Integration of Leveling and GNSS Data to Develop Relative Vertical Movements of the Earth’s Crust Using Hybrid Models

Abstract

This study compared two approaches to integrating leveling and GNSS data to develop relative vertical movements of the Earth’s crust. Novel approaches were tested using transformation and hybrid grid adjustment. The results from double-leveling measurements in Poland were used as test data, and GNSS measurements developed using the PPP technique were used as Supplementary Data. The least squares method was used for the adjustment, and the isometric, conformal and affine methods were used for the transformation, with and without Hausbrandt correction. So-called pseudo-nodal points, i.e., points identified as common in both networks, whose weight was determined according to the assumptions of scale-free network theory, were used as integration points. Both integration methods have similar results and are suitable for integrating leveling and GNSS data to determine the relative vertical movements of the Earth’s crust. The average unit error m₀ of the transformation was 0.1 mm/yr and the average error after adjustment of the hybrid network was 0.1 mm/yr. The use of the Hausbrandt correction does not significantly improve the transformation results. A 12-parameter affine transformation is recommended as the transformation method.

1. Introduction

The subject of this article is relevant to the integration of vertical movements of the Earth’s crust determined from various geodetic survey techniques, both terrestrial and satellite. The aim of this paper is to study the possibility of integrating the vertical crustal movements developed from two independent sets (leveling and GNSS PPP—Precise Point Positioning data) by using a hybrid network adjustment (nonlinear method) and mathematical transformation combined. This paper hypothesizes that the combined adjustment of vertical motion networks from hybrid data is comparable to the use of yield transformations.

The leveling network consists of the benchmarks between which the elevation measurements are made. It forms a leveling network. The leveling network is subject to adjustment with respect to a point or points of known elevation. Re-measurements of elevations are carried out at intervals (measurement campaigns) form a double-leveling network [1]. This allows for the calculation of the movement of the benchmarks identified with the vertical crustal movements. Two approaches are implemented: static—based on the difference in the height of the benchmarks—and non-linear—based on the adjustment of the double-leveling network (adjustment of vertical movement between the benchmarks). Each method gives good results under certain conditions, i.e., it depends on the existing leveling material; for example, both campaigns were carried out along the same leveling lines; successive campaigns were carried out in different parts of the area [2].

The use of nonlinear methods makes it possible to determine vertical movements associated with earthquakes and movements of a local nature. It also allows for the incorporation of tide gauges data and data from other surveying methods into this study. The incorporation is carried out as vertical movement at a point (sea level change—tide gauges data) or movements between points (e.g., GNSS permanent stations) [1].

The advantage of nonlinear methods is the greater ability to influence the “observation” data through observation errors. The disadvantage is the transformation of the hybrid network (triangulation) each time when GNSS stations are added or subtracted.

The advantage of transformation is the rapid preparation of new sets in case of GNSS station additions or subtractions. The disadvantage is that it is difficult to evaluate the choice of location and number of adjustment points and the reliability of vertical movements determined on them [3,4].

These solutions are not commonly used to combine vertical crustal motions (deformations) determined from dual leveling and GNSS coordinate time series. The Polish area is not tectonically active, and the GNSS time series do not always reflect the vertical motion of the crustal surface [5]. If GNSS stations reflecting the station’s own movements and those indicating the influence of underground mining are eliminated, it can be presumed that the resulting vertical movement of the station is compatible with the deformation of its crust. In the case of repeated leveling measurements, dozens of papers published (even from before the era of satellite measurements) clearly indicated consistency between leveling measurements and crustal movements. In areas with low tectonic activity (e.g., Poland), vertical movements determined from GNSS data are linear [5]; so, despite the low temporal resolution of leveling measurements, it can be assumed that the same relationship exists for leveling data.

Models of the Earth’s vertical crustal movements (VCM) or vertical land movements (VLM) are developed as absolute models and relative models. Absolute models refer to a rotational ellipsoid or quasi-geoid, while relative models refer to a specific level, e.g., the sea level, defined by a point or points (tide gauges). Other Global Navigation Satellite Systems (GNSS) references or stations, whose vertical motion is considered constant over time or have a known ‘velocity’, can also be adopted as reference points [6]. For relative models, leveling measurements are often used in combination with measurements on tide gauges as well as GNSS.

Due to the way it is developed, this group also includes data acquired from Synthetic Aperture Radar (SAR) [7]. The vertical crustal movements absolute models are mainly based on GNSS data [8,9]. The vertical movements determined from GNSS data mainly depend on the duration of station operation, the way the data are processed [10] (e.g., Precise Point Positioning—PPP or double difference), and the methods adopted for time series analysis and trend determination, e.g., noise analysis [11], detecting offsets [12], and seasonal signals [13]. Permanent GNSS stations are not linked to each other by a fixed network structure and the distances between them range from a few up to almost a hundred kilometers. This may cause the vertical movements determined on them to show local or own station movements. The quality of leveling data is more predictable and depends mainly on the measurement technique adopted (usually precision leveling), which makes it possible to limit the influence of local ground deformations.

