# Evaluation and Improvement of the Method for Selecting the Ridge Parameter in System Differential Response Curves

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}, with an average annual rainfall of 1702 mm and an annual streamflow coefficient ranging from 0.5 to 0.6. It experiences a subtropical monsoon humid climate, characterized by substantial vegetation coverage and shallow groundwater depths. Rainfall within the basin exhibits significant interannual variability and a spatial distribution that generally decreases from north to south. Earlier studies have effectively employed the SWAT model for runoff simulation in this basin [24]. For this study, meteorological data from nearby stations, including Shaowu, Qixianshan, Wuyishan, Pucheng, and Jianou meteorological stations were utilized. Furthermore, rainfall data were collected from 17 rainfall stations located within the basin, as depicted in Figure 2.

#### 2.2. Data

#### 2.3. Least Squares Method

#### 2.4. System Differential Response Curve (SRC)

#### 2.5. The Ideal Model

#### 2.6. Method for Selecting Ridge Parameters

#### 2.6.1. Ridge Trace Method

#### 2.6.2. L-Curve

- (1)
- Selecting different ridge parameters and calculating the corresponding $\left|\right|A\widehat{X}-b\left|\right|$ and $\left|\right|\widehat{X}\left|\right|$. In this study, the minimum ridge parameter is set to 0.001, the maximum value is set to 200, and the iteration step size is set to 0.01.
- (2)
- Plotting the residual item ($\left|\right|A\widehat{X}-b\left|\right|$) on the horizontal axis and the estimated norm ($\left|\right|\widehat{X}\left|\right|$) on the vertical axis. Subsequently, fitting the points into a curve will yield the computed L-curve.
- (3)
- Calculating the curvature of the curve and the ridge parameter corresponding to the point with the maximum curvature of the curve is the optimal ridge parameter.

#### 2.6.3. Constrained L-Curve (CL-Curve)

- (1)
- If all the constraint terms ${P}_{i}$ are greater than or equal to the absolute value of the negative of ${\widehat{X}}_{i}$, which means that the constraints are not in effect, then the L-curve can be obtained, and the result of the optimal ridge parameter is consistent with the traditional L-curve.
- (2)
- When the constraint term comes into effect and $\lambda $ approaches 0, the left side is no longer a line close to being perpendicular. Instead, as $\lambda $ gradually increases,$\left|\right|A{\widehat{X}}_{1}-b\left|\right|$ and $\left|\right|{\widehat{X}}_{1}\left|\right|$ decrease until a turning point appears in the fitted curve. This is different from the traditional L curve. Subsequently, as $\lambda $ continues to increase, $\left|\right|{\widehat{X}}_{1}\left|\right|$ keeps decreasing while $\left|\right|A{\widehat{X}}_{1}-b\left|\right|$ starts to increase.
- (3)
- The graph of the constrained L-curve is no longer an “L” shape. Instead, it assumes a U-shaped form that tilts towards the right.

#### 2.7. SWAT Model

#### 2.7.1. The Theory of Model

#### 2.7.2. Configuration of the SWAT Model

^{2}. The outlet was identified in sub-basin number 28, as shown in Figure 3. By applying a threshold of 10%, this paper overlaid and analyzed the spatial data, resulting in the division of the basin into 274 Hydrologic Response Units (HRUs). Subsequently, a SWAT model was constructed to simulate daily streamflow.

#### 2.7.3. Parameter Calibration of the SWAT Model

#### 2.7.4. Evaluation Criteria

## 3. Result

#### 3.1. Validation of the Ideal Model

#### 3.2. The Simulated Results of the SWAT Model

#### 3.3. The Corrected Results of SRC

#### 3.4. The Corrected Results of RT-SRC

#### 3.5. The Corrected Results of L-SRC

#### 3.6. The Corrected Results of CL-SRC

## 4. Discussion

## 5. Conclusions

- (1)
- The SRC method establishes a functional relationship between rainfall and streamflow through the analysis of the streamflow’s response to minor perturbations in rainfall. By inverting this relationship, it becomes possible to extract error series in rainfall with precision. As a result, updating streamflow by reducing the error in rainfall proves to be both feasible and effective.
- (2)
- The ridge trace method proves to be effective in the system differential response curve, with a corrected average NSE of 0.84 and an average RE of 14.7% for runoff depth over multiple years. However, the subjective nature of selecting the optimal ridge parameter makes precise localization challenging. Additionally, the complexity of ridge trace plots increases as the number of variables increases.
- (3)
- The traditional L-curve is not suitable for the system differential response curve as it tends to fall into numerically optimal points, which leads to insufficient constraints on the regularization term and degradation of the fitting accuracy of the model.
- (4)
- By introducing constrained terms, the constrained L-curve demonstrates better applicability and accuracy in the selection of the ridge parameter. It accurately identifies the actual optimal ridge parameter and effectively limits excessive correction, resulting in an average NSE of 0.88 and an average runoff depth RE of 11.0% over multiple years. Compared to the L-curve, the constrained L-curve exhibits better applicability.

