# Error Modeling of Fiber Optic Gyroscope Universal Time Measurement
## 3 UT1 and the Instantaneous Angular Velocity of the Earth's Rotation  
UT1 refers to the axial rotation angle (normally expressed in time units) of the earth relative to the conventional celestial (inertial) reference frame. Due to the inhomogeneity of the earth's rotation velocity, the UT1 parameters vary irregularly from day to day and cannot be predicted accurately. As a result, UT1 must be measured in real time. In practice, the UT1 value is expressed in the form of a corrected value (UT1-UTC). This value describes the difference between uniform time, provided by atomic clocks contributing to the coordinated universal time (UTC) timescale, and the variable earth rotation angle, which is related to UT1 [34]. It is common to use \(\Delta\)LOD (length of day, LOD) to reflect the change in the angular rate of rotation \(\Delta\Omega\) of the earth during a day. Since UT1-UTC and \(\Delta\)LOD are first derivative relations of time, the following relationships exist:  
\[\Delta\mathrm{LOD}=-\frac{\Delta\Omega}{\Omega_{0}}\times 86400\mathrm{s} \tag{12}\]  
\[\Delta\mathrm{LOD}=\frac{d(\mathrm{UT1-UTC})}{dt} \tag{13}\]  
\[(\mathrm{UT1-UTC})(t)=K_{k}*\int_{t_{0}}^{t}\Delta\Omega(t)dt \tag{14}\]  
Based on the observation equations model outlined in Equation (3) for FOG measurements, the error suppression is achieved through the application of the appropriate model and instrumentation. A corresponding FOG measurement of the angular velocity variation can then be established and expressed as:  
\[ \begin{array}{c}\Delta\Omega=\frac{1}{(1+K_{a})}\frac{\Omega^{\prime}}{\sin \varphi_{0}}-\left(\frac{K_{pl}\Omega_{bias}}{\sin\varphi_{0}}-\frac{\Omega_{0 }\cos\varphi_{0}}{\sin\varphi_{0}}dT_{nsbias}\right)\\ -\left(\frac{K_{pl}\sigma_{FOG}t}{\sin\varphi_{0}}-\frac{\Omega_{0}\cos\varphi _{0}a_{pl}}{\sin\varphi_{0}}\right)\\ -\left(\frac{\Omega_{0}\cos\varphi_{0}}{\sin\varphi_{0}}dT_{ns}+\varepsilon_{ m1m2}-\varepsilon_{d\varphi}+\Omega_{0}\right)\end{array}  \tag{15}\]  
where \(K_{a}\) is the FOG scale factor deviation. Let \(K_{p1}=1/(1+K_{a})\), \(K_{p2}\) is the second and third terms on the right side of the equal sign of Equation (15), \(K_{e}\) is the fourth term on the right side of the equal sign of Equation (15), \(\Omega_{bias}\) and \(dT_{nsbias}\) are the instrumental zero deviation of the FOG and the leveling device, respectively, and can be considered as constants, \(a_{FOG}\) and \(a_{ns}\), respectively, represent the first-order drift over time for the FOG and tiltmeter. Therefore, the final equation of UT1 solution using the angular velocity measured by FOG can be expressed as:  
\[(\mathrm{UT1-UTC})(t)=K_{k}*T_{s}*\int_{t_{0}}^{t}\left[\left(K_{p1}\frac{ \Omega^{\prime}}{\sin\varphi_{0}}-K_{p2}-K_{e}\right)\right] \tag{16}\]  
However, the fiber optic ring is affected by the fabrication process, environment or other factors, resulting in small changes in the geometric parameters, so that there is a small difference between the actual scale factor of the gyroscope and the theoretical scale factor. Before the solution of UT1, an initial calibration of the FOG's scale factor is required; for this purpose, we employ the VLBI observations to calibrate the parameters and eliminate the error due to inaccurate scale factor at the initial moment. The initial coefficients \(K_{p1}\) and \(K_{p2}\) need to be calculated using the FOG measured data; thus, the segment of the initial correction data needs to be in the same cycle of the FOG measurements as the UT1 solution data in order to keep the parameters unchanged. The initial coefficients are determined by using the UT1-UTC data of the previous two days of the corresponding measurementperiod of EOP 14C04 published by IERS as a reference standard. After calibrating the initial coefficients of the model, the UT1 solution can be achieved independently in subsequent measurement periods solely using the FOG.