**7.1 General information**

An almost 200 year long era of precise time determination with conventional astronomical methods ended in 2010. New satellite and radio interferometric techniques with about 2-3fold better measurements' resolution have substituted them. However, astrometric observations of stars, moon and planets positions accumulated through over the past centuries are invaluable for an overall picture of the Earth's rotation from the middle of the XVII century to the present. It is impossible to create adequate prediction models of Earth Rotation Parameters (ERP), such as universal time and pole coordinates, without use of these observations over a large enough time periods. A number of applied and fundamental Earth sciences have a need for ERP accurate data and predicted values for different periods. A study of fine structures in Earth rotation speed variations and its poles movements is the most urgent challenges of geodesy, astronomy and geodynamics.

According to a data cited in (Finkelshtein, 2007) the accuracy of ERP in IERS is about 50-60 microseconds of arc for pole coordinates and nutation angles, and about 4-10 microseconds for Universal Time. Achievement of such a high accuracy in comparison to classical methods, where the same estimation was usually made with a tolerance of approximately 1- 2 ms, is now possible due to the comprehensive use of new means and techniques of the Measurement Assurance and Earth Rotation Parameters Prediction (MAERPP). The MAERPP Complex functionality includes the following measurement assurance technologies:


Each of the methods for Earth rotation measurement has its own advantages, disadvantages and specific sources of systematic errors. Therefore, combining ERP series obtained by different methods and their combination in a single solution is an effective way to minimize systematic and random errors in ERP definitions.

#### **7.2 Definition of Earth rotation parameters means VLBI**

The VLBI is the most high-precision and independent ERP determination technique among the ones stated above, because stationary objects of the Universe named quasars are being observed. This method's principle of the geodynamics parameters determination consists in a measurement of the delay time of the same radio signal received by radio telescopes located at a distance from each other. Figure 3 shows the operation principle of the VLBI.

Fig. 3. The measurement principle of the VLBI.

fold better measurements' resolution have substituted them. However, astrometric observations of stars, moon and planets positions accumulated through over the past centuries are invaluable for an overall picture of the Earth's rotation from the middle of the XVII century to the present. It is impossible to create adequate prediction models of Earth Rotation Parameters (ERP), such as universal time and pole coordinates, without use of these observations over a large enough time periods. A number of applied and fundamental Earth sciences have a need for ERP accurate data and predicted values for different periods. A study of fine structures in Earth rotation speed variations and its poles movements is the

According to a data cited in (Finkelshtein, 2007) the accuracy of ERP in IERS is about 50-60 microseconds of arc for pole coordinates and nutation angles, and about 4-10 microseconds for Universal Time. Achievement of such a high accuracy in comparison to classical methods, where the same estimation was usually made with a tolerance of approximately 1- 2 ms, is now possible due to the comprehensive use of new means and techniques of the Measurement Assurance and Earth Rotation Parameters Prediction (MAERPP). The MAERPP Complex functionality includes the following measurement assurance

Each of the methods for Earth rotation measurement has its own advantages, disadvantages and specific sources of systematic errors. Therefore, combining ERP series obtained by different methods and their combination in a single solution is an effective way to minimize

The VLBI is the most high-precision and independent ERP determination technique among the ones stated above, because stationary objects of the Universe named quasars are being observed. This method's principle of the geodynamics parameters determination consists in a measurement of the delay time of the same radio signal received by radio telescopes located at a distance from each other. Figure 3 shows the operation principle of the VLBI.

most urgent challenges of geodesy, astronomy and geodynamics.

 Very Long Baseline Interferometers (VLBI); Global Navigation Satellite Systems (GNSS); Satellite Laser Ranging Systems (SLR).

systematic and random errors in ERP definitions.

Fig. 3. The measurement principle of the VLBI.

