[silence]
Hi, everyone. Welcome to the Earthquake Science Center
seminar series for January 27th, 2021.
As a reminder, please turn off your cameras and mute your microphones.
All of the functions are available through the menu bar that pops up
when you hover over the bottom of your Teams window, or, at this point,
everyone’s probably updated, and it’s up at the top.
Live captioning is also available. Click the three-dot More button,
and choose “turn on live captions.”
Today’s speaker is Andy Michael.
But first, we have a few announcements.
Next week’s seminar will be by Leah Langer from the USGS,
giving a talk entitled, Impact of 3D Structure
on Static Green’s Functions for Slip Inversion.
If you have recommendations for future seminar speakers
this season – and remember, our pool of possible speakers
has been greatly increased by the new virtual world that we live in.
Feel free to email myself and/or Tamara Jeppson to nominate speakers.
The next all-hands meeting for the Earthquake Science Center
is on February 12th. Those involved will have gotten
a calendar invite for that. The big news is that next week,
the Earthquake Science Center is hosting the Northern California
Earthquake Hazards Workshop. It will take place from Tuesday through
Thursday, February 2nd through 4th. This will take the place of next week’s
seminar. Registration for that is still open until Friday this week.
And I think Susan or one of us may insert a link
to the registration in the chat.
Okay. So today’s speaker is Andy Michael from the
Earthquake Science Center. If you have questions for Andy,
you can either type them into the chat, and Tamara and I will field them
and decide whether they’re appropriate to interrupt the speaker.
Or you can raise your hand using the function in Teams.
We will be monitoring the chat and reading out the questions, either,
if they’re urgent, at the time, or we’ll save them for the end.
When we read the questions at the end, feel free to ask the question yourself.
You can either raise your hand or unmute yourself and
turn on video to chime in. We all like seeing faces these days.
With that, I will hand it over to Jeanne Hardebeck,
who will be introducing Andy. - Hi. Thank you.
It’s my pleasure to introduce Andy Michael to a crowd that
probably mostly knows him already, but – so Andy did his undergrad at MIT
and his Ph.D. at Stanford. And he came over to the Survey
right after Stanford and has been with the Survey ever since.
And, as a Ph.D. student, Andy developed a method for inverting
earthquake focal mechanisms for stress, which is now a standard and widely
used method for inverting for stress. And he’s also applied that method to
a number of important problems, such as understanding how a large
earthquake can change the stress around the area where it occurred.
After that, he’s turned his attentions more towards earthquake forecasting,
testing earthquake forecasting and predictions, and working on PSHA –
probabilistic seismic hazard assessment. He and I are currently leading the
development of the USGS Operational Aftershock Forecast product – a product
that’s been live on the USGS website for a couple of years now, where we
issue aftershock forecasts following magnitude 5 events in the U.S.
On the more research-y side, he’s been working recently with Andrea Llenos
on how to do forecasting and PSHA in areas of induced seismicity in swarms.
And that’s the sort of work he’s going to be talking about today.
Andy also serves on NEPEC, the National Earthquake Prediction
Evaluation Council, which advises the USGS director on kind of
all matters related to earthquake forecasts and earthquake predictions.
And he’s had a number of service roles in – important roles in
Seismological Society of America, including being the editor in chief
of BSSA for many years, serving on the SSA board of directors
and publication committee.
One of Andy’s other interests is music, and he’s actually found ways to
combine those interests in music and seismology and put together
and gave a widely enjoyed talk called The Music of Earthquakes,
that hopefully some of you have had the chance to see at some point.
So today he’s going to be talking about his work on probabilistic seismic hazard
assessment, and particularly a new method that he’s developed for dealing
with situations where earthquakes are very clustered, which breaks down
some of the usual assumptions that go into PSHA.
So I will turn it over to Andy.
- Okay. Thank you, Jeanne. Appreciate the very kind introduction.
And I also want to thank Susan Garcia, Tamara Jeppson, and Austin Elliott
for their work to lead the seminar and organize all of this.
So, with that, I’ll share my screen and admit that I’m a little bit nervous about
giving this talk because I think the audience is a combination of people
who know far more about volcanoes, and particularly Kilauea, than I do
and people who know far more about PSHA than I do.
But I think this talk is an example of how we can start with a, you know,
applied problem, which was developing the update to the National Seismic
Hazard Model for Hawaii and come across some really interesting
science to do in it. And so my pleasure – while I’m giving the talk,
this is really a collaboration – very equal collaboration between
myself and Andrea Llenos and is actually the subject of two papers –
one on forecasting just the – really the – focused on the collapse
of Kilauea, which you can see in the background of this slide.
