# Corrected Tilt Calculation for Atmospheric Pressure-Induced Seismic Noise
## 3 Results  
In this section, representative examples are presented for the amplitude and decay with depth of the effective horizontal displacement and the vertical displacement. Equation (15a,b) are solved for \(\varphi_{0}\) and \(\psi_{0}\) using MATLAB [7], then Equations (12b), (13b) and (23) are used to evaluate the vertical displacements at the surface and at depths and the effective horizontal displacement at the surface and at depths. For comparison with [PERSON], the period \(T=\frac{1}{f}\) is used as the independent variable in place of frequency. These are exact results without any of the approximations used by [PERSON] [3]. Figure 3 displays the surface values of the effective horizontal displacements (solid line) and the vertical displacements (dotted line) per \(\upmu\)bar of pressure amplitude versus period for three cases from [PERSON] [3]. Model 1 represents the case for unconsolidated soils with low values of \(\alpha\), \(\beta\), and \(\rho\), which explains the comparative maximum displacements among the three models. Values of \(\alpha\), \(\beta\), and \(\rho\) for different geology are shown in Table 1.  
Figure 3: Exact calculation of total horizontal (solid line) and vertical (dotted line) displacement amplitude per microbar at the surface of an elastic half-space as a function of time period of a plane pressure wave moving at a speed of 5 m/s for Models 1, 2b, and 4. Examination of Equations (27)-(30) shows that the principal factor effecting the magnitude of the displacement terms is the shear modulus \(\mu=\rho\beta^{2}\), which varies by a factor of order 1000 between Model 1 and Model 4. For a given pressure amplitude, both effective horizontal displacement and vertical displacement increase with the increase in period but the vertical displacement is dominant for periods below 5 s. For periods greater than \(\sim\)5 s, the effective horizontal displacement is greater than the vertical displacement. Figure 3 presents results explicitly for the surface of an elastic half-space and corresponds to results provided by [PERSON] [3]. Figure 4 shows apparent ground motion caused by harmonic pressure fluctuations on the ground for different periods and depths. The total horizontal displacement ratio and vertical displacement ratio in Figure 4 are expressed as a function of depth (z);  
\[\text{Horizontal displacement ratio }(u)=20\log_{10}\biggl{[}\frac{u_{ \text{effective}}(z)}{w(0)}\biggr{]} \tag{35}\]  
\[\text{Vertical displacement ratio }(w)=20\log_{10}\biggl{[}\frac{w(z)}{w(0)} \biggr{]}. \tag{36}\]  
The results shown in Figure 4 are based on Model 4 and use the tilt expression corrected from Sorrells for horizontal displacement measured by a sensor deployed under the ground. Vertical displacements observed in Figure 4 across a wide range of periods of pressure variations associated with the atmosphere are comparable to [PERSON] [3] and continually attenuate with an increase in depth, as suggested by Equation (28). [PERSON]'s [3] expression for horizontal displacement (Equation (30)) shows that tilt attenuation is directly dependent on depth, and displays similar behavior to the attenuation of vertical displacement with depth. Since horizontal displacement has a tilt correction, from Equation (29) it can be seen that the size of tilt attenuates rapidly with increase in depth. This attenuation is different from [PERSON]'s [3] expression because it does not contain the term \(\frac{\omega\omega}{c}\) in the square brackets. Due to this, the total horizontal displacement exhibits different asymptotic behavior than the vertical displacement. The correct total displacement displays much larger attenuation at large depths. From Figure 4a, it can be seen that at a short period (1.0 s) the size of tilt is small, and as a result the vertical displacement is larger than the effective horizontal displacement. For longer periods (Figure 4b-d), the effective horizontal displacement is larger.