## 6. Impacts and Limitations

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bogner, K.; Pappenberger, F. Multiscale error analysis, correction, and predictive uncertainty estimation in a flood forecasting system. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Si, W.; Bao, W.M.; Wang, H.Y.; Qu, S. The Research of Rainfall Error Correction Based on System Response Curve. Appl. Mech. Mater.
**2013**, 368, 335–339. [Google Scholar] [CrossRef] - Bao, W.; Si, W.; Qu, S. Flow Updating in Real-Time Flood Forecasting Based on Runoff Correction by a Dynamic System Response Curve. J. Hydrol. Eng.
**2014**, 19, 747–756. [Google Scholar] - Bao, W.; Li, Q. Estimating Selected Parameters for the XAJ Model under Multicollinearity among Watershed Characteristics. J. Hydrol. Eng.
**2012**, 17, 118–128. [Google Scholar] - Bao, W.; Zhao, L. Application of Linearized Calibration Method for Vertically Mixed Runoff Model Parameters. J. Hydrol. Eng.
**2014**, 19, 04014007. [Google Scholar] [CrossRef] - Si, W.; Zhong, H.; Jiang, P.; Bao, W.; Shi, P.; Qu, S. A dynamic information extraction method for areal mean rainfall error and its application in basins of different scales for flood forecasting. Stoch. Environ. Res. Risk Assess.
**2021**, 35, 255–270. [Google Scholar] [CrossRef] - Sun, Y.; Bao, W.; Jiang, P.; Ji, X.; Gao, S.; Xu, Y.; Zhang, Q.; Si, W. Development of Multivariable Dynamic System Response Curve Method for Real-Time Flood Forecasting Correction. Water Resour. Res.
**2018**, 54, 4730–4749. [Google Scholar] [CrossRef] - Liu, K.; Bao, W.; Hu, Y.; Sun, Y.; Li, D.; Li, K.; Liang, L. Improvement in Ridge Coefficient Optimization Criterion for Ridge Estimation-Based Dynamic System Response Curve Method in Flood Forecasting. Water
**2021**, 13, 3483. [Google Scholar] [CrossRef] - Tikhonov, A.N. On solving incorrectly posed problems and method of regularization. DokI. Acad. Nauk USSR
**1963**, 151, 501–504. [Google Scholar] - Rong, Q.; Shi, J.; Ceglarek, D. Adjusted least squares approach for diagnosis of ill-conditioned compliant assemblies. J. Manuf. Sci. Eng.
**2001**, 123, 453–461. [Google Scholar] [CrossRef] - Salahi, M. Regularization tools and robust optimization for ill-conditioned least squares problem: A computational comparison. Appl. Math. Comput.
**2011**, 217, 7985–7990. [Google Scholar] [CrossRef] - Tikhonov, A.N.; Arsenin, V.Y. Solutions of Ill-Posed Problems; John Wile and Sons: New York, NY, USA, 1977. [Google Scholar]
- Chen, G.Y.; Gan, M.; Chen, C.P.; Li, H.X. A Regularized Variable Projection Algorithm for Separable Nonlinear Least Squares Problems. IEEE Trans. Autom. Control
**2018**, 64, 526–537. [Google Scholar] [CrossRef] - Novati, P.; Russo, R.M. A GCV based Arnoldi-Tikhonov regularization method. BIT Numer. Math.
**2014**, 54, 501–521. [Google Scholar] [CrossRef] - Rezghi, M.; Hosseini, S.M. A new variant of L-curve for Tikhonov regularization. J. Comput. Appl. Math.
**2009**, 231, 914–924. [Google Scholar] [CrossRef] - Antoni, J.; Idier, J.; Bourguignon, S. A Bayesian interpretation of the L-curve. Inverse Probl.
**2023**, 39, 065016. [Google Scholar] [CrossRef] - Mitrouli, M.; Roupa, P. Estimates for the generalized cross-validation function via an extrapolation and statistical approach. Calcolo
**2018**, 55, 24. [Google Scholar] [CrossRef] - Hansen, P.C. Analysis of Discrete Ill-Posed Problems by Means of the L-Curve. SIAM Rev.
**2006**, 34, 561–580. [Google Scholar] [CrossRef] - Hansen, P.C.; O’Leary, D.P. The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems. SIAM J. Sci. Comput.
**2006**, 14, 1487–1503. [Google Scholar] [CrossRef] - Calvetti, D.; Reichel, L.; Shuibi, A. L-Curve and Curvature Bounds for Tikhonov Regularization. Numer. Algorithms
**2004**, 35, 301–314. [Google Scholar] [CrossRef] - Kindermann, S.; Raik, K. A simplified L-curve method as error estimator. arXiv
**2019**, arXiv:1908.10140. [Google Scholar] [CrossRef] - Anaraki, M.V.; Achite, M.; Farzin, S.; Elshaboury, N.; Al-Ansari, N.; Elkhrachy, I. Modeling of Monthly Rainfall–Runoff Using Various Machine Learning Techniques in Wadi Ouahrane Basin, Algeria. Water
**2023**, 15, 3576. [Google Scholar] [CrossRef] - Anaraki, M.V.; Farzin, S.; Mousavi, S.F.; Karami, H. Uncertainty analysis of climate change impacts on flood frequency by using hybrid machine learning methods. Water Resour. Manag.
**2021**, 35, 199–223. [Google Scholar] [CrossRef] - Wang, Y.; Chen, Y.; Lu, W.; Tian, Y. The Hydrological Response to Different Land Use Scenarios in the Minjiang River Basin. J. Soil Water Conserv.
**2020**, 34, 30–36. [Google Scholar] - Arnold, J.G.; Srinivasan, R.; Muttiah, R.S.; Williams, J.R. Large area hydrologic modeling and assessment, part I: Model development. JAWRA J. Am. Water Resour. Assoc.
**1998**, 34, 73–79. [Google Scholar] [CrossRef] - Gassman, P.W.; Reyes, M.R.; Green, C.H.; Arnold, J.G. The soil and water assessment tool: Historical development, applications, and future research directions. Trans. ASABE
**2007**, 50, 1211–1250. [Google Scholar] [CrossRef] - Yang, J.; Reichert, P.; Abbaspour, K.C.; Xia, J.; Yang, H. Comparing uncertainty analysis techniques for a SWAT application to the Chaohe Basin in China. J. Hydrol.
**2008**, 358, 1–23. [Google Scholar] [CrossRef]