This is the post – a post-collapse picture of the central part of Kilauea caldera.
And that paper, which is Llenos and Michael, is already in review in GRL.
And another paper on the methodology, which I’m supposed to be writing,
is still in preparation. So Andrea is ahead of me.
We have a lot of acknowledgements to give.
From HVO, people like Jefferson Chang in seismology,
and Tina Neal, who is leading as their head, and Paul Okubo,
who is now at University of Hawaii. Brian Shiro really gave us a lot of
help understanding the volcano. And I put an et al. there because,
for instance, pictures like this, which are looking from over the collapse,
which is sort of a little bit behind the camera, back at HVO, which is actually
under the word “Acknowledgments” here, were taken by many, many people
working on the 2018 eruption and the volcano before and afterwards.
I also want to point out that, while HVO is here, and you can see
its sort of parking lot here, we’re going to be looking at data
from a seismometer that’s in this sort of open area here.
So a little bit – not quite at HVO, a little further away, but just
by a couple hundred meters. From the Volcano Science Center,
gotten a lot of help from Kyle Anderson, Ed Brown,
Alicia Hotovec-Ellis, and Emily Montgomery-Brown.
And I think one of the things I really miss about being out of the office is
not being across the hall [chuckles] from Kyle, Alicia, and Emily.
I see a lot of my closer colleagues on Teams a lot,
but I haven’t seen you guys for a while.
The Geologic Hazards Science Center – basically, a long list of people from
the National Seismic Hazard Model. So Ned, Nico, Dan McNamara,
who is now a consultant, Morgan Moschetti, Chuck Mueller,
Mark Petersen, Peter Powers, and Allison Shumway have been key.
We decided to put Fred Klein and Tom Wright on a line by themselves
because their truly voluminous and exhaustive works on Kilauea were
critical to our starting to understand the volcano at all.
And I’m really glad – so I noticed that Tom is on here.
And finally, Jack Baker at Stanford, who has put some great notes online
to help me really understand how PSHA actually works.
And also lent me some chapters from his upcoming textbook
that also helped me along the way.
So let’s just locate ourselves in the 2018 eruptions.
This is from a Tina Neal et al. Science paper on the overall eruption.
So the island of Hawaii is here. And we’re going to look – this larger
inset is what is sometimes called sort of the south – well, the south flank is here.
But this – we’re looking at this part of the island. Kilauea Caldera is here,
and the lava comes up, and it then travels down through –
underground through the Upper, Middle, and East Rift Zones.
And then most of the great photographs you saw of lava spewing out of the
ground and fountaining come from the lower East Rift Zone here,
so fairly far away from Kilauea Caldera. And there’s a map of the eruptions here.
What what we’re going to look at are actually the explosive earthquakes
and the collapse of Kilauea Caldera, or at least part of the caldera,
and the damage it caused.
So the next thing is going to be a time lapse.
We’re looking from HVO.
This is in April – this is April 14th, 2018.
So this is the inner caldera, or Halemaʻumaʻu.
The full caldera is much larger than this.
And you can see, there was already lava and some steaming going on.
This had been going on for quite a while.
And we’re now going to do a time lapse in 23 seconds.
It’s going to take us through August 20th, and you’ll see this area
start dropping down and, on some of these scarps out here, a very large
amount of displacement. The total down-dropping is 500 meters
in a few months. Actually, it mostly takes place in a month.
So now we’re getting to the beginning of May when things are going to
start really happening. There’s explosions.
Ash hiding things a lot.
And you can start seeing [laughs] – stuff is starting to drop out.
If you keep your eye over here, you’ll see this huge white scarp open up.
And, by the end – so now we’re at August 20th, the scenery has changed
a lot, and, as Andrea likes to say, you can’t watch this often enough.
So let’s do it one more time.
[silence]
So we’re in June now when the collapse accelerates.
And the end of July. And it’s basically over by a few days into July.
Whoops. [laughs] We didn’t mean to play it a third time,
but sometimes control of PowerPoint gets beyond me.
Okay, so let’s talk about the overall caldera and the collapse again.
So this is a Lidar image taken out of Tina Neal’s paper again.
HVO is here. The seismometer I mentioned, UWE, is here.
And you can see the outline, in this Lidar image,
of the overall caldera of Kilauea.
This caldera actually formed about 500 years ago in a very large collapse
and then largely has filled in. And then here is the – sort of the
inner caldera. And this is a really interesting place to me.
