# A Simple Time-Varying Sensitivity Analysis (TVSA) for Assessment of Temporal Variability of Hydrological Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. TVSA Function Description

#### 2.1. Code Description

_{iQ}) and the Kling–Gupta Efficiency (KGE) were included in the code. The threshold value is required for the application of the RSA model (Figure 1b,c) since it groups the parameter sets as B and NB. A parameter set is considered behavioral if the value of the objective function is greater (or less, depending on the interpretation of the objective function) than a threshold value defined by the modeler and required in the graphical user interface; otherwise, the parameter set is considered nonbehavioral. For example, this is seen in Figure 1b, where the values of parameter X

_{i}within the gray area correspond to models considered B. The cumulative distribution functions, ${\mathrm{F}}_{\mathrm{i}}^{\mathrm{B}}\left({X}_{i}\right)$ and ${\mathrm{F}}_{\mathrm{i}}^{\mathrm{NB}}\left({X}_{i}\right)$ of the two groups are compared and the discrepancy between them quantified using, as a sensitivity index, the Kolmogorov–Smirnov statistic or the maximum vertical distance (MVD) between the curves (Figure 1c). This indicator is expressed as:

#### 2.2. Objective Functions

#### 2.2.1. Nash–Sutcliffe Efficiency (NSE)

#### 2.2.2. Root Mean Square Error (RMSE)

#### 2.2.3. Nash–Sutcliffe Efficiency Calculated on Inverse Streamflows (NSEiQ)

_{iQ}values vary between 1 and −∞, and the optimal value is 1.

#### 2.2.4. Kling–Gupta Efficiency (KGE)

#### 2.3. Convergence Analysis

## 3. Application Example

#### 3.1. Study Area and Data

^{2}), located in the south of Chile (Figure 2).

#### 3.2. Description of the Model

_{a}), which is equal to potential evapotranspiration (PET

_{d}) if the relationship between soil moisture (SM) and field capacity (FC) is above a threshold value for potential evapotranspiration (LP). However, for soil moisture values below LP, the actual evapotranspiration will be linearly reduced.

_{d}) from daily mean air temperature and the long-term PET and monthly temperature averages.

_{p}).

_{0}and Q

_{1}), whereas the lower deposit has one (Q

_{2}). When the water level in the upper deposit exceeds a threshold value (L), runoff is produced quickly in its upper part (Q

_{0}). The response of the other outlets is relatively slow. The streamflows are controlled by recession coefficients k0, k1, and k2, which represent the response functions of the upper and lower deposits. The constant infiltration rate (Q

_{p}) is controlled by a coefficient kp.

_{2}) must be slower than that of the second one (Q

_{1}); therefore, k2 must be lower than k1 [27]. For a better understanding of the model, see Bergström [42], Lindström et al. [43], and Seibert [44].

#### 3.3. TVSA Implementation

## 4. Results and Discussion

#### 4.1. Computational Cost

#### 4.2. Temporal Variability of Hydrological Processes

_{2}in dry years.

_{1}parameters than for a 5-year moving window analysis. Meanwhile, during the summer months (January–March), the model is more sensitive to the K

_{2}, K

_{p}, and Lp parameters and slightly sensitive to Cmelt.

_{1}and less sensitive to β, whereas in summer, it is more sensitive to Lp, K

_{2}, K

_{p}, and Cmelt.

_{1}and Q

_{1}), which also indicate that β favors soil storage over runoff.

_{2}and Q

_{2}) and percolation response (K

_{p}and Q

_{p}). Temperatures in this period also increase; therefore, evapotranspiration (Lp and c) and snowmelt (Cmelt) take on more importance in the time of greatest precipitation. These results confirm the behavior identified through the model and proposed analysis. The results are consistent with the behavior of the watershed and findings of Abebe et al. [50], Zelelew and Alfredsen [51], and Pianosi and Wagener [20], where the model performance proves to be highly influenced by parameter β. Pianosi and Wagener [20] performed a TVSA based on PAWN for a 31-day moving window. They showed that the relative importance of model parameters over time can be distinguished. The TVSA shown in this study helps distinguish influential periods across data as well as for different aggregation times in the analysis (i.e., size of the time window). Therefore, it complements the existing TVSA options, allowing the same analyses as, e.g., those shown by Pianosi and Wagener [20], to be performed.

