# Corrected Tilt Calculation for Atmospheric Pressure-Induced Seismic Noise
## 2 Theory
### Apparent Horizontal Displacement Due to Tilt  
Tilt noise is a pseudo-signal measured by a mass-based seismic sensor when the instrument is tilted from its level position by an angle \(\theta\), thereby changing the direction of gravitational force on the seismic mass. Variations in the local atmospheric pressure are a prime cause of tilt in the ground, which results in increased horizontal seismic noise in the long-period band. This tilt noise restricts the use of horizontal data for sensitive seismological studies. Figure 2 shows a schematic sketch of a two-component sensor (geophone), represented by masses on springs, that is affected by tilt as the ground undergoes a period of deformation. The sensor shown in the figure is experiencing a positive tilt by an angle \(\theta\) with respect to the vertical, and g is the acceleration due to gravity. The horizontal (x) component experiences an increase in force, whereas the vertical (z) component experiences a reduction in force due to tilt. The horizontal displacement at a single angular frequency (\(\omega\)), corresponding to the apparent acceleration, is given by  
\[\mathrm{u}_{\mathrm{tilt}}=-\frac{\mathrm{g}}{\omega^{2}}\mathrm{sin}\ \theta\approx-\frac{\mathrm{g}}{\omega^{2}}\theta. \tag{16}\]  
The negative sign indicates that the apparent displacement is negative for a positive rotation and the effect goes to zero when the tilt is zero. The vertical displacement from equilibrium, w, at a single angular frequency (\(\omega\)), corresponding to the apparent acceleration, is given by  
\[\mathrm{w}_{\mathrm{tilt}}=-\frac{\mathrm{g}}{\omega^{2}}(1-\cos\theta). \tag{17}\]  
Figure 2: Mass-spring representation of a horizontal and vertical component geophone. The negative sign indicates that the tilt effect is an upward displacement (reduced elongation of the spring) due to reduced force acting along the direction of the spring. The effect of tilt on the vertical component, w, is second-order. The expression for the tilt angle, \(\theta\), appropriate to a sensor on the surface is  
\[\tan\theta=-\frac{\Delta\mathrm{w}}{\Delta\mathrm{x}}\approx-\frac{\partial \mathrm{w}}{\partial\mathrm{x}}\approx\theta \tag{18}\]  
for small \(\theta\). We note that [PERSON] [3] has a similar equation with the opposite sign. Since he only discusses magnitude and does not combine the tilt contribution with the actual horizontal displacement, this introduces no error. We present [PERSON]'s result converted to our sign convention below. Since we are interested in investigating the combined response over a wide range of frequencies and as a function of depth, we calculate the effective horizontal displacement as  
\[\mathrm{u}_{\mathrm{effective}}=\mathrm{u}+\mathrm{u}_{\mathrm{tilt}}. \tag{19}\]  
where \(\mathrm{u}\) is the actual displacement, \(\mathrm{u}_{\mathrm{tilt}}\) is the effect of tilt, and \(\mathrm{u}_{\mathrm{effective}}\) is the effective displacement measured by a mass-based sensor. In elastic media, the angle of tilt inside the medium induced by displacements is given by [6]  
\[\theta=\frac{1}{2}\left(\
abla\mathrm{x}\overset{\rightarrow}{\mathrm{u}}\right) \tag{20}\]  
which in our coordinate system is  
\[\theta=\frac{1}{2}\bigg{(}\frac{\partial\mathrm{u}}{\partial\mathrm{z}}-\frac {\partial\mathrm{w}}{\partial\mathrm{x}}\bigg{)}. \tag{21}\]  
Combining Equations (16), (19) and (21) gives the effective horizontal displacements as  
\[\mathrm{u}_{\mathrm{effective}}=\mathrm{u}-\frac{\mathrm{g}}{2\omega^{2}} \bigg{(}\frac{\partial\mathrm{u}}{\partial\mathrm{z}}-\frac{\partial\mathrm{w }}{\partial\mathrm{x}}\bigg{)}. \tag{22}\]  
Substituting in the expressions for the displacements in terms of potentials, Equations (12b) and (13b), into Equation (22), the generalized expression for the effective horizontal displacement is  
\[\mathrm{u}_{\mathrm{effective}}=\mathrm{u}-\frac{\mathrm{g}}{2\omega^{2}}\bigg{(} \frac{\omega}{\beta}\bigg{)}^{2}\psi=\mathrm{u}-\frac{\mathrm{g}}{2\beta^{2}}\psi \tag{23}\]  
which is valid throughout the elastic half-space.