SME
Physical Geology
Earthquakes and volcanoes
Use this for “A fault ruptures 100 km long and 20 km deep and slips 2 m in crust of rigidity 30 GPa. Find M₀ and Mw.”
Diagram
The patch is 100 km by 20 km, drawn to scale (5 : 1).
M₀ = μLWD
M₀
=
30 ×
10
9
Pa
×
100,000
m
×
20,000
m
×
2
m
M₀
=
1.2 ×
10
20
N·m
Mw
=
(
2
3
)(log₁₀
M₀
−
9.1)
=
(
2
3
)(20.08
−
9.1)
=
7.319,
about
95.32
times
the
energy
of
an
M
6.
This is an example. Type your own values to start a new problem.
A (km²)
=
L (km)
×
W (km)
M₀ (N·m)
=
μ (GPa)
×
A (km²)
×
D (m)
Mw
=
(2
÷
3)(log₁₀
M₀
−
9.1)
log₁₀ M₀
Log of the moment
worked out (not typed)
Formulas
A
=
L
×
W
2,000 = 100 × 20
M₀
=
μ
×
10
9
×
A
×
10
6
×
D
1.2 ×
10
20
=
30 ×
10
9
×
2,000 ×
10
6
×
2
log₁₀
M₀
=
log₁₀(
M₀
)
20.0792
=
log₁₀(1.2
×
10
20
)
Mw
=
(
2
3
)
×
(log₁₀
M₀
−
9.1)
7.3195
=
(
2
3
)
×
(20.0792
−
9.1)
Assumptions
•
Mw measures the work of slip on the fault, so it does not saturate the way local magnitudes do.
•
Rigidity μ is about 30 GPa in the crust; M₀ = μAD needs μ in Pa, A in m² and D in m.
•
Each whole step of Mw is about 32 times the energy; the largest quakes recorded are near Mw 9.5.
Step-by-step
WE KNOW
D = 2 m, L = 100 km, W = 20 km, μ = 30 GPa
FIND
rupture area (A), seismic moment (M₀), log of the moment (log₁₀ M₀), moment magnitude (Mw)
Step 1 · Find rupture area (A)
Use A = L × W
The patch that slipped is the rupture’s length times its width down the fault.
A
=
100
×
20
A
=
2,000
km²
Step 2 · Find seismic moment (M₀)
Use M₀ = μ × 10⁹ × A × 10⁶ × D
The
moment
is
rigidity
(Pa)
×
area
(
m
2
)
×
slip
(m):
GPa
to
Pa
is
×
10
9
,
km²
to
m
2
is
×
10
6
.
M₀
=
30 ×
10
9
×
2,000 ×
10
6
×
2
M₀
=
(6
×
10
19
)
×
2
M₀
=
1.2 ×
10
20
N·m
Step 3 · Find log of the moment (log₁₀ M₀)
Use log₁₀ M₀ = log₁₀(M₀)
Take the log of the moment: the power of 10 it is.
log₁₀
M₀
=
log₁₀(1.2
×
10
20
)
log₁₀
M₀
=
20.0792
Step 4 · Find moment magnitude (Mw)
Use Mw = (2 ÷ 3) × (log₁₀ M₀ − 9.1)
The
scale
is
set
so
each
step
of
Mw
is
10
1.5
,
about
32
times
the
moment.
Mw
=
(
2
3
)
×
(20.0792
−
9.1)
Mw
=
2
3
×
10.9792
Mw = 0.6667 × 10.9792
Mw = 7.3195
Check
Put the numbers back in. Both sides should match.
2,000 = 100 × 20 ✓
1.2 ×
10
20
=
30 ×
10
9
×
2,000 ×
10
6
×
2
✓
20.0792
=
log₁₀(1.2
×
10
20
)
✓
7.3195
=
(
2
3
)
×
(20.0792
−
9.1)
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
Distance to a quake from the S − P lag
Use this for “The S wave arrives 25 s after the P wave. With vP = 6 km/s and vS = 3.5 km/s, how far away is the focus?”
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?”
Volcano types from magma and shape
Use this for “A steep cone of andesite layers erupts explosively. What type of volcano is it, and why?”