarity of fine-resolution small parts and coarse-120 resolution large parts of data. Hausdorff dimension is defined and calculated based on the self-affinity of data series. The periodicity and long-term memory of time series referred by its fractality are highly related. Long-term memory is the feature that the effect of an event in a series may persist for a relatively long time. Long-term memory of hydrological time series is usually studied with rescaled range analysis. The irregularity of a fractal series refers to the unpredictable changes in a time series, 125 which is a feature of chaos system. Generally, the Hausdorff dimension of streamflow series represents the magnitude of fluctuation, i.e., the fluctuations in river flow are large for large river flow and small for small river flow (Movahed and Hermanis, 2008). Such feature is also called as long-term correlation, which can be described with the Hausdorff dimension (Onyutha et al., 2019). However, applications of fractal theory in hydrology are limited in simple streamflow analysis, mostly only 130 using the Hurst index (Katsev and L'Heureux, 2003). Also, some studies mentioned other indices based on fractal theory Zhou et al., 2014;Zhang et al., 2010), but again, the research 删除了: (Jain and Sudheer, 2008) 删除了: Overall, there is still large vacancy for calibration criteria which can give consideration to individual data and 135 whole hydrograph. ! 删除了: (Hurst, 1951) 删除了: Davis et al., 1994)

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删除了: (Movahed and Hermanis, 2008) 145 删除了: (Onyutha et al., 2019) 删除了: were 删除了: (Katsev and Lheureux, 2003) 删除了: literatures 删除了: indexes 150 删除了: Yu et al., 2014;Zhang et al., 2010) 4 objects were only observed hydrological data. Recent studies made a progress to hydrological modelling based on fractal theory , but the model only reconstructed flood/drought grades series. As demonstrated by all these studies, the fractality of observed streamflow series (as well as other hydro-meteorological data) is inherent and represents some peculiarity of their study 155 cases. However, few studies have tried to explore the applications of fractal theory in hydrological model calibration. To our best knowledge, the only exception is Onyutha et al. (2019), who utilized the Hurst-Kolomogorov framework to evaluate the performance of climate models (GCM and RCM) rather than to calibrate hydrological models. In their study, the Hurst exponent was used to represent the long-range dependence and evaluate the reproductivity of variability (Onyutha et al., 2019). 160 However, the benefits of using fractal theory in model building and calibration have not been discussed.
Unlike typical statistical evaluation of fluctuation (such as standard deviation and distribution function), the Hausdorff dimension takes the order of data into account. Therefore, on the basis of classical criteria who compare observed and simulated water balances, the Hausdorff dimension can 165 offer useful insight into mechanisms controlling the extreme hydrological events (including floods, droughts and low flows) (Radziejewski et al., 1997). Another difference between fractal dimension and classical criteria is the influence of individual (or a small number of) data. While approaching of every simulated individual data to observed data makes the coefficient better, it may make the Hausdorff dimension of simulated data closer or farther away from that of observed data. That means,170 to reproduce all characteristics of observed streamflow, simulated streamflow and observed streamflow should have similar Hausdorff dimensions, as well as other traditional metrics. Given all that, the Hausdorff dimension is proposed in this study in hydrological model calibration.
Since the fractal dimension describes the fractality of streamflow series and two different series may have the same fractal dimension, the fractal dimension could not be used to calibrate hydrological 175 model independently. Multi-objective optimization approaches are widely used by hydrological community (Harlin, 1991;Yapo et al., 1998;Liu et al., 2017Liu et al., , 2019Pan et al., 2017;Shafii and Tolson, 2015). This set the stage of using some uncomprehensive but effective criteria as targets, such as aforementioned hydrological signatures (Shafii and Tolson, 2015;Westerberg and McMillan, 2015), 删除了:    (Pushpalatha et al., 2012). Most of the criteria are based on the squared residuals or absolute errors 245 (Pushpalatha et al., 2012). Krause et al. (2005) compared nine efficiency criteria including correlation coefficient (r 2 ), Nash-Sutcliffe efficiency ( ), index of agreement (d) and their variants, but none of them show overall dominance.
Kling-Gupta efficiency was developed by Gupta et al. (2009)