The length of operation of permanent stations currently allows vertical movements to be determined with an accuracy comparable to that of those determined from leveling data. Merging these collections will allow for verification of the local nature of vertical movement at GNSS stations and increase the spatial resolution of the data. It is not only sets based on different measurement methods that can be combined but also sets of the same type but differing in processing, coverage and measurement epochs. Different approaches to this topic can be found in the literature [10]. The predominant methods are mainly based on a statistical–mathematical approach: velocity averaging, network (joint adjustment of several networks of vertical motions based on, for example, a triple or quadruple leveling network) [11], transformation [12] and interpolation (if the data used are developed as relative motions to the same datum) [13].

Table 1. Disadvantages of methods for modeling vertical movements of the Earth’s crust.

Leveling	low time resolution	hard-to-find archive material	labor-intensive calculations and data preparation	no stable points of reference over time	relative measurement only
GNSS	low spatial resolution	no physical network structure	accuracy depends on the length of station operation	sensitivity to local geotechnical effects or inherent movements	many variables affecting the result
Hybrid network-velocity averaging	sensitivity to outliers	additional analysis of variance required	statistical grouping and weighting of values	not very reliable accuracy
Hybrid network-adjustment	sensitivity to incorrect input data	requires statistical assumptions	time required for data preparation	determining observation weights	high dependence on network configuration and ‘fixed’ points
Hybrid network-transformation	depends on the quality of input data	the need for even distribution of common points	non-modeling of non-linear deformations of the system
Hybrid network-interpolation	sensitivity to outliers	overfitting	value estimation	sensitivity to point location and density	edge effect

and

Table 2. Advantages of methods for modeling vertical movements of the Earth’s crust.

Leveling	high accuracy	direct measurement	error control	high measurement repeatability	high spatial resolution
GNSS	high time resolution	large spatial coverage (mountains, forests, cities)	high availability of collections
hybrid network-	reduction in the impact of random errors	one value representative for the area
velocity averaging	elimination of random errors	data control, possibility of weighting observations	statistical evaluation of results	one consistent frame of reference	geometric network consistency check
hybrid network-adjustment	no need to import data into a single reference system beforehand	allows one to combine archival data with contemporary data	increases spatial consistency	not very complicated preparation of input data
hybrid network-transformation	completes values at any point	enables continuous modeling of the phenomenon	enables the creation of numerical models	wide range of interpolation methods

summarize the advantages and disadvantages of individual methods of modeling vertical movements of the Earth’s crust.

The selection of connections between GNSS stations can take place in any way. In [6], the connections were formed from one point (GNSS stations) to the others. In other work [14], GNSS stations were connected in a network structure based on the Delaunay triangulation algorithm, where the observations are vertical movements between GNSS stations constituting the vertices of the triangles. In [15], stations were connected into triangles based on criteria such as the closest location between the combined stations and the normal geometry condition. To perform the transformation process or simultaneously align two or more vertical motion networks from leveling, GNSS or other geodetic data, it is also necessary to define combined points. In [3,16], some general principles as to their location were indicated, while [17,18] present how to select relevant common points in hybrid networks based on the scale-free network theory using the example of the European leveling network (UELN) and the European GNSS permanent station network (EPN) [19,20,21].

GNSS stations and leveling benchmarks are generally not physically the same points. Vertical tectonic deformations are usually characterized by a slow linear trend of subsidence or uplift, while seismically active regions may experience sudden changes (or events) in vertical movements [22]. If vertical deformation only concerns tectonics, tectonic movements should have similar movements for individual tectonic blocks [23]. It should then be assumed that two (or more) measured points are some distance apart (distance criterion) but are a single point (pseudo-nodal point) [18]. This study showed that for a hybrid network based on UELN and EPN points, up to 90% of the points are pseudo-nodal points. The determination of the distance criterion is arbitrary. Its selection can be influenced by the tectonics and seismicity of the area, geology or terrain. The authors of [24] state that when combining networks based on multiple measurement techniques, the distances between individual measurement points should be close enough, but do not give a specific distance criterion. An analysis of vertical movements based on leveling data showed that such a criterion could be 10 km, while [25] stated that vertical movements of the Earth’s crust are consistent within a radius of 40 km. The selection of such a distance should be based on detailed analyses at each selected location. In [26], based on SAR data, it was shown that crustal surface motions are not consistent even at a close ~100 m distance from a GNSS station. The analyses carried out in this work showed that even some IGS stations located in a tectonically and seismically stable area should not be accepted as adjustment points [5]. Studies [17] have shown that 100% of common points are not required for hybrid networks. Approximately 3% is sufficient but the only condition is the selection of the most relevant points in the network.

Both, the choice of transformation and the method of aligning the vertical motion network are individual for each area and depend on the number of common points, spatial resolution, data quality, etc. A parametric method was used to align vertical movement networks from leveling data in [11] and a robust method was used to align vertical movement networks from GNSS data in [27]. Depending on the study, tide gauges on the southern Baltic coast [28] or a selected point in central Poland [11] were taken as reference points. The choice of reference point(s) is an authoritative choice; its location has a strong influence on the error distribution in the network. The adopted vertical movement at the reference point only affects the magnitude of relative movements. However, if several points are adopted as a reference and their vertical movements are known, then each point affects the determined relative movements to some extent.

