Locating an Earthquake from S–P Times at Three Stations
Three seismic stations in Kyushu-A, B, and C-record the same shallow earthquake. The crustal P-wave speed is taken as 6.0 km/s and the S-wave speed as 3.5 km/s, both constant along the paths. The measured S-minus-P arrival-time differences are: station A, 5.0 s; station B, 8.0 s; station C, 6.0 s. A student uses the S–P time at station A alone to estimate the epicentral distance, then uses all three stations to locate the epicentre on a map. The student assumes the focus is at the surface for the first estimate. A news report later states that because station A recorded the smallest S–P time, the earthquake must have occurred directly beneath station A at a depth of about 10 km. The class is asked to judge that claim and to explain what additional information would be needed to determine the focal depth.
1. Derive the relationship between S–P time and distance, and calculate the distance for each station. State the main assumptions.
Read the worked answer
Let the hypocentral distance be (d). The P wave arrives after time (d/v_P) and the S wave after (d/v_S), so
[\Delta t_{\text{S-P}}=\frac{d}{v_S}-\frac{d}{v_P} =d\left(\frac{1}{3.5}-\frac{1}{6.0}\right).]
[\frac{1}{3.5}-\frac{1}{6.0}=0.2857-0.1667=0.1190\ \text{s/km}.]
Hence
[d=\frac{\Delta t_{\text{S-P}}}{0.1190}\approx8.4,\Delta t_{\text{S-P}}\ \text{km}.]
For station A: (d\approx8.4(5.0)=42\ \text{km}).
For station B: (d\approx8.4(8.0)=67\ \text{km}).
For station C: (d\approx8.4(6.0)=50\ \text{km}).
The main assumptions are that the waves travel in straight lines at constant speeds, that the same earthquake origin applies to all stations, and-for using these distances as epicentral distances-that the focus is shallow compared with the station spacing.
2. Explain why three stations are used rather than one or two, and why the first estimate treats the distance as epicentral.
Read the worked answer
A single S–P time gives only one distance, so the epicentre could lie anywhere on a circle of that radius around the station. Two stations give two circles, which generally intersect at two points; one of those intersections may be the epicentre, but the ambiguity remains. A third station gives a third circle that normally passes through only one of the two intersections, resolving the location. For a shallow earthquake, the hypocentral distance is nearly equal to the epicentral distance, so drawing circles with the S–P distances on a map is a good first approximation. If the focus were deep, those circles would systematically overestimate the surface distance, because the hypocentral distance includes the depth. A fourth station, or a set of arrival times and a velocity model, is needed to estimate depth rather than assuming it.
3. Evaluate the news report's claim that the earthquake occurred directly beneath station A at a depth of about 10 km.
Read the worked answer
The claim confuses 'smallest S–P time' with 'directly beneath.' The smallest S–P time means station A is closest to the hypocentre, not that the hypocentre is vertically below it. If the earthquake were directly beneath station A at 10 km depth, the hypocentral distance would be 10 km, giving
[\Delta t_{\text{S-P}}=\frac{10}{3.5}-\frac{10}{6.0}\approx2.86-1.67=1.19\ \text{s},]
not the observed 5.0 s. The observed 5.0 s corresponds to a hypocentral distance of about 42 km. Even if the depth were 10 km, the epicentral distance from A would be
[\sqrt{42^2-10^2}\approx40.8\ \text{km},]
so the epicentre would still be far from directly beneath A. One station cannot determine depth; at least several stations with good azimuthal coverage, a crustal velocity model, and preferably depth-sensitive phases such as pP or sP are needed.