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Okay, good morning, everyone.
Today I’m going to be talking about

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some of the methods we’ve
developed to do earthquake

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forecasting during swarms.
Before we begin, I just wanted to

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thank my collaborators – Andy,
Nicholas, Morgan, Jeanne, and Sara,

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whom I’ve worked with on various
aspects of swarms over the years.

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And a special shout-out
goes to Sara, who is actually

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the one who came up
with this pretty great title.

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So earthquake swarms are sequences
that are thought to be driven primarily

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by some external transient aseismic 
process, such as fluid flow –

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either natural or anthropogenic, 
or slow slip. And while swarms in

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southern California, particularly in the
Salton Trough, have been causing

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more of a fuss recently, swarms
happen in the Bay Area, too,

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particularly in the East Bay,
shown here in this map,

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near the cities of San Ramon,
Danville, and Alamo.

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At least 11 swarms have happened
here since 1970, which you can

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see color-coded on this map by –
from a paper by Xue and others.

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And you can see some of their
characteristics here in this table,

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also from their paper.
So the largest earthquakes in these

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swarms tend to be in
the mid-4s – about 4.2, 4.4.

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And the swarms typically last
from weeks to months.

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These swarms seem to be occurring
in an extensional step-over where slip

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is transferring from the northern
Calaveras Fault to the Mount Diablo

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Fault and then further north
to the Concord Fault.

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Swarms present unique challenges
for earthquake forecasting.

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Unlike aftershocks, which can be
modeled with empirical laws like

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Omori decay, swarms are typically
modeled as time-varying background

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rates, particularly in the framework
of the Epidemic-Type Aftershock

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Sequence, or ETAS, model,
shown here, which is a commonly

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used earthquake forecasting model.
So, in this model, the seismicity rate

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in a region at some time is equal to
the sum of a background rate plus the –

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some of the aftershock rates that are
triggered by all the earthquakes that

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occur in that region prior to time, t.
So the problem is that the lack of

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direct knowledge or observation
of the underlying driving process

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really makes it difficult to
predict how long a swarm,

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or how long this high 
background rate is going to last.

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And earthquake forecasts are
very sensitive to the assumed

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swarm duration. So this figure here
is a comparison of two end member

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forecasts that were made during the
2016 Bombay Beach swarm in the

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Salton Trough in southern California.
These forecasts were made by using

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the ETAS model to simulate
a bunch of catalogs and then

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see how many earthquakes occur
during this forecast interval.

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So these plots are showing cumulative
number of earthquakes over time.

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And the forecast interval is
shaded by the – by the gray.

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So the top panel shows the
forecasted number of events,

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assuming that the high background
rate of the swarm continues

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throughout the
one-week forecast period.

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And the bottom shows a forecast using
an end member model that assumes

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that the high background rate stops
as soon as the forecast was made.

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And this ultimately leads to almost
an order of magnitude difference

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in the number of earthquakes
that each of the models forecast.

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So, in this talk, I’m going to go
over some of the basic building blocks

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that we’ve developed and started
to implement for earthquake

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forecasting during swarms.
These include a model for

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swarm duration based on
actuarial statistics,

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region-specific swarm models,
and ensemble forecasts.

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And the – much of what I’m going
to be covering in this talk come from

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two papers that we published
in 2019 in BSSA.

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So if you’d like further details,
I can refer you to those papers.

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So, to start off with, to build
a simple swarm duration model,

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we took a page from actuarial statistics.
And this is the kind of math that’s used

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to compute things like life expectancy
tables for a given population

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by agencies like the Social
Security Administration.

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Regions like the Salton Trough,
which has a high level of swarm

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activity, we can consider the observed
swarms there as a population

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and then put together what’s essentially
a swarm life expectancy table.

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And so that’s what shown here.
So, using a data set of about 20

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Salton Trough swarms that were
identified by Chen and Shearer,

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we can calculate the number of swarms
that last to a particular age or duration,

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and then, from there,
compute the probability

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that a swarm that’s lasted that
particular age ends in the next

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24 hours, and then the expected
time remaining in the swarm.

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So, as it turns out, the swarm
durations in the Salton Trough

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are exponentially distributed.
This figure shows, in the black

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solid line, the probability that
a swarm that has lasted to

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a particular duration has ended.
And the gray solid line –

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the probability that it will
end in the next 24 hours.

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And, again, these – so these solid
lines are computed directly from

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the observed swarms in this table.
And the dashed lines show the fit

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from a Poisson model – basically
a Poisson model of termination,

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which applies an exponential
model for a duration.

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The bottom plot here shows
the average time remaining –

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again, observed in black.
And then, once a correction

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has been applied to account for
the  relatively small sample size,

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that’s shown here by the dotted line.
And you can see that it’s again

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well fit by an exponential duration
or Poisson termination model.

