Sensitivity Analysis of a Conceptual, Lumped Model Using VARS-TOOL Applied to Western Ghats Catchments of India

The present work considers the application of Variogram Analysis of Response Surfaces Toolbox (VARS-TOOL) to identify the sensitive parameters of a rainfall-runoff model in the Netravati river basin of Karnataka, India using the global sensitivity analysis method. The statistical bootstrapping method is used to obtain the confidence intervals around each of the sensitivity indices. The VARS-TOOL generates results based on the different approaches considering the desired sampling technique and the rankings and reliability estimates of the rankings of the different parameters can be obtained. The sensitive parameters from most influential to least influential parameters are grouped and represented in the form of a dendrogram. The precipitation multiplier, fraction of soil entering fast reservoir and slow reservoir coefficients were found to be most influential parameters for the basin. The air temperature threshold for melting/freezing and base melt factor was least influential. The results of the present study can prove to be helpful in further understanding of the application of VARS-TOOL and can be used for further development of the toolbox for sensitivity analysis.


Introduction
Earth system models play a major role in decision making within a confidence interval, by offering predictive capacity and support for scenario assessment. Different hydrological models provide distinct perspectives of modelling and approximate several governing processes expressed in terms of parameters. By understanding the processes and their heterogeneity, these models improve in their complexities. This has led to computationally intensive models with various parameters which have an effect on the models. Hence, the effects of these parameters are uncertain and it must be understood and characterised. Therefore, it is necessary to consider these uncertainties in model. The sensitivity analysis involves determining the contribution of each input of the model to the uncertainties in the output. A systematic classification of sensitivity analysis methods used in environmental modelling and their application is reported elsewhere (Gan et al. 2014;Pianosi et al. 2016). The analysis of variance and Sobol's method were found to be superior to the regional sensitivity analysis and parameter estimation software (Tang et al. 2007).
The local sensitivity was the basis for early sensitivity analysis studies, which is focused on the effects of uncertain inputs around a point, which is found to be potentially false and also incomplete (Saltelli and Annoni 2010). It has led to a high standard known as global sensitivity analysis (GSA) (Saltelli et al. 2008;Sheikholeslami et al. 2018) which is increasingly used in environmental modelling (Pianosi et al. 2015). The GSA methods estimate the influence of all the inputs or their aggregated effect on the change in output.
The VARS-TOOL is a multi-approach toolbox which is based on the theory of Variogram Analysis of Response Surfaces (VARS) (Razavi and Gupta 2019). It uses the directional variogram and co-variograms to define sensitivity which results in less computational cost. Considering the challenges involved in global sensitivity analysis methods like more computationally intensive models or cost effective and the conflicting assessments of sensitive parameters when using different approaches, VARS was developed. The VARS has a variogram-based paradigm for GSA which bridges the normally used gradient-based approach and the variance-based approach. The VARS could uniquely characterise the perturbation-scale dependency and generate sensitivity measures applicable to all the perturbation scales (Haghnegahdar and Razavi 2017).
The VARS yields an original set of sensitivity metrics called IVARS (Integrated Variogram Across a Range of Scales). These metrics will show the rate of change in the model response within a scope, which is known as perturbation scale in the parameter space. The VARS also produces the Sobol (variance-based) total-order effect and the Morris (derivative-based) elementary effects. In the present work, the STAR (space-time autoregressive model)-VARS was utilised which is statistically robust as well as highly efficient and offers stable outcomes when compared to other approaches (Razavi and Gupta 2016b).
The Hydrologiska Byrans Vattenbalansavdelning (HBV) model was introduced in 1972 (Bergström and Forsman 1973) by the Swedish Meteorological and Hydrological Institute (SMHI). The HBV model is a conceptual hydrological model mainly used for runoff simulation and hydrological forecasting which was proved to be reasonably accurate (Lindstrom et al. 1997). It was originally developed to assist hydropower operations by providing the hydrological forecasts. In data scarce regions data from regional climate models may be used as input in HBV models for discharge simulations which proved good in terms of both robustness and uncertainty ranges (Akhtar et al. 2009). The HBV model in a semi-humid catchment in Mississippi, USA was applied by Abebe et al. (2010). The analysis of sensitivity of each of the parameters was carried out by calibrating the model using MOSCEM (Multi-Objective Shuffled Complex Evolution) algorithm and was found that, identifiability and sensitivity of parameters were quite different for the HBV hydrologic model with the objective functions of NS, RMSE and BIAS. The HBV-Light model applied to Narayani river, Nepal was able to simulate the peak flows correctly except a few sharp peaks but the low flows were underestimated (Bhattarai et al. 2018) which was proved to be the other way for another basin in Nepal (Normand et al. 2010).
In the present study, the VARS-TOOL has been applied to a rainfall-runoff model (HBV) in the Netravati basin. The main objective of the work is to understand the application of the newly developed toolbox to the basin and to identify the most sensitive parameters. It is necessary to establish a strategic sensitivity analysis technique to estimate parameters and to comprehend the behaviour of the hydrological model to more representative parameter changes and identify the dependencies of these parameters in the model solution. Therefore, the evaluation of a distributed, conceptual model like HBV is necessary to determine the uncertainty and sensitivity of the rainfall-runoff model using different GSA methods. The HBV is mainly used in the regions where snowfall is involved but its application in other regions is to be explored and no studies have been reported with HBV model application in India.

