of 15
Geophys.
J.
Int.
(2024)
236,
889–903
https://doi.org/10.1093/gji/ggad459
Advance
Access
publication
2023
November
28
GJI
Seismology
Finite
source
properties
of
large
strike-slip
earthquakes
James
Atterholt
and
Zachary
E.
Ross
Seismological
Laboratory,
California
Institute
of
Technology,
Pasadena,
CA
91125
,
USA.
E-mail:
atterholt@caltech.edu
Accepted
2023
November
20.
Received
2023
October
12;
in
original
form
2023
July
10
S
U
M
M
A
R
Y
Ear
thquake
r
uptures
are
complex
physical
processes
that
may
var
y
with
the
str
ucture
and
tectonics
of
the
region
in
which
they
occur.
Characterizing
the
factors
controlling
this
variability
would
provide
fundamental
constraints
on
the
physics
of
earthquakes
and
faults.
We
investigate
this
by
determining
finite
source
properties
from
second
moments
of
the
stress
glut
for
a
global
data
set
of
large
strike-slip
earthquakes.
Our
approach
uses
a
Bayesian
inverse
formulation
with
teleseismic
body
and
surface
waves,
which
yields
a
low-dimensional
probabilistic
description
of
r
upture
proper
ties
including
the
spatial
de
viation,
directi
vity
and
temporal
de
viation
of
the
source.
This
technique
is
useful
for
comparing
events
because
it
makes
only
minor
geometric
constraints,
avoids
bias
due
to
rupture
velocity
parametrization
and
yields
a
full
ensemble
of
possible
solutions
given
the
uncertainties
of
the
data.
We
apply
this
framework
to
all
great
strik
e-slip
earthquak
es
of
the
past
three
decades,
and
we
use
the
resultant
second
moments
to
compare
source
quantities
like
directivity
ratio,
rectilinearity,
average
moment
density
and
vertical
deviation.
We
find
that
most
strike-slip
earthquakes
have
a
large
component
of
unilateral
directi
vity,
and
man
y
of
these
earthquakes
show
a
mixture
of
unilateral
and
bilateral
behaviour.
We
notice
that
oceanic
intraplate
earthquakes
usually
rupture
a
much
larger
width
of
the
seismogenic
zone
than
other
strike-slip
earthquakes,
suggesting
these
earthquakes
may
often
breach
the
expected
ther
mal
boundar
y
for
oceanic
ruptures.
We
also
use
these
second
moments
to
resolve
nodal
plane
ambiguity
for
the
large
oceanic
intraplate
earthquakes
and
find
that
the
rupture
orientation
is
usually
unaligned
with
encompassing
fossil
fracture
zones.
Key
words:
Fault
zone
rheology;
Probability
distributions;
Earthquake
source
observations;
Intraplate
processes;
Fractures
faults
and
high
strain
deformation
zones.
1
INTRODUCTION
Large
earthquakes
involve
complex
ruptures
that
can
vary
strongly
between
events.
The
characteristics
of
these
ruptures
may
be
con-
trolled
by
the
structural
and
tectonic
characteristics
of
the
fault
zone,
and
understanding
patterns
in
these
ruptures
may
improve
our
understanding
of
the
interplay
between
source
phenomenology
and
the
rupture
zone.
In
particular,
large
strik
e-slip
earthquak
es
are
known
to
show
considerable
variability
in
rupture
properties
be-
tween
events
(e.g.
Hayes
2017
;
Yin
et
al.
2021
;
Bao
et
al.
2022
).
Systematically
characterizing
this
variability
has
the
potential
to
yield
insights
into
the
underlying
controls
on
the
rupture
process.
These
insights
are
of
societal
and
scientific
interest
because
these
earthquakes
present
significant
global
hazard
and
provide
unique
windows
into
the
structure
and
rheology
of
the
lithosphere.
Sev-
eral
faults
known
to
host
large
strike-slip
earthquakes
are
in
close
proximity
to
dense
population
centres.
