Ⓔ
Combining Multiple Earthquake Models in Real
Time for Earthquake Early Warning
by Sarah E. Minson, Stephen Wu, James L. Beck, and Thomas H. Heaton
Abstract
The ultimate goal of earthquake early warning (EEW) is to provide local
shaking information to users before the strong shaking from an earthquake reaches
their location. This is accomplished by operating one or more real-time analyses that
attempt to predict shaking intensity, often by estimating the earthquake
’
s location and
magnitude and then predicting the ground motion from that point source. Other EEW
algorithms use finite rupture models or may directly estimate ground motion without
first solving for an earthquake source. EEW performance could be improved if the
information from these diverse and independent prediction models could be combined
into one unified, ground-motion prediction. In this article, we set the forecast shaking
at each location as the common ground to combine all these predictions and introduce
a Bayesian approach to creating better ground-motion predictions. We also describe
how this methodology could be used to build a new generation of EEW systems that
provide optimal decisions customized for each user based on the user
’
s individual
false-alarm tolerance and the time necessary for that user to react.
Electronic Supplement:
Animations of the waveform envelope fits and predicted
shaking intensity for both the 2014
M
w
6.0 Napa earthquake and the 1 July 2015 false
alarm, along with the details of all reports from all earthquake early warning (EEW) algo-
rithms for both events, as well as for the 2014
M
w
6.8 off Cape Mendocino earthquake.
Introduction
The goal of earthquake early warning (EEW) is to pro-
vide users with an estimate of the ground motion that they
will feel and the time that shaking will occur at their location.
With this information, people and automated systems can
take action to minimize the impact of the earthquake. Many
countries and regions around the world, including Japan,
Mexico, Taiwan, and the United States, are operating, or
are in the process of building, EEW systems. Sometimes,
these systems employ multiple real-time analyses for predict-
ing local ground shaking; in these cases, a more robust warn-
ing system could be built if the information from these
different analyses could be combined into one unified shak-
ing forecast. In this article, we outline a strategy to accom-
plish precisely that.
Our examples focus on ShakeAlert, the EEW system for
the west coast of the United States. ShakeAlert is being
built by the U.S. Geological Survey in cooperation with the
States of California, Oregon, and Washington, as well as the
California Institute of Technology, University of California
(Berkeley), University of Oregon, and University of
Washington. ShakeAlert
’
s design is to run multiple analyses
simultaneously and then to synthesize information from these
independent algorithms to issue one unified forecast of local
shaking (
Böse
et al.
, 2014
).
Currently, ShakeAlert receives warnings from three algo-
rithms: (1) Onsite (e.g.,
Kanamori, 2005
;
Böse
et al.
,2009
),
Earthquake Alarm Systems (ElarmS; e.g.,
Allen and Kana-
mori, 2003
;
Kuyuk
et al.
, 2013
), and Virtual Seismologist
(VS; e.g.,
Cua and Heaton, 2007
;
Cua
et al.
, 2009
). Each of
these algorithms independently uses real-time measurements
of seismic data to detect an earthquake and determine its
point-source description (magnitude, location, and origin time
[OT]). Collectively, these three algorithms provide a spectrum
of behavior, from faster response (at the potential cost of less
accurate source parameters) to more accurate source informa-
tion (at the potential cost of less timeliness). The location and
magnitude information from these algorithms are used as in-
put to ground-motion prediction equations (GMPEs) that are
used to predict the ground motion that a user at a particular
location may expect. However, if the earthquake is larger than
what can be reasonably modeled as a point source, the ex-
pected shaking intensity (SI) at a certain station will also be
dependent on the distance to that fault plane, which can be
different from the hypocentral distance. Thus, these point-
1868
Bulletin of the Seismological Society of America, Vol. 107, No. 4, pp. 1868
–
1882, August 2017, doi: 10.1785/0120160331
source models may underpredict ground motions for large
earthquakes with finite rupture extents. (Of course, other
sources of ground-motion variability may cause under- or
overprediction of expected ground motion as well.) Therefore,
in addition to these three point-source algorithms, it is antici-
pated that future generations of the ShakeAlert system will
include real-time finite-fault source models as well. For exam-
ple, currently under development and testing are a finite seis-
miclinesourcesolution(FinDer;
Böse
et al.
,2012
,
2015
)and
a number of finite-fault source models based on real-time,
high-rate Global Positioning System data (e.g.,
Grapenthin
et al.
