Introduction

During the era of traditional geodetic methods, dissemination of a unified geodetic coor­dinate system throughout the entire territory of the Russian Federation was one of the priority tasks in the construction of each regional fragment of the state geodetic network [6]. Currently, with the use of satellite methods in geodesy, the national geodetic coordinate system is extended to the entire territory of the country, albeit through the relatively sparse Fundamental Astronomi­cal Geodetic Network (FAGN) [9]. Its subsequent densification is usually carried out fragmenta­rily [5], often taking into account the practical needs of specific regions. The densification the national geodetic reference network with regional fragments results in a separate task that requi­res not only a well-established unification mechanism, but also its mathematical justification in the view of preserving the qualitative characteristics of both the national geodetic reference net­work and the embedded fragments [7, 8, 10]. As the quality of regional network construction improves, the internal accuracy of embedded fragments can significantly exceed the accuracy of reference station coordinates, and failure to take this circumstance into account can introduce distortions into regional fragments. Thus, to achieve proper densification quality, it is necessary to get adequate accuracy estimates for both the reference geodetic networks and the embedding regional fragments.

Despite the intensive development of absolute methods for determining the coordinates of reference network stations, to date, relative methods allow obtaining the best accuracy of the determined coordinates in comparison with other methods of geodetic processing of GNSS measurements. However, an adequate assessment of the accuracy of determining coordinates using purely relative methods is difficult, so this is usually done within the framework of the procedure of geodetic tying to reference stations [14]. With this approach, as already noted, the quality of the embedded regional fragments may decrease due to errors in the reference stations' coordinates [10].

In this paper, an analytical algorithm has been developed that allows, based on a relative solution linking the coordinates of all stations of a certain regional geodetic network, to obtain for these stations sufficiently representative estimates of the accuracy of their absolute positions in a coordinate system that is uniform to the given geodetic network.

Designations

We will assume that a geodetic network is being developed on a certain territory, consis­ting of (n+1) stations, which we will designate with numbers 0, 1, 2, ..., n. The work carried out in this network is aimed at establishing geodetic connections between all these stations.

We will denote by Ι the identity matrix of size 3×3, and by Ο the zero matrix of size 3×3, implying that these matrices will interact mainly with spatial position vectors whose dimension is equal to 3:

By Ι_n we will denote the identity matrix, which contains n matrices Ι on the main diago­nal, J_n is for the block column matrix, consisting of n matrices Ι, and Ο_n means the block column matrix, consisting of n matrices Ο:

In the course of the presentation, when representing matrices element by element, we will adhere to the following rule. If the block size of the matrix is n, then its representation will look like in the previous formulas (1) – two elements at the beginning, then an ellipsis and the last ele­ment of the matrix. If the matrix size is n+1, we will represent it as in the formulas (2) – three elements at the beginning, then an ellipsis and the last element of the matrix:

It is assumed that the matrices of block size n+1 interact with the vectors of absolute po­sitions of stations (i.e. positions of stations relative to a reference point that is not combined with any station of a given network), with indices i = 0, 1, 2, ..., n, and the matrices of block size n in­teract with the vectors of relative positions of stations, with indices 0i = 01, 02, ..., 0n (relative to station number 0), in which the position of the zero station is absent. In addition to being more vi­sual, this rule prevents gross errors when performing cumbersome matrix operations.

We denote the spatial coordinates X-Y-Z of a geodetic station with number i in the refe­rence coordinate system by the vector R_i, and the column matrix of spatial (absolute) positions of all network stations in this coordinate system by Φ:

If one of the stations is designated as a reference point (in this work, this will always be the station with the zero number), then the spatial positions of the remaining stations of the geo­detic network relative to the reference one will be designated by R_0i, and the column matrix of all relative spatial positions – by Ψ:

where, obviously, R_0i = R_i – R₀. The analytical matrix operation implementing the transition from absolute positions of stations to relative ones is performed using the matrix operator Ω and is re­presented by the following expressions

Please note that the element-wise representation of the matrix Ω follows the rule establi­shed above.

If the vector Ψ of relative positions is obtained through mathematical processing of field geodetic measurements, then it will correspond to the covariance matrix C^Ψ of errors in the va­lues of coordinates Ψ, obtained from the same processing. We will call both of these objects to­gether a vector solution. Thus, this term means the values of the coordinates of the stations relative to one of them, designated as the reference, and the covariance matrix of their errors – (Ψ, C^Ψ).