There
is
also
wide
specu-
lation
that
the
propagation
behaviour
and
rupture
dimensions
are
dictated
by
the
structural
(Ben-Zion
&
Andrews
1998
;
Wesnousky
2008
)
and
rheological
properties
(Abercrombie
&
Ekstr
̈
om
2001
;
Boettcher
et
al.
2007
)
of
the
host
fault
zone.
Intraplate
oceanic
earthquakes
are
particularly
enigmatic,
because
the
explanation
for
the
weakening
of
the
oceanic
lithosphere
that
accommodates
these
events
remains
elusive
(Lay
2019
).
A
general
quantity
for
describing
the
space–time
kinematics
of
ear
thquake
r
uptures
is
the
so-called
stress
glut
(Backus
&
Mulc-
ahy
1976a
),
which
quantifies
the
breakdown
of
linear
elasticity
in
space
and
time
(Dahlen
&
Tromp
1998
).
Finite-fault
slip
distri-
butions,
w
hich
appro
ximate
the
stress
glut
as
discretized
slip
on
a
predefined
fault
plane,
are
routinely
computed
for
large
events
(Mai
&
Thingbaijam
2014
).
These
solutions
provide
a
high
di-
mensional
view
of
fault
ruptures
but
in
practice
are
challenging
to
compare
between
events
due
to
pervasive
non-uniqueness
in
the
inverse
problem,
a
priori
fault
plane
parametrization,
poor
rupture
velocity
sensitivity
and
regularization.
An
alternative
technique
for
characterizing
earthquake
source
properties
is
the
second-moment
formulation
(Backus
&
Mulcahy
1976a
,
b
).
Instead
of
approximat-
ing
the
stress
glut
as
a
superposition
of
assigned
subevents,
this
approach
involves
solving
for
the
second
order
polynomial
mo-
ments
of
the
stress
glut,
yielding
a
source
covariance
matrix
that
approximates
the
spatiotemporal
extent
of
the
source.
This
tech-
nique
has
been
successfully
applied
in
the
past
(Bukchin
1995
;
C

The
Author(s)
2023.
Published
by
Oxford
University
Press
on
behalf
of
The
Royal
Astronomical
Society.
This
is
an
Open
Access
article
distributed
under
the
terms
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the
Creative
Commons
Attribution
License
(
https://creati
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y/4.0/
),
which
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unrestricted
reuse,
distribution,
and
reproduction
in
any
medium,
provided
the
original
work
is
properly
cited.
889
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890
J.
Atterholt
and
Z.E.
Ross
McGuire
et
al.
2000
,
2001
,
2002
;
Cl
́
ev
́
ed
́
e
et
al.
2004
;
Chen
2005
;
Meng
et
al.
2020
),
but
has
received
far
less
attention
than
slip
in
versions.
This
lo
w-dimensional
framew
ork
makes
only
minor
as-
sumptions
about
rupture
geometries
and
has
the
advantages
of
not
requiring
an
explicitly
parametrized
rupture
velocity
and
avoiding
the
discretization
challenges
that
arise
when
performing
slip
in-
versions.
The
low
dimensionality
of
the
solution
also
facilitates
computation
of
these
moments
with
a
Bayesian
approach
(Atterholt
&
Ross
2022
),
which
can
provide
uncertainty
estimates
crucial
for
comparing
the
source
processes
of
different
earthquakes.
Our
contributions
to
this
study
are
as
follows.
We
compute
sec-
ond
moments
for
all
of
the
M
w
7.5
strike-slip
earthquakes
of
the
past
three
decades
using
a
Bayesian
inference
approach.
We
use
this
catalogue
to
establish
baselines
for
the
range
of
values
observed
globally
and
compare
values
between
events,
subgroups
and
other
tectonic
features.
From
these
analyses
we
conclude
that
(i)
large
strik
e-slip
earthquak
es
almost
al
wa
ys
show
unilateral
or
a
comparable
amount
of
unilateral
and
bilateral
directivity
behaviour,
(ii)
that
large
intraplate
oceanic
earthquakes
usually
rupture
over
a
much
wider
depth
range
and
(iii)
that
oceanic
intraplate
strike-
slip
earthquakes
are
not
systematically
aligned
with
fossil
fracture
zones.
