# Improving the Positional Accuracy of Traditional Cadastral Index Maps with Membrane Adjustment in Slovenia

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## Abstract

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## 1. Introduction

- The positional integration of vector LAS datasets (cadasters, planning zones, agriculture, and forestry land use units) to improve and optimize maintenance of the system;
- The existence of low-quality traditional cadastral maps of large regions at a scale of 1:2880 (~80% of Slovenian territory) and higher-quality cadastral maps of small enclaves;
- The heterogeneous accuracy of cadastral maps and other land administration datasets;
- The improvement in spatial accuracy and geometric shapes of the cadastral parcel network so as to ensure a GIS user interface for matching the overlap of cadastral boundaries and approximated real world spatial situations, presented via aerial imagery and other datasets;
- Investigation of the quality of local cadastral measurement data, recorded in fieldbooks, systematically used for the maintenance of cadastral index maps.

## 2. Data and Methods

#### 2.1. Input Data

#### 2.2. Methods

#### 2.2.1. Proximity Fitting with a Mechanical Membrane Model Based on Hooke’s Law

- ΔL Displacement of body in the direction of the force F;
- L Length of the unloaded body;
- $\sigma $ Tension;
- E Elasticity modulus (stiffness of the body material);
- f Force;
- A Cross-sectional area of the body (bar).

- $m$ Scale;
- $\Delta m$ Scale change;
- $F$ Surface area (area of membrane triangles);
- ${W}_{F}$ Deformation energy.

- $p$ Weights;
- $v$ Residuals of observations.

- Scale in the x-direction;
- Scale in the y-direction;
- Shearing between the x- and y-axis.

#### 2.2.2. Adjustment of Geodetic Networks

- l Matrix of observations (measurements);
- v Residuals of observations;
- A Functional matrix (description of network geometry);
- x Unknowns (e.g., coordinates).

#### 2.3. The Positional Accuracy Improvement Process Description

## 3. Processing and Results

#### 3.1. Calculation Process Description

- Coordinate introduction and subject and reference point grouping;
- Fieldbook data treatment and geometrical constraint observations;
- Identity observations and the connection of subject and reference points;
- Calculation context setup with general and group observation weights;
- Topological net construction (Delaunay triangulation);
- Calculation of approximate coordinates (adjustment with the conjugate gradient);
- Gross error detection with observation elimination (Baarda data snooping [19]);
- Indirect adjustment by the Cholesky algorithm;
- Neighborhood adjustment with the Gaussian least-square method (proximity fitting with Hooke’s membrane model);
- Coordinate comparison at control points.

#### 3.2. Interpretation of the Results of the Experimental Improvements of the Cadastral Index Map

#### 3.3. Verification of the Results by the Control Survey

## 4. Discussion

- Random mapping errors;
- Systematic distortions due to the fact that both the calculation and the mapping were carried out according to the principle of neighborhood. These distortions result from the associated error propagation.

- The neighborhood adjustment has fully modeled the systematic distortion;
- The remaining residuals are due to random mapping errors (higher accuracy is impossible in principle, from existing data resources);
- The differences of the achieved standard deviations in the three calculations are random;
- If higher accuracies than the random mapping error are required, they can only be achieved by introducing observations, which have a higher accuracy than map coordinates.

- Criterion 1: The distribution of the residuals must be dependent on the distance between the new points and the connection points;
- Criterion 2: The proximity fitting relations are not allowed to interfere with the introduction of geometric constraints into the fitting model;
- Criterion 3: The fitting model must be independent of the distribution of connection points;
- Criterion 4: The fitting model must be independent of the distribution of new points.

## 5. Conclusions

- The membrane method offers an option to introduce geometric conditions as observations (orthogonality, straightness, distance, etc.). These observations are adjusted in one shot with the coordinate difference observations of the membrane triangles. The deformation of the membrane is not only caused by displacements in the connection points, but also by the effect of geometric conditions. The fact that the environment of geometric conditions is also deformed in one and the same adjustment step ensures that the topological integrity is maintained;
- The membrane method is formulated as a linear adjustment problem. As long as no additional nonlinear observations are introduced, the calculation needs only one iteration step and leads to an unambiguous result;
- The membrane method offers the option to connect membranes of different stiffness (maps of different accuracy) arbitrarily by connection points and to adjust them in one step;
- Even with an irregular point distribution, the membrane method produces a result that is invariant to the distribution of the subject points;
- The membrane method does not require a regular grid. However, if necessary, e.g., to minimize discretization errors or to generate a grid (for instance NTv2 grid), such a point grid can also be calculated.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Deformation energy ${W}_{F}$ at a surface strain in L-direction (

**a**) and decomposition of a surface into differentially narrow bars interrelating with the triangle and the points (

**b**).

**Figure 2.**Adjusted fieldbook geometric data of cadastral case and proximity fitted neighborhood of cadastral case—subject digital cadastral map (DCM) (gray) and shift vectors (black).