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So, once we have a duration model,
we can now go back to the 2016

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Bombay Beach forecasts and see how
its implementation affects the forecasts.

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So we again make forecasts by
using the ETAS model to simulate

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a large number of catalogs.
But now, for each synthetic catalog,

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we can randomly draw a duration over
which to apply a high background rate.

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So this figure shows the results
from the different models.

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On the left are one-week forecasts.
On the right are one-month forecasts.

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The top row are forecasts that are made
using this exponential duration model.

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The middle row are forecasts that
are made using the end member

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model where we assume that the
swarm continues throughout

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the entire forecast interval.
And then the bottom is where

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we assume that the
swarm has ended.

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And on each panel, the black line
shows the forecast number of

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earthquakes with the shading
indicating 95% confidence interval.

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And the red is the observed number
in each of the forecast windows.

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So you can see that the duration model
does indeed improve the forecasts.

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And perhaps the improvement
is only slight here in the case of

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the one-week forecasts.
And this is likely because the

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average number of – average length
of a swarm in the Salton Trough

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is about a week, which is on the same
order as the forecast interval itself.

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But you can see that the improvement
is much more significant for the

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one-month forecasts. So our results
suggest that a duration model is

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definitely a useful improvement to
implement for swarm forecasting.

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However, this approach still assumes
that the background rate is constant

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throughout the entire swarm.
But, because swarms are often driven

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by some process that varies over time,
we might also expect that the

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background rates during
a swarm would vary as well.

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So, to explore the best way to estimate
and update the background rate

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for a swarm forecast,
we turn to northern California,

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specifically the East Bay, which,
as I’ve shown – showed in your

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introduction, has seen a number of
swarms in the last few decades.

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So there again,
shown here in this map,

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here’s a cumulative number of
earthquakes over time for the region.

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So, unlike the Salton Trough,
where the average duration of

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the swarms was less than a week,
the average swarm here lasts

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well over a month, which you
can also see in this table,

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again from the paper
by Xue and others.

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A particularly significant
swarm occurred in 2015

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near San Ramon over here.
And it lasted over a month.

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Here’s the cumulative number over
time and the magnitude over time.

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And it produced almost
100 magnitude 2 to 3-1/2 earthquakes.

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And so this is the swarm that
we’ll be using as our test case.

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So, to determine what background rate
produces the best forecasts during the

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swarm, we ran a suite of different
forecast models, where we estimated

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the background rates from either
all the previous seismicity in the

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San Ramon area, only the previous
swarms, or only the current swarm –

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the 2015 swarm using different
look-back windows of 2, 5, or 10 days

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over which to estimate
the background rate.

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So this figure here shows how
the different – the different

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background rates compare.
In the first two models, the Fixed/All –

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all previous seismicity model
and the only previous swarms model,

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you can see the background rates
are fixed throughout the

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entire swarm
at the same rate.

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For the other models,
the background rate is

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updated at the start
of each forecast window.

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So, at the start of each
three-day forecast window,

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we simulate 100,000 catalogs using the
different models to forecast the number

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of earthquakes that are likely to
occur over the next three days.

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And here are the results.
So this plot shows the number of

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earthquakes in each three-day
time window during the course

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of the swarm, predicted by the
different models, which are shown

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by the different symbols. The observed
number is shown by the squares.

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And the legend also contains the
log-likelihood score of each of the

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different models, with the higher
scores being the better models.

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So, of the two fixed models –
the Fixed/All model and the

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Fixed/Swarm model,
the Fixed/All model almost always

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underestimates the rates because
it has the lowest background rate.

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So you can see that
by the green triangles here.

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And the Fixed/Swarm model,
with its higher background rate,

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does a better job, but it tends to
start over-predicting towards the

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end of the swarm here because it
doesn’t know that the swarm is ending.

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The updated models all generally
perform better than the fixed models,

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but, as we look – as the swarm
progresses, you can see that

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no single model clearly provides
the best forecast throughout

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the entire swarm. So how do we
decide which model to use,

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especially at the beginning
of the swarm, before we know –

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we have very
much information?

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To avoid having to make that
arbitrary choice, we turn to

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ensemble earthquake forecast
methods – in particular,

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those developed by Marzocchi
and others in 2012.

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Ensembles are made by weighting
and combining individual models

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according to their
past performance.

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We tested two weighting schemes –
a Bayesian model averaging scheme,

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where the weight assigned to a model
is equal to the exponential of its

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likelihood, and then a score model
averaging scheme, where the weight

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is the inverse of its likelihood.
So these figures here show the relative

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weighting of each of the models get –
each of the individual models get

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within the ensemble
as the swarm progresses.

