# Random Forest Ability in Regionalizing Hourly Hydrological Model Parameters

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Why and How Do We Regionalize Hydrological Model Parameters?

#### 1.2. Random Forest: A Potentially Useful Tool for Regionalization

#### 1.3. Application of RF for Model Regionalization in Multiple Land-Use Environments

#### 1.4. Context and Scope of the Study

## 2. Data

#### 2.1. Sample Selection

^{2}to 110,000 km

^{2}, with a median value of 222 km

^{2}. Characterization in terms of urbanization was carried out using the Catchment Percent Developed (CPD) measure [54], which informs about the fraction of the catchment that is occupied by an urban landscape. Hence, CPD varies from 0% for a completely rural catchment to 100% for a completely urbanized catchment.

#### 2.2. Catchment Descriptors

- Climate: The catchment’s response inherits most of its variability from the catchment’s climate [66]. Many climate characteristics were computed over each catchment’s record period in order to limit their dependency on the record period. As climate descriptors, we considered mean hourly precipitations P (mm/h), mean hourly potential evapotranspiration PE (mm/h), humidity index HI (-), and flashiness of precipitations (-).
- Morphology: The catchment’s morphology is essential in predicting the catchment’s response timing and the repartition of precipitations into infiltration and runoff. For this reason, we used the catchment drained area (km
^{2}), drainage density (km/km^{2}), and the median compound topographic index (-) as morphological descriptors. - Land use: The catchment’s water yield and evapotranspiration losses depend on the catchment land use. Also, it is in our case of a central interest as we are dealing with the catchment’s level of urbanization. Thus, three land-use metrics were assessed: the CPD (%), the fraction of forest (%), and the fraction of open water (%).
- Geopedology: The catchment’s water transfers to and from the subjacent aquifers are modulated by the catchment’s geological and pedological characteristics. Hence, mean porosity (-), mean of log-transformed values of intrinsic permeability (m
^{2}), mean soil and subsoil content of gravel (%), silt (%), and clay (%) were considered as geopedological characteristics.

## 3. Methods

#### 3.1. Model Parameters and Calibration

_{1}(mm) is the production store maximum capacity. It controls the amount of water that can be stored and then lost either through actual evapotranspiration or percolation. X

_{2}(mm/h) is the groundwater–surface water exchange rate, which is positive when importing water to the river stream and negative otherwise. X

_{3}(mm) is the nonlinear routing store maximum capacity. It plays a role in determining the low-frequency component of the simulated hydrograph. X

_{4}(h) controls the routing unit hydrograph time base. It represents the characteristic time of the catchment response. Ficchi [85] and Perrin et al. [56] give a more detailed description of the model equations.

#### 3.2. Estimating the Model Parameters at Ungauged Locations Using RF

_{i}} for the remaining 120 urban catchments (treated as ungauged) using their descriptors {D

_{j}}. RF construction requires a specification about the number of trees to be grown (here Ntree = 500) and the number of variables to be selected randomly at each tree growing (here mtry was fixed at 5 by using the tuneRF function in the randomForest R package). While growing a tree, about one third of the cases—called out-of-bag (OOB) data—are left out of the sample [92,93], and then used to compute errors at each grown tree, e.g., the mean squared errors (MSE).

_{j}was determined. If MSE

_{0}is the MSE of the RF computed during the construction, i.e., with non-permuted values for the OOB data, then IncMSE represents the difference between MSE

_{0}and MSE

_{j}, after being computed and then scaled by the standard deviation of the differences over the trees. The greater the IncMSE, the more important the predictor variable.

#### 3.3. Benchmark Regionalization Techniques

- The RF-estimated parameters using the catchment descriptors.
- The transferred parameters from the closest neighbor catchment. Spatial closeness was computed by weighting the distances between the catchment centroids (80%) and outlets (20%) [97]. Close catchments were selected either from the whole 2105 catchments used to train the RF_ALL (Figure 4) or from the 119 urban catchments used to grow RF_URB.
- The transferred parameters from the most similar catchment with respect to the descriptors used to construct the RF. For each descriptor, the catchment ranks were determined. Then, the Euclidean distance between ranks was computed in the hyperspace of descriptors [15]. Similar catchments were selected either from the whole 2105 catchments used to construct the RF_ALL (Figure 4) or from the 119 urban catchments used to construct RF_URB.