**Figure 4.**(

**a**) Theoretical geometric setting and (

**b**) the effect of the proximity fitting with membrane model based on Hooke’s law.

**Figure 5.**Situation of the theoretical geometric setting with the distances between connecting and new points, with missing connections between new points (dashed lines).

**Figure 6.**Faults of the inverse distance weighted (IDW) method presented with irregular point shifts and triangles deformed.

**Table 1.**Frequency distribution of the accuracy of the digital cadastral index map points in 2003 (Data Source: Geodetic Institute of Slovenia).

Accuracy Interval [m] | No. of Cadastral Municipalities Sub-Regions | Share of the Total [%] |
---|---|---|

<0.5 | 1235 | 32 |

0.5–1.0 | 104 | 3 |

1.0–2.0 | 725 | 19 |

2.0–5.0 | 1256 | 32 |

5.0–10.0 | 508 | 13 |

10.0–30.0 | 50 | 1 |

30.0–70.0 | 8 | 0 |

Total | 3886 | 100 |

Experiment | 1 | 2 | 3 | |
---|---|---|---|---|

INPUT DATA | Cadastral index map points (digitized) | 21,290 | 21,290 | 21,290 |

Number of proposed cadastral points (LCPs) | 4674 | 4674 | 4674 | |

Number of used cadastral points (LCPs) (a) | 3815 | 3815 | 3815 | |

Additionally surveyed boundary stones (b) | 0 | 129 | 224 | |

Number of edges | 24,981 | 24,981 | 24,981 | |

Number of cadastral parcels | 4135 | 4135 | 4135 | |

Rectangularity constraints | 2080 | 2080 | 2080 | |

Straight line constraints | 134 | 134 | 299 | |

Number of useful archived fieldbooks | 0 | 0 | 76 | |

Points from fieldbook relative geometry | 0 | 0 | 1655 |

Experiment | 1 | 2 | 3 | |
---|---|---|---|---|

INDIRECT ADJUSTMENT | Reference points (a + b) | 3815 | 3944 | 4039 |

Unknowns | 42,584.0 | 42,738.0 | 46,644 | |

Redundancy | 9537.1 | 9724.0 | 19,382.8 | |

σ of adjusted reference coordinates [m] | 0.099 | 0.095 | 0.103 | |

Number of adjusted cadastral index map points | 17,475 | 17,346 | 17,251 | |

σ of index map coordinates [m] | 2.313 | 2.237 | 2.259 | |

σ of unit weight | 1.1 | 1.1 | 1.2 | |

PROXIMITY FITTING | Triangles | 64,065 | 64,065 | 77,096 |

Unknowns | 42,778 | 43,036 | 47,214 | |

Redundancy | 25,890 | 27,224 | 40,265 | |

σ of fitted land cadastral points ((LCP)) reference coordinates [m] | 0.038 | 0.038 | 0.046 | |

Number of fitted cadastral index map points | 17,475 | 17,346 | 17,251 | |

σ of fitted index map coordinates [m] | 0.795 | 0.788 | 0.836 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Čeh, M.; Gielsdorf, F.; Trobec, B.; Krivic, M.; Lisec, A.
Improving the Positional Accuracy of Traditional Cadastral Index Maps with Membrane Adjustment in Slovenia. *ISPRS Int. J. Geo-Inf.* **2019**, *8*, 338.
https://doi.org/10.3390/ijgi8080338

**AMA Style**

Čeh M, Gielsdorf F, Trobec B, Krivic M, Lisec A.
Improving the Positional Accuracy of Traditional Cadastral Index Maps with Membrane Adjustment in Slovenia. *ISPRS International Journal of Geo-Information*. 2019; 8(8):338.
https://doi.org/10.3390/ijgi8080338

**Chicago/Turabian Style**

Čeh, Marjan, Frank Gielsdorf, Barbara Trobec, Mateja Krivic, and Anka Lisec.
2019. "Improving the Positional Accuracy of Traditional Cadastral Index Maps with Membrane Adjustment in Slovenia" *ISPRS International Journal of Geo-Information* 8, no. 8: 338.
https://doi.org/10.3390/ijgi8080338