## 1. Introduction

Geodetic control networks are widely used for investigating and monitoring any geographical features and phenomena. They are of great priority to many human actions, like geodetic and geophysical measurements, surveying, civil and environmental engineering, GIS development, and gathering spatial data. These networks provide a series of accurate positional measures that cover the analyzed area, thus delivering a time-continuous infrastructure enabling studying and tracking of crustal deformation [

1], as well as seismic and geomorphologic landform activities (e.g., [

2,

3,

4]). The ground stability of geodetic control points provides quality to the results of geometric measures, enables an accurate and detailed description of land topography, and changes relative to Earth processes, natural hazards, and climate change [

5]. Control surveys serve as the basis for initiating or reviewing subordinate surveys for property boundary delineation [

6], route and construction planning [

7,

8], engineering object displacements [

9], cadastral and topographic mapping [

10], as well as many environmental applications and cultural heritage preservation and monitoring [

11,

12]. They are indispensable as a reference framework for land administration systems, especially for georeferencing spatial objects [

13,

14,

15]. Many scholars point out that the production and quality control of any geographical data and products requires references to the base geodetic network that realizes the CRS (Coordinate Reference System) and provides the consistent frame for GNSS users [

5,

16,

17]. Hence, the base geodetic control points (thereinafter refed as BGCPs) are of utmost importance for georeferencing of manuscripts of historical maps [

11,

18,

19] or images from unmanned aerial vehicles [

20].

Geodetic control points should cover an area relatively evenly to enable accurate and cost-effective measurements [

21,

22,

23]. Although there is a variety of studies considering the design and densification of geodetic control networks [

16,

24,

25] as well as investigating the influence of topographic objects that hinder the visibility of the horizon, interfere with satellite signals, and, finally, affect the quality of geodetic control stations’ positioning [

26,

27,

28], a profound analysis of the spatial pattern of geodetic control points still requires investigation. The problem of geospatial distribution of geodetic control points was previously discussed in several publications [

29,

30,

31]. However, these studies were concerned with the detail (third-order) of geodetic network point analyses in relatively small (less than 200 sq. km) rural areas. The results, related to surveying units and then grouped according to land use types, showed that geodetic control points are scattered with significantly visible groupings along roads, railways, and built-up areas. Moreover, the number and density of geodetic control points depend on the development of the area in question, and 35% to 50% depend on the land cover, mainly in locations of built-up areas, roads, and railways [

30,

31].

The density of geodetic control points is specified by the National Mapping Agencies. In Poland, all requirements for geodetic control networks are included in the Regulation of the Ministry of Administration and Digitization [

32] implementing the Geodetic and Cartographic Law of the Polish Parliament. The document, like many other national regulations [

33,

34,

35,

36], assumes that in order to ensure an appropriate accuracy of survey measurements, the country’s territory should be covered evenly by high-order geodetic control networks. The anticipated network density is, in general, expressed by the inter-point distance or the density of the geodetic control points, and varies between counties, depending on their level of development [

37].

The objective of this study was to investigate the spatial pattern of the geodetic control points of the Base National Geodetic Network (BNGN) in Poland, in particular, by performing an ex-post analysis of the uniformity of spatial distribution and determination of factors that disturb the assumed evenness. This is the first comprehensive study covering the entire country using GIScience methods, i.e., incremental spatial statistics, cluster and density analysis, as well as first and second order statistics. This study makes contributions in both research and applications. Based on a GIScience theoretical background, it introduces a comprehensive methodology for ex post analysis of geodetic control points’ spatial pattern as well as the quantification of geodetic network uniformity into regularly dense and regularly thinned. The conducted tests of the first and second order statistics provide, in the form of graphs and maps, exhaustive information on the spatial distribution type (regular, random, or clustered) of the geodetic points. The study is intended to be a methodological resource and reference for the National Mapping Agency and the Head Office of Geodesy and Cartography for the maintenance, as well as the further densification or modernization, of the base geodetic network in Poland. Furthermore, the results give surveyors the ability to quickly assess the availability of geodetic points, as well as identify environmental obstacles that may hamper measurements. Moreover, the method of investigating the spatial pattern of the BGCPs could be applied in any country or region, and after a small modification, on any man-made point feature of which the location is assumed to not be distributed randomly.