#### 4.3. Size of the Window of Analysis

## 5. Conclusions

_{2}and K

_{p}) and snowmelt (Cmelt). Therefore, not only the choice of objective function, hydrological model, SA method, or study area but also the moving window size influences the estimation of sensibility indices in TVSA. To avoid errors in sensitivity estimation, a TVSA based on different moving window sizes is recommended.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Pianosi, F.; Beven, K.; Freer, J.; Hall, J.W.; Rougier, J.; Stephenson, D.B.; Wagener, T. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environ. Model. Softw.
**2016**, 79, 214–232. [Google Scholar] [CrossRef] - Sarrazin, F.; Pianosi, F.; Wagener, T. Global Sensitivity Analysis of environmental models: Convergence and validation. Environ. Model. Softw.
**2016**, 79, 135–152. [Google Scholar] [CrossRef][Green Version] - Norton, J. An introduction to sensitivity assessment of simulation models. Environ. Model. Softw.
**2015**, 69, 166–174. [Google Scholar] [CrossRef] - Pianosi, F.; Sarrazin, F.; Wagener, T. A Matlab toolbox for Global Sensitivity Analysis. Environ. Model. Softw.
**2015**, 70, 80–85. [Google Scholar] [CrossRef][Green Version] - Devak, M.; Dhanya, C.T. Sensitivity analysis of hydrological models: Review and way forward. J. Water Clim. Chang.
**2017**, 8, 557–575. [Google Scholar] [CrossRef] - Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D.; Saisana, M.; Tarantola, S. Global Sensitivity Analysis; Wiley: Chichester, UK, 2008. [Google Scholar]
- Borgonovo, E.; Plischke, E. Sensitivity analysis: A review of recent advances. Eur. J. Oper. Res.
**2016**, 248, 869–887. [Google Scholar] [CrossRef] - Campolongo, F.; Saltelli, A.; Cariboni, J. From screening to quantitative sensitivity analysis. A unified approach. Comput. Phys. Commun.
**2011**, 182, 978–988. [Google Scholar] [CrossRef] - Razavi, S.; Gupta, H.V. A new framework for comprehensive, robust, and efficient global sensitivity analysis: 2. Application. Water Resour. Res.
**2016**, 52, 440–455. [Google Scholar] [CrossRef] - Razavi, S.; Gupta, H.V. A new framework for comprehensive, robust, and efficient global sensitivity analysis: 1. Theory. Water Resour. Res.
**2016**, 52, 423–439. [Google Scholar] [CrossRef][Green Version] - Khorashadi Zadeh, F.; Nossent, J.; Sarrazin, F.; Pianosi, F.; van Griensven, A.; Wagener, T.; Bauwens, W. Comparison of variance-based and moment-independent global sensitivity analysis approaches by application to the SWAT model. Environ. Model. Softw.
**2017**, 91, 210–222. [Google Scholar] [CrossRef][Green Version] - Cukier, R.I.; Fortuin, C.M.; Shuler, K.E.; Petschek, A.G.; Schaibly, J.H. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I Theory. J. Chem. Phys.
**1973**, 59, 3873–3878. [Google Scholar] [CrossRef] - Spear, R.C.; Hornberger, G.M. Eutrophication in peel inlet-II. Identification of critical uncertainties via generalized sensitivity analysis. Water Res.
**1980**, 14, 43–49. [Google Scholar] [CrossRef] - Morris, M.D. Factorial sampling plans for preliminary computational experiments. Technometrics
**1991**, 33, 161–174. [Google Scholar] [CrossRef] - Sobol, M.I. Sensitivity analysis for non-linear mathematical models. Math. Model. Comput. Exp.
**1993**, 1, 407–414. [Google Scholar] - Pianosi, F.; Wagener, T. A simple and efficient method for global sensitivity analysis based oncumulative distribution functions. Environ. Model. Softw.
**2015**, 67, 1–11. [Google Scholar] [CrossRef][Green Version] - Razavi, S.; Sheikholeslami, R.; Gupta, H.V.; Haghnegahdar, A. VARS-TOOL: A toolbox for comprehensive, efficient, and robust sensitivity and uncertainty analysis. Environ. Model. Softw.