**7.2 Definition of Earth rotation parameters means VLBI** 

technologies:

The radio telescopes are installed in the points *А* and *В* at the distance D from each other; the interferometer's bases receive the radiation of the same quasar *К* in the centimetric spectrum synchronously*.* Processing of radio signals recorded allows determination of the time delay of moments of radio wavefronts' arrival at the radio telescope *A* relative to the *B*, as well as interference frequency *f* . The difference between the distances from the radio telescopes to the quasar at the moment of observation *s* , where - radio propagation velocity. As a result of coprocessing of multiple values of and *f* the following parameters can be determined: the difference in geocentric coordinates between the ends of the radio interferometer's base, which is used for determination of the *D* length and *Ө* angle, source's and the Earth's pole coordinates; the Earth's rotation instantaneous velocity; the elements of precession and nutation; the Greenwich apparent time of and *f* determination moments; and other parameters.

#### **7.3 Navigation and laser methods of the ERP determination**

By the navigation and satellite laser ranging methods the ERP are being determined jointly with the spacecrafts' orbits parameters subsequent to the results of aggregate measurements from a large number of observation stations. The parameters, which have to be defined more precisely, may also include parameters of the models of atmosphere, Earth's gravity field, solar radiation, etc. The principles and methods of determination of the parameters named above are described quite full in a number of books (Urmaev, 1981), (Duboshin, 1983). However, the descriptions in literature are mostly general in nature. In order to implement them by means of software an additional work on calculation algorithms compilation has to be done first. The algorithm for joint determination of ERP and spacecrafts' orbits parameters, which can be used as a basis for ERP and spacecrafts' orbits parameters software compilation, is shown below.

If the earth referenced coordinates *XYZ* , , of a station tracking the passages of the spacecraft are known and the preliminary approximate values of the , *X Y P P* pole coordinates and the differences *UT t UT UTC* () 1 are known also, then the station position vector *XYZ* , , at the time t in the middle equatorial coordinate system can be calculated by the following formula:

$$
\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} = R\_{\mathbf{x}} \begin{pmatrix} -\boldsymbol{\varepsilon} \end{pmatrix} R\_{\mathbf{z}} \begin{pmatrix} \boldsymbol{\Delta} \boldsymbol{\Psi} \end{pmatrix} R\_{\mathbf{x}} \begin{pmatrix} \boldsymbol{\varepsilon} + \boldsymbol{\Delta} \boldsymbol{\varepsilon} \end{pmatrix} R\_{\mathbf{z}} \begin{pmatrix} -\boldsymbol{\varepsilon} \\ \boldsymbol{X} \\ \boldsymbol{Y} \\ \bar{\boldsymbol{Y}} \end{pmatrix} \bar{\boldsymbol{Y}} \begin{pmatrix} \bar{\boldsymbol{X}} \\ \bar{\boldsymbol{Y}} \\ \bar{\boldsymbol{Z}} \\ \bar{\boldsymbol{Z}} \end{pmatrix} \tag{21}
$$

where the ERP orientation matrixes are specified as usual:

$$R\_x(a) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos a & \sin a \\ 0 & -\sin a & \cos a \end{pmatrix} \cdot R\_z(a) = \begin{pmatrix} \cos a & \sin a & 0 \\ -\sin a & \cos a & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot R\begin{pmatrix} \bar{-}\_p \\ \bar{X}\_p, \bar{Y}\_p \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 & -\bar{X}\_p \\ 0 & 1 & \bar{Y}\_p \\ \bar{X}\_p & -\bar{Y}\_p & 1 \end{pmatrix} \cdot \begin{pmatrix} \bar{X}\_p \\ \bar{Y}\_p \\ \bar{X}\_p \end{pmatrix}$$

 - is the obliquity of the movable equator plane to the plane of the instantaneous ecliptic; – is the nutation in longitude;


*S* is the Greenwich apparent sidereal time calculated at the moment *t UT t* .

The calculated topocentric distance:

$$
\rho\_c(t) = \sqrt{\left(x - X\right)^2 + \left(y - Y\right)^2 + \left(z - Z\right)^2} \,\,\,\,\tag{22}
$$

where , , *x y z* are the coordinates of the spacecraft in the middle equatorial coordinate system.

Discrepancies *t tt* <sup>0</sup> *<sup>c</sup>* are caused both by pseudorange measurement errors 0() ( ) *rec rad t ct t* and by errors of an accepted computation model for the *<sup>c</sup> t* value, which depends on the spacecraft coordinates *xyz* , , errors and station coordinates errors , , *XYZ* , as it follows from the equation (22).