Several years back, Stephanie and I – a lot of you know my wife,
Stephanie Ross, in the marine program – works on tsunamis.
So we were – we did a large hike around and across Kilauea Caldera.
And we came out of Kilauea Iki – another caldera that’s right here –
sort of an offshoot. And you can come down, and there’s a trail that
starts about here and goes sort of north of this blue transect line.
And this is a very long, desolate hike that goes slightly uphill when we did it.
And we got up to, eventually, here. Hiked around – there’s another trail
that will take you around to this sort of public overlook.
This is actually a parking lot.
And then, you know, finished the transect across and then went back
to hiking around the outside of the – so here’s what it looked like –
that’s a 2009 Lidar image. Here it is at the end of the collapse.
And it’s dramatically different.
Our hike would now take us down a – at this point –
let me add the cross-sections. So, as our hike – we got to this point,
you go down about an 85- to 100-meter scarp.
Now, instead of going uphill, you’re going downhill.
I’m sorry. I suppose I should turn on the laser pointer.
Okay, now you should be able to see my pointer better.
You’re now going downhill, and our hike would have taken us
to about this point, at which point, you just drop down another
100 or 200 meters, and hopefully not more, before the old hike would
have taken us around in a circle. The parking lot and the overlook
from Halemaʻumaʻu is down – also dropped down into the collapse,
which took place on these ring faults.
So the – this is a collapse of – total volume of about 0.83 cubic kilometers.
So, while there’s a lot of the caldera that you can see is still undisturbed –
it’s not as big as the collapse 500 years ago – it’s a pretty big event
and probably the largest collapse in the last 200 years.
But, of course, I’m not, you know, professionally interested in the change
in the topography as the earthquakes interest me.
And I remember that there was a really great video of the earthquakes that was
done by Jefferson Chang from HVO, and he first showed it at SSA.
At least, the first time I saw it was at SSA in 2019.
And I borrowed it from him. He was very gracious to let me
use it for this video. And not only are we going to see
map animation of the earthquakes at Kilauea Caldera and down the entire
East Rift Zone, and there’s a magnitude 6.9 that will happen here that was
part of this eruptive sequence – that’s this earthquake here,
but he also sonofied the seismogram – a continuous seismogram from this
station, RIMD. And so we’ll be hearing the earthquakes.
And one thing I really want you to listen for as we start playing
through it – as this red line moves through – is these two earthquakes here,
which is a magnitude 5.4 foreshock to the 6.9, and the 6.9 are extremely sharp,
impulsive earthquakes. And, when we get to these caldera
collapse events, of which there are many, listen to the fact that they’re
not as impulsive, even though they’re very close to the station.
You don’t get all the high-frequency content.
And that’s a very interesting part of these earthquakes.
Now, to show you this, I’m going to have to jump out of the PowerPoint
and go to QuickTime because this video does not play well in …
So you should be able to hear it, and I will also be talking over it.
And I’m going to fast-forward at times because it’s a three-minute video,
and we don’t need to watch all of it today.
But particularly remember to listen for these very sharp earthquakes
and the difference in the caldera collapse magnitude 5 events.
[static sounds]
[explosions]
All right, so you can get that sort of sound in your head –
those very sharp events. And we will come up here
on the first of the caldera collapse events.
[periodic explosion sounds]
Around May 17.
[static and periodic explosion sounds]
[more frequent explosion sounds]
So a lot of magnitude 5 events happening.
As we start looking at these, notice the seismicity, which is
showing – total number of earthquakes in shown in light blue –
ramps up to magnitude 5 and then ends.
[explosion sounds]
So, unlike aftershocks, there’s a pause after the biggest earthquakes
in this sequence within the caldera.
[explosion sounds]
And they start happening basically daily now.
[explosion sounds]
And I’m actually going to speed ahead because there’s many, many of these.
So you lose the sound for a while.
Just imagine being a few kilometers from these sort of magnitude 5
explosive earthquakes on a daily basis. Let’s listen to a few more of them.
[explosion sounds]
They haven’t changed.
[explosion sounds]
[inaudible] pattern of seismicity leading up these events.
And then finally, early in August, we’re going to have the last two of these.
[explosion sounds]
And then the sequence dies away at this point.
[quieter explosion sounds]
[video stops] And notice that there was a total of
[chuckles] 58,000 earthquakes in this animation over a few months.
It’s really an impressive sequence and also an impressive job of data analysis
by the seismologists such as Jefferson and Brian Shiro at HVO.