**Figure 6.**The optimal ridge parameters selected by the ridge trace method (Please note that in the legend, ${X}_{i}$ ($i$ = 1, …, 10) represent different components of the regression coefficients).

Type of the Data | Resolution of the Data | Source of the Data |
---|---|---|

Hydrological data | Daily Scale | Hydrological Yearbook |

Meteorological Data | Daily Scale | Chinese Surface Climate Daily Data Set |

Data of Land Use | 1 km × 1 km | 2015 Data from the Chinese Academy of Sciences Resource and Environmental Science Data Center |

Data of soil | 1 km × 1 km | Second National Land Survey Data from the Nanjing Institute of Soil Science |

Parameters | Meaning of Parameters | Range of Parameters | Optimal Values of Parameters |
---|---|---|---|

r__CN2.mgt | SCS runoff curve number | −0.2~0.2 | 0.198700 |

v__ALPHA_BF.gw | Baseflow alpha factor | 0.0~1.0 | 0.949000 |

v__GW_DELAY.gw | Groundwater delay | 30~450 | 269.656342 |

v__GWQMN.gw | Treshold depth of water in the shallow aquifer required for return flow to occur | 0.0~5000 | 88.500000 |

v__GW_REVAP.gw | Groundwater revap coefficient. | 0.0~0.2 | 0.194300 |

v__ESCO.hru | Soil evaporation compensation factor | 0.8~1.0 | 0.885584 |

v__CH_N2.rte | Manning’s n value for the main channel. | 0.0~0.3 | 0.298234 |

v__CH_K2.rte | Effective hydraulic conductivity in main channel alluvium. | 5.0~130 | 129.000000 |

r__SOL_AWC(1).sol | Available water capacity of the soil layer. | −0.2~0.4 | 0.097000 |

Scale/Day | ${\mathit{P}}_{\mathit{o}\mathit{b}\mathit{s}}$ (mm) | $\mathit{E}$ (mm) | ${\mathit{P}}_{\mathit{s}\mathit{i}\mathit{m}}$ (mm) | ${\mathit{Q}}_{\mathit{o}\mathit{b}\mathit{s}}$ (m³/s) | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ (m³/s) | ${\mathit{E}}_{1}$ (mm) | ${\mathit{P}}_{\mathit{c}}$ (mm) | ${\mathit{Q}}_{\mathit{c}}(\mathbf{m}\xb3/\mathbf{s})$ |
---|---|---|---|---|---|---|---|---|

