# A Review of Tank Model and Its Applicability to Various Korean Catchment Conditions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}. In addition, three optimization algorithms—dynamically dimensioned search (DDS), robust parameter estimation (ROPE), and shuffled complex evolution (SCE)—are selected to test whether the model parameters can be optimized consistently within a narrower range than the uncertainty bounds. From the uncertainty analysis, it is found that there is limited success in refining the priori distributions of the model parameters, indicating there is a high degree of equifinality for some parameters or at least there are large numbers of parameter combinations leading to good solutions within model’s uncertainty bounds. Out of the three optimization algorithms, SCE meets the criteria of the consistency best. It is also found that there are still some parameters that even the SCE method struggles to refine the priori distributions. It means that their contribution to model results is minimal and can take a value within a reasonable range. It suggests that the model may be reconceptualized to be parsimonious and to rationalize some parameters without affecting model’s capacity to replicate historical flow characteristics. Cross-validation indicates that sensitive parameters to catchment characteristics can be transferred when geophysical similarity exists between two catchments. Regionalization can be further improved by using a regression or geophysical similarity-based approach to transfer model parameters to ungauged catchments. It may be beneficial to categorize the model parameters depending on the level of their sensitivities, and a different approach to each category may be applied to regionalize the calibrated parameters.

## 1. Introduction

^{2}. In Section 4, the calibrated sets of parameters and their corresponding results are viewed in conjunction with the model uncertainty and the sensitivity of the model’s parameters. Model validation and cross-validation are also presented in this section. Key findings and future work are discussed in Section 5.

## 2. Tank Model

## 3. Model Uncertainty and Optimization Algorithms

#### 3.1. Model Uncertainty Estimation

#### 3.2. Dynamically Dimensioned Search (DDS)

- Define a maximum number of iterations, $I$.
- Populate an initial set of $N$ parameters, ${X}_{N}=\left({x}_{1},{x}_{2},\cdots ,{x}_{N}\right)$.
- Calculate hydrologic performance $F\left({X}_{N}\right)$ and allocate the best performing set to ${F}^{\mathrm{best}}$ and ${X}^{\mathrm{best}}$.
- Generate a random number for each parameter space and select all parameter sets for perturbation when their random numbers are bigger than $P\left(i\right)=1-\mathrm{ln}\left(i\right)/\mathrm{ln}\left(I\right)$, where $i$ is the current iteration.
- Generate ${x}_{j}^{*}$ by perturbate ${x}_{j}^{best}$ for the selected parameters from Step 4 with a standard normal random variable of $N\left(0,1\right)$ as ${x}_{j}^{*}={x}_{j}^{best}+{\delta}_{j}N\left(0,1\right)$, where $j=1,2,\cdots ,M$, ${\delta}_{j}=r\left({x}_{j}^{\mathrm{max}}-{x}_{j}^{\mathrm{min}}\right)$, $r$ is the parameter determining the perturbation range and $M$ is the total number of the parameter sets selected in Step 4.
- ${F}^{\mathrm{best}}=F\left({X}^{*}\right)$ and ${X}^{\mathrm{best}}={X}^{*}$, if $F\left({X}^{*}\right)>{F}^{best}$.
- Go to Step 4 until the predefined maximum iteration is reached.

#### 3.3. Robust Parameter Estimation (ROPE)

- Select random $N$ data sets, ${X}_{N}$.
- Measure their hydrological performances.
- Select best performing sets (10% of the initial sets), ${X}_{N}^{*}$.
- Calculate the depths of every point in ${X}_{N}$ with respect to ${X}_{N}^{*}$.
- Generate $M$ random parameter sets, ${Y}_{M}$ such that the sets have higher depths with respect to ${X}_{N}^{*}$.
- Replace ${X}_{N}$ with ${Y}_{M}$.
- Repeat Steps 2–6 until the performance from two samples are not significantly different or specified maximum iteration numbers are exceeded.

