# A Modified Two-Parameter Monthly Water Balance Model for Runoff Simulation to Assess Hydrological Drought

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Modified TPMWB Model

_{t}, PE

_{t}, P

_{t}, and Q

_{t}represent the monthly actual evapotranspiration, potential evapotranspiration, rainfall, and runoff at the t-th month, respectively. NS

_{t}is the generalized effective/net water storage for generating runoff in the month, which can be calculated as (GS

_{t}

_{−1}+ P

_{t}− E

_{t}) with GS

_{t}

_{−1}being the generalized water content at the end of the (t − 1)-th month and the beginning of the t-th month, all with a unit of millimeter. tanh(∙) is the hyperbolic tangent function. After the runoff generation, the water content at the end of the t-th month was calculated according to the water conservation law (Equation (3)). C is a dimensionless parameter used to account for the effect of time-scale transformation, while SC is the second parameter representing the maximum soil water capacity of the catchment with a unit of millimeter. Note that water holding capacities of groundwater and unsaturated zone storage were not separated in order to keep the model structure simple without deviation from the runoff formation mechanism at a monthly time scale.

_{t}and runoff Q

_{t}from water availability NS

_{t}, as follows:

_{obs,i}and Q

_{sim,i}is the observed and simulated runoff, respectively, ${\overline{Q}}_{obs}$ is the mean value of Q

_{obs,i}, and n is length of calibration/validation period.

#### 2.2. Calculation of Standardized Drought Index

_{sim,t}(t = 1, 2,..., 12) is of concern and hereafter denoted as Q for short. In this paper, the traditional curve-fitting method is applied to model the theoretical PDF of simulated monthly runoff. Considering the significant skewness of monthly runoff series, especially in the dry season, together with the potential huge difference among runoff distributions in different months [22], a uniform distribution derived based on the principle of maximum entropy (POME) is adopted to fit the empirical frequencies of monthly runoff, without presupposing any PDF of underlying distributions [47,48].

_{Q}(q), then the Shannon entropy H(q) is defined as follows:

^{i}(i = 1, 2, 3) are the first to third power of q, while $\overline{{q}^{i}}$ (i = 1, 2, 3) are corresponding sampling original moments of q.

_{0}, λ

_{1}, λ

_{2}, and λ

_{3}are the Lagrange multipliers, ${\lambda}_{0}=\mathrm{ln}{\displaystyle \underset{a}{\overset{b}{\int}}\mathrm{exp}\left(-{\lambda}_{1}q-{\lambda}_{2}{q}^{2}-{\lambda}_{3}{q}^{3}\right)}dq.$

^{i}] (i = 1, 2, 3) are the expectations of q

^{i}obtained from observations. The Newton–Raphson method was then employed to iteratively calculate the Lagrange multiplier.

_{Q}(q) is constructed by making maximum use of the information collected from the simulated runoff data while avoiding redundant information as much as possible. Once the PDF is determined, the cumulative distribution function (CDF) can then be estimated by integrating the PDF as follows:

^{−1}(∙) is the inverse of a standard normal CDF.

#### 2.3. Identification of Drought Events and Characteristics

#### 2.4. Correlation Analysis between Drought Characteristics and Watershed Features

^{2}is used to investigate the strength of coupling.

## 3. Study Area and Data

#### 3.1. Study Area

^{2}(see Figure 2). The Danjiangkou (DJK) reservoir divides the whole basin into an upper and a mid-lower sub-basin. A sub-tropical monsoon climate and the varying topography from high mountains in the upper reach to relatively flat plains in the lower give rise to dramatic spatial–temporal diversity of water resources distribution in this area.

#### 3.2. Data

## 4. Results and Discussion

#### 4.1. Performance of the Modified TPMWB Model

_{t}, PE

_{t}, and P

_{t}, in consideration of a time-scale transformation from year to month. For the SC parameter, very robust SC values are obtained through the optimization procedure with almost all calibrated SC values distributing around 300 mm, except for a few outliers such as the Jiajiafang and Jiangwan sub-basins. It is shown that the SC parameter contains a wealth of information related to underlying surface properties of the catchment. Given the acceptable performance of the modified TPMWB model, it can be used to reproduce or predict the hydrographs of calibrated sub-basins with confidence.

