of 25
www.sciencemag.org/content/
357/6357/1277/suppl/DC1
Supplementary
Material
s for
The hidden simplicity of subduction megathrust earthquakes
M.
-A. Meier,* J. P. Ampuero, T. H. Heaton
*Corresponding author. Email: menandrin@gmail.com
Published 22 September
2017,
Science
357
, 1277 (2017)
DOI:
10.1126/science.aan5643
This PDF file includes:
Supplementary
Text
S1 to S7
Figs. S1 to S13
References
Other Supplementary
Material
s for this manuscript include
the following
:
(available at
www.sciencemag.org/content/357/6357/1277/suppl/DC1)
Data File
S1
(i STF
_vs_
medSTF
.zip): A set of 116 png files showing the individual
source time functions and how they compare to the median source time functions in
seven
magnitude bins
Supplementary Text
S1: Kinemat
ic source inversion by Ye et al., 2016a
Ye et al., 2016a have applied a uniform kinematic source inversion method to a large
number of large subduction zone thrust earthquakes of the past three decades. They use P
-wave
(and in a few cases SH
-) displacement
records
in the frequency band 0.005-
0.9Hz
, and
theoretical Greens functions assuming a 1D velocity model. Each earthquake is described as a
rectangular plane array of sub
-faults. On each sub
-fault slip rate is parameterized as a sequence
of overlapping tr
iangles whose amplitudes are optimized to best explain the observed waveforms
in a least
-squares sense. The method imposes the following constraints: slip rate positivity
, a
higher bound on rupture velocity and rise time, and linear spatial smoothing (
18
). T he STF of the
2004 M
w
9.1 Sumatra earthquake
is an exception in that it has not been inferred with the same
code
, but has been taken
from (
48
)
.
S2: Sampling log-
normal distributions
Supplementary
Fig.
S1 shows how well synthetic input STFs are recovered using different
distribution center estimates if residuals are log
-normally distributed. We construct 1e4 synthetic
STFs for each
of the seven magnitude bins using the functional form y
fit
= μ*t*exp(
-1/2*(
λ
t)^2).
We choose μ=1.25e18Nm/s
2
and optimize
λ
to fit the functional form to the median observed
STF in each bin. We multiply each STF with a random log
-normal error term, 10^
ε
, with
ε
=
N(0,
σ
2
), and where
σ
2
has been measured from the real STF relat
ive to the fitted curve of each
bin. We then estimate the distribution center at each point in time by taking the median, the
arithmetic mean and the arithmetic mean of the log
-transformed amplitudes (Fig
. S1). As
expected, the median and the mean of the l
og-transformed values give an unbiased estimate of
the distribution center, while the arithmetic mean is strongly biased towards higher amplitudes.
This is because log
-normal distributions are
skewed towards large values
, i.e. the deviations
above the cent
er are larger then the ones below the center
. This bias is stronger for larger
magnitude bins, which then
result
s in an apparent magnitude
-dependence of STF growth
(Fig
.
S1
B). In this study we use median statistics because they are reliable estimates independent of
the distribution, and are particularly suitable for log
-normal distributions.
Fig. S1. Synthetic test for sampling log
-normal STF amplitude distributions.
At each
point in time the distribution center is estimated using (
A
) medians,
(
B
) arithmetic means and
(
C
) arithmetic means of log
-transformed values (right).
While the median and log
-means
accurately recover the synthetic input function, the arithmetic me
an leads to a biased estimate
with over
-estimated amplitudes and an apparent magnitude dependence of the initial slope.
S3: Comparison of STF from three independent data set
s
Earthquake rupture s
ource inversions are
generally
ill-
posed and non
-unique
, and
source
inversion results can be affected by method-
specific biases that are difficult to quantify.
In order
to evaluate if the STF models we use are affected in this way we make use of two alternative
data sets that contain sufficiently large numbers o
f STFs
, and that were also obt
ained by applying
a consistent method to a large number of events
. The two data sets are those
of
Hayes 2017, (
20
,
in t
he following referred to as "Hayes17"), and that of Vallée et al., 2011 (
21
, "Vallée11")
. W
e
refer to the data set of Ye et al., 2016a
as
"Ye16
".
We f
irst identified matching STFs in the three data sets in order to do a one
-to-one STF
comparison. We then repeated the main part of our STF analysis with the two alternative
data
sets
, for a systematic robustness analysis
of our results
. This analysis reveals that the
observations we report are stable across data sets
and inference methods
. In particular, all three
data sets have i) linear and magnitude-
independent moment rate growth, ii) a simple near
-
triangular a
verage shape that
- after normalization
- is virtually identical for all magnitudes in the
studied range
, and iii) log
-normal deviations from a best fit model
.
Data sets
All three STF data sets are derived from teleseismic data, but use different methods to
infer the STFs.
