# Development of a Deep Learning Emulator for a Distributed Groundwater–Surface Water Model: ParFlow-ML

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Integrated Hydrologic Model, ParFlow

^{−1}), ${S}_{w}\left(h\right)$ is the relative saturation (–), $\phi $ is the porosity (–), ${K}_{s}\left(x\right)$ is the saturated hydraulic conductivity tensor (LT

^{−1}), ${k}_{r}\left(h\right)$ is the relative permeability (–), ${q}_{r}$ is a sink/source term (T

^{−1}), and ${\theta}_{x}$ is the local angle of a topographic slope (S). The overland flow equation is given as [36,38]:

^{−1}), and $\lambda $ is a constant equal to the inverse of the grid scaling (L

^{−1}) [38].

^{−1}) and n (–) are soil parameters, s

_{sat}(–) is the relative saturated water content, and s

_{res}(–) is the relative residual saturation.

#### 2.2. The Emulator Version of ParFlow, ParFlow-ML

#### 2.3. Experiment Design

#### 2.3.1. Study Areas

^{2}and a mean elevation of 3500 m (Figure 3). The outlet of the basin is located at Almont, Colorado, USA. The Little Washita basin, located in the Southwestern Oklahoma, has an area of 600 km

^{2}and is characterized by rolling terrain. The outlet of the basin is located at Smithville, Oklahoma, USA.

#### 2.3.2. The Emulator Setup

#### 2.3.3. Model Setup

#### 2.3.4. Rainfall–Runoff Scenarios

^{3}/s (scenario 8) to 22,000 m

^{3}/s (scenarios 15). Scenarios included in the validation and test sets also have multiple peak flows whose magnitudes vary from 7000 m

^{3}/s (scenario 18) to 21,000 m

^{3}/s (scenario 23).

#### 2.4. Training Process

^{−3}. After 2000 iterations, loss (mean squared error) of the model compared with both the train and the validation sets decreased to 2.55 × 10

^{−3}and 2.57 × 10

^{−3}, respectively (Figure 5). Since both training and validation losses did not change over the last 200 iterations, we decided to impose early stopping after 2000 iterations.

#### 2.5. Performance Metrics

## 3. Results

^{3}/s), water table depth (m), and total water storage (m

^{3}). Below is an evaluation for each of the variables.

