A Numerical Model of the SEIS Leveling System Transfer Matrix and Resonances: Application to SEIS Rotational Seismology and Dynamic Ground Interaction

Abstract

Both sensors of the SEIS instrument (VBBs and SPs) are mounted on the mechanical leveling system (LVL), which has to ensure a level placement on the Martian ground under currently unknown local conditions, and provide the mechanical coupling of the seismometers to the ground. We developed a simplified analytical model of the LVL structure in order to reproduce its mechanical behavior by predicting its resonances and transfer function. This model is implemented numerically and allows to estimate the effects of the LVL on the data recorded by the VBBs and SPs on Mars. The model is validated through comparison with the horizontal resonances (between 35 and 50 Hz) observed in laboratory measurements. These modes prove to be highly dependent of the ground horizontal stiffness and torque. For this reason, an inversion study is performed and the results are compared with some experimental measurements of the LVL feet’s penetration in a martian regolith analog. This comparison shows that the analytical model can be used to estimate the elastic ground properties of the InSight landing site. Another application consists in modeling the 6 sensors on the LVL at their real positions, also considering their sensitivity axes, to study the performances of the global SEIS instrument in translation and rotation. It is found that the high frequency ground rotation can be measured by SEIS and, when compared to the ground acceleration, can provide ways to estimate the phase velocity of the seismic surface waves at shallow depths. Finally, synthetic data from the active seismic experiment made during the HP³ penetration and SEIS rotation noise are compared and used for an inversion of the Rayleigh phase velocity. This confirms the perspectives for rotational seismology with SEIS which will be developed with the SEIS data acquired during the commissioning phase after landing.

Appendix: Rotation Sensitivities of the VBB Sensors

As shown by Forbriger (2009), pendulum and therefore VBBs have a rotational sensitivity in addition of their acceleration sensitivity along their sensitivity direction. VBBs are in addition oblique sensors and use gravity to lower their eigenfrequency and increase their mechanical sensitivity, which must also be taken into account.

Let us consider one of the VBB sensor and note $G$ its center of mass and $M$ a part of the proof mass. The acceleration sensed by the point $M$ can be expressed as

$$ \frac{d^{2} \overrightarrow{OM} }{d t^{2}} = \overrightarrow{\gamma } + \overrightarrow{\ddot{\varOmega }} \times \overrightarrow{OM} + \overrightarrow{ \dot{\varOmega }} \times [ \overrightarrow{\dot{\varOmega }} \times \overrightarrow{OM} ] $$

(23)

where $\overrightarrow{\gamma }$ is the acceleration of the VBB pivot and $\overrightarrow{\dot{\varOmega }}$ is the absolute rotation rate with respect to a fixed frame and $\overrightarrow{\ddot{\varOmega }}$ the rotation acceleration. $\overrightarrow{\varOmega }$ is therefore the sum of the LVL rotation, noted hereafter $\overrightarrow{\varOmega _{F}}$ and of the VBB pendulum rotation, noted $\theta \overrightarrow{\pi }$ where $\overrightarrow{\pi }$ the pivot directed vector (in trigonometric direction) and $\theta$ is the rotation of the pendulum with respect to the equilibrium and recentered position. If we limit this expression to only the first order linear term, assuming the rotation to remain small due to the pendulum restoring forces, we can express the acceleration of $M$

$$ \frac{d^{2} \overrightarrow{OM} }{d t^{2}} = \overrightarrow{\gamma } + \overrightarrow{\ddot{\varOmega }} \times \overrightarrow{OM_{0}} = \ddot{\theta } \overrightarrow{\pi } \times \overrightarrow{OM_{0}} + \overrightarrow{\gamma _{G}} + \overrightarrow{\ddot{\varOmega }_{F}} \times \overrightarrow{G_{0}M_{0}} $$

(24)

where $G_{0}$ and $M_{0}$ are now the center of gravity of the mobile mass and point $M$ both taken in their equilibrium positions. $\ddot{\theta }$ is the second derivative of $\theta$, $\ddot{\varOmega }_{F}$ the rotation acceleration along the pivot direction and $\overrightarrow{\gamma _{G}}$ is the full acceleration (including frame rotation) at the equilibrium center of gravity $G_{0}$. By assuming that the pendulum of the VBB has one of its inertial moment axis along the pivot and computing the equation of angular momentum with respect to rotation around the pivot, we can finally write the pendulum equation of movement as:

$$ J \ddot{\theta } =\mathcal{{M}}_{0} - C \theta -C_{R} \theta +m ( \overrightarrow{OG} \times \overrightarrow{g}). \overrightarrow{ \pi }- \bigl[m ( \overrightarrow{OG_{0}} \times \overrightarrow{\gamma } _{G}). \overrightarrow{\pi } + \bigl(J - mD^{2}\bigr) \ddot{ \varOmega }_{F} \bigr] , $$