Relative vertical movements of the Earth’s crust from repeated precision leveling measurements [11], and absolute vertical movements determined from coordinates of permanent GNSS stations of the Polish ASG-EUPOS network [5] were used as data. The area of Poland lies in central Europe, influenced by glacial isostatic adjustment, with a minor influence of tectonic and geological movements, mainly caused by mineral extraction [5]. Such conditions limit the possibility of generalizing the results to seismically active regions. Nevertheless, they provide greater certainty regarding the results obtained. One of the main crustal and lithospheric boundaries in Europe, known as the Teisseyrre-Tornquist Zone—TTZ [29,30]—passes through Poland. It separates the Precambrian crystalline crust of the East European platform and its oldest cover from the crust of the Palaeozoic platform. A very detailed description of this zone can be found in [29].

Relative vertical movements vary in Poland from 0 mm/yr near the Carpathian Mountains to −3.5 mm/yr in the rest of the area. Point-wise, they are higher, e.g., −6 mm/yr—caused by the strong impact of the flooded salt mine in the city of Inowrocław. Relative vertical movements refer to changes in the level of the Baltic Sea, where average changes in sea level are from 2 mm/yr to 3 mm/yr. Ellipsoidal (absolute) movements determined from the PPP measurement technique range from −3 mm/yr to +3 mm/yr.

2. Materials and Methods

The first type of data used for this analysis is leveling data. The elaboration of relative vertical movements of the Earth’s crust from leveling data requires extensive measurement. There are several approaches to this subject [31,32,33,34,35,36]. With these methods, separate segments of leveling networks and data from tide gauges and GNSS stations can be combined. Each method gives good results under specific conditions. One universal method is based on repeated measurement campaigns: double and triple with reference to a chosen reference level: local sea level, a point assumed to be ‘constant in time’ or a fixed velocity [11]. Nodal benchmarks on which measurements were made in both campaigns are used as binding points between the two networks. The double-leveling networks form a so-called vertical motion network in which the observations are the vertical motion between the same benchmarks. These motions are calculated based on unleveled elevations devoid of normal corrections. The time difference is calculated from the measurement epochs assigned to individual sections or leveling lines. Movement is assumed to be constant over time.

In Poland, four campaigns of precision leveling (

Table 3. Documented results from 4 leveling campaigns in Poland from 1926 to 2003 (source: own study).

Number of Campaign	Line Length [km]	Number			Mean Length [km]			Mean Error After Adjustment
		Traverses nF	Lines nL	Sections nR	Traverse Perimeter F	Line L	Sections R	$\left[\mathbf{m}\mathbf{m}\right]$
#1 (1926–1937)	10,046	36	121	5907	445	83	1.7	±1.04
#2 (1953–1955)	5778	12	60	about 4500	609	96	1.3	±0.78
#3 (1974–1982)	17,015	135	371	15,827	221	46	1.1	±0.84
#4 (1997–2003)	17,516	138	382	16,150	217	45.8	1.085	±0.88

) were carried out every 20 years on average (#1 1926–1937, #2 1953–1955, #3 1974–1982, #4 1997–2003). Due to the poorly documented measurement records from campaign #1, it is not used to study vertical movements of the Earth’s crust. The leveling from the last two leveling campaigns (#3 and #4) is complete, and measurements were made along almost the same lines, which is not the case in #2. The Polish area is practically seismically inactive; so, we assumed that despite the measurement periods not overlapping, the data used are suitable for the purpose of this research.

Leveling data are organized in the form of a database [2]. Calculated vertical movements between common nodal benchmarks from campaigns #3 and #4 were used in this study. These vertical movements were determined when creating the VCM2006 model of vertical movements of the Earth’s crust [ using Formula (1):

$$V_i={\displaystyle \frac{{∆h}_{ij}}{{∆T}_{ij}}}$$

(1)

where:

• V_i—vertical movement of the line;
• Δh_ij—height difference in epochs i and j;
• ΔT_ij—difference between epochs i and j expressed in decimal years.

The errors (1) used for weighting the observations were calculated with the assumption that the height differences are uncorrelated [11]:

$$m_{V_i}=\sqrt{{\displaystyle \frac{2L^2}{{∆T}_{ij}^2}}+{\displaystyle \frac{{∆h}_{ij}^2}{{∆T}_{ij}^2}}·{\left(0.5{\displaystyle \frac{\mathrm{m}\mathrm{m}}{\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}}\right)}^2}$$

(2)

where:

• L—length of the line in km;
• T—date of measurement of the leveling line;
• h—uncompensated elevation gain between the beginning and the end of the leveling line;
• 0.5—assumed maximum accuracy error in determining vertical movements.

