# Spatial Modelling and Prediction Assessment of Soil Iron Using Kriging Interpolation with pH as Auxiliary Information

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}.

#### 2.2. Soil Sampling and Laboratory Analysis

#### 2.3. Semivariograms and Semivariogram Modelling

_{0}), which represents measurement errors and the short distance unexplained or random spatial variation; the total ‘sill’ (C

_{0}+ C

_{1}), which is the semivariance value at which the semivariogram levels off and represents the variance of the variable; and, finally, the ‘range’ (a), which is the value of distance at which the semivariogram reaches the sill and, beyond that, distance at which the data are no longer correlated. The semivariogram and its parameters can only be positive.

_{0}/C

_{0}+ C

_{1}). A ratio less than 25% indicates strong spatial dependence, a ratio between 25% and 75% indicates a moderate spatial dependence, and a ratio over 75% indicates a weak spatial dependence [28].

_{0}is the nugget, α the range, and C

_{0}+ C

_{1}is the sill. Semivariogram modelling and estimation is extremely important for structural analysis and spatial interpolation [31].

#### 2.4. Kriging Interpolation

_{0}with the following equation:

_{i}that minimize the variance of the estimator:

#### 2.4.1. Ordinary Kriging

#### 2.4.2. Universal Kriging

_{0}is defined by Equation (7) in OK; however the weights ${\lambda}_{i}$ are optimized to minimize the kriging variance for k = 0, 1, ..., K:

#### 2.4.3. Co-Kriging

#### 2.5. Cross-Validation Methods and Error Assessment

#### 2.6. Software

## 3. Results and Discussion

#### 3.1. Exploratory Data Analysis

#### 3.2. Semivariograms Analysis

_{0}OK: Fe = 0.51, pH = 0.58, C

_{1}OK: Fe = 0.82, pH = 0.74, range OK: Fe = 3795 m, pH = 3458 m); for 2014, there was a difference in sill (C

_{0}OK: Fe = 0.30, pH = 0.36, C

_{1}OK: Fe = 0.70, pH = 0.46, range OK: Fe = 4251 m, pH = 4303 m); and, for 2015, there was a nugget deviation (C

_{0}OK: Fe = 0.23, pH = 0.41, C

_{1}OK: Fe = 0.94, pH = 0.85, range OK: Fe = 3397 m, pH = 3148 m). However, the overall similar semivariogram parameters supported that these co-variables exhibited a spatial relationship (in addition to the strong value correlation presented in the regression analysis) that might increase the prediction accuracy in Co-Kriging. The spatial dependence of pH for OK was rather moderate, from 32% to 43%.

#### 3.3. Prediction Maps and Prediction Error Maps

#### 3.4. Cross Validation

## 4. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**Frequency distributions of natural logarithms of soil Fe for each year. The blue line represents the normal distribution.

**Figure 7.**Experimental semivariograms and their fitted models of ln(Fe) for every interpolation method each year. Regarding Co-Kriging interpolation (last line of the semivariograms), apart from the Ordinary Kriging (OK) ln(Fe), the OK pH semivariogram and the ln(Fe)-pH cross-semivariogram are also presented for each year.

**Figure 10.**Root mean square error (RMSE) of the predictions between the training set and the test set for each year for ln(Fe).

Fe (ppm) | 2013 | 2014 | 2015 | pH | 2013 | 2014 | 2015 |
---|---|---|---|---|---|---|---|

mean | 47.73 | 46.86 | 46.16 | mean | 6.42 | 6.47 | 6.25 |

median | 34.44 | 36.91 | 35.19 | median | 6.61 | 6.50 | 6.42 |

sd | 45.74 | 41.00 | 38.00 | sd | 1.09 | 0.84 | 1.06 |

max | 201.80 | 232.60 | 187.20 | max | 8.14 | 7.74 | 7.97 |

min | 2.22 | 4.40 | 2.97 | min | 3.93 | 3.75 | 4.18 |

measurements | 177 | 109 | 114 | measurements | 177 | 109 | 114 |

skewness | 1.57 | 1.94 | 1.38 | skewness | −0.45 | −0.50 | −0.17 |

kurtosis | 5.01 | 8.09 | 4.92 | kurtosis | 2.22 | 2.83 | 1.90 |

2013 | 2014 | 2015 | |
---|---|---|---|

Pearson’s r | −0.827 | −0.842 | −0.887 |

**Table 3.**The semivariogram models and parameters for the three interpolation methods for each year along with the cross-semivariogram parameters for Co-Kriging. The C

_{0}/C

_{0}+ C

_{1}is the nugget to total sill ratio. Models were estimated from the training set.

