Keywords

1 Introduction

The advent of GNSS, particularly after 1990, revolutionized the everyday geodetic and surveying workflow worldwide. The time-demanding and often cumbersome terrestrial measurements were replaced by GNSS occupations in static and real-time modes. However, until today, the majority of current high-level geodetic networks, which were established through observations collected during conventional terrestrial surveying campaigns as far as a century ago, are in 2D. Therefore, the combination of 3D network (mainly now from GNSS, earlier e.g., from Doppler measurements) and classical 2D networks should be materialized. Such combination schemes have been extensively studied (Gargula 2021; Ilie 2016; Kadaj 2016; Peterson 1974; Weiss et al. 2022). In addition, the local ties used to establish the International Terrestrial Reference Frame (ITRF, Altamimi et al. 2023) combine both space and terrestrial observations (Abbondanza et al. 2009; Lösler et al. 2023).

The main scope of the combination is to align the combined network to a unified global reference frame, -e.g., the ITRF, which may be a version of or even a regional one, e.g. ETRS89 or SIRGAS (Kenyeres et al. 2019; Sánchez and Drewes 2020). We may categorize the combination methodologies as follows:

  1. a.

    Common Adjustment (CA): The GNSS observations are introduced as 3D baselines (dX, dY, dZ) and the 2D/1D conventional surveying observations are treated as they are observed. In most cases, these observations are incorporated into an appropriate Least Squares (LS) adjustment software.

  2. b.

    Helmert transformation (HLMT): Through common points, one set of stations is transformed to the other’s reference frame.

Despite the fact that these two methodologies are dominant throughout the geodetic literature, there are some limitations for each. For the case of CA, the main problem arises when the System Independent Exchange (SINEX, Blewitt et al. 1994) file is used. It is not clear how observations such as baselines, spatial distances or angles can be derived, since the SINEX file format focuses on sets of coordinates or/and velocities and their associated Normal Equation or Covariance Matrix. Additionally, typically, the CA method is realized to a well-designed network with favourable geometry and a substantial number of stations are occupied using both classical and GNSS observations. However, this is not always the case.

On the other hand, the HLMT methodology exhibits limitations when it is applied in cases of poor networks’ geometry, e.g., the common stations do not cover the whole area or when there are sparse areas in the network and in cases of small areas, where the correlations of the estimated parameters could become notable. Finally, when 2D information is used, the vertical information is lost. In that case, even though the existing methods are widely and successfully used, they have some pitfalls which potentially can lead to less accurate results. A common pitfall is the lack of the full Covariance matrix (CV matrix) of the estimated coordinates (most of the times only the standard deviations of the points are known) which leads to non-rigorous results.

The present study deals with the description of an alternative strategy in the direction of the optimal combination of modern 3D and classical networks, exploiting existing zenith angle measurements and mathematical models. The proposed strategy is based on the combination of Normal Equations (NEQs) which are properly converted, added, and restored according to existing methodologies. Special treatment is applied to remedy datum-related information. In addition, two different schemes are presented for the combination of a modern 3D GNSS network and a classical one. These two schemes pertain to the use of the vertical part of the classical networks at the combination process. The alternative strategy is implemented in the region of Drama, Greece using two local networks (GNSS and classical).

2 Combination Strategies

The aim of our study is to build rigorous yet easily-applied algorithms to combine 3D (GNSS) networks from a SINEX file format, and 2D classical ones including spatial distances, horizontal and zenith angles (vertical circular reading of the instrument with respect to the plumb line) and/or azimuths (astronomical azimuths reduced to geodetic and grid ones, respectively). The proposed approach focuses on the following aspects:

  1. 1.

    Utilization of both 3D (GNSS) and 2D (classical) information.

  2. 2.

    Explicit definition of the Reference System, including datum specification.

  3. 3.

    Adaption to various scenarios based on observation accuracy and requirements.

The observed zenith angles and the deflections of the vertical play a key role in our approach, as we elaborate below. Zenith angles are first corrected due to refraction and curvature of the earth (as e.g., in Torge and Müller 2012). The usage of the zenith angles is a matter of great importance, as they practically enable the extension of the 2D networks to complete 3D representation. Initially we present the common steps for all different strategies.