All right. So back to the PowerPoint.
Let’s see if I can do all these things smoothly.
And we’re on the laser pointer. Yeah.
Okay, so what happened at HVO?
So here’s HVO. Here’s the observation tower
where the previous video was taken from – the time lapse.
And you – and coming to work in the morning, and you have explosive
eruptions going on not far away. So that gives you – actually this makes
it looks almost like HVO was even closer than it – than it might be
if you look at it from above. But this is pretty menacing looking.
There was damage at HVO from the magnitude 6.9 earthquake,
which was further away. But looking at photos that
Ed Brown from the Volcano Science Center lent me,
clearly there’s progressive damage and additional damage from
the sequence of magnitude 5s. So some of the damage
were these very large floor offsets and tilts.
This is a photo that – there was apparently some cracking in these
columns from the 6.9, but it got worse. So this is a support column.
You can see it’s spalling away.
A table here that collapsed. Some of the things that were on it
and attached to it in one of the labs. And things thrown off the shelves.
And this is their library. There’s a lot of discussion –
and these earthquakes did not have the highest frequencies in them,
but let’s be clear that they had – they still produced strong shaking
that threw stuff around at HVO. I think it’s a little hard for me to know
fully exactly what happened in the 6.9 and what happened later,
but for instance, this picture of the library, there are other pictures where
stuff hadn’t been thrown away and – thrown off, and I saw pictures
from sometime in June. I like to point out that BSSA
seems to largely still be on the shelf.
All right. So earthquakes like this at caldera collapses are not unique.
One of the really famous collapses is in the Galapagos,
island of Fernandina, in 1968. There were 75 magnitude 4.5 events,
and, in a paper that John Filson led in JGR 1973 – so here’s the earthquakes.
Magnitudes up to 5 – well, I think a little above that,
but this is the magnitude 5 line. This is for a week in June 1968
showing essentially the same thing. A lot of little earthquakes leading up
to 5s. And he suggested that there was some sort of tidal periodicity going on.
These are both water and Earth tides.
Nearby – I don’t know if that really holds up or not.
The problem with studying this sequence is that the
closest seismometer was on South America and –
I believe in Ecuador, and so you don’t have any close-in records.
Here’s another collapse more recently from Iceland.
And it had 77 magnitude greater than 5 events from this paper
by Gudmundsson in Science. And here is a plot of M-w
versus time for several months. And we can see that there’s
a lot of magnitude 5 events. The blue and the red symbols
have to do with the locations around the caldera.
And they also had caldera subsidence measured by GPS.
So you can see that some of the subsidence events – now we can
see they happen with the earthquakes happen.
The view of this isn’t as good as the view at Kilauea.
Because here is an aerial photo of the volcano.
The caldera is shown here, and it’s ice-filled.
So [chuckles] we’re having to put the GPS station on top of the ice in
the middle of the caldera. We can’t just see exactly what was happening.
So the Kilauea collapse is really unique in terms of the quality of the data.
So we want to assess the hazard from these collapse earthquakes
for the 2021 NSHM update. Our first goal was to try to build
a physical model of these events. And we spent a lot of time looking at
the data that went into, for instance, this figure from the Neal et al. paper,
such as a GPS record of the collapse from a station that’s very close to it,
and actually was then lost once the collapse enveloped it, and then
another station that was further out. There’s also tilt meters that you can see
the individual collapse events each day. And the earthquake counts.
The problem we ran into is, while you might want to tie these earthquakes to,
say, the slip on the ring faults that actually brought the caldera down,
that’s not what was happening here. These earthquakes are negative
implosions, or negative vector – comes in with linear vector dipoles.
They’re essentially the trigger at the beginning of the collapse.
Most of the slip on the ring faults seems to be aseismic.
And so this would be like trying to do a physical model of magnitude, you
know, 7 or 8 or 9 earthquakes with only the first few seconds of that earthquake.
And so we spent a long time trying to reprocess the data in varied ways to
see if we could tie, for instance, the size of the daily collapse,
the size of the earthquake, and there’s really no good relationship there.
And so we had to move on to trying to build a statistical model.
And so I want to go over how we did that and then
how we applied it to PSHA.
So HVO, nicely, puts on a updated table of Kilauea historical activity.
They updated it – last time we looked at it was in June.
And this takes into account a lot of previous work, including, you know,
compilations by Fred Klein and Tom Wright.
And so we went through this table and we – you can’t read it here, I’m sure,
but what we did was we looked for summit subsidence events.
There’s a lot of language in this table.
