**6.1 General information**

It is worth using software simulators of navigation measurements for development of model systems designed for registry of effects of various natural factors on spacecraft (SC) GNSS traffic and for improvement of navigation measurement information processing technology. Such simulators help to solve the task of parameter definition of a SC GNSS traffic mathematical model, as well as the task of some nuisance parameter part definition. They also serve to determine metrological estimations of effects of these parameters on navigation measurements results. In order to solve these tasks, a software simulator should fulfil the following functions:


Fulfilment of the first item is a theoretical task related to application of the equation of SC traffic in the Earth's gravitational field for the purpose of determination of the selected SC coordinates expected values at preset time points. The second and the third items set parameters describing conditions and an observing station. The fourth item solves tasks of nuisance factors simulation on the basis of the navigation measurements equation:

$$S\_{ui} = D\_{2i} \left( X\_{\text{NA}} \left( t\_{2i} \right), X\_{\text{MII}} \left( t\_{3i} \right) \right) + \delta \mathcal{D}\_{\text{B49}} + \delta \mathcal{D}\_{\text{IP}} + \delta \mathcal{D}\_{\text{MOH}} + \delta \mathcal{D}\_{\text{F}} + + \delta \mathcal{D}\_{\text{LQM}} + \delta \mathcal{D}\_{\text{AMI}} + \delta \mathcal{D}\_{\text{PET}} + \delta \mathcal{D}\_{\text{SI}'} \tag{14}$$

where *X t КА* <sup>2</sup>*i* are coordinates of a SC position as of the time of radiation of signal *<sup>i</sup> t*<sup>2</sup> ;

*X t ИП* <sup>3</sup>*i* are meter vector coordinates as of the time of receiving signal *<sup>i</sup> t*<sup>3</sup> ;

*DX t X t* 22 3 *<sup>i</sup> КА <sup>i</sup>* , *ИП <sup>i</sup>* is the geometric value of a range to the SC;

*DБФЗ* is the correction for phase delays of vehicle-borne equipment;

*DTP* is the correction for tropospheric refraction;

*DИОН* is the correction for ionospheric refraction;

*DF* is the correction for discrepancy of phases and frequencies of SC generators and the meter;

*DЦМ* is the correction for shift of the antenna phase centre with reference to SC centre of mass;

*DАИП* is the correction for shift of the meter antenna phase centre;

*DРЕЛ* is the relativistic correction;

*DSI* is the correction for an instrumentation error.

Selection of a particular mathematical model for imitation of natural factors when you calculate the right side of the equation (14) is determined by requirements to precision of necessary parameters definition. In case of navigation non-requesting measurements the following parameters should be defined: orbital parameters; clock parameters and models

second. Only the VLBI technology can fulfil such requirements at long distances 2000 km or

It is worth using software simulators of navigation measurements for development of model systems designed for registry of effects of various natural factors on spacecraft (SC) GNSS traffic and for improvement of navigation measurement information processing technology. Such simulators help to solve the task of parameter definition of a SC GNSS traffic mathematical model, as well as the task of some nuisance parameter part definition. They also serve to determine metrological estimations of effects of these parameters on navigation measurements results. In order to solve these tasks, a software simulator should fulfil the

1. computation of motion of a navigation satellites orbit group under conditions that

2. building of non-requesting measuring stations (NMS) network and observed SC

Fulfilment of the first item is a theoretical task related to application of the equation of SC traffic in the Earth's gravitational field for the purpose of determination of the selected SC coordinates expected values at preset time points. The second and the third items set parameters describing conditions and an observing station. The fourth item solves tasks of

more (Finkelshtein, А. (2007).

**6.1 General information** 

following functions:

constellation;

*DБФЗ* 

*DTP* 

*DF* 

meter; 

mass; *DАИП* 

*DРЕЛ* 

*DSI* 

**6. Simulation of navigation measurements** 

produce an effect of perturbation on satellites;

3. estimation of a geometric range from stations to a SC;

is the correction for tropospheric refraction;

is the relativistic correction;

*DИОН* is the correction for ionospheric refraction;

is the correction for an instrumentation error.

4. imitation of factors that influence precision of trajectory measurements.

 

*X t ИП* <sup>3</sup>*i* are meter vector coordinates as of the time of receiving signal *<sup>i</sup> t*<sup>3</sup> ; *DX t X t* 22 3 *<sup>i</sup> КА <sup>i</sup>* , *ИП <sup>i</sup>* is the geometric value of a range to the SC;

is the correction for phase delays of vehicle-borne equipment;

is the correction for shift of the meter antenna phase centre;

nuisance factors simulation on the basis of the navigation measurements equation:

*S DX t X t D D D D D D D D <sup>и</sup>i i* 22 3 *КА <sup>i</sup>*, *ИП <sup>i</sup> БФЗ TP ИОН <sup>F</sup> ЦМ АИП РЕЛ SI*

where *X t КА* <sup>2</sup>*i* are coordinates of a SC position as of the time of radiation of signal *<sup>i</sup> t*<sup>2</sup> ;

is the correction for discrepancy of phases and frequencies of SC generators and the