Maximum loop disclosure ${\phi }_{∆v}$ and their average errors $m$ were calculated using Formulas (3) and (4) [36]:

$${\phi }_{∆v}=0.15\sqrt{\Sigma L\left(km\right){\displaystyle \frac{\mathrm{m}\mathrm{m}}{\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}}}}$$

(3)

$$m=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i-1}^N{\displaystyle \frac{{\phi }^2}{\Sigma L\left(km\right)}}}$$

(4)

where: ${\phi }_{∆v}$—maximum loop disclosure; $\phi$—calculated loop disclosure; and N—number of loops.

Failure to close the loops in double-leveling nets #3 and #4 exceeded the limits in two loops. The total number of loops was 128, and the average “not fully closed” error was +− 0.15 mm/year.

The model (VCM2006) is related to the tide gauge in Władysławowo, where the average velocity of change in the level of the Baltic Sea was determined based on tide gauge data. A total of 366 common leveling lines were defined and 222 common nodal points (Figure 1) were used in this research. The vertical movements between benchmarks are shown in Figure 2. Errors in the determination of vertical displacements on the 1 km double-leveling line were 99% less than 0.1 mm/yr.

Permanent GNSS stations do not form a network structure similar to a leveling network. The development of relative vertical movements of the Earth’s crust from GNSS data based on the simultaneous adjustment method requires the determination of vertical movements between permanent GNSS stations. This designation can be carried out in two ways: The first, presented in [6,10], involves selecting one GNSS station as the main one and calculating the vertical movement between this station and neighboring stations; in a slightly modified form, vertical movements are counted between all stations used. A different approach was used in [27], where the vertical movements between stations were determined based on the difference in the time series of coordinates of neighboring stations. Connections between stations were based on Delaunay triangulation [14]. Delaunay triangulation is the triangulation of a set of points P in the plane, such that no point from the set P is inside the circle described on the triangle belonging to the triangulation. Vertical movements at permanent GNSS stations are determined as a linear trend based on coordinate time series, which are subject to decomposition [38].

At this stage, outliers, spikes, seasonal components, signal noise, the influence of tide models, etc., are removed [39]. There are many computational strategies for each decomposed factor [40,41,42,43,44,45,46,47]. Station coordinates can come from double-difference solutions [10] or Precise Point Positioning (PPP) solution [48,49].

In Poland, the vertical movement was determined at all permanent stations of the ASG-EUPOS network from 2011 to 2021 (11 years) based on the PPP solution [5]. This is the latest study of this type in this area (Figure 3). Its advantage is the use of long coordinate time series data [50].

The largest movements were observed in the mining area (southern Poland) located on the Western European plate. Small variable movements are observed in the remaining area—the Eastern European plate. Single larger peaks are observed near the northern border—influence of isostatic glacial adjustment.

Time series were created from RINEX files—Receiver Independent Exchange Format (raw observation obtained from the Polish Central Office of Geodesy)—using GipsyX [48]. The vertical movements were determined as a linear trend. The vertical movements and their standard deviations are shown below (Figure 4).

Combining the results from [6] and the solution of connecting GNSS stations into a network structure [14], the vertical movements between permanent GNSS stations were determined in two variants. The first assumed the use of all stations developed in the work [27] (Figure 5) and the second only those adopted for the model of vertical movements.

The obtained vertical movements between stations are shown in Figure 6. Errors in the determination of vertical displacements on the 1 km double-leveling line were 99% less than 0.1 mm/yr.

2.1. GNSS and Leveling Data Integration

Based on the literature [6,7,8,9,10,11,12], data integration methods for determining the vertical movements of the Earth’s crust can be divided into five approaches. In all approaches, the data are vertical movements at points (GNSS stations, nodal benchmarks) or between points. A brief description of the main requirements and capabilities of these methods is shown below:

• Data averaging: Vertical movements must be determined at the same points with known uncertainty in their determination, the same reference level used and measurement in similar time intervals; under certain assumptions, vertical movements can be determined at close points. Possibilities—vertical movements determined from GNSS data (from various calculation strategies) as well as vertical movements determined from double leveling from different measurement epochs can be used as data.
• Transformation: Vertical movements must be determined at the same points, but not for the entire set—only for adjustment points—and one reference system should be adopted—secondary (reference); under certain assumptions, vertical movements can be determined at close points, similarly to adjustment points (pseudo-nodal points). Possibilities—vertical movements determined from GNSS data (from different calculation strategies), vertical movements at nodal and intermediate benchmarks, and hybrid data (GNSS vertical movements and vertical movements at benchmarks) can be used as data, the data can have different temporal and spatial resolution, and adjustment points can be individually selected.
• Interpolation: Vertical movements should be related to the same reference level. Possibilities—vertical movements determined from GNSS data (from various computational strategies), vertical movements at nodal and intermediate benchmarks, and hybrid data (GNSS vertical movements and vertical movements at benchmarks) can be used as data; the data may have different temporal and spatial resolutions. The vertical movements do not have to be determined at the same points. It is possible to weight observations.
• Adjustment of the hybrid network: The sets should have a network structure, vertical movements should be determined between adjacent points for each adopted set separately with known uncertainty, common points should be defined, and a defined reference level should be defined. Possibilities—vertical movements determined from GNSS data (from different calculation strategies), and vertical movements between nodal benchmarks and hybrid data (vertical movements between GNSS stations and vertical movements between nodal benchmarks) can be used as data, the data can have different temporal and spatial resolution, adjustment points can be selected individually, and the reference level may be adopted individually.
• A fifth approach may combine all four of the above assumptions in various configurations.