Study Area Characteristics
The river Netravati is one of the west flowing rivers originating from the Western Ghats of India falling in the range 12ºN -13º11ʹ N and 74º54ʹ E -75º47ʹ E (figure 1a).

Data Used
The current study uses daily meteorological data for the period 1971-2007 which is procured from the India Meteorological Department (IMD). The IMD gridded rainfall data (0.25 0 x0.25 0 ) and 1 0 x1 0 temperature data were collected for the period 1971-2011. The 1º x 1º temperature was converted to 0.25 degree using MATLAB and the values of rainfall and temperature were extracted. The long term average precipitation, temperature and potential evapotranspiration were estimated for the required time period for each of the study area from the daily data. The rainfall, streamflow and temperature records were available for 25 years  for Bantwal and Uppinangadi stations and for 8 years (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003) for Sarve station. The Thiessen polygon method was used to get the average value of rainfall over the basin with 14 grid points.

Methods
The entire problem is run in MATLAB environment, including the HBV model. There are 12 parameters considered in this model the details of which are available elsewhere (Razavi et al. 2019). Of this, the first 11 model parameters are related to hydrologic processes, whereas the 12 th parameter is considered to explain for error in precipitation. Out of the above, the main parameters used in the HBV model are discussed here. The bifurcation of precipitation into rain and snow was revealed to notably influence the simulation of water and energy balance (Wen et al. 2013). Therefore, the air threshold temperature (TT) in ºC for freezing/melting and separation of rain and snow was considered. The temperature deviation correction in 1/ºC of potential evapotranspiration. A positive departure indicates that, the measured temperature is warmer than the baseline temperature, averaged over the data, while a negative departure indicates the measured temperature is cooler than the baseline. The limit for potential evapotranspiration (LP) is a multiplier to the field capacity (FC), i.e, a soil moisture value beyond which evapotranspiration reaches its permissible value. β is a dimensionless shape parameter which is an exponent for the soil release equation which controls the contribution to the response function or the increase in soil moisture storage from rainfall. The fraction of soil moisture entering fast reservoir (FRAC) and K1 and K2 are the fast and slow reservoir coefficients which determine the proportion of storage released in a day. α is a shape parameter that is used for fast reservoir equation. UBAS is the base of the unit hydrograph used for watershed routing in day and PM is a precipitation multiplier to address any uncertainties in precipitation. For accounting of input uncertainty, spatiallydistributed hydrological modelling requires precipitation multiplier approaches. The lower and upper bounds of the parameters are listed in table 1. in VARS_out_XX.txt. Here, XX is from 10 to 100 as 100 stars were taken for the model run. Figure 8 shows the methodology of the present work.

VARS Results for Netravati basin
The ranking of parameters on the basis of IVARS50 and the reliability estimates of the rankings during each STAR run are shown in figures 9 and 10 respectively. The VARS has the advantage that the results during each STAR run can be seen in the form of graphs. In figure 9, the final result including the rankings and reliability estimates after 100 STAR runs can be observed. In figure 10, the directional variogram and the sensitivity metric vs input factors plot showing confidence intervals for the sensitive parameters are shown.
These plots are helpful to the user to monitor how a change in sample size affects the estimates of factor sensitivities and rankings (as more model evaluations become available).
The Sobol' sensitivity analysis could be successfully applied for factor fixing and factor prioritization with respect to the input parameters of a SWAT model (Nossent et al. 2011).
From the graphs and outputs containing RMSE sensitivity metrics generated by running the program, it is evident that, the PM (Precipitation multiplier to reduce error in precipitation), K2 (Slow reservoir coefficient) and FRAC are the most sensitive indices affecting the output. Figure 11, shows the dendrogram for factor grouping created for Netravati basin parameters represented by the HBV model. This assemblage is created on the sensitivity of RMSE metric on the modelled and recorded stream flows over the measured record to the 12 parameters.
The parameters are arranged from most dominant (to the left-hand), i.e., K2, PM and FRAC to the least dominant (to the right-hand), i.e., TT and C0. The ideal grouping by elbow method is linked to coloured sets. The factors are clustered into a unit of clusters which can be entered by the user or the VARS-TOOL will recommend an ideal unit of clusters, based on maximising the differences between the groups by using "elbow method". Figure 11. Dendrogram for factor grouping The first three sensitive parameter rankings (K2, PM and FRAC) are the same after both VARS-TO and IVARS executions. These parameters are grouped as the most influential parameters. The VARS-TO groups the parameters into 3 groups, while, IVARS groups them into 4 groups based on their influence on the simulated value. Table 2 lists the factor rankings based on VARS-TO and IVARS for the Netravati basin.

