# Multi-Gene Genetic Programming Regression Model for Prediction of Transient Storage Model Parameters in Natural Rivers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Methods

#### 2.1. Transient Storage Model

#### Remarks for the TSM

#### 2.2. Multi-Gene Genetic Programming

## 3. Formulation of Empirical Equations

#### 3.1. Dimensional Analysis and Data Collection

#### 3.2. Formulated Equations

#### 3.2.1. Formulation by MGGP

#### 3.2.2. Formulation by PCR-Based Regression

#### 3.3. Statistical Performance of the Models

## 4. In-Stream Application

#### 4.1. Tracer Test Description

^{3}/s, and the cross-sectional area was calculated, dividing measured discharge by mean velocity (Table 4). Sinuosity can be calculated with plan view. The calculated sinuosity of the specified reaches S1-S2, S2-S3, and S3-S4 are 1.0562, 1.0671, and 1.1207, respectively.

#### 4.2. Simulation Results

^{2}, and the calibrated ${A}_{f}$ was smaller than the actual cross-section. The defect of the calibrated ${A}_{f}$ contributed 5.4298 m

^{2}to the storage zone area, but ${A}_{f}+{A}_{s}$ was greater than the measured cross-section area due to the sand bar HTS. Moreover, the measured velocity at S1 (0.39 m/s) was more than three times faster than that at U2 (0.13 m/s) and S2 (0.11 m/s), so that the estimate of the effective area was low. Keeping in mind the ${A}_{f}$s value used in the F2019, the MGGP model better predicted ${A}_{f}$ than the PCR model, in the sub-reach S1-S2. The F2019 and PCR models over-estimated ${K}_{f}$; the values in those models were 4.31 times, and 2.5 times higher, respectively. Every model under-estimated the storage parameters (${A}_{s}$ and $\alpha $).

## 5. Sensitivity Analysis

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1D-ADE | 1-Dimension Advection-Dispersion Equation |

BTC | Breakthrough Curve |

DR | Discrepancy Ratio |

GA | Genetic Algorithm |

GP | Genetic Programming |

HTS | Hyporheic Transient Storage |

MGGP | Multi-Gene Genetic Programming |

MSE | Mean Squared Error |

MSL | Mean Sea Level |

OAT | One-At-a-Time |

OTIS | One-Dimensional Transport with Inflow and Storage |

PCR | Principal Components Regression |

RMSE | Root Mean Squared Error |

RTD | Residence Time Distribution |

RWT | Rhodamine WT |

SC-SAHEL | Shuffled Complex-Self Adaptive EvoLution |

SCE-UA | Shuffled Complex Evolution-University of Arizona |

SI | Sensitivity Index |

STS | Surface Transient Storage |

TSM | Transient Storage Model |

VIF | Variance Inflation Factor |

## Appendix A. Description of the Gam-Creek Tracer Test

**Figure A2.**Observed BTCs in Gam-Creek tracer test and calibrated BTCs; (

**a**) section 1–2. (

**b**) section 2–3. (

**c**) section 3–4. (

**d**) All stations (only meausred curves).

Variables | Reach | |||
---|---|---|---|---|

S1-S2 | S2-S3 | S3-S4 | ||

Hydraulic Features | ${L}_{reach}$ (m) | 1200 | 830 | 2000 |

Q (cms) | 11.06 | 11.06 | 11.06 | |

W (m) | 57.36 | 58.86 | 53.00 | |

h (m) | 0.36 | 0.36 | 0.43 | |

${S}_{0}$ | 0.0007 | 0.0024 | 0.0007 | |

$\overline{U}$ (m/s) | 0.53 | 0.52 | 0.48 | |

${S}_{n}$ | 1.082 | 1.028 | 1.078 | |

TSM Parameters | $Kf$$({m}^{2}/s)$ | 0.568 | 0.596 | 4.926 |

${A}_{f}$$({m}^{2})$ | 18.279 | 17.175 | 31.135 | |

${A}_{s}$$({m}^{2})$ | 4.1473 | 2.6932 | 10.4883 | |

$\alpha \times {10}^{4}$ (1/s) | 3.758 | 2.920 | 1.533 |

## Appendix B. Derived PCR Equations Using Total Dataset

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**Figure 2.**Calibrated BTCs of Cheongmi Creek experiment data from [16] using 1D-ADE and TSM.