There are occasionally things called “caldera collapses.”
Those are usually also called “summit subsidence events.”
But a lot of subsidence events are not called collapses.
So what we decided was we were going to call something a collapse
if it was a summit subsidence event that had non-elastic deformation
as shown by the existence of collapse earthquakes,
or at least earthquakes in the caldera during the collapse.
From 1906 onwards, from when we really have good sort of continuous
observations of Kilauea, there were 16 summit subsidence events in that table.
And we have an earthquake catalog that’s been compiled, both by people
at HVO and then also was re-processed to try to make it as consistent as possible
by Chuck Mueller for the NSHM. And, in 114 years, there were
six seismogenic collapses – in 1924, ’55, ’60, ’69, ’77, and 2018 –
that occurred with seismicity that was likely to be in the caldera.
Obviously, if we go back further in time, some of the
locations are less certain. For instance, here is, for the 1924
collapse, a magnitude time plot – magnitude here and time across here.
And we see that these inner dotted lines are the actual time of the collapse,
and there were a lot of earthquakes occurring up to – not up to –
almost up to magnitude 5. But all these earthquakes were
given the same location here on sort of the rim of the caldera.
So we don’t necessarily have good locations, but still, it seems consistent
this was some sort of collapse. And, actually, this one is definitely
called a collapse in everyone’s writing about it.
So what we can do is we can model the six collapses as a Poisson rate –
a Poisson distribution of 2.6 collapses every 50 years.
And we have tested the sequence of six collapses,
and we cannot reject the Poisson model for them.
So they may be random, independent collapses.
The next thing we need to know – because what we really want are the
number of earthquakes due to collapses, is we need to know the – you know,
the forecasted number of earthquakes per collapse.
So how many magnitude 4 or 5 earthquakes occur per collapse?
So often, we would try to infer this from extending things out
on a magnitude-frequency plot, such as we have here, where we
have the number of earthquakes. And this is in the 2018 collapse
versus magnitude. And this does not fit the sort of nice,
straight line we like to see usually on Gutenberg-Richter plots for PSHA.
So it’s hard to extrapolate. What we really have here is a lot of
earthquakes just over magnitude 5 – around magnitude 5.3.
And, of course, the magnitudes of the earlier earthquakes
are much more uncertain. So what we did was, we just tabulated
the number of earthquakes produced in each of the seismic collapses,
and then we modeled that with a negative binominal distribution.
We like it – use this distribution. First of all, it’s a discrete distribution.
It models – it gives us probabilities of the numbers of integers from
zero and greater, which is the same thing the Poisson distribution does.
And it’s often used to model earthquake clustering because it provides more
variability – a wider range of outcomes than you get from a Poisson model.
So in 1924 – we already showed this magnitude-time plot.
Notice that all the earthquakes were fixed,
except for one or two, to a zero depth
in fixed locations. There were 24 magnitude 4 and aboves. Two of them
were greater than magnitude 4.5, and none were above magnitude 5.
In 2018, we have much more exact locations in terms of depths,
and locations, but there were 85 4s and greater, 54 5s and greaters.
And I’m just going to tell you – not show all the maps – that, in all the
other seismic collapses, there were no magnitude 4 or greater earthquakes.
They were magnitude 2s and 3s.
So now, to get the probability of the number of events per collapse,
we tabulate the number of events – and so, for magnitude 4s,
we have 24, four zeroes, and 85. And, for magnitude 5,
we have five zeroes and 54. And we use the negative
binomial distribution. And so, in each of these plots, starting here
with magnitude 4, we can see the number of magnitude 4 events
in a collapse versus the cumulative probability.
So four of the six times, or two-thirds, there were zero.
And then we have the one time with 24.
And then the one time with 85. And then the model – the negative
binomial model, you know, comes close to this and smooths it out
so that we could have the possibility of having all the intervening numbers,
or even more earthquakes.
Obviously, we don’t have a lot of data, so this is uncertain, but we do
carry through that uncertainty in the distributions to our results.
And here is the fit for the magnitude 5 and greater earthquakes.
So, as I said, we have the possibility of more earthquakes, but we wanted to
put some maximum number on here. And what we did was we looked at,
what was the biggest collapse that could happen if the magma body
under Kilauea doesn’t radically change.
So basically, the idea of these collapses is, you have a
magma body here. This is depth. And this is distance across.
And the collapse is – you know, there’s some sort of withdrawal
of magma here – you know, the eruption that was taking place –
and basically is a piston that this drops down.