From the above, two methods are most suitable for this article: hybrid network transformation and adjustment. The same reference point was adopted. The following computational strategy was adopted:

• Test adjustment of the DLN (double-leveling network) and GTN (GNSS triangulation network) networks separately.
• Adjustment of the hybrid network (DLN+GTN), common points (nodal points) 50 adopted from [17]—version 1.
• Transformation of absolute GNSS vertical movements [5] at ASG-EUPOS stations to relative vertical movements determined from the DLN adjustment, common points (nodal points) 50 adopted from [17]—version 1.
• Statistical evaluation of hybrid network adjustment and transformation.
• Re-adjustment of the hybrid network based on common points with the best match between the hybrid network and transformation vertical movements, without points showing self-motions or VLM movements [5]—(version 2).
• Transformation based on common points as in point 5.
• Statistical evaluation of the results.

2.2. Hybrid Network Adjustment

To align the hybrid network of vertical movements of the Earth’s crust from leveling and GNSS data, it is necessary to establish several parameters: selection of the adjustment method, weighting of observations, identification of connective points and their spatial distribution, and establishment of reference points. During the development of VCM 2006 [37], several adjustment methods were tested and all of them gave the same results. The least squares method was adopted for strict adjustment [51]. The observations are unbalanced vertical movements between GNSS stations and unbalanced vertical movements between nodes of the double-leveling network. Uncompensated vertical movements errors were calculated as stated in the “Materials and Methods” section. The weighting of observations was uniform for both types of observations (vertical movements from double leveling and vertical movements from GNSS data) as the inverse of the square of the error. The tide gauge in Władysławowo was used as a reference point [37]. GNSS stations do not physically coincide with the nodal points of the basic height network [17]; therefore it is necessary to identify the connecting points as the so-called pseudo-nodal points [18]. These are points separated by a certain distance that represent the same movement. This assumption results from the fact that the vertical movements of the Earth’s crust are not point-like, but coherent in a certain area [17,52]. A distance criterion of 10 km was adopted, as in [18]. For the first adjustment, 50 identified common points of Figure 7 were adopted.

The second adjustment was performed based on 11 common points (Figure 7), which were identified based on the criterion of matching the relative vertical movements at the nodal points (50) between the hybrid network adjustment and the transformation. The other criteria were not changed.

2.3. Transformation of GNSS Absolute Vertical Movements into Relative Vertical Movements

Coordinate transformation between systems involves modifying the transformed coordinates using [53] translation, scaling and rotation. Similarly to the adjustment of a hybrid network, it is important in the transformation to identify common points in both systems (primary and secondary). There are no specific rules regarding the location of these points in the network. According to [15,16,17]:

• They should be distributed evenly in the area (and its surroundings) subject to transformation, both inside and outside.
• There should be such a subset of adjustment points in the set of adjustment points, which, connected by a closed sequence, creates a figure containing completely transformed points.
• These should satisfy Inequality (5):

$$n=n_1+{\displaystyle \frac{n_2}{2}}\ge 4$$

(5)

where:

• k—means the conventional indicator of the accuracy class of transformed points, e.g., k = 1, 2, 3, 4 (k = 1—the highest accuracy class);
• n₁ means the number of class adjustment points with an index < k − 1;
• n₂—the number of adjustment points grades k − 1.

3. Results

Hybrid Network Adjustment and Transformation Results

In the first stage, control adjustment of the DLN and GTN networks was carried out using a robust method to identify possible outlier observations (

Table 4. Characteristics of the approach in version 1 and version 2 (source: own study).

	Version 1	Version 2
Total number of observations	632	632
Redundancy observations	362	323
Network points	270	309
Reliability index z	0.57278	0.51108
Otrębski parameter p	0.42722	0.48892
Mean observation error reduction factor q	0.65362	0.69923
number of common points between DLN—GTN networks	50	11

). Two outlier observations were identified in the DLN network, identical to Figure 1, which shows two unclosed loops of the double-leveling network with the permissible error exceeded. These observations were removed from the set. Then, both DLN, GTN and hybrid networks were aligned using the least squares method. The error distribution is shown in Figure 8.

The average errors obtained from the equalization of the hybrid network in version 2 are closest to the normal distribution. For the DLN network, there is a left-sided skewness, and for the DTN network, there is a clear kurtosis.