**Figure 7.**Boxplots of dimensionless variables; (

**a**) $\frac{U}{{U}_{*}}$ (

**b**) $\frac{W}{h}$ (

**c**) ${S}_{n}$ (

**d**) $\frac{{K}_{f}}{h{U}_{*}}$. (

**e**) $\frac{{A}_{f}}{Wh}$. (

**f**) $\frac{{A}_{s}}{Wh}$. (

**g**) $\frac{\alpha}{{U}_{*}/h}$.

**Figure 9.**Observed parameter values versus predicted values by empirical equations for the training set; (

**a**) ${K}_{f}$. (

**b**) ${A}_{f}$. (

**c**) ${A}_{s}$. (

**d**) $\alpha $.

**Figure 10.**Observed parameter values versus predicted values by empirical equations for the test set; (

**a**) ${K}_{f}$. (

**b**) ${A}_{f}$. (

**c**) ${A}_{s}$. (

**d**) $\alpha $.

**Figure 11.**DR histograms of each empirical equations for TSM parameters for the training set; (

**a**) ${K}_{f}$. (

**b**) ${A}_{f}$. (

**c**) ${A}_{s}$. (

**d**) $\alpha $.

**Figure 12.**DR histograms of each empirical equations for TSM parameters for the test set; (

**a**) ${K}_{f}$. (

**b**) ${A}_{f}$. (

**c**) ${A}_{s}$. (

**d**) $\alpha $.

**Figure 14.**Plots of measured hydraulic data in Cheong-mi Creek; (

**a**) Width and depth; (

**b**) Velocity and shear velocity.

**Figure 15.**Simulated and observed breakthrough curves in Cheongmi Creek experiment case: (

**a**) section 1–2. (

**b**) section 2–3. (

**c**) section 3–4. (

**d**) All stations (only measured curves).

**Figure 16.**Spider plots and calculated indices for sensitivity analysis of the MGGP model; (

**a**) $\frac{{K}_{f}}{h{U}_{*}}$. (

**b**) $\frac{{A}_{f}}{Wh}$. (

**c**) $\frac{{A}_{s}}{Wh}$. (

**d**) $\frac{\alpha}{{U}_{*}/h}$.

Parameter | Training Set (90 Sets) | Test Set (38 Sets) | ||||
---|---|---|---|---|---|---|

Mean | Minimum | Maximum | Mean | Minimum | Maximum | |

$\frac{W}{h}$ | 46.83 | 3.80 | 331.86 | 36.51 | 3.80 | 114.51 |

$\frac{U}{{U}_{*}}$ | 5.07 | 1.02 | 17.59 | 5.20 | 1.02 | 15.58 |

${S}_{n}$ | 1.36 | 1.00 | 2.27 | 1.34 | 1.00 | 2.00 |

$\frac{{K}_{f}}{h{U}_{*}}$ | 335.26 | 0.60 | 5558.20 | 295.79 | 3.42 | 1841.00 |

$\frac{{A}_{f}}{Wh}$ | 1.02 | 0.26 | 7.00 | 1.01 | 0.47 | 2.78 |

$\frac{{A}_{s}}{Wh}$ | 0.22 | 0.00 | 3.09 | 0.14 | 0.01 | 0.44 |

$\frac{\alpha}{{U}_{*}/h}\times {10}^{4}$ | 19.50 | 0.04 | 256.80 | 14.14 | 0.34 | 164.99 |

Parameter | Settings |
---|---|

Function set | $+,-,\times ,\xf7,\sqrt{}$, square, cube, exp, tanh, power |

Population size | 500 |

Number of generations | 500 |

Runs | over 200 |

Maximum number of genes allowed in an individual | 4 |

Maximum tree depth | 6 |

Tournament size | 15 |

Elitism | 0.01 % of population |

Crossover events | 0.84 |

High level crossover | 0.2 |

Low level crossover | 0.8 |

Mutation events | 0.14 |

Sub-tree mutation | 0.9 |

Replacing input terminal with another random terminal | 0.05 |

Criteria | TSM Parameter | Training Set (90 Sets) | Test Set (38 Sets) | ||||
---|---|---|---|---|---|---|---|