#### 3.4. Shuffled Complex Evolution (SCE)

- Generate $s$ samples using a uniform probability distribution from users defined bounds.
- Sort performances of the samples in increasing order.
- Divide the samples into $p$ partitions with $m$ points in each partition in a way that the n
^{th}partition contains every $\left(p\left(k-1\right)+n\right)$ ranked point, where $k=1,2,\cdots ,m$. - Evolve each complex based on the competitive complex evolution (CCE, [35]).
- Combine the evolved points into a single sample then repeat Steps 2–5 until convergence criteria are satisfied.

## 4. Application

^{2}where annual average rainfall of 1200~1600 mm per year is received on mountainous terrain leading to a high yield of runoffs.

#### 4.1. Parameter Sensitivity and Uncertainty

^{2}, Bias and NSE) mentioned earlier are used to develop posterior distributions for the GLUE approach. R

^{2}measures the deviation of modeled and observed data sets and is defined as:

^{6}parameter sets are created randomly in a uniform distribution. Simulated results using the initial parameter sets are compared with observed discharges by using the three performance measures. All parameter sets resulting in the performance measures within acceptable ranges are kept for developing posterior distribution. The acceptable ranges in this study are defined as ${R}^{2}>0.7$, $-15{\mathrm{m}}^{3}/\mathrm{sec}\le \mathrm{Bias}\le 15{\mathrm{m}}^{3}/\mathrm{sec}$ and $\mathrm{NSE}>0.5$.

#### 4.2. Comparison of the Optimization Algorithms

^{2}values indicate that the algorithms can sufficiently match historical data, it seems that modeled results are biased toward to underprediction given BIAS being negative and MAE higher than the other stations.

