In particular, self-similar pulse and crack models
predict that moment rate onset evolves as ~th,
with h = 2 [e.g., (5, 40, 41)], in contrast to the
linear growth that we observed. The implications
for rupture models can be understood through
simple scaling relations. For a rupture with area
growing as SðtÞºta and average slip growing as
;DðtÞºtb, seismic moment M0ðtÞºSðtÞ;DðtÞ
scales as M0ðtÞºtaþb and moment rate grows
with an exponent of h ¼ a þ b ; 1. The exponent that we observed is hobs ¼ 1 such that
a þ b ¼ 2. This constraint is incompatible with
standard self-similar rupture models, which have
a ¼ 2 and b ¼ 1. One admissible end-member
model is a circular pulse growing with constant
speed and constant slip (i.e., a ¼ 2 and b ¼ 0).
Such behavior emerges in steady-state pulse
models (42, 43) and in models of rupture over
disconnected slip patches with areas of systematic slip deficits (22–25, 39). If slip is not constant
(i.e., b > 0), the rupturing area must grow more
slowly than predicted by these models (a < 2).
An admissible model with b > 0 is that of a
rupture expanding in one predominant direction
[e.g., (36, 44)] with linearly increasing slip (i.e.,
a ¼ 1 and b ¼ 1). Combined with independent
scaling relations (e.g., between moment and slip),
the condition a þ b ¼ 2 derived here from STFs
is a valuable constraint on rupture models.
Further insight into the deviation from self-similar rupture can be gained by combining our
observations with peak ground motion displacement observations from short-distance recordings of shallow crustal earthquakes (12). Peak
displacements are a proxy for moment rate and
were observed to be consistent with self-similar
two-dimensional growth (h = 2) for ≤1 s after
rupture onset, after which growth was substantially slower. The shallow crustal earthquakes
also had the same magnitude-independent onset behavior in that small and large earthquakes
grow with equal rates at first until the smaller
ones branch out as they approach their peak
moment rates. The STFs analyzed here, which
become reliable only after several seconds (19),
have a well-resolved linear trend (h = 1). This
suggests that the ruptures at some point transition from initially self-similar to systematically
slower growth. This transition must occur before the linear trend is well constrained, at
source dimensions much smaller than the seismogenic width, which is ~200 km for subduction earthquakes (45). Therefore, whereas small
earthquakes with durations <1 s (i.e., M ≲ 5.0)
may be accurately described as self-similar ruptures, the nature of larger ruptures seems to
change after they have reached a critical size
that is not controlled by seismogenic width.
The change could occur, e.g., because plausible
weakening mechanisms, such as fault melting or
thermal pressurization, need a minimum energy
level to be triggered.
The multiplicative nature of the STF residuals
has direct consequences for long-period (T > 1 s)
strong earthquake ground motion. Large events
with high moment rates have proportionally high
moment rate fluctuations around the smooth
model. Moment rates are directly proportional
to long-period ground motion, which explains
why ground motion variability at such periods
grows strongly with magnitude, when measured
on a linear scale. In ground motion prediction
equations, this increasing variability is commonly
captured with a log-normal residual description
[e.g., (46, 47)]. The multiplicative nature of STF
residuals implies that large-magnitude events
can feature extreme values of long-period ground
motion, not as outliers, but as random realiza-
tions of a multiplicative process.
The shape that we observed for typical mo-
ment rate evolution precludes strong rupture
predictability as smaller and larger earthquakes
are statistically indistinguishable until they reach
their peak moment rate. This result has implica-
tions for EEW. Rupture-onset observations can-
not diagnose the final rupture size. However, the
fairly symmetrical nature and the gradual decay
of STFs imply a weak rupture predictability: At
any point during moment rate growth, the final
seismic moment is statistically at least twice as
large as the moment already generated. This
makes a large difference for large ruptures. For
instance, an earthquake that reaches Mw = 7.2
during its growing phase will at least double its
size to Mw = 7.4. This strongly increases the area
experiencing dangerous ground motion. Estimat-
ing moment rates in real time with EEW algo-
rithms could therefore substantially lengthen
Individual source time functions of large shal-
low subduction zone earthquakes are often com-
plicated and have a diverse range of forms,
reflecting complex physical rupture processes. De-
spite this large diversity, however, there is a well-
defined temporal pattern that such earthquakes
typically follow. This pattern attests to the uni-
versality of the physical mechanisms that dom-
inate rupture process in such earthquakes. The
simple base model and the multiplicative devi-
ations from it constitute a comprehensive char-
acterization of first- and second-order universal
features of the macroscopic behavior of large
earthquakes. They provide powerful observational
constraints for our attempts to understand the
physics of earthquake rupture.
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The STF data used in this study were generated by Ye et al.
(2016), and can be accessed from https://sites.google.com/site/
linglingye001/home. The finite fault models and STFs from Hayes
(2017) can be obtained through the U.S. Geological Survey
National Earthquake Information Center Combined Catalog (https://
earthquake.usgs.gov/earthquakes/search/). The STFs from
Vallée et al. (2011) can be downloaded from http://scardec.projects.
sismo.ipgp.fr/. This study has been partially funded by the Swiss
National Science Foundation and the Gordon and Betty Moore
Foundation. J.P.A. acknowledges funding from NSF CAREER award
EAR-1151926. We thank L. Ye, H. Kanamori, K. Dahmen, N. Lapusta,
H. Owhadi, L. Rivera, D. Sornette, and J.-P. Avouac for discussions,
and M. Vallée and G. Hayes for providing their STF data sets.
Figs. S1 to S13
2 May 2017; accepted 23 August 2017