MGGP | PCR | F2019 | MGGP | PCR | F2019 | ||

Accuracy (%) | ${K}_{f}$ | 53.33 | 41.11 | 46.67 | 47.37 | 42.11 | 42.11 |

${A}_{f}$ | 95.56 | 92.22 | - | 97.37 | 97.37 | - | |

${A}_{s}$ | 56.67 | 57.78 | 56.67 | 47.37 | 52.63 | 47.37 | |

$\alpha \times {10}^{4}$ | 36.67 | 34.44 | 34.44 | 21.05 | 28.95 | 28.95 | |

RMSE | ${K}_{f}$ | 28.20 | 83.04 | 475.91 | 75.83 | 32.00 | 43.80 |

${A}_{f}$ | 345.60 | 391.75 | - | 56.92 | 44.86 | - | |

${A}_{s}$ | 41.96 | 44.80 | 21.43 | 9.54 | 2.60 | 10.58 | |

$\alpha \times {10}^{4}$ | 8.52 | 14.63 | 15.74 | 10.34 | 8.67 | 8.46 | |

${R}^{2}$ | ${K}_{f}$ | 0.49 | −3.42 | −144.06 | 0.20 | 0.86 | 0.73 |

${A}_{f}$ | 0.93 | 0.91 | - | 0.84 | 0.90 | - | |

${A}_{s}$ | −0.07 | −0.21 | 0.72 | −3.34 | 0.68 | −4.34 | |

$\alpha \times {10}^{4}$ | 0.67 | 0.04 | −0.11 | −0.29 | 0.09 | 0.14 | |

$\rho $ | ${K}_{f}$ | 0.89 | 0.78 | 0.74 | 0.47 | 0.96 | 0.96 |

${A}_{f}$ | 0.99 | 0.99 | - | 0.92 | 0.96 | - | |

${A}_{s}$ | 0.62 | 0.61 | 0.89 | 0.97 | 0.97 | 0.62 | |

$\alpha \times {10}^{4}$ | 0.83 | 0.32 | 0.08 | 0.35 | 0.48 | 0.55 |

Station | ${\mathit{L}}_{\mathit{IP}}$ (m) | W (m) | h (m) | U (m/s) | ${\mathit{U}}_{*}$ (m/s) |
---|---|---|---|---|---|

I.P | 0 | 17.1 | 0.72 | 0.19 | 0.023 |

U1 | 380 | 32.5 | 0.45 | 0.15 | 0.020 |

S1 | 940 | 17.5 | 0.33 | 0.39 | 0.055 |

U2 | 1300 | 32.6 | 0.53 | 0.13 | 0.017 |

S2 | 1690 | 31.7 | 0.63 | 0.11 | 0.014 |

U3 | 2050 | 34 | 0.59 | 0.11 | 0.014 |

U4 | 2410 | 16.5 | 0.35 | 0.39 | 0.055 |

U5 | 2730 | 34.6 | 0.18 | 0.37 | 0.057 |

S3 | 3080 | 14.1 | 0.39 | 0.41 | 0.056 |

S4 | 3550 | 24.25 | 0.36 | 0.26 | 0.036 |

Average | 1810 | 25.48 | 0.42 | 0.21 | 0.028 |

Sub-Reach | Methods | TSM Parameters | $\mathit{DaI}$ | $\mathit{\u03f5}$ | |||
---|---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{f}}$ (m^{2}/s) | ${\mathit{A}}_{\mathit{f}}$ (m^{2}) | ${\mathit{A}}_{\mathit{s}}$ (m^{2}) | $\mathit{\alpha}\times {10}^{4}$ (1/s) | ||||

S1-S2 | Calibrated | 1.3335 | 9.6377 | 5.4298 | 2.4187 | 2.1467 | 0.5634 |

F2019 | 5.7524 | 14.6009 | 1.0905 | 0.1282 | 0.8939 | 0.0747 | |

MGGP | 1.1441 | 15.1014 | 2.3081 | 0.4206 | 1.5898 | 0.1528 | |

PCR | 3.2945 | 15.4021 | 1.5045 | 0.1633 | 0.9379 | 0.0977 | |

S2-S3 | Calibrated | 1.2135 | 9.0384 | 2.9589 | 1.2481 | 2.8132 | 0.3274 |

F2019 | 7.2714 | 11.1621 | 1.2463 | 0.2499 | 1.7079 | 0.1117 | |

MGGP | 2.1512 | 11.5136 | 2.0088 | 0.8116 | 3.8688 | 0.1745 | |

PCR | 4.3786 | 11.9084 | 1.2275 | 0.2918 | 2.2867 | 0.1031 | |

S3-S4 | Calibrated | 2.0850 | 7.5056 | 1.5380 | 1.5573 | 1.4293 | 0.2049 |

F2019 | 5.9005 | 7.1906 | 1.2024 | 0.4659 | 0.4863 | 0.1672 | |

MGGP | 1.6943 | 7.3859 | 1.0414 | 1.0338 | 1.2850 | 0.1410 | |

PCR | 4.3206 | 7.4839 | 0.6643 | 0.3746 | 0.7151 | 0.0888 |

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## Share and Cite

**MDPI and ACS Style**

Noh, H.; Kwon, S.; Seo, I.W.; Baek, D.; Jung, S.H.
Multi-Gene Genetic Programming Regression Model for Prediction of Transient Storage Model Parameters in Natural Rivers. *Water* **2021**, *13*, 76.
https://doi.org/10.3390/w13010076

**AMA Style**

Noh H, Kwon S, Seo IW, Baek D, Jung SH.
Multi-Gene Genetic Programming Regression Model for Prediction of Transient Storage Model Parameters in Natural Rivers. *Water*. 2021; 13(1):76.
https://doi.org/10.3390/w13010076

**Chicago/Turabian Style**

Noh, Hyoseob, Siyoon Kwon, Il Won Seo, Donghae Baek, and Sung Hyun Jung.
2021. "Multi-Gene Genetic Programming Regression Model for Prediction of Transient Storage Model Parameters in Natural Rivers" *Water* 13, no. 1: 76.
https://doi.org/10.3390/w13010076