2
PRELIMINARIES
The
stress
glut
is
a
tensor
field
representing
the
expected
stress
due
to
the
application
of
Hooke’s
law
to
inelastic
strain
in
a
body
(Backus
&
Mulcahy
1976a
,
b
).
The
stress
glut
is
a
useful
source
characteri-
zation
quantity,
because
the
stress
glut
is
identically
zero
outside
the
source
region
and
can
be
used
to
compute
displacements
anywhere
on
Earth
resulting
from
an
arbitrary
source
(Dahlen
&
Tromp
1998
).
The
dimensionality
of
the
stress
glut
can
be
significantly
reduced
by
assuming
the
source
mechanism
does
not
change
throughout
the
rupture:

ij
(
ξ
ξ
ξ,
τ
)
=
ˆ
M
ij
f
(
ξ
ξ
ξ,
τ
)
.
(1)
Here,



is
the
stress
glut,
ˆ
M
is
the
normalized
mean
seismic
moment
tensor
and
f
is
a
scalar
function
of
position
ξ
ξ
ξ
and
time
τ
.
The
second
moment
formulation
is
defined
by
taking
the
second
central
moment
of
the
scalar
stress
glut
rate
function
(
̇
f
)
with
terms
for
the
spatial
and
temporal
components.
The
equation
for
these
moments
is
given
by:
̇
f
(
ξ
ξ
ξ
c
,
τ
c
)
(
m
,
n
)
=
̇
f
(
ξ
ξ
ξ,
τ
)(
ξ
ξ
ξ
ξ
ξ
ξ
c
)
m
(
τ
τ
c
)
n
d
ξ
ξ
ξ
d
τ,
(2)
where
ξ
ξ
ξ
and
τ
are
position
and
time,
ξ
ξ
ξ
c
and
τ
c
are
the
centroid
position
and
centroid
time,
which
are
the
first
moments
of
the
dis-
tribution,
and
m
and
n
are
the
spatial
order
and
temporal
order
of
the
moment.
The
central
moments
of
order
m
+
n
=
2
correspond
to
the
covariance
of
the
source.
Specifically,
f
(
ξ
ξ
ξ
c
,
τ
c
)
(2
,
0)
is
the
spatial
covariance,
f
(
ξ
ξ
ξ
c
,
τ
c
)
(0
,
2)
is
the
temporal
variance
and
f
(
ξ
ξ
ξ
c
,
τ
c
)
(1
,
1)
is
the
spatiotemporal
covariance.
The
distribution
̇
f
is
defined
by
the
superposition
of
its
polynomial
moments,
and
at
low-enough
frequencies,
the
contribution
of
moments
of
order
three
or
greater
may
be
approximated
as
zero.
Under
this
approximation,
the
second
moments
can
be
linearly
related
to
displacement:
d
=
Fp
,
(3)
where
d
is
a
vector
of
the
difference
between
the
measured
dis-
placements
and
the
theoretical
Green’s
functions,
F
is
a
forward
propagation
matrix
of
spatial
and
temporal
integrals
and
deri
v
ati
ves
of
the
Green’s
tensor
weighted
by
the
components
of
the
moment
tensor
M
,
and
p
is
a
vector
that
contains
the
independent
parameters
of
the
second
order
stress-glut
moments.
Since
the
standard
deviation
of
a
distribution
gives
a
low-
dimensional
estimate
of
the
width
of
a
distribution,
these
second
moments,
which
are
the
covariance
of
the
stress-glut,
can
be
used
to
compute
low-dimensional
measures
of
the
volume,
duration
and
directivity
of
a
source
distribution.
In
particular,
we
define
dimen-
sional
quantities
of
the
source
that
describe
the
shape
of
the
stress-
glut
distribution
about
the
centroid.