The remainder of the article is structured as follows:

Section 2 provides an overview of the National Geodetic Control Network in Poland,

Section 3 presents the area and data used, the fundamentals of the point pattern analysis, and steps in data analyzing,

Section 4 and

Section 5 describe the obtained outcomes in detail and provide discussion with the results, and finally,

Section 6 concludes this study.

## 2. Overview of the National Geodetic Control Network in Poland

The control networks in Poland, maintained by the Head Office of Geodesy and Cartography, are designed and established according to the Geodetic and Cartographic Law [

38] and the pertinent implementation rules established by the Ministry of Administration and Development [

32]. Information about geodetic control points is stored in spatial databases, and includes the point number and unique identifier, coordinates, survey uncertainty, and information about their source, type of monumentation, and topographic details (e.g., location, access details, sketch plans); enabling the recovery or reconstruction of the point in terrain, and allowing the maintenance of the basic control network.

The geodetic network in Poland consists of points of the horizontal and the vertical networks. The horizontal network is hierarchical and comprises three orders of geodetic control networks. The main criterion for assigning a geodetic control point to the appropriate network order is the accuracy of determining its position, expressed as the mean error. The fundamental (first-order) network consists of 127 GNSS stations that belong to the Active Geodetic Network–European Position Determination System (ASG-EUPOS) out of which 24 are located in the neighboring countries [

39]. These stations meet the criteria established by the Subcommittee EUREF and provide an average density of one point per 20,000 km

^{2}. The base geodetic control is the second-order geodetic network providing the density of one point per 50 km

^{2}. The first and second-order geodetic networks, according to the Ministry Regulation [

32], form the so called base (main) geodetic network in Poland [

40]. All points belonging to these networks are monumented and have precisely measured positions (0.01 m for horizontal position and 0.02 m for geodesic height [

32]) as they form the basis for determining the positions of other points. All BGCPs should be situated on stable ground, far from sources of electromagnetic interference or surface vibrations (e.g., caused by heavy traffic), objects that cause reflection of satellite signals (e.g., water reservoirs, flat metal surfaces, walls, wire fences, high antenna towers), as well as terrain barriers above 10 degrees over the horizon. The detailed geodetic network is a third-order control with 1,378,383 points and the average density of control points of ca. one point per 22.5 ha. The mean error of the third-order point’s location is in the range from 0.05 to 0.10 m, however for newly established points it should be less than 0.07 m. The control points are accurately tied to at least three reference geodetic controls. The geodetic control networks are the backbone of the Polish Spatial Information Infrastructure [

40,

41].

## 4. Results

#### 4.1. General Characteristic of BGCPs’ Location

The Base Geodetic Control Network in Poland comprises 6723 points, including 103 ASG-EUPOS stations. The geographical points location highly (with the coefficient of determination equals to 0.92) corresponds to the main land cover type, namely: Artificial areas, agriculture, forest, and bushes (see

Table 2 and

Figure 3a). As many as 57.37% of BGCPs are situated on agricultural land, 30.78% in forest, and 5.65% in urban areas. There are relatively few points on pastures and meadows. This could be explained by the difficulty of maintaining a point’s monumentation stability in lands with high flood risk and variable groundwater levels, which, according to [

68], is typical for Polish meadows and pastures, located mainly in river valleys.

The average density of the BGCPs in Poland amounts to 1.08 point per 50 km

^{2} and is in line with the national regulation [

32].

A total of 6256 points are situated no further than 200 m from the transportation network (

Table 3,

Figure 3b), of which as many as 343 (5.1%) are within a 20 m buffer around paved roads, and two points are closer than 20 m from the railway.

The relatively close (up to 200 m) distance of as much as 93.05% of geodetic control points to paved roads makes them easily accessible to surveyors, which is undoubtedly a great advantage of the Base Geodetic Control Network in terms of its use and maintenance.