2.1 Common Steps of the Alternative Strategies

The alternative strategies are based on particular algorithmic steps. The first three steps are common and described below. Assuming that there is a file in SINEX format containing a Covariance Matrix of the solution:

  1. 1.

    Convert the 3D cartesian coordinates (XYZ) to the topocentric (ENU) system. This holds for both the approximate and estimated coordinates. The 3D network refers to a modern 3D Terrestrial Reference Frame (TRF). The conversion is realized through the following pointwise formula:

$$ {\mathbf{q}}_{\mathrm{i}}=\mathbf{R}{\mathbf{x}}_{\mathrm{i}},\vspace*{-3pt} $$
(1)

where q i = [Ei Ni Ui]T represents the topocentric coordinates (East, North, Up components), \( \mathbf{R}=\left[\begin{array}{ccc}-\sin {\uplambda}_{\mathrm{m}}& \cos {\uplambda}_{\mathrm{m}}& 0\\ {}-\sin {\upvarphi}_{\mathrm{m}}\cos {\uplambda}_{\mathrm{m}}& -\sin {\upvarphi}_{\mathrm{m}}\sin {\uplambda}_{\mathrm{m}}& \cos {\upvarphi}_{\mathrm{m}}\\ {}\cos {\upvarphi}_{\mathrm{m}}\cos {\uplambda}_{\mathrm{m}}& \cos {\upvarphi}_{\mathrm{m}}\sin {\uplambda}_{\mathrm{m}}& \sin {\upvarphi}_{\mathrm{m}}\end{array}\right] \) the orthogonal conversion matrix, \( {\mathbf{x}}_{\mathrm{i}}={\left[\begin{array}{ccc}{\mathrm{X}}_{\mathrm{i}}& {\mathrm{Y}}_{\mathrm{i}}& {\mathrm{Z}}_{\mathrm{i}}\end{array}\right]}^{\mathrm{T}} \) the Cartesian coordinates, φm, λm the average/reference geodetic latitude and longitude, respectively, of the area. We may refer that for our case study (see Sect. 3 ibid.), the average latitude and longitude are estimated for an area not larger than 10 × 10 km.

  1. 2.

    Transform the Covariance matrix (CV) and the Right-Hand Side (RHS) of the 3D network from geocentric to topocentric system, using error propagation theory:

$$ {\mathbf{C}}_{\mathbf{q}}^{\mathbf{3D}}=\mathbf{J}{\mathbf{C}}_{\mathbf{x}}^{\mathbf{3D}}{\mathbf{J}}^{\mathbf{T}}, $$
(2a)

and

$$ {\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}={\mathbf{J}}^{\mathbf{T}}{\mathbf{u}}_{\mathbf{x}}^{\mathbf{3D}}, $$
(2b)

where \( {\mathbf{C}}_{\mathbf{q}}^{\mathbf{3D}} \), \( {\mathbf{C}}_{\mathbf{x}}^{\mathbf{3D}} \) the full 3D CV matrices of the topocentric and geocentric coordinates, respectively, \( {\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}} \), \( {\mathbf{u}}_{\mathbf{x}}^{\mathbf{3D}} \) the full 3D RHS of the topocentric and geocentric coordinates, respectively, \( \mathbf{J}=\left[\begin{array}{ccc}\mathbf{R}&\ &\ \\ {}\ & \boldsymbol{\ddots}&\ \\ {}\ &\ & \mathbf{R}\end{array}\right] \) the total transition matrix (Jacobian) from geocentric to topocentric coordinates. The RHS for the geocentric coordinates can be computed by the following equation:

$$ {\mathbf{u}}_{\mathbf{x}}^{\mathbf{3D}}={\mathbf{C}}_{\mathbf{x}}^{\mathbf{3D}}\left({\mathbf{x}}^{\mathbf{est}}-{\mathbf{x}}^{\mathbf{apr}}\right), $$
(3)

where x est, x apr the vectors of the estimated and approximate geocentric coordinates, respectively (expect to be found in a typical SINEX file). The RHS is the part of the Normal Equation related to the observations of the 3D (GNSS) network.