(25)

where $\mathcal{{M}}_{0}$ is the spring moment at recentered position, $J$ is the VBB moment of inertia with respect to the pivot, $C$ the spring/pivot stiffness, $C_{R}$ the feedback force (as a function of the pendulum rotation), $\overrightarrow{g}$ the Mars gravity and $D$ the distance from pivot to center of gravity. Note that when the VBB is perturbed, $G$ is moving such that:

$$ \overrightarrow{OG} = \overrightarrow{OG}_{0} + \theta \overrightarrow{ \pi } \times \overrightarrow{OG}_{0} = D [ \overrightarrow{p} + \theta \overrightarrow{n} ] , $$

(26)

where $\overrightarrow{n}$ is the direction sensitivity vector. When the infinetesimal rotation $\overrightarrow{\varOmega }$ is perturbing the VBB in addition to ground acceleration, the local gravity is changing. If we rewrite the previous equation in the moving frame of the VBB sensor, the gravity can then be written as

$$ \overrightarrow{g} = \overrightarrow{g}_{0} - \overrightarrow{ \varOmega } \times \overrightarrow{g}_{0} . $$

(27)

These two equations leads to the pendulum dynamic equation. When we note that

$$ {\mathcal{{M}}}_{0} + m (\overrightarrow{OG}_{0} \times \overrightarrow{g}_{0}). \overrightarrow{\pi } = 0 , $$

(28)

this can be rewritten as:

$$\begin{aligned} &J \ddot{\theta } + \bigl[ C - m D (\overrightarrow{n} \times \overrightarrow{g}_{0}) . \overrightarrow{\pi } + C_{R} \bigr] \theta \\ & \quad = - m D \overrightarrow{p} \times [\overrightarrow{\varOmega } \times \overrightarrow{g}_{0} + \overrightarrow{\gamma }_{G}] . \overrightarrow{ \pi } - \bigl(J - mD^{2}\bigr) \ddot{\varOmega }_{F} , \end{aligned}$$

(29)

$$\begin{aligned} &J \ddot{\theta } + \bigl[ C - m D (\overrightarrow{n} \times \overrightarrow{g}_{0}) . \overrightarrow{\pi } + C_{R} \bigr] \theta \\ &\quad = - m D [\overrightarrow{p} . \overrightarrow{g_{0}} \ \overrightarrow{ \varOmega }_{F} + \overrightarrow{\gamma }_{G} . \overrightarrow{n}] - \bigl(J - mD^{2}\bigr) \ddot{ \varOmega }_{F} , \end{aligned}$$

(30)

$$\begin{aligned} &J \ddot{\theta } + [ C - m D \cos \alpha + C_{R}] \theta \\ &\quad = - m D \biggl[ \overrightarrow{\gamma }_{G} . \overrightarrow{n} + \frac{ J - mD^{2} }{mD^{2}} D \ddot{\varOmega }_{F} - g_{0} \cos \alpha {\varOmega } _{F} \biggr] , \end{aligned}$$

(31)

where $\alpha$ is the angle of $\overrightarrow{p}$ with vertical (see Fig. 37 of Lognonné et al. 2018 for VBB pendulum geometry). The amplitude of the first rotation term is related to the non-point character of the pendulum. For the VBBs family, we have $J = 2.56 \times 10^{-4}~\mbox{kg}\,\mbox{m}^{2}$, $D= 0.0256\mbox{ m}$ and $m=190\mbox{ g}$ and therefore $\frac{J - mD^{2} }{mD^{2}}$ is about 1.06. For a rotation acceleration of about $\ddot{\varOmega }_{F} = \omega \frac{\gamma _{z} }{c}$, where $\omega$ is the angular frequency and $c$ the wave phase velocity, the ratio of the second term to acceleration is $\frac{ D \omega }{c \sin \alpha } = 2\%$ for 10 Hz and $c=150\mbox{ m}/\mbox{s}$. This is therefore significant but does not request more than 5% of moment of inertia or location of center of gravity errors to generate errors of less than 0.1%. The third term is small and for seismic waves detection the ratio between rotation term and acceleration is of the order of $\frac{g_{0} }{\tan \alpha \omega c}$. On Mars, the last term will be smaller than $10^{-3}$ for frequencies larger than 7 Hz and phase velocity of 150 m/s. This is expected to be significantly lower than the calibration error of the VBBs and SPs and we can therefore neglect this third rotation sensitivity term of the VBBs for the HP³ surface waves signals in our simulation, which are expected in the 10–50 Hz bandwidth.