**2019**, 112, 95–107. [Google Scholar] [CrossRef] - Wagener, T.; McIntyre, N.; Lees, M.J.; Wheater, H.S.; Gupta, H.V. Towards reduced uncertainty in conceptual rainfall-runoff modelling: Dynamic identifiability analysis. Hydrol. Process.
**2003**, 17, 455–476. [Google Scholar] [CrossRef] - Guse, B.; Pfannerstill, M.; Strauch, M.; Reusser, D.E.; Lüdtke, S.; Volk, M.; Gupta, H.; Fohrer, N. On characterizing the temporal dominance patterns of model parameters and processes. Hydrol. Process.
**2016**, 30, 2255–2270. [Google Scholar] [CrossRef] - Pianosi, F.; Wagener, T. Understanding the time-varying importance of different uncertainty sources in hydrological modelling using global sensitivity analysis. Hydrol. Process.
**2016**, 30, 3991–4003. [Google Scholar] [CrossRef][Green Version] - Ghasemizade, M.; Baroni, G.; Abbaspour, K.; Schirmer, M. Combined analysis of time-varying sensitivity and identifiability indices to diagnose the response of a complex environmental model. Environ. Model. Softw.
**2017**, 88, 22–34. [Google Scholar] [CrossRef][Green Version] - Zhao, Y.; Nan, Z.; Yu, W.; Zhang, L. Calibrating a hydrological model by stratifying frozen ground types and seasons in a cold alpine basin. Water
**2019**, 11, 985. [Google Scholar] [CrossRef][Green Version] - Wagener, T.; Kollat, J. Numerical and visual evaluation of hydrological and environmental models using the Monte Carlo analysis toolbox. Environ. Model. Softw.
**2007**, 22, 1021–1033. [Google Scholar] [CrossRef] - Herman, J.D.; Kollat, J.B.; Reed, P.M.; Wagener, T. From maps to movies: High-resolution time-varying sensitivity analysis for spatially distributed watershed models. Hydrol. Earth Syst. Sci.
**2013**, 17, 5109–5125. [Google Scholar] [CrossRef][Green Version] - Herman, J.D.; Kollat, J.B.; Reed, P.M.; Wagener, T. Technical note: Method of Morris effectively reduces the computational demands of global sensitivity analysis for distributed watershed models. Hydrol. Earth Syst. Sci.
**2013**, 17, 2893–2903. [Google Scholar] [CrossRef][Green Version] - Herman, J.D.; Reed, P.M.; Wagener, T. Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior. Water Resour. Res.
**2013**, 49, 1400–1414. [Google Scholar] [CrossRef] - Aghakouchak, A.; Habib, E. Application of a Conceptual Hydrologic Model in Teaching Hydrologic Processes. Int. J. Eng. Educ.
**2010**, 26, 963–973. [Google Scholar] - Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. J. Hydrol.
**1970**, 10, 282–290. [Google Scholar] [CrossRef] - Gupta, H.V.; Kling, H.; Yilmaz, K.K.; Martinez, G.F. Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. J. Hydrol.
**2009**, 377, 80–91. [Google Scholar] [CrossRef][Green Version] - Medina, Y.; Muñoz, E. Analysis of the Relative Importance of Model Parameters in Watersheds with Different Hydrological Regimes. Water
**2020**, 12, 2376. [Google Scholar] [CrossRef] - Pfannerstill, M.; Guse, B.; Fohrer, N. Smart low flow signature metrics for an improved overall performance evaluation of hydrological models. J. Hydrol.
**2014**, 510, 447–458. [Google Scholar] [CrossRef] - Moriasi, D.N.; Arnold, J.G.; Van Liew, M.W.; Bingner, R.L.; Harmel, R.D.; Veith, T.L. Veith Model Evaluation Guidelines for Systematic Quantification of Accuracy in Watershed Simulations. Trans. ASABE
**2007**, 50, 885–900. [Google Scholar] [CrossRef] - Parra, V.; Fuentes-Aguilera, P.; Muñoz, E. Identifying advantages and drawbacks of two hydrological models based on a sensitivity analysis: A study in two Chilean watersheds. Hydrol. Sci. J.
**2018**, 63, 1831–1843. [Google Scholar] [CrossRef] - Wagener, T.; van Werkhoven, K.; Reed, P.; Tang, Y. Multiobjective sensitivity analysis to understand the information content in streamflow observations for distributed watershed modeling. Water Resour. Res.
**2009**, 45. [Google Scholar] [CrossRef][Green Version] - Singh, J.; Knapp, H.V.; Demissie, M. Hydrologic Modeling of the Iroquois River Watershed Using HSPF and SWAT ISWS CR 2004-08; Illinois State Water Survey: Champaign, IL, USA, 2004; Available online: http://hdl.handle.net/2142/94220 (accessed on 20 July 2019).
- Le Moine, N. Le Bassin Versant de Surface vu par le Souterrain: Une Voie D’amélioration des Performances et du Réalisme des Modèles Pluie-Débit? Ph.D. Thesis, Université Pierre et Marie Curie-Paris VI, Paris, France, 2008. [Google Scholar]
