of 17
1
Geophysical
Research
Letters
Supporting
Information
for
The
2021
South
Sandwich
Island
M
w
8.2
earthquake:
a
slow
event
sandwiched
between
regular
ruptures
Zhe
Jia
1*
,
Zhongwen
Zhan
1
,
Hiroo
Kanamori
1
1
Seismological
Laboratory,
California
Institute
of
Technology,
Pasadena,
CA
91125,
USA.
Contents
of
this
file
Texts
S1
S2
Figures
S1
to
S10
Table
S1
S2
Introduction
This
supporting
information
provides:
two
additional
texts
for
the
methods
(Text
S1
S2),
10
figures
(Figure
S1
S10),
and
two
tables
(Table
S1
S2)
to
support
the
main
text.
2
Text
S1.
W
phase
inversion.
NEIC
PDE
listed
2
events:
8/12/2021
18:32:52
(UTC),
depth=47.2
km,
M
w
=7.5
8/12/2021
18:35:17
(UTC),
depth=22.8
km,
M
w
=8.1
These
2
events
are
only
145
s
apart,
and
the
standard
W
phase
inversion
for
the
1
st
event
could
not
converge
at
any
reasonable
solution
because
the
W
phases
from
the
2
events
interfered.
The
standard
inversion
for
the
2
nd
event
did
not
work
either
for
the
same
reason.
For
the
2
nd
event,
only
after
several
trials
using
different
frequency
bands,
we
could
achieve
a
reasonable
waveform
fit
with
a
relatively
narrow
low
frequency
band
0.00125
to
0.002
Hz
(
i.e
.,
500
to
800
s).
This
solution
(Fig.
S2)
can
fit
the
W
phase
waveforms
reasonably
well
for
45
phases
at
40
stations
(Fig.
S2).
However,
because
of
the
very
narrow
band
and
the
complex
interference
of
phases,
the
centroid
location,
especially
the
depth,
is
poorly
constrained
(Fig.
S3).
The
overall
mechanism
and
size
of
the
event
seem
to
be
reasonably
well
constrained.
The
centroid
time
shift,
t
c
,
is
20
s
from
18:35:17(UTC)
which
means
that
the
centroid
time
of
this
event
is
18:35:37(UTC)
which
is
165
s
after
the
origin
time
of
the
1
st
event.
Thus
we
consider
that
this
event
approximately
corresponds
to
the
GCMT
event
202108121832A
(referred
to
as
GCMT1,
centroid
time
18:35:25
UTC,
M
w
=8.3).
However,
the
quality
of
the
solution
is
not
up
to
the
standard
W
phase
solution
because
of
the
complex
waveform
interferences
of
the
multiple
events
of
the
sequence.
3
Text
S2.
Subevent
inversion
for
the
M
w
8.2
South
Sandwich
Island
sequence.
We
applied
our
subevent
inversion
method
to
simultaneously
estimate
the
source
parameters
of
5
subevents.
Each
subevent
has
10
point
source
parameters,
including
3
parameters
for
the
subevent
horizontal
location
and
depth,
a
centroid
time,
a
source
time
duration,
and
5
deviatoric
moment
tensor
elements.
For
the
long
period
subevent
E3,
we
added
two
finiteness
parameters,
rupture
velocity
and
rupture
direction,
to
accommodate
a
Haskell
unilateral
rupture
source
with
a
constant
rupture
velocity.
In
this
Haskell
model,
the
dependence
of
the
apparent
source
duration
on
rupture
direction
can
be
given
by
the
following
equation
(local
rise
time
is
ignored),
퐷ൌ퐷
1െ
∙cos
휃െ휑
,
where
is
the
apparent
source
duration
for
a
station
of
azimuth
,
is
the
rupture
duration,
is
the
rupture
velocity,
is
the
phase
velocity,
and
is
the
rupture
direction.
In
this
study,
we
calculate
phase
velocities
for
teleseismic
P
and
SH
waves
with
ray
tracing,
using
the
IASPEI91
model.
Because
the
regional
full
waveforms
are
dominated
by
surface
waves,
we
assume
to
be
4
km/s
as
approximate
Rayleigh
and
Love
phase
velocities
at
a
dominant
period
of
50s
and
300s.
Overall,
this
hybrid
parameterization
of
point
source
and
Haskell
subevents
has
52
unknown
parameters.
To
improve
the
searching
efficiency,
we
divide
our
inversion
procedure
into
two
stages,
where
we
search
a
part
of
these
parameters
nonlinearly
and
invert
the
data
for
other
parameters
in
a
linear
way.
The
outer
stage
has
a
Markov
Chain
Monte
Carlo
(MCMC)
inversion
sampler
that
searches
nonlinear
parameters
(subevent
locations,
centroid
times,
source
durations,
rupture
velocity
and
rupture
direction).
Its
random
walk
process
to
propose
new
models
is
driven
by
a
Metropolis
Hasting
algorithm.
