Approximate national spatial coordinates of the ground control points C052, C022, C045, C033, C004 in VN2000–3D.

## Abstract

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.

### Keywords

- method of recurrent adjustment
- combined adjustment of terrestrial geodetic and GNSS networks
- recurrent adjustment method with rotation
- method of Givens rotation
- national spatial reference system

## 1. 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 techno-scientific 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

where

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.

The passive GNSS points may have a 20–100 km density [1, 2, 6, 7, 8]. The passive GNSS networks have been built in many countries, for example High Accuracy Reference Network (USA) [33], Passive Control Network (Canada) [3, 38], Auscope GNSS Network (Australia), and so on.

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

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 ±4 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

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

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

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

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

In common case, for the GNSS points, the spatial coordinates

Now we symbolize

where vector

With above presented notations, for the np ground control points from formula (5) yields

where block matrix **G** with dimension NP × 7 has form:

additionally sub-block matrix

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:

where the third condition equation in the abovementioned model is inferred from the relation (9) accounting for formulas (6), (7), (8); vector of misclosures

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

where the subvector

In this case, a difference between the vector

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

Vector of spatial coordinate correction

With the purpose of decrease in influence of the errors in the vector of approximate seven coordinate transformation parameters

where

From the relation

where the block matrix

additionally sub-block matrix

where the vector of free components has the 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

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.

### 2.2. 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

First, we will investigate the method of recurrent adjustment of geodetic networks containing n independent measurements and k unknown parameters. For the *i*th measured value*i*th measured value

After performing the Taylor linear expansion, we obtain the observation equation of the *i*th measurement

according to weight

For the every *i*th measured value

To start the recurrent adjustment process, we obtain:

the initial inverse matrix

initial vector of corrections

initial value

where the number m is equal to 6 and

It is assumed that after performing the recurrent adjustment process for (i − 1) first measured values, we have obtained the inverse matrix*i*th measurement

where the vector

the free component

The number

The *i*th measurement

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

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

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 “T – recurrent algorithm” with an initial matrix of

where the number m is equal to 6;

It is necessary to underline that for the method of Givens rotation, a transformation of every element of the row vector of coefficients

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

For starting the recurrent adjustment process, we get the initial matrix

For sequential insertion of the *i*th measured value

We symbolize*j*th row of matrix*j*th row

where*j*th element of row*j*th element of the

The elements*j*th and*j*th and

Multiplying matrix

By such a way after the accomplishment of rotation transformation of all k rows of the matrix

which has the following form:

When*i*th measurement

The outlier detection will then be performed by a way, analogous to the Q – recurrent algorithm. After the accomplishment of the recurrent adjustment process for the n measured values, the vector of corrections

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

Solving the system of observation equations (29) by the least squares method, we obtain the system of normal equations:

where

After the Cholesky decomposition, the system of normal equations (30) has been transformed into a system of equivalent equations:

where

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 *i*th measured value

By an analogous way to formula (33), we have

On an account of the relation

Because the rotation matrix

Substituting*i*th measured value

In the case outliers exist in the geodetic network, we will determine the corrections vector

### 2.3. Method for finding outliers in the geodetic network

In case the dispersion_{1} - norm condition:

where*i*th measurement*i*th measurement and i = 1,2,…,n.

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:

where weight

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*i*th measurement

with weight*i*th measurement*i*th measurement

The observation equation (40) will be sequentially inserted in the recurrent adjustment process by the T- recurrent algorithm. After the accomplishment of

A process of the iterative recurrent adjustment of the geodetic network will be ended, if in two (

where

### 2.4. 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

For the GNSS network comprising NP GNSS points, the components

with variance-covariance matrix

where

additionally**(−1 0 0 … 1 0 0),****(0 –1 0 … 0 1 0),****(0 0 –1… 0 0 1)**;*i*th vector of pseudo-observations

where**s** and **h**.

A weight matrix

where

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

where

By such a way, the system of observation equations (44) has a unit weight matrix**C** of length 6. After the performance of operations sequentially by the below represented procedure:

we will obtain corresponding elements of the upper triangular 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

where number m is equal to 6,

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

The recurrent adjustment process will be realized by the T- recurrent algorithm sequentially for every observation equation from the system of observation equations (44). The outlier detection will be performed if the *i*th vector of pseudo-observations

After the accomplishment of the separate adjustment of the GNSS network with the insertion of all N vectors of pseudo-observations in the form (41) in the recurrent adjustment process by the T-recurrent algorithm, if outliers are encountered in the network, we will perform outlier detection using the method represented in Subsection 2.3.