These
are:
r
c
(
ˆ
n
)
=
ˆ
n
T
·
[
̇
f
̇
f
̇
f
(2
,
0)
(
ξ
ξ
ξ
c
,
τ
c
)
/
̇
f
(0
,
0)
(
ξ
ξ
ξ
c
,
τ
c
)]
·
ˆ
n
,
t
c
=
2
̇
f
(0
,
2)
(
ξ
ξ
ξ
c
,
τ
c
)
/
̇
f
(0
,
0)
(
ξ
ξ
ξ
c
,
τ
c
)
,
v
0
=
̇
f
̇
f
̇
f
(1
,
1)
(
ξ
ξ
ξ
c
,
τ
c
)
/
̇
f
(0
,
2)
(
ξ
ξ
ξ
c
,
τ
c
)
.
(4)
Here,
r
c
(
ˆ
n
)
is
the
distance
from
the
centroid
in
the
direction
of
a
unit
vector
ˆ
n
that
defines
a
spatial
deviation
ellipsoid
in
which
most
of
the
moment
was
released.
The
characteristic
spatial
devia-
tion
of
the
source
is
gi
ven
b
y
L
c
=
2
r
c
(
η
η
η
)
,
where
η
η
η
is
the
principal
eigenvector
of
̇
f
̇
f
̇
f
(2
,
0)
.
t
c
is
the
characteristic
temporal
deviation
that
captures
a
time
interval
in
which
most
of
the
moment
was
re-
leased.
v
0
is
the
average
instantaneous
velocity
of
the
rupture
cen-
troid.
These
quantities
together
provide
a
physically
interpretable,
low-dimensional
estimate
of
a
rupture’s
spatiotemporal
behaviour
(Backus
1977
;
Silver
&
Jordan
1983
).
From
the
aforementioned
quantities,
we
can
compute
ensembles
of
parameters
that
may
further
illuminate
potential
differences
be-
tween
ruptures.
In
particular,
we
inspect
four
derived
parameters
in
this
study:
rectilinearity
(
R
),
directivity
ratio
(
α
),
average
moment
density
(
̄
m
)
and
vertical
deviation
(
Z
).
Rectilinearity
is
a
measure
of
the
degree
of
elongation
along
an
axis
and
has
been
used
in
seis-
molo
gy
to
e
v
aluate
particle
motion
(Vidale
1986
;
Jurke
vics
1988
).
We
instead
use
this
measure
to
e
v
aluate
the
elongation
of
ruptures
along
the
principal
axis,
and
define
it
so
the
values
are
bounded
between
0
(spherical
source)
and
1
(linear
source):
R
=
1
1
2
(
λ
2
+
λ
3
)
λ
1
.
(5)
The
variables
λ
1
,
λ
2
and
λ
3
are
the
eigenvalues
of
the
spatial
second
moment,
in
order
of
largest
to
smallest
and
yield
estimates
of
the
dimensions
of
the
source
along
its
principal
axes.
The
directivity
ratio
provides
an
estimate
of
the
degree
of
directivity
of
a
rupture
by
comparing
the
average
centroid
velocity
to
the
maximum
possible
av
erage
centroid
v
elocity,
and
has
been
used
in
the
second
moment
literature
pre
viousl
y
(McGuire
et
al.
2002
):
α
=
||
v
0
||
(
L
c
/
t
c
)
.
(6)
The
average
moment
density
comes
from
the
moment
tensor
den-
sity
formulation
(Aki
&
Richards
2002
)
and
compares
the
scalar
moment,
M
0
,
to
the
volume
of
the
ellipsoid
representation
of
the
spatial
second
moments:
̄
m
=
M
0
(2)
3
4
3
πλ
1
λ
2
λ
3
.
(7)
As
is
apparent,
the
equation
for
̄
m
is
very
similar
to
analytic
solutions
for
the
stress
drop
on
planar
sources
(e.g.
Eshelby
1957
)
and
has
units
of
stress.
Finally,
we
define
a
measure
of
the
vertical
extent
of
the
source:
Z
=
2
γ
z
r
c
(
γ
γ
γ
)
.
(8)
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