Problem: The three (3) seismic travel times observations were reported at seismic stations AMX05, GEO11, and SEIS01 as illustrated in the table 1, below after passing various processing. The day during which these seismic events were recorded was January ,25 of 2022. Assuming that you were the seismologist on duty at that time, also the event was local and involved first arrivals. What was the origin time of these events?
Table 1: Seismic observations from three (3) stations.
Wadati Diagram Technique
Solution
You can approach this task in different ways however the fast and simple technique is to use Wadati Diagram Technique. As this method has been proven many times to give the best solution and it does not require you to have much knowledge on seismic earthquake.
I will show you how to attempt this task by just using Microsoft Excel, you know what Microsoft Excel is powerful spreadsheet software and simple to grasp as many of us have already installed it in our computers. So this can make it easy for you to learn at your pace! Remember I approached this problem by using my own data range, you can adjust this to fit your needs.
Our approach
1: Complete the Lag time column in the given data: This can be done by simply inserting a given function f(x),
Lag - time (Lag)
Lag time can be defined as the difference of travel times for Shear (S) and Primary (P) waves. It is also known as S-P time, Look the screenshot below how I have done this,
2: Plot the scatter plots for S-P time Versus P - time
The plot is adapted from the equations,
tap = t0 + tf and tas = t0 + tf (Vp/Vs)
By eliminating the tf , then the equation is modified as,
tas - tap = [(Vp/Vs) - 1] (tap - t0)
Where
tap - P - Arrival time, tas - S - Arrival time, Vp - Velocity of P - wave, Vs - Velocity of S - wave,
tf - travel time function, t0 - Origin time.
If we compare the equation above with the linear gradient equation, y = mx + c.
slope (m) = Vp/Vs - 1
The tas - tap Versus tap can be plotted as shown below
3: Extract y - intercept, As seen from the graph above, On extrapolating the linear line (Blue) by dashed line (Black), we can extract the tas - tap time = 3.5 sec at which tap = 0. This means at the y - intercept point.
4: Compute Slope: Let us compute the slope between the marked points (By red line), as shown in the graph here below.
5: Determine Crustal Parameter: Then slope = 0.81, we can then compare this value by Vp/Vs - 1, then the crustal parameter can be determined.
Vp/Vs - 1 = 0.81
Then, Vp/Vs = 1.81.
6: Check for the Validity: Let's check for validity of this parameter, as an ideal value should be √3 = 1.73 as computed by Poison Equation., But for any variation of conditions also it may change from an idea, however the Quality check is +-0.1 OK, means we are still on the RIGHT TRACK!
7: Extract the Origin time: The Remained Task is to obtain the Origin time (Time of Rupture)
Recall, tas - tap = [(Vp/Vs) - 1] (tap - t0)
If, we Put tas - tap = 0, as the Origin time (t0) is obtained at this condition,
Then, 0 = [(Vp/Vs) - 1] (tap - t0), we can find that tap = t0
If tas - tap = 3.5, then 3.5 = 0 + 0.81t0
t0 = 3.5/0.81
t0 = 4.27
We have done it!
Let us report this event as : The Origin Time was 4.27sec.
On reporting: Origin time, 2022 0125 4:27
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