# Calculation of Theoretical Travel Time and Automatic Picking of Actual Travel Time in Seismic Data
## 3 Automatic Picking of the Seismic Phase Actual Travel Time  
In the same seismic event, there is a certain degree of similarity between stations, and the degree of similarity can be judged by the magnitude of the correlation coefficient, so the correlation coefficient is used as a measure of the degree of correlation between two random variables. Let two columns of random signals \(x(t)\) and \(s(t)\) be accepted by \(N\) stations, and the correlation coefficient between \(x(t)=\{x_{1},x_{2},\ldots,x_{n}\}\) and \(s(t)=\{s_{1},s_{2},\ldots,s_{n}\}\) is  
\[R_{i}(\tau)=\frac{\sum x_{i}(t+\tau)s(t)}{\sqrt{\sum x_{i}^{2}(t)\sum s^{2}(t)}} \tag{18}\]  
The range of the correlation coefficient definition's value for \(R_{i}\) is 0 to 1. When \(R_{i}=1\), it indicates that the two series are identical; when \(R_{i}=0\), it indicates that the two series have no connection at all. The higher the value of \(R_{i}\), the more similar the two waveforms are to one another. The traditional correlation method is to determine the first arrival point by finding the delay time of the first arrival. However, because of the influence of the similarity between seismic traces, the delay time finding is prone to great deviation, and even the delay time cannot be obtained. When the waveform similarity between the reference channel and the other seismic channels is good, a good pickup effect can be achieved by directly using the reference channel to find the correlation coefficient with each seismic channel. However, the waveform information received by the seismic recording instruments is affected by various factors such as different propagation paths, noise, and receiving instruments, and the similarity of the actual data waveforms obtained is not ideal, so the pickup effect of the first arrivals is less accurate. Many scholars pick by the similarity between the reference tract and each tract, and the correlation coefficient is found by the reference tract and each tract, and the correlation coefficient discrete form  
\[R_{i}(k)=\frac{\sum_{j=k}^{k+N_{s}-1}x_{ij}s_{j}}{\sqrt{\sum_{j=k}^{k+N_{s}-1}x _{ij}^{2}\sum_{j=k+N_{s}}^{2}s_{j}^{2}}} \tag{19}\]  
The discrete signal is \(x_{ij}\) (\(i=1,2,3,\ldots,N,j=1,2,3,\ldots,N_{x}\)); \(s_{k}\) (\(k=1,2,3,\ldots,N_{s}\)), \(t=(k-1)\Delta t\), \(k=1,\,2,\,3,\ldots,\,N_{x}\), \(\Delta t\) is the seismic waveform's sampling interval. The arrival time of the specified seismic phase received by the station is given by \(t=(k-1)\Delta t\) if the correlation coefficient between the seismic record of the \(i\)th station after moment t and the specified wavelet is maximized. The seismic wave travel time \(t_{i}\) of the phase can then be determined using the seismic event's onset moment \(t_{0}\). In Figure 3, the time signal sequences \(x(t)\) and \(s(t)\) were created by artificially shifting the same signal. \(x(t)\) was acquired after \(s(t)\) was artificially shifted to the right for 10 s, at the period when the value of \(R_{i}\) was at its highest. The data used in near-sound tomography are absolute travel times, while in far-sound tomography, the data used are relative travel time residuals. Suppose that the \(i\)th earthquake is observed by \(N\) stations after its occurrence.