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and Kling et al. (2012) to provide a diagnostically interesting decomposition of the Nash-Sutcliffe efficiency, which facilitates the analysis of the relative importance of its different components (correlation, bias and variability) in the context of hydrological modelling. Apart from criteria which 255 are used to calculate model errors over the entire test period, there are also many criteria which focus on a certain period of interests. For example, criteria mentioned above are calculated over flood periods (Liu et al., 2017;Liu et al., 删除了: (Harlin, 1991;Yapo et al., 1998;Liu et al., 2017;Liu et al., 2019;Pan et al., 2017;Shafii et al., 2015;Ye et al., 2014) 删除了: (Shafii et al., 2015;Westerberg and McMillan,  statistical targets and fractal criteria. Nonetheless, the strategy to use the Hausdorff dimension to 260 calibrate hydrological models has not been studied. In this study, a new criterion defined as ratio of fractal dimension ( ) is introduced, as well as a calibration strategy. The criterion and calibration strategy should be able to consider the self-affinity, periodicity, long-term memory and irregularity of hydrograph during model calibration. Three catchments with different climate and geography are used as case studies. The aim of this study is to 265 examine the applicability of using as one of the targets of multi-objective calibration and explore the performance of hydrological models when is considered. Section 2 describes differences between and classical criteria, and how is used in calibration ( -strategy). Section 3 contains the brief information of study areas and methods used in this study to investigate the advantages of . Section 4 provides the results and Section 5 provides the discussion. Section 6 is 270 the summary and conclusion. In this study, our goal is to answer the following questions: (1) Is a proper criterion for hydrological modelling, even if the reflection of is not as direct as classical criteria? (2) Could -strategy explicitly improve the performance of hydrological models? (3) Why can be used to improve calibration?

Ratio of fractal dimensions and -strategy
The box-counting method used to calculate the Hausdorff dimension is based on the idea of separating data into boxes and count the number of boxes (Mandelbrot, 2004). When adopted to analyze time series, the box-counting method sums adjacent data up (put adjacent individuals into boxes) and compares the treated data of various resolutions (different sizes of boxes). Fig. 1 graphically shows 280 how the box-counting method works with time series. Fig. 1 (a) shows how the number of boxes needed to cover all data (N) changes when the size of boxes changes (resolution, δ). Fig. 1 (b) shows the log-linear relationship between N and δ. The definition of the Hausdorff dimension D is: Where δ is the size of boxes and N is the number of boxes (Evertsz and Mandelbrot, 1992).

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Most of the coefficients are based on the squared residuals (Pushpalatha et al., 2012). According to squared-residualsbased coefficients' calculation formula, approaching of every simulated individual data to observed data makes the coefficients better. However, this doesn't make models more

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As stated before, the observed and simulated streamflow series shall have the same Hausdorff dimension. In this study, a new criterion named as ratio of dimension ( ) is defined as follow: where D s is the Hausdorff dimension of simulated streamflow series and D o is the Hausdorff dimension of observed streamflow series. The range of is from 0 to +∞. When RD=1, the 525 simulated streamflow series has the same Hausdorff dimension with that of observed streamflow series, which means that the model is the best in terms of fractals. The relevant examination of models' performance under the supervision of has not been studied either.
Obviously, , as a metric of self-affinity deviation of simulated streamflow series from observed series, is not a criterion capable of evaluating the performance of hydrological models by itself. An 530 immediate thought is to combine and another statistical criterion in model calibration.
Three features are demanded for the statistical criterion to be combined with . Firstly, the statistical criterion shall be able to evaluate the performance of models in terms of water balance to some extent.
Secondly, the statistical criterion shall evaluate the response of streamflow to meteorological forcing.
Thirdly, the criterion shall calculate model errors over the entire test period. These features make sure 535 that the strategy meets basic needs. An additional requirement for the statistical criterion used in this study is the popularity of this criterion within the hydrological community. Therefore in this study, we choose E as the statistical criterion. Another reason to choose is that the pros and cons of are more familiar for hydrologists than other metrics, and this original version is still mostly often used in hydrological model calibration. In this manner, the advantages of emerge as well as the 540 benefits of multi-objective calibration based on .
Nash-Sutcliffe efficiency coefficient ( ), a commonly used criterion since initially proposed (Nash and Sutcliffe, 1970), is calculated is: Where * is the simulated flow, + is the observed flow and + ,,,, is the mean value of the observed 545 flow. The difference usually implicates that self-affinity of the time series changes as the resolution changes 560 and in hydrology specifically, dominant driver of hydrological processes changes. For example, the dominant drivers of daily and annual circle of temperature are different. The Hausdorff dimension of joint data series (also called as joint multifractal spectrum) verifies the freezing-thawing process of soil moisture in a quantitative and solid way which unfolds the complex nonlinear relationship among three hydrological variables . According to this idea, the Hausdorff dimension 565 determines whether the streamflow components are reasonably simulated. In this study, the largest temporal resolution is set as 365 days (1 year), to leave the inter-annual drivers out. It is believed that the range of resolution is enough for the Hausdorff dimension to reflect drivers of hydrological processes.

