# Corrected Tilt Calculation for Atmospheric Pressure-Induced Seismic Noise
## 2 Theory
### Analytic Approximations  
[PERSON] presents approximate results for \(\mathrm{u}\), \(\mathrm{w}\), and \(\mathrm{u}_{\mathrm{tilt}}\). Two approximations are employed. First, he notes that \(\mathrm{c}\), the wind speed, is much lower than the seismic speeds, so the vertical attenuations \(\gamma\) and \(\delta\) can be expanded to first-order. Next, [PERSON] notes the following for low frequency:  
\[|\omega|z\ll\frac{2\beta^{2}}{c}<\frac{2\alpha^{2}}{c} \tag{26}\]  
and the exponential terms \(\mathrm{e}^{-\delta z}\) and \(\mathrm{e}^{-\gamma z}\) can be expanded about \(\mathrm{e}^{\frac{-\mathrm{wz}}{c}}\). The exact term, Equation (23), has been approximated to the same order. The figures in this paper are calculated exactly from the theory using MATLAB [7]. The approximate forms useful for discussing the general behavior are presented below. The signs have been adjusted to conform with our use of the \(\mathrm{e}^{-\mathrm{i}\omega t}\) notation and stress due to pressure as \(-\mathrm{P_{o}}\). These terms are:  
Horizontal displacement:  
\[\mathrm{u}=\frac{\mathrm{i}\mathrm{c}\mathrm{P_{o}}}{2\mu\omega}\bigg{[}\frac {\mu}{\lambda+\mu}-\frac{\omega z}{c}\bigg{]}\mathrm{e}^{\frac{-\omega z}{c}} \mathrm{e}^{\mathrm{i}(\mathrm{k}\chi-\omega t)} \tag{27}\]  
Vertical displacement:  
\[\mathrm{w}=\frac{\mathrm{c}\mathrm{P_{o}}}{2\mu\omega}\bigg{[}\frac{\lambda+2 \mu}{\lambda+\mu}+\frac{\omega z}{c}\bigg{]}\mathrm{e}^{\frac{-\omega z}{c}} \mathrm{e}^{\mathrm{i}(\mathrm{k}\chi-\omega t)} \tag{28}\]  
Tilt contribution:  
\[\mathrm{u_{tilt}}=\frac{\mathrm{i}\mathrm{g}}{\omega^{2}}\frac{\mathrm{P_{o}} }{2\mu}\bigg{[}\frac{\lambda+2\mu}{\lambda+\mu}\bigg{]}\mathrm{e}^{\frac{- \omega z}{c}}\mathrm{e}^{\mathrm{i}(\mathrm{k}\chi-\omega t)} \tag{29}\]  
Tilt contribution for [PERSON]'s [3] expression:  
\[\mathrm{u_{tilt}^{S}}=\frac{\mathrm{i}\mathrm{g}}{\omega^{2}}\frac{\mathrm{P_{ o}}}{2\mu}\bigg{[}\frac{\lambda+2\mu}{\lambda+\mu}+\frac{\omega z}{c} \bigg{]}\mathrm{e}^{\frac{-\omega z}{c}}\mathrm{e}^{\mathrm{i}(\mathrm{k}\chi -\omega t)}. \tag{30}\]  
We note that the effect of tilt, Equations (29) and (30), is the same at the surface of the medium (z = 0), as expected from the discussion above. However, Equation (29) for the complete solution does not contain the \(\frac{\omega z}{c}\) dependent term, so the predicted tilt will decay more quickly with depth than the tilt predicted using [PERSON]'s expression, (Equation (30)) or the actual displacements (Equations (27) and (28)). When \(\frac{\omega z}{c}\) is large, i.e., has a large depth, the tilt correction for the horizontal displacement will be negligible. Furthermore, the horizontal and vertical displacements will be of equal magnitude. The required depth for this transition will be greater for the lower-frequency (longer-period) signals. [PERSON]'s expression for the tilt contribution is proportional to \(\mathrm{w}\) irrespective of depth in the medium, and will overpredict the tilt contribution as a function of depth. The relative magnitude of the tilt correction with respect to \(\mathrm{u}\) and \(\mathrm{w}\) at the surface z = 0 are  
\[\frac{\mathrm{u_{tilt}}(0)}{\mathrm{u}(0)}=\frac{\mathrm{g}}{\omega c}\bigg{[} \frac{\alpha^{2}}{\beta^{2}}\bigg{]} \tag{31}\]  
and  
\[\frac{\mathrm{u_{tilt}}(0)}{\mathrm{w}(0)}=\frac{\mathrm{g}}{\omega c}. \tag{32}\]  
The importance of tilt is more pronounced in soft grounds. For \(\omega<\frac{\mathrm{g}}{c}\), which by assumption is of order one or less, the horizontal displacement associated with tilt at the surface will be larger than both the actual horizontal and vertical displacements. However, given the decay of the effect of tilt with depth discussed above, the sum of the actual and tilt contributions must be analyzed. For measurements below the ground surface, one cannot assume that if tilt dominates at the surface, it will dominate at all depths.