Development of Recurrent Method with Rotation for Combined Adjustment of Terrestrial Geodetic and GNSS Networks in National Spatial Reference System Development of Recurrent Method with Rotation for Combined Adjustment of Terrestrial Geodetic and GNSS Networks in National Spatial Reference System

A construction of national spatial reference systems (NSRS) is promoted in many countries due to modern achievements of Global Navigation Satellite System (GNSS) methods and results of building of high accurate geoid/quasi-geoid models at centimeter level of accuracy. One of the most popular methods used for the construction of the NSRS is related to Helmert block adjustment method, by which we ought to solve techno-scientific task of a separate adjustment of GNSS network in International Terrestrial Reference Frame (ITRF) and next combination of a results of adjustment of the terrestrial geodetic and GNSS networks in the NSRS. In this chapter, we carry out a research on the usage of a recurrent adjustment method with Givens rotation for solving the abovementioned task on an account of its advantages of being effective for application of a technique of sparse matrix, outlier detection and very simple for solving the subsystem of observation equations, created based on the transformation of the results of the separate adjustment of the GNSS network from the ITRF into the NSRS. The experiment results of solving the abovementioned task for the GPS network in the North Vietnam had shown that the horizontal and vertical position accuracy of the GPS points in VN2000 – 3D had reached the few centimeter level.


