Gutenberg-Richter Law
logN(M)=a-bM
Introduction
Earthquake magnitude
>9.0
8~8.9
7~7.9
6~6.9
5~5.9
4~4.9
3~3.9
2~2.9
number/per year
0
1
18
108
800
6200
49000
300000
Dashed line: N is the number of earthquake
per year of magnitude= M ± ∆M / 2
Solid line: N is the number of earthquake per
year of magnitude ≥ M
Gutenberg - Richter Relation
logN vs. M are typically modeled with a
Gutenberg - Richter relation:
logN(M)=a-bM
M is the earthquake magnitude
N is the number of earthquake per
year of magnitude.
a is called the “productivity”
b is called the “b-value”, and is typically
in the range of 0.8-1.1.
The b-value has served as a kind of
tectonic parameter.
1.8-1.0 oceanic ridge
1.0-0.7 interplate
0.7-0.4 intraplate
Using the GR Relation
Suppose b=1, and you are given that there is
one M 5.0+ earthquake per year in the
region. How often does an M 7.0+ occur?
logN(M)=a-bM
log(1)=a-(1)(5) --> a=5.0
logN(7)=5-(1)(7)=-2
N(7)=10-2=0.01/year
Some subtleties with GR
Need to distinguish between N(M) and n(M)
– N(M) is a cumulative curve, giving the number of
earthquakes of magnitude M or larger per year.
– n(M) is incremental, such that n(M)dM gives the
number of earthquakes in a magnitude range of
width dM, centered on M
dN ( M )
– They are related: n ( M ) = −
dM
– Watch notation: some authors are not careful
with which symbol they use.
How to find the b-value?
Two ways.
– Count n(M) or N(M), then fit a least-squares line
to logN or logn vs M.
– The alternative is the “maximum likelihood”
method:
log10 e
0.434
b=
(M − M min )
=
(M − M min )
– Mmin is the smallest earthquake in the catalog.
– M is the average.
Minimum magnitude is 4.8.
Average magnitude: 5.15
Application to find the b-value:
1966 data
Least squares: We had b=0.84
Maximum likelihood:
0.434
0.434
=
=1.24
b=
(M − Mmin) (5.15− 4.80)
Don’t be surprised if the two methods disagree.
– Maximum likelihood puts more weight on smaller
magnitudes.
– But the two approaches usually give closer results
for larger data sets.