# Global Sensitivity Analysis and Calibration by Differential Evolution Algorithm of HORTSYST Crop Model for Fertigation Management

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## Abstract

**:**

^{−2}. Air temperature and relative humidity were measured with an S-THB-M008 model sensor. Global solar radiation was measured with an S-LIB-M003 sensor connected to a U-30-NRC datalogger. In the sensitivity analysis run in the two growth stages, it was observed that a greater number of parameters were more important at the beginning of fructification than at the end of crop growth for 10% and 20% of the variation of the parameters. The sensitivity analysis came up with nine parameters ($RUE$, $a$, $b$,${c}_{1}$,${c}_{2}$,$A$, ${B}_{d}$, ${B}_{n}$, and $PTIini$) as the most important of the HORTSYST model, which were included in the calibration process with the DE algorithm. The best fit, according to $RMSE$, was for $LAI$, followed by $Nup$, $DMP$, and $E{T}_{c}$ for both crop seasons; the $RMSE$ was close to zero, indicating a good prediction of the model’s performance.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Experiments

^{−2}. For the first experiment, tomato seeds were sown on 18 July 2015, and the seedlings were transplanted on 21 August 2015, in an 8 × 8 m chapel-type glasshouse. In the second experiment, the seeds were sown on 24 March 2016, and transplanted on 24 April 2016, in a plastic-covered 8 × 15 m greenhouse with natural ventilation. The plants were set in polyethylene bag pots of 35 × 35 cm (12 L).

#### 2.2. Model Description

^{−1}), dry matter production ($DMP,$g m

^{−2}), and nitrogen uptake ($Nup,$g m

^{−2}) as the state variables, while the leaf area index ($LAI,$ m

^{2}m

^{−2}) and crop transpiration ($E{T}_{c}$, kg m

^{−2}) are considered as output variables. The model structure is summarized in Table 1. Figure 1 shows the general structure of the model using a Forrester diagram. The model structure is based on the VegSyst model developed by Gallardo et al. [25,28,29,30].

^{−1}), as has been proposed by several researchers [34,35]. The fraction of light intercepted $\left({f}_{i-PAR}\right)$ is the fraction of global solar radiation that enters through the canopy of a crop characterized by LAI.

#### 2.3. Global Sensitivity Analysis of the HORTSYST Model

#### 2.4. Sobol Sensitivity Analysis Method

#### 2.5. Differential Evolution Algorithm

^{−8}, generation number of 1000, the minimum values were taken from the mean of 25 runs, and the strategy of the DE/rand/1/bin algorithm was implemented during the analysis [18,48,49]. F is a constant, which affects the differential variation between two solutions and was set to 0.6 in our experiments. The crossover rate (CR) controls the change in the population’s diversity, and a value of 0.9 was taken.

#### 2.6. Optimization Problem Description

#### 2.7. Goodness of Fit Performance of Simulations

## 3. Results and Discussion

#### 3.1. Sobol’s Sensitivity Analysis Method

_{1}for ETc output, whereas the other parameters kept their order of importance found with 10% uncertainty. In the case of 119 DAT, the sum of the first-order effects and the total indices were for PTI (0.90, 1.00), DMP (0.91, 1.01), Nup (0.95, 1.02), LAI (0.99, 1.00), and ETc (0.96, 1.02).

#### 3.2. Calibration of HORTSYST Model by Differential Evolution Algorithm

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Forrester’s relational diagram for the HORTSYST model of a greenhouse tomato crop: inputs, outputs, state variables, and parameters of the crop model. State variables are represented by rectangles, rate variables by valves, parameters with a horizontal line, input variables with a circle and a horizonal line, and auxiliary variables with circumferences. Flows of material are represented by normal arrows and information flows with dashed lines.

