# 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  
For IFOG solidly strapped down to earth, the measurement principle is based on the Sagnac effect. IFOG achieves rotation observations by detecting Sagnac [22] phase differences \(\Delta\phi\). \(\Delta\phi\) is expressed as:  
\[\Delta\phi=\frac{4\pi LR}{\lambda c}\Omega \tag{1}\]  
where \(\Omega\) is the angular velocity of rotational motion, \(\lambda\) is the central wavelength of the light, \(c\) denotes the speed of light in a vacuum, \(L\) and \(R\) are the length and radius of the fiber optic ring, respectively. Since the FOG is horizontally installed, the measured angular velocity represents the projection of the instantaneous earth rotation angular velocity vector in the direction normal to the plane of the fiber optic ring, thus exhibiting latitude dependence. Figure 1 illustrates the principle of using a FOG to sense the earth's angular velocity. Let the angular velocity of earth's rotation at latitude \(\varphi\) be \(\Omega_{e}\), which can be decomposed into a horizontal component \(\Omega_{eh}\) and a vertical component \(\Omega_{ev}\). Since the FOG is positioned horizontally on the measurement platform, the sensitive axis is perpendicular to the local horizontal plane. According to the projection principle, \(\Omega_{ev}=\Omega_{e}\sin\varphi\). Considering the influence of each error term and the noise during the measurement process, the earth's rotation parameters, orientation parameters and instrument noise are modeled based on the Sagnac effect, as described in Equation (1) [23]. The sources of errors are illustrated in Figure 2.  
Based on the Sagnac effect equation, the parameters were modeled, respectively. These include the sensor model, the rotation model for the rotation vector, and the orientation model for the normal of the FOG (see Figure 2). The sensor model includes the scale factor, bias error and random error [1]. The orientation and rotation models can be divided into two parts: global and local. Tidal and geophysical effects cause tilt errors in the direction of the FOG normal [18]. Local rotations due to local geophysical effects and the influence of external celestial bodies cause variations in the angular rate and position of the earth's instantaneous rotation axis [16], which affects the FOG measurements. Since the FOG is sensitive to vertical upward rotation, the vector of the instantaneous rotational angular velocity of the earth detected in the o-xyz coordinate system is denoted as \(\overset{\rightarrow}{\Omega}=(\Omega_{1},\Omega_{2},\Omega_{3})=\Omega_{0}( \mu_{1},m_{2},1+m_{3})\). The normal vector of the plane of the fiber optic ring placed horizontally on the measurement platform is \(\overset{\rightarrow}{\Omega}=(0,0,1)\). According to  
Figure 1: Principle of using a FOG to sense the earth’s angular velocity. Figure 2: Schematic representation of the instrumental, orientation and rotation models in FOG data analysis. the coordinate system change relationship, in the o-x'y'z' coordinate system, based on the Sagnac effect equation, bringing in variations of rotation matrix and measurement platform, it can be expressed as:  
\[ \begin{array}{c}\Delta\phi=\Omega_{1}\sin\theta\cos\lambda+\Omega_{2}\sin \theta\sin\lambda\\ \qquad\qquad+\Omega_{3}(\sin T_{ns}\sin\theta+\cos T_{ns}\cos T_{ew}\cos \theta)\end{array}  \tag{2}\]  
where \(T_{ns}\) and \(T_{ew}\) represent the north-south and east-west tilt, respectively. Variations in north-south orientation change the projection of the earth's rotation vector onto the ring fiber area. Tilt in the east-west orientation is not visible on a single horizontally orientated gyroscope component, because they enter as negligible small offsets to a cos(0\({}^{\circ}\)) term [10]. From the above equation, it can be seen that the FOG output value depends on the orientation of the instrument normal vector with respect to the instantaneous rotation vector of the earth.