And so the maximum collapse that can take place is essentially the volume
that’s above the magma chamber, which we estimated at being
about 2.2 cubic kilometers. The 2018 collapse was about 0.825 –
that’s a pretty exact number. There’s certainly uncertainty in that.
And so, if we just do a ratio of the number of earthquakes to the number –
the volume of possible collapses, and then round up a bit, we concluded
that we would allow up to 300 magnitude 4-plus events
happen in any single collapse. This is an extremely uncertain number.
It’s a back-of-the-envelope sort of calculation.
But we have looked at the sensitivity of our results to this number,
and they’re not very sensitive to it all. It is a parameter that has the
least control over the hazard. And so we’re not too worried
about it, but we do think it’s a good idea not just to let there
be an infinite number of events.
Okay, so we have a input collapse rate.
That was the Poisson model at 2.6 for 50 years.
And we have these negative binomial distributions for the event rates.
So we have the Poisson distribution of collapses.
That’s going to be the probably, sub u, of u. And the reason I use u is it’s for
the probability of underlying processes. That’s what’s driving the system.
So the p-sub-u is that probability. We’re using a Poisson model in this
case – it doesn’t always have to be Poissonian – with a rate, lambda.
And then we need to have the probability of the number of
earthquakes – and this is the probability of the response to the underlying
process, p-sub-r for response. The number of earthquakes
due to one collapse. And that’s the negative binomials
that we saw in the previous slides. And the parameters are r and p,
which are shape parameters for the negative binomial distribution.
And then, Nmax, which we went over as a cap that fortunately
doesn’t matter too much. So we have to combine these two
to get an overall model. Because what we want
is the combined probability. The probability combined that this
gives us a probability of e earthquakes occurring during the time period
of interest – say, the next 50 years. We have those two input distributions –
just here so you can look at them if you need them.
And then the combined probability would just be a sum over the sum –
up to some total number of collapses. We just go up to some number that’s,
like, very unlikely to happen. So anything we’re leaving out
is negligible. And we have the probability of the number of collapses
happening times the probability of earthquakes given that number
of collapses. So the problem is, we don’t actually have this.
What we have is the probability of earthquakes due to one collapse.
But we can take basically, kind of with a recursive process,
over the number of collapses, where we can get this p-r, e, comma, u.
And so, basically, we can always – if we have u minus 1 –
so here’s p-r of u minus 1.
And we have p-r of 1 – of k, comma, 1 – so we are using
a k here because we have sums going on. That’s the summation here.
We can always get the next one. So, if we have one, we can put
p-r-k-1 here and p-r-j-1 here. We can get p-r-e, comma, 2.
And we can just add, then, build 3, 4, 5, and 6 with a special case
for the zero, where it’s basically just that the probability – if there’s
no collapses, that the probability of collapse earthquakes is zero.
So, basically, this recursive process allows us to analytically
derive this combined distribution that forecasts the number of earthquakes
in the next 50 years due to collapses.
Okay, so here’s what the overall thing – overall distribution looks like.
So here we have magnitude 4s – the number of magnitude 4
earthquakes in 50 years. And then, in the right-hand plot,
we have magnitude 5 earthquakes in 50 years.
And the vertical axes are the cumulative probability.
And the black lines show what happened in the 2018 eruption.
So 85 here, and I think this is about 54 events here.
Our preferred model using the best-fitting parameters for the
Poisson distribution and the negative binomial is this blue line.
And so, for instance, here, we can see that, if we get to
the observed number of earthquakes in 2018, there’s
an 85% chance that we’d have a lower number in the next 50 years
and about a 15% chance that we’d have a higher number.
So this is a somewhat unlikely outcome, what happened
in 2018, on a 50-year basis. But that was the largest collapse in at least
200 years. So it doesn’t surprise us that we get a result like this.
The green lines – dashed lines show the range of uncertainties derived
from these gray distributions are from sampling the range of
uncertainty in all of our parameters.
So this is plus or minus one sigma – the dashed green lines.
Here’s the Poisson model in red. So the Poisson model in red,
if we go out to the observed number of earthquakes, basically says that
it was certain, or virtually certain – it’s 99.999-something – that we would
have fewer earthquakes and a very small probability that we would
have more than what we observed. So the Poisson model essentially rules
out the possibility of what we observed in 2018, or says, at least,
that it was uncertain on the scale of many millennia.
It’s extremely unlikely. And we get the same thing –
patterns if we looked at the magnitude 5 earthquakes.
Again, the Poisson model basically cannot explain those earthquakes.