**Figure 2.**The total and main sensitivity indices estimated using Sobol’s method for (

**A**) PTI = photo–thermal time, (

**B**) DMP = dry matter production, (

**C**) Nup = nitrogen uptake, (

**D**) LAI = leaf area index, and (

**E**) ETc = crop transpiration for 20% of parameter variation after 40 days of growth.

**Figure 3.**The total and main sensitivity indices estimated using Sobol’s method for (

**A**) PTI = photo–thermal time, (

**B**) DMP = dry matter production, (

**C**) Nup = nitrogen uptake, (

**D**) LAI = leaf area index, and (

**E**) ETC = crop transpiration for 20% of parameter variation at the end of the growth cycle (119 DAT).

**Figure 4.**The total and main sensitivity indices estimated using Sobol’s method for (

**A**) PTI = photo–thermal time, (

**B**) DMP = dry matter production, (

**C**) Nup = nitrogen uptake, (

**D**) LAI = leaf area index, and (

**E**) ETc = crop transpiration for 20% of parameter variation integrating the daily values of the outputs during the entire growth cycle (119 DAT).

**Figure 5.**(

**a**,

**c**) PTI = photo–thermal time estimated, (

**b**) PTI = photo–thermal time estimated and LAI = leaf area index simulated data for autumn–winter, 2015, and (

**d**) for spring–summer season, 2016, after calibration. DAT: days after transplant.

**Figure 6.**Measured and simulated data after calibration for (

**A**) DMP = dry matter production, (

**C**) Nup = nitrogen uptake, (

**E**) LAI = leaf area index and (

**G**) ETc = crop transpiration during autumn–winter, 2015, and the (

**B**,

**D**,

**F**,

**H**) 1:1 plots for each variable, respectively. Measured of all variables are indicated by circles. DAT: days after transplant.

**Figure 7.**Measured and simulated data after calibration for (

**A**) DMP = dry matter production, (

**C**) Nup = nitrogen uptake, (

**E**) LAI = leaf area index, and (

**G**) ETc = crop transpiration during spring–summer, 2016, and the (

**B**,

**D**,

**F**,

**H**) 1:1 plots for each variable, respectively. Measured of all variables are indicated by circles. DAT: days after transplant.

Variable | Definition | Equation | Units |
---|---|---|---|

$PTI$ | Photo–thermal time | $PTI\left(j+1\right)=PTI\left(j\right)+\u2206PTI$ | ${\mathrm{MJ}\mathrm{m}}^{-2}$ |

$DMP$ | Dry matter production | $DMP\left(j+1\right)=DMP\left(j\right)+\u2206DMP$ | ${\mathrm{g}\mathrm{m}}^{-2}$ |

${N}_{up}$ | Nitrogen uptake | ${N}_{up}\left(j+1\right)={N}_{up}\left(j\right)+\u2206{N}_{up}$ | ${\mathrm{g}\mathrm{m}}^{-2}$ |

$ETc$ | Daily crop transpiration | $ETc\left(j+1\right)=ETc\left(j\right)+\u2206ETc$ | ${\mathrm{kg}\mathrm{m}}^{-2}$ |

$\u2206PTI$ | Daily photo–thermal time | $\u2206PTI\left(j\right)=\left({\displaystyle {\displaystyle \sum}_{i=1}^{24}}TT\left(i,j\right)\right)PAR\left(j\right)\times {f}_{i-PAR}\left(j\right)$ | ${\mathrm{MJ}\mathrm{m}}^{-2}{\mathrm{d}}^{-1}$ |

$TT$ | Normalized thermal time | $TT=\{\begin{array}{ll}0\hfill & ({T}_{a}<{T}_{min})\hfill \\ \left({T}_{a}-{T}_{min}\right)/\left({T}_{ob}-{T}_{min}\right)\hfill & ({T}_{min}\le {T}_{a}<{T}_{ob})\hfill \\ 1\hfill & \left({T}_{ob}\le {T}_{a}\le {T}_{ou}\right)\hfill \\ \left({T}_{max}-{T}_{a}\right)/\left({T}_{max}-{T}_{ou}\right)\hfill & ({T}_{ou}<{T}_{a}\le {T}_{max})\hfill \\ 0\hfill & ({T}_{a}>{T}_{max})\hfill \end{array}$ | $\left[\begin{array}{c}dimension\\ less\end{array}\right]$ |