- Pushpalatha, R.; Perrin, C.; Le Moine, N.; Andréassian, V. A review of efficiency criteria suitable for evaluating low-flow simulations. J. Hydrol.
**2012**, 420–421, 171–182. [Google Scholar] [CrossRef] - Pechlivanidis, I.G.; Jackson, B.; McMillan, H.; Gupta, H. Use of an entropy-based metric in multiobjective calibration to improve model performance. Water Resour. Res.
**2014**, 50, 8066–8083. [Google Scholar] [CrossRef] - Patil, S.D.; Stieglitz, M. Comparing spatial and temporal transferability of hydrological model parameters. J. Hydrol.
**2015**, 525, 409–417. [Google Scholar] [CrossRef][Green Version] - Sheffield, J.; Goteti, G.; Wood, E.F. Development of a 50-year high-resolution global dataset of meteorological forcings for land surface modeling. J. Clim.
**2006**, 19, 3088–3111. [Google Scholar] [CrossRef][Green Version] - Thornthwaite, C.W. An Approach toward a Rational Classification of Climate. Geogr. Rev.
**1948**, 38, 55. [Google Scholar] [CrossRef] - Bergström, S. The HBV Model—Its Structure and Applications; SMHI RH No. 4; SMHI: Norrköping, Sweden, 1992. [Google Scholar]
- Lindström, G.; Johansson, B.; Persson, M.; Gardelin, M.; Bergström, S. Development and test of the distributed HBV-96 hydrological model. J. Hydrol.
**1997**, 201, 272–288. [Google Scholar] [CrossRef] - Seibert, J. Multi-criteria calibration of a conceptual runoff model using a genetic algorithm. Hydrol. Earth Syst. Sci.
**2000**, 4, 215–224. [Google Scholar] [CrossRef][Green Version] - Muñoz, E.; Rivera, D.; Vergara, F.; Tume, P.; Arumí, J.L. Identifiability analysis: Towards constrained equifinality and reduced uncertainty in a conceptual model. Hydrol. Sci. J.
**2014**, 59, 1690–1703. [Google Scholar] [CrossRef] - Kollat, J.B.; Reed, P.M.; Wagener, T. When are multiobjective calibration trade-offs in hydrologic models meaningful? Water Resour. Res.
**2012**, 48, W3520. [Google Scholar] [CrossRef] - Medina, Y.; Muñoz, E. Estimation of Annual Maximum and Minimum Flow Trends in a Data-Scarce Basin. Case Study of the Allipén River Watershed, Chile. Water
**2020**, 12, 162. [Google Scholar] [CrossRef][Green Version] - Seibert, J.; Vis, M.J.P. Teaching hydrological modeling with a user-friendly catchment-runoff-model software package. Hydrol. Earth Syst. Sci.
**2012**, 16, 3315–3325. [Google Scholar] [CrossRef][Green Version] - World Meteorological Organization. Standardized Precipitation Index User Guide. Available online: https://public.wmo.int/en/resources/library/standardized-precipitation-index-user-guide (accessed on 24 June 2020).
- Abebe, N.A.; Ogden, F.L.; Pradhan, N.R. Sensitivity and uncertainty analysis of the conceptual HBV rainfall-runoff model: Implications for parameter estimation. J. Hydrol.
**2010**, 389, 301–310. [Google Scholar] [CrossRef] - Zelelew, M.B.; Alfredsen, K. Sensitivity-guided evaluation of the HBV hydrological model parameterization. J. Hydroinform.
**2013**, 15, 967–990. [Google Scholar] [CrossRef] - Massmann, C.; Wagener, T.; Holzmann, H. A new approach to visualizing time-varying sensitivity indices for environmental model diagnostics across evaluation time-scales. Environ. Model. Softw.
**2014**, 51, 190–194. [Google Scholar] [CrossRef]