#### 4.2. Hydrological Drought Assessment Based on Standardized Drought Indices

#### 4.2.1. Calculation of Standardized Drought Indices

#### 4.2.2. Drought Characteristics (DC) Analysis

_{SRI}− DC

_{SDI}]/DC

_{SDI}) of drought characteristics between the SRI and SDI are shown in Table 3. It is found that the occurrence of drought events estimated by SRI (43.13 times on average) is less frequent compared with that estimated by SDI (48.19 times on average), with a mean RB of −9.7%. We can also notice that the average drought duration (E[D]) for most sub-basins are, however, overestimated by the SRI as the mean E[D] calculated from SRI for the 30 sub-basins (4.03 months) is obviously longer than that from SDI (3.44 months) with a mean RB of 18.1%. Moreover, the maximum relative error of E[D] reaches 54.6%. This is similar for the average inter-arrival times (E[L]) with a mean and maximum RB of 11.7% and 41.5%, respectively. In fact, a few short and minor droughts would be removed, while several dependent droughts would be combined to a larger independent drought event when extracting drought events from the SRI series, since continuously simulated runoff series may eliminate the random fluctuations in observation series.

#### 4.3. Impacts of Watershed Features on Drought Characteristics

_{t}/NS

_{t}= (NS

_{t}− Q

_{t})/SC], when the watershed is experiencing a meteorological drought, which begins with a decrease in precipitation or a rise in temperature, more available water is used to refill the water shortage in soil and underground aquifers, so hydrological droughts may last longer. The smaller the Runoff Coefficient RC, the less precipitation that can be transferred into runoff; when a meteorological drought occurs in the watershed, it would be necessary to accumulate precipitation not lower than the normal level for a long time to restore the hydrological drought to a normal level, resulting in longer drought duration. The larger the Aridity Index AI, the drier the watershed; when the temperature increases, the evaporation capacity increases and the precipitation decreases, thus the watershed with a large AI may further amplify the drought. Since parameter C is the ratio of E

_{t}/PE

_{t}, and the compression transformation tanh(P

_{t}/PE

_{t}), when the precipitation is small and evapotranspiration is large, that is, in the dry period, in the basin with large C, the actual evapotranspiration accounts for a higher proportion of evapotranspiration capacity, which results in a relatively small runoff.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Observed and simulated monthly runoff hydrographs at: (

**a**) Sheqi; (

**b**) Taikou; (

**c**) Wanyugou; (

**d**) Youshuijie.

**Figure 4.**Standardized hydrological index series calculated from simulated runoff (SRI) and observed streamflow (SDI) at: (

**a**) Sheqi; (

**b**) Taikou; (

**c**) Wanyugou; (

**d**) Youshuijie.

No. | Watershed | Area (km ^{2}) | Data Period | P (mm∙yr ^{−1}) | PE (mm∙yr ^{−1}) | Q (mm∙yr ^{−1}) | AI (-) | RC (-) |
---|---|---|---|---|---|---|---|---|

