# How does PTF Interpret Soil Heterogeneity? A Stochastic Approach Applied to a Case Study on Maize in Northern Italy

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## Abstract

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_{0}). The analysis showed sensitivity of the simulated process to the parameter n being significantly higher than to k

_{0}, although the former was much less variable. The PTFs showed a smoothing effect of the output variability, even though they were previously validated on a set of measured data. Interesting positive and significant correlations were found between the n parameter, from measured water retention curves, and the NDVI (Normalized Difference Vegetation Index), when using multi-temporal (2004–2018) high resolution remotely sensed data on maize cultivation. No correlation was detected when the n parameter derived from PTF was used. These results from our case study mainly suggest that: (i) despite the good performance of PTFs calculated via error indexes, their use in the simulation of hydrological processes should be carefully evaluated for real field-scale applications; and (ii) the NDVI index may be used successfully as a proxy to evaluate PTF reliability in the field.

## 1. Introduction

^{2}), the root mean square error (RMSE), the model efficiency (EF) calculated on the residuals [10,16,17,18], or on observed average or variance [19]. Integrated indexes have also been developed [14,20].

- (i)
- Evaluating to what extent the variability of the soil hydraulic properties parameters is reflected in the hydrological processes observed at the field scale. To do that, a preliminary analysis of the sensitivity of a physically-based agro-hydrological model to the measured variability of both water retention and hydraulic conductivity parameters will be carried out. This analysis will be based on a stream tube approach used in a stochastic (Monte Carlo) framework;
- (ii)
- Evaluating the effectiveness of selected PTFs in reproducing the field scale hydrological pattern described by the measured hydraulic properties through using independent and spatially distributed information (Normalized Difference Vegetation Index ‒ NDVI) as data quality control.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Hydraulic Properties

_{0}was determined by the variable head technique [34]. Soil water retention curve θ(h) was measured starting from the saturation by using the suction table (9 points) and the pressure plate apparatus (3 points) [35].

#### 2.3. Evaluation of PTFs’ Performance

_{i}is the ith measured value, E

_{i}is the estimated ith value, and n is the number of soil water content pairs. $\overline{\mathrm{M}}$ is the mean of measured soil water content.

#### 2.4. Soil-Plant-Atmosphere Model

_{r})/(θ

_{0}−θ

_{r}), θ

_{r}and θ

_{0}being the residual water content and the water content at h = 0, respectively, and in which α (cm

^{−1}), n, and m are curve-fitting parameters.

_{r}[39]. Assuming m = 1-1/n, a closed-form analytical solution to predict k

_{r}at a specified volumetric water content was obtained:

_{0}is the hydraulic conductivity at θ

_{0}, and τ is a parameter which accounts for the tortuosity and partial correlation between adjacent pores.

_{p}is calculated using the Penman–Monteith equation [41] based on daily data (i.e., air temperature and humidity, net radiation, wind speed). A simplified approach of the Penman-Monteith equation can be used on a large scale, due to the lack of detailed daily information required for its application. In this study, specifically, potential evapotranspiration was calculated from reference evapotranspiration (ET

_{0}), estimated by means of the Hargreaves and Samani formula [42], and Kc, a crop-specific factor (ET

_{p}= Kc * ET

_{0}).

_{p}is partitioned into T

_{p}and E

_{p}on the basis of Beer’s law [43]:

_{r}(cm) being the thickness of the root zone and α(h) a semi-empirical function of pressure head h, varying between 0 and 1. The shape of the function α(h) depends on four critical values of h, which are related to crop type and to potential transpiration rates. The actual transpiration rate T

_{act}(cm d

^{−1}) is computed by the integration of S over the root layer.

#### 2.5. Monte Carlo Simulations

#### 2.6. Remote Sensing Data

^{2}:0.83) between these MODIS NDVI data and total dry biomass data [31].

- An NDVI map at high resolution from a visible RGB and near IR Quickbird image for 21 July 2004 (spatial resolution 2.4 m).
- NDVI maps derived from free Landsat 5 TM and Landsat OLI-8 scenes Collection 1 Level-2 on-Demand—path 193–194/row 28–29—at spatial resolution of 30 m, atmospherically corrected [50,51], including a cloud, shadow, water, and snow mask produced using CFMASK [52], as well as a per-pixel saturation mask for years 2009, 2010, 2013, 2014, 2015, 2016, 2017, and 2018 ranging from 1 to 25 July, in order to strengthen our assumptions on the relationship between NDVI and hydraulic properties. The selection of these layers (years and periods) was done considering their availability, the need to observe NDVI data within July as was done for 2004, and the presence of masking clouds during specific days.