Nevertheless, there is a set of 383 BGCPs located no further than 20 m from environmental objects that could provide significant disruption of an electromagnetic signal, and finally, decrease the accuracy of positioning (see

Table 3 and

Figure 4).

Local density of BGCPs varies from 0.2 to 3.65 points over 50 km

^{2} and shows places of higher and lower point density, which are underlined in red and orange in the kernel density map (

Figure 5a). A significant increase in the density of geodetic points is observed in major metropolitan agglomerations and heavily industrialized regions, such as mining areas (

Figure 5b).

The highest density is observed in the Upper Silesian Polycentric Metropolitan Area (from 3 to 3.5 points per 50 km

^{2}) followed by Warsaw, Lodz, Wroclaw, Szczecin, Bydgoszcz-Torun-Grudziadz, tri-cities metropolitan areas, where the density of BGCPs varies from 1.2 to 2.85 points per 50 km

^{2}. The lowest values (less than 0.5 over 50 km

^{2}) are mainly found in the Southern Pomerania Lakelands, Sandomierz and Nida Basins, as well as the North Polesie Plain, which are mostly agricultural and forested regions with a high share of natural vegetation and relatively sparsely populated areas, and the percentage of urbanized areas lower than 1.17 (

Figure 5b).

#### 4.2. Point Pattern Analysis

The null hypothesis presuming Complete Spatial Randomness (CSR) for BGCPs’ pattern analysis was rejected, using Average Nearest Neighbor tools. The

${z}_{ANN}$ score amounts to 1.39, with a 99% confidence level, which indicates that there is less than 1% likelihood that the dispersed pattern of the BGCPs could be the result of random chance and that the points are dispersed over the territory of Poland. The observed mean nearest inter-point distance equals 5.583 km, while the expected amounts to 4.013 km. Moreover, the coefficient of variation

CV amounts to 21.66%, which means that the nearest distances between BGCPs do not vary significantly (see

Figure 6).

Ripley’s K function (

Figure 7) is estimated for distances up to 20 km, in 1 km increments, and indicates two different departures from randomness.

At the distances from 2 up to 7 km, typical for base geodetic control networks, L(d) lies below the expected distance value (higher than expected approximately 2000 m) and indicates statistically significant (with the level of confidence of 99%) evidence of regular dispersion of geodetic points. At longer distances (greater than 7 km) L(d) lies above the expected distance, indicating spatial clustering. In general, at near distances, the BGCPs are expected to be dispersed, as the number of points within a given distance of each individual point is small, as the distance increases, each BGCP would typically have more neighbors, hence they are clustered.

#### 4.3. Thiessen Polygons Morphology and Land Cover Structure

The analysis of the Thiessen polygons area and shape also emphasizes that the distribution of BGCPs in Poland is generally dispersed. The shape of polygons, expressed as the shape index (Eq.4), is almost equal. The SI values vary from 0.095 to 0.935 (range 0.840), with the mean of 0.826, and median of 0.840 (see

Table 4, and

Figure 8a). Moreover, 50% of polygons are characterized by the SI index that fluctuates from 0.812 (lower quartile) to 0.863 (upper quartile). H-spread (IQR) amounts to 0.051, showing very low dispersion, which is also visible in the CV index value, equal to 8.45. The exclusion of polygons neighboring the country border entails even smaller diversification of the SI, which underlines the almost double reduction of the CV index from 8.45% to 4.48%.

The analysis of the Thiessen polygons’ size distribution indicates little deviation from the normal distribution. The mean equals 46.48 km

^{2}, and the median is 46.64km

^{2}. Skewness (0.051) shows the fairly symmetric character, with a little longer right tail, while kurtosis (0.896) indicates the fewer and less extreme outliers than in the normal distribution. The area of 50% polygons ranges from 3.685 to 46.636 km

^{2}, and 90% does not exceed 64.4 km

^{2} (

Figure 8b). The coefficient of disparity (CV) of the polygons’ area amounts to 32.4% and emphasizes their medium differentiation.