Advantages and disadvantages of vector solutions

Up to the present time, relative coordinate determinations allow achieving the highest ac­curacy of coordinate determination in comparison with other existing methods of processing of geodetic measurements. However, direct practical use of relative coordinates is difficult, therefore they are reduced in one way or another to absolute coordinates in a given reference system, usually by means of geodetic tying to reference stations. In this case, external errors, both random and systematic, introduced by errors in the coordinates of the reference stations are added to the intrinsic errors of the vector solution [11]. Since the internal accuracy of vector solutions can be significantly higher than the accuracy of the reference stations coordinates, this can appreciably distort the results presented by vector solutions.

There is interest in such a transformation of the vector solution from relative positions to absolute ones, which does not use the coordinates of the reference stations. However, the reverse transition in the formula Ψ = ΩΦ is impossible – the transition matrix Ω is not invertible, since it is not square.

However, throughout the use of satellite geodesy, attempts have been made from time to time to overcome this obstacle. For example, one of the trends is to use the procedure of the so-called pseudo-inversion of a non-square matrix [1, 2, 8, 12, 13, 15, 17, 18]. Another approach is to introduce additional conditions-restrictions (minimal and inner constraints) when processing geodetic measurements data and combining both measurement information and additional conditions into a common solution using the geodetic adjustment procedure [1, 16, 17].

Without going into detail, we note that these methods involve the inclusion of additional conditions in the system of coupling equations to overcome the rank defect of the transforming matrix. Obviously, additional conditions can work similarly to errors in reference coordinates (which are one of the forms of external constraint conditions) – introducing additio­nal distortions into the vector solution, both random errors and systematic deformations.

In this paper, we explore a method for solving this problem that does not use additional conditions.

Initial assumptions

In previous works [3, 4], variants of constructing vector solutions based on complete and, in a more general case, incomplete sets of vector measurements were studied. Since in both cases the same session of satellite geodetic measurements is used, the choice of option is determined only by the possibility or impossibility of obtaining a complete set of vector measurements in a given geodetic network. And this possibility depends primarily on the size of this network. If the size is too large, it may turn out that for very long baselines it is impossible to obtain measured vectors, which makes us limit ourselves to incomplete sets of varying degrees of filling.

This work is focused on the needs of regional permanent geodetic networks, which are characterized by sizes ranging from tens to the first hundreds of kilometers. In such conditions, obtaining a complete set of measured vectors from processing satellite measurements in modern conditions does not present any difficulties. Therefore, the main studies were carried out on the vector solution obtained using the full set of measured vectors. However, additional studies were performed on the vector solution obtained on the basis of an incomplete set in order to make an opportunity of assessing the nature and extent of the impact of this circumstance on the main results.

As shown in [3, 4], the covariance matrix of a vector solution constructed using the full set of measured vectors has the following structure:

where d₀, d₁, …, d_n are 3×3 matrices representing the variances of spatial position errors. It is also shown that the covariance matrix of such a structure can be considered as the result of transforming some absolute solution (Φ, C^Φ) to a relative one (Ψ, C^Ψ) by means of the transformation operator Ω declared in (3):

in the case where the covariance matrix of the absolute solution Φ has a diagonal form:

In the presented form, the reconstructed absolute solution (Φ, C^Φ) is ideally suited for servicing practical problems with reference geodetic information, because all the stations deter­mined in this solution equally and independently of each other provide a geodetic basis for the territory covered by this network. However, the initial information (Ψ, C^Ψ), obtained on the basis of real geodetic work in a given territory, does not have such properties, and there is no direct, immediate transition (Ψ, C^Ψ) → (Φ, C^Φ) from a vector solution to an absolute one.

The fundamental difference between these solutions is that in the vector one there is no equality between the stations. One of the stations is considered as a reference point; it repre­sents the origin of the coordinate system for a given vector solution, and thus in this coordinate system its position is considered to be error-free and equal to zero. Even if the entire vector solution is shifted by a given value, this will actually mean only a shift in the origin of the coordinate system used – the reference station will have a given error-free position in the new coordinate system, and the remaining stations will be connected to it by the solution (Ψ, C^Ψ). It is not possible to make the transition to a solution of the form (Φ, C^Φ) in this way. What is needed is a transition that would lead to a solution in which all stations are equal.

One of such transitions can be the displacement of the origin of the used coordinate sys­tem to the center of gravity of the vector solution. In relation to the center of gravity, all stations are equal, since each of them takes an equal part in calculating the position of the center. It is obvious that such an auxiliary coordinate system is not connected in any way with the existing reference coordinate systems and it should be understood as the proper coordinate system of the vector solution [3, 4]. In this work, this particular variant of transition from a relative, vector solution to an absolute one was investigated.