1 | Baitugang | 1134 | 1963–1989 | 923 | 864 | 389 | 0.94 | 0.42 |

2 | Baiyan | 690 | 1965–1989 | 779 | 708 | 286 | 0.91 | 0.37 |

3 | Bandian | 425 | 1955–1989 | 742 | 896 | 176 | 1.21 | 0.24 |

4 | Caodian | 683 | 1960–1987 | 1142 | 1205 | 443 | 1.06 | 0.39 |

5 | Chadianzi | 1683 | 1967–1989 | 868 | 624 | 319 | 0.72 | 0.37 |

6 | Chaiping | 2364 | 1969–1989 | 845 | 880 | 369 | 1.04 | 0.44 |

7 | Dazhuhe | 2651 | 1968–1989 | 1229 | 735 | 899 | 0.60 | 0.73 |

8 | Guihuayuan | 1275 | 1964–1989 | 859 | 721 | 446 | 0.84 | 0.52 |

9 | Houhui | 816 | 1958–1989 | 902 | 849 | 318 | 0.94 | 0.35 |

10 | Huayuan | 2601 | 1964–1987 | 1113 | 1207 | 356 | 1.08 | 0.32 |

11 | Jiajiafang | 1281 | 1960–1989 | 839 | 885 | 279 | 1.05 | 0.33 |

12 | Jiangwan | 781 | 1959–1989 | 849 | 890 | 309 | 1.05 | 0.36 |

13 | Kouzihe | 421 | 1963–1989 | 863 | 865 | 316 | 1.00 | 0.37 |

14 | Lianghekou | 2816 | 1967–1989 | 889 | 716 | 438 | 0.81 | 0.49 |

15 | Miping | 1404 | 1967–1989 | 794 | 875 | 209 | 1.10 | 0.26 |

16 | Nankuanping | 3936 | 1965–1989 | 773 | 869 | 264 | 1.12 | 0.34 |

17 | Nanshahe | 243 | 1967–1989 | 1208 | 585 | 705 | 0.48 | 0.58 |

18 | Pingshi | 748 | 1960–1989 | 975 | 1010 | 336 | 1.04 | 0.34 |

19 | Qingfeng | 2082 | 1963–1989 | 931 | 875 | 298 | 0.94 | 0.32 |

20 | Qingniwan | 1377 | 1965–1989 | 798 | 869 | 293 | 1.09 | 0.37 |

21 | Shengxiancun | 2143 | 1961–1988 | 890 | 738 | 414 | 0.83 | 0.47 |

22 | Sheqi | 1044 | 1966–1989 | 796 | 981 | 200 | 1.23 | 0.25 |

23 | Shuhe | 581 | 1969–1989 | 881 | 851 | 391 | 0.97 | 0.44 |

24 | Taikou | 2073 | 1965–1989 | 883 | 888 | 466 | 1.01 | 0.53 |

25 | Tiesuoguan | 433 | 1966–1989 | 1106 | 632 | 611 | 0.57 | 0.55 |

26 | Wanyugou | 560 | 1965–1989 | 1013 | 996 | 494 | 0.98 | 0.49 |

27 | Wuguan | 724 | 1959–1989 | 772 | 888 | 227 | 1.15 | 0.29 |

28 | Xianhekou | 772 | 1966–1989 | 974 | 733 | 489 | 0.75 | 0.50 |

29 | Xinzhou | 4660 | 1964–1989 | 1055 | 830 | 598 | 0.79 | 0.57 |

30 | Youshuijie | 911 | 1961–1989 | 939 | 734 | 536 | 0.78 | 0.57 |

No. | Watershed | C (-) | SC (-) | NSE (%) | RE (%) |
---|---|---|---|---|---|

1 | Baitugang | 0.94 | 300.63 | 0.83. | 10.4 |

2 | Baiyan | 0.98 | 301.01 | 0.75 | 12.8 |

3 | Bandian | 1.14 | 301.11 | 0.80 | 4.1 |

4 | Caodian | 0.85 | 302.08 | 0.84 | 15.5 |

5 | Chadianzi | 1.23 | 300.61 | 0.85 | 21.9 |

6 | Chaiping | 0.76 | 302.35 | 0.86 | 10.2 |

7 | Dazhuhe | 0.36 | 301.84 | 0.84 | 7.7 |

8 | Guihuayuan | 0.62 | 300.63 | 0.74 | 11.0 |

9 | Houhui | 1.03 | 302.78 | 0.82 | 7.8 |

10 | Huayuan | 0.95 | 302.08 | 0.81 | 15.2 |

11 | Jiajiafang | 1.03 | 390.16 | 0.80 | 1.8 |

12 | Jiangwan | 1.00 | 512.38 | 0.78 | 2.7 |

13 | Kouzihe | 1.02 | 301.83 | 0.86 | 14.4 |

14 | Lianghekou | 1.22 | 302.35 | 0.83 | 18.7 |

15 | Miping | 1.15 | 300.93 | 0.81 | 5.7 |

16 | Nankuanping | 0.93 | 300.63 | 0.84 | 12.2 |

17 | Nanshahe | 0.91 | 301.81 | 0.86 | 17.8 |

18 | Pingshi | 1.00 | 300.63 | 0.75 | 13.1 |

19 | Qingfeng | 1.09 | 303.66 | 0.80 | 2.1 |

20 | Qingniwan | 0.90 | 305.02 | 0.85 | 12.3 |

21 | Shengxiancun | 0.86 | 301.01 | 0.76 | 30.4 |

22 | Sheqi | 1.13 | 306.95 | 0.88 | 5.9 |

23 | Shuhe | 0.89 | 300.63 | 0.86 | 4.6 |

24 | Taikou | 0.73 | 305.02 | 0.81 | 1.2 |

25 | Tiesuoguan | 0.85 | 300.63 | 0.86 | 22.1 |

26 | Wanyugou | 0.75 | 300.93 | 0.86 | 5.8 |

27 | Wuguan | 1.09 | 302.08 | 0.82 | 6.7 |

28 | Xianhekou | 0.67 | 302.08 | 0.81 | 13.2 |

29 | Xinzhou | 0.69 | 303.13 | 0.82 | 3.8 |

30 | Youshuijie | 0.65 | 300.31 | 0.86 | 10.6 |

Ave. | 0.91 | 311.91 | 0.82 | 10.7 | |

Min. | 0.36 | 300.31 | 0.74 | 1.2 | |

Max. | 1.23 | 512.38 | 0.88 | 30.4 |

No. | Watershed | Number (Times) | E[D] (Month) | E[S] (-) | E[I] (-/Month) | E[L] (Month) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SRI | SDI | RB(%) | SRI | SDI | RB(%) | SRI | SDI | RB(%) | SRI | SDI | RB(%) | SRI | SDI | RB(%) | ||