The analysis of land cover structure in the Thiessen polygons reveals high diversity, emphasized by the CV coefficient, which varied from 285.18% for water bodies to 47.76% for agriculture areas. The land cover dispersion in dense urban areas (CV = 166.16) is higher than in sparsely urbanized ones (CV=123.97) and was followed by forested lands (CV = 76.35). On average, the number of CLC classes (NC=9) is nearly four times lower than the number of patches (NP = 34), which suggests that the landscape heterogeneity is lower than its fragmentation. Moreover, there were some polygons covered entirely by one land cover patch (NP amounts to one). On average, 53.4% of the area of Thiessen polygons is agricultural, followed by forests (32.4%). Urban areas occupy about 6.6%, and dispersed settlement 2.8%. Water bodies (lakes and rivers) cover only 2.1%.

Artificial land (heavily and sparsely urbanized), on average, occupies 9.44% of the Thiessen polygons. However, the smallest polygons, with an area of less than 16.5 km

^{2} (mean − 2.0 std. dev.), are covered by artificial surfaces in 32.06%, while for very large polygons (bigger than 77 km

^{2}, mean − 2.0 std. dev.) artificial land covers only 4.64% (see

Figure 9). The Pearson’s correlation coefficient (PCC), equal to −0.28 (significant at

p < 0.05000), indicates the weak downhill linear correlation between the size of the polygons and acreage of urbanized areas. The level of urbanization, however, is moderately correlated with road density (PCC equals 0.50). Nevertheless, in small polygons, the average share of urbanized and agricultural areas amounted to 30% and the average share of forests amounted to 20%, which shows a fairly balanced land cover structure.

A moderate positive linear correlation (with the significant at p < 0.05) was observed between a polygon’s size and the number of patches (PCC = 0.56), while, the relationship between a polygon’s size and the area of agricultural land was very weak (PCC = 0.14). Likewise, the linear relationship between the shape index the land cover structure was similarly very weak (PCC lower than 0.15).

#### 4.4. Quantification of Geodetic Network Uniformity

The Thiessen polygons’ variability in size and shape analysis supported by cluster and outlier analysis (Anselin Local Moran’s I), and preceded by a global autocorrelation assessment, made it possible to delineate the places of more or less densified geodetic networks. Global Moran’s I statistics, based on Thiessen polygons’ locations and attributes values, such as area, shape, and nearest distance to neighbor polygons’ centroids, returned positive I and z-score values (I = 1.06/z = 204.43, I = 0.24/z − 45.18, and I = 1.08/z = 207.78, correspondingly) with a 99% level of significance. This empowers rejection (with a probability greater than 1%) of the null hypothesis, assuming that the Thiessen polygons characterised by the attribute being analyzed are randomly distributed in a study area that covered all of Poland.. Therefore, the spatial distributions of high and low area, shape, and NN distance (NNd) values are clustered. Such characteristics of the spatial distribution of polygons was the basis for further analysis, namely determining the type of the polygon clusters, due to the high/low values of the attribute of surrounding polygons (see

Figure 10a–d).

Out of 6723 polygons, 3914 (58.2%) created statistically significant clusters of autocorrelated polygons, characterized by the HH/LL or HL/LH area values. As much as 1234 Thiessen polygons (18.3%) create HH clusters (light red), and 1130 create LL clusters (light blue) (

Figure 10a). The number of autocorrelated Thiessen polygons is less if the nearest distance between neighboring polygons is analyzed; a total of 2054, which is 30.6% of all polygons. As much 46.0% (945 points) form LL clusters and 870 (42.4%) form HH clusters. These Thiessen polygons comprise 14.1% and 12.9% of all polygons, respectively. If the shape (i.e., the SI) of polygons is considered (

Figure 10c), polygons of LL and HL are mainly located along the country border. They constitute 43.4% of autocorrelated polygons, and just 10% of all of them. A further 746 polygons form HH clusters. However, LL and HL clusters are rather small, with an average of 11 polygons, so they were eventually excluded from further analysis aimed at finding local BGCP Thiessen polygon clusters.