  1. 3.

    Re-calculate the approximate topocentric coordinates of classical network’s stations, through analytical geometry formulation. This could be easily estimated for the horizontal components E and N utilizing the well-known formulas for the traverse solution (latitudes and departures), considering at least two stations as fixed. The approximate up-components can be determined by trigonometric calculations, fixing at least one station. The fixed stations are the common stations of the two networks (3D and classical) whose topocentric coordinates are straightforwardly computed from the geocentric coordinates, as we already mention in Step 1. The new approximate values of the classical network refer now to the modern 3D TRF. Hence, the two different networks refer to a common reference system (datum).

  2. 4.

    Invert the CV matrix of the topocentric coordinates to obtain the associated Normal Equation Matrix (NEQ):

$$ {\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}={\left({\mathbf{C}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{-1}. $$
(4)

2.2 First Alternative Strategy: Transition from a 2D to a 3D Network Employing Zenith Angles

Normally, the classical network includes observations of zenith angles referring to the physical surface (plumbline). Through the Deflections of Vertical (DoV), they are reduced to an ellipsoid (Barzaghi et al. 2016). The corrected and reduced zenith angles (with respect to the vertical of geodetic system) are estimated as follows (Rossikopoulos 1999):

$$ \begin{array}{ll}{z}_{ij}&\displaystyle ={\zeta}_{ij}+\underset{refraction\ term\ correction}{\underbrace{\frac{k_{ij}}{2R}{\left({\rho}_{ij}\ {sin}\ {\zeta}_{ij}\right)}^2}}\\[24pt]&\quad \displaystyle -\underset{Eart{h}^{\prime }s\ cruvature\ term\ correction}{\underbrace{\frac{1}{2R}{\left({\rho}_{ij}\ {sin}\ {\zeta}_{ij}\right)}^2}}\\[24pt]&\quad \displaystyle +\underset{DoV}{\underbrace{\ {sin}\ {a}_{ij}{\eta}_i+ {cos}\ {a}_{ij}{\xi}_i}}, \vspace*{-3pt}\end{array}$$
(5)

where z ij is the reduced zenith angle (from station i to station j) with respect to the vertical, ζ ij the observed zenith angle, k ij the refraction term, ρ ij the spatial distance, R the Earth’s mean radius for the area, α ij the azimuth and ξ i, η i the deflections of the vertical. If the refraction term is unknown, then a mean value of 0.13 can be assumed for Greece (Labrou and Pantazis 2010).

Continuing from 2.1

  1. 5.

    Solve the classical network in 3D (topocentric system), using the original observations which can include: spatial distances horizontal angles, directions and azimuths (geodetic or grid ones). The zenith angles are reduced according to Eq. (5). Now the solution offers a complete 3D dataset. The observation equations of the spatial distance and the zenith angle with respect to the topocentric coordinates, are as follows:

$$ {\rho}_{ij}=\sqrt{\left(\Delta {E}_{ij}^2+\Delta {N}_{ij}^2+\Delta {U}_{ij}^2\right),}\vspace*{-3pt} $$
(6a)

and

$$ {z}_{ij}={arctan}\ \frac{\sqrt{\left(\Delta {E}_{ij}^2+\Delta {N}_{ij}^2\right)}}{\Delta {U}_{ij}}. \vspace*{-3pt}$$
(6b)

There are some major issues regarding the reliability of the measured zenith angles: (a) their accuracy is strongly dependent on the atmospheric refraction and (b) the requisite knowledge of the deflections of the vertical (DoV), is many times either not applicable or it is rather problematic. Practically, the first alternative strategy imposes a “three-dimensionalization” of the classical network through the corrected and reduced zenith angles. The solution of the classical network results the NEQ and the RHS, \( {\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{classical}} \) and \( {\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{classical}} \), respectively.

  1. 6.

