Comparative analysis of local angular rotation between the ring laser gyroscope GINGERINO and GNSS stations

Abstract

The study of local deformations is a hot topic in geodesy. Local rotations of the crust around the vertical axis can be caused by deformations. In the Gran Sasso area, the ring laser gyroscope GINGERINO and the GNSS array are operative. One year of data of GINGERINO is compared with the ones from the GNSS stations, homogeneously selected around the position of GINGERINO, aiming at looking for rotational signals with period of days common to both systems. At that purpose the rotational component of the area circumscribed by the GNSS stations has been evaluated and compared with the GINGERINO data. The coherences between the signals show structures that even exceed 60% coherence over the 6–60 days period; this unprecedented analysis is validated by two different methods that evaluate the local rotation using the GNSS stations. The analysis reveals that the shared rotational signal’s amplitude in both instruments is approximately 10⁻¹³ rad/s, an order of magnitude lower than the amplitudes of the signals examined. The comparison of the ring laser data with GNSS antennas provides evidence of the validity of the ring laser data for very low frequency investigation, essential for fundamental physics test.

Appendix A: Error propagation

Following are all the equations to derive angular velocity in the rotation vector method. Starting from the linear velocities of the individual stations, the angles defined previously to derive the perpendicular component of the distance vector and the absolute value of the distance itself.

$$\begin{aligned} \omega _1 = \frac{|v_1|}{|r_1|}\sin {(\alpha _1-\theta _1)} \end{aligned}$$

(12)

In order to do the weighted sum of the individual stations, we must propagate the error in the appropriate way:

$$\begin{aligned}{} & {} \sigma _{\omega _1}^2 = \left( \frac{\partial \omega _1}{\partial v_1}\right) ^2\sigma _{v_1}^2+\left( \frac{\partial \omega _1}{\partial r_1}\right) ^2\sigma _{r_1}^2 +\nonumber \\{} & {} \quad +\left( \frac{\partial \omega _1}{\partial \alpha _1}\right) ^2\sigma _{\alpha _1}^2+\left( \frac{\partial \omega _1}{\partial \theta _1}\right) ^2\sigma _{\theta _1}^2 \end{aligned}$$

(13)

$$\begin{aligned}{} & {} \left( \frac{\partial \omega _1}{\partial v_1}\right) ^2 = \left( \frac{\sin {(\alpha _1-\theta _1)}}{|r_1|}\right) ^2 \end{aligned}$$

(14)

$$\begin{aligned}{} & {} \left( \frac{\partial \omega _1}{\partial r_1}\right) ^2 = \left( -\frac{|v_1|\sin {(\alpha _1-\theta _1)}}{|r_1|^2}\right) ^2 \end{aligned}$$

(15)

$$\begin{aligned}{} & {} \left( \frac{\partial \omega _1}{\partial \alpha _1}\right) ^2 = \left( \frac{|v_1|\cos {(\alpha _1-\theta _1)}}{|r_1|}\right) ^2 \end{aligned}$$

(16)

$$\begin{aligned}{} & {} \left( \frac{\partial \omega _1}{\partial \theta _1}\right) ^2 = \left( -\frac{|v_1|\cos {(\alpha _1-\theta _1)}}{|r_1|}\right) ^2 \end{aligned}$$

(17)

$$\begin{aligned}{} & {} \sigma _{v_1} = \sqrt{\frac{v_x^2\sigma _{v_x}^2+v_y^2\sigma _{v_y}^2}{v_x^2+v_y^2}} \end{aligned}$$

(18)

$$\begin{aligned}{} & {} \sigma _{\alpha _1} = \frac{\sqrt{v_y^2\sigma _{v_x}^2+v_x^2\sigma _{v_y}^2}}{v_x^2+v_y^2} \end{aligned}$$

(19)

$$\begin{aligned}{} & {} \sigma _{v_x} = \sqrt{\sigma _{x_i}^2+\sigma _{x_{i+1}}^2} \end{aligned}$$

(20)

all other parts can be obtained analytically from GNSS measurements, instead, $\sigma _{r_1}$ and $\sigma _{\theta _1}$ are evaluated with a parametric bootstrap method [25], because they are obtained with the “distance” function of MATLAB.

In our approach, we began by generating normal functions that were centered around the positions of the stations. The width of these functions was determined by the sigma value obtained from our measurements. Using the ’distance’ function, we then calculated the distance to GINGERINO as well as the corresponding angle. We repeated this procedure a million times, randomly extracting values from the aforementioned normal distribution at each iteration. This allowed us to obtain distributions of the measurements, and we associated the standard deviations of these distributions with the errors in the distance $r_1$ and angle $\theta _1$.