1 | 13.8 | −0.2 | 13.6 | 125 | 124 | 0.2 | 13.8 | 125 |

2 | 28.4 | −2.2 | 26.2 | 239 | 223 | 2.2 | 28.4 | 239 |

3 | 15.7 | 0.7 | 16.4 | 302 | 287 | −0.7 | 15.7 | 302 |

4 | 15.7 | −0.1 | 15.6 | 336 | 324 | 0.1 | 15.7 | 336 |

5 | 23.7 | 3.4 | 27.1 | 399 | 417 | −3.4 | 23.7 | 399 |

6 | 22 | 0.5 | 22.5 | 456 | 483 | −0.5 | 22.0 | 456 |

7 | 20.6 | −1.2 | 19.4 | 490 | 504 | 1.2 | 20.6 | 490 |

8 | 14.9 | −0.3 | 14.6 | 468 | 474 | 0.3 | 14.9 | 468 |

9 | 25.2 | 5.4 | 30.6 | 517 | 567 | −5.3 | 25.3 | 518 |

10 | 21.4 | −0.2 | 21.2 | 541 | 589 | 0.4 | 21.6 | 544 |

11 | 13.6 | −1 | 12.6 | 498 | 528 | 1.0 | 13.6 | 500 |

12 | 25.4 | 1.9 | 27.3 | 539 | 575 | −1.9 | 25.4 | 541 |

13 | 19 | 2.1 | 21.1 | 540 | 585 | −2.1 | 19.0 | 541 |

14 | 17.5 | 1.2 | 18.7 | 521 | 568 | −1.1 | 17.6 | 523 |

15 | 24 | 1.5 | 25.5 | 554 | 603 | −1.6 | 23.9 | 554 |

Data’s Time | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |
---|---|---|---|---|---|---|---|---|

NSE | 0.79 | 0.80 | 0.72 | 0.71 | 0.74 | 0.43 | 0.75 | 0.80 |

RE(%) | −1.8 | 6.4 | −4.5 | 16.1 | −5.4 | −9.2 | −7.9 | 1.9 |

Data’s Time | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |
---|---|---|---|---|---|---|---|---|

The rank of the response matrix | 362 | 364 | 363 | 366 | 365 | 365 | 365 | 366 |

Whether the matrix is full rank | N | N | N | Y | Y | Y | Y | Y |

Basin’s Name | Data’s Time | NS | RE of Streamflow Depth (%) | MSE | |||
---|---|---|---|---|---|---|---|

${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{S}\mathit{R}\mathit{C}}$ | ||

Jianyang | 1993 | 0.79 | / | −1.8 | / | 24,869 | / |

1994 | 0.80 | / | 6.4 | / | 14,268 | / | |

1995 | 0.72 | / | −4.5 | / | 37,009 | / | |

1996 | 0.71 | −221.43 | 16.1 | 783.7 | 7521 | 5,847,849 | |

1997 | 0.74 | −51.26 | −5.5 | 452.7 | 23,464 | 4,829,644 | |

1998 | 0.43 | −30.42 | −9.2 | 403.9 | 21,840 | 11,784,026 | |

1999 | 0.75 | −1106.98 | −7.9 | 823.9 | 14,245 | 6,376,510 | |

2000 | 0.80 | −120.00 | 1.9 | 517.7 | 14,764 | 8,946,160 | |

Average value | 0.71 | −306.02 | 6.7 | 596.38 | 19,747 | 7,556,838 |

Data’s Time | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |
---|---|---|---|---|---|---|---|---|

Optimal Ridge Parameters | 70 | 80 | 70 | 60 | 40 | 70 | 100 | 100 |

Basin’s Name | Data’s Time | NSE | RE of Streamflow Depth (%) | MSE | |||
---|---|---|---|---|---|---|---|

${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{R}\mathit{T}\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{R}\mathit{T}\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{R}\mathit{T}\mathit{S}\mathit{R}\mathit{C}}$ | ||