1 | Baitugang | 50 | 53 | −5.6 | 3.42 | 3.29 | 4.0 | 2.53 | 2.34 | 8.0 | 0.53 | 0.67 | −21.3 | 6.42 | 6.04 | 6.3 |

2 | Baiyan | 44 | 48 | −8.8 | 3.70 | 3.23 | 14.6 | 2.69 | 2.40 | 11.9 | 0.51 | 0.64 | −20.6 | 6.75 | 6.13 | 10.0 |

3 | Bandian | 42 | 48 | −12.9 | 4.76 | 4.94 | −3.6 | 3.81 | 4.28 | −10.9 | 0.42 | 0.50 | −15.9 | 9.71 | 8.56 | 13.5 |

4 | Caodian | 46 | 58 | −20.2 | 4.22 | 3.15 | 33.7 | 2.97 | 2.86 | 3.8 | 0.54 | 0.55 | −0.5 | 6.93 | 5.53 | 25.5 |

5 | Chadianzi | 40 | 40 | 0.0 | 3.48 | 3.68 | −5.6 | 2.81 | 2.75 | 2.2 | 0.61 | 0.59 | 3.8 | 6.50 | 6.83 | −4.8 |

6 | Chaiping | 38 | 34 | 11.8 | 3.82 | 4.09 | −6.7 | 2.65 | 3.03 | −12.3 | 0.60 | 0.62 | −4.1 | 6.53 | 7.18 | −9.1 |

7 | Dazhuhe | 52 | 52 | 0.0 | 2.90 | 2.70 | 7.5 | 2.02 | 1.94 | 4.2 | 0.58 | 0.66 | −12.1 | 5.00 | 4.98 | 0.4 |

8 | Guihuayuan | 47 | 51 | −7.9 | 3.83 | 2.83 | 35.5 | 2.53 | 2.35 | 7.7 | 0.58 | 0.71 | −17.7 | 6.38 | 6.03 | 5.9 |

9 | Houhui | 49 | 59 | −16.4 | 4.53 | 3.61 | 25.4 | 3.02 | 3.27 | −7.5 | 0.51 | 0.61 | −16.3 | 7.78 | 6.48 | 20.0 |

10 | Huayuan | 41 | 44 | −7.7 | 3.85 | 3.38 | 14.1 | 2.88 | 3.05 | −5.7 | 0.52 | 0.50 | 4.3 | 6.59 | 6.30 | 4.6 |

11 | Jiajiafang | 44 | 61 | −27.3 | 4.32 | 3.21 | 34.4 | 3.19 | 3.01 | 5.8 | 0.48 | 0.53 | −8.9 | 8.02 | 5.90 | 36.1 |

12 | Jiangwan | 43 | 44 | −3.2 | 4.60 | 4.07 | 13.1 | 3.39 | 2.95 | 15.1 | 0.48 | 0.51 | −5.6 | 8.49 | 8.21 | 3.4 |

13 | Kouzihe | 46 | 53 | −13.1 | 4.02 | 3.67 | 9.5 | 2.73 | 2.97 | −8.1 | 0.49 | 0.53 | −8.3 | 6.96 | 6.02 | 15.5 |

14 | Lianghekou | 34 | 49 | −30.8 | 4.18 | 2.70 | 54.6 | 3.32 | 2.89 | 15.0 | 0.61 | 0.77 | −20.3 | 7.82 | 5.53 | 41.5 |

15 | Miping | 30 | 35 | −14.2 | 4.67 | 4.10 | 13.9 | 3.63 | 4.02 | −9.6 | 0.51 | 0.57 | −11.3 | 8.90 | 7.77 | 14.5 |

16 | Nankuanping | 39 | 37 | 5.4 | 4.49 | 4.48 | 0.1 | 3.07 | 3.32 | −7.5 | 0.51 | 0.56 | −9.9 | 7.59 | 7.89 | −3.8 |

17 | Nanshahe | 50 | 48 | 4.2 | 2.78 | 2.81 | −1.1 | 2.25 | 2.30 | −2.4 | 0.69 | 0.72 | −5.0 | 5.46 | 5.69 | −4.0 |

18 | Pingshi | 40 | 51 | −21.7 | 5.08 | 3.79 | 34.0 | 3.57 | 3.59 | −0.5 | 0.48 | 0.57 | −14.9 | 8.78 | 6.99 | 25.6 |

