Okay, good morning, everyone. Today I’m going to be talking about
some of the methods we’ve developed to do earthquake
forecasting during swarms. Before we begin, I just wanted to
thank my collaborators – Andy, Nicholas, Morgan, Jeanne, and Sara,
whom I’ve worked with on various aspects of swarms over the years.
And a special shout-out goes to Sara, who is actually
the one who came up with this pretty great title.
So earthquake swarms are sequences that are thought to be driven primarily
by some external transient aseismic process, such as fluid flow –
either natural or anthropogenic, or slow slip. And while swarms in
southern California, particularly in the Salton Trough, have been causing
more of a fuss recently, swarms happen in the Bay Area, too,
particularly in the East Bay, shown here in this map,
near the cities of San Ramon, Danville, and Alamo.
At least 11 swarms have happened here since 1970, which you can
see color-coded on this map by – from a paper by Xue and others.
And you can see some of their characteristics here in this table,
also from their paper. So the largest earthquakes in these
swarms tend to be in the mid-4s – about 4.2, 4.4.
And the swarms typically last from weeks to months.
These swarms seem to be occurring in an extensional step-over where slip
is transferring from the northern Calaveras Fault to the Mount Diablo
Fault and then further north to the Concord Fault.
Swarms present unique challenges for earthquake forecasting.
Unlike aftershocks, which can be modeled with empirical laws like
Omori decay, swarms are typically modeled as time-varying background
rates, particularly in the framework of the Epidemic-Type Aftershock
Sequence, or ETAS, model, shown here, which is a commonly
used earthquake forecasting model. So, in this model, the seismicity rate
in a region at some time is equal to the sum of a background rate plus the –
some of the aftershock rates that are triggered by all the earthquakes that
occur in that region prior to time, t. So the problem is that the lack of
direct knowledge or observation of the underlying driving process
really makes it difficult to predict how long a swarm,
or how long this high background rate is going to last.
And earthquake forecasts are very sensitive to the assumed
swarm duration. So this figure here is a comparison of two end member
forecasts that were made during the 2016 Bombay Beach swarm in the
Salton Trough in southern California. These forecasts were made by using
the ETAS model to simulate a bunch of catalogs and then
see how many earthquakes occur during this forecast interval.
So these plots are showing cumulative number of earthquakes over time.
And the forecast interval is shaded by the – by the gray.
So the top panel shows the forecasted number of events,
assuming that the high background rate of the swarm continues
throughout the one-week forecast period.
And the bottom shows a forecast using an end member model that assumes
that the high background rate stops as soon as the forecast was made.
And this ultimately leads to almost an order of magnitude difference
in the number of earthquakes that each of the models forecast.
So, in this talk, I’m going to go over some of the basic building blocks
that we’ve developed and started to implement for earthquake
forecasting during swarms. These include a model for
swarm duration based on actuarial statistics,
region-specific swarm models, and ensemble forecasts.
And the – much of what I’m going to be covering in this talk come from
two papers that we published in 2019 in BSSA.
So if you’d like further details, I can refer you to those papers.
So, to start off with, to build a simple swarm duration model,
we took a page from actuarial statistics. And this is the kind of math that’s used
to compute things like life expectancy tables for a given population
by agencies like the Social Security Administration.
Regions like the Salton Trough, which has a high level of swarm
activity, we can consider the observed swarms there as a population
and then put together what’s essentially a swarm life expectancy table.
And so that’s what shown here. So, using a data set of about 20
Salton Trough swarms that were identified by Chen and Shearer,
we can calculate the number of swarms that last to a particular age or duration,
and then, from there, compute the probability
that a swarm that’s lasted that particular age ends in the next
24 hours, and then the expected time remaining in the swarm.
So, as it turns out, the swarm durations in the Salton Trough
are exponentially distributed. This figure shows, in the black
solid line, the probability that a swarm that has lasted to
a particular duration has ended. And the gray solid line –
the probability that it will end in the next 24 hours.
And, again, these – so these solid lines are computed directly from
the observed swarms in this table. And the dashed lines show the fit
from a Poisson model – basically a Poisson model of termination,
which applies an exponential model for a duration.
The bottom plot here shows the average time remaining –
again, observed in black. And then, once a correction
has been applied to account for the relatively small sample size,
that’s shown here by the dotted line. And you can see that it’s again
well fit by an exponential duration or Poisson termination model.
So, once we have a duration model, we can now go back to the 2016
Bombay Beach forecasts and see how its implementation affects the forecasts.
So we again make forecasts by using the ETAS model to simulate
a large number of catalogs. But now, for each synthetic catalog,
we can randomly draw a duration over which to apply a high background rate.
So this figure shows the results from the different models.
On the left are one-week forecasts. On the right are one-month forecasts.
The top row are forecasts that are made using this exponential duration model.
The middle row are forecasts that are made using the end member
model where we assume that the swarm continues throughout
the entire forecast interval. And then the bottom is where
we assume that the swarm has ended.
And on each panel, the black line shows the forecast number of
earthquakes with the shading indicating 95% confidence interval.
And the red is the observed number in each of the forecast windows.
So you can see that the duration model does indeed improve the forecasts.
And perhaps the improvement is only slight here in the case of
the one-week forecasts. And this is likely because the
average number of – average length of a swarm in the Salton Trough
is about a week, which is on the same order as the forecast interval itself.
But you can see that the improvement is much more significant for the
one-month forecasts. So our results suggest that a duration model is
definitely a useful improvement to implement for swarm forecasting.
However, this approach still assumes that the background rate is constant
throughout the entire swarm. But, because swarms are often driven
by some process that varies over time, we might also expect that the
background rates during a swarm would vary as well.