$PAR$ | PAR | $PAR\left(j\right)=0.5\times {R}_{g}$ | ${\mathrm{MJ}\mathrm{m}}^{-2}$ |

$\u2206DMP$ | Daily dry matter production | $\u2206DMP\left(j\right)=RUE\times {f}_{i-PAR}\left(j\right)\times PAR\left(j\right)$ | ${\mathrm{g}\mathrm{m}}^{-2}$ |

${f}_{i-PAR}$ | Intercepted PAR fraction | ${f}_{i-PAR}=1-\mathrm{exp}\left(-k\times LAI\left(j\right)\right)$ | $\left[\begin{array}{c}dimension\\ less\end{array}\right]$ |

$LAI\left(j\right)$ | Leaf area index | $LAI\left(j\right)=\left[\frac{{c}_{1}\times \u2206PTI\left(j\right)}{{c}_{2}+\u2206PTI\left(j\right)}\right]\times d$ | ${\mathrm{m}}^{2}{\mathrm{m}}^{-2}$ |

$\%N\left(j\right)$ | Nitrogen content | $\%N\left(j\right)=a\times {\left(\u2206DMP\right)}^{-b}$ | $\left[\begin{array}{c}dimension\\ less\end{array}\right]$ |

$\u2206{N}_{up}$ | Daily nitrogen uptake | ${N}_{up}\left(j\right)=\left(\%N\left(j\right)/100\right)\times DMP\left(j\right)$ | ${\mathrm{g}\mathrm{m}}^{-2}$ |

$ETc\left(i\right)$ | Hourly transpiration | $ETc\left(i\right)=A\times \left(1-\mathrm{exp}\left(-k\times LAI\left(j\right)\right)\right)\times Rg\left(i\right)+LAI\left(DPV\right){B}_{\left(d,n\right)}$ | ${\mathrm{kg}\mathrm{m}}^{-2}{\mathrm{h}}^{-1}$ |

$ETc\left(j\right)$ | Daily evapotranspiration | $\u2206ETc={\displaystyle {\displaystyle \sum}_{i=1}^{24}}ETc\left(i\right)$ | ${\mathrm{kg}\mathrm{m}}^{-2}$ |

**Table 2.**HORTSYST model parameters with 10% and 20% of the variation of their nominal value, used for sensitivity analysis under the experimental condition for the spring–summer crop cycles.

No | Parameter | Symbol | Range 10% | Range 20% | Reference |
---|---|---|---|---|---|

1 | Top upper temperature (°C) | T_{max} | 31.50–38.50 | 28.40–42.00 | [40] |

2 | Top bottom temperature (°C) | T_{min} | 9.00–11.00 | 8.00–12.00 | [40] |

3 | Optimum minimum temperature (°C) | T_{ob} | 15.30–18.70 | 13.60–19.80 | [41] |

4 | Optimum maximum temperature (°C) | T_{ou} | 21.60–26.40 | 19.80–28.40 | [41] |

5 | Radiation use efficiency (g MJ^{−1}) | RUE | 2.79–3.41 | 2.48–3.72 | [30,42] |

6 | Extinction coefficient | k | 0.58–0.70 | 0.51–0.77 | |

7 | N concentration in the dry biomass at the end of the exponential growth period (g m^{−2}) | a | 6.79–8.31 | 6.04–9.06 | [30] |

8 | Is the slope of the nitrogen uptake vs. dry biomass production function | b | −0.17–(−0.14) | −0.18–(0.12) | [30] |

9 | Slope of the curve (m^{−2}) | c_{1} | 2.76–3.38 | 2.46–3.68 | Estimated |

10 | Intersection coefficient | c_{2} | 158.08–193.2 | 140.51–210.77 | Estimated |

11 | Radiative coefficient | A | 0.44–0.54 | 0.39–0.59 | [43] |

12 | Aerodynamic coefficient during day (W m^{−2} kPa^{−1}) | B_{d} | 10.08–12.32 | 8.96–13.44 | [43] |

13 | Aerodynamic coefficient during night (W m^{−2} kPa^{−1}) | B_{n} | 7.45–9.11 | 6.62–9.94 | [43] |

14 | Initial photo–thermal time (MJ m^{−2}) | PTIini | 0.06–0.07 | 0.05–0.07 | Measured |

15 | Initial dry matter production (g m^{−2}) | DMPIni | 0.22–0.27 | 0.20–0.29 | Measured |

16 | Plant density (plants m^{−2}) | d | 3.15–3.85 | 2.8–4.2 | Established |

**Table 3.**Values of global solar radiation (R

_{g}), air temperature (T

_{a}), and relative humidity (RH) during the autumn–winter and spring–summer crop seasons.