**Figure 1.**Code flow diagram. (

**a**) Input window. (

**b**) Objective function values. (

**c**) Cumulative distribution functions of B and NB groups. (

**d**) Graph of the MVD index at each window.

**Figure 3.**Conceptual diagram of the Hydrologiska Byråns Vattenbalansavdelning (HBV) model [47].

**Figure 4.**Convergence analysis performed for the (

**a**) Regional Sensitivity Analysis (RSA), (

**b**) PAWN, (

**c**) Morris, and (

**d**) Sobol methods. Lines indicate the sensitivity index related to each model parameter as a function of number of evaluations.

**Figure 5.**(

**a**) Standardized precipitation index (SPI) calculated for the 1952–2010 period. The left vertical axis shows the SPI values and the horizontal axis, time in years. The right vertical axis indicates the categories. The orange areas correspond to dry years, blue to wet years, and gray to normal years. (

**b**) Maximum vertical distance (MVD) index for each year (5-year window). The left vertical axis shows the parameters, the horizontal axis, shows the time in years, and the right vertical axis shows the streamflow in m

^{3}/s as a reference. The dotted red line is the daily mean streamflow series. The MVD values are seen in grayscale.

**Figure 6.**MVD calculated in windows of (

**a**) 3 months (1990–1999) and (

**b**) 31 days (1992–1994). The left vertical axis shows the parameters, the horizontal axis shows the time, and the right vertical axis shows the streamflow in m

^{3}/s. The dotted red line is the daily mean streamflow series (1990–1999). The MVD values are seen in grayscale. The white areas indicate that the index cannot be calculated.

Parameter | Description | Range |
---|---|---|

Mass balance | ||

A | Precipitation modification parameter | 0.8–2.5 |

Snow Module | ||

TT (°C) | Threshold temperature that indicates the initiation of snowmelt (normally, 0 °C) | 0 |

Cmelt $\left({\mathrm{mm}}^{\xb0}{\mathrm{C}}^{-1}{\mathrm{day}}^{-1}\right)$ | Fraction of snow that melts above the threshold temperature (TT) from the beginning of snowmelt. | 0.5–7 |

Moisture module | ||

FC (mm) | Field capacity (storage in the soil layer) | 0–2000 |

$\mathsf{\beta}$ | Empirical coefficient that represents the soil moisture variation in the area | 0–7 |

LP | Fraction of field capacity to calculate the permanent wilting point (PWP = LP × FC) | 0.3–1 |

C (${}^{\xb0}{\mathrm{C}}^{-1}$) | Correction factor for potential evapotranspiration | 0.01–0.3 |

Response module | ||

L (mm) | Threshold for quick runoff response | 0–100 |

$\mathrm{k}0$ (${\mathrm{day}}^{-1}$) | Quick response coefficient (upper reservoir) | 0.3–0.6 |

$\mathrm{k}1$ (${\mathrm{day}}^{-1}$) | Slow response coefficient (upper reservoir) | 0.1–0.2 |

$\mathrm{k}2$ (${\mathrm{day}}^{-1}$) | Lower reservoir response coefficient | 0.01–0.1 |

$\mathrm{kp}$ (${\text{}\mathrm{day}}^{-1}$) | Maximum flow coefficient for percolation | 0.01–0.1 |

SPI Values | Category |
---|---|

2 and above | Extremely wet (EM) |

1.5 to 1.99 | Severely wet (SM) |

1.0 to 1.49 | Moderately wet (MM) |

−0.99 to 0.99 | Normal or near normal (NN) |

−1.0 to −1.49 | Moderate drought (MD) |

−1.5 to −1.99 | Severe drought (SD) |

−2 and below | Extreme drought (ED) |

**Table 3.**Number of evaluations and computing costs associated with performing a single sensitivity analysis using different methods.

Method | Evaluations to Reach Convergence | Time of Model Evaluation (s) | Time for SI Calculation and Objective Functions (s) | Memory (%) |
---|---|---|---|---|

RSA | 10,000 | 122 | 5.55 | 75 |

PAWN | 19,000 | 231 | 7.36 | 98 |

Morris | >22,000 * | 246 | – | Out of memory |

Sobol (TE) | >22,000 * | 246 | – | Out of memory |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Medina, Y.; Muñoz, E.
A Simple Time-Varying Sensitivity Analysis (TVSA) for Assessment of Temporal Variability of Hydrological Processes. *Water* **2020**, *12*, 2463.
https://doi.org/10.3390/w12092463

**AMA Style**

Medina Y, Muñoz E.
A Simple Time-Varying Sensitivity Analysis (TVSA) for Assessment of Temporal Variability of Hydrological Processes. *Water*. 2020; 12(9):2463.
https://doi.org/10.3390/w12092463

**Chicago/Turabian Style**

Medina, Yelena, and Enrique Muñoz.
2020. "A Simple Time-Varying Sensitivity Analysis (TVSA) for Assessment of Temporal Variability of Hydrological Processes" *Water* 12, no. 9: 2463.
https://doi.org/10.3390/w12092463