## 4. Results

#### 4.1. Model Performances and Estimated Parameters

_{1}, X

_{3}, and X

_{4}, while for X

_{2}, none of the methods was satisfactory. Re-estimating the calibrated parameters with respect to transformed flows was systematically easier than when non-transformed flows were considered for calibration. The statistical test results indicated that the correlations were highly significant, even in the case of X

_{2}where the values were hardly above 0.1. However, the R

^{2}values were very moderate and in line with the values obtained in previous studies [10]. These values suggest that the calibrated parameters are difficult to estimate, meaning that there are still some remaining issues concerning the descriptors used and/or concerning model structure/parametrization uncertainties. Also, the clear superiority of the RF in estimating each parameter individually did not translate to model performances, especially in the case when KGESR and NSESR were considered. This may be due to the relative sensitivity of the model to its four free parameters and/or the fact that RF considers each parameter independently. Indeed, an RF was constructed for each model parameter, which means that the estimation is independent from one parameter to another. This might diminish the power of possible interactions between the parameters, i.e., compensation effects, whereas these effects were kept in the transferred sets.

^{2}, mean CPD: 59.6%) between 1 March 2013 and 31 August 2013. This period was chosen as it belongs to the catchment’s wettest year between 2010 and 2017 (i.e., P2). This catchment was recently investigated by Diem et al. [98] who found it to be hydrologically altered due to rapid shifts in the catchment’s land cover. For this example, only the case when KGESR was used is shown. Table 3 details the values of the five sets of parameters used to compute these hydrographs: estimated from RF_ALL and RF_URB, transferred from calibration over the catchment’s first period P1, and transferred from the closest catchment (Dick Creek at Old Atlanta Road, near Suwanee, Georgia, USGS code: 02334620, area: 17.8 km

^{2}, mean CPD: 55.3%) and the most similar catchment (Reedy River near Greenville, South Carolina, USGS code: 02164000, area: 125.1 km

^{2}, mean CPD: 68.2%).

#### 4.2. Descriptor Importance

_{1}. Land-use characteristics, in particular CPD, also exhibited moderate weights in deciding X

_{1}. However, they were not highly decisive compared with the remaining descriptors. This seems coherent, as X

_{1}modulates the soil–atmosphere interactions, thus relying on a large number of descriptors. In the case of X

_{2}, the drainage density yielded a remarkable score, followed by mean potential evapotranspiration PE

_{m}and some soil characteristics (mean content of clay, mean porosity) and other morphology descriptors (area and CTI). The influence of PE

_{m}on X

_{2}may stem from the possible interaction with X

_{1}, as both play a role in matching the catchment water budget. The drainage density, the mean content of clay, the porosity, and CTI give implicit or explicit information about the soil permeability, hence, their influence over X

_{2}was expected as X

_{2}characterizes the interactions between the groundwater and the surface water. For X

_{3}and X

_{4}, the most influential descriptors were very easily distinguishable. PE

_{m}had the heaviest weight on X

_{3}, the parameter that shapes the component of slow flow; this may be due to the fact that in recession periods, PE

_{m}plays a major role in conditioning the recession flow curvature. The mean content of clay, PE

_{m}, and the drainage density played a major role in determining X

_{4}. This parameter is coherent with the characteristic time of the catchment response, and drives majorly the correlation at the hourly time step between the observed and simulated flows. Besides, X

_{4}is correlated with the catchment area, which was revealed to be the fifth most influential descriptor.

_{3}and X

_{4}, which represents the transfer function of the model. Their values were lower for the urban catchments, meaning that the time response of the catchments was relatively shortened. The results of the Mann–Whitney–Wilcoxon (MWW) statistical equality test indicated that there were also differences in the budget parameters, i.e., X

_{1}and X

_{2}.

_{1}and X

_{3}), meaning that the RF approach succeeded in adapting the model parameters depending on the urbanization stage, if the training sample was adapted to the target catchments. Exceptions were made for X

_{2}and X

_{4}, where the statistical test results indicated that RF-estimated parameters for the urban catchments were different from the calibrated ones. For these two parameters, CPD had the lowest ranks (12th and 14th out of 15), which means that for urban catchments with relatively low CPD values (i.e., close to 20%), the estimation of these two parameters was driven more by other descriptors than by CPD.

_{1}where the parameters exhibited a trend toward higher values as witnessed in the urban sample (confirmed by the MWW test), which suggests an overall weak CPD sensitivity of the RF_ALL.