Climatic Variable | Autumn–Winter Season | Spring–Summer Season | ||||
---|---|---|---|---|---|---|

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

${R}_{g}$ (MJ m^{−2}) | 0.88 | 3.99 | 8.89 | 5.40 | 10.59 | 14.18 |

${T}_{a}$ (°C) | 14.12 | 18.31 | 21.83 | 15.31 | 17.84 | 21.94 |

$RH$ (%) | 62.59 | 78.58 | 93.98 | 29.47 | 76.82 | 93.16 |

**Table 4.**Sensitivity of HORTSYST parameters in descending order of importance obtained with Sobol’s method applied to two tomato crop stages.

Output Response | At the Beginning of Fructification | At the End of Crop Growth |
---|---|---|

Parameters (10% of variation) | ||

PTI | ${\mathrm{T}}_{\mathrm{ob}}$, ${\mathrm{c}}_{1}$, $\mathrm{d}$,$\mathrm{k}$, ${\mathrm{c}}_{2}$, ${\mathrm{T}}_{\mathrm{ou}}$ | ${\mathrm{T}}_{\mathrm{ob}}$, ${\mathrm{T}}_{\mathrm{min}}$ |

DMP | ${\mathrm{c}}_{1}$, $\mathrm{d}$, $\mathrm{k}$, ${\mathrm{c}}_{2}$, $\mathrm{RUE}$ | $\mathrm{RUE}$, ${\mathrm{c}}_{1}$, $\mathrm{d}$, $\mathrm{k}$ |

Nup | $\mathrm{d}$, ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$, $\mathrm{k}$, $\mathrm{a}$, $\mathrm{RUE}$ | $\mathrm{a}$, $\mathrm{b}$, $\mathrm{RUE}$ |

LAI | ${\mathrm{c}}_{1}$, $\mathrm{d}$, ${\mathrm{c}}_{2}$, ${\mathrm{T}}_{\mathrm{ob}}$, $\mathrm{k}$ | ${\mathrm{c}}_{1}$, $\mathrm{d}$, ${\mathrm{c}}_{2}$ |

ETc | ${\mathrm{c}}_{1}$,$\mathrm{d}$, ${\mathrm{c}}_{2}$,$\mathrm{k}$ | ${\mathrm{c}}_{1}$, $\mathrm{A}$, $\mathrm{d}$, ${\mathrm{c}}_{2}$, $\mathrm{k}$ |

Parameters (20% of variation) | ||

PTI | $\mathrm{k}$, $\mathrm{d}$, ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$, ${\mathrm{T}}_{\mathrm{ob}}$, ${\mathrm{T}}_{\mathrm{ou}}$, ${\mathrm{T}}_{\mathrm{max}}$ | ${\mathrm{T}}_{\mathrm{ob}}$, ${\mathrm{T}}_{\mathrm{min}}$ |

DMP | $\mathrm{k}$, $\mathrm{d}$, ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$, $\mathrm{RUE}$ | $\mathrm{RUE}$, ${\mathrm{c}}_{1}$, $\mathrm{d}$, $\mathrm{k}$ |

Nup | $\mathrm{k}$, $\mathrm{d}$, ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$, $\mathrm{a}$, $\mathrm{RUE}$ | $\mathrm{a}$, $\mathrm{b}$, $\mathrm{RUE}$ |