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So, for BMA weighting, you can see
that, because of the exponent here,

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BMA is a strong weighting scheme,
so the best-fitting model –

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the best-fitting individual model
tends to dominant the ensemble,

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which has definitely
happened here.

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In this case, the best-fitting model is
the LB2 – the Look-Back 2 model.

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The individual models get more
representation in the SMA ensemble.

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So here are the resulting
forecasts using the ensembles.

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This is the same plot that I showed
earlier with the observed and forecast

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number of earthquakes during the
swarm for the different models.

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And now I’ve added the BMA forecast
in gold and the SMA forecast in blue.

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And here on the right, I’m showing
the cumulative likelihood of

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each of the models over time
throughout – so you can see

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how it evolves during the swarm.
And you can see that the two ensemble

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models have the highest likelihood
throughout the entire swarm.

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Both perform about equally
well within uncertainty.

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We just have a preference
for the SMA ensemble

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because it’s less dominated
by a single model.

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So we also did the same analysis for
some of the older East Bay swarms,

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and the results are shown here.
And these include the 1976 Danville

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swarm, the 1990 Alamo swarm, and
then also the 2018 Danville swarm.

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In general, the Fixed/Swarm model
performs worse for the earlier swarms

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because there are fewer data to
constrain the swarm parameters.

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But the ensemble forecasts still perform
best overall, which you can see

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in the cumulative likelihood
plots here on the right.

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So this suggests that this method
can work, even in regions

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that don’t have a very
extensive swarm history.

00:13:22.080 --> 00:13:25.920
So, all in all, our results show
that ensemble forecasts can

00:13:25.920 --> 00:13:30.893
be a valuable tool in developing
swarm earthquake forecasting.

00:13:32.320 --> 00:13:35.760
So, to summarize, earthquake swarms
present challenges to current forecast

00:13:35.760 --> 00:13:39.576
models, which are really geared
more towards aftershock sequences.

00:13:39.600 --> 00:13:41.920
But we’ve developed some building
blocks for swarm earthquake

00:13:41.920 --> 00:13:45.680
forecasting, and these include
a swarm duration model,

00:13:45.680 --> 00:13:49.760
which is actually now being
implemented in USGS swarm forecasts.

00:13:49.760 --> 00:13:54.160
So a couple of screenshot examples are
shown below from the forecasts we put

00:13:54.160 --> 00:13:59.736
out for the Salton Sea and Westmorland
swarms from last summer and fall.

00:13:59.760 --> 00:14:03.040
We’ve also shown the usefulness
of having a regional swarm model

00:14:03.040 --> 00:14:06.160
tuned to prior swarm seismicity.
And we’ve shown that including

00:14:06.160 --> 00:14:08.320
time-varying background rates
and ensemble methods

00:14:08.320 --> 00:14:11.496
can also be used to improve
forecasts during swarms.

00:14:11.520 --> 00:14:15.756
Nevertheless, there’s still
a lot more work to be done.

00:14:16.800 --> 00:14:20.160
So, for my last slide, I just wanted
to end with a few of what I see

00:14:20.160 --> 00:14:22.160
are some of the next steps
towards a comprehensive

00:14:22.160 --> 00:14:25.096
swarm operational
earthquake forecast model.

00:14:25.120 --> 00:14:28.640
These include folding the actuarial
duration model into the ensemble

00:14:28.640 --> 00:14:32.729
approach and also investigating how
widely applicable the exponential

00:14:32.729 --> 00:14:38.288
swarm duration model is across
different regional and tectonic settings.

00:14:39.360 --> 00:14:42.400
We also need to figure out how to
detect when a swarm is happening

00:14:42.400 --> 00:14:44.800
and when a swarm forecast
model would be more appropriate

00:14:44.800 --> 00:14:48.685
to use than an aftershock
model, or vice versa.

00:14:49.600 --> 00:14:52.400
We also need to consider how
might messaging need to differ

00:14:52.400 --> 00:14:56.536
for swarm-specific forecasts
versus aftershock forecasts.

00:14:56.560 --> 00:14:59.200
And finally, how might a better
understanding of the physics of

00:14:59.200 --> 00:15:02.000
swarms, which our next speaker,
Alicia Hotovec-Ellis, is going to

00:15:02.000 --> 00:15:06.560
discuss, and how might external
information about the physical

00:15:06.560 --> 00:15:09.840
driving process, such as
fluid flow, strain, or GPS,

00:15:09.840 --> 00:15:12.936
be used to improve our
forecasts in the future.

00:15:12.960 --> 00:15:16.560
So, with that, I’ll end there,
and thank you for your attention.