Introduction
In the past, in different countries, national horizontal and vertical reference systems had been constructed independently from each other; in addition, horizontal control points almost did not coincide with vertical control points. A national first-and second-order astro-geodetic network constructed by traditional geodetic methods did not allow to obtain horizontal positioning accuracy of the horizontal control points at the centimeter level. Because of an accumulation of measurement errors in the national horizontal control network, a coordinate transmission from one origin point led to the more horizontal positioning error of the distant horizontal control points. For example, the horizontal position accuracy in NAD83 (1986) reached the level of 1 m [4,34]. Such analogical situation had also been happened to the vertical control network. For the national first-and second-order astro-geodetic networks of the former Soviet Union in SK95, the maximal RMS of horizontal position of horizontal control points reached the level of 1.5 m [9].
Nowadays, traditional geodetic methods cannot satisfy the accuracy requirements of the national horizontal and vertical reference systems at the centimeter level according to modern technoscientific achievements. The abovementioned accuracy requirements only will be satisfied by the construction of the NSRS based on modern achievements of the GNSS methods, the construction of the highly accurate national geoid/quasi-geoid model and the geopotential vertical datum.
Present-day worldwide and rapid development of GNSS methods, especially the construction of Continuously Operating Reference Station (CORS) networks of GNSS base stations and mathematical processing of GNSS data in the ITRF with usage of International GNSS Service (IGS) products, and construction of national hybrid geoid/quasi-geoid models with an accuracy at the level of few centimeters had created favorable conditions for building of the NSRS in many countries, for example, ETRS89/DREF91/2016 (Germany), GDA2020 (Australia) [14], NSRS2022 (USA CONUS, Canada, Caribbean Islands. Hawaii and Greenland) [35], and so on.
In case of processing the GNSS data in the ITRF, highly accurate spatial coordinates of geodetic points will be converted from the ITRF to the NSRS by the seven parameter Bursa-Wolf formula. Next, we symbolize For geodetic purposes, the NSRS contains an ellipsoidal surface used as the reference surface for the determination of an ellipsoidal coordinate system and a national plane coordinate system. A Geoid/quasigeoid surface is used for the reference surface of the national vertical reference system. In addition, the national geoid/quasigeoid model creates relationship of the geoid/quasi-geoid surface to the ellipsoidal surface and satisfies the connection of the spatial coordinates of geodetic points with the national vertical reference system.
In practice of the construction of GNSS network by the static relative positioning technique, the components ΔZ ΔY, ΔX, of baseline vector between two GNSS points obtained from the processing of GNSS observations have been used as measured values in the GNSS network. Using IGS products for processing GNSS observations in the ITRF, the components ΔZ ΔY, ΔX, of the baseline vectors have very high accuracy and have been used for the adjustment of the GNSS network. In [19], formulas for apriori assessment of relative horizontal position accuracy xy M between two GNSS points and accuracy of ellipsoidal height H m had been proposed in the following forms: .b, . From formulas (1) and (2), we see that for the GNSS network, constructed by the static relative positioning technique, using IGS products for processing of GNSS observations in the ITRF enables very high relative horizontal position accuracy between two any GNSS points and the very high accuracy of ellipsoidal heights. The highly accurate GNSS network can be used for maintenance and improvement of the accuracy of the national horizontal and vertical reference systems. The construction of the NSRS will satisfy the abovementioned demands. For the construction of the NSRS, we can solve for the three of the following main techno-scientific tasks: • Construction of the passive GNSS network, covering whole national territory; • Construction of the national geoid/quasigeoid model with the accuracy at centimeter level; • Combined adjustment of the terrestrial geodetic and passive GNSS networks in the NSRS.
With the purpose of the maintenance and the improvement of the accuracy of the national horizontal and vertical reference systems, apart from some of the CORS stations, the passive GNSS network still consists of horizontal and vertical control points which are called as ground control points and have been selected by the following criteria [1,7]: • Their location must satisfy requirements of a good satellite geometry and a sky visibility.
• Quick and easy access to them.
• Selected points may be located on geologically stable positions.
Highly accurate ellipsoidal heights at the vertical control benchmarks which are derived from the processing of co-located GNSS observations in the ITRF, especially for the countries at the low and mid-latitudes, enable determining highly accurate geoid/quasi-geoid heights at those benchmarks. Those are very important data source for the determination of GNSS-leveling geoid/quasigeoid heights used for the improvement of accuracy of the national gravimetric geoid/quasi-geoid models.
Over the last decade, countries in Europe, South America, Canada, the United States of America, and so on had developed the geoid-based vertical reference systems (geopotential datum) [36,37,40]. An initial surface of the geopotential datum is the geoid surface with the geopotential . 0 W With usage of the geopotential datum, we have determined geopotentials of the vertical control benchmarks that will be used for the construction of the geopotential field model on the national territory or in a region. In addition to this, highly accurate ellipsoidal heights determined by the GNSS methods at the vertical control benchmarks allow calculating anomalous geopotentials of those control benchmarks which are additional data source for making more precision of spherical harmonic coefficients of the Earth Gravitational Model [21].