#### 4.3. Validation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- MLIT. Long Term Water Resource Plan (2001-2020), 3rd ed.; Ministry of Land, Infrastructure and Transport (MLIT); Government of South Korea: Seoul, Korea, 2016. [Google Scholar]
- Sugawara, M.; Fuyuki, M. A Method of Revision of River Discharge by Means of a Rainfall Model. In A Collection of Research Papers about Forecasting Hydrologic Variables; The Geosphere Research Institute of Saitama University: Saitama, Japan, 1956; pp. 14–18. [Google Scholar]
- Yokoo, Y.; Kazama, S.; Sawamoto, M.; Nishimura, H. Regionalization of lumped water balance model parameters based on multiple regression. J. Hydrol.
**2001**, 246, 209–222. [Google Scholar] [CrossRef] - Powell, M.J.D. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Comput. J.
**1964**, 7, 155–162. [Google Scholar] - Chen, R.-S.; Pi, L.-C.; Hsieh, C.-C. Application of parameter optimization method for calibrating tank model. J. Am. Water. Resour. Assoc.
**2005**, 41, 389–402. [Google Scholar] [CrossRef] - Duan, Q.; Sorooshian, S.; Gupta, V.K. Effective and efficient global optimization for conceptual rainfall-runoff models. Water Resour. Res.
**1992**, 28, 1015–1031. [Google Scholar] [CrossRef] - Kim, T.; Jung, I.W.; Koo, B.Y.; Bae, D.H. Optimization of Tank model parameters using multi-objective genetic Algorithm (I): Methodology and model formulation. J. Korea Water Resour. Assoc.
**2007**, 40, 677–685. [Google Scholar] [CrossRef] [Green Version] - Park, C.I.; Baek, C.W.; Jun, H.D.; Kim, J.H. Parameter estimation of Tank model by data interval and rainfall factors for dry season. J. Korean Soc Water Qual.
**2006**, 22, 856–864. [Google Scholar] - Beven, K.J.; Binley, A.M. The future of distributed models: model calibration and uncertainty prediction. Hydrol. Process.
**1992**, 6, 279–298. [Google Scholar] [CrossRef] - Vrugt, J.A.; ter Braak, C.J.F.; Clark, M.P.; Hyman, J.M.; Robinson, B.A. Treatment of input uncertainty in hydrologic modeling: doing hydrology backward with Markov chain Monte Carlo simulation. Water Resour. Res.
**2008**, 44, W00B09. [Google Scholar] [CrossRef] [Green Version] - Vrugt, J.A.; Ter Braak, C.J.F.; Diks, C.G.H.; Robinson, B.A.; Hyman, J.M.; Higdon, D. Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int. J. Nonlin. Sci. Num.
**2009**, 10, 273–290. [Google Scholar] - Kuczera, G.; Renard, B.; Thyer, M.; Kavetski, D. There are no hydrological monsters, just models and observations with large uncertainties. Hydrolog. Sci. J.
**2010**, 55, 980–991. [Google Scholar] [CrossRef] [Green Version] - Kang, S.U.; Lee, D.R.; Lee, S.H. A study on calibration of Tank model with soil moisture structure. J. Korea Water Resour. Assoc.
**2004**, 37, 133–144. [Google Scholar] [CrossRef] - Lee, S.H.; Kang, S.U. A parameter regionalization study of a modified Tank model using characteristic factors of watersheds. J. Korean Soc. Civ. Eng.
**2007**, 27, 379–385. [Google Scholar] - Kang, M.G.; Lee, J.H.; Park, K.W. Parameter regionalization of a Tank model for simulating runoffs from ungauged watersheds. J. Korea Water Resour. Assoc.
**2013**, 46, 519–530. [Google Scholar] [CrossRef] [Green Version] - Sugawara, M. Tank model. In Computer Models of Watershed Hydrology; Singh, V.P., Ed.; Water Resources Publishers: Berlin/Heidelberg, Germany, 1995; pp. 165–214. [Google Scholar]
- Stedinger, J.R.; Vogel, R.M.; Lee, S.U.; Batchelder, R. Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resour. Res.
