# Physics-Informed Data-Driven Models to Predict Surface Runoff Water Quantity and Quality in Agricultural Fields

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## Abstract

**:**

## 1. Introduction

## 2. Database Preparation

#### 2.1. Physically-Based Model

_{b}is a vertical coordinate of the soil surface, t is time (T; where T denotes units of time), h is the surface water depth (L), R is rainfall or the irrigation rate (LT

^{−1}), and I is the infiltration rate (LT

^{−1}), S is the mean local slope (defined as dz

_{b}/dx) (-), and k is a unit conversion factor (L

^{1/3}T

^{−1}). The parameter n

_{man}is a Manning’s roughness coefficient for overland flow, and it is dimensionless when k = 1 m

^{1/3}s

^{−1}or has units of TL

^{−1/3}when k is not equal to unity. See the graphical abstract for a schematic of some of these variables.

^{3}L

^{−3}), K

_{S}is the saturated hydraulic conductivity (LT

^{−1}), and F is the cumulative depth of infiltration (L). The value of ∆θ is defined by using the following equations:

_{e}is the effective saturation (-), θ

_{e}is the effective water content (L

^{3}L

^{−3}), n is the porosity (L

^{3}L

^{−3}), θ

_{i}is the initial water content (L

^{3}L

^{−3}), and θ

_{r}is the residual water content (L

^{3}L

^{−3}).

^{−3}; where M denotes units of mass), c

_{r}is the concentration in rainfall water (ML

^{−3}); s is the sorbed solute concentration at the soil surface area (ML

^{−2}), D is the effective dispersion coefficient accounting for both molecular diffusion and hydrodynamic dispersion (L

^{2}T

^{−1}), ϕ is a sink/source term that accounts for various zero- and first-order or other reactions (ML

^{−3}T

^{−1}), and Q is the runoff flow rate (L

^{2}T

^{−1}). The parameter Q is given as:

^{−1}) calculated using the Manning–Strickler uniform flow formula [41]. The effect of diffusion on the dispersion coefficient can often be ignored and in this case the value of D can be defined as the product of the dispersivity (λ, (L)) and $U$. Kinetic sorption/desorption between the solid and aqueous phases can be described using the following equation:

_{D}is the linear equilibrium partition coefficient (L) and ω is the first-order desorption rate coefficient (T

^{−1}). The product of K

_{D}and ω is proportional to the sorption rate coefficient. Details about the numerical solutions of flow and transport models are provided in the HYDRUS-1D manual [45].

#### 2.2. Numerical Simulations

_{r}, 0.1), 0.2, 0.3, and the saturated water content (θ

_{s}), for each soil type. Selected Green-Ampt infiltration parameters (soil porosity, residual water content, suction head, and hydraulic conductivity) were taken from Rawls et al. [46] (Table 2). In all water flow simulations, cumulative water volumes at the bottom outlet at 25 evenly spaced times were outputs. The resulting water flow database thus contained 2,310,000 entries (22 soils × 7 rainfall rates × 150 field conditions × 4 initial water contents × 25 print times).

_{i}= 1 g/cm

^{2}) of solute was initially distributed along the soil profile. Since the solute transport equation is a linear equation, the results obtained for a unit initial solute concentrations can be simply multiplied by other initial concentrations to get corresponding results. Runoff water generated by rainfall events “mobilized” solutes from the land surface and transported them to the bottom outlet of the field. The “mobilization” process was described using different sorption and desorption rates. The exchange of solute concentrations between solid and liquid phases is controlled by the overland water flow rate and kinetic sorption/desorption parameters (ω and K

_{D}) in Equation (8). The kinetic sorption model approaches equilibrium conditions when ω is large, whereas non-equilibrium conditions prevail when ω is small and solute release is slow. In this study, the database considered transport processes and solute desorption for near-equilibrium conditions (ω = 8640 day

^{−1}) to mitigate the complexity of these transport processes. Hence, two additional input parameters, variable K

_{D}shown in Table 1 and constant ω = 8640 day

^{−1}, were accounted for in these simulations. Solute transport simulations generated the output that included the cumulative solute mass at the bottom outlet at 25 evenly spaced times. The inclusion of two additional parameters to describe transport of mobilized solutes (K

_{D}and ω) resulted in a database that was five times bigger than the water flow database. Furthermore, the variance in the solute transport database was higher than the water flow database.