In [25], it is shown that in the NSRS, the relative accuracy of spatial coordinates may reach the level of 10 -9 . Based on this criterion, in [20], it had been proved that the accuracy of the national geoid/quasi-geoid model can be improved to a level of higher than AE4 cm. At present, the national geoid/quasigeoid models in many countries, for example, AUSGeoid09 (Australia), USGG2012 (USA), CGG2013 (Canada), OSGM15 (UK), GCG2016 (Germany), and so on, have the accuracy higher than the abovementioned limitation, which guarantees to obtain orthometric/ normal heights of points of interest with accuracy at the centimeter level based on the highly accurate national geoid/quasi-geoid model and results of GNSS data processing in the ITRF.
With the purpose of the maintenance and the improvement of accuracy of the national horizontal and vertical reference systems, in this chapter, we research on methods used for combined adjustment of terrestrial geodetic and passive GNSS networks, especially on a recurrent method with rotation for a combined adjustment of terrestrial geodetic and GNSS networks in the NSRS.
Although we use the terminology "combined adjustment of terrestrial geodetic and GNSS networks in the NSRS," the terrestrial geodetic network comprising horizontal and vertical control networks had been adjusted previously. Therefore, in this chapter, we understand this terminology as "combination of the results of separate adjustment of terrestrial geodetic and GNSS networks in the NSRS." 2. Methodology 2.1. Methods for combined adjustment of terrestrial geodetic and passive GNSS networks A terrestrial geodetic network contains the national horizontal and vertical control networks that had been adjusted separately in the national horizontal and vertical reference systems. For the ground control points selected from the national horizontal and vertical control points and used for the construction of the passive GNSS network, their national ellipsoidal coordinates play very important role in solving the task of the combined adjustment of the terrestrial geodetic and the passive GNSS networks. The accuracy improvement of the national ellipsoidal coordinates (or corresponding spatial coordinates) of the abovementioned ground control points in the NSRS is the purpose of solving of the abovementioned task.
In common case, it is assumed that the national reference ellipsoid and the global reference ellipsoid are different form each other. For the national horizontal control points from results of processing of colocated GNSS observations in the ITRF according to the global reference ellipsoid, we will create relationship between the global geodetic latitudes, longitudes and the national geodetic latitudes, longitudes of these points by the Molodensky formula. This allows to obtain the national geodetic latitude, longitude of the vertical control benchmarks on which GNSS observations had been performed.
The orthometric/normal height of the national horizontal control points can be obtained by precise spirit (geometric) leveling or using a national geopotential field model with determined geopotential 0 W of the national geoid. The first national geopotential field model in Vietnam had been declared in [23].
Such national ellipsoidal heights of the ground control points fully can be derived based on the highly accurate national geoid/quasigeoid model and the GNSS method. By such ways, we will obtain the national ellipsoidal coordinates of the ground control points, which will then be used for the calculation of approximate spatial coordinates Z Y, X, of these points in the NSRS.
In geodetic practice have been created two different directions related to development of methods for the combined adjustment of the terrestrial geodetic and GNSS networks. In the first direction, the components ΔZ ΔY, ΔX, of baseline vectors in the GNSS network have been used as pseudo-observations for the combined adjustment with different terrestrial observations on the national reference ellipsoid and for them in observation equations unknown parameters are ellipsoidal coordinate corrections and coordinate transformation parameters Δm , ε , ε ε Z Y , X [26]. In case the seven coordinate transformation parameters by Bursa-Wolf formula are known, the components Z Y X , , of baseline vectors will be transformed from the ITRF to the NSRS. After that, those components ΔZ ΔY, ΔX, of baseline vectors can be transformed to Δh, α, s, where α s, -is length and azimuth of the geodesic; Δhis the difference of ellipsoidal heights. The values Δh α, s, will be used as pseudo-observations for the combined adjustment with various terrestrial observations on the national reference ellipsoid [26,27].
The second direction is related to the development of methods for the combined adjustment of the terrestrial geodetic and GNSS networks based on the Helmert block method by principle: a separate adjustment of the terrestrial geodetic and GNSS networks and their next combination. The separate adjustment of the passive GNSS network will be performed with two following purposes: • Outlier detection and their removal (if they exist) in the passive GNSS network.
• Determination of highly accurate spatial coordinates of the GNSS points in the ITRF.
We will continue the research of the second direction in the following contents of this chapter.
It is assumed that the passive GNSS network consists of NP points, in which np common points (np ≤ NP) are the ground control points. In addition, these points have the approximate spatial coordinates in the NSRS presented in the form of the national spatial coordinate vector: with variance-covariance matrix , 3 -order of matrix; τ μ -RMS of the unit weight determined apriori.
Without the loss in generality, we arrange ground control points in the first orders. After the separate adjustment of the passive GNSS network in the ITRF, we obtain the adjusted spatial coordinate vector of the NP GNSS points in following form: with variance-covariance matrix , .R μ K 1 S