Study area 570
A small catchment located in Tibet named Dong, a medium sized catchment located in southeastern China named Jinhua and a large catchment located in the middle reach of Yangtze River named Xiang are used in this study.
Dong is a small tributary of the Yarlung Zangbo River, with elevations ranging from 3512 to 5869 m.

HBV model
The HBV model is a conceptual rainfall-runoff model originally developed by Swedish Meteorological and Hydrological Institute (SMHI) Bergstrom, 1992;Lindström et al., 1997). The model has been successfully used in many cases (Seibert and Vis, 2012;Tian et al., 2015Tian et al., , 2016. The HBV model is composed of precipitation and snow accumulation routines, a soil 610 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.  Bergström, 685 1992;Lindström et al., 1997). The HBV model has been successfully used in many cases (Seibert and Vis, 2012;Tian et al., 2015;Tian et al., 2016). The HBV model is composed of precipitation and snow accumulation routines, a soil moisture routine, a quick runoff routine, a baseflow routine 690 and a transform function. The HBV model takes into account the effect of snow melting and accumulation, which is significant in the Dong catchment. In this study, the HBV model utilized is provided by Tian et al. (2015), compiled with MATLAB. !

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The HBV model has 14 parameters to control all abovementioned hydrological processes. A simple description of seven parameters mentioned later is given here. !

Multi-objective genetic algorithm
A controlled and elitist genetic algorithm (a variant of NSGA-II) ) is applied in model calibration. A controlled and elitist GA favors individuals with better fitness value (rank) as well as 705 individuals that can help increase the diversity of the population even if they have a lower fitness value. An important behavior of this genetic algorithm is that, the individual with the best performance according to anyone of the criterion would be retained with the lowest rank. This makes sure that with multi-objective genetic algorithm, parameter set with the best possible could be found and the following comparison between -and is reasonable. In this study, | 1 − | is used as one 710 of the criteria.
Since HBV has 14 parameters to calibrate, the number of generations is 2800. Each generation has 600 population. The crossover fraction is set as 0.8 (meaning). The Pareto fraction is set as 0.2 (meaning). The population migrates every 20 generations, and the migration fraction is set as 0.5.
These settings make sure that population will not trap in local optimum, which is important because 715 varies in a wider range than traditional criteria. Most of these numbers are the default settings, which is applicable to most of the problems. Only the number of the population of each generation (600) is larger than default (200) for finer presentation of Pareto front of the optimization. The meanings of settings can be found in .
All 600 Pareto-optimized solutions of the last generation are used in the following analysis. GA 720 optimization with the -calibration strategy (described in 2.1) will not drop population with perfect (=1) and unsatisfactory . Several representative selected parameter sets and corresponding simulated streamflow series are deeply studied.

Approach for model evaluation
To investigate the 's effects in hydrological model evaluation, several tools are utilized. To understand how the model is adjusted when is used as one of the objectives, the relationship 740 between parameters and is analyzed. The distance correlation , is used to determine whether the variations of model's parameters and are related. The distance correlation, as a multivariate measure of dependence, calculates the correlation of distances between points to means. The distance correlation is believed to have better performance when solving problems with non-linear data or extreme values (Székely et al., 2007). The relationship between parameters and may not be linear, 745 which brings the necessity of using a nonlinear analysis approach rather than Pearson's linear correlation coefficient. Distance correlation is also more robust to data outliers, than rank correlations. present the significant difference of simulated streamflow from the aspect of fractal. In this study, 775 is often lower than 1 and sometimes only slightly higher than 1, which agrees with the smooth hydrograph and simple structure of HBV model. Because that RD=1 means that the model is best in terms of fractals (see Section 2.2), the models whose is larger than 1 need discussion, too. cases, the trend of bias in this study is regarded to be random. In addition, in Xiang case, there is a break in Fig. 4. On two sides of the break, is close by and is significantly different.
A single-objective calibration is operated to support the assumptions made in Section 3.3 that, in this study, the NSGA II algorithm can find the best . The comparison between results of single-objective calibration and multi-objective calibration ( -strategy) is listed in Table 1. Besides, to get rid of 790 the possible influence of the lengths of time series, a comparison of the multi-objective calibration with the same length of data is made. The results show that, at least in the cases of this study, thestrategy would not change its behavior with the lengths of data.  A precondition of adopting the -strategy is the irrelevancy or weak correlation between two criteria. This precondition could be simply verified by looking into their calculation schemes or by 815 examining during the multi-objective calibration. In this study, the calculation of two metrics ( and ) are totally different, and the results of multi-objective calibration also show that the significant change of only leads to minor difference of (see Fig. 4). The best and worst are close according to the result of multi-objective calibration. Figs. 4 and 5 further imply that only little decrease of happens when pursuing better . In this study, the equifinality of using only 820 emerges. Table 2 lists the values of typical models selected by the -calibration strategy and optimized model. Table 2 confirms the assumption that, in this study, directly analyzing the models calibrated by -calibration strategy is reasonable and efficient. Hydrological signatures including relative variance, lag-1 auto-correlation, percentage bias and maximum/minimum monthly flow are used to 825 show the effect of .