,2014
;
Minson
et al.
, 2014
;
Crowell
et al.
,2016
).
In the current ShakeAlert prototype production system,
the outputs from the three point-source algorithms (Onsite,
ElarmS, and VS) are averaged together to obtain an average
latitude and longitude for the earthquake
’
s epicenter, average
magnitude, and average OT; these averaged parameters are
then input into a GMPE for predicting each user
’
s expected
ground motion. This framework is not optimal because it is
not known how to map uncertainties in the separate predictions
of the source parameters to uncertainties in the ground-motion
predictions; more importantly, there is no way to objectively
suppress false alarms. Further, this framework cannot treat
non-point-source algorithms: it is nonphysical to attempt to
average the source properties of three point sources, one line
source, and a collection of various distributed slip models, each
of which is built upon a different fault geometry. Additionally,
we anticipate that in the future there will be a need to combine
earthquake source models with algorithms that forecast future
ground motion directly from current ground motion without an
assumed earthquake source (e.g.,
Hoshiba, 2013
;
Hsu
et al.
,
2013
;
Hoshiba and Aoki, 2015
). In this article, we propose
a new approach that allows us to probabilistically combine in-
formation from multiple rupture models in real time (regardless
of how each rupture model is parameterized) to provide a sin-
gle unified and accurate ground-motion forecast, along with
associated uncertainties and better suppression of false alarms.
Although different early warning algorithms may use very
different source parameterizations (or no source parameteriza-
tion), which makes it impossible to combine their source mod-
els, we can still combine their gr
ound-motion predictions. We
accomplish this by using Bayesian analysis to obtain what is
known as the posterior hyper-robust predictive (PHRP) proba-
bility density function (PDF; e.g.,
Beck, 2010
). This PDF is so
named because it accounts not only for uncertainties in the
model parameters that any EEW algorithm may use (such as
earthquake magnitude, location, and OT) but also for uncertain-
ties in the choice of which EEW algorithm to use to predict
ground motion. Further, when we evaluate this PDF, we also
include an extra virtual EEW algorithm that predicts no ground
motion. (We refer to this as the
“
No Event
”
algorithm because it
is the algorithm that always thinks there is no earthquake hap-
pening, and thus always predicts zero ground motion.) This al-
lows us to calculate the probability that the output from the
EEWalgorithm(s) is a false alarm by comparing to the observed
ground motion both the ground-motion predictions from the
EEWalgorithm(s) and the prediction of zero ground motion that
we would expect if there actually were no event.
We envision that this analysis would be done in a mod-
ule that accepts input from EEW algorithms and then sends
appropriate ground-motion warnings to users. We call this
module the central decision module (CDM). Each EEW
algorithm would send its output to the CDM, which would
then construct the PHRP PDF from all reporting algorithms
and send users a probabilistic description of the expected
ground motion at their locations.
Methodology
A detailed derivation of the mathematical underpinnings
of our methodology is given in the
Theoretical Foundation
for the Method
section. Readers who are particularly inter-
ested in probabilistic inference may wish to read that section
now. Other readers may prefer simply to read this section and
skip the mathematical details in the
Appendix
.
An EEW system such as ShakeAlert may operate any
number of EEW algorithms, each of which produces an inde-
pendent prediction of ground motion. Our goal is to combine
these ground-motion predictions into a single unified shaking
forecast. We accomplish this by estimating the probability that
each EEW algorithm is correct, and then use these probabil-
ities to determine how to appropriately blend the ground-mo-
tion predictions from each algorithm. The resulting probability
distribution describing the expected future ground motion is
known, as stated earlier, as the PHRP PDF (
Beck, 2010
). Pos-
terior (as opposed to prior) simply means that the probability
distributions have been updated based on observations of
ground motions. Robust means that the ground-motion predic-
tion is based on all potential values of an algorithm
’
smodel
parameters (e.g., earthquake magnitude, location, and OT)
rather than a single estimate. Finally, hyper-robust means that
it also accounts for uncertainty concerning which EEW algo-
rithm to use for predicting ground motion. (This terminology
originates in the fact that the space of ground-motion predic-
tions from all EEW algorithms is a hyperspace encompassing
the space of ground-motion predictions for all possible values
of the model parameters of a single EEW algorithm.)