Assuming that these values are small, the equation for discrepancies can be expanded in a Taylor series with the first term of the series only:

$$
\Delta \rho\_c = A \begin{pmatrix} \Delta \mathbf{x} \\ \Delta \mathbf{y} \\ \Delta z \end{pmatrix} + B \begin{pmatrix} \Delta X \\ \Delta Y \\ \Delta Z \end{pmatrix} \tag{23}
$$

where *A* and *B* vectors from the ground station to a satellite and back are as follows:

$$A = \left(\frac{x - X}{\rho\_c}, \frac{y - Y}{\rho\_c}, \frac{z - Z}{\rho\_c}\right), \ B = -A \ . \tag{24}$$

The transition from the formula (22) to the equation (23) is called linearization.

The spacecraft's coordinates , , *x y z* have a complex dependence on mean orbital elements in Т0 epoch; is the ascending node-perigee angle; *n* is the mean motion;  *is the*  longitude; *i is the* obliquity; *e* is the eccentricity; *M* is the mean anomaly.

The coordinates of the station depend on the position of the pole. , *X Y p p* .

Since the value *UT* belongs to the sidereal time calculation formula *S t UT t* , the following substitute can be made: *UT t S t* in rad.

Linear relations between the coordinates , , *XYZ* and orbital elements of the spacecraft can be represented as a matrix:

$$
\begin{pmatrix}
\Delta x\\ \Delta y\\ \Delta z\\ \Delta z
\end{pmatrix} = \mathbf{C} \begin{pmatrix}
\Delta a\\ \Delta \Omega\\ \Delta i\\ \Delta e\\ \Delta \mathbf{M}\\ \Delta \mathbf{n}
\end{pmatrix},
\tag{25}
$$

2 2 <sup>2</sup>

where , , *x y z* are the coordinates of the spacecraft in the middle equatorial coordinate system.

which depends on the spacecraft coordinates *xyz* , , errors and station coordinates

Assuming that these values are small, the equation for discrepancies can be expanded in a

where *A* and *B* vectors from the ground station to a satellite and back are as follows:

, , *ccc xX zZ y Y <sup>A</sup>* 

The spacecraft's coordinates , , *x y z* have a complex dependence on mean orbital elements

Since the value *UT* belongs to the sidereal time calculation formula *S t UT t* , the

Linear relations between the coordinates , , *XYZ* and orbital elements of the spacecraft can

*i*

 

*e*

*n*

 

*x*

*z*

*y С*

 

is the ascending node-perigee angle; *n* is the mean motion;  *is the* 

 

The transition from the formula (22) to the equation (23) is called linearization.

longitude; *i is the* obliquity; *e* is the eccentricity; *M* is the mean anomaly.

The coordinates of the station depend on the position of the pole. , *X Y p p* .

following substitute can be made: *UT t S t* in rad.

*x X Ay BY z Z*

 

*t xX y Y zZ* , (22)

, (23)

, *B A* . (24)

, (25)

*<sup>c</sup> t* value,

*t tt* <sup>0</sup> *<sup>c</sup>* are caused both by pseudorange measurement errors


*S* is the Greenwich apparent sidereal time calculated at the moment *t UT t* .

*t ct t* and by errors of an accepted computation model for the

*c*

– is the nutation in longitude;

The calculated topocentric distance:

Discrepancies

0() ( ) *rec rad*

in Т0 epoch;

be represented as a matrix:

*c* 

> 

errors , , *XYZ* , as it follows from the equation (22).

Taylor series with the first term of the series only:


where *C* is the sensitivity matrix of the spacecraft coordinates to errors of the orbital elements:

$$\mathbf{C} = \begin{pmatrix} \frac{\partial \mathbf{x}}{\partial \boldsymbol{\alpha}} & \frac{\partial \mathbf{x}}{\partial \boldsymbol{\Omega}} & \frac{\partial \mathbf{x}}{\partial \boldsymbol{i}} & \frac{\partial \mathbf{x}}{\partial \boldsymbol{e}} & \frac{\partial \mathbf{x}}{\partial \mathbf{M}} & \frac{\partial \mathbf{x}}{\partial \boldsymbol{u}} \\ \frac{\partial \mathbf{y}}{\partial \boldsymbol{\alpha}} & \frac{\partial \mathbf{y}}{\partial \boldsymbol{\Omega}} & \frac{\partial \mathbf{y}}{\partial \boldsymbol{i}} & \frac{\partial \mathbf{y}}{\partial \boldsymbol{e}} & \frac{\partial \mathbf{y}}{\partial \boldsymbol{M}} & \frac{\partial \mathbf{y}}{\partial \boldsymbol{u}} \\ \frac{\partial \mathbf{z}}{\partial \boldsymbol{\alpha}} & \frac{\partial \mathbf{z}}{\partial \boldsymbol{\Omega}} & \frac{\partial \mathbf{z}}{\partial \boldsymbol{i}} & \frac{\partial \mathbf{z}}{\partial \boldsymbol{e}} & \frac{\partial \mathbf{z}}{\partial \boldsymbol{\mathbf{M}}} & \frac{\partial \mathbf{z}}{\partial \boldsymbol{u}} \end{pmatrix}. \tag{26}$$

Linear relations for the station coordinates are as follows:

$$
\begin{pmatrix} \Delta X \\ \Delta Y \\ \Delta Z \end{pmatrix} = D \cdot \begin{pmatrix} \Delta x\_p \\ \Delta y\_p \\ \Delta y\_\oplus \end{pmatrix}' \tag{27}
$$

where *D* is the sensitivity matrix of the spacecraft coordinates to ERP errors:

$$D = \begin{pmatrix} \frac{\partial X}{\partial \boldsymbol{\alpha}\_p} & \frac{\partial X}{\partial \boldsymbol{y}\_p} & \frac{\partial X}{\partial S\_{\oplus}} \\ \frac{\partial Y}{\partial \boldsymbol{\alpha}\_p} & \frac{\partial Y}{\partial \boldsymbol{y}\_p} & \frac{\partial Y}{\partial S\_{\oplus}} \\ \frac{\partial Z}{\partial \boldsymbol{\alpha}\_p} & \frac{\partial Z}{\partial \boldsymbol{y}\_p} & \frac{\partial Z}{\partial S\_{\oplus}} \end{pmatrix}. \tag{28}$$

Since the partial derivatives of along the spacecraft's ascending node longitude and the sidereal time *S* are calculated as follows:

$$\frac{\partial \mathbf{x}}{\partial \Omega} = \frac{\partial \mathbf{X}}{\partial \mathbf{S}\_{\oplus}} = -y \; \text{/} \; \frac{\partial y}{\partial \Omega} = \frac{\partial Y}{\partial \mathbf{S}\_{\oplus}} = \mathbf{x} \; \text{/} \; \frac{\partial z}{\partial \Omega} = \frac{\partial Z}{\partial \mathbf{S}\_{\oplus}} = \mathbf{0} \; \text{/} \tag{29}$$

the last columns of the matrixes *C* and *D* are linearly dependent. Hence it is impossible to determinate and *UT* at the same time, and therefore the value *UT t*( ) at small time intervals shall be approximated by the linear function:

 <sup>0</sup> <sup>0</sup> 86400 *DR t t UT t UT t* , (30)

where *DR* (sec.day) is variation length of the day.

Combining the equations (27) and (28) we get:

$$
\begin{pmatrix}
\Delta X\\ \Delta Y\\ \Delta Z
\end{pmatrix} = \begin{pmatrix}
\frac{\partial X}{\partial x\_p} & \frac{\partial X}{\partial y\_p} & \frac{\partial X}{\partial S\_{\oplus \oplus}}\\ \frac{\partial Y}{\partial x\_p} & \frac{\partial Y}{\partial y\_p} & \frac{\partial Y}{\partial S\_{\oplus \oplus}}\\ \frac{\partial Z}{\partial x\_p} & \frac{\partial Z}{\partial y\_p} & \frac{\partial Z}{\partial S\_{\oplus \oplus}}\\ \frac{\partial Z}{\partial x\_p} & \frac{\partial Z}{\partial y\_p} & \frac{\partial Z}{\partial S\_{\oplus \oplus}}
\end{pmatrix} \cdot \begin{pmatrix}
\Delta x\_p\\ \Delta y\_p\\ \Delta S\_{\oplus}
\end{pmatrix} \tag{31}
$$

where *Sn S* ; <sup>4</sup> *n* 0,72921 10 radian in sec - mean motion; 