The largest clusters of the large and regular Thiessen polygons (HH COType values of the attributes area and shape), named the regularly thinned clusters (RT), are located in the North-West (Pomerania Lakeland, Northern Podlasie Plain), East (east part of Northern Masovia Lowland), south-east (Lublin and Roztocze Upland), South (Sandomierz Basin, Nida Basin, Wester Beskidy Foothills), and the central part (Masovia Lowland and Hills) of the country.

The RT regions (shown in green

Figure 10e) are mainly covered by forest, agriculture, or agricultural-forest lands, and occupy 92,644 km

^{2} (29.7% of Poland). The regions of regularly densified geodetic control points (RD, presented in red,

Figure 10e) are created by small, regularly shaped Thiessen polygons. They are composed of a smaller number of polygons than the RT regions (

Table 5), and are more dispersed throughout the country territory, covering mainly urbanized areas, i.e., large agglomerations and accompanying industrial areas. Tiny RD clusters are located along the country’s border, mainly West (Odra river bands), North (along Baltic coastal zone), and South (the Sudetes, the Tatra Mountains and the Western and Eastern Carpathians). The two largest clusters are located in Silesia Upland and Oświęcim Basin as well as in the central part of the Masovia and Podlasie Lowlands. RG regions cover just 9.9% of the country and comprise 16.8% of all Thiessen polygons. When it comes to the polygons’ size, RD regions are more homogeneous than RT clusters, because Thiessen polygons’ area ranges from 3.68 to 46.44 km

^{2} with an average value of 27.35 km

^{2}, while, in RT, is takes values from 9.29 to 120.80 km

^{2} (see

Table 5).

Base Geodetic Network densification and thinning also emphasize the shortest distances between adjacent (near neighbor) points, which ranges from 5840 to 20,580 m in RT clusters, and from 145 to 5560 in RD clusters. Distances below 500 m occur in Upper Silesia, in areas threatened by land surface deformation related to hard coal mining.

## 5. Discussion

New technological advances in GIScience have made it easier to justify Tobler’s laws of geography. Now, we are able to measure almost everything, but the choice of technology and methods has a significant impact on the result of the analysis and final conclusions. Tobler’s laws give the background for GIScience theory, while GIS software provides tools for spatial analysis. In our study, Tobler’s laws of geography are read as a statement about the form of geodetic control points’ spatial arrangement, particularly in the sense of similarity. Regardless of the nature and type of geodetic control networks, their task is to create a basis for determining the position of various objects on the Earth’s surface. Preferably, the entire area concerned should be covered at a uniform density with a simultaneously adjusted network of control survey stations. Both ANN and Ripley’s K-function are best suited to analyzing the type of spatial distributions. They help avoid the modifiable areal unit problem [

69] as they treat space as continuous, without constraining data within any regions or units, and, contrary to the quadrat method, they are free from statistical bias caused by data aggregation [

70]. Nevertheless, Ripley’s K and NN functions describe different aspects of a point process. In particular, processes with the same K(t) function may have different nearest-neighbor distribution functions, and vice versa.

As stated in many national regulations and standards, the spacing of the higher-order stations can vary from 5 to 16 km [

23,

33,

71]. The conducted research proved that, in Poland, such requirements are fulfilled, with the mean observed interstation distance equal to 5.6 km. Moreover, each geodetic control point should be monumented with substantial stable survey monuments, and situated far from sources of electromagnetic interference and ground vibration [

5,

25,

32]. The analysis revealed that these requirements are fulfilled, although, 5.7% of the base geodetic control points, which are located in the near vicinity of geographical features that could provide significant disruption of electromagnetic signal, have to be monitored on a regular, short-term basis in order to ensure a high accuracy of their position.