The PHRP PDF for a quantity
y
, given the predictions
from all of
N
EEWalgorithms, follows directly from the total
probability theorem:
EQ-TARGET;temp:intralink-;df1;313;209
p
y
j
D
Σ
N
i
1
p
y
j
D
;A
i
P
A
i
j
D
1
(
Beck, 2010
), in which
D
is the observed seismic data, and
A
i
is the
i
th EEW algorithm issuing a report for this potential
earthquake detection (
i
1
;
...
;N
). The PDF p
y
j
D
;A
i
is
the robust posterior prediction for
y
from the algorithm
A
i
,
and P
A
i
j
D
is the posterior probability of algorithm
A
i
,
based on data
D
.(
Σ
N
i
1
P
A
i
j
D
1
, by definition, because
probabilities must sum to one.)
For EEW, the quantity being predicted
y
will typically be
peak ground acceleration (PGA), peak ground velocity (PGV),
or SI, such as modified Mercalli intensity (MMI). Some users
Combining Multiple Earthquake Models in Real Time for Earthquake Early Warning
1869
may be interested in response spectra or computing a particular
spectral acceleration (SA; e.g.,
Convertito
et al.
,2008
). Thus,
p
y
j
D
;A
i
, the PDF describing PGA (or PGV or SI or SA)
predicted from each algorithm can be obtained by inputting
each algorithm
’
s earthquake source model into a GMPE.
Equation
(1)
shows that the PHRP PDF of the ground-
motion parameter (e.g., PGA or PGVor SI or SA)
y
is simply
the combination of the predicted value of
y
for each algorithm
p
y
j
D
;A
i
, weighted by the posterior probability of each
algorithm P
A
i
j
D
. This approach properly propagates the
uncertainties from the predictions of all algorithms
A
i
into
the PHRP PDF. The PHRP PDF is robust to parameter uncer-
tainty within a model class (i.e., uncertainty in the model
parameters estimated by an individual EEW algorithm, typi-
cally earthquake magnitude, location, and OT) and hyper-
robust to model class uncertainty (i.e., uncertainty concerning
which
A
i
to choose to make probabilistic predictions).
According to Bayes
’
theorem
EQ-TARGET;temp:intralink-;df2;55;517
P
A
i
j
D
∝
p
D
j
A
i
P
A
i
;
2
in which p
D
j
A
i
is known as the marginal likelihood or evi-
dence in favor of the model class (i.e., algorithm)
A
i
,andthe
prior probability P
A
i
is a measure of the plausibility of algo-
rithm
A
i
as a predictor of the quantity
y
. If we take an unbiased
stance in which all algorithms are considered equally plausible
before observing any data (i.e., P
A
i
= constant), then
EQ-TARGET;temp:intralink-;df3;55;420
P
A
i
j
D
∝
p
D
j
A
i
:
3
Let
D
be the observed waveform data and
θ
i
represent the
source parameters for EEWalgorithm
A
i
. For example, for On-
site, ElarmS, and VS,
θ
i
{magnitude, location, OT},
whereas for FinDer,
θ
i
{location, OT, fault strike, rupture
length}. Then, applying the total probability theorem
EQ-TARGET;temp:intralink-;df4;55;333
p
D
j
A
i
Z
p
D
j
θ
i
;A
i
p
θ
i
j
A
i
d
θ
i
:
4
Each EEW algorithm
A
i
should report its marginal like-
lihood p
D
j
A
i
via equation
(4)
as part of its real-time earth-
quake analysis. However, because none of the current EEW
algorithms in ShakeAlert employ Bayesian analysis or
compute their formal probabilities, this information is
unavailable. We tested several methods of estimating the
marginal likelihood p
D
j
A
i
, which yields the probability
of each algorithm P
A
i
j
D
through equation
(3)
. In the
end, we chose to approximate P
A
i
j
D
using a maximum
likelihood estimate that we obtained by a constrained
least-squares (LSQ) fit of the observed waveforms to the pre-
dictions from all reporting EEW algorithms. We view this as
an effective but somewhat
ad hoc
fix to overcome the current
inability of the EEW algorithms to provide formal posterior
probabilities computed via equations
(3)
and
(4)
. However,
we can justify our choice by noting that using this approach
ensures that the mean of the PHRP PDF, and thus the mean
of the CDM
’
s predicted ground motion, matches the ob-
served ground motions. (See the
Appendix
for proof.)
The framework in equation
(1)
is completely general
and applies to any quantity,
y
, which we might want to ro-
bustly predict and any PDF describing an individual EEW
algorithm
’
s prediction of
y
,p
y
j
D
;A
i
. For EEW, the quan-
tities that we want to predict are measures of ground motion.