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删除了: In this study, the former is apparent and the latter is conducted and briefly shown in Fig. 4. 上移了 [7]: In this study, the former is apparent and the latter is conducted and briefly shown in Fig. 4.

Effect of on model parameters
All parameter sets in the Pareto front of three cases vary. The distance correlations ( , -) of parameters and are used to determine whether the change of parameters is stable. In addition, high value of , indicates the significant relationship between the Hausdorff dimension and these parameters. In this study, the relation between GA-selected parameter sets and is not shown because and 860 in the Pareto front are highly related and the variance of is small (see Fig. 4). Table 3 lists the determinative parameters of three cases respectively. Distance correlation ( , -) is used to illustrate the non-linear relationship between and in the Pareto's optimal. The parameter effective precipitation exponent ( β ) and degree-day factor are also listed in Table 3. Effective precipitation exponent is listed in Table 3 because of two reasons: 1) the range of , of β in Jinhua 865 case is from 0.709 to 0.739 in Xiang case, which is better than all unlisted parameters; 2) β, as well as determinative parameters , , , is a runoff-generation-related parameter. The degree-day factor is listed in Table 3 because of two reasons: 1) distance correlation between the degree-day factor 删除了: 4 删除了: ier 870 删除了: In this study, a value 0.8 of $ # is used as the threshold of being determinative.

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The parameters with $ # < 0.8 in all cases are not listed in Table 3 and is close to 0.8 in Xiang case; 2) in Dong case, distance correlation analysis does not show 875 the significance of ablation of snow to hydrograph. Capillary transport is not determinative parameters to in Dong and Jinhua and therefore no further discussion of them is given afterwards. The , of (0.512) in Dong catchment is smaller than those in other cases. Fig. 6 shows the relationship between and .
*: , -≥ 0.8 Fig. 6. and relationship in three cases. 885 An explicit relationship between parameters and criteria confirms that the effect of is not random.
Six parameters (β, , fast flow factor, baseflow factor, percolation, degree-day factor) are selected by distance correlation analysis for further discussion.  the best model and best model decreases from more than 20% to less than 5%. In Xiang case, the relative difference between the best model and best model decreases from about 16% to 8%. The difference between the best model and best model is important during the dry period and reduces as water storage of upper reservoir increases (the wet period). The relative difference is greater in Jinhua case than in Xiang case in low flow periods but smaller in Jinhua case in high flow 915 periods. That difference between the best model and best model will finally lead to the greater variation of fast flow in low flow periods than in high flow periods. There are break points in Xiang case (Fig. 6, 7 and 8), but no evident effects shown in Fig. 5 and 9. The percolation is significantly related to in all cases. The range of percolation in Dong case is larger than the others. Fig. 11 shows the relationship between percolation and in three cases.
Percolation increases in Dong case and decreases in other two cases when increases. The range 935 of percolation in Dong is larger than in the others. and percolation determine the way HBV models baseflow. The percolation in Dong case is larger than the others, which is the reflection of 删除了: KS 删除了: the Dong catchment's arid climate. The percolation in Jinhua case is larger than the percolation in Xiang 940 case, because the slope in Jinhua catchment is larger. Fig. 11. Percolation and relationship in three cases.
The degree-day factor is significantly related to in Xiang case. However, the relationship 945 between the degree-day factor and in Dong and Jinhua is weak. Fig. 12 shows the relationship between the degree-day factor and . The degree-day factor of most selected models of Dong case is smaller than 0.05, indicating that these models barely have any snow-melt runoff. When >0.9, several models have degree-day factors larger than 7. When is around 1, the range of degree-day factor is 8.18 to 11.76, indicating that somehow detects the snow-melt runoff in the hydrograph 950 and makes the HBV model simulate the snow-melt runoff more reasonably. Notably, the -selected degree-day factor in Dong case is too large according to the guidance of HBV (1.5 to 4 mm/day, in Sweden) (HBV light version 2, user's manual), which may result from the unsuitable lumped model structure of HBV in rugged mountainous catchment.
The degree-day factors in all selected models of Jinhua are large, but the temperature in Jinhua is too 955 high to have snow accumulation. The distance correlation between the degree-day factor and is weak in Dong and Jinhua case. The range of degree-day factor of most models in Xiang case is from 2.8 to 3.4. The range is small and so as the difference of snow-melt runoff of selected models. By checking the temperature series in the Xiang catchment, we find there are 61 days (out of 27 years) when the average temperature is below 0°C. Actually, since the Xiang catchment is large, there are 960 snow events somewhere in the catchment almost every year. The low temperature may be covered by averaging, but the -strategy captured it and illustrated this by noticeable value of degree-day factor. Fig. 12. Degree-day factor and relationship in three cases. 965 As illustrated in Fig. 7, 8, 10 and 11, the three runoff-generation-routine parameters, namely baseflow 删除了: (Seibert, 2005)