S S
where S is the RMS of the unit weight and S R is the normal matrix of the order K obtained from the process of the separate adjustment of the passive GNSS network. In addition, the order K = 3.NP; 1 S is a subvector of the spatial coordinates of the np ground control points in the ITRF; 2 S is a subvector of the spatial coordinates of the remaining (NPnp) GNSS points in the ITRF.
In common case, for the GNSS points, the spatial coordinates Z , Y , X in the ITRF are related to the spatial coordinates in the NSRS by Bursa-Wolf formula in the following form: as seven coordinate transformation parameters from the ITRF to the NSRS; τ as vector of the adjusted spatial coordinates of the ground control points in the NSRS, which will be obtained after the combined adjustment of the terrestrial geodetic and passive GNSS networks and has the following form: .
where vector τ is represented in form (3); δτis vector of spatial coordinate corrections.

δS S S
as vector of the adjusted spatial coordinates of the GNSS points in the ITRF obtained after the combined adjustment of the terrestrial geodetic and passive GNSS networks.
In addition, vector S and vector of spatial coordinate corrections have following forms with respect to vector S represented in form (4): With above presented notations, for the np ground control points from formula (5) yields where block matrix G with dimension NP Â 7 has form: When the seven coordinate transformation parameters of Bursa-Wolf formula are unknown, the mathematical model of the combined adjustment of the terrestrial geodetic and GNSS network had been proposed in Ref. [31] in the following form: System of observation equations (10) has K + k + 7 unknown parameters, in which there are K + k spatial coordinate corrections.
In case the approximate values of the seven coordinate transformation parameters of Bursa-Wolf formula had been determined, we fully can convert vector S (4) from the ITRF to the NSRS and get a vector of the transformed spatial coordinates θ of the all GNSS points in the NSRS in the following form: where the subvector 1 θ corresponds to the np ground control points; subvector 2 θ refers to the remaining (NPnp) GNSS points.
In this case, a difference between the vector τ (3) and the subvector 1 θ in (11)   We will carry out a research on the method of the combined adjustment of the terrestrial geodetic and passive GNSS networks in the NSRS proposed in [16]. In this method, the subvector 2 θ in the form of (11) will be used for the subvector of approximate spatial coordinate of the remaining (NPnp) GNSS points in the NSRS. Then taking into account vector τ (3), the vector of the approximate spatial coordinate τ of the all GNSS points in the NSRS has the following form: Vector of spatial coordinate correction τ δˆand vector of last spatial coordinate τ δ τ τˆa re represented in the following forms: additionally sub-block matrix i E is an unit matrix of the order of 3x3 (i = 1,2,..,NP), taking into account the formulas (11), (12), (13), (14), (15). We obtain the system of observation equations in the following form: Finally, we obtain the mathematical model of the combined adjustment of the terrestrial geodetic and passive GNSS networks in the NSRS in the following form [16,20]: where the vector of spatial coordinate corrections τ δˆhas the form (13).
It should be underlined that at present we can determine the seven coordinate transformation parameters of Bursa-Wolf formula System of observation equations (17) has all K + 3 unknown parameters. A study of the method of Givens rotation for solving this system of observation equations is performed in Subsection 2.4.

Brief description of the method of recurrent adjustment of geodetic network with Givens rotation
To obtain the best linear unbiased estimate of unknown parameters by the least squares method, we must adopt an outlier detection method for geodetic observations in geodetic networks. In [29], a method of recurrent adjustment of geodetic networks had been developed, which allows for the detection of outliers in the calculation process and is realized by the following procedure: A recurrent adjustment process is performed sequentially for every measured value in combination with outlier detection method for redundant measurements.
Because the method of recurrent adjustment is working with an inverse matrix Q related to a normal matrix R by the formula , 1 R Q this method is called as "Qrecurrent algorithm." First, we will investigate the method of recurrent adjustment of geodetic networks containing n independent measurements and k unknown parameters. For the ith measured value After performing the Taylor linear expansion, we obtain the observation equation of the ith measurement i y in the following form: according to weight

PV] T [V Φ
To start the recurrent adjustment process, we obtain: where the number m is equal to 6 andd kxk E is the identity matrix of the order of k Â k.
It is assumed that after performing the recurrent adjustment process for (i À 1) first measured values, we have obtained the inverse matrix The recurrent adjustment process for the ith measurement i y with the observation equation (18) will be performed by the following way: where the vector The number i g is an inverse weight of the free component i l and is calculated by the formula: The ith measurement i y will be recognized as the redundant measurement, if number i g satisfies the condition i i p 100 g [30]. When i y is the redundant measurement, the outlier detection will be performed based on the comparison of the free component i l with its limitation , i l then we have base to accept an assumption that in the first i measured values outliers exist.
In the case of the absence of any outliers in the geodetic network, after accomplishment of the recurrent adjustment process for n measurements, the vector of adjusted parameters X and the RMS error of weight unit μ after adjustment of the geodetic network have been calculated by the following formulas: Although the recurrent algorithm Q has the ability to detect outliers in recurrent adjustment process, the inverse matrix Q is a full matrix that leads to a decrease in the efficiency of the adjustment of a large geodetic network. The method of Givens rotation becomes efficient in case of using a sparse matrix technique [12]. In [13], the usage of Givens rotation method had been proposed for the adjustment of large geodetic networks. The method of Givens rotation allows the transformation of the elements of the coefficients matrix nxk A in the system of observation equations to the elements of an upper triangular matrix kxk T related to the normal matrix R by the formula . T T R T On an account of abilities of the method of recurrent adjustment for outlier detection in recurrent adjustment process and the method of Givens rotation for using the technique of a sparse matrix, in [15], a method of recurrent adjustment with rotation that had been constructed based on the method of Givens rotation had been proposed by using the technique of sparse matrix and has been performed in the procedure of recurrent adjustment process with outlier detection. This method is called as "Trecurrent algorithm" with an initial matrix of 0 T of the recurrent adjustment process represented in the following form: , kxk m 0 where the number m is equal to 6; kxk Eis identity matrix of order k; k is number of unknown parameters.
It is necessary to underline that for the method of Givens rotation, a transformation of every element of the row vector of coefficients i a in the observation equation (18)  In this chapter, we carry out a research on the usage of T -recurrent algorithm for the recurrent adjustment of geodetic networks containing n independent values of measurements. We symbolize Y as the vector of transformed free components related to the vector of corrections δX by the system of equations. .