    Combine GNSS and classical network in full 3D by NEQ stacking. The combined NEQ and RHS yield:

$$ {\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}={\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{GNSS}}+{\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{classical}}, $$
(7a)

and

$$ {\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}={\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{GNSS}}+{\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{classical}}. $$
(7b)
  1. 7.

    Define a stable and accurate reference system, by implementing the Controlled Datum Removal (CDR, (Kotsakis and Chatzinikos 2017)). Using CDR, the user selects which of the fundamental datum quantities (origin, scale and orientation) should be externally imposed with minimum constraints. The CDR-related NEQ and RHS now yield:

$$ {\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{CDR}}=\left[\begin{array}{cc}{\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}& {\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}{\mathbf{E}}^{\mathrm{T}}\\ {}\mathbf{E}{\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}& \mathbf{E}{\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}{\mathbf{E}}^{\mathrm{T}}\end{array}\right], $$
(8a)

and

$$ {\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{CDR}}=\left[\begin{array}{c}{\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}\\ {}\mathbf{E}{\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}\end{array}\right], $$
(8b)

where E is a properly selected transformation matrix (with respect to the Helmert parameters), defining the datum parameters that should be externally defined. The \( {\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{CDR}} \) has a rank deficiency which corresponds to the number of the rows of the E matrix. The solution is then achieved by imposing minimum constraints.

  1. 8.

    Convert the coordinates and CV matrix from the topocentric to a geocentric system, respectively.

2.3 Second Alternative Strategy: Transition from a 2D to a Quasi-3D Network Through the Zenith Angles

This scenario is practically a special case of the first one: In order to mitigate—as much as possible—the effect of the low accuracy of the observed zenith angles, we divide the classical network into two components:

  1. a.

    Classical 2D solution where the horizontal part (for E, N components) NEQ and RHS (\( {\mathbf{N}}_{\mathbf{q}}^{\mathrm{hor}},{\mathbf{u}}_{\mathbf{q}}^{\mathrm{hor}} \)) are estimated.

  2. b.

    The spatial distances and the reduced zenith angles used for geometric height differences observations Δh ij (trigonometric levelling, Rossikopoulos 1999):

$$ \Delta {\mathrm{U}}_{\mathrm{i}\mathrm{j}}=\Delta {\mathrm{h}}_{\mathrm{i}\mathrm{j}}={\uprho}_{\mathrm{i}\mathrm{j}}\cos {\mathrm{z}}_{\mathrm{i}\mathrm{j}}+{\mathrm{s}}_{\mathrm{i}}-{\mathrm{t}}_{\mathrm{j}}, $$
(9)

where s and t are the heights of the instrument and the target, being derived from the record of the classical observations. The height difference observations lead to vertical (Up) NEQs and RHS (\( {\mathbf{N}}_{\mathbf{q}}^{\mathrm{vertical}},{\mathbf{u}}_{\mathbf{q}}^{\mathrm{vertical}} \)). The geometric height differences (on an ellipsoid) correspond to the Up-component differences of the topocentric system (Vanicek and Krakiwsky 1986, p. 334).

Continuing from 2.1

  1. 5.

    Stack NEQs and RHS. The stacking (GNSS-derived and classical one) is realized separately for horizontal and vertical NEQs and RHS parts of the contributing networks, forming finally a consistent 3D NEQ and RHS for the topocentric coordinates. The combined NEQ and RHS are formulated as follows:

$$ \begin{array}{ll}\displaystyle {\left({\mathbf{N}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}&\displaystyle ={\left[\begin{array}{cc}{\mathbf{N}}_{\mathbf{q}}^{\mathrm{hor}}& {\mathbf{N}}_{\mathbf{q}}^{\mathrm{hor},\mathrm{vertical}}\\ {}{\left({\mathbf{N}}_{\mathbf{q}}^{\mathrm{hor},\mathrm{vertical}}\right)}^{\mathrm{T}}& {\mathbf{N}}_{\mathbf{q}}^{\mathrm{vertical}}\end{array}\right]}^{\mathrm{GNSS}}\\[18pt]&\displaystyle \quad +{\left[\begin{array}{cc}{\mathbf{N}}_{\mathbf{q}}^{\mathrm{hor}}& \\ {}& {\mathbf{N}}_{\mathbf{q}}^{\mathrm{vertical}}\end{array}\right]}^{\mathrm{classical}}, \end{array}$$
(10a)