## 3. Data-Driven Models

#### 3.1. Linear Regression

_{i}, x

_{ij}), where i =1, 2, 3, …, N and j =1, 2, 3, …, k. In here, i is the index for training samples in the data set, j is the index for the input parameters. A general linear regression problem can be developed by assuming that output variables y

_{i}are influenced by input parameters x

_{ij}as:

_{j}are regression coefficients, β

_{0}is intercept, and e

_{i}represent a deviation between actual and predicted values. The ordinary least squares procedure seeks to minimize the total sum of the residuals. However, since this approach treats data as a matrix, the process will be very computational intensive when we have large data sets. Therefore, gradient-based optimization approaches, which optimize the values of the coefficients by iteratively minimizing the sum of the squared errors for each pair of input and output values are used here [49].

#### 3.2. Support Vector Machine

_{ij}) that has at most a deviation ε from actual y

_{i}for all training points x

_{ij}:

_{j}are weights and b

_{i}is the bias term that should be estimated from the training data. One great benefit of using ε-SVR is the use of kernels, which inherently map the data into a nonlinear space depending on the chosen kernel function. Therefore, the above regression can be formulated as [52]:

_{ij}) denotes nonlinear transformation (kernel functions). Several nonlinear kernel functions are available, such as polynomial, radial basis, and sigmoid functions. In this study, we tested both SVR with linear (SVM-L) and nonlinear (SVM-NL) sigmoid kernel functions.

_{j}and b

_{i}are estimated by minimizing the following objective function:

_{ij}) and the amount up to which deviations larger than ε are tolerated. The slack variables ${\xi}_{i}$ and ${\xi}_{i}{}^{*}$ determine the degree to which data points will be penalized if the error is larger than ε.

#### 3.3. K-Nearest Neighbor Regression

_{nn}) which are most similar to a new instance. Once the nearest-neighbor list is obtained, the prediction of output y is achieved by assigning weights to the contributions of the neighbors, so that the nearer neighbors contribute more to the average than the more distant ones. For example, the Euclidean distance d between lth training sample x

_{lj}and ith test sample x

_{ij}is defined as [53]:

_{d}can be defined by a selected K

_{nn}value and a distance between lth and ith samples:

_{i}can be predicted by

#### 3.4. Deep Feed-Forward Network

_{ij}are weighted and summed up to produce the hidden neurons S

_{p}(p = 1, 2, …, P):

_{p}), according to an activation function (f(S

_{p})). Finally, an activation function, which is applied to S

_{p}, provides the final output from this logistic sigmoid [10] (see Figure 1). The sigmoid activation function is given as:

## 4. Model Training and Evaluation

_{c}) for data-driven models and the PBM. The calculated R

^{2}and the best-fitted regression line were shown in these scatter plots to quantify the goodness of the data-driven model prediction.

_{i}and O

_{i}are the output values predicted by data-driven models and observed data obtained from the PBM, respectively, ${\overline{O}}_{i}$ is the average of observed data, and N is the number of observations. It should be noted that data with zero values of surface runoff were removed from this analysis for model inter-comparison.

^{−1}) and different K

_{D}values. The solute transport database had a similar structure as the water flow database but was five times bigger in size.

_{c}, K

_{D}, and other input parameters associated with the water flow database. The remaining 20% of the solute transport database were employed for testing the model. Two simulation examples were chosen to further investigate the predictive ability of the trained data-driven models. These simulation scenarios considered the same input variables as for the water flow model (e.g., differences in initial water contents or soil texture) with K

_{D}= 2 cm and ω = 8640 day

^{−1}.

## 5. Results and Discussion

#### 5.1. Surface Runoff Quantity

#### 5.1.1. Linear Regression

_{c}using the LR model versus observed Q

_{c}for the PBM test data set are shown in Figure 3a. We can see that R

^{2}is equal to 0.439, which indicates that the LR model failed to accurately describe the PBM data. The LR algorithm usually works well with high-bias and low variance data. However, the PBM describes real-world scenarios with complex, nonlinear relationships between input and output variables that generate high variance data sets.

_{c}for early times and underestimate Q

_{c}for later times. Other authors (e.g., [18]) similarly found that the linear regression model does not provide enough flexibility to represent complex nonlinear hydrological phenomena.

#### 5.1.2. Support Vector Machine

^{2}(R

^{2}= 0.386) than the LR model (R

^{2}= 0.439). Similarly, simulation examples shown in Figure 4b (different initial water contents) and 4c (different textures) indicate that the SVM-L model completely failed to capture the large values of Q

_{c}at later times. Linear models (LR and SVM-L) are therefore not well suited to describe complex nonlinear runoff problems.