19 | Qingfeng | 34 | 49 | −30.8 | 4.79 | 3.21 | 49.3 | 3.78 | 3.42 | 10.6 | 0.53 | 0.63 | −16.9 | 9.18 | 6.50 | 41.1 |

20 | Qingniwan | 42 | 41 | 3.3 | 4.21 | 3.93 | 7.3 | 2.86 | 2.85 | 0.4 | 0.58 | 0.62 | −5.6 | 7.05 | 7.15 | −1.5 |

21 | Shengxiancun | 51 | 62 | −18.3 | 3.25 | 2.70 | 20.6 | 2.63 | 2.68 | −1.7 | 0.68 | 0.67 | 1.8 | 6.47 | 5.32 | 21.7 |

22 | Sheqi | 29 | 39 | −25.2 | 5.52 | 4.02 | 37.2 | 3.87 | 3.57 | 8.3 | 0.48 | 0.50 | −4.7 | 9.41 | 7.32 | 28.6 |

23 | Shuhe | 31 | 32 | −3.6 | 4.77 | 3.83 | 24.6 | 3.36 | 3.09 | 8.9 | 0.48 | 0.69 | −29.9 | 7.94 | 7.59 | 4.6 |

24 | Taikou | 46 | 48 | −4.6 | 3.83 | 2.97 | 28.7 | 2.52 | 2.34 | 7.7 | 0.54 | 0.72 | −25.5 | 6.43 | 6.07 | 6.0 |

25 | Tiesuoguan | 53 | 51 | 3.9 | 3.02 | 3.09 | −2.4 | 2.19 | 2.28 | −3.9 | 0.60 | 0.67 | −10.3 | 5.36 | 5.59 | −4.1 |

26 | Wanyugou | 49 | 52 | −5.8 | 3.43 | 2.85 | 20.3 | 2.40 | 2.87 | −16.4 | 0.56 | 0.70 | −19.9 | 6.06 | 5.65 | 7.3 |

27 | Wuguan | 42 | 52 | −19.2 | 4.62 | 3.74 | 23.6 | 3.51 | 3.62 | −3.0 | 0.49 | 0.59 | −18.0 | 8.67 | 7.05 | 22.9 |

28 | Xianhekou | 45 | 49 | −8.5 | 3.38 | 2.82 | 19.8 | 2.63 | 2.97 | −11.5 | 0.68 | 0.68 | 0.0 | 6.31 | 5.75 | 9.7 |

29 | Xinzhou | 50 | 55 | −8.8 | 3.38 | 2.72 | 24.3 | 2.45 | 2.14 | 14.1 | 0.62 | 0.72 | −14.1 | 6.02 | 5.61 | 7.3 |

30 | Youshuijie | 47 | 50 | −6.2 | 3.98 | 3.57 | 11.4 | 2.98 | 2.63 | 13.5 | 0.64 | 0.63 | 1.5 | 7.28 | 6.88 | 5.8 |

Ave. | 43.13 | 48.19 | −9.7 | 4.03 | 3.44 | 18.1 | 2.94 | 2.93 | 1.2 | 0.55 | 0.62 | −10.9 | 7.23 | 6.48 | 11.7 | |

Min. | 29 | 32 | −30.8 | 2.78 | 2.70 | −6.7 | 2.02 | 1.94 | −16.4 | 0.42 | 0.50 | −29.9 | 5.00 | 4.98 | −9.1 | |

Max. | 53 | 62 | 11.8 | 5.52 | 4.94 | 54.6 | 3.87 | 4.28 | 15.1 | 0.69 | 0.77 | 4.3 | 9.71 | 8.56 | 41.5 |

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## Share and Cite

**MDPI and ACS Style**

Hong, X.; Guo, S.; Chen, G.; Guo, N.; Jiang, C. A Modified Two-Parameter Monthly Water Balance Model for Runoff Simulation to Assess Hydrological Drought. *Water* **2022**, *14*, 3715.
https://doi.org/10.3390/w14223715

**AMA Style**

Hong X, Guo S, Chen G, Guo N, Jiang C. A Modified Two-Parameter Monthly Water Balance Model for Runoff Simulation to Assess Hydrological Drought. *Water*. 2022; 14(22):3715.
https://doi.org/10.3390/w14223715

**Chicago/Turabian Style**

Hong, Xingjun, Shenglian Guo, Guiya Chen, Na Guo, and Cong Jiang. 2022. "A Modified Two-Parameter Monthly Water Balance Model for Runoff Simulation to Assess Hydrological Drought" *Water* 14, no. 22: 3715.
https://doi.org/10.3390/w14223715