and

$$ {\left({\mathbf{u}}_{\mathbf{q}}^{\mathbf{3D}}\right)}^{\mathrm{combined}}={\left[\begin{array}{c}{\mathbf{u}}_{\mathbf{q}}^{\mathrm{hor}}\\ {}{\mathbf{u}}_{\mathbf{q}}^{\mathrm{vertical}}\end{array}\right]}^{\mathrm{GNSS}}+{\left[\begin{array}{c}{\mathbf{u}}_{\mathbf{q}}^{\mathrm{hor}}\\ {}{\mathbf{u}}_{\mathbf{q}}^{\mathrm{vertical}}\end{array}\right]}^{\mathrm{classical}}. $$
(10b)
  1. 6.

    Apply CDR and solve the NEQ system. The procedure is identical to the 7th step of the first alternative strategy.

  2. 7.

    Identical to Step 8 of the first alternative strategy.

The second alternative strategy brings two sets of different NEQs and RHS which artificially “de-correlate” the horizontal and the vertical parts. This approach serves two primary objectives (a) ensuring the 3D nature of the final network and (b) mitigating the impact of the relatively larger errors of the zenith angle measurements on the 3D result.

We may also underline that, even though our present study is dealing with GNSS networks, the aforementioned strategies can be easily applied for other space techniques (VLBI, SLR and DORIS), since the core of the method is the use of the SINEX format. This can be useful for some applications such as local ties.

3 Case Study

The alternate strategies are implemented over two geodetic networks, located in Drama Prefecture in Greece (Fig. 1). In 1998 Drama’s Municipal Enterprise of Water Supply and Sewerage (DEYAD) established a classical 2D geodetic network, occupied with spatial distances (accuracy: 0.5 mm + 5 ppm), directions and zenith angles (measurement accuracy for the angular quantities: 1 mgon). In total, eight stations have been occupied, with two of them being part of the National Triangulation Network (NTN-state’s benchmarks). The classical network was aligned to the Hellenic Geodetic Reference System of 1987 (HGRS 1987; Veis 1996).

Fig. 1
figure 1

The location of the networks. The red polygon defines the study area in Drama (Kallifitos village) (from [Open Street Map], licensed under (CC BY-SA 2.0))

Full size image

In 2014, a GNSS campaign was conducted for the needs of DEYAD. In total, nine sites were occupied with static GNSS observation (for at least 2 h). Four of them belong to the NTN. The GNSS network was aligned to ITRF2008, epoch 2014.35. The solution of the GNSS solution is expressed in SINEX file format. Figure 2 visualizes the two described networks.

Fig. 2
figure 2

The classical and the GNSS networks established in Drama. Each symbol corresponds to different type of network (GNSS and classical one). T1 and T3 are the common stations of the two networks

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As shown in Fig. 2, only two benchmarks are common between the two networks, both located only in the middle of the study area. In fact, the geometry of the networks does not support either the implementation of the HLMT methodology (only two common sites, not enclosing the area) or CA (the 3D network solution is expressed in SINEX format, providing coordinates and their associated CV matrix. The sites M0, M1 and M4 are now covered with dense canopy (probably was not the case back in 1998) and there is no mutual visibility between the 3D and 2D network sites, thus there is no way to connect them with classical observations. To proceed with the Alternative Strategies application, the DoV values for the area need to be estimated (see Eq. 5, ibid). Since we do not have any information from any local agency regarding the DoVs, we employ those calculated from the XGM2019e model (Zingerle et al. 2020) complete to degree and order 2190. The CDR was applied for all datum parameters (origin, scale, and orientation). The rank deficiency (after the application of CDR) was compensated by the application of minimum constraints to the set of the four NTN stations. Table 1 shows the results of (a) Each individual 3D and 2D networks solution and (b) after the implementation of the Alternative Strategies.

