SME
Grade 12 Science · Problem type
Locating an epicenter from three stations
Use this for “Why are three seismic stations needed to find an epicenter?” and for the distances from each lag.
Diagram
The three circles meet at one point, 168 km east and 126 km north: the epicenter.
This is an example. Type your own values to start a new problem.
L₁
Lag at station 1
example
L₂
Lag at station 2
example
L₃
Lag at station 3
example
vₚ
P-wave speed
example · km/s
vₛ
S-wave speed
example · km/s
k
Km of distance per second of lag
worked out (not typed) · km/s
d₁
Distance from station 1
calculated
d₂
Distance from station 2
calculated
d₃
Distance from station 3
calculated
Formulas
k
=
1
1 ÷ vₛ − 1 ÷ vₚ
8.4
=
1
1 ÷ 3.5 − 1 ÷ 6
d₁
=
k
×
L₁
210 = 8.4 × 25
d₂
=
k
×
L₂
84 = 8.4 × 10
d₃
=
k
×
L₃
210 = 8.4 × 25
Assumptions
•
Each distance is a circle round its station.
•
Two circles meet at two points; the third picks one.
•
The stations must not lie in one line.
•
With the crust’s speeds of 6 and 3.5 km/s, each second of lag is 8.4 km.
•
k is not a wave speed: it is the km of distance for each second of lag.
Step-by-step
WE KNOW
L₁ = 25 s, L₂ = 10 s, L₃ = 25 s, vₚ = 6 km/s, vₛ = 3.5 km/s
FIND
km of distance per second of lag (k), distance from station 1 (d₁), distance from station 2 (d₂), distance from station 3 (d₃)
Step 1 · Find km of distance per second of lag (k)
Use k = 1 ÷ (1 ÷ vₛ − 1 ÷ vₚ)
Each
km
adds
1
vₛ
−
1
vₚ
seconds
of
lag;
flip
it
to
get
km
per
second
of
lag.
k
=
1
1 ÷ 3.5 − 1 ÷ 6
k
=
1
0.28571 − 0.16667
k
=
1
0.11904
k
=
8.4
km/s
Step 2 · Find distance from station 1 (d₁)
Use d₁ = k × L₁
Each
second
of
lag
puts
the
station
k
km
farther
from
the
quake.
d₁
=
8.4
×
25
d₁
=
210
km
Step 3 · Find distance from station 2 (d₂)
Use d₂ = k × L₂
Each
second
of
lag
puts
the
station
k
km
farther
from
the
quake.
d₂
=
8.4
×
10
d₂
=
84
km
Step 4 · Find distance from station 3 (d₃)
Use d₃ = k × L₃
Each
second
of
lag
puts
the
station
k
km
farther
from
the
quake.
d₃
=
8.4
×
25
d₃
=
210
km
Check
Put the numbers back in. Both sides should match.
8.4
=
1
1 ÷ 3.5 − 1 ÷ 6
✓
210 = 8.4 × 25 ✓
84 = 8.4 × 10 ✓
210 = 8.4 × 25 ✓
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