As an example, we consider estimating the expected PGA.
The most likely PGA estimate from each EEW algorithm is
obtained using a GMPE. However, ground-motion variabil-
ity about the mean predicted by the GMPE is quite large and
a significant contributor to uncertainty in EEW predictions of
expected ground motion (e.g.,
Allen, 2007
;
Iervolino
et al.
,
2009
). Observed ground-motion variability about the log of
the PGA predictions from GMPEs is typically modeled as
Gaussian-distributed with a standard deviation
σ
of about
a factor of 2. (Specifically,
σ
1
:
55
−
1
:
8
for
M
8to
M
5 earthquakes in
Chiou and Youngs, 2014
;
σ
2
:
2
for
M
<
4
and 1.8 for
M
>
5
:
5
in
Boore
et al.
, 2014
;
σ
ranges
from 2.3 at
M
3 to 1.8 at
M
8in
Campbell and Bozorgnia,
2008
; and
σ
goes from 2.45 at
M
3 to 1.9 at
M
8in
Abra-
hamson
et al.
, 2014
.) Thus, the PDF describing the ground-
motion prediction from a single EEW algorithm
A
i
is
EQ-TARGET;temp:intralink-;df5;313;481
N
log PGA
j
log PGA
i
A
i
;
log
σ
;
5
in which log PGA
i
A
i
is the log of the GMPE-based
ground-motion prediction from the
i
th EEW algorithm
A
i
,
and
N
log PGA
j
log PGA
i
A
i
;
log
σ
denotes a Gaussian
distribution with mean log PGA
i
A
i
and standard deviation
log
σ
. Equivalently, we can write
EQ-TARGET;temp:intralink-;df6;313;394
p
y
j
D
;A
i
p
PGA
j
D
;A
i
log
N
PGA
j
log PGA
i
A
i
;
log
σ
;
6
in which log
N
PGA
j
log PGA
i
A
i
;
log
σ
denotes the (base
10) lognormal distribution, such that log PGA is Gaussian dis-
tributed with mean log PGA
i
A
i
and standard deviation log
σ
.
For simplicity, we assume
σ
2
. Then, the PHRP PDF
for PGA is
EQ-TARGET;temp:intralink-;df7;313;290
p
PGA
j
D
P
No event
×
δ
PGA
Σ
N
i
2
f
log
N
PGA
j
log PGA
i
A
i
;
log
2
P
A
i
j
D
g
;
7
in which we take
A
1
as the No Event virtual algorithm men-
tioned in the
Introduction
(i.e., the algorithm that always believes
there is no earthquake happening and so always predicts there is
zero expected ground motion), and
δ
is the Dirac delta function
(equal to one at PGA
0
and zero otherwise).
The cumulative density function (CDF) associated with
the PDF in equation
(7)
is
EQ-TARGET;temp:intralink-;df8;313;138
P
PGA
<x
j
D
P
No event
Σ
N
i
2
f
P
A
i
j
D
Z
x
0
log
N
PGA
j
logPGA
i
A
i
;
log
2
d
PGA
g
:
8
1870
S. E. Minson, S. Wu, J. L. Beck, and T. H. Heaton
The median predicted ground motion and associated
confidence intervals can be determined from the CDF (equa-
tion
8
). For example, the median and upper and lower 95%
confidence bounds are found by solving for
x
, such that
P
PGA
<x
j
D
equals 0.5, 0.975, and 0.025, respectively.
Examples
In the following examples, at each one second increment
of time, we follow these analysis steps: first, we take as our
data
D
envelopes of acceleration (
Cua, 2005
) observed over
the last five minutes at all regional strong-motion stations con-
tributing to ShakeAlert. Second, we generate predicted enve-
lopes of acceleration from each reporting EEW algorithm
A
i
(including the predictions of zero ground motion from the No
Event virtual algorithm that always thinks there is no earth-
quake happening) using the
Cua (2005)
relations between
ground-motion envelopes and the distance and magnitude
of an earthquake. Third, in the absence of probabilities pro-
vided by the EEW algorithms, we estimate the probability of
each algorithm (including the No Event algorithm) P
A
i
j
D
,
using a constrained LSQ fit between the observed and pre-
dicted ground-motion envelopes. (In this way, we obtain
not only the probability associated with each EEW algorithm
but also P
No event
, the probability that the earthquake report is a
false alarm.) Finally, we obtain the PHRP
prediction of ground motion by combining
the ground-motion predictions of each al-
gorithm (including the No Event algorithm)
according to equation
(1)
.