In the transformation process (version 1), three methods were used: isometric (6 parameters), conformal (7 parameters) and affine (12 parameters), all in the variant without and with Hausbrandt correction. As a result of the transformation from the primary system to the secondary system, all adjustment points received new coordinates. The purpose of the Hausbrandt correction is to leave the coordinates of the adjustment in the secondary system unchanged. For the remaining points, corrections were calculated using interpolation formulas. The number of common points was 50, and the transformation errors are shown in

Table 5. Errors after transformation [mm/yr] (version 1 and 2) (source: own study).

	Isometric		Conformal		Affine
	Ver 1	Ver 2	Ver 1	Ver 2	Ver 1	Ver 2
Average unit error m0	0.35	0.31	0.35	0.32	0.27	0.11
Transformation error mt	0.60	0.49	0.60	0.49	0.45	0.15

. In version 2, the average errors did not exceed 0.1 mm/yr, except for in one case, where 0.11 mm/yr was observed. In the transformation process (version 2), the same transformation methods were used as in version 1. The number of common points was 11, and the transformation errors are shown in Table 5.

The differences between the values from the hybrid network adjustment and individual transformation variants from versions 1 and 2 were calculated (

Table 6. Difference between hybrid network vertical movements and transformation (version 1) [mm/yr] (source: own study).

	Isometric				Conformal				Affine
	Transformation with Correction		Transformation		Transformation with Correction		Transformation		Transformation with Correction		Transformation
Version	1	2	1	2	1	2	1	2	1	2	1	2
Median	0.72	0.95	0.72	0.96	0.72	0.96	0.72	0.95	0.55	0.27	0.57	0.26
Max. difference	4.30	4.11	4.30	4.60	4.3	4.11	4.30	4.11	3.23	0.91	2.98	0.91
Min. difference	0.02	0.01	0.02	0.01	0.02	0.01	0.01	0.01	0.01	0.02	0.01	0.01
Median errors of common points	0.13	0.25	0.13	025	0.13	0.25	0.13	0.25	0.12	0.12	0.12	0.12

).

In the second version, the hybrid network was aligned in the same way as in the first version based on 11 connecting points (pseudo-nodal points); these points were not identified as suspicious, i.e., those that show their vertical movements [5], and, at the same time, in the transformation process, these showed the smallest differences compared to the vertical movements from the hybrid network: max. 0.2 mm/yr. The adjustment errors determined (Figure 9) did not differ significantly from those obtained based on 50 common points.

4. Discussion

A comparative analysis of hybrid network adjustment and transformation was carried out.

Table 7. Basic descriptive statistics of differences between vertical movements in networks: DLN, GTN, hybrid and from transformation [mm/yr].

	DLN Hybrid		GTN Hybrid		Hybrid Ver 1–Ver 2	Affine Transformation Hybrid
Version	1	2	1	2	1–2	1	2
Median	−0.5	−0.2	−0.1	−0.3	0.0	0.5	0.3
MAX	1.3	0.1	0.6	−0.2	2.6	3.0	0.9
MIN	−2.6	−0.8	−0.5	−0.5	−1.5	0.0	0.0

shows the statistical evaluation of the obtained results from the hybrid network adjustment and transformation.

In total, 50 node points, as shown in Figure 7, were used as connecting points. To transform the absolute vertical movements at the points of the ASG-EUPOS network to the relative vertical movements of the DLN network, these were used similarly to that observed with isometric transformation and conformal and affine transformation.

In the second variant of the transformation, the same 11 points as in the second adjustment of the hybrid network were used as connective points.

Figure 10 shows the spatial distribution of vertical movements differences between DLN and hybrid networks in version 1 and version 2.

The biggest differences between the vertical movements in the DLN and hybrid networks (version 1) are at GNSS stations—ELBL, SIPC, WODZ, WRON, NYSA, TRNW, WLOC, SOCH, which are in the vicinity of Żuławy (Vistula delta plain rising just above the Baltic Sea level—Figure 1), on the border of the TT zone, and parts of the Mountains Sudetes and Western Mountains Carpathians (south of Wrocław—Figure 3), an area of influence of mining areas. The maximum vertical movements’ differences at double-leveling nodes are smaller than for the vertical movements at permanent GNSS stations, and in terms of area, they largely coincide with the vertical movements of GNSS stations, except for the border areas with the Kaliningrad Oblast (Russia). The maximum relative vertical movements (Figure 3) near Inowrocław (disused salt mine) and near the border with Ukraine (natural gas mines—Rzeszów area) were reduced and are no longer outliers.

Figure 11 shows the differences between the vertical movements at the GTN network points and the hybrid network in version 1 and version 2. The differences between the vertical movements at the GTN network points and the hybrid network in version 1 do not exceed 1 mm/yr, while between the hybrid network in version 2, they do not exceed 0.5 mm/yr.

Figure 12 shows the spatial arrangement between the two versions of the hybrid network.

Between the two versions of the hybrid network, the biggest differences are at the ELBL, SIPC and WODZ stations. The differences are slightly smaller at the TRNW, WROC, KROT and NYSA stations and double-leveling nodal points in the vicinity of these stations, and in the Żuławy region. From this, we can see that nodal points with different vertical movements in DLN and GTN networks play an important role.