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factor ( ), fast flow factor ( ) and fast flow exponent ( ), have the same change patterns in three cases, suggesting a consistent preference of in all cases. Fig. 9 shows the visual difference of fast flow caused by introducing . However, other parameters have different change patterns along with because of different features of catchments ( Fig. 6 and 12). For example, the soil parameter , as illustrated by Equation (6), redistributes the precipitation and divides it into effective precipitation 980 and infiltration.
in Dong case and Xiang case increases and in Jinhua decreases when is getting better.

Analysis of separated streamflow
Separated simulated and observed streamflow series further reveal how influences model calibration results. The simulated total flow is also separated with the WETSPRO tool to make the 985 principle of separation of simulation and observation same. Table 4   cases, the fast flow of best model is smaller than that of best model and the difference is greater in low flow periods, which is consistent with Fig. 9.
In all three cases, the best models simulated baseflow well. Dong are 0.87 and 0.79 respectively. The best models in Jinhua and Xiang case, however, are close to the best models. According to Fig. 10 and 11, in Jinhua and Xiang cases, smaller and percolation (of best model) make smaller recharge and outflow (baseflow) of lower reservoir and smaller fluctuation of baseflow. In Dong case, bigger percolation increases the recharge and total baseflow and smaller extends the period of baseflow recession, making simulated baseflow more 1040 consistent with observation (Fig. 15).
Two reasons exist for the unsatisfactory simulation of fast flow in Dong case. The first one is that the HBV model is not capable of accurately simulating mountainous catchment with snowpack and little gauge data are available for the Dong catchment. The second one is that WETSPRO may fail to correctly separate the short streamflow series of Dong catchment. This needs to be further verified. 1045 Another visual demonstrator of the preference of is Fig. 14 and 15. Fast flow generation based on is more immediate while baseflow generation based on is smoother. Both of them are visually better than that when is not taken into account.
Above results reveal the benefits of using and the slight decrease of . The selection principle based on multi-objective calibration is therefore suggested following two steps: 1) sieving out all 1050 parameter sets whose is around 1 (in this case, considering the data precision of MATLAB, =1); 2) Choosing the parameter set with best among the sets in Step 1. It is determined that thestrategy using this selection principle improves the reliability of streamflow components simulation.
That is, selects responsive fast flow (confirmed in Fig. 14) and smooth baseflow (confirmed in 1055 Fig. 15) in all cases.

Conclusion
This study targeted at examining the possibility of using fractal theory to improve the performance of hydrological models. The definition of ratio of fractal dimension ( ) was proposed and used as a fractal criterion (against traditional statistical criteria). A scheme which combined and Nash-1060 Sutcliffe efficiency coefficient ( ) to calibrate hydrological models was developed and examined.
Three study cases named Dong, Jinhua and Xiang were included in the examination. This is the first 删除了: to 删除了: simulate 删除了: ' 1065 删除了: 5 time (to our best knowledge) that fractal theory was applied to calibrate hydrological models.
The main conclusions of this study are as follows: 1) The varying patterns of parameters of runoff generation routine (namely fast flow factor, fast flow exponent and baseflow factor) are similar in all cases of our study. 1070 2) Several parameters were found to be related to . For instance, the -strategy selected the degree-day factors with relatively high value in Dong case, which is not seen when only was considered.
3. The -strategy is innovative in hydrological modelling. That is, the -calibration strategy is a potential way to take the fractality of observed streamflow series into consideration in 1075 model calibration. Since fractal (also regarded as self-affinity) widely exists in nature, the as a criterion can be a good supplement for hydrological model calibration.
The -strategy introduced in this study needs more case studies to corroborate its capability further. The combination of other traditional statistical criteria and shall also be examined. More studies are also needed to dig out more benefits of applying fractal theory in hydrological modelling.