The Hayes17 STFs were generated with a kinematic source inversion procedure
(REF) similar to that applied by Ye et al., 2016a,
but
Hayes17 used a different slip
parameterization with smaller spatial grid cells and a single cosine slip ra
te function on each sub
-
cell, leading to a non
-linear inverse problem
. This is in contrast to the linear inverse problem
based on multi
-window source parameterization used for the Ye16 data set. 62 of the events in
the Ye16 data set are also contained in H
ayes17 data set. The missing events are mostly events
with Mw<7.5, which is a range of magnitudes outside the focus of the Hayes17 data set. The
STFs from Vallée11
, on the other hand,
are computed via de-
convolution of synthetic Greens
functions
, and in a fully automated fashion. For each recording station
an apparent STF is
computed, which is then averaged to obtain the actual
STF.
Of the 116 STFs in the Ye16 data set
we found 99 matching
events in the Vallée11 data
set
.
Comparison of individual STFs
Despite fundamental differences in how the STF
s are computed
, they only differ in terms
of second-
order details
for a majority of cases
. Fig
. S
2 show
s all individual 116 STFs of the
Ye
16 data set, along with the matching STFs from the other two data sets
. Apart from a small
number of cases with large differences, such as the Mw7.2, 2007 Santa Cruz earthquake, or the
failure of the Vallée11 method for the Mw9.1 2004 Sumatra earthquake
, the STF from the
different methods are very similar. For most earthquakes
, the first
-order STF shape is consistent
between the three methods,
and
the
y have comparable amounts of
high frequency
oscillation
s. In
some cases, the Vallée11 method seems to o
versimplify the STF, in that it suggests a very simple
STF while the other tw
o methods
suggest a more
complicated rupture evolution (e.g. Mw7.2
2003 New Zealand, Mw8.6 2005 Sumatra). The Hayes17 method often seems to produce
long
tails (e.g. Mw8.8 2010 Chile, Mw7.8 2010 N
-Sumatra). Since the termination of the ruptures is
typically
not well constrained (see discussion in main manuscript), these tails are rather
unrealistic
. Other than that, the STFs are very similar.
The fact that kinematic source inversions lead to highly similar STFs to those from a de
-
convolution approach is
remarkable, and speaks to the general stability of source time function
estimates. This supports the general perception that the STF is one of the most robust features of
finite source descriptions because it is
a relatively direct mapping of the displacem
ent
waveforms. This is particularly true for situations where the dip is well constrained, such as in
the large subduction earthquakes considered in this study. The fundamental ill
-posedness of
source inversion problems may manifest itself more strongly in
other aspects of source
inversions, in particular in the spatial distribution
of slip.
Fig
. S2
: Comparison of individual STFs from 3 different methods.
O
f the 116 STF
s from the
Ye16 data set, 99 are also contained in the Vallée11 data set, and 62 in t
he Hayes17 data set. The
STF
s generally agree very well across methods and show the same first order shape. No method
appears to be producing systematically steeper onsets, or systematically more skewed shapes.
The Hayes17 method often produces rather long
low amplitude tails
that are not supported by the
other two methods
.
Fig
. S2
, continued.
Fig
. S2
, continued.
Inferring the median STF with the three data sets
We repeat
ed the main body of our STF analysis with the matching STFs of the Vallée11
(99 STFs)
and
the
Hayes17 (62 STFs) data sets. The median STFs in the seven magnitude bins
for all 3 data sets lead to the near
-triangular typical STF shape
(Fig. S
3A
-C). The largest
difference is that the median Hayes17
STFs
(Fig
. S3C)
are more skewed towards early tim
es and
feature longer tails
than the other two data sets. Apart from this discrepancy, all three data sets
show the same main features that our analysis reports:
Larger magnitudes start neither more nor less impulsively
t han smaller ones
. Large
earthquakes started as smaller ones do, but do not stop as quickly. This means, there will not
be strong rupture predictability.
The moment rate growth is approximately linear all the way until peak moment rate is
reached; this is i
n strong contradiction to often-
used standard models, which predict
quadratic growth.
After normalization, the median STF for different magnitudes collapse onto a single near
-
triangular shape, suggesting scalability between small and large ruptures.
The l
argest difference is that, w
hile the median STFs from the Ye16 and the Vallée11
data sets are fairly symmetric, those of
Hayes17 are somewhat more skewed
towards early times
and feature longer tails
(Fig. S
3 E&F
). This asymmetry
likely stems from the prominent low
-
amplitude tails we have observed with the individual STFs of Hayes17 (Fig
. S 2). In order to
verify this presumption we truncated the Hayes17 STF
s when the corresponding STFs from the
other two data sets ended. We measure the time of the last non-
zero sample of the Ye16 and, if
available,
Vallée11 STFs and use the l
onger of the two
for the truncation.
In ot
her words, we
suppress the tail
s of the Hayes17 STF
s if they are not supported by at least one of the other two
methods
. Note t
hat for the truncation we use the duration estimate of when the STF
s reach
amplitude 0, rather than the centroid times
, to be conservative and truncate as little as possible.