In
each
step,
the
model
is
proposed
by
perturbing
one
of
the
nonlinear
parameters
while
keeping
the
other
nonlinear
parameter
at
their
current
values.
This
approach
ensures
a
high
acceptance
rate
and
improves
the
efficiency
of
converging
to
the
optimum.
For
each
set
of
nonlinear
parameters,
we
can
linearly
invert
the
data
for
the
moment
tensors
of
subevents
as
the
inner
stage,
because
the
observed
time
series
can
be
linearly
related
by
subevent
moment
tensors
and
their
Green’s
functions
when
subevent
locations
and
timing
are
available.
In
practice,
we
predict
apparent
source
time
functions
at
all
stations
using
the
subevent
locations
and
timings,
then
convolve
them
with
the
corresponding
Green’s
functions,
and
eventually
invert
for
deviatoric
subevent
moment
tensors
by
extending
the
linear
framework
to
multiple
sources.
In
this
way,
only
27
nonlinear
parameters
are
searched
through
the
MCMC
inversion,
and
it’s
much
easier
to
extensively
explore
the
model
space.
We
generated
72
Markov
Chains
and
eventually
kept
24
of
them
to
avoid
being
trapped
in
local
minima.
The
initial
sample
for
each
chain
is
randomly
generated
from
bounded
uniform
distributions.
Our
MCMC
inversion
incorporates
a
Bayesian
framework
that
propagates
the
data
error
and
prior
knowledge
to
the
model
error.
We
set
the
prior
of
all
unknown
parameters
to
be
uniform
distributions.
We
also
empirically
set
the
data
error
to
be
10%
to
accommodate
the
inaccurate
assumptions
of
the
wave
propagation
processes,
even
though
the
true
data
error
(noise
and
instrumental
error)
of
the
seismic
waves
are
very
small.
This
data
error
eventually
turns
to
the
width
of
the
Markov
Chain
sample
distributions,
which
reflects
the
posterior
probability
density
functions.
We
used
58
vertical
component
teleseismic
(epicentral
distance
of
30°
90°)
P
waves
in
both
displacement
and
velocity,
43
transverse
component
teleseismic
SH
waves
in
displacement,
12
three
component
regional
(epicentral
distance
within
40°)
full
waveforms
in
displacement
in
4
our
subevent
inversion
from
the
Global
Seismic
Network
and
the
International
Federation
of
Digital
Seismograph
Networks.
The
weighting
of
these
three
datasets
is
set
to
be
20:10:1
for
similar
final
misfit
contributions.
For
the
inversion
of
teleseismic
waves,
we
calculate
the
Green’s
functions
with
a
hybrid
method
that
combines
propagator
matrix
and
ray
theory,
and
use
a
combination
of
the
CRUST2.0
velocity
model
at
the
source
location
with
an
IASPEI91
model
in
the
deeper
earth.
A
limitation
of
this
forward
simulation
method
is
that
it
does
not
consider
PP
or
SS
phases,
but
since
the
M
w
7.5
foreshock
is
much
smaller
than
the
M
w
8.1
mainshock,
the
PP
and
SS
amplitudes
of
the
foreshock
do
not
overwhelm
the
P
and
S
of
the
mainshock,
the
waveform
interferences
are
limited.
We
also
compute
the
regional
full
waveform
synthetics
with
a
frequency
wavenumber
integration
algorithm
using
the
PREM
model
as
an
average
structure
from
sea
to
land.
We
used
the
P
and
S
arrival
times
predicted
from
ray
tracing
with
the
IASPEI91
model,
and
allowed
maximum
time
shifts
of
4s,
6s
and
10s
for
the
P,
SH,
and
regional
full
waves.
Because
all
subevents
could
move
their
horizontal
locations
together
with
the
seismograms
shifting
simultaneously,
we
need
to
fix
the
horizontal
location
of
one
subevent.
Therefore,
we
anchored
the
location
of
the
first
subevent
E1
at
the
hypocenter
of
the
M
w
7.5
foreshock,
assuming
the
rupture
dimension
between
the
initiation
and
centroid
of
E1
is
small
compared
with
the
full
sequence.
5
Figure
S1.
Tsunami
of
the
South
Sandwich
Island
sequence
observed
by
tide
gauges.
(a)
Distributions
of
the
tide
gauge
stations
(black
triangles).
The
red
star
indicates
the
hypocenter
of
the
South
Sandwich
Island
sequence.
(b)
Waveforms
recorded
at
the
tide
gauges
in
(a).
The
waveforms
are
high
pass
filtered
to
periods
shorter
than
5
hours.
The
number
above
each
trace
shows
the
peak
absolute
amplitude
of
the
tsunami.
Below
each
trace
the
tide
gauge
station
name
and
the
distance
from
the
South
Sandwich
Island
earthquake
source
are
shown.