Okay. So we have a model of the number of earthquakes.
And we now want to compute the hazard due to those earthquakes –
the ground shaking hazard. And we go back to the 1968 paper
by Allin Cornell on engineering seismic risk analysis –
a seminal paper for PSHA. And, near the end, he says the
assumption that the occurrences of earthquakes follow the behavior
of the Poisson process model can only be removed at a great penalty.
And what he meant by that was that it would make the solutions
very inefficient or not analytic. And he wanted efficient,
analytic solutions so that you could explore parameter choices.
And that’s critical. And that’s what we do with logic trees in PSHA.
And so, I mean, he was using the Poisson model, not because he thought
it explained everything – he was specific that it didn’t explain elastic
rebound and it didn’t explain aftershocks – but because it got to
a tool that was very useful. But a lot of the rest of the paper,
I’ll point out, was actually about developing ground motions at a site
due to earthquakes, say, in some uniform area or due to
uniform earthquakes happening along some fault.
There’s a lot of development of other parts of PSHA that are incredibly
important in this paper. It’s really a very, you know, broad and
beautiful paper. But, when we have distributions of events that can’t
be modeled by the Poisson process, we have the urge to see if we
need to try something else. So there are some existing
approaches for non-Poissonian PSHA.
You can do simulations. So, for instance, the UCERF3-ETAS
model that Ned Field led and many of us participated in,
is a simulation model. However, Boyd’s 2012 paper included
some simulation approaches. And simulations are great.
They allow for a very wide range of effects.
I would almost say they were great enough to be considered
the best possible method.
But they can be extremely computationally expensive.
And so they may not be usable for very, very large problems.
And so here is an example of aftershock hazard from a HayWired hypothetical
earthquake on the Hayward Fault that was published in 2018 in SRL.
So that’s a simulation model.
Another approach are these deterministic combinations.
This is Memphis. There are the New Madrid
earthquakes, so that’s 1, 2, and 3. And both Oliver and then, before him,
Toro and Silva in the Technical Report, looked at the ground motions due to,
say, the union of all of these events. And this allows for correlations
between sources – like, these three events will always happen
when one of them happens. But it tends to be a very specific
approach and doesn’t really allow us to model a range of behaviors
such as we want to see in Kilauea Caldera.
I’m going to do a quick review of Poisson PSHA so you know
how it works. I had to learn a lot of this recently.
So the first thing we need is a ground motion prediction equation.
I’m going to use one from Allin Cornell in 1979.
And what we have in these equations are – this one’s very simple.
Just a function of magnitude and distance, R.
We get a mean ground motion due to some earthquake, and we can plug in a
magnitude 6, for instance, 10 kilometers away from our site of interest.
And we also have some sort of sigma. And basically, we get a log normal
distribution of ground motions due to one earthquake.
So, if one earthquake happens, then our distribution of
ground motions looks like this.
We have ground motion as a PGA on this scale plotted in log form.
And then the probability of exceedance. So we can, say, read off, you know,
here there’s a 10% chance that the ground motions will exceed something
like 0.6 g, if that earthquake happens. And that’s sort of our basic ingredient,
and we’re used to plotting it in this way.
These probability of exceedance given one earthquake can include
variations in magnitude, distance – like, a lot of what was in Allin’s paper –
focal mechanisms, stress drop – anything you want.
So here is our one-earthquake curve. Here are a suite of curves
from magnitude 5 to 7.5. And then the red line is the
combination of those using the probability distribution of
a Gutenberg-Richter distribution with a b value of 1.
So I could use just this one curve to represent the possibility that our
next earthquake comes from a range of possible earthquakes.
I’m not actually going to do that. I’m going to use this magnitude 6 curve,
but it really doesn’t matter. The brilliance of this is that we
can use – this curve can contain a lot of information about the
earthquake source, and we can just carry it along as a single distribution.
If we want to factor in the Poisson rate of earthquakes,
so we start with that exceedance curve – probability of ground motion greater
than x, given one earthquake and a Poisson rate of earthquakes.
And the rate of exceedance given that is simply the rate of ground
motion greater than x, given – instead of one earthquake, given
a Poisson distribution, is just lambda times the original distribution.
So, on this graph here, we see the original distribution
here given one earthquake. And, if the rate if 0.5,
it just drops down by a factor of 2. The dashed line is then the rate curve.
And, to be honest, a lot of PSHA just stops here. This stops,
not with probability of exceedance, but with rate of exceedance.