#### 2.7. Climate Data

## 3. Results

#### 3.1. Variability of Soil Hydraulic Properties

_{r}=0. The mean of the parameters along with standard deviation, skewness, and coefficient of variation are reported in Table 1. Furthermore, textural information is also reported.

_{0}. θ

_{s}and n follow a normal distribution, and α and k

_{0}a log-normal distribution.

#### 3.2. Model Validation

_{0}parameters. In addition, in Figure 2, the soil water content measurements and the simulations coming from the calibrated parameters are also shown (red squares and red dashed line, respectively). What comes up is the good agreement between measured and simulated soil water content. The agreement is rather good during the drying period after the rain, with the exception of the period lasting from DOY 335 to DOY 342. The differences can be attributed to the amount of the rain inputted in the model. The gauge at the Caviaga site was located less than 1 km away from the experimental station, therefore we presume that the spatial variability of the rain played a role in our simulation. Therefore, we attribute the underestimation of soil water content simulation to the underestimation of few millimeters of rain in the experimental farm with respect to the measurement gauge.

#### 3.3. Predictive Capability of PTFs

#### 3.4. Soundness of PTF Estimations Based on NDVI Data

^{2}= 0.63 at 0.01 level of significance); all the other parameters were uncorrelated (data not shown). Then, in order to verify if this finding was consistent over time, the same procedure was applied using the NDVI data derived from free Landsat 8 images (30m x 30m spatial resolution) for the year 2004 again, and also for years 2009, 2010, 2013, 2014, 2015, 2016, 2017, and 2018. The use of the multitemporal Landsat images was justified because of their free availability as against the Quickbird ones that are available for a fee, thus making the procedure more easily repeatable. In Table 3 the relationships between measured and estimated parameters vs NDVIs are shown. The estimated parameters are all uncorrelated to the NDVI, while at this coarser spatial resolution, positive and statistically significant correlations were also found, again on n parameter derived from measured water retention curves for the years 2004, 2010, 2013, 2016, and 2017. For the same years, significant negative correlation was also found for the α parameter. Only for 2015 was k

_{0}positively correlated with NDVI.

## 4. Discussion

#### 4.1. Variability of Soil Hydraulic Properties Parameters and its Impact on the Process under Study

_{0}). Indeed, despite the fact that variability of the parameter n is considerably lower than that shown by k

_{0}[CV (n) = 7.74%; CV (k

_{0}) = 185%] (see Table 1), its variation affects the variability of the simulated process to an extent equal to or even larger than for k

_{0}. This is evident in Figure 2, where the width of the ± 1 standard deviation vertical bars is displayed. Although on average they show similar variability [SD (n) = 0.04; SD (k

_{0}) = 0.03], the model is more sensitive to the variability of the n parameter during the drainage processes (i.e., from day 310 to 345), whereas it is more dependent on k

_{0}variability during the infiltration / redistribution processes (i.e., from day 290 to 310). This agrees with the findings in Reference [29], who observed a similar behavior for a field process simulated with bimodal hydraulic parameters, and with Reference [55], who found during an infiltration process that predictive uncertainty is most sensitive to uncertainty in the saturated hydraulic conductivity k

_{0}.

#### 4.2. Effectiveness of PTFs in Representing the Actual Variability of Hydraulic Properties and its Impact on the Process under Study

## 5. Conclusions

_{0}? The study area is characterized by intensive agriculture—mainly livestock and maize farming—with old irrigation systems where over-irrigation practices are widely applied (i.e., flooding irrigation). From a hydrological point of view, k

_{0}and n are the two parameters mainly affecting the drainage process [27], i.e., the higher their value, the more hydraulically conductive the soil. Therefore, for this case study conductive Ap horizons—where the majority of the water uptake takes place—provided better conditions for crop growth, reducing the effect of saturation (e.g., lack of oxygen for root uptake functioning [60]). Therefore, despite several factors that could influence crop responses, such as slopes, soil depth, rainfall, temperatures, and management practices, in this case study (plain territory, not very extended, similar management) the contribution of spatial variability of soil hydraulic properties is considered the main factor.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Study area and sampling sites (white crosses). The white circle identifies the representative soil profile. Adapted from [31].