**2008**, 44, W00B06. [Google Scholar] [CrossRef] - Choi, H.T.; Beven, K. Multi-period and multi-criteria model conditioning to reduce prediction uncertainty in an application of TOPMODEL within the GLUE framework. J. Hydrol.
**2007**, 332, 316–336. [Google Scholar] - Houska, T.; Multsch, S.; Kraft, P.; Frede, H.-G.; Breuer, L. Monte Carlo-based calibration and uncertainty analysis of a coupled plant growth and hydrological model. Biogeosciences
**2014**, 11, 2069–2082. [Google Scholar] [CrossRef] [Green Version] - Hellweger, F.L.; Lall, U. Modeling the effect of algal dynamics on arsenic speciation in Lake Biwa. Environ. Sci. Technol.
**2004**, 38, 6716–6723. [Google Scholar] [CrossRef] - Smith, R.M.S.; Evans, D.J.; Wheater, H.S. Evaluation of two hybrid metric-conceptual models, for simulating phosphorus transfer from agricultural land in the river enborne, a lowland UK catchment. J. Hydrol.
**2005**, 304, 366–380. [Google Scholar] [CrossRef] - Blasone, R.-S.; Vrugt, J.A.; Madsen, H.; Rosbjerg, D.; Robinson, B.A.; Zyvoloski, G.A. Generalized likelihood uncertainty estimation (GLUE) using adaptive Markov Chain Monte Carlo sampling. Adv. Water. Resour.
**2008**, 31, 630–648. [Google Scholar] - Vrugt, J.A. Markov chain Monte Carlo simulation using the DREAM software package: Theory, concepts, and MATLAB implementation. Environ. Model. Softw.
**2016**, 75, 273–316. [Google Scholar] [CrossRef] [Green Version] - Houska, T.; Kraft, P.; Chamorro-Chavez, A.; Breuer, L. SPOTting Model Parameters Using a Ready-Made Python Package. PLoS ONE
**2015**, 10, e0145180. [Google Scholar] [CrossRef] [PubMed] - Tolson, B.A.; Shoemaker, C.A. Dynamically dimensioned search algorithm for computationally efficient watershed model calibration. Water Resour. Res.
**2007**, 43, W01413. [Google Scholar] [CrossRef] - Duan, Q.; Gupta, V.K.; Sorooshian, S. Shuffled complex evolution approach for effective and efficient global minimization. J. Optimiz. Theory App.
**1993**, 76, 501–521. [Google Scholar] [CrossRef] - Jungnickel, D. The Greedy Algorithm. In Graphs, Networks and Algorithms. Algorithms and Computation in Mathematics; Springer: Berlin/Heidelberg, Germany, 1999; Volume 5. [Google Scholar]
- Bárdossy, A.; Singh, S.K. Robust estimation of hydrological model parameters. Hydrol. Earth Syst. Sci.
**2008**, 12, 1273–1283. [Google Scholar] - Tukey, J. Mathematics and picturing data. Proc. Int. Ernation. Cong. Math.
**1975**, 2, 523–531. [Google Scholar] - Bárdossy, A. Calibration of hydrological model parameters for ungauged catchments. Hydrol. Earth Syst. Sci.
**2007**, 11, 703–710. [Google Scholar] - Serfling, R. Generalized quantile processes based on multivariate depth functions, with applications in nonparametric multivariate analysis. J. Multivar. Anal.
**2002**, 83, 232–247. [Google Scholar] [CrossRef] [Green Version] - Cheng, A.Y.; Liu, R.Y.; Luxhøj, J.T. Monitoring multivariate aviation safety data by data depth: control charts and threshold systems. IIE Trans.
**2002**, 32, 861–872. [Google Scholar] [CrossRef] - Liu, R.Y. Control charts for multivariate processes. J. Am. Stat. Assoc.
**1995**, 90, 1380–1387. [Google Scholar] [CrossRef] - Chebana, F.; Ouarda, T.B.M.J. Depth and homogeneity in regional flood frequency analysis. Water Resour. Res.
**2008**, 44, W11422. [Google Scholar] [CrossRef] [Green Version] - Nelder, J.A.; Mead, R. A simplex method for function minimization. Comput. J.
**1965**, 7, 308–313. [Google Scholar] [CrossRef] - Duan, Q.; Sorooshian, S.; Gupta, V.K. Optimal use of the SCE-UA global optimization method for calibrating watershed models. J. Hydrol.
**1994**, 158, 265–284. [Google Scholar] [CrossRef] - Sorooshian, S.; Duan, Q.; Gupta, V.K. Calibration of rainfall-runoff models’ application of global optimization to the Sacramento soil moisture accounting model. Water Resour. Res.
**1993**, 29, 1185–1194. [Google Scholar] [CrossRef] - Abudulla, F.A.; Lettenmaier, D.P. Development of regional parameter estimation equations for a macroscale hydrologic model. J. Hydrol.
**1997**, 197, 230–257. [Google Scholar] [CrossRef] - Vrugt, J.A.; Gupta, H.V.; Bastidas, L.A.; Bouten, W.; Sorooshian, S. Effective and efficient algorithm for multiobjective optimization of hydrologic models. Water Resour. Res.
**2003**, 39, 1–19. [Google Scholar] [CrossRef] [Green Version] - Arsenault, R.; Brissette, F.P. Continuous streamflow prediction in ungauged basins: The effects of equifinality and parameter set selection on uncertainty in regionalization approaches. Water Resour. Res.
**2014**, 50, 6135–6153. [Google Scholar] [CrossRef]