Y X T. ð24Þ
For starting the recurrent adjustment process, we get the initial matrix 0 T form (23), initial vector of transformed free components 0 Y 0 and initial value .
It is assumed that after performing the recurrent adjustment process for the first (i À 1) values of measurements, we have obtained an upper triangular matrix

PV] T [V Φ
For sequential insertion of the ith measured value i y with the observation equation (18) in the recurrent adjustment process, we will create auxiliary matrix (0) B with dimensions (k + 1) Â (k + 1) in the following form ( [15], [18]): We symbolize ) ,.. we build a rotation matrix j H in underrepresented form (26). The elements j C aand j S of the rotation matrix j H are calculated by the following formulas: The elements j C and j S of the rotation matrix j H are located on the jth and ) 1 (k th rows as well as on the jth and ) 1 (k th columns as represented in form (26).

Multiplying matrix
which has the following form: When .
With the purpose of the outlier detection for the ith measurement , i y which is a redundant measured value, we will calculate a vector i t from the system .
The free component i l (19) and its inverse weight i g (20) are calculated by the following formulas [15,18]: The outlier detection will then be performed by a way, analogous to the Qrecurrent algorithm. After the accomplishment of the recurrent adjustment process for the n measured values, the vector of corrections is calculated from the system of equations (24). The vector of adjusted parameters X and the RMS error of weight unit μ after the adjustment of the geodetic network are then calculated by formulas (21), (22).
The correctness of the form (28), obtained from Givens rotation, can be checked by the following way [15,18]. It is assumed that for the first (i-1) measured values in a geodetic network, we have a system of observation equations in the following form: with a weight matrix   After the Cholesky decomposition, the system of normal equations (30) has been transformed into a system of equivalent equations: , where 1 i Y is the vector of transformed free components,

.T T R
From formula (31), we can obtain the following value: On an account of the formulas (30) and (32) from (29), we will obtain a following value: Now after insertion of the ith measured value i y with the observation equation (18) in the adjustment process, we will obtain some known relations: . , By an analogous way to formula (33), we have On an account of the relation ,  (33) and (35) will be inferred the following value: Because the rotation matrix j H (26) is the orthogonal matrix that satisfies the condition is the unit matrix of the order of (k + 1) Â (k + 1), from formula (27), we obtain the following relationship: Substituting (0) B (25) and B (28) into (37), we obtain the known formulas (34) and (36). That proved the correctness of the form (28), obtained from Givens rotation after the insertion of the ith measured value i y with the observation equation (18) in the recurrent adjustment process.
In the case outliers exist in the geodetic network, we will determine the corrections vector (0) V for n measurements that will be used for finding outliers. A method for finding outliers is investigated in Subsection 2.3.

Method for finding outliers in the geodetic network
In case the dispersion of measurements has not been derived confidently and has been changed in whole measurement process, i. e. 0 ≤ < ∞, errors of measurements obey a Laplace distribution [32]. In this case apart from random errors, errors of measurements still consist of gross errors and as the maximum likelihood estimate, the least absolute residuals (LAR) estimate will be established under the following L 1 -norm condition: The LAR method is more efficient in estimating the parameters of the regression model; in the case, the data are contaminated with gross errors. The LAR method has the ability of resisting against blunders (outliers) [39]. Accounting for the popularity of the calculation schema by the least squares method, in [11] had been proposed an iteratively reweighted least squares (IRLS) method, through which condition (38) is represented in the form: In [5], a convergence of the iterative calculation process by the IRLS method and a diminution of amplitude of absolute residuals after every iteration under the condition had been proven (39). The experiments show that the IRLS method allows outliers to be found reliably only for such dense geodetic networks with large number of redundant measurements such as traditional triangulation, the GNSS network and the vertical network created by leveling lines between nodal benchmarks [18].
First, we symbolize m as the number of iterations ,...). 2 , 1 , 0 (m As presented in Subsection 2.2, after adjusting the geodetic network by the T-recurrent algorithm with the discovery of existence of outliers in the geodetic network, we have calculated the vector (0) V of corrections to n measurements that will be used for the iterative adjustment of the geodetic network by the IRLS method in order to find outliers. In the m th iteration, based on the condition (39) for the ith measurement i y the observation equation (18) will be expressed in the following form: with weight  A process of the iterative recurrent adjustment of the geodetic network will be ended, if in two ( m -1)th and m th adjacent iterations for all residuals satisfy the following condition: where ε is a small positive number. The outliers can be found from the measured values which have the largest residuals (corrections).