So, to explore the best way to estimate and update the background rate
for a swarm forecast, we turn to northern California,
specifically the East Bay, which, as I’ve shown – showed in your
introduction, has seen a number of swarms in the last few decades.
So there again, shown here in this map,
here’s a cumulative number of earthquakes over time for the region.
So, unlike the Salton Trough, where the average duration of
the swarms was less than a week, the average swarm here lasts
well over a month, which you can also see in this table,
again from the paper by Xue and others.
A particularly significant swarm occurred in 2015
near San Ramon over here. And it lasted over a month.
Here’s the cumulative number over time and the magnitude over time.
And it produced almost 100 magnitude 2 to 3-1/2 earthquakes.
And so this is the swarm that we’ll be using as our test case.
So, to determine what background rate produces the best forecasts during the
swarm, we ran a suite of different forecast models, where we estimated
the background rates from either all the previous seismicity in the
San Ramon area, only the previous swarms, or only the current swarm –
the 2015 swarm using different look-back windows of 2, 5, or 10 days
over which to estimate the background rate.
So this figure here shows how the different – the different
background rates compare. In the first two models, the Fixed/All –
all previous seismicity model and the only previous swarms model,
you can see the background rates are fixed throughout the
entire swarm at the same rate.
For the other models, the background rate is
updated at the start of each forecast window.
So, at the start of each three-day forecast window,
we simulate 100,000 catalogs using the different models to forecast the number
of earthquakes that are likely to occur over the next three days.
And here are the results. So this plot shows the number of
earthquakes in each three-day time window during the course
of the swarm, predicted by the different models, which are shown
by the different symbols. The observed number is shown by the squares.
And the legend also contains the log-likelihood score of each of the
different models, with the higher scores being the better models.
So, of the two fixed models – the Fixed/All model and the
Fixed/Swarm model, the Fixed/All model almost always
underestimates the rates because it has the lowest background rate.
So you can see that by the green triangles here.
And the Fixed/Swarm model, with its higher background rate,
does a better job, but it tends to start over-predicting towards the
end of the swarm here because it doesn’t know that the swarm is ending.
The updated models all generally perform better than the fixed models,
but, as we look – as the swarm progresses, you can see that
no single model clearly provides the best forecast throughout
the entire swarm. So how do we decide which model to use,
especially at the beginning of the swarm, before we know –
we have very much information?
To avoid having to make that arbitrary choice, we turn to
ensemble earthquake forecast methods – in particular,
those developed by Marzocchi and others in 2012.
Ensembles are made by weighting and combining individual models
according to their past performance.
We tested two weighting schemes – a Bayesian model averaging scheme,
where the weight assigned to a model is equal to the exponential of its
likelihood, and then a score model averaging scheme, where the weight
is the inverse of its likelihood. So these figures here show the relative
weighting of each of the models get – each of the individual models get
within the ensemble as the swarm progresses.
So, for BMA weighting, you can see that, because of the exponent here,
BMA is a strong weighting scheme, so the best-fitting model –
the best-fitting individual model tends to dominant the ensemble,
which has definitely happened here.
In this case, the best-fitting model is the LB2 – the Look-Back 2 model.
The individual models get more representation in the SMA ensemble.
So here are the resulting forecasts using the ensembles.
This is the same plot that I showed earlier with the observed and forecast
number of earthquakes during the swarm for the different models.
And now I’ve added the BMA forecast in gold and the SMA forecast in blue.
And here on the right, I’m showing the cumulative likelihood of
each of the models over time throughout – so you can see
how it evolves during the swarm. And you can see that the two ensemble
models have the highest likelihood throughout the entire swarm.
Both perform about equally well within uncertainty.
We just have a preference for the SMA ensemble
because it’s less dominated by a single model.
So we also did the same analysis for some of the older East Bay swarms,
and the results are shown here. And these include the 1976 Danville
swarm, the 1990 Alamo swarm, and then also the 2018 Danville swarm.
In general, the Fixed/Swarm model performs worse for the earlier swarms
because there are fewer data to constrain the swarm parameters.
But the ensemble forecasts still perform best overall, which you can see
in the cumulative likelihood plots here on the right.
So this suggests that this method can work, even in regions
that don’t have a very extensive swarm history.
So, all in all, our results show that ensemble forecasts can
be a valuable tool in developing swarm earthquake forecasting.
So, to summarize, earthquake swarms present challenges to current forecast
models, which are really geared more towards aftershock sequences.
But we’ve developed some building blocks for swarm earthquake
forecasting, and these include a swarm duration model,
which is actually now being implemented in USGS swarm forecasts.
So a couple of screenshot examples are shown below from the forecasts we put
out for the Salton Sea and Westmorland swarms from last summer and fall.
We’ve also shown the usefulness of having a regional swarm model
tuned to prior swarm seismicity. And we’ve shown that including
time-varying background rates and ensemble methods
can also be used to improve forecasts during swarms.
Nevertheless, there’s still a lot more work to be done.
So, for my last slide, I just wanted to end with a few of what I see
are some of the next steps towards a comprehensive
swarm operational earthquake forecast model.
These include folding the actuarial duration model into the ensemble
approach and also investigating how widely applicable the exponential
swarm duration model is across different regional and tectonic settings.
We also need to figure out how to detect when a swarm is happening
and when a swarm forecast model would be more appropriate
to use than an aftershock model, or vice versa.
We also need to consider how might messaging need to differ
for swarm-specific forecasts versus aftershock forecasts.
And finally, how might a better understanding of the physics of
swarms, which our next speaker, Alicia Hotovec-Ellis, is going to
discuss, and how might external information about the physical
driving process, such as fluid flow, strain, or GPS,
be used to improve our forecasts in the future.
So, with that, I’ll end there, and thank you for your attention.