If the GNSS network does not contain outliers, the obtained upper triangular matrix

where

The usage of the T-recurrent algorithm for solving the system of observation equations (17) has the remarkable advantage of being very simple for solving the subsystem of observation equations (46), created based on the transformation of the results of the separate adjustment of the GNSS network from the ITRF into the NSRS.

The subsystem of observation equations (46) has a unit weight matrix

where an upper triangular matrix

The task of the combined adjustment of the terrestrial geodetic and GNSS networks in the NSRS will be performed by the T-recurrent algorithm based on a sequential insertion of observation equations from the subsystem of observation equations (46) in the recurrent adjustment process with the usage of the initial matrix

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

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

Solving the system of observation equations (48) under condition

where matrix

For the strict separate adjustment of the GNSS network in the ITRF and avoiding the singularity of the normal matrix

where the matrix

As mentioned in Subsection 2.1, the normal matrix

On an account of (49), the product

Therefrom we infer the equality:

Now performing the combined adjustment of the terrestrial geodetic and GNSS networks in the NSRS with solving the system of observation equations (17) under the condition

Additionally, the matrix

For the system of normal equations (54), substituting

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

## 3. Experimental results

### 3.1. 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 GPS data had been processed in the ITRF2008 in the period 2009–2010. The coordinate transformation parameters from the ITRF to the VN2000–3D have the following values:

In [24], the results of the construction of the initial national quasigeoid model VIGAC2017 with the accuracy level of ±5.8 cm had been presented.

From 11 to 14 November 2013, Vietnam Institute of Geodesy and Cartography (VIGAC) had accomplished four sessions of 24 h GPS observations at 11 points of the GPS network in the North Vietnam (see 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 considered equal to

No | Common (ground control) points | Approximate spatial coordinates in VN2000–3D | ||
---|---|---|---|---|

X_{τ}^{(0)} (m) | Y_{τ}^{(0)} (m) | Z_{τ}^{(0)} (m) | ||

1 | C052 | −1513714.136 | 5735121.344 | 2337092.916 |

2 | C022 | −1472179.244 | 5771490.833 | 2274632.893 |

3 | C045 | −1538604.244 | 5750184.813 | 2283824.080 |

4 | C033 | −1439254.798 | 5758082.515 | 2328258.441 |

5 | C004 | −1355466.287 | 5762595.502 | 2367026.391 |

On the basis of the algorithm of transformation of the variance-covariance matrix to the upper triangular matrix, represented in Subsection 2.4, we had got the upper triangular matrices for five common points in the NSRS in the following forms:

These upper triangular matrices will be used for creating the submatrix

### 3.2. 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.

No | Points | |||
---|---|---|---|---|

1 | C052 | −1513714.150 | 5735121.372 | 2337092.873 |

2 | C022 | −1472179.207 | 5771490.916 | 2274632.850 |

3 | C045 | −1538604.253 | 5750184.910 | 2283824.046 |

4 | C033 | −1439254.784 | 5758082.567 | 2328258.392 |

5 | C004 | −1355466.267 | 5762595.567 | 2367026.370 |

6 | C049 | −1473387.532 | 5720475.185 | 2397685.386 |

7 | C065 | −1576881.025 | 5710639.642 | 2355075.670 |

8 | C056 | −1592783.012 | 5745126.934 | 2259055.888 |

9 | C014 | −1564014.818 | 5782717.991 | 2183130.973 |

10 | C075 | −1723353.458 | 5702825.780 | 2270214.971 |

11 | C070 | −1710135.062 | 5667162.086 | 2367393.020 |

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

## 4. 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 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 VN2000–3 D, the horizontal and vertical position accuracy of the GPS points had reached the few centimeter level. The mean values of the RMS of national ellipsoidal coordinates of GPS points after solving task of the combined adjustments of the GPS network in VN2000–3D are equal to

## Acknowledgments

The author is thankful to InTech Open for invitation and helps to write this chapter in book project “Positioning Accuracy of GNSS methods”.