#### 3.1. Streamflow Evaluation

#### 3.2. Water Table Depth (WTD) Evaluation

#### 3.3. Total Water Storage Evaluation

#### 3.4. Execution Time

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Wood, E.F.; Roundy, J.K.; Troy, T.J.; van Beek, L.P.H.; Bierkens, M.F.P.; Blyth, E.; de Roo, A.; Döll, P.; Ek, M.; Famiglietti, J.; et al. Hyperresolution global land surface modeling: Meeting a grand challenge for monitoring Earth’s terrestrial water. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Bierkens, M.F.P.; Bell, V.A.; Burek, P.; Chaney, N.; Condon, L.E.; David, C.H.; de Roo, A.; Döll, P.; Drost, N.; Famiglietti, J.S.; et al. Hyper-resolution global hydrological modelling: What is next?: “Everywhere and locally relevant”. Hydrol. Process.
**2015**, 29, 310–320. [Google Scholar] [CrossRef] - Burstedde, C.; Fonseca, J.A.; Kollet, S. Enhancing speed and scalability of the ParFlow simulation code. Comput. Geosci.
**2018**, 22, 347–361. [Google Scholar] [CrossRef] [Green Version] - Kollet, S.J.; Maxwell, R.M.; Woodward, C.S.; Smith, S.; Vanderborght, J.; Vereecken, H.; Simmer, C. Proof of concept of regional scale hydrologic simulations at hydrologic resolution utilizing massively parallel computer resources. Water Resour. Res.
**2010**, 46, 4201. [Google Scholar] [CrossRef] - Hokkanen, J.; Kollet, S.; Kraus, J.; Herten, A.; Hrywniak, M.; Pleiter, D. Leveraging HPC accelerator architectures with modern techniques—Hydrologic modeling on GPUs with ParFlow. Comput. Geosci.
**2021**, 25, 1579–1590. [Google Scholar] [CrossRef] - Le, P.V.V.; Kumar, P.; Valocchi, A.J.; Dang, H.V. GPU-based high-performance computing for integrated surface-sub-surface flow modeling. Environ. Model. Softw.
**2015**, 73, 1–13. [Google Scholar] [CrossRef] [Green Version] - Gentine, P.; Pritchard, M.; Rasp, S.; Reinaudi, G.; Yacalis, G. Could Machine Learning Break the Convection Parameterization Deadlock? Geophys. Res. Lett.
**2018**, 45, 5742–5751. [Google Scholar] [CrossRef] - Rasp, S.; Pritchard, M.S.; Gentine, P. Deep learning to represent subgrid processes in climate models. Proc. Natl. Acad. Sci. USA
**2018**, 115, 9684–9689. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zanna, L.; Brankart, J.M.; Huber, M.; Leroux, S.; Penduff, T.; Williams, P.D. Uncertainty and scale interactions in ocean ensembles: From seasonal forecasts to multidecadal climate predictions. Q. J. R. Meteorol. Soc.
**2019**, 145, 160–175. [Google Scholar] [CrossRef] [Green Version] - Hsu, K.-L.; Gupta, H.V.; Sorooshian, S. Artificial Neural Network Modeling of the Rainfall-Runoff Process. Water Resour. Res.
**1995**, 31, 2517–2530. [Google Scholar] [CrossRef] - Hsu, K.L.; Gupta, H.V.; Gao, X.; Sorooshian, S.; Imam, B. Self-organizing linear output map (SOLO): An artificial neural network suitable for hydrologic modeling and analysis. Water Resour. Res.
**2002**, 38, 38-1–38-17. [Google Scholar] [CrossRef] [Green Version] - Tao, Y.; Gao, X.; Hsu, K.; Sorooshian, S.; Ihler, A. A deep neural network modeling framework to reduce bias in satellite precipitation products. J. Hydrometeorol.
**2016**, 17, 931–945. [Google Scholar] [CrossRef] - Tao, Y.; Gao, X.; Ihler, A.; Sorooshian, S.; Hsu, K. Precipitation identification with bispectral satellite information using deep learning approaches. J. Hydrometeorol.
**2017**, 18, 1271–1283. [Google Scholar] [CrossRef] - Wang, C.; Tang, G.; Gentine, P. PrecipGAN: Merging Microwave and Infrared Data for Satellite Precipitation Estimation Using Generative Adversarial Network. Geophys. Res. Lett.
**2021**, 48, e2020GL092032. [Google Scholar] [CrossRef] - Fang, K.; Shen, C. Full-flow-regime storage-streamflow correlation patterns provide insights into hydrologic functioning over the continental US. Water Resour. Res.
**2017**, 53, 8064–8083. [Google Scholar] [CrossRef] - Feng, D.; Fang, K.; Shen, C. Enhancing Streamflow Forecast and Extracting Insights Using Long-Short Term Memory Networks With Data Integration at Continental Scales. Water Resour. Res.
**2020**, 56, e2019WR026793. [Google Scholar] [CrossRef] - Ha, S.; Liu, D.; Mu, L. Prediction of Yangtze River streamflow based on deep learning neural network with El Niño–Southern Oscillation. Sci. Rep.
**2021**, 11, 11738. [Google Scholar] [CrossRef] - Kratzert, F.; Klotz, D.; Herrnegger, M.; Sampson, A.K.; Hochreiter, S.; Nearing, G.S. Toward Improved Predictions in Ungauged Basins: Exploiting the Power of Machine Learning. Water Resour. Res.
**2019**, 55, 11344–11354. [Google Scholar] [CrossRef] [Green Version] - Le, X.H.; Ho, H.V.; Lee, G.; Jung, S. Application of Long Short-Term Memory (LSTM) neural network for flood forecasting. Water
**2019**, 11, 1387. [Google Scholar] [CrossRef] [Green Version] - Shen, C. A Transdisciplinary Review of Deep Learning Research and Its Relevance for Water Resources Scientists. Water Resour. Res.
**2018**, 54, 8558–8593. [Google Scholar] [CrossRef] - Afzaal, H.; Farooque, A.A.; Abbas, F.; Acharya, B.; Esau, T. Groundwater estimation from major physical hydrology components using artificial neural networks and deep learning. Water
**2020**, 12, 5. [Google Scholar] [CrossRef] [Green Version] - Huang, X.; Gao, L.; Crosbie, R.S.; Zhang, N.; Fu, G.; Doble, R. Groundwater recharge prediction using linear regression, multi-layer perception network, and deep learning. Water
**2019**, 11, 1879. [Google Scholar] [CrossRef] [Green Version] - Lähivaara, T.; Malehmir, A.; Pasanen, A.; Kärkkäinen, L.; Huttunen, J.M.J.; Hesthaven, J.S. Estimation of groundwater storage from seismic data using deep learning. Geophys. Prospect.
**2019**, 67, 2115–2126. [Google Scholar] [CrossRef] [Green Version] - Ma, Y.; Montzka, C.; Bayat, B.; Kollet, S. Using Long Short-Term Memory networks to connect water table depth anomalies to precipitation anomalies over Europe. Hydrol. Earth Syst. Sci.
**2021**, 25, 3555–3575. [Google Scholar] [CrossRef] - Malakar, P.; Mukherjee, A.; Bhanja, S.N.; Ray, R.K.; Sarkar, S.; Zahid, A. Machine-learning-based regional-scale groundwater level prediction using GRACE. Hydrogeol. J.