_{c}using the SVM-NL model versus observed Q

_{c}from the test set. The R

^{2}(R

^{2}= 0.952) for the SVM-NL model was much higher than for the LR and SVM-L models. The SVM-NL model also provided a much-improved description of the PBM results for the two examples shown in Figure 5b,c. However, there were still considerable deviations between the SVM-NL model and the PBM, especially for larger print times, higher initial water contents (Figure 5b), and for the Silt Loam soil (Figure 5c).

#### 5.1.3. K-Nearest Neighbor Regression

^{2}(R

^{2}= 0.957) for the kNN-5 model was slightly higher than that for the SVM-NL model (R

^{2}= 0.952). However, there were more predicted outliers when the observed Q

_{c}was relatively small. Although the overall runoff trends were correctly captured in Figure 6b, the kNN-5 model underestimated the cumulative water flux at every print time and incorrectly predicted that Q

_{c}stopped increasing when rainfall ended. Furthermore, this deviation tended to increase for smaller initial water contents. However, Figure 6c shows that the kNN-5 model did a much better job of predicting large values of Q

_{c}than the SVM-NL model.

#### 5.1.4. Deep Feed-Forward Neural Network

^{2}and the Root Mean Squared Error (RMSE) were not improved when neuron numbers were increased beyond 16. Therefore, the number of neurons in each hidden layer was set to 16.

_{c}using DNN models with three different layers versus observed Q

_{c}for the PBM data set. Values of R

^{2}were equal to 0.978, 0.981, and 0.959 for the DNN-1, DNN-2, and DNN-3 models, respectively. The DNN models provided a description of runoff that was fairly accurate and obtained higher R

^{2}values than the other ML models. However, the top and bottom rows of Figure 7 indicate that the DNN-1 and DNN-3 models tended to underestimate and overestimate observed values of Q

_{c}, respectively, whereas the DNN-2 model provided the best prediction. This result suggests that the one hidden layer model was not “deep” enough, whereas a three-layer model probably was too complicated and results in overfitting. A more quantitative comparison of various ML models is given below.

#### 5.1.5. Comparison of Data-Driven Models

_{c}was over 100 L, whereas the DNN-3 model slightly underestimated large Q

_{c}values. The DNN-2 model shows the narrowest box plot of relative errors and the best performance statistics for the considered ML models. This indicates that the DNN-2 model provided the best prediction of runoff water volumes.

#### 5.2. Surface Runoff Quality

^{2}= 0.992 in Figure 9a, and the agreement between observed (PBM) and predicted results for the example simulations for various water contents (Figure 9b) and soil types (Figure 9c).

## 6. Conclusions and Outlook

^{2}) less than 0.5. In particular, both linear models tended to overestimate the runoff water volume for early times and underestimate it for later times. Both SVM-NL and kNN models performed much better than linear models, having R

^{2}of 0.93 and 0.96, respectively. However, there were still considerable deviations between the SVM-NL and kNN models and PBM predictions, especially for larger times, higher initial water contents, and for some textures (such as Silt Loam). Finally, the DNN models with one, two, and three hidden layers (DNN-1, DNN-2, and DNN-3) were tested to determine the optimum number of hidden layers and to minimize overfitting of output variables. The best performance was obtained by the DNN model with two hidden layers (DNN-2) (R

^{2}= 0.98). The DNN-1 and DNN-3 models tended to underestimate, and overestimate observed values of runoff water volumes, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**A feed-forward neural network (a single hidden layer,

**left**) and a deep feed-forward neural network (multiple hidden layers,

**right**).

**Figure 3.**Comparison of cumulative water fluxes (Q

_{c}in liters, L) calculated by the physically-based models PBM and the Linear Regression (LR) model ((

**a**) overall, (

**b**) different initial water contents, (

**c**) different textures)).

**Figure 4.**Comparison of cumulative water fluxes (Q

_{c}in liters, L) calculated by the PBM and the Linear Support Vector Machine (SVM-L) model ((

**a**) overall, (

**b**) different initial water contents, (

**c**) different textures)).

**Figure 5.**Comparison of cumulative water fluxes (Q

_{c}in liters, L) calculated by the PBM and the Nonlinear Support Vector Machine (SVM-NL) model ((