2013 | 2014 | 2015 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model | Nugget (C_{0}) | Par. Sill (C_{1}) | Range (a) in Metres | C_{0}/C_{0} + C_{1} | Model | Nugget (C_{0}) | Par. Sill (C_{1}) | Range (a) in Metres | C_{0}/C_{0} + C_{1} | Model | Nugget (C_{0}) | Par. Sill (C_{1}) | Range (a) in Metres | C_{0}/C_{0} + C_{1} | |

OK ln(Fe) | Sph. | 0.51 | 0.82 | 3795 | 38% | Sph. | 0.30 | 0.70 | 4251 | 30% | Sph. | 0.23 | 0.94 | 3397 | 19% |

UK ln(Fe) | Sph. | 0.52 | 0.19 | 1892 | 73% | Sph. | 0.29 | 0.22 | 1401 | 56% | Sph. | 0.22 | 0.43 | 2351 | 33% |

OK pH | Sph. | 0.58 | 0.74 | 3458 | 43% | Sph. | 0.36 | 0.46 | 4303 | 43% | Sph. | 0.41 | 0.85 | 3148 | 32% |

CO-Kr | Sph. | −0.28 | −0.89 | 3795 | 23% | Sph. | −0.25 | −0.43 | 4251 | 36% | Sph. | −0.2 | −0.89 | 3396 | 18% |

**Table 4.**Cross-validation holdout method results of soil Fe predictions. The root mean square error (RMSE), mean error (ME), and mean squared deviation ratio (MSDR) are referring to ln(Fe), for which Fe is in ppm.

2013 | 2014 | 2015 | |||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | ME | MSDR | RMSE | ME | MSDR | RMSE | ME | MSDR | |

OK | 0.777 | 0.162 | 0.885 | 0.680 | 0.363 | 1.108 | 0.618 | 0.069 | 1.003 |

UK | 0.801 | 0.158 | 1.028 | 0.678 | 0.336 | 1.135 | 0.649 | 0.053 | 1.210 |

Co-Kriging | 0.772 | 0.170 | 0.871 | 0.638 | 0.319 | 1.097 | 0.618 | 0.068 | 0.995 |

**Table 5.**The cross-validation results with the leave-one-out method of soil Fe predictions. The root mean square error (RMSE), mean error (ME), and mean squared deviation ratio (MSDR) are referring to ln(Fe), where Fe is in ppm.

2013 | 2014 | 2015 | |||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | ME | MSDR | RMSE | ME | MSDR | RMSE | ME | MSDR | |

OK | 0.800 | −0.0006 | 1.009 | 0.632 | −0.006 | 0.945 | 0.611 | −0.005 | 0.973 |

UK | 0.798 | 0.0001 | 1.046 | 0.625 | −0.007 | 0.954 | 0.601 | −0.006 | 1.062 |

Co-Kriging | 0.601 | 0.001 | 0.880 | 0.437 | 0.0004 | 0.970 | 0.393 | −0.001 | 0.937 |

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**MDPI and ACS Style**

Tziachris, P.; Metaxa, E.; Papadopoulos, F.; Papadopoulou, M.
Spatial Modelling and Prediction Assessment of Soil Iron Using Kriging Interpolation with pH as Auxiliary Information. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 283.
https://doi.org/10.3390/ijgi6090283

**AMA Style**

Tziachris P, Metaxa E, Papadopoulos F, Papadopoulou M.
Spatial Modelling and Prediction Assessment of Soil Iron Using Kriging Interpolation with pH as Auxiliary Information. *ISPRS International Journal of Geo-Information*. 2017; 6(9):283.
https://doi.org/10.3390/ijgi6090283

**Chicago/Turabian Style**

Tziachris, Panagiotis, Eirini Metaxa, Frantzis Papadopoulos, and Maria Papadopoulou.
2017. "Spatial Modelling and Prediction Assessment of Soil Iron Using Kriging Interpolation with pH as Auxiliary Information" *ISPRS International Journal of Geo-Information* 6, no. 9: 283.
https://doi.org/10.3390/ijgi6090283