## 5. Discussion and Conclusions

#### 5.1. Regionalization with RF: What Is Appreciated and What Is Depreciated?

_{3}and X

_{4}, RF_ALL had a moderate maximum score in estimating X

_{3}(R

^{2}= 0.45, objective function: KGESR, Table 2) but failed to maintain the same score for X

_{4}(R

^{2}= 0.29, Table 2), and when the estimation of X

_{4}was satisfactory (R

^{2}= 0.61, objective function: NSE, Table 2), the corresponding X

_{3}estimation score was low (R

^{2}= 0.32, Table 2). RF_URB appeared to be globally more efficient in estimating these routing parameters (R

^{2}= 0.41–0.6 for X

_{4}), perhaps due to the homogeneity in catchment size between the two urban-catchments samples; knowing that X

_{4}is, to first order, dependent on the catchment area, (3) the descriptors might not be relevant in regionalizing the model parameters, or the methods of their determination were not suitable. For the time scale, the descriptors were averaged over the whole simulation period, and also aggregated spatially up to the catchment scale, under the assumption of stable catchment characteristics over the study period. Intra-temporal and intra-catchment variability representation could yield more information to explain the variance of the parameters.

#### 5.2. Weak Sensitivity of the RF-Derived Relationships with the Urbanization Measure

_{1}was the only parameter that responded to the CPD shifts, displaying values similar to the ones estimated for the urban catchments. The reasons why this shift was not visible on the remaining parameters could be (1) the insensitivity of the model parameters to the CPD measure and (2) the multidimensional aspect of the parameter–descriptor relationship, which makes the parameters less sensitive to one descriptor itself but wholly dependent on all the descriptors, as shown in the variable importance, where almost no variable was remarkably influential. Perhaps this parameter insensitivity was amplified by the rural catchment insensitivity to the CPD shifts, as the natural descriptors (i.e., all the descriptors except CPD) predominate, and (3) the unsuitability (or insufficiency) of the CPD measure to describe the urbanization features, which may be viewed as a need for other urbanization descriptors to be included.

#### 5.3. Conclusions and Perspectives

## Author Contributions

## Funding

## Acknowledgments

^{®}product provided by the IGN (http://professionnels.ign.fr/bdcarthage). Soil properties were described using the GLHYMPS dataset (available at http://www.groundwaterscienceandsustainability.org/data.html) and the HWSD dataset extracted using the hwsd R package. Finally, we thank Jordan Read (USGS), David Blodgett (USGS), David Watkins (USGS), and William Watkins (USGS) for their help concerning the geoknife R package, Tammy Walker (ORNL) and Michele Thornton (ORNL) for their help about the Daymet dataset.

## Conflicts of Interest

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**Figure 3.**GR4H model structure [85]. The water fluxes are specified in blue and red. The four free model parameters X

_{1}to X

_{4}are in green.

**Figure 4.**Random forest (RF) construction using the 2105 rural and urban catchments. Validation is carried on the remaining set of 120 urban catchments (i.e., with CPD ≥ 20%).

**Figure 5.**Validation scores on P2 in terms of (

**a**) KGESR (

**b**) Kling–Gupta efficiency (KGE) (

**c**) NSESR and (

**d**) Nash–Sutcliffe efficiency (NSE) using calibrated parameters on P1 (CALIB ON P1), estimated parameters using random forest (RF_ALL and RF_URB), transferred parameters from the closest catchments (CLOSE_ALL and CLOSE_URB) and from the most similar catchments (SIMILAR_ALL and SIMILAR_URB). The values indicate the minimum, the median, and the maximum scores. Some minimum values were not shown as they were less than −4 for the considered metric of evaluation. The letters indicate the statistical equality at 10% risk between the performances, estimated using the Mann–Whitney–Wilcoxon test.

**Figure 6.**Observed hourly flow time series (in m

^{3}/s) and simulated flow using parameters transferred from calibration over P1, estimated parameters using RF_ALL and RF_URB, and transferred parameters from the closest and the most similar catchment to the Suwanee Creek catchment (U.S. Geological Survey (USGS) code: 02334885), as response to measured precipitations (in mm/h) between 1 March and 31 August 2013. The sets of parameters correspond to KGESR.

**Figure 7.**IncMSE scores (%) of the catchment descriptors for (

**a**) X

_{1}, (

**b**) X

_{2}, (

**c**) X

_{3}, and (

**d**) X

_{4}. Importance scores were extracted from the RF_ALL.

**Figure 8.**Distribution of parameters (

**a**) X

_{1}, (

**b**) X

_{2}, (

**c**) X

_{3}, and (

**d**) X

_{4}with values indicating the minimum, the median, and the maximum. CAL_RUR are the rural calibrated parameters (with respect to KGESR), RF_RUR are the RF-estimated rural parameters, RF_UPDATED are the RF-estimated parameters over the rural sample with transferred CPD value from the corresponding urban catchment, RF_URB are the RF-estimated parameters over the urban sample, and CAL_URB are the calibrated parameters (with respect to KGESR). The letters indicate statistical equality at 5% risk between the different parameter distributions, estimated using the Mann–Whitney–Wilcoxon test.