Application of the recurrent adjustment method with Givens rotation for separate adjustment of GNSS network in the ITRF and next its combination to the NSRS
where i V is a vector of corrections (residuals) to the measured values A weight matrix i P of the order 3 is assigned to the vector of pseudo-observations i Y (41) and represented in form: where 0 μ is the RMS of unit weight determined apriori.
As we had seen in Subsection 2.2, with the purpose of outlier detection, the recurrent adjustment method is effectively realized for independent observations. The components are the dependent observations. Therefore, for the application of the recurrent adjustment method, we must transform the dependent observations By such a way, the system of observation equations (44) has a unit weight matrix , 3x3 i

E P
where 3x3 E is the unit matrix of the order of 3 x 3. In [20], an algorithm for a transformation of a matrix
Before the separate adjustment of the GNSS network, we ought to choose one GNSS point to be "a fixed point" that has spatial coordinates in both the ITRF and the NSRS. Without losing generality, this fixed point is numbered with the number sign 1. Based on a method of a temporary fixation of an initial point, proposed in [18], an inverse weight matrix F Q of the spatial coordinates of the fixed point in the ITRF is accepted to be , 3x3 2m .E 10 that is. where number m is equal to 6, 3x3 E -unit matrix of the order of 3 Â 3.
The choice of a fixed point guarantees the nonsingularity of normal matrix obtained in a process of the adjustment of the GNSS network. Below, we will prove that after the combined adjustment of terrestrial geodetic and GNSS networks, the temporary fixation of the initial point will be automatically eliminated.
To start the separate adjustment of the GNSS network in the ITRF, on an account of formula (45), we obtain an initial upper triangular matrix 0 ) S (T of the order of K for the recurrent adjustment process in the following form: . For the end of this subsection, we prove that performing the separate adjustment of the GNSS network in the ITRF, the temporary fixation of an initial point by assigning the inverse matrix F Q (45) to the spatial coordinates of the fixed point will be automatically eliminated after the combined adjustment of the terrestrial geodetic and GNSS networks.
It is assumed that for all N baseline vectors in the GNSS network, a system consisting of 3.N observation equations has been created in the following form: with weight matrix P. If in the GNSS network there is not any fixed point, that is, the GNSS network becomes the free network, then the normal matrix S R will be singular due to the rank defect d = 3. In this case, the matrix of coefficients A with dimension 3.N Â K has the rank defect d = 3 and satisfies the condition: where matrix has the form (16) with K = 3.NP rows and 3 columns. As mentioned in Subsection 2.1, the normal matrix S R (50) is used as the weight matrix S P assigned to the second subsystem of observation equations in (17).
On an account of (49), the product .

.Ω R S
When we get relationship from (50): On an account of the formulas (16), (50), (51), (52), (53) we obtain: Finally, substituting (56) into (55), we obtain the following system of normal equations: in which the effect of the temporary fixation of an initial point, made in the process of the separate adjustment of the GNSS network in the ITRF, fully has been eliminated.
It can be concluded that the usage of the method of the temporary fixation of initial point for the strict separate adjustment of the GNSS network in the ITRF and avoiding the singularity of the normal matrix S R does not cause any influence on the results of the combined adjustment of the terrestrial geodetic and GNSS networks in the NSRS. Moreover, this method allows the spatial coordinates of the initial point be corrected after the abovementioned combined adjustment. We will lose valuable priori information regarding the spatial coordinates of the initial point of the GNSS network for the accuracy improvement of the national spatial coordinates of GNSS points in the NSRS, if the spatial coordinates of the abovementioned initial point of the GNSS network are considered to be nonerroneous.