*DЦМ* is the correction for shift of the antenna phase centre with reference to SC centre of

Selection of a particular mathematical model for imitation of natural factors when you calculate the right side of the equation (14) is determined by requirements to precision of necessary parameters definition. In case of navigation non-requesting measurements the following parameters should be defined: orbital parameters; clock parameters and models

     

, (14)

    parameters. Orbital parameters include six orbital units, three light pressure scaling factors and empiric speed-up factors, and among the latter ones cyclical factors are the most important. Clock parameters include: current parameters of the clock on board SC and nonrequesting measuring stations (NMS). Models parameters include corrections for: radio waves propagation delays in ionosphere and troposphere; non-uniqueness of phase modifications; position of SC centre of mass with reference to phase centres of SC and NMS antennas, instrumental noise, etc. In terms of this chapter's subject analysis of the task of clock current parameters simulation is of the utmost interest.

#### **6.2 Models of instability of quantum frequency standards**

According to the classical concept of clock rate instability, there are long-term and shortterm components of its deviation from the uniform time scale. For example, in the book by (Tryon, 1983) a system of linear stochastic difference equations, describing the process of atomic time deviation in time domain *S k*( ) and frequency domain *q k*( ) , as well as of frequency drift *W k*( ) is given:

$$\begin{aligned} S(k+1) &= S(k) + q(k)h + 0.5w(k)h^2 + V\_s(k), \\ q(k+1) &= q(k) + w(k)h + V\_q(k), \\ w(k+1) &= w(k) + V\_w(k), \end{aligned} \tag{15}$$

Where ( ) *V k <sup>s</sup>* , ( ) *V k <sup>q</sup>* , ( ) *V k <sup>w</sup>* are centred Gaussian processes by type of white noise with spread characteristics: , , *s <sup>q</sup> <sup>w</sup>* ; *k k* <sup>1</sup> *ht t* discretization interval of processes.

The given stochastic equations include both regular long-term and short-term stochastic components of atomic clock instabilities. They are used in recurrent procedures by Kalman type for estimation of amounts S(t), q(t), w(t) in tasks of time scales formation of group keepers, in synchronization tasks and so on.

In the book by (Oduan & Gino, 2002) a mathematical model of quantum clock is reviewed, which describes the relation of the noise power spectral density at different frequencies by type of:

$$S\_y \left( f \right) = \sum\_{a=-2}^{2} h\_a f^a \,, \tag{16}$$

where *h* are factors that determine power density of some noise components with frequencies *f* .

Depending on integral number value it is assumed that there are 5 types of noise processes. For example, when changes from -2 to +2 with interval equal to 1, formula (16) describes respectively: white phase noise; flicker phase noise; white frequency noise and frequency random walk noise. Mean square two-sample dispersion (Alan variations) is connected to noise power *S f <sup>y</sup>* by the relation (Oduan & Gino, 2002):

$$
\sigma\_y^2(\tau) = \bigcap\_{0}^{\infty} H\_A(f) \Big|^2 \left| H\_f(f) \right|^2 S\_y(f) df \,\,\,\,\,\tag{17}
$$

where <sup>2</sup> <sup>4</sup> <sup>2</sup> sin <sup>2</sup> *f f <sup>A</sup> fH* is a transfer function square modulus of a frequency digital filter in divergent integral (17).

 <sup>2</sup> <sup>1</sup> 0 *h f h для f f H f для f f* is a transfer function square modulus of a low-frequency filter with cutoff frequency *hf* .

Formula for Alan dispersion computation is the result of integration of formula (17) with account of formula (16) and in view of condition: 2 1 *<sup>h</sup> f* ,

$$
\sigma\_y^2 \left( \tau \right) = \frac{3h\_2 f\_h}{4\pi^2 \tau^2} + \frac{h\_1}{4\pi^2 \tau^2} \left[ 1, 04 + 3 \ln \left( 2\pi f\_h \tau \right) \right] + \frac{h\_0}{2\tau} + 2h\_{-1} \ln 2 + \frac{2}{3} \pi^2 h\_{-2} \tau \,. \tag{18}
$$

Factors *h* included into (18) are determined by preset Alan variations' values <sup>2</sup> *y i* in the left side of equation. In order to receive a unique definition of all five factors *h* in the right side of equation (18), it is necessary to set up a system of five equations. Five values of Alan variations are determined by statistical processing of results of atomic clock rate measurements during five intervals *<sup>i</sup>* of different sizes. Normally intervals of 1, 10, 100 seconds, 1 hour and 1 day are chosen. As a result of solution of the system of equations set up in this way, five values of factors *h* are determined. By inserting the determined values *h* into equation (18), we receive a model of an observed clock instability, which is further used to make forecast of rate of scale deviation from a nominal value. However, the given classical model is not applicable to description of instabilities of some types of quantum clock. In particular, in terms of interval characteristics of instabilities typical for rubidium frequency standards, solution of the system of equations composed of (18) includes negative values of some factors *h* , which is theoretically impossible . This model works only in case of clock, for which white frequency noise is predominant. The reason of restrictions on application of the classical model to description of instabilities of some kinds of quantum clock is that it does not determine extent of influence of some noise components on actual characteristics of instabilities of an observed clock.