*S* is the correction for the accepted value *n* .

$$\overline{D} = \begin{pmatrix} \frac{\partial X}{\partial \mathbf{x}\_p} = -z \cos S\_{\oplus \} & \frac{\partial X}{\partial y\_p} = -z \sin S\_{\oplus \} & \frac{\partial X}{\partial S\_{\oplus \}} = -y \left(t - t\_0\right) \\ \frac{\partial Y}{\partial \mathbf{x}\_p} = -z \sin S\_{\oplus \} & \frac{\partial Y}{\partial y\_p} = z \cos S\_{\oplus \} & \frac{\partial Y}{\partial S\_{\oplus \}} = x \left(t - t\_0\right) \\ \frac{\partial Z}{\partial \mathbf{x}\_p} = \overline{X} & \frac{\partial Z}{\partial y\_p} = -\overline{Y} & \frac{\partial Z}{\partial S\_{\oplus \}} = 0 \end{pmatrix}. \tag{32}$$

Then the following formula will be true for calculation of the *DR* value:

$$DR = -\frac{2\pi}{n\_{\oplus}} \cdot \frac{\stackrel{\bullet}{\Delta S\_{\oplus}}}{n\_{\oplus}} \, , \tag{33}$$

In the above expressions for the partial derivatives (32) 0*t* is a fixed time, at which the ERP is determined.

Combining the equations (23), (25) and (31) we get the following conditional equation of the least-squares method:

$$\begin{aligned} \rho\_0 \left( t \right) - \rho\_c \left( t \right) = A \cdot \mathbf{C} \begin{pmatrix} \Delta \alpha \\ \Delta \Omega \\ \Delta i \\ \Delta e \\ \Delta \Theta \\ \Delta \mathbf{M} \\ \Delta \pi \end{pmatrix} + B \cdot \bar{D} \begin{pmatrix} \Delta x\_p \\ \Delta x\_p \\ \Delta y\_p \\ \Delta S\_{\oplus} \end{pmatrix}, \end{aligned} \tag{34}$$

in the matrixes *А*,,, *BC D* all the partial derivatives are known, since they are calculated on the accepted initial values of mean orbital elements at epoch *T*<sup>0</sup> : , ,, , , *ieMn* , Earth Rotation Parameters *x y UT t p p* , , and coordinates of observatories *XYZ* , , .

Corrections , , , , , , , , *p p ieMnx y S* are calculated on a number of the spacecraft's constellation observations from several stations at different points of time.

N conditional equations can be worked out on the aggregate data obtained. Discrepancies in the conditional equations can be minimized by the least-squares method by working out the M normal equations corresponding to the number of the parameters to be determined:

$$\sum\_{i=1}^{N} \frac{\partial \rho\_0(t)}{\partial P\_k} \Delta \rho(t) = \sum\_{j=1}^{M} \left[ \sum\_{i=1}^{N} \frac{\partial \rho\_c(t)}{\partial P\_k} \frac{\partial \rho\_c(t)}{\partial P\_j} \right] \Delta P\_j \tag{35}$$

where *k M* 1,... .

cos sin

*x y <sup>S</sup>*

sin cos

*YYY D z S z S xt t x y <sup>S</sup>*

*ZZ Z X Y x y <sup>S</sup>*

> <sup>2</sup> *<sup>S</sup> DR n n*

In the above expressions for the partial derivatives (32) 0*t* is a fixed time, at which the ERP

Combining the equations (23), (25) and (31) we get the following conditional equation of the

\_

*c p*

*n*

 

in the matrixes *А*,,, *BC D* all the partial derivatives are known, since they are calculated on

N conditional equations can be worked out on the aggregate data obtained. Discrepancies in the conditional equations can be minimized by the least-squares method by working out the M normal equations corresponding to the number of the parameters to be determined:

<sup>0</sup>

 

*P P P*

*t t t*

*ieMnx y S* are calculated on a number of the

*t P*

(35)

 

*i t t AC BD y <sup>e</sup>*

0

 

the accepted initial values of mean orbital elements at epoch *T*<sup>0</sup> :

Rotation Parameters *x y UT t p p* , , and coordinates of observatories *XYZ* , , .