Jianyang | 1993 | 0.79 | 0.88 | −1.8 | 21.1 | 24,869 | 13,361 |

1994 | 0.80 | 0.90 | 6.4 | 17.4 | 14,268 | 6710 | |

1995 | 0.72 | 0.84 | −4.5 | 18.8 | 37,009 | 20,827 | |

1996 | 0.71 | 0.82 | 16.1 | 21.4 | 7521 | 4608 | |

1997 | 0.74 | 0.92 | −5.5 | 14.5 | 23,464 | 7602 | |

1998 | 0.43 | 0.54 | −9.2 | −3.4 | 21,840 | 17,109 | |

1999 | 0.75 | 0.88 | −7.9 | 9.9 | 14,245 | 6830 | |

2000 | 0.80 | 0.91 | 1.9 | 11.5 | 14,764 | 6503 | |

Average value | 0.71 | 0.84 | 6.7 | 14.7 | 19,747 | 10,443 |

Data’s Time | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |
---|---|---|---|---|---|---|---|---|

Ridge Parameters | 3.0 | 1.21 | 3.33 | 12.22 | 0.48 | 0.40 | 0.02 | 0.29 |

Basin’s Name | Data’s Time | NSE | RE of Runoff Depth (%) | MSE | |||
---|---|---|---|---|---|---|---|

${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{L}\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{L}\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{L}\mathit{S}\mathit{R}\mathit{C}}$ | ||

Jianyang | 1993 | 0.79 | −1.12 | −1.8 | 123.7 | 24,869 | 250,624 |

1994 | 0.80 | −0.40 | 6.4 | 114.9 | 14,268 | 103,147 | |

1995 | 0.72 | −1.65 | −4.5 | 118.3 | 37,009 | 352,420 | |

1996 | 0.71 | 0.65 | 16.1 | 45.4 | 7521 | 9100 | |

1997 | 0.74 | −1.27 | −5.5 | 120.8 | 23,464 | 209,820 | |

1998 | 0.43 | −9.92 | −9.2 | 151.2 | 21,840 | 4,094,890 | |

1999 | 0.75 | −32.2 | −7.9 | 292.2 | 14,245 | 1,908,889 | |

2000 | 0.80 | −4.70 | 1.9 | 158.5 | 14,764 | 421,600 | |

Average value | 0.71 | −6.33 | 6.7 | 140.6 | 19,747 | 918,811 |

Data’s Time | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 |
---|---|---|---|---|---|---|---|---|

Ridge Parameters | 62.0 | 192.8 | 99.6 | 59.8 | 39.5 | 42.3 | 90.7 | 82.3 |

Basin Name | Data’s Time | NSE | RE of Runoff Depth (%) | MSE | |||
---|---|---|---|---|---|---|---|

${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{C}\mathit{L}\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{C}\mathit{L}\mathit{S}\mathit{R}\mathit{C}}$ | ${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m}}$ | ${\mathit{Q}}_{\mathit{C}\mathit{L}\mathit{S}\mathit{R}\mathit{C}}$ | ||

Jianyang | 1993 | 0.79 | 0.88 | −1.8 | 16.4 | 24,869 | 14,349 |

1994 | 0.80 | 0.90 | 6.4 | 12.7 | 14,268 | 7123 | |

1995 | 0.72 | 0.84 | −4.5 | 10.0 | 37,009 | 20,891 | |

1996 | 0.71 | 0.82 | 16.1 | 16.7 | 7521 | 4677 | |

1997 | 0.74 | 0.94 | −5.5 | 7.0 | 23,464 | 5764 | |

1998 | 0.43 | 0.84 | −9.2 | −2.1 | 21,840 | 58,536 | |

1999 | 0.75 | 0.88 | −7.9 | 10.7 | 14,245 | 6754 | |

2000 | 0.80 | 0.92 | 1.9 | 12.7 | 14,764 | 6084 | |

Average value | 0.71 | 0.88 | 6.7 | 11.0 | 19,747 | 15,522 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xiao, H.; Qu, S.; Zhang, X.; Shi, P.; You, Y.; Li, F.; Yang, X.; Chen, Q.
Evaluation and Improvement of the Method for Selecting the Ridge Parameter in System Differential Response Curves. *Water* **2023**, *15*, 4205.
https://doi.org/10.3390/w15244205

**AMA Style**

Xiao H, Qu S, Zhang X, Shi P, You Y, Li F, Yang X, Chen Q.
Evaluation and Improvement of the Method for Selecting the Ridge Parameter in System Differential Response Curves. *Water*. 2023; 15(24):4205.
https://doi.org/10.3390/w15244205

**Chicago/Turabian Style**

Xiao, Hao, Simin Qu, Xumin Zhang, Peng Shi, Yang You, Fugang Li, Xiaoqiang Yang, and Qihui Chen.
2023. "Evaluation and Improvement of the Method for Selecting the Ridge Parameter in System Differential Response Curves" *Water* 15, no. 24: 4205.
https://doi.org/10.3390/w15244205