Reducing a vector solution to the center of gravity

To begin, we will form a generalized vector solution, which will include both the original vector solution and the reference point. This is necessary to perform analytical operations to calculate the center of gravity and bring the vector solution to it. The generalized vector of sta­tions' positions will be denoted by Λ, the initial position in it is intended for the reference station under the zero number. Obviously, both the spatial position R₀₀ of this station, and the variances of its errors, and the covariances with the errors of the positions of other stations have zero va­lues:

The spatial position of the center of gravity relative to the reference point is denoted by R_0g; it is calculated in the usual way, as the arithmetic mean of the spatial positions of all stations of the vector network:

Further forming the differences between the positions of all stations of the vector network Λ and the position of the center of gravity R_0g, we obtain the set Υ of spatial positions of these stati­ons relative to the center of gravity:

Solution Υ can be considered as an emulation of the absolute solution Φ – all the stations in solution Υ are also equal, there is no reference station, and an external point is taken as the refe­rence one. The only difference between them is that the solution Υ is presented in an auxiliary reference frame, associated only with the given vector solution. Analytically, the equivalence is expressed in the fact that for the solution Υ, similarly to (3), the equality ΩΥ = Ψ is also satisfied.

Next, we move on to calculating the covariance matrix C^Υ of the errors in the positions of all the stations relative to the center of gravity, which, taking into account (7), in general form is re­presented by the following expression:

The first step, taking into account the block structure of the matrix C^Λ, we also split the covariance matrix C^Υ into blocks of the corresponding sizes:

and, bearing in mind expressions (2, 6), we perform calculations of individual blocks in ge­neral form. Upon completion of these calculations we obtain analytical representations of the blocks:

Next, considering the block structure (1, 4) of the matrices included in these ex­pressions, we substitute their values in a block form and after performing the necessary intermedi­ate transformations we arrive at the following expressions:

where

Finally, combining these blocks into a common matrix C^Υ and decomposing it into terms of different levels of significance, we obtain the final expression:

Thus, we have obtained the covariance matrix of the errors of all the stations’ positions of the original vector network in some external reference system. This coordinate system is not aligned with any station of this geodetic network and has no established connections with refe­rence coordinate systems, i.e. it is of an auxiliary nature. The transformation (Ψ, C^Ψ) → (Υ, C^Υ) of the relative solution to the absolute one is performed without involving additional information or additional constraints, using only the transformation operator of the trivial structure (i.e. the operator formed from unit and zero matrix blocks). Note that in the final expressions (7), (11) for the transformed solution (Υ, C^Υ) there are only those components that are included in the original vector solution. This guarantees the objectivity and representativeness of the results obtained.

Analysis of the result

Expression (11) allows us to identify a number of properties of the covariance matrix C^Υ.

Firstly, in this matrix, as in the solution Υ itself, all stations of the vector network are equal, the one with the zero number does not stand out from the rest.

Secondly, the main influence of the errors of the obtained solution is again described by a diagonal matrix, see the first term in (11), as in the ideal reconstruction (5) of the covariance matrix of the absolute solution C^Φ, which significantly facilitates and simplifies the analysis of the quality of the constructed geodetic network.

The second term in (11) shows that in this solution the errors in the positions of an arbitrary pair of stations are correlated through the errors in the vector measured between the stations of the pair. Due to the reduction factor 1/(n+1), these covariances are small, and if the network contains even a few stations or more, their influence becomes negligible.

The third term reflects the impact on the absolute coordinates of all these stations at once of errors in the position of the center of gravity (and, consequently, the origin of the coordinate system, which in this case will also be a random variable). These errors do not affect the relative positions of network stations.

Thus, it can be stated that, without noticeable distortions, the diagonal elements of the matrix C^Υ, obtained from the initial vector solution (Ψ, C^Ψ) as a result of the transformation ac­cording to formulas (6, 8), can serve as an objective and representative assessment of the stati­ons' positions accuracy in a uniform coordinate system throughout the territory covered by a given geodetic network.

An additional assessment of the quality of the solution can be provided by analyzing the correlations between the errors in the positions of individual stations. As follows from the obtai­ned results, correlations can have a value approximately equal to 2/(n+1). If, however, a specific solution (Υ, C^Υ) contains correlations of a significantly larger magnitude, this may indicate its heterogeneity and, thus, significant heterogeneities in the initial measurement data caused by violations of the field measurement regime at some stations.

From the point of view of the information provided both the original vector solution (Ψ, C^Ψ) and its variant (Υ, C^Υ), transformed to its proper coordinate system, are equally representa­tive. However, it should be borne in mind that the absolute solution, unlike the vector one, has a rank defect, and its covariance matrix C^Υ is degenerate. The magnitude of the rank defect is equal to 3 – the difference between the dimensions of the vector Ψ and absolute Υ solutions.