**2021**, 29, 1027–1042. [Google Scholar] [CrossRef] - Su, Y.; Sen, N.C.F.; Li, W.C.; Lee, I.H.; Lin, C.P. Applying deep learning algorithms to enhance simulations of large-scale groundwater flow in IoTs. Appl. Soft Comput. J.
**2020**, 92, 106298. [Google Scholar] [CrossRef] - Pan, B.; Hsu, K.; AghaKouchak, A.; Sorooshian, S. Improving Precipitation Estimation Using Convolutional Neural Network. Water Resour. Res.
**2019**, 55, 2301–2321. [Google Scholar] [CrossRef] [Green Version] - Vandal, T.; Kodra, E.; Ganguly, S.; Michaelis, A.; Nemani, R.; Ganguly, A.R. DeepSD: Generating high resolution climate change projections through single image super-resolution. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Halifax, NS, Canada, 13–17 August 2017; pp. 1663–1672. [Google Scholar] [CrossRef]
- Shi, X.; Chen, Z.; Wang, H.; Yeung, D.Y.; Wong, W.K.; Woo, W.C. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. In Proceedings of the 28th International Conference on Neural Information Processing Systems, NIPS’15, Montreal, QC, Canada, 7–12 December 2015; Volume 1, pp. 802–810. [Google Scholar]
- Miao, Q.; Pan, B.; Wang, H.; Hsu, K.; Sorooshian, S. Improving monsoon precipitation prediction using combined convolutional and long short term memory neural network. Water
**2019**, 11, 977. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Gao, Z.; Long, M.; Wang, J.; Yu, P.S. PredRNN++: Towards a Resolution of the Deep-in-Time Dilemma in Spatiotemporal Predictive Learning. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholm, Sweden, 15 July 2018; Volume 11, pp. 8122–8131. [Google Scholar]
- Wang, Y.; Wu, H.; Zhang, J.; Gao, Z.; Wang, J.; Yu, P.S.; Long, M. PredRNN: A Recurrent Neural Network for Spatiotemporal Predictive Learning. arXiv
**2021**, arXiv:2103.09504. [Google Scholar] - Kuffour, B.N.O.; Engdahl, N.B.; Woodward, C.S.; Condon, L.E.; Kollet, S.; Maxwell, R.M. Simulating Coupled Surface-Subsurface Flows with ParFlow v3.5.0: Capabilities, Applications, and Ongoing Development of an Open-Source, Massively Parallel, Integrated Hydrologic Model. Geosci. Model Dev.
**2020**, 13, 1373–1397. [Google Scholar] [CrossRef] [Green Version] - Ashby, S.F.; Falgout, R.D. A Parallel Multigrid Preconditioned Conjugate Gradient Algorithm for Groundwater Flow Simulations. Nucl. Sci. Eng.
**1996**, 124, 145–159. [Google Scholar] [CrossRef] - Jones, J.E.; Woodward, C.S. Newton-Krylov-Multigrid Solvers for Large-Scale, Highly Heterogeneous, Variably Saturated Flow Problems. Adv. Water Resour.
**2001**, 24, 763–774. [Google Scholar] [CrossRef] [Green Version] - Kollet, S.J.; Maxwell, R.M. Integrated Surface-Groundwater Flow Modeling: A Free-Surface Overland Flow Boundary Condition in a Parallel Groundwater Flow Model. Adv. Water Resour.
**2006**, 29, 945–958. [Google Scholar] [CrossRef] [Green Version] - Maxwell, R.M. A Terrain-Following Grid Transform and Preconditioner for Parallel, Large-Scale, Integrated Hydrologic Modeling. Adv. Water Resour.
**2013**, 53, 109–117. [Google Scholar] [CrossRef] - Maxwell, R.M.; Condon, L.E.; Kollet, S.J. A High-Resolution Simulation of Groundwater and Surface Water over Most of the Continental US with the Integrated Hydrologic Model ParFlow V3. Geosci. Model Dev.
**2015**, 8, 923–937. [Google Scholar] [CrossRef] [Green Version] - Richards, L.A. Capillary Conduction of Liquids through Porous Mediums. J. Appl. Phys.
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - van Genuchten, M.T. A Closed-Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef] [Green Version] - Srivastava, R.K.; Greff, K.; Schmidhuber, J. Training Very Deep Networks. In NIPS’15, Proceedings of the 28th International Conference on Neural Information Processing Systems, Montreal, QC, Canada, 7–12 December 2015; Neural Information Processing Systems Foundation: Montreal, QC, Canada, 2015; Volume 2, pp. 2377–2385. [Google Scholar]
- Condon, L.E.; Maxwell, R.M. Modified Priority Flood and Global Slope Enforcement Algorithm for Topographic Processing in Physically Based Hydrologic Modeling Applications. Comput. Geosci.
**2019**, 126, 73–83. [Google Scholar] [CrossRef] - Schaap, M.G.; Leij, F.J. Database-Related Accuracy and Uncertainty of Pedotransfer Functions. Soil Sci.
**1998**, 163, 765–779. [Google Scholar] [CrossRef] - Gleeson, T.; Smith, L.; Moosdorf, N.; Hartmann, J.; Dürr, H.H.; Manning, A.H.; van Beek, L.P.H.; Jellinek, A.M. Mapping Permeability over the Surface of the Earth. Geophys. Res. Lett.
**2011**, 38. [Google Scholar] [CrossRef] [Green Version] - Maxwell, R.M.; Condon, L.E. Connections between Groundwater Flow and Transpiration Partitioning. Science
**2016**, 353, 377–380. [Google Scholar] [CrossRef] [Green Version] - Kingma, D.P.; Ba, J.L. Adam: A Method for Stochastic Optimization. In Proceedings of the 3rd International Conference on Learning Representations, ICLR 2015—Conference Track Proceedings, San Diego, CA, USA, 7–9 May 2015. [Google Scholar]
- Gupta, H.V.; Kling, H.; Yilmaz, K.K.; Martinez, G.F. Decomposition of the Mean Squared Error and NSE Performance Criteria: Implications for Improving Hydrological Modelling. J. Hydrol.
**2009**, 377, 80–91. [Google Scholar] [CrossRef] [Green Version] - Kling, H.; Fuchs, M.; Paulin, M. Runoff Conditions in the Upper Danube Basin under an Ensemble of Climate Change Scenarios. J. Hydrol.
**2012**, 424–425, 264–277. [Google Scholar] [CrossRef] - Aghakouchak, A.; Mehran, A. Extended Contingency Table: Performance Metrics for Satellite Observations and Climate Model Simulations. Water Resour. Res.
**2013**, 49, 7144–7149. [Google Scholar] [CrossRef] [Green Version] - Beucler, T.; Pritchard, M.; Rasp, S.; Ott, J.; Baldi, P.; Gentine, P. Enforcing Analytic Constraints in Neural Networks Emulating Physical Systems. Phys. Rev. Lett.
**2021**, 126, 098302. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**From Wang et al. [31], schematic of the causal LSTM, in which the temporal and spatial memories are connected in a cascaded way through gated structures. Colored parts are newly designed operations, concentric circles denote concatenation, and σ is the element-wise Sigmoid function.