**Table 1.**Description of the different climatic, topographic, land-use, and geopedological characteristics estimated for each catchment.

Notation | Index Name | Computation | Unit | Data Source |
---|---|---|---|---|

P_{m} | Mean hourly precipitation | Total depth of precipitations over the recorded period (8–16 years) divided by the number of hours, aggregated spatially to the catchment scale | mm/h | COMEPHORE product of Meteo-France, 1-km resolution [67] and NEXRAD Stage IV dataset, 4-km resolution, extracted using the geoknife R Package [68,69,70] |

PE_{m} | Mean hourly potential evapotranspiration | Total depth of potential evapotranspiration over the recorded period (8–16 years) divided by the number of hours, aggregated spatially to the catchment scale | mm/h | Evaluated using temperature-based formula [71]. Daily temperature was extracted from SAFRAN product of Meteo-France, 8-km resolution [72] and Daymet dataset, 1-km resolution [73] |

HI | Humidity index | $\mathrm{HI}=\frac{{\mathrm{P}}_{\mathrm{m}}}{{\mathrm{PE}}_{\mathrm{m}}}$ | — | P_{m} and PE_{m} data sources |

FP | Flashiness of precipitation | $\mathrm{FP}=\frac{{{\displaystyle \sum}}_{\mathrm{i}}\left|{\mathrm{P}}_{\mathrm{i}}-{\mathrm{P}}_{\mathrm{i}-1}\right|}{{{\displaystyle \sum}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{i}}}$, with P_{i} the precipitation depth (mm) at hour i [74] | — | P_{i} data source |

A | Catchment area | — | km^{2} | [57,75] |

DD | Drainage density | $\mathrm{DD}=\frac{{{\displaystyle \sum}}_{\mathrm{i}}{\mathrm{L}}_{\mathrm{i}}}{\mathrm{A}}$, with L_{i} length of stream i (km) and A the catchment area (km^{2}) | km/km^{2} | The hydrographic networks were extracted from the BD Carthage dataset (France) and the National Hydrography Dataset NHD (USA) using the FedData R Package [76] |

CTI | Median compound topographic index | $\mathrm{CTI}=\mathrm{median}\text{}(\mathrm{log}\left(\frac{{\mathrm{A}}_{\mathrm{s},\mathrm{i}}}{\mathrm{tan}{\mathsf{\beta}}_{\mathrm{i}}}\right))$, with A_{s,i} the ith cell’s specific area and β_{i} its slope angle | — | [77] |

CPD | Catchment percent developed | Sum of the pixels attributed to urbanization classes divided by the total number of pixels | % | National Land Cover Database (NLCD) 2001, 2006, and 2011 (USA) and CLC 1990, 2000, 2006, and 2012 (France) |

fW | Fraction of open water | Sum of pixels occupied by open water class divided by the total number of pixels | % | NLCD 2001, 2006, and 2011 (USA) and CLC 1990, 2000, 2006, and 2012 (France) |

fFOR | Fraction of forest | Sum of pixels occupied by forest classes divided by the total number of pixels | % | NLCD 2001, 2006, and 2011 (USA) and CLC 1990, 2000, 2006, and 2012 (France) |

POROSITY | Mean porosity of the catchment’s soil and subsoil geologic units | Volume of voids divided by the total volume | — | GLobal HYdrogeology MaPS (GLHYMPS) [78] |

PER | Mean of logarithm values of soil and subsoil permeability | — | log(m^{2}) | GLobal HYdrogeology MaPS (GLHYMPS) [78] |

M_GRAVEL | Mean gravel content of soil and subsoil geologic units | — | % | Harmonized World Soil Database HWSD (Version 1.2) [79,80] |

M_SILT | Mean silt content of soil and subsoil geologic units | — | % | Harmonized World Soil Database HWSD (Version 1.2) [79,80] |

M_CLAY | Mean clay content of soil and subsoil geologic units | — | % | Harmonized World Soil Database HWSD (Version 1.2) [79,80] |

**Table 2.**R

^{2}computed between the calibrated parameters with respect to different objective functions (KGESR, KGE, NSESR, and NSE) and the estimated parameters via RF, transferred parameters from the close and from the similar catchments for the 120-urban-catchments sample. Statistical significance of the correlation is indicated by asterisks, where