From the vertical movements’ comparison of the affine transformation and hybrid network adjustment (Figure 13), the biggest differences are in the vertical movements for version 1; in version 2, there is much better consistency in the results. The largest vertical movements differences for version 1 of the affine transformation and the hybrid network are at stations identified as outliers [5]: ELBL, SIPC, WLBR, WODZ, KATO, and PRZM. The are slightly smaller at the following stations: KLDZ, NYSA, WROC, WLOC, SOCH, NODW, CCHN, ILAW, DRWP, TRNW, KROS, BILG, TABG, and HOZD.

In version 1 of the hybrid network adjustment, the post-adjustment errors for the DLN network nodes decreased by an average of 0.02 mm/yr, while the errors for the GTN network points increased by 0.02 mm/yr. These values are not significant from the point of view of the movements of the Earth’s crust. In the hybrid network, the errors did not exceed 0.1 mm/yr.

In version 1 of the transformation, the average unit error m₀ and the transformation error are the smallest for the affine transformation, and the isometric and conformal transformations are characterized by a similar error. For isometric and conformal transformations, the median is the same regardless of the application of the Hausbrandt correction. For the affine transformation, the median was 0.55. The median errors of common points in the isometric and conformal transformations were 0.13 mm/y, and for the affine transformation, they were 0.12 mm/yr. From the point of view of vertical movements of the Earth’s crust, all transformation methods can be considered suitable. However, the differences (Table 6) are smallest when using a 12-parameter affine transformation. At 50 adjustment points, the differences in vertical movements did not exceed 0.2 mm/yr compared to the equalized values.

Version 2 of the hybrid network equalization showed that the average errors after equalization do not differ significantly from those from the adjustment in version 1, but their distribution is more similar to the normal distribution.

In version 2 of the transformation, the average unit error m₀ is the smallest for the affine transformation. The isometric and conformal transformations are characterized by the same error 3x greater than the m₀ of the affine transformation. Similarly, the affine transformation error is 3x smaller than in the other cases. For the isometric and conformal transformations, the median is the same regardless of the application of the Hausbrandt correction, similarly to version 1. For the affine transformation, the median was 0.27 mm/yr. The median errors of common points in the isometric and conformal transformations were 0.25 mm/yr (worse compared to version 1), and 0.12 mm/yr for the affine transformation (they did not change compared to version 1). Regardless of the use of the Hausbrandt correction, statistically, the results are the same for both conformal and affine transformations.

The smallest differences between the DLN network and the hybrid network were obtained in version 2 (11 common points), from 0.1 mm/yr to −0.8 mm/yr.

The differences (Figure 11) between the GTN network and the hybrid network version 1 do not exceed 1 mm/yr, while between the hybrid network version 2, they do not exceed 0.5 mm/yr. The smallest differences between the GTN network and the hybrid network were obtained in version 2.

Between the two versions of the hybrid network, the maximum vertical movements differences range from 2.6 mm/yr to 1.5 mm/yr.

Table 5, Table 6 and Table 7 show that the smallest transformation errors and consistency with the hybrid network (versions 1 and 2) are obtained using the 12-parameter affine transformation method with Hausbrandt correction. The comparison of vertical movements from the affine transformation and hybrid network adjustment (Figure 13) shows that the biggest differences are in vertical movements for version 1; in version 2, there is much better consistency in the results. The largest vertical movements differences for version 1 of the affine transformation and the hybrid network are at stations identified as outliers.

An analysis of the impact of the distribution of adjustment points on the example of affine transformation showed that reducing the number of adjustment points did not increase the transformation errors, even though the adjustment points were not evenly distributed across the study area. The transformation errors increased most for the ELBL station, SIPC, and WLBR (approximately 0.2 mm/yr), while in the remaining area, they ranged from 0.0 mm/yr to 0.1 mm/yr. The application of the theory of scale-free networks allowed for the selection of points that have the most significant impact on the structure of the network and are not random. Reducing the number of adjustment points did not affect the quality of the transformation or cause a local adjustment effect.

The hybrid network adjustment of version 2 with 11 connection points had smaller errors than version 1 with 50 points. The distribution of errors after equalization in the hybrid network in version 2 is closer to the normal distribution than in version 1. Transformation using isometric, conformal and affine methods showed that the affine method had the smallest transformation error m₀. Other parameters of the affinity transformation are shown in

Table 8. Affine, isometric and conformal transformation parameters.

Parameters	Isometric		Conformal		Affine
	Ver 1	Ver 2	Ver 1	Ver 2	Ver 1	Ver 2
Displacement parameters (center of gravity coordinates)—secondary system (ZS2)	−2.67746	−2.49576	−2.67746	−2.49576	−2.67746	−2.49576
Axial scale control:
Sx	1	1	1	1	1	1
Sy	1	1	1	1	1	1
Sz	1	1	9.9999 × 10−0001	9.9999 × 10−0001	2.9474 × 10−0001	6.7190 × 10−0001

.