The basis for our comparison between
observed and predicted ground motion is
the
Cua (2005)
envelope relationships. Fol-
lowing
Cua (2005)
, the observed envelopes
of ground motion are defined as the maxi-
mum of the absolute value of the ground
motion in a 1 s window.
Cua (2005)
used
30,000 three-component seismograms from
stations within 200 km of 70
M
w
2.0
–
7.3
southern California earthquakes to deter-
mine relationships for the expected wave-
form envelope as a function of source
magnitude and distance from the source.
These relationships allow us to compare the
time evolution of the observed acceleration
waveforms to those predicted by each algo-
rithm
’
s source model using only the point-
source location and magnitude information
that the EEW algorithms provide. Further,
although in this article we have restricted
ourselves to considering strong-motion ac-
celeration records, the
Cua (2005)
relation-
ships will allow us to, in the future, utilize
velocity and filtered displacement seismo-
grams as well. Alternatively, these envelope
relations could someday be replaced by a more advanced set
of envelope relationships that include a PDF describing the
uncertainty in the ground-motion envelopes themselves.
We present a simple synthetic test plus two real exam-
ples from ShakeAlert. In our synthetic test, we assume that
two algorithms (Alg1 and Alg2) are operating and issue an
alert 3.5 s after the initiation of an earthquake and that data
are available for analysis within 1 s, making the first solution
available at 5 s after origin (in integer seconds). The earth-
quake is located on the southern San Andreas fault and is
surrounded by stations spaced 20 km apart. Both algorithms
produce correct magnitude estimates, but the locations from
the two algorithms have errors of 15 and 5 km, respectively.
Although a case in which two EEW algorithms produce er-
rors up to 15 km and no magnitude error is somewhat ex-
treme, these are not implausible values. In a retrospective
test of ShakeAlert EEW algorithms using large historic
events,
>
75%
of events have epicentral errors less than
20 km, and
>
80%
of events have magnitude errors less than
0.5 units (E. Cochran, personal comm., 2017).
Figure
1
shows a comparison of the observed waveform
envelopes of acceleration to the corresponding waveform
envelopes predicted from the two algorithms. This compari-
son allows us to observe that the Alg2 solution (which has
the smaller location error) is more probable than the Alg1
solution, and that both are more probable than the possibility
118°W
117.5°W
117°W
34°N
34.5°N
35°N
t
= 5 s
Algorithm probabilities
Alg1
Alg2
No event
Data
1
2
3
4
t
= 8 s
Algorithm probabilities
Alg1
Alg2
No event
Data
100 cm/s/s
10 s
10 km
Figure
1.
Waveform comparisons for a synthetic test of two algorithms that accu-
rately estimate the earthquake
’
s magnitude but mislocate the event. The envelopes of
acceleration for the scenario source are plotted in black. The envelopes of acceleration
predicted by the two algorithms are shown in red and blue. Triangles are station loca-
tions. The black star is the location of the input source, whereas the blue and red stars are
locations estimated by the algorithms. The yellow and red circles are the spatial extent
after 5 s (left) and 8 s (right) of the
P
and
S
waves, assuming velocities of 6 and
3
:
5
km
=
s,
respectively. Numbers in circles identify locations for which probability density func-
tions (PDFs) of expected ground motion are plotted in Figure
2
. A least-squares (LSQ)
fit between the observations and the algorithms
’
predictions yields the probabilities for
each algorithm shown in the pie chart at the top of each subplot.
Combining Multiple Earthquake Models in Real Time for Earthquake Early Warning
1871
that this is a false alarm. As time goes on and more ground
motions are available for comparison with the predictions
from the algorithms, this divergence increases, with the prob-
ability that this is a false alarm going to 0% and the solution
increasingly favoring Alg2 over Alg1.
Figure
2
shows the process by which the CDM creates
its PDF of predicted ground motion from the probabilities
computed in Figure
1
. The source parameters from Alg1
and Alg2 are each used as input into a GMPE and each
GMPE output is used to define the center of a lognormal
PDF (red and blue lines in Fig.
2
). These PDFs, along with
a prediction of zero ground motion from the No Event algo-
rithm, are weighted by the probabilities in Figure
1
and then
summed, yielding the CDM prediction of ground motion
(magenta line in Fig.