**Figure 1.**Conceptual rainfall-runoff model with four serially connected tanks (A, B, C and D) with soil moisture stores at the top Tank A.

**Figure 2.**Map of South Korea at the top panel and zoomed area showing the location of the selected five stations for model calibration and their upstream catchments.

**Figure 3.**GLUE derived posterior distribution of parameters for Station 1,002,640 (

**a**) ${O}_{A1}$ (outlet constraint of the lower outlet at Tank A, initially ranged from 0~0.5), (

**b**) ${K}_{2}$ (soil moisture exchange rates between the top two tanks, initially ranged from 0~100), (

**c**) ${H}_{A2}$ (storage level to the upper outlet at Tank A, initially ranged from 0~150 m) and (

**d**) ${H}_{c}$ (storage level to the outlet at Tank C, initially ranged from 0~20 m).

**Figure 4.**DREAM derived posterior distribution of parameters for Station 1,002,640 (

**a**) ${O}_{A1}$ (outlet constraint of the lower outlet at Tank A, initially ranged from 0~0.5), (

**b**) ${K}_{2}$ (soil moisture exchange rates between the top two tanks, initially ranged from 0~100), (

**c**) ${H}_{A2}$ (storage level to the upper outlet at Tank A, initially ranged from 0~150 m) and (

**d**) ${H}_{c}$ (storage level to the outlet at Tank C, initially ranged from 0~20 m).

**Figure 5.**Uncertainty ranges of 95% simulations derived from the GLUE and DREAM approaches using the tank model with observed discharges (∙) and areal averaged precipitation by the Thiessen polygon network at Station 1002640.

**Figure 6.**GLUE derived ranges of the four selected parameters across the five calibration stations ((

**a**) ${O}_{A1}$, (

**b**) ${K}_{2}$, (

**c**) ${H}_{A2}$ and (

**d**) ${H}_{C}$).

**Figure 7.**DREAM derived ranges of the four selected parameters across the five calibration stations ((

**a**) ${O}_{A1}$, (

**b**) ${K}_{2}$, (

**c**) ${H}_{A2}$ and (

**d**) ${H}_{C}$).

**Figure 8.**Comparison of convergence rates measured by Nash–Sutcliffe efficiency (NSE) between the three optimization algorithms at Station 1,002,640 ((

**a**) 20,000 iterations and (

**b**) 5000 iterations).

**Figure 9.**Comparison of calibrated results using the three optimization algorithms with their performance measure of NSE values at Station 1002640.

**Figure 10.**Comparison of the NSE measures developed from 100 sets of randomly selected initial parameters for the three optimization algorithms across the five calibration stations ((

**a**) Station 1002640, (

**b**) Station 1003630, (

**c**) Station 1011690, (

**d**) Station 1303680 and (

**e**) Station 3009650).

**Figure 11.**Ranges of two most sensitive parameters (

**a**,

**b**) and one least sensitive parameter (

**c**) developed from 100 different initial parameters using the dynamically dimensioned search (DDS) algorithm.

**Figure 12.**Ranges of two most sensitive parameters (

**a**,

**b**) and one least sensitive parameter (

**c**) developed from 100 different initial parameters using the robust parameter estimation (ROPE) algorithm.

**Figure 13.**Ranges of two most sensitive parameters (

**a**,

**b**) and one least sensitive parameter (

**c**) developed from 100 different initial parameters using the shuffled complex evolution (SCE) algorithm

**Figure 14.**Cross-validation of three sites (Station 1: 1002640, Station 2: 1003630 and Station 5: 3009650) where calibrated parameters at one site are applied to the other sites. It is indicated by the shaded background when the calibrated parameters are reapplied to the same site. Model performance is reporting by the median and variation of NSE values from 100 simulations.

**Table 1.**Five calibration stations used in this study with their key geographical characteristics (catchment size, average slop, and maximum of standard deviation (SD) of altitude), annual total rainfall and years used for calibration and verification.