Displacement parameters (center of gravity coordinates)—secondary system (ZS2)—are the same in any transformations. There are differences in the axial scale control parameter in axial Z. The introduction of the Hausbrandt correction in the transformation provided a better fit to the results of the hybrid network. Using fewer adjustment points in the transformation resulted in smaller transformation errors. It has been demonstrated that Hausbrandt’s correction is important not only when transforming from ellipsoidal heights to normal heights, but also when transforming vertical movements determined on the basis of coordinates from these two height systems. This is normal behavior of the sets as the networks are less rigid. However, this suggests that the adjustment points should not only be taken in terms of the distance criterion but also the vertical movements adjustment criterion; so, the transformation should be performed in several iterations.

The largest differences in vertical movements in all tested variants were at adjustment points that were identified as having their own movements or those not related to the movements of the Earth’s crust. These points should be removed before the final version of the VCM model is developed. Adjustment of the hybrid network also showed that some of these points had been leveled (Inowrocław, Rzeszów).

Figure 14 shows examples of relative VCM using leveling data, GNSS data, hybrid data (equalization) and hybrid (transformation) data. Triangulation with linear interpolation was used for modeling. They show dominating points mainly due to uneliminated GNSS stations (vertical movements that differ from the others) and local surface movements from leveling data.

A visual comparison of the figures ‘Hybrid model adjustment’ and ‘Hybrid model transformation’ shows a greater number of local extreme values compared to the figures ‘Data alignment—DLN’ and ‘GNSS data—GTN’. Due to the lower spatial resolution, the model from GNSS data is less differentiated than the model from leveling data.

There is a general convergence of hybrid models. The differences that arise are influenced by the lack of current leveling data (measurement 1997–2003), the GNSS-PPP method used to determine coordinates, and vertical anthropogenic movements.

5. Conclusions

This study compares two approaches to integrating leveling and GNSS data for the relative study of vertical movements of the Earth’s crust. Theoretical approaches were tested using transformation and hybrid network adjustment.

The results can be summarized as follows:

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The smallest unit error m₀ (0.27 mm/yr (version 1) and 0.11 mm/yr (version 2)) and transformation error m_t (0.45 mm/yr (version 1), 0.15 mm/yr (version 2)) are 3x smaller for the affine transformation compared to the other transformations.

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An error distribution close to the normal distribution was obtained for the hybrid network adjustment version 2.

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The average error after adjustment of the hybrid network in both versions is 0.1 mm/yr.

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The smallest differences in vertical movement between affine transformation and hybrid network adjustment were obtained for version 2: 0.91 mm/yr.

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The use of the Hausbrandt correction does not significantly improve the transformation results.

The results obtained show that both methods are suitable for the above purpose. Integration using the hybrid network adjustment method is more difficult to automate than transformation. Each new point in the network generates a change in its structure and properties. Of the transformation methods, the smallest errors were obtained for the 12-parameter affine transformation. Therefore, it is recommended to use this transformation.

The results of this work can be used to develop terrain deformation models from hybrid data over any size of area. There is no need to create artificial links between GNSS stations and carry out time-consuming adjustment of the hybrid network. The same effect is obtained using an affine transformation.

It has been proven that the use of the transformation allows for the transition from an absolute reference to a relative reference, which is one of the main problems when combining leveling and GNSS data for vertical crustal variations.

The largest differences in vertical movements in all tested variants were at the adjustment points. These points were identified as having intrinsic or unrelated crustal motions.

In both the adjustment and transformation of the hybrid network, local vertical motions in excess of regional motions are canceled out.

Not all common points in the hybrid network are suitable as connecting points. The selection of joint points in adjustment as well as adjustment points in transformation should be controlled and outlier points should be verified geologically and tectonically.

Scale-free network theory has been shown to be suitable for verifying the utility and location of connection points in a hybrid network.

The typing of connection points using scale-free network theory makes it possible to identify the connection points of greatest importance in the network. The same connection points can be used in hybrid network adjustment, as well as transformation.

It has been confirmed that not all possible connection points in the hybrid network should be adopted for transformation or adjustment (they must be relevant in the network).

The area of Poland shows little vertical movement. The isolines are strongly influenced by local underground mining and the pressure of urban agglomerations on the Earth’s crust.

GNSS stations are subject to vertical movements caused by local factors; therefore, it is necessary to analyze the geological location of stations, especially those adopted as adjustment points.

Once the model of vertical movements from hybrid data is developed, geodynamic correction is also necessary.

The proposed approach can be extended with new data, e.g., vertical movements from SAR, SLA—Sea Level Anomaly, gravimetry. SAR data are available through the Copernicus Land Monitoring Service as the European Ground Motion Service (EGMS). It can be used, among other things, to verify the location of permanent GNSS stations and to correct the model of vertical movements of the Earth’s crust.

Due to the occurrence of small crustal deformations and local/small scale effects (marker instabilities, local subsidence, mining effects) in the area of Poland, an attempt should be made in subsequent studies to incorporate another measurement technique which would provide high spatial and temporal resolution, e.g., SAR.