LAI | $\mathrm{d}$, ${\mathrm{c}}_{1}$,${\mathrm{c}}_{2}$, $\mathrm{k}$, ${\mathrm{T}}_{\mathrm{ob}}$ | $\mathrm{d}$, ${\mathrm{c}}_{1}$, ${\mathrm{c}}_{2}$ |

ETc | $\mathrm{d}$, ${\mathrm{c}}_{1}$,${\mathrm{c}}_{2}$, $\mathrm{k}$ | $\mathrm{d}$, ${\mathrm{c}}_{1}$, $\mathrm{A}$, ${\mathrm{c}}_{2}$,$\mathrm{k}$ |

**Table 5.**Parameter values and standard deviations after the differential evolution (DE) calibration process.

Parameters | Autumn–Winter | Spring–Summer | ||
---|---|---|---|---|

Nominal Values | Standard Deviations | Nominal Values | Standard Deviations | |

PTIini | 0.03 | 0.01 (2.05 × 10^{−9}) | 0.06 | 0.031 (4.58 × 10^{−9}) |

RUE | 4.01 | 4.79 (3.81 × 10^{−7}) | 3.10 | 2.99 (2.10 × 10^{−7}) |

a | 7.55 | 5.89 (1.23 × 10^{−5}) | 7.55 | 5.68 (7.34 × 10^{−6}) |

b | −0.15 | −0.19 (4.06 × 10^{−7}) | −0.15 | −0.17 (2.23 × 10^{−7}) |

c1 | 2.82 | 2.65 (4.02 × 10^{−8}) | 3.07 | 2.97 (3.52 × 10^{−8}) |

c2 | 74.66 | 63.46 (1.26 × 10^{−9}) | 175.64 | 167.99 (8.85 × 10^{−13}) |

A | 0.30 | 0.63 (4.58 × 10^{−9}) | 0.49 | 0.56 (2.40 × 10^{−9}) |

Bd | 18.70 | 28.57 (1.99 × 10^{−7}) | 11.20 | 15.69 (2.18 × 10^{−7}) |

Bn | 8.50 | 4.73 (4.45) | 8.28 | 16.51 (6.13 × 10^{−7}) |

**Table 6.**Goodness of fit statistics resulting from calibration of the model for autumn–winter and spring–summer.

Outputs | Autumn–Winter | Spring–Summer | ||||
---|---|---|---|---|---|---|

Bias | RMSE | EF | Bias | RMSE | EF | |

DMP | 0.41566 | 13.3133 | 0.9970 | −1.5437 | 14.7602 | 0.9989 |

Nup | −0.0708 | 0.5004 | 0.9909 | 0.0287 | 0.3583 | 0.9980 |

LAI | 0.0249 | 0.0989 | 0.9979 | −0.0007 | 0.1564 | 0.9962 |

ETc | 3.6465 | 39.3297 | 0.8153 | 1.2918 | 28.2060 | 0.9581 |

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**MDPI and ACS Style**

Martínez-Ruiz, A.; Ruiz-García, A.; Prado-Hernández, J.V.; López-Cruz, I.L.; Valencia-Islas, J.O.; Pineda-Pineda, J.
Global Sensitivity Analysis and Calibration by Differential Evolution Algorithm of HORTSYST Crop Model for Fertigation Management. *Water* **2021**, *13*, 610.
https://doi.org/10.3390/w13050610

**AMA Style**

Martínez-Ruiz A, Ruiz-García A, Prado-Hernández JV, López-Cruz IL, Valencia-Islas JO, Pineda-Pineda J.
Global Sensitivity Analysis and Calibration by Differential Evolution Algorithm of HORTSYST Crop Model for Fertigation Management. *Water*. 2021; 13(5):610.
https://doi.org/10.3390/w13050610

**Chicago/Turabian Style**

Martínez-Ruiz, Antonio, Agustín Ruiz-García, J. Víctor Prado-Hernández, Irineo L. López-Cruz, J. Olaf Valencia-Islas, and Joel Pineda-Pineda.
2021. "Global Sensitivity Analysis and Calibration by Differential Evolution Algorithm of HORTSYST Crop Model for Fertigation Management" *Water* 13, no. 5: 610.
https://doi.org/10.3390/w13050610