**Figure 2.**Soil water content measured and simulated (red squares and red dashed line) in the reference soil. Black and grey lines are the average soil water content applying 500 Monte Carlo runs of the n and k

_{0}parameters, respectively. Error bars represent the standard deviation of the two set of 500 Monte Carlo runs.

**Figure 3.**Average and error bars (standard deviation) of both measured (black lines) and estimated (grey lines) SWRCs.

**Figure 4.**Measured (•) and estimated (Δ) values of n parameter vs NDVI values estimated by Landsat 8 images.

**Figure 5.**Average daily temperature versus cumulated rain recorded during the period May–June for nine different years. Dashed line identifies years with high correlation; continuous line identifies years with low correlation. Black dots represent years reported in Figure 4.

θ_{s} | α | n | k_{0} (cm d^{−}^{1}) | Clay (%) | Silt (%) | Sand (%) | |
---|---|---|---|---|---|---|---|

Mean | 0.42 | 0.02 | 1.32 | 486 | 16.4 | 34.0 | 49.6 |

Standard deviation | 0.06 | 0.02 | 0.10 | 897 | 3.8 | 7.1 | 8.5 |

Coefficient of variation (%) | 13.9 | 109 | 7.74 | 185 | 23.0 | 20.8 | 17.1 |

Type of distribution (n: normal; Log-n: Lognormal) | n | Log-n | n | Log-n | n | n | n |

Root Mean Error (RME) | Root Mean Square Error (RMSE) | Model Efficiency (EF) | |
---|---|---|---|

HYPRES | 0.09 | 5.51 | −1.56 |

VERECKEEN | −2.86 | 5.47 | −1.25 |

ROSETTA | −7.03 | 5.76 | −2.56 |

**Table 3.**Square of the correlation coefficients (r

^{2}) between measured and estimated hydraulic parameters vs NDVI estimated by Landsat-8 images.

Hydraulic Parameters | ||||||||
---|---|---|---|---|---|---|---|---|

Estimated | Measured | |||||||

Years | θ_{s} | k_{0} (cm/d) | α (1/cm) | n | θ_{s} | k_{0} (cm/d) | α (1/cm) | n |

2004 | 0.16 | 0.15 | 0.03 | 0.04 | 0.13 | 0.04 | 0.30 * | 0.48 ** |

2009 | 0.00 | 0.04 | 0.00 | 0.09 | 0.05 | 0.09 | 0.01 | 0.01 |

2010 | 0.00 | 0.04 | 0.01 | 0.11 | 0.00 | 0.02 | 0.18 * | 0.22 * |

2013 | 0.15 | 0.02 | 0.08 | 0.04 | 0.02 | 0.18 | 0.40 ** | 0.32 ** |

2014 | 0.08 | 0.02 | 0.01 | 0.04 | 0.16 | 0.08 | 0.00 | 0.01 |

2015 | 0.18 | 0.02 | 0.06 | 0.00 | 0.02 | 0.62 ** | 0.18 | 0.03 |

2016 | 0.01 | 0.00 | 0.08 | 0.04 | 0.01 | 0.03 | 0.32 ** | 0.48 ** |

2017 | 0.00 | 0.00 | 0.08 | 0.00 | 0.02 | 0.08 | 0.26 ** | 0.31 ** |

2018 | 0.00 | 0.00 | 0.04 | 0.05 | 0.09 | 0.00 | 0.17 * | 0.12 |

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**MDPI and ACS Style**

Basile, A.; Bonfante, A.; Coppola, A.; De Mascellis, R.; Falanga Bolognesi, S.; Terribile, F.; Manna, P. How does PTF Interpret Soil Heterogeneity? A Stochastic Approach Applied to a Case Study on Maize in Northern Italy. *Water* **2019**, *11*, 275.
https://doi.org/10.3390/w11020275

**AMA Style**

Basile A, Bonfante A, Coppola A, De Mascellis R, Falanga Bolognesi S, Terribile F, Manna P. How does PTF Interpret Soil Heterogeneity? A Stochastic Approach Applied to a Case Study on Maize in Northern Italy. *Water*. 2019; 11(2):275.
https://doi.org/10.3390/w11020275

**Chicago/Turabian Style**

Basile, Angelo, Antonello Bonfante, Antonio Coppola, Roberto De Mascellis, Salvatore Falanga Bolognesi, Fabio Terribile, and Piero Manna. 2019. "How does PTF Interpret Soil Heterogeneity? A Stochastic Approach Applied to a Case Study on Maize in Northern Italy" *Water* 11, no. 2: 275.
https://doi.org/10.3390/w11020275