advances.sciencemag.org/cgi/content/full/6/27/eaaz5548/DC1
Supplementary Materials for
The predictable chaos
of slow earthquakes
A. Gualandi*, J.-P. Avouac, S. Michel, D. Faranda
*Corresponding author. Email: adriano.geolandi@gmail.com
Published 1 July 2020,
Sci. Adv.
6
, eaaz5548 (2020)
DOI: 10.1126/sciadv.aaz5548
This PDF file includes:
Figs. S1 to S45
Fig. S1:
As Fig. 2, but for segments #1 and #2. The red dashed lines are shown only for these
two segments because they are the only ones exhibiting a significant low dimension (see Fig. 3).
Fig. S
2
:
As
Fig. S1, but for segments #3 and #4.
Fig. S
3
:
As
Fig. S1, but for segments #5 and #6.
Fig. S
4
:
As
Fig. S1, but for segments #7 and #8.
Fig.
S5
:
As
Fig. S1, but for segments #9 and #10.
Fig. S
6
:
As
Fig. S1, but for segments #11 and #12.
Fig. S
7
:
As
Fig. S1, but for
segment #13.
Fig. S
8
:
ET results.
Left: Correlation dimension ν vs Log(
r
) calculated on non
-
causally filtered
slip potency rate time series
(filter
EF
1
/
35
1
/
21
)
. Small value of
r
are dominated by noise, but a
plateau becomes visible at around Log(
r
) 8.2, 7.8, and 7.2 for segments #1, #2, and #3,
respectively. Right: False neighbors’ metrics
E
1
and
E
2
calculated on causally filtered slip
potency rate time series.
Fig. S
9
:
Same as Fig. S
8
, but for segments #4, #5, and #6.
Fig. S
10
:
Same as Fig. S
8
, but for segments #7, #8, and #9.
Fig. S
11
:
Same as Fig. S
8
, but for segments #10, #11, and #12.
Fig. S
12
:
Same as Fig. S
8
, but for segment #13.
Fig. S
1
3
:
Norm and quantile
q
effects for non
-
causally filtered time series
(filter
EF
1
/
35
1
/
21
)
. Top
-
left: L1 norm,
q
= 0.98; top
-
right: L2 norm,
q
= 0.98; bottom
-
left: L1 norm,
q
= 0.99; bottom
-
right: L2 norm,
q
= 0.99.
Fig. S
14
:
Same as Fig. S
13
, but for causally filtered time series.
Fig. S
15
:
Instantaneous dimension
d
and instantaneous extremal index
θ
. The statistics are
shown for non
-
causally filtered
(filter
EF
1
/
35
1
/
21
)
time series for segments #1, #2, and #3. For each
statistics, the top panel is the histogram of the instantaneous values, and the bottom panel shows
the temporal evolution.
Fig. S
16
:
Same as Fig. S
15
, but for segments #4, #5, and #6.
Fig. S
17
:
Same as Fig. S
15
, but for segments #7, #8, and #9.
Fig. S
18
:
Same as Fig. S
15
, but for segments #10, #11, and #12.
Fig. S
19
:
Same as Fig. S
15
, but for segment #13.
Fig. S
20
:
NFA results
on
all segments
time series
unfiltered (black lines) and
filtered
using the
EF
1
/
35
1
/
21
filter
(red and blue lines)
.
Th
ese
plots refer to embedding
s
with
휏
=
7
days
and
푚
=
9
.
Top panel:
Normalized prediction error
휖
as a function of
the prediction time
푇
푝
.
Equation
(S1
4
)
to estimate
H
is valid for
휖
≪
1
, and
푡
∗
=
1
/
퐻
values are calculated
using
points
below the
green dashed line
휖
∗
=
0
.
3
.
Bottom panel:
Correlation
휌
as a function of prediction time
푇
푝
.
We
use the
points for which
휌
≥
0
.
98
to
estimate
H
from equation
(S15)
.
B
oth statistics (
휖
and
휌
)
are almost constant for unfiltered time series and degrade
when increasing the prediction time for
filtered time series. This indicates that the
high
-
frequency
noise dominates in unfiltered time
series,
resulting in
statistics characteristic of a stochastic system, while
the filtering step let the
dynamics of the
system emerge clearly.
Fig. S
21
:
Same as Fig. S
20
but for
only segment #1 and different filters. Left column:
Normalized prediction error
휖
as a function of the prediction time
푇
푝
. Right column: Correlation
휌
as a function of prediction time
푇
푝
.
The estimate of
H
from the
right column plots is now
performed using the first 3 data points
for all the
tested
filters
.
Top
, central and bottom rows
refer to the EF, HWF with constant
푓
푐푢푡표푓푓
=
1
/
28
and HWF with constant
푁
푏
=
60
,
respectively
.
Fig. S
22
:
Surrogate data test on the estimate of
D
via EVT
(using an L2 norm and a
quantile
q
=
0.98)
for segment #1 for different filters and filter parameters.
The point with abscissa 0
corresponds to the case of unfiltered data
:
the value estimated from the data (green dot)
is not
distinguishable from the
surrogate data estimates, and
we cannot reject t
he null hypothesis for
which the data were generated by a linear stochastic model
.
The reported
p
-
values
indicate the
degree of confidence at which the null hypothesis can be rejected for various
filter paramters.
They
are
sorted from left to right,
i.e. with increasing abscissa.
Fig. S
23
:
Surrogate data test as in
Fig. S
22
using a Hamming window filter of the 60
th
order
, but
for segment
s
#2
, #3, #4, and #5
.
Fig. S
24
:
Same as Fig. S
23
, but for segment
s #6, #7
, #8,
and
#9
.
Fig. S
25
:
Same as Fig. S
23
, but for segments #10, #11, #12, and #13.
Fig.
S
26
:
Filter effects on the estimation of
d
(left column) and
θ
(right column)
for segment #1.
Top row: non
-
filtered case. Middle row: non
-
causally filtered case. Bottom row: causally filtered
case.
Fig. S
27
:
Filter effects on the estimation of
E
2
for segment #1. Left: Case for non
-
filtered time
series. Right: Case for causally filtered time series.