# Modeling of the Vertical Movements of the Earth’s Crust in Poland with the Co-Kriging Method Based on Various Sources of Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- (a)
- Isotropic: On isotropic surfaces, changes in attribute values are influenced by distance but not by direction; therefore, attribute values change identically in all directions.
- (b)
- Anisotropic: On anisotropic surfaces, changes in attribute values are influenced by both distance and direction; therefore, attribute values change irregularly in space and they differ in various directions.

- (a)
- Geometric anisotropy: Attribute values change similarly in all directions but vary with distance; therefore, the same variability is achieved in different directions when points are separated by a varied distance.
- (b)
- Zonal anisotropy: Variability is not regularly distributed in space; this type of anisotropy results from data trends.

## 3. Calculation

#### 3.1. Variogram Maps for Evaluating Dataset Coherence: Anisotropy and Isotropy of Data

^{2}. The results of the coherence analysis are presented in the semivariogram maps in Figure 3.

#### 3.2. Empirical Variograms and the Selection of Theoretical Variograms

#### 3.3. Calculation of the Interpolation Parameters

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Leveling network (blue triangles) and distribution of GNSS stations (red frames). Source: mapy.geoportal.gov.pl.

**Figure 4.**Selected examples of directional variograms (empirical and theoretical), AB—fluctuations in the variance, G—greatest differences between the minimum and maximum values, BD—nugget effect, D—isotropic collection.

**Figure 5.**Analysis of the relationships between the lengths of semi-major and semi-minor axes of the ellipse and the standard deviation (datasets A, B, C, D, and E).

**Figure 6.**Analysis of the relationships between the lengths of semi-major and semi-minor axes of the ellipse and the root mean square (RMS) (datasets A, B, C, D, and E).

**Figure 7.**Analysis of the variation in the anisotropic direction, partial sill, predicted value, RMS, and standard deviation in dataset A.

Authors | Data | Type | Type of Data Processing | Form Maps (Models) | Interpolation | Determination of Anisotropy | Additional Isoline Corrections |
---|---|---|---|---|---|---|---|

[25] | Precise leveling | Point | Adjustment | Analog | Linear | No | Geological |

[26] | Precise leveling | Point | Adjustment | Analog | Linear | No | Geological |

[27] | Precise leveling | Point | Adjustment | Analog, numerical (grid: 20′ × 20′) | Collocation using the Hirvonen analytical function | No | No |

[28] | Data from GNSS stations | Point | Development of time series | Analog | Kriging method with the linear semivariogram | No data | No |

[29] | Data from GNSS stations | Point | Development of time series, adjustment | Analog | Kriging method with the linear semivariogram | No | No |

**Table 2.**Characteristics of datasets used in the analysis. GNSS: Global Navigation Satellite System.

Data Set | Leveling Data | GNSS Stations Data | Number of Nodal Points | Number of Leveling Campaing |
---|---|---|---|---|

A | + | − | 98 | #2, #3 |

B | + | − | 222 | #3, #4 |

C | + | − | 228 | #2, #3, #4 |

D | − | + | 123 | − |

E | + | + | 345 | #3, #4 |

**Table 3.**Values of the nugget effect, anisotropy, direction, and partial sill in the analyzed datasets.

Data Collections | Nugget Efect | Anisotropy | Direction | Partial Sill | |||
---|---|---|---|---|---|---|---|

[1][0] | [1][1] | [1][0] | [0][1] | [1][1] | |||

A | 0.01 | 2.44 | 10.01 | 0.39 | |||

B | 0.05 | 1.83 | 142.73 | 0.21 | |||

C | 0.11 | 1.65 | 139.74 | 0.17 | |||

D | 0.59 | 1 | 0 | 0 | |||

AB | 0 | 0.24 | 1.64 | 21.79 | 0.44 | −0.01 | 0.0003 |

BD | 0.23 | 0.59 | 1 | 0 | 0 | 0 | 0 |

DB | 0.60 | 0.16 | 1 | 0 | 0 | 0 | 0 |

E | 0.26 | 2.46 | 146.60 | brak | brak | 0.28 |

A | B | C | D | E | F | G | AB | BA | BD | DB | |
---|---|---|---|---|---|---|---|---|---|---|---|

Mean standard error | 0.69 | 0.85 | 1.60 | 4.66 | 1.23 | 2.31 | 2.05 | 2.11 | 0.94 | 1.35 | 4.71 |

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**MDPI and ACS Style**

Kowalczyk, K.; Kowalczyk, A.M.; Chojka, A.
Modeling of the Vertical Movements of the Earth’s Crust in Poland with the Co-Kriging Method Based on Various Sources of Data. *Appl. Sci.* **2020**, *10*, 3004.
https://doi.org/10.3390/app10093004

**AMA Style**

Kowalczyk K, Kowalczyk AM, Chojka A.
Modeling of the Vertical Movements of the Earth’s Crust in Poland with the Co-Kriging Method Based on Various Sources of Data. *Applied Sciences*. 2020; 10(9):3004.
https://doi.org/10.3390/app10093004

**Chicago/Turabian Style**

Kowalczyk, Kamil, Anna Maria Kowalczyk, and Agnieszka Chojka.
2020. "Modeling of the Vertical Movements of the Earth’s Crust in Poland with the Co-Kriging Method Based on Various Sources of Data" *Applied Sciences* 10, no. 9: 3004.
https://doi.org/10.3390/app10093004