# Calculation of Theoretical Travel Time and Automatic Picking of Actual Travel Time in Seismic Data
## 2 Seismic Phase Theory Travel Time Calculation  
The IASP91 Earth model is shown in Figure 1 and Table 1.  
Figure 1: IASP91 Earth model. Using the IASP91 standard Earth velocity model for ray tracing [30], seismic waves propagate in a three-dimensional medium satisfying the equation function equation  
\[(\
abla\tau)^{2}=\frac{1}{v^{2}} \tag{1}\]  
\(\tau\) is the travel time of the seismic wave propagating from the source \(S\) to any point \((x,y,z)\) in space. \(v\) is the velocity of the medium at any point in space. Equation (1) can be expressed in the Cartesian coordinate system as  
\[\left(\frac{\partial\tau}{\partial x}\right)^{2}+\left(\frac{\partial\tau}{ \partial y}\right)^{2}+\left(\frac{\partial\tau}{\partial z}\right)^{2}=\frac{ 1}{v^{2}} \tag{2}\]  
Now, a variable \(s\) is introduced, which denotes the arc length of the ray as the elastic wave propagates. Since the ray is orthogonal to the wave front \(t=\tau\)\((x,y,z)\), the derivative of the ray at the wave front thus has  
\[\frac{d\tau}{ds}=\frac{1}{v} \tag{3}\]  
Any spatial position (\(x\), \(y\), \(z\)) during the propagation of the elastic wave can be described by a parametric equation with the ray length \(s\) as a variable, i.e., \(x=x(s)\), \(y=y(s)\), \(z=z(s)\). For simplicity, the spatial position of the ray, i.e., the ray trajectory, can be expressed by a vector equation as  
\[\widetilde{x}=\widetilde{x}(s) \tag{4}\]  
Since the ray is orthogonal to the wavefront through which it passes, it follows that the ray path \(\widetilde{x}\) must be parallel to \(\
abla\tau\), i.e.,  
\[\frac{d\widetilde{x}}{ds}=\alpha\
abla\tau \tag{5}\]  
where \(\alpha\) is the scaling factor. From Equations (1) and (5), we can get  
\[(\
abla\tau)^{2}=\frac{1}{\alpha^{2}}\left(\frac{d\widetilde{x}}{ds}\right)^{ 2}=\frac{1}{\alpha^{2}}\left[\left(\frac{dx}{ds}\right)^{2}+\left(\frac{dy}{ ds}\right)^{2}+\left(\frac{dz}{ds}\right)^{2}\right]=\frac{1}{v^{2}} \tag{6}\]  
Moreover, since \(ds^{2}=dx^{2}+dy^{2}+dz^{2}\) and thus we have \(\alpha=v\), it is possible to write the differential equation for the ray (5) in the following representation:  
\[\frac{1}{v}\frac{d\widetilde{x}}{ds}=\
abla\tau \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Depth (km)** & **Radius (km)** & **V\({}_{\mathbf{P}}\) (km/s)** & **V\({}_{\mathbf{S}}\) (km/s)** \\ \hline
0.0–20.0 & 6351–6371 & 5.8 & 3.36 \\
20.0–35.0 & 6336–6351 & 6.5 & 3.75 \\
35.0–120.0 & 6251–6336 & 8.78541 – 0.749530r & 6.706231 – 2.248585x \\
120.0–210.0 & 6161–6251 & 25.41389 – 17.69722x & 5.75020 – 1.274200x \\
210.0–410.0 & 5961–6161 & 30.78765 – 23.25415x & 15.242130 – 11.085520x \\
410.0–660.0 & 5711–5961 & 29.38896 – 21.40656x & 17.707320 – 13.506520x \\
660.0–760.0 & 5611–5711 & 25.96984 – 16.93412x & 20.768900 – 16.531470x \\
760.0–2740.0 & 3631–5611 & 25.14860 – 41.15380x + 51.99320x2 – & 12.930300 – 21.259000x + 27.89880 x 2\({}^{-}\) 14.10800 x3 \\
2740.0–2889.0 & 3482–3631 & 14.49470 – 1.47089x & 8.166160 – 1.582060x \\
2889.0–5153.9 & 1217.1–3482 & 10.03904 – 3.75665x – 13.67046 x2 & 0 \\
5153.9–6371.0 & 0–1217.1 & 11.24094 – 4.09689 x2 & 3.564540 – 3.452410 x2 \\ \hline \end{tabular} Note: \(x=r/R\), where \(r\) is the radius from the Earth center, \(R\) is the Earth’s radius (\(R=6371.00\) km).  
\end{table} 
Table 1: The parameters of IASP91 Earth model. From Equation (3), we get  
\[\frac{d(\
abla\tau)}{ds}=\
abla\bigg{(}\frac{1}{v}\bigg{)} \tag{8}\]  
Finding the derivative with respect to \(s\) for both sides of Equation (7) and taking into account Equation (8), we have  
\[\frac{d}{ds}\bigg{(}\frac{1}{v}\frac{d\widetilde{x}}{ds}\bigg{)}=\frac{d}{ds}( \
abla\tau)=\
abla\bigg{(}\frac{1}{v}\bigg{)}\]  
i.e.,  
\[\frac{d}{ds}\bigg{(}\frac{1}{v}\frac{d\widetilde{x}}{ds}\bigg{)}=\
abla\bigg{(} \frac{1}{v}\bigg{)} \tag{9}\]  
It can be seen that Equation (9) is a differential equation for the ray path that does not depend on the phase function \(\tau\). This equation can be expressed as three differential equations corresponding to the three coordinate components:  
\[