**Figure 2.**From Wang et al. [31], the final architecture (

**top**) with the gradient highway unit (

**bottom**), where concentric circles denote concatenation, and σ is the element-wise Sigmoid function. The blue parts indicate the gradient highway connecting the current timestep directly with previous inputs, while the red parts show the deep transition pathway.

**Figure 4.**Rainfall–runoff scenarios in the Taylor River basin for the train (

**a**), validation (

**b**), and test (

**c**) sets. Rain is represented by an upside-down horizontal bar; outflow is represented by a line with a corresponding color.

**Figure 6.**(

**a**) shows the streamflow timeseries at the outlet of the Taylor River basin for ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for scenarios 2 and 3 of the testing set. (

**b**,

**c**) show a snapshot of the stream network of ParFlow simulations (left) and ParFlow-ML predictions (right) for scenario 2 at timestep # 30 (

**b**) and scenario 3 at timestep # 65 (

**c**).

**Figure 7.**(

**a**) shows the streamflow timeseries at the outlet of the Little Washita River basin for ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for scenarios 1 and 4 of the testing set. (

**b**,

**c**) show a snapshot of the stream network of ParFlow simulations (left) and ParFlow-ML predictions (right) for scenario 1 at timestep # 42 (

**b**) and scenario 4 at timestep # 65 (

**c**).

**Figure 8.**(

**a**) Water table depth from ParFlow simulations for scenario 3 of the testing set at timestep # 20 for the Taylor River basin. (