Additional research

This section briefly describes the steps and presents the results of the research concerning the differences in the variant of constructing a vector solution, which uses an incomplete set of measured vectors. In the article [4] it is shown that a factor depending on the distance between stations comes into play – as the distance increases, the effect of this factor becomes stronger, and errors in relative position can increase. As can be seen from formula (8) of article [4], in this case the covariance matrix of the vector solution is the sum of two terms. The first one is exact­ly equal to the covariance matrix, which is represented in this paper by expression (4), and the second (let us denote it as C^Ψ+) looks like this:

where r_ij is the distance between stations i and j, and the 3×3 matrix Θ declared in the expression (5) of the article [4] characterizes the rate of increase in the variance of errors in the relative position of pairs of network stations with increasing the distance between the stations. Thus, in order to obtain what differences a vector solution of a given type has as a result of the influence of the second term, it is necessary to perform the operations presented in formulas (8, 9, 10) for the additional term (12). After performing the necessary transformations, we obtain a matrix additional to C^Υ (let us denote it as C^Υ+), which must be added to expression (11) to obtain the covariance matrix of a vector solution of this type, reduced to the center of gravity, i.e. transformed into absolute coordi­nates:

This expression uses the notations 

The first term in expression (13), its impact on the quality of the obtained coordinate so­lution were considered in the article [4, pp. 10-11, Fig. 2] using the example of an experimental network of stations, and therefore is not discussed here. The influence of the second term de­pends on the average distance between each station and all other stations in the network - the greater this distance, that is, the further a given station is removed from all the rest in the net­work, the greater the error in its position. The third term makes an equal contribution to the po­sitions of all the network stations and reflects the influence of possible errors in determining the position of the center of gravity. If we write it in a more compact form: –SJ_n+1Θ(J_n+1)^T/(n+1)², it is easily seen from this notation that the influence is proportional to the average distance S/(n+1)² between all the stations in the network – the greater the distances in the network, the more this factor affects.

The degree of the impact of the additional covariance matrix C^Υ+ on the accuracy of the constructed network in comparison with the main matrix C^Υ is difficult to estimate a priori; it is necessary to know the relationship between d_i and Θ. However, the influence of the additional matrix C^Υ+ can be suppressed or at least reduced, as was shown in [4], if in each specific case of constructing a fragment of the reference geodetic network the most complete set of measured vectors is used.

If the first terms in expressions (11) and (13) are designated C⁰ and C⁰⁺, respectively, then their sum C⁰+C⁰⁺ represents the covariance matrix of the reconstructed absolute solution, given in [4] by expressions (9-11). Now forming the sum of expressions (11) and (13), it is easy to verify after performing the necessary block matrix transformations that the covariance matrix C^Υ+C^Υ+ of the general vector solution reduced to the center of gravity can be represented as follows:

This relationship is consistent with the obvious rule that the transformation of a coordina­te solution from an absolute to a relative form does not change the position of its center of gra­vity within the framework of a given coordinate solution. Schematically, this rule can look like this: (Φ, C^Φ) → (Ψ, C^Ψ) → (Υ, C^Υ) = (Φ, C^Φ) → (Υ, C^Υ), i.e. as a result of a two-stage trans­formation, first from an absolute solution to a vector one, and then its reduction to the center of gravity, the same result is obtained with the same qualitative characteristics as in the case of direct reduction of the absolute solution to the center of gravity. Therefore, the obtained relation simply confirms that the previously performed cumbersome matrix transformations were carried out without errors.

Conclusion

An analytical matrix relationship is constructed between the relative, vector solution (Ψ, C^Ψ), connecting all stations of a certain geodetic network, and the equivalent absolute solution (Υ, C^Υ). The relationship is represented by expressions (6, 7, 8). The obtained result implements the method of linear transformation of the relative coordinate solution to the absolute one, pre­sented in some auxiliary coordinate reference system, uniform and universal for all the stations of the defined network. The resulting covariance matrix of errors of the stations’ absolute coordinates is practically diagonal, which provides easy and convenient opportunities for visualization and control of the achieved accuracy of geodetic coordinate determinations in vector solutions con­structed on the basis of relative methods of processing GNSS satellite measurements. For ex­ample, based on the results of applying this method, it becomes possible to perform adequate quality control of the solution, to evaluate:

– whether the coordinate solution as a whole turned out to be of sufficient quality in the view of the requirements established for the problem being solved;

– whether the newly created geodetic network is sufficiently uniform in terms of accuracy, or if it contains any stations whose coordinates are of insufficient quality;

– are there any inflated correlations between the coordinate errors of different stations, which may indicate some violations in the execution or processing of field geodetic measurements.

The developed method can be a convenient tool for assessing the quality of construction of regional geodetic networks using the relative satellite method. Obtaining representative esti­mates based directly on the results of a vector solution or on the results of geodetic tying to reference stations seems much more difficult.