SME
Physical Geology · Problem type
How often quakes strike: Gutenberg–Richter
Use this for “A region has a = 5 and b = 1. How many M 6 or larger quakes a year, and what is the chance of one in 30 years?”
Chart
log₁₀ N(6) = −1
This is an example. Type your own values to start a new problem.
log₁₀
N
=
a
−
b
×
M
log₁₀ N
Log of the yearly count
worked out (not typed)
T
Recurrence interval
calculated · yr
t
Time window
example · yr
P
Chance of one or more
calculated · %
Formulas
log₁₀
N
=
a
−
b
×
M
−1 = 5 − 1 × 6
N
=
10
log₁₀
N
0.1
=
10
-1
T
=
1
N
10
=
1
0.1
P
=
100
×
(1
−
e
−
N
×
t
)
95.0213
=
100
×
(1
−
e
−0.1 × 30
)
Assumptions
•
log₁₀ N = a − bM counts the quakes a year at or above magnitude M; b is usually near 1.
•
Quakes strike at random at a steady rate, so the chance of at least one in t years is 1 − e−Nt.
•
The recurrence interval T = 1 ÷ N is an average gap, not a schedule.
Step-by-step
WE KNOW
M = 6, t = 30 yr, a = 5, b = 1
FIND
log of the yearly count (log₁₀ N), quakes per year (N), recurrence interval (T), chance of one or more (P)
Step 1 · Find log of the yearly count (log₁₀ N)
Use log₁₀ N = a − b × M
Each
step
up
in
magnitude
cuts
the
count
by
a
factor
of
10
b
.
log₁₀
N
=
5
−
1
×
6
log₁₀
N
=
−1
Step 2 · Find quakes per year (N)
Use N = 10log₁₀ N
Undo the log: raise 10 to that power.
N
=
10
-1
N
=
0.1
yr⁻¹
Step 3 · Find recurrence interval (T)
Use T = 1 ÷ N
N
quakes
a
year
means
one
every
1
N
years
on
average.
T
=
1
0.1
T
=
10
yr
Step 4 · Find chance of one or more (P)
Use P = 100 × (1 − e−N × t)
e
−
Nt
is
the
chance
of
no
quake
in
t
years;
the
rest
is
the
chance
of
one
or
more.
P
=
100
×
(1
−
e
−0.1 × 30
)
P
=
100
×
(1
−
e
-3
)
P
=
100
×
(1
−
0.04979)
P
=
100
×
0.9502
P
=
95.0213%
Check
Put the numbers back in. Both sides should match.
−1 = 5 − 1 × 6 ✓
0.1
=
10
-1
✓
10
=
1
0.1
✓
95.0213
=
100
×
(1
−
e
−0.1 × 30
)
✓
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Distance to a quake from the S − P lag
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Topic check
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