The review of world-wide scientific literature concerning geodetic points’ location, as well as international and national regulation on network design and monumentation, revealed that intertwining between global, high-, and low-order networks is clearly visible. This is due to the growing demand for high-precision location measurements of objects, especially engineering, which is ensured by GNSS-based measurement techniques and methods [

71]. Therefore, there is a clear tendency to place detailed geodetic control points (low-order) in a position that maximizes the use of various measurement techniques, especially those that are intended for use in GNSS observations [

21,

28,

72]. High-order geodetic stations are dispersed regularly [

25,

39,

73,

74], and this evenness is only slightly disturbed in densely urbanized areas [

23] or those with challenging environmental conditions [

75]. At the local level, based on Poland as a case study, the clustered distribution and its relation to land cover/land use is evident and well documented [

22,

29,

30], and the analysis revealed high clustering along roads, water courses, and in dense urban areas. High-order geodetic control networks, with regularly dispersed reference stations, define the geodetic reference system in Poland [

16]. To the contrary, local, detailed networks, that are a part of land administration systems, are tailor-made to the covered area, and are suitable for performing local measurements, including surveying deformations and the necessity to monitor engineering structure [

28,

72]. They are inevitably designed to match local purposes and scale.

Scale is, de facto, a fundamental concept in any analysis concerned with spatial pattern and processes [

48,

66] as the patterns and processes of spatial objects have an implicit scale of variation. As noticed by Goodchild [

76,

77], scale is often perceived as an alternative dimension to interpret the patterns and processes. Scale also implicite relates to land policy, especially in land administration domain systems and models [

15,

78]. Scale dependent variation of BGCPs’ spatial pattern is shown by Ripley’s K-function and spatial autocorrelation. Furthermore, the spatial pattern of geodetic control networks is locally hampered by environmental features or processes (e.g., surface deformation as a result of mining, vicinity of water reservoirs, sky obstruction by a forest canopy) and changes from regular to densified or thinned.

## 6. Conclusions

The novelty of this study relied on quantifying the observed dispersed spatial pattern of the base geodetic control points (BGCPs) into regularly dense and regularly thinned, depending on a cluster and outlier analysis. The rules and principles of designing geodetic control networks are well recognized and described in the literature, but mainly as theoretical, numerical examples. Our research examines the spatial distribution of geodetic control points monumented within the frame of the base geodetic network in Poland. Hence, it has a broader context and provides some results concerning the spatial pattern of the BGCPs in Poland, which slightly changes within scales. The spatial dispersion of the BGCPs was documented by an ANN analysis, one of the oldest spatial statistics. The z-score of 1.39 for the ANN analysis, placed in the tail of normal distribution, provides clear evidence for dispersion. This dispersion was confirmed by the second-order spatial statistic, namely the Ripley’s K-function, which shows that at the near distances from 2 to 7 km, which are typical lengths in base geodetic networks. This leads to the conclusion that the assumed uniform distribution of BGCPs over the country’s territory has been achieved, however, in regions where the inter-point distance equals 2 km on average, a considerable densification of BGCPs is observed.

Only 6% of the BGCPs could be influenced by environmental obstacles, such as close vicinity to express roads, water reservoirs, or electricity lines. Moreover 452 points are located in artificial areas, where the satellite signals may be disturbed in multiple ways. All these points should be monitored regularly to ensure that their coordinates are determined accurately.

The presented research faces challenges in examining the configuration of Polish base geodetic control points by means of advanced spatial pattern methods, especially in analyzing the distribution of basic control network points. The achieved results could serve the scientific community and also land surveyors. They could be helpful at the survey planning stage because they show the accessibility of points in terms of inter-point distance, geometric distribution, and environmental conditions in the vicinity of stations.

The originality of the research lies in its complex approach to analyzing the density of geodetic control points and the evenness of their spatial distribution. Although the methods used are well known and widely described in the literature, their combination is an added value, especially for the National Mapping Agency that is responsible for geodetic control networks maintenance as well as for surveyors. Ultimately, the goal of the present research is not only to quantify the uniformity of spatial distribution of the base geodetic control points in Poland, but also to find environmental obstacles that hamper the assumed regular dispersed distribution and, finally, to provide methodological fundamentals enabling to conduct the analytical procedure for geodetic control points spatial pattern assessment.