Table 2. Factor Rankings based on VARS-TO and IVARS for RMSE metric (Netravati basin)
For the Netravati basin, PM and FRAC were the most sensitive parameters, of which PM is the most sensitive. This may be due to uncertainties in the precipitation input provided. Table   3 lists the reliability estimates of the factor rankings.  For VARS-TO, the reliability estimates are varying from 0.9 -1, which is highly reliable. In IVARS50, the reliability shows a range of 0.6 to1, which is also fairly reliable. Table 4 shows the grouping of factors based on their influence on the simulated results (in this case, streamflow). The factor assemblage is created on the sensitivity of RMSE performance index on the modelled and measured stream flows over the chronological period to the 12 model factors.

VARS Results for Sarve and Uppingadi basins
The VARS was also run for Sarve and Uppinangadi sub-basins of Netravati river. As Netravati is a basin of large area with varying topography and climate, the rankings and sensitivity of the various parameters may be different in different regions. In order to compare the results, two smaller sub-basins were selected at different locations and results were analysed on the basis of RMSE, MAE and NSE metrics. For both stations, MAE, RMSE and NSE metrics were generated and related graphs are plotted. Table 5 to table 10 show the factor rankings, reliability estimates of the rankings and the factor groupings based on the RMSE metrics for Serve and Uppinangadi stations.  The reliability estimates of the rankings vary from 0.55 to 1 for VARS-TO method and 0.58 to 1 for IVARS50.   Often, in most of the hydrological models, these parameters are not generally considered.
Therefore, application of the HBV model considering the sensitive parameters can improve and help in effective hydrological modelling of the area.

Conclusions
There is a need for the sensitivity and error investigation to be an essential portion of any model progress, expectation, and decision-making process. This gives an intuition into various issues like uncertainty apportionment, diagnostic testing, planning and management, and policy prioritization. Computational difficulties and lack of interpretability and transparency hamper some of the best practices in modelling. The VARS-TOOL is designed such that, complicated, multi-dimensional and computationally expensive models can be simplified using the computationally efficient toolbox.
From the study, it is evident that, the application of VARS-TOOL is very efficient for the sensitive parameter rankings and also for the reliability estimation.
 In the case of Netravati, using RMSE metric, PM (Precipitation multiplier to address uncertainties in rainfall), FRAC (fraction of soil release entering fast reservoir) and K2 (Slow reservoir coefficient) are the most sensitive indices with ranks 1, 2 and 3 respectively.
 The reliability estimates for these three rankings were 1, ie. 100% reliability. The estimates ranged from 0.9-1 for VARS-TO and 0.6-1 for IVARS estimates.
 For Sarve basin, MAE, RMSE and NSE metrics were used. With MAE metric, K2 was ranked 1, followed by FRAC and PM. In both the other metrics, it was same as that of Netravati basin.
 For Uppinangadi basin, using MAE, RMSE and NSE metrics, has the same sensitive parameters as Netravati basin.
 The reliability estimates showed a range from 0.6 to 1 for all the metrics generated. It indicates that, the rankings of the parameters are reliable.
 Grouping mechanism in VARS toolbox has grouped these 3 parameters (PM, K2 and FRAC) as group 1 (most influential parameters) and the rest into group 2 and 3 with decreasing sensitivity.
With the help of VARS, clear visualisation and easier interpretation of the results is possible. As reliability estimates of the parameter rankings are also available, it is easier to judge whether the results obtained are completely reliable. The conventional methods like Sobol, Morris approaches can be replaced with VARS, since it gives results with less computational cost. Also, VARS acts as a bridge between the conventional methods. As different methods consider different philosophies, it is necessary to have a method which considers all these different theories.
Some of the limitations of the study are:  The main parameters in HBV model are snow related parameters. Though it is suggested that HBV can be applied in India, some parameters could not be considered since the study area is not a snow region.
 The time period of input data considered is different for the study areas. If the same time period is taken, a more reliable result may be obtained.
The HBV model can be effectively applied in a snow region in India and VARS can be executed. Also, VARS can be applied to other models in other programming languages other than HBV or MATLAB.