Station No (Station Name) | Longitude/Latitude | Catchment Area (km^{2}) | Average Slop (%) | Altitude Max/ SD (m) | Rainfall (mm/yr) | Calibration Year | Validation Year |
---|---|---|---|---|---|---|---|

1002640 (Sangbangrim) | 128.42/ 37.43 | 527.9 | 47.9 | 1574.7/183.2 | 1336 | 2011 | 2012 |

1003630 (Osa Ri) | 128.51/ 37.10 | 4786.2 | 49.6 | 1574.6/238.8 | 1246 | 2012 | 2013 |

1011690 (Wolhak Ri) | 128.21/ 38.12 | 301.1 | 63.3 | 1701.5/262.0 | 1587 | 2011 | 2014 |

1303680 (Osipcheon Br.) | 129.23/ 37.70 | 371.7 | 58.1 | 1353.8/252.6 | 1249 | 2018 | 2012 |

3009650 (Youngchon Br.) | 127.32/ 36.25 | 83.4 | 44.2 | 872.0/126.6 | 1313 | 2011 | 2016 |

**Table 2.**Comparison of various performance measurements for the calibration stations with the number of iterations set to 20,000. The performance measurements include Nash–Sutcliffe efficiency (NSE), bias, coefficient of determination (${R}^{2}$) and mean absolute error (MAE).

Station | NSE | $\mathbf{BIAS}\text{}\left({\mathbf{m}}^{3}/\mathbf{sec}\right)$ | R^{2} | $\mathbf{MAE}\text{}\left({\mathbf{m}}^{3}/\mathbf{sec}\right)$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DDS | ROPE | SCE | DDS | ROPE | SCE | DDS | ROPE | SCE | DDS | ROPE | SCE | |

1002640 | 0.90 | 0.90 | 0.90 | 0.5 | 2.1 | 0.8 | 0.92 | 0.90 | 0.92 | 9.1 | 9.4 | 9.1 |

1003630 | 0.91 | 0.89 | 0.90 | −0.2 | −15.4 | −0.3 | 0.91 | 0.89 | 0.90 | 35.1 | 44.0 | 36.7 |

1011690 | 0.78 | 0.82 | 0.82 | 0.7 | 0.8 | 0.8 | 0.79 | 0.87 | 0.86 | 7.6 | 8.3 | 8.0 |

1303680 | 0.82 | 0.77 | 0.81 | 5.7 | 6.2 | 5.7 | 0.85 | 0.83 | 0.85 | 8.8 | 9.6 | 9.1 |

3009650 | 0.85 | 0.80 | 0.82 | 1.3 | 1.9 | 1.2 | 0.87 | 0.83 | 0.86 | 2.5 | 3.1 | 2.7 |

**Table 3.**Validation of calibrated parameters to different events and its performance measured by NSE values (median and variation from 100 simulations with different initial parameter sets).

Algorithm | NSE | Station | ||||
---|---|---|---|---|---|---|

1002640 | 1003630 | 1011690 | 1303680 | 3009650 | ||

DDS | Median | 0.78 | 0.82 | 0.62 | 0.46 | 0.68 |

95%ile–5%ile | 0.08 | 0.07 | 0.10 | 0.51 | 0.25 | |

ROPE | Median | 0.80 | 0.82 | 0.51 | 0.50 | 0.70 |

95%ile–5%ile | 0.09 | 0.07 | 0.10 | 0.42 | 0.29 | |

SCE | Median | 0.79 | 0.81 | 0.53 | 0.66 | 0.77 |

95%ile–5%ile | 0.01 | 0.01 | 0.02 | 0.01 | 0.19 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 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**

Lee, J.W.; Chegal, S.D.; Lee, S.O.
A Review of Tank Model and Its Applicability to Various Korean Catchment Conditions. *Water* **2020**, *12*, 3588.
https://doi.org/10.3390/w12123588

**AMA Style**

Lee JW, Chegal SD, Lee SO.
A Review of Tank Model and Its Applicability to Various Korean Catchment Conditions. *Water*. 2020; 12(12):3588.
https://doi.org/10.3390/w12123588

**Chicago/Turabian Style**

Lee, Jong Wook, Sun Dong Chegal, and Seung Oh Lee.
2020. "A Review of Tank Model and Its Applicability to Various Korean Catchment Conditions" *Water* 12, no. 12: 3588.
https://doi.org/10.3390/w12123588