Data
In [22], the results of the construction of the initial national spatial reference system VN2000-3D on the base of the orientation of the WGS84 ellipsoid to best fit it to the Hon Dau local quasigeoid at tide gauge Hon Dau with using the most stable 164 colocated GPS observations performed at the first-and second-order benchmarks had been presented. The In [24], the results of the construction of the initial national quasigeoid model VIGAC2017 with the accuracy level of AE5.8 cm had been presented.  Figure 1). Average distance between GPS points is 105 km. The GPS data had been processed in the ITRF2008 by the software Bernese v. 5.2 using IGS service products.
The GPS network has five common (ground control) points C052, C022, C045, C033, C004, that have the approximate national spatial coordinates in VN2000-3D (see Table 1) and have been numbered sequentially from 1 to 5. In Vietnam, horizontal coordinates of geodetic points are determined in VN2000-2D, and their normal heights are determined in national the vertical reference system Haiphong1972 (HP72). On an account of the national quasigeoid model VIGAC2017, the RMS of the national ellipsoidal coordinates of the geodetic points had been These upper triangular matrices will be used for creating the submatrix 15 15x T in the initial upper triangular matrix in the form (36) with the purpose of the combined adjustment of the GPS network, shown in Figure 1, into VN2000-3D.

Results
In [28], the experiments of the combined adjustment of the GPS network, shown in Figure 1, in VN2000-3D had been accomplished. The GPS network had been adjusted separately in the ITRF2008 by the T-recurrent algorithm with the temporary fixation of an initial point for GPS point C052. The adjusted spatial coordinates of all 11 GPS points had been transformed from the ITRF2008 to VN2000-3D (see Table 2).
The last spatial coordinates of all 11 GPS points in VN2000-3D obtained after the combined adjustment of the GPS network in VN2000-3D based on insertion of the system of observation equations in the recurrent adjustment process by the T-recurrent algorithm are shown in Table 3.   Table 2. Spatial coordinates of all 11 GPS points had been transformed from the ITRF2008 to VN2000-3D.

No
Common (ground control) points Approximate spatial coordinates in VN2000-3D The mean values of the RMS of national ellipsoidal coordinates of GPS points after solving the task of the combined adjustments of the GPS network in VN2000-3D are equal to

Conclusions
A tendency of construction of the NSRS strongly is promoted in many countries in the world due to development of the passive GNSS networks, comprising the ground control points and some CORS stations, based on the GNSS methods and results of building of the highly accurate national geoid/quasigeoid models at the centimeter level of accuracy thanks to detailed gravimetric data and the Earth gravitational models with high resolution.
From demands of usage of the high accurate spatial coordinates of GNSS points in the ITRF for different geodetic applications and next their usage for the construction of the national spatial reference frame has been arisen techno-scientific task of the separate adjustment of the passive GNSS network in the ITRF and next its combined adjustment with the terrestrial geodetic network in the NSRS.
In this chapter, a recurrent adjustment method with Givens rotation had been represented for solving the above mentioned task on an account of its abilities to use the technique of sparse matrix, to detect outliers in the recurrent adjustment process and to find them, especially to use effectively results of the separate adjustment of the passive GNSS network in the ITRF for  Table 3. Final spatial coordinates of all 11 GPS points in VN2000-3D after the combined adjustment of the GPS network.
creating the system of observation equations (46) and its realization in the process of the combined adjustment of the passive GNSS network with the terrestrial geodetic network in the NSRS.
In this chapter, the method of the temporary fixation of an initial point used for the separate adjustment of the passive GNSS network in the ITRF had been represented. The abovementioned temporary fixation of an initial point allows not only to perform the strict adjustment of the passive GNSS network in the ITRF and to avoid the singularity of transformed matrix but also to correct the spatial coordinates of fixed point after the combined adjustment of the GNSS network in the NSRS. Additionally, the temporary fixation of the initial point does not cause any influence to the results of the above represented combined adjustment.
The results of experiments performed on the basis of the usage of the T-recurrent algorithm for the separate adjustment of the GPS network in the North Vietnam and the its combined adjustment into VN2000-3 D confirmed the significant improvement of positional accuracy of the GPS points in VN2000-3 D and effectivity of the T-recurrent algorithm in mathematical processing of the GPS network for the construction of the national spatial reference frame. Apart from that, after the combined adjustment of the GPS network in VN2000 ./.