# Error Modeling of Fiber Optic Gyroscope Universal Time Measurement
## 2 Analysis Model for FOG Measurements of Earth's Rotation
### Measurement Principle and Modeling of FOG
Additionally, the physical effects affecting the earth's rotation, and hence the observed Sagnac frequency, vary depending on locations [24]. Variations in the rate of rotation occur when the FOG is subjected to local tilt or when the earth's axis of rotation changes direction with respect to the earth. To complete the list of influencing factors, localized tilt effects, gravitational effects of celestial bodies and physical effects within the earth must be considered for inclusion. Celestial influences include polar motion, precession and nutation, according to the equinox-based transformation relationship [25] between the International Terrestrial Reference System (ITRS) and the Geocentric Celestial Reference System (GCRS) in Figure 3, which can be transformed into the ITRS coordinate system by the matrix transformation relationship of polar motion, precession and nutation. Figure 3: Celestial to terrestrial coordinate transformations. Here, P03 refers to the P03 precession model. The four precession angles are as follows. \(\varepsilon_{a}\) is the obliquity of the ecliptic at J2000.0, \(\psi_{a}\) and \(\omega_{a}\) are the precession quantities in longitude and obliquity, respectively, referred to as the ecliptic of epoch, \(\chi_{a}\) represents the precession of the ecliptic along the equator, \(\Delta\psi_{2000a}\) is nutation in longitude, and \(\Delta\varepsilon_{2000a}\) is the change in obliquity due to nutation. By accounting for the combined effects of the multiple factors influencing the angular velocity measured by the FOG, the observation equation for the Sagnac effect in the geodesic coordinate system O-XYZ can be obtained as:  
\[ \begin{array}{c}\Delta\phi=K\Omega_{0}\left[\begin{array}{c}\cos\varphi_{0 }\cos\lambda_{0}\\ \cos\varphi_{0}\sin\lambda_{0}\\ \sin\varphi_{0}\end{array}\right]\cdot\left[\begin{array}{c}m_{1}\\ m_{2}\\ 1+m_{3}\end{array}\right]\\ +K\left[\begin{array}{c}\cos\varphi_{0}\cos\lambda_{0}\cos\theta_{A}+\cos \varphi_{0}\sin\lambda_{0}\sin\theta_{A}\\ \cos\varphi_{0}\sin\lambda_{0}\cos\theta_{A}-\cos\varphi_{0}\cos\lambda_{0} \sin\theta_{A}\\ \sin\varphi_{0}\end{array}\right]\cdot\left[\begin{array}{c}\Omega_{1}\\ \Omega_{2}\\ \Omega_{3}\end{array}\right]\\ +K\Omega_{0}(dT_{ns}\cos\varphi_{0}-d\varphi\cos\varphi_{0})\ +\Delta\phi_{instr} \end{array}  \tag{3}\]  
where \(\Omega_{0}\) is the mean rotational angular velocity of the earth, \(m_{1},m_{2}\) denote the change in polar motion, \(m_{3}\) denotes instantaneous angular velocity variations, \(dT_{ns}\) is the tilt of the normal in the north-south direction, \(d\varphi\) is the change in latitude of the station, \(\Delta\phi_{instr}\) is the error due to instrumental defects, and \(\theta_{A}\)[25] is the apparent sidereal time:  
\[ \begin{array}{c}\theta_{A}=\theta_{E}+(0.014506^{\prime\prime}+4612.1573996 6^{\prime\prime}T_{TT}\\ +1.39667721^{\prime}T_{TT}^{2}-0.00009344^{\prime\prime}T_{TT}^{3}\\ +0.00001882^{\prime\prime}T_{TT}^{4})+\Delta\psi\cdot\cos\varepsilon\end{array}  \tag{4}\]  
where \(\theta_{E}=2\pi\times(0.779057+1.002737 dJ|D_{UT1})\) is the angle of rotation of the earth, and \(dJ|D_{UT1}=JD_{UT1}-JD_{00}\) is the Julian day interval from the Julian epoch (January 2000 1.5) reference calendar element to the moment of the \(JD_{UT1}\) calendar element of the Julian day. The right end of the polynomial in Equation (4) includes the projection of the total precession of the ecliptic on the celestial equator, \(\Delta\psi\cdot\cos\varepsilon\) represents the projection of the ecliptic nutation on the celestial equator, i.e., the equinoctial nutation. \(T_{TT}=JD_{UT1}-JD_{00}/36525\) is the number of Julian centuries between the Julian epoch (January 2000 1.5) reference calendar element (\(JD_{00}=2451545.0\)) and the Julian day \(JD_{TT}\) calendar element moment. Continuous monitoring of the earth's rotational variations by large optical gyroscope currently complements the VLBI technique in geodesy.