Indeed, a
fter the truncation, the median STFs (Fig. S3 G&H
) are no longer more
asymmetri
c than th
ose of the other data sets
, and share the same near
-triangular median STF
shape. This confirms
that it is indeed the badly constrained low amplitude tails that caused the
stronger asym
metry
. The truncated Hayes17 STF on average reach peak moment rates between
35-
55% of total rupture durations, a
s is the case with the other two data sets
. For normalizing the
STFs in Fig. S3H we use
d twice the centroid time, rather than the truncation duration, in order to
be consistent across all
data sets.
Note that, owing to the different sizes
of the three data sets
, and their different magnitude
ranges, the median STFs
cannot always be compared directly
. The
smaller number of STFs in
the Hayes17 data set
(roughly half the size of the Ye16 data set)
means that the 20 nearest
neighbors of our binning scheme cover a wider magnitude range. Furthermore, the lower
magnitude bins are not well represented in the Hayes17 data set, because the data set has few
events with
Mw<7.5. One should consider the median Mw for the bins, rather than the target Mw
(both are given in the figure legends
).
Fig.
S 3: Median STFs for the different
data sets
. Equivalent plots to Fig. 2 in main
manuscript. The number of STFs in each data set is given in title
. (A &
B) Main data set used in
this study, by Ye et al., 2016a
. (C & D)
Data set of Vallée et al., 2011, GJI. (E & F)
Data set of
Hayes et al., 2017, EPSL
. ( G & H
). Data set of Hayes et al., 2017, EPSL with badly constrained
low
-amplitude tails removed
. All data sets exhibit linear and magnitude independent STF
growth, near
-triangular STF shapes,
and magnitude
-independent normalized shapes. For detailed
caption, see Fig. 2 in main manuscript.
Parameterization of individual STF with the three data sets
When we fit the functional form
y
fit
= μ*t*exp(
-
1/2*( λ
t)
2
)
to each individual STF, we
find very similar optimal parameters
across the three data sets (Fig
. S
4) . After seeing the
general
similarity of the individual STFs (Fig.
S 2) this is
not surprising
. This exercise shows that the fits
between all three data sets are not just similar, but
indeed
statistically consistent in that the
difference between individual bootstrap realizations are larger than the differences between data
sets. All three data sets are consistent with the hypothesis that the initial moment rate growth is
not different for smaller and larger events. Furthermore, in all data sets
the scaling exponent
between seismic moment and duration is higher
than 1/3 (Ye16: 0.41, Vallée11: 0.46, Hayes17:
0.48, Hayes17, truncated: 0.46).
Fig
. S4: Optimized p
arameter
s for th
e different data sets
: Ye
16 (
A-C), Vallée11 (
D-
F), Hayes17 (
G-I) and Hayes17 with truncated tails (J
-L). The trends from all data sets are
consistent with each other in that the variability between data sets is smaller than the bootstrap
variability for a
ny individual data set. For detailed caption, see Fig. 3 in main manuscript.
Deviations from best
-fit model
Similarly, if we compare the deviations of the individual STFs
from
the best
-fit model
(Fig.
S 5) , the results f
rom the three data sets are consistent. In particular, the deviations are
multiplicative and Gaussian
, with standard deviations that do not change substantially between
magnitude bins. The
average standard deviati
ons are slightly larger for the Vallée11 (0.
49) and
the Hayes17 data (0.
42, 0.42) than those of the Ye16 data (0.38)
.
Most of t
he Hayes17 STFs have negative residuals
at the very beginning
, suggesting that
the observed STFs have low
er amplitudes than the best
-fit STF. This likely is a consequence of
the cosine slip rate parameterization employed by Hayes17. While this parameterization requires
only a single parameter per sub-
fault, this comes at the cost of reduced flexibility
(
49
). The shape
of the slip rate function is prescribed and cannot be adapted to best match the observed shape.
The multiple
-time
-window approach of Ye et al., 2016a, on the other hand, comes with more
flexibility to match a wide range of observed shape
s. Since both the m
ultiple
-time
-window
approach of Ye16, as well
the free form approach of Vallée11
agree well with the initial part of
the functional form, we
conclude that those descriptions are more reliable
for the purpose of
analyzing STF onsets
.
Summary
The systematic comparison of the STFs from the three different data sets shows that the
STFs are well constrained, with very similar properties across the data sets. This supports the
common perception that the STF
s are among
the best
-resolved aspects of earth
quake source
descriptions. In particular the fact that fundamentally different methodologies (kinematic source
inversion vs. Greens function deconvolution) lead to very similar STF demonstrates the
robustness of the estimated STFs. We therefore conclude that the observations we report (near
-
triangular STF shape, linear and magnitude
-independent moment rate growth, and multiplicative
Gaussian
moment rate deviations) are stable and are not systematically affected by the properties
of the inversion methodology
of Ye et al. 2016a.