1 1 1

spacecraft's constellation observations from several stations at different points of time.

*<sup>N</sup> M N c c*

*i j k k i j*

Corrections , , , , , , , , *p p*

where *k M* 1,... .

radian in sec - mean motion;

*XXX z S z S yt t*

0

 

*p*

*x*

*S*

  0

, (33)

0

. (32)

, (34)

, ,, , , *ieMn* , Earth

*j*

where *Sn S*

*S* 

is determined.

least-squares method:

; <sup>4</sup> *n* 0,72921 10

is the correction for the accepted value *n* .

*p p*

*p p*

*p p*

Then the following formula will be true for calculation of the *DR* value:

Calculated from equation (35) corrections for *Pj* add a parameter *Pj* to the initial values and the process is repeated again until the value of discrepancies get below a specified value .

The algorithm of the differential method for joint correction of the ERP series and spacecraft's orbits parameters is applicable both to navigation and to laser ranging measuring instruments. The main difference between the Satellite Laser Ranging (SLR) method and the GNSS technology is that SLR uses not a radio range of wave length, but an optical spectrum for measurements. This reduces atmospheric effects, eliminates uncertainty of the multiline radio signal propagation and provides a potentially higher resolution of measurements. In addition, the laser ranging of satellites allows calculating a change rate of RLS positions coordinates in the global velocity field. Precision geodetic systems are created on the basis of SLR location points, which monitor the geodetic satellites. WGS-84 is the most famous of these, which consists of dozens of points, located on the Earth's surface. The coordinates of these points are determined constantly, that allows to register tectonic processes. Information processing is performed by specialized centres, where it goes through operational communication channels, for example, the Internet. It implements the main advantage of the laser ranging in comparison to the radio positioning i.e. the possibility to determine a displacement of SLR points on the surface and in height during the geophysical researches. Such networks have a zero accuracy grade according to the geodetic classification. They are a basis for development of other levels of engineering networks integrated into the global system WGS-84. However, despite these advantages the SLR means are less influential in the ERP determination compared to the GNSS and VLBI techniques, so they inferior to them in mass and immediacy of the observational data supply.

In order to compare the quality of the different methods for ERP determination let us present data obtained from the bulletin of the Russian main metrology centre State service of time, frequency and IERS's bulletin. Table 2 shows the systematic deviations by the ERP determination from the IERS data, obtained in 2010 in the leading Russian processing centres according to the: Glonass/GPS; Satellite Laser and VLBI within the network of points of the domestic and world stations (Bulletin E-141-144, 2010).


Table 2. Systematic deviations by the ERP determination using different techniques in 2010 according to the Russian processing centres.

The following abbreviations are used in the table 2:

SSTF - State service of time, frequency and the Earth rotation parameters determination; IAA - Institute for applied astronomy of the Russian Academy of Sciences (RAS);

IAC - Information-analytical centre TSNIIMASH.

Table 3 shows root-mean-square errors (RMS) of the ERP determinations obtained in 2010 by IERS according to the result of GPS, VLBI and SLR measurements processing in the leading processing centres (Bulletin B 285, 2011). Since there are more than 30 processing centres in the IERS Bulletins, we will state only the RMS values spread limits for ERP determinations in these centres.


Table 3. Estimated accuracies of the ERP determinations by using different techniques in 2010 according to the IERS data.

Analyzing the data in the Tables 2 and 3 we can conclude that the VLBI technique is the most accurate for the universal time parameter definition and is inferior to the GNSS in accuracy of the pole coordinates determination. The SLR method shows the results with a similar accuracy of the pole coordinates determinations in comparison with VLBI and is also inferior to the GNSS technique. However, the SLR and GNSS techniques have similar results by determination of the length of the day. It should also be noted that the data obtained from the Russian processing centres (see Table 2.) conform to an international standard for ERP determination accuracy level.