**a**) overall, (

**b**) different initial water contents, (

**c**) different textures)).

**Figure 6.**Comparison of cumulative water fluxes (Q

_{c}in liters, L) calculated by the PBM and the k-Nearest Neighbor Regression (kNN-5) model ((

**a**) overall, (

**b**) different initial water contents, (

**c**) different textures)).

**Figure 7.**Comparison of cumulative water fluxes (Q

_{c}in liters, L) calculated by the PBM and the Deep Feed-Forward Neural Network (DNN-1 (

**top row**), DNN-2 (

**middle row**), and DNN-3 (

**bottom row**)) models ((

**a**) overall, (

**b**) different initial water contents, (

**c**) different textures)).

**Figure 8.**Statistical parameters associated with the trained data-driven models of runoff quantity and quality. The root mean square error (RMSE), mean bias error (MBE), mean absolute error (MAE), model efficiency (EF), and a box plot of the relative error distribution for data-driven models are given. The units of RMSE, MBE, and MAE of the runoff quantity and quality are liters and g, respectively. The upper and lower boundaries of the boxes show the 75th and 25th percentile, the whiskers of the box plot show the maximum and minimum values, and the red line within the box is the median value. Blue dot symbols indicated the outliers.

**Figure 9.**Comparison of the cumulative solute mass (in grams, M) calculated by the PBM and the DNN-2 model for the near equilibrium training data set ((

**a**) overall, (

**b**) different initial water contents, (

**c**) different textures)).

Input | Values |
---|---|

Rainfall intensity (cm day^{−1}) | 21.6, 28.8, 40.8, 52.8, 69.6, 86.4, 105.6 |

Manning’s roughness coefficient | 0.01, 0.02, 0.04, 0.07, 0.15, 0.24 |

Slope (-) | 0.01, 0.02, 0.04, 0.08, 0.16 |

Length (m) | 30, 60, 120, 240, 480 |

K_{D} (cm) | 0.1, 0.5, 1, 1.5, 2 |

Initial water content, θ_{i} | max(θ_{r}, 0.1), 0.2, 0.3, θ_{s} |

Texture | θ_{s} | θ_{r} | θ_{e} | ψ (cm) | K (cm s^{−1}) |
---|---|---|---|---|---|

Sand | 0.437 | 0.020 | 0.417 | 4.950 | 3.272 × 10^{−3} |

Loamy Sand | 0.437 | 0.036 | 0.401 | 6.130 | 8.306 × 10^{−4} |

Sandy Loam | 0.453 | 0.041 | 0.412 | 11.010 | 3.028 × 10^{−4} |

Loam | 0.463 | 0.029 | 0.434 | 8.890 | 9.444 × 10^{−5} |

Silt Loam | 0.501 | 0.015 | 0.486 | 16.680 | 1.806 × 10^{−4} |

Sandy Clay Loam | 0.398 | 0.068 | 0.330 | 21.850 | 4.167 × 10^{−5} |

Clay Loam | 0.464 | 0.155 | 0.309 | 20.880 | 2.778 × 10^{−5} |

Silty Clay Loam | 0.471 | 0.039 | 0.432 | 27.300 | 2.778 × 10^{−5} |

Sandy Clay | 0.430 | 0.109 | 0.321 | 23.900 | 1.667 × 10^{−5} |

Silty Clay | 0.470 | 0.047 | 0.423 | 29.220 | 1.389 × 10^{−5} |

Clay | 0.475 | 0.090 | 0.385 | 31.630 | 8.333 × 10^{−6} |

© 2019 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**

Liang, J.; Li, W.; Bradford, S.A.; Šimůnek, J. Physics-Informed Data-Driven Models to Predict Surface Runoff Water Quantity and Quality in Agricultural Fields. *Water* **2019**, *11*, 200.
https://doi.org/10.3390/w11020200

**AMA Style**

Liang J, Li W, Bradford SA, Šimůnek J. Physics-Informed Data-Driven Models to Predict Surface Runoff Water Quantity and Quality in Agricultural Fields. *Water*. 2019; 11(2):200.
https://doi.org/10.3390/w11020200

**Chicago/Turabian Style**

Liang, Jing, Wenzhe Li, Scott A. Bradford, and Jiří Šimůnek. 2019. "Physics-Informed Data-Driven Models to Predict Surface Runoff Water Quantity and Quality in Agricultural Fields" *Water* 11, no. 2: 200.
https://doi.org/10.3390/w11020200