Table 1 The results of the individual solutions and the Alternative Strategies [Units: cm]
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Next, we proceed with an additional quality check criterion. We compare the results of the combined solution (for both alternative strategies) with the individual ones (as they solved solely). We estimate the following discrepancies, pointwise:

$$ {\fontsize{8pt}{10pt}\selectfont\begin{array}{l}\updelta {\mathrm{s}}_{\mathrm{i}}=\sqrt{{\left({\mathrm{X}}_{\mathrm{i}}^{\mathrm{GNSS}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{comb}}\right)}^2+{\left({\mathrm{Y}}_{\mathrm{i}}^{\mathrm{GNSS}}-{\mathrm{Y}}_{\mathrm{i}}^{\mathrm{comb}}\right)}^2+{\left({\mathrm{Z}}_{\mathrm{i}}^{\mathrm{GNSS}}-{\mathrm{Z}}_{\mathrm{i}}^{\mathrm{comb}}\right)}^2}\end{array}} $$
(11)

for the GNSS network (points labelled with S according to Fig. 2).

$$ \updelta {\mathrm{d}}_{\mathrm{i}}=\sqrt{{\left({\mathrm{E}}_{\mathrm{i}}^{\mathrm{classical}}-{\mathrm{E}}_{\mathrm{i}}^{\mathrm{comb}}\right)}^2+{\left({\mathrm{N}}_{\mathrm{i}}^{\mathrm{classical}}-{\mathrm{N}}_{\mathrm{i}}^{\mathrm{comb}}\right)}^2} $$
(12)

for the classical network (points labelled with M with according to Fig. 2). For this, test, the classical network was solved with respect to the ITRF2008 (topocentric coordinates, see Sect. 2.1 ibid.). Table 2 refers to the GNSS network comparisons, while Table 3 to the classical network, respectively.

Table 2 Statistics of the discrepancies between the individual GNSS and the combined solutions, respectively [Units: cm]
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Table 3 Statistic of the discrepancies between the individual classical and the combined solutions, respectively [Units: cm]
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The solutions from the two alternative strategies yield some notable findings. First, the second alternative strategy (NEQs are separated into horizontal and vertical part and combined), performs better than the first alternative strategy (1.01 vs 1.85 cm mean spherical error, respectively). It seems that the discarding the zenith angles uncertainty leads to large errors. On the other hand, the first alternative strategy gives worse results as the zenith angles play indeed a significant role, contaminating the achieved accuracy for both the horizontal and vertical components. Even though the accuracy is slightly worse in the second alternative strategy after the combination, compared to the individual 3D network solution (1.01 cm compared to 0.67 cm), the results are still suitable for surveying applications and the existing infrastructure could directly refer to the combined solution aligned to a modern global TRF.

Furthermore, Tables 2 and 3 show that (a) the combined solutions perform at good level of consistency with the individual ones (better than 1 cm for the mean discrepancies for both GNSS and classical networks) and (b) the second alternative strategy provides better results compared to the first one (as mean average and as maximum values, respectively). These findings confirm that for the tested network the alternative strategies can be beneficial towards the alignment of a classical network to an accurate global TRF.

4 Conclusions

The suggested strategies for the optimal combination between 3D and 2D networks can stand as alternative scenarios for the cases were the Common Adjustment (CA) and the Helmert Transformation (HLMT) methodologies encounter challenges. A usual problem is the poor geometry of the combined networks (few common stations, not well-designed observations, gaps). Furthermore, the alternative strategies can be easily applied under the existence of a SINEX format file since there is no need of special treatment of the observations.

The second alternative strategy (separating the horizontal and the vertical part of the NEQ) emerges as superior option compared to the first one. Finally, the alternative strategies can be applied in combination cases involving space geodetic techniques (VLBI, SLR, and DORIS) following the same conceptual manner. This can be useful for, e.g., the co-location sites which contribute to the inter-system ITRF construction.