*****is significant at 5%,

******at 1%, and

*******at 0.1% risk.

Parameter | Objective Function | Using the 2105 Catchments | Using the 119 Urban Catchments | ||||
---|---|---|---|---|---|---|---|

RF | CLOSE | SIMILAR | RF | CLOSE | SIMILAR | ||

X_{1} | KGESR | 0.476 *** | 0.361 *** | 0.367 *** | 0.448 *** | 0.367 *** | 0.33 *** |

KGE | 0.144 *** | 0.077 ** | 0.066 ** | 0.08 ** | 0.047 * | 0.085 ** | |

NSESR | 0.53 *** | 0.353 *** | 0.332 *** | 0.434 *** | 0.301 *** | 0.25 *** | |

NSE | 0.152 *** | 0.084 ** | 0.066 ** | 0.111 *** | 0.034 * | 0.069 ** | |

X_{2} | KGESR | 0.022 | 0.037 * | 0.062 ** | 0.009 | 0.014 | 0.064 ** |

KGE | 0.054 * | 0.053 * | 0.098 *** | 0.064 ** | 0.011 | 0.045 * | |

NSESR | 0.109 *** | 0.026 | 0.056 ** | 0.064 ** | 0.023 | 0.002 | |

NSE | 0.11 *** | 0.085 ** | 0.047 * | 0.067 ** | 0.105 *** | 0.036 * | |

X_{3} | KGESR | 0.449 *** | 0.275 *** | 0.213 *** | 0.346 *** | 0.194 *** | 0.173 *** |

KGE | 0.222 *** | 0.147 *** | 0.094 *** | 0.245 *** | 0.094 *** | 0.069 ** | |

NSESR | 0.408 *** | 0.444 *** | 0.333 *** | 0.442 *** | 0.374 *** | 0.202 *** | |

NSE | 0.318 *** | 0.225 *** | 0.277 *** | 0.405 *** | 0.227 *** | 0.245 *** | |

X_{4} | KGESR | 0.287 *** | 0.082 ** | 0.207 *** | 0.438 *** | 0.077 ** | 0.207 *** |

KGE | 0.301 *** | 0.076 ** | 0.201 *** | 0.415 *** | 0.09 *** | 0.167 *** | |

NSESR | 0.417 *** | 0.064 ** | 0.355 *** | 0.578 *** | 0.1 *** | 0.396 *** | |

NSE | 0.613 *** | 0.121 *** | 0.284 *** | 0.604 *** | 0.096 *** | 0.262 *** |

**Table 3.**Sets of parameters transferred from calibration over P1, estimated using RF_ALL and RF_URB, and transferred from the closest catchment and the most similar catchment for the Suwanee Creek catchment. Score Whole is the KGESR over the period 2 (i.e., between 2010 and 2017) and Score Period is computed over the 6-months period for which the resulting hydrographs are shown in Figure 6. MaxSim/MaxObs is the ratio of the simulated and observed peak flows over the same 6-month period.

Source | Parameters | Score Whole | Score Period | MaxSim/MaxObs Period | |||
---|---|---|---|---|---|---|---|

X_{1} (mm) | X_{2} (mm/h) | X_{3} (mm) | X_{4} (h) | ||||

Calibration over P1 | 973.92 | 0.13 | 9.36 | 19.7 | 0.824 | 0.818 | 0.72 |

RF_ALL | 1258.52 | 0.1 | 32.92 | 7.71 | 0.872 | 0.819 | 1.21 |

RF_URB | 1250.69 | 0.08 | 22.87 | 7.44 | 0.876 | 0.821 | 1.36 |

Close | 1269.62 | 0.14 | 13.27 | 3.34 | 0.719 | 0.6 | 2.5 |

Similar | 1394.09 | 0.19 | 25.53 | 3.86 | 0.807 | 0.722 | 1.98 |

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**MDPI and ACS Style**

Saadi, M.; Oudin, L.; Ribstein, P. Random Forest Ability in Regionalizing Hourly Hydrological Model Parameters. *Water* **2019**, *11*, 1540.
https://doi.org/10.3390/w11081540

**AMA Style**

Saadi M, Oudin L, Ribstein P. Random Forest Ability in Regionalizing Hourly Hydrological Model Parameters. *Water*. 2019; 11(8):1540.
https://doi.org/10.3390/w11081540

**Chicago/Turabian Style**

Saadi, Mohamed, Ludovic Oudin, and Pierre Ribstein. 2019. "Random Forest Ability in Regionalizing Hourly Hydrological Model Parameters" *Water* 11, no. 8: 1540.
https://doi.org/10.3390/w11081540