A model of instability of quantum frequency standards (QFS) offered in researches by (Tissen, Tolstikov, 2004, 2011) enables to simulate by means of a computer a random process of QFS scale deviation from a nominal value for all probability distributions. It enables in its turn to check adequacy of application of a particular QFS physical model to describe deviation of the observed scale from the nominal value. In this context such a model can be regarded as software tool for metrological control of QFS physical models that are already worked out and just being developed. The algorithm development is based on the supposition of clock deviation from the nominal value in the form of a random recurrent process:

$$\mathbf{x}\_{i} = \mathbf{x}\_{i-1} + f\_{0}\mathbf{r} + \delta\mathbf{x}\_{i\prime} \tag{19}$$

where *<sup>i</sup> x* is clock rate variation as of the time point *it* ;

<sup>0</sup>*f* is generator rated frequency;

is discretization interval;

*i x* is a random variable, determining root-mean-square error of clock deviation.

Formula for Alan dispersion computation is the result of integration of formula (17) with

<sup>2</sup> <sup>2</sup> <sup>1</sup> <sup>0</sup> <sup>2</sup>

the left side of equation. In order to receive a unique definition of all five factors *h*

*h f h h*

deviation from the nominal value in the form of a random recurrent process:

 *f* ,

22 22 1 2 <sup>3</sup> <sup>2</sup> 1,04 3ln 2 2 ln 2 4 4 2 3

included into (18) are determined by preset Alan variations' values <sup>2</sup>

 *f* 

right side of equation (18), it is necessary to set up a system of five equations. Five values of Alan variations are determined by statistical processing of results of atomic clock rate

seconds, 1 hour and 1 day are chosen. As a result of solution of the system of equations set

of clock, for which white frequency noise is predominant. The reason of restrictions on application of the classical model to description of instabilities of some kinds of quantum clock is that it does not determine extent of influence of some noise components on actual

A model of instability of quantum frequency standards (QFS) offered in researches by (Tissen, Tolstikov, 2004, 2011) enables to simulate by means of a computer a random process of QFS scale deviation from a nominal value for all probability distributions. It enables in its turn to check adequacy of application of a particular QFS physical model to describe deviation of the observed scale from the nominal value. In this context such a model can be regarded as software tool for metrological control of QFS physical models that are already worked out and just being developed. The algorithm development is based on the supposition of clock

> *i i* 1 0 *<sup>i</sup> xx f x*

*x* is a random variable, determining root-mean-square error of clock deviation.

 into equation (18), we receive a model of an observed clock instability, which is further used to make forecast of rate of scale deviation from a nominal value. However, the given classical model is not applicable to description of instabilities of some types of quantum clock. In particular, in terms of interval characteristics of instabilities typical for rubidium frequency standards, solution of the system of equations composed of (18) includes negative

*<sup>A</sup> fH* is a transfer function square modulus of a frequency digital

is a transfer function square modulus of a low-frequency filter

, which is theoretically impossible . This model works only in case

. (18)

of different sizes. Normally intervals of 1, 10, 100

are determined. By inserting the determined values

, (19)

 *h h*

 *y i* in

> in the

where

0

with cutoff frequency *hf* .

 

values of some factors *h*

*i* 

*для f f H f для f f* 

<sup>2</sup> <sup>1</sup>

*f*

Factors *h*

*h*

filter in divergent integral (17).

<sup>4</sup> <sup>2</sup> sin <sup>2</sup>

<sup>2</sup>

*h*

*h*

*h*

measurements during five intervals *<sup>i</sup>*

up in this way, five values of factors *h*

characteristics of instabilities of an observed clock.

where *<sup>i</sup> x* is clock rate variation as of the time point *it* ; <sup>0</sup>*f* is generator rated frequency;

is discretization interval;

account of formula (16) and in view of condition: 2 1 *<sup>h</sup>*

 

*y h*

*f f* Variable *<sup>i</sup> x* in each interval is calculated according to the formula:

$$\delta \mathfrak{X}\_i = \int\_{t\_{i-1}}^{t\_i} Y\_i dt \tag{20}$$

where 1 *N i j j Yt y t* is relative variation of generator frequency, composed of *<sup>N</sup>*

components of frequency variations <sup>0</sup> 0 *j j f f t <sup>y</sup> <sup>f</sup>* ;

> <sup>0</sup> , *<sup>j</sup> f f t* is an initial value and a current value of generator frequency; *i* is number of time interval; *j* is number of component of frequency variation.

In Figure 2 a block scheme which illustrates method of the described time scale formation algorithm in case 6 *N* is given.

A standard programmable random number generator (RNG) synthesizes 6 random number groups r1, r2,... r6 with preset statistical characteristics of spread in 6 intervals of the following sizes: 1,10,…105 sec. Time scale formation is the result of superposition of six random number groups generated by RNG. Accumulated groups of clock indications are analyzed according to the statistical Alan method to determine statistical characteristics of the time scale being simulated.

Fig. 2. Block scheme of quantum clock instability model