2
). This process successfully combines
the predictions from Alg1 and Alg2 to create a new PDF that
better predicts the peak ground motion that will eventually be
observed in these synthetic seismograms.
The first of two real examples from the ShakeAlert EEW
system is the 2014
M
w
6 South Napa earthquake. For this
earthquake, ElarmS produced a timely and accurate source
model. Onsite had identified the stations closest to the hypo-
center as having high background noise, and accordingly set a
high triggering threshold for these stations. Because of this,
the earliest Onsite triggers came from stations located signifi-
cantly south of the earthquake, resulting in an Onsite solution
that was issued late and that inaccurately located the earth-
quake
’
s epicenter. VS did not issue a relevant warning for this
earthquake.
One of the advantages of our CDM analysis is that it can
combine the strengths and help overcome the weaknesses of
each EEW algorithm, without having to know the details of
how each EEW algorithm operates. It is neither necessary to
know why VS did not issue a warning nor necessary to know
the particulars of ElarmS and Onsite
’
s triggering approaches
or their relative effectiveness overall. (In this case, Onsite
’
s
higher triggering threshold was a weakness, but perhaps in
other cases it has successfully suppressed false alarms.) What
is important is to identify that, for this earthquake, the
ElarmS ground-motion prediction is more probable than the
Onsite ground-motion prediction, as well as more probable
than the prediction that there is no earthquake, so that we can
then use these probabilities to produce a probabilistic predic-
tion of the expected ground motion.
We now apply our CDM analysis to the output from the
ShakeAlert EEW algorithms for the 2014 South Napa earth-
quake. As time after origin increases,
N
in equation
(1)
ranges from 1 (No Event) to 2 (No Event, ElarmS) to 3 (No
Event, ElarmS, Onsite). Until the first EEW algorithm re-
ports, the CDM has no choice but to issue a 100% probability
of No Event and thus zero predicted ground motion. The
ideal CDM behavior would be to identify, as soon as the
ElarmS solution is released, that the ElarmS solution is a
good predictor of the observed ground motion and immedi-
ately push the probabilities to heavily favor the ElarmS
model over No Event and to estimate ground motions in
accordance with the ElarmS predictions. Then, when the On-
site solution is issued, the ideal CDM would recognize that it
is a poor solution and severely downweight both Onsite and
No Event in favor of the ElarmS solution. Replaying our pro-
posed CDM analysis on the waveform data for the Napa
earthquake and the evolution of alerts issued by ElarmS
and Onsite produces exactly this behavior (Figs.
3
–
5
). At
6 s after the OT, the first second for which a solution from
ElarmS exists, comparison with the observed envelopes of
ground motion yields a probability of
∼
97
:
6%
in favor of
the ElarmS report to 2.4% probability of a false alarm
(Fig.
3
). The first Onsite report is available at 12 s after
OT but fits the data so poorly that it received 0% probability,
with 98.4% probability to ElarmS and 1.6% probability to
No Event. This means that the resulting PHRP PDFs for
PGA, PGV, and SI (which are based on combining the pre-
dictions of ElarmS, Onsite, and No Event with those prob-
abilities) is heavily weighted toward the ElarmS solution,
1
3
t
= 8 s
Algorithm probabilities
Alg1
Alg2
No event
CDM
Data
2
4
Location
10
20
50
100
200
500
1000
Figure
2.
PDFs of expected peak ground acceleration (PGA) at
the four locations marked in Figure
1
. The red and blue lines are the
PDFs of expected ground motion for the two earthquake early warn-
ing (EEW) algorithms, assuming that the distribution of expected
ground motion is given by a lognormal distribution with mean given
by the ground-motion prediction equation output for that algo-
rithm
’
s source model and standard deviation
σ
2
(equation
5
or, equivalently, equation
6
). The magenta line is the posterior hy-
per-robust predictive (PHRP) PDF calculated by combining these
PDFs (along with a prediction of zero ground motion), weighted
by the probabilities given in the pie chart at top. Medians and
95% confidence bounds are given by the circles and brackets below
the PDFs. Thick black bars show the actual PGA that will eventu-
ally arrive at each location. This figure shows how the central de-
cision module (CDM) combines the predictions from the two EEW
algorithms to produce a PGA forecast (magenta circle) that is closer
to the PGA that will be observed (black line) than either of the origi-
nal EEW algorithms (red and blue circles).
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S. E. Minson, S. Wu, J. L. Beck, and T. H. Heaton