**b**) Water table depth from ParFlow-ML predictions for scenario 3 of the testing set at timestep # 20 for the Taylor River basin. Locations for three points: A, B, and C are denoted in (

**a**,

**b**). (

**c**) Water table depth from ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set at Point A. (

**d**) Water table depth from ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set at Point B. (

**e**) Water table depth from ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set at Point C.

**Figure 9.**(

**a**) Water table depth from ParFlow simulations for scenario 2 of the testing set at timestep # 10 for the Little Washita River basin. (

**b**) Water table depth from ParFlow-ML predictions for scenario 2 of the testing set at timestep # 10 for the Little Washita River basin. Locations for three points: A and B are denoted in (

**a**,

**b**). (

**c**) Water table depth from ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set at Point A. (

**d**) Water table depth from ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set at Point B.

**Figure 10.**Total water storage in the Taylor River basin evaluation between ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set.

**Figure 11.**Total water storage in the Little Washita River basin evaluation between ParFlow simulations (solid lines) and ParFlow-ML predictions (dash lines) for all the scenarios of the testing set.

**Figure 12.**Execution time comparison between ParFlow simulations (blue points) and ParFlow-ML predictions (red points). Linear regression lines for ParFlow execution time and PredRNN execution time are in blue and red, respectively.

**Table 1.**Intensity, length, and recession for the rainfall scenarios. Note that the scenarios are color-coded for train (1–16), validation (17–20), and test (21–24).

Scenarios | Rain Intensity (m/h) | Rain Length (h) | Recession Length (h) |
---|---|---|---|

1 | 0.0619 | 14 | 22 |

2 | 0.0557 | 28 | 30 |

3 | 0.0283 | 22 | 17 |

4 | 0.0631 | 7 | 33 |

5 | 0.0334 | 18 | 36 |

6 | 0.0569 | 28 | 21 |

7 | 0.0532 | 21 | 54 |

8 | 0.0119 | 7 | 52 |

9 | 0.0331 | 12 | 35 |

10 | 0.0668 | 29 | 21 |

11 | 0.0344 | 25 | 30 |

12 | 0.0161 | 10 | 47 |

13 | 0.0389 | 16 | 24 |

14 | 0.0775 | 13 | 41 |

15 | 0.0797 | 22 | 57 |

16 | 0.0213 | 12 | 56 |

17 | 0.0677 | 26 | 36 |

18 | 0.0451 | 12 | 15 |

19 | 0.0765 | 13 | 26 |

20 | 0.0792 | 10 | 26 |

21 | 0.0474 | 11 | 25 |

22 | 0.0215 | 28 | 16 |

23 | 0.0357 | 29 | 54 |

24 | 0.0539 | 29 | 38 |

Testing Scenarios | KGE | Relative Bias | Spearman’s Rho | |||
---|---|---|---|---|---|---|

Taylor | LW | Taylor | LW | Taylor | LW | |

21 | 0.747 | 0.975 | 0.177 | 0.020 | 0.947 | 0.994 |

22 | 0.960 | 0.882 | 0.022 | 0.082 | 0.976 | 0.992 |

23 | 0.970 | 0.854 | 0.005 | 0.103 | 0.776 | 0.872 |

24 | 0.787 | 0.973 | 0.135 | 0.019 | 0.937 | 0.970 |

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**MDPI and ACS Style**

Tran, H.; Leonarduzzi, E.; De la Fuente, L.; Hull, R.B.; Bansal, V.; Chennault, C.; Gentine, P.; Melchior, P.; Condon, L.E.; Maxwell, R.M.
Development of a Deep Learning Emulator for a Distributed Groundwater–Surface Water Model: ParFlow-ML. *Water* **2021**, *13*, 3393.
https://doi.org/10.3390/w13233393

**AMA Style**

Tran H, Leonarduzzi E, De la Fuente L, Hull RB, Bansal V, Chennault C, Gentine P, Melchior P, Condon LE, Maxwell RM.
Development of a Deep Learning Emulator for a Distributed Groundwater–Surface Water Model: ParFlow-ML. *Water*. 2021; 13(23):3393.
https://doi.org/10.3390/w13233393

**Chicago/Turabian Style**

Tran, Hoang, Elena Leonarduzzi, Luis De la Fuente, Robert Bruce Hull, Vineet Bansal, Calla Chennault, Pierre Gentine, Peter Melchior, Laura E. Condon, and Reed M. Maxwell.
2021. "Development of a Deep Learning Emulator for a Distributed Groundwater–Surface Water Model: ParFlow-ML" *Water* 13, no. 23: 3393.
https://doi.org/10.3390/w13233393