For instance, pulling out of the NSHM model for 2014 for Building 19 at
Moffett Field, we see that what they plot up is actual annual frequency –
or, which is another term for rate – annual frequency exceedance.
This is not a probability plot.
And this is – the black line, for instance, is the PGA curve,
and then we have a couple spectral levels.
And then the line through here is the time horizon of 475 years or 10%.
The rate that – it implies 10% in 50 years, as we talk about.
This is extremely convenient to work in rates.
Rates are linear. So if we have, say, two source zones at a site, and we have
exceedance rate curves for both of them, we can just add them together.
If we want to go from annual frequency to 50 years, we can just multiply by 50.
So, I mean, we don’t want to give up on Poisson because
it does a lot, but it doesn’t do everything.
We want to – by the way, compute probability of exceedance
of Poisson rate – go another step. So we want to get the rate.
We start with the rate. And we want to say, you know,
the number of times – k is number of exceedances, so we want to know
the probability of getting more than one ground motion.
We can take 1 minus the probability of getting zero ground motions,
and that just drops out very nicely in Poisson statistics for –
to be 1 minus e to the minus rate.
And so, on this plot here, the black dashed line is the rate, and then
the red dashed line is the probability. And notice that they become
very similar – almost equal for small probabilities, where we’re often
working for engineering applications. So, again, you can get away with
working in rate because, for small probabilities,
they’re essentially the same.
I want to note one thing I do do here that’s not usually done.
I’m showing the probability of no ground motion to basically
be the probability of no event in the class we’re studying.
Of course, this is a log-log plot.
I just artificially stick it down here with a zero.
This may bother you to see zero on a log scale, but, for some of
what I’ll discuss, I want you to note that I’m doing it because there is a
probability of no caldera collapse earthquakes in the next 50 years.
Okay, so that’s how we do Poisson.
But what about the caldera collapses – the probability of a number of caldera
collapses of – collapse earthquakes in 50 years?
As we said before, it cannot be explained
by the Poisson model, so we need something else.
So we’re going to do non-Poissonian PSHA from order statistics.
I have not been able to find this in the literature anywhere,
so I’m hoping we didn’t miss something. Because I think it’s a unique application
of order statistics. So we have the probability of events –
the probability of an event in our time period.
And we have the probability of exceedance for one event.
Those are the things we’ve developed so far.
So, if we know the probability of exceedance for not one event,
but for any n – this is what we don’t know yet – then the probability
of ground motion exceeding x, given the probability distribution of n,
would just be the sum over n of the probability of n events times
the probability of ground motion, given exactly n events.
And then order statistics are going to give us a way to calculate
this term here that we don’t know yet.
So order statistics are basically an approach to say, let’s say,
instead of taking one sample, we take two samples or three samples,
and we want to look at, say, the smallest of those samples
or the largest or the middle – any number within them.
And an important part of order statistics if that order statistics for
samples that were taken originally from a uniform distribution.
So from zero to 1, we have equal probabilities of picking any number.
And so, if we picked one sample, and we could only look at
what’s called rank – the level of sample we’re going to look at
after we sort them – there’s only one to look at.
So rank 1, sample 1, we get the uniform distribution back.
Let’s say we look at five samples, and we – each time we do
a random draw, we take five samples, and we always look at the smallest one.
That’s rank equals 1. Then obviously, we would
have a tendency to more likely be picking the smaller values –
our smallest value of 5 will likely be small.
And if we pick rank 5, the largest of five samples,
then we’re likely to pick a large one. And that’s really what we do
in PSHA is we care about the larger ground motions.
But, if we pick three samples out of five, we get something
with a central tendency.
So the actual math here is that the probability of various values of x
on the uniform distribution, given picking n samples or r is this equation.
The equation itself is not particularly important.
Just want you to know it exists. It’s actually a form of what’s called
a beta distribution, which is available quickly calculated in packages,
such as R, which is where I do all my calculations.
So I want to do some transformations first to make things easier for the
way I think, at least, and I think the way most statisticians think.
We’re used to using exceedance curves. But I’m going to reverse this and
make it a cumulative curve. Instead of the probability of
ground motion being greater than x,
it’s now the probability of ground motion less than x.
And I’m now going to – we usually plot things on a log-log plot.
I’m now going to plot them on a linear plot.
So we have cumulative probability here. Linear ground motion here.
And this actually makes it easy to see that we have a log normal distribution.
So we’re going to work this way, but it’s just a plotting thing.
Okay, so what we want to do is we want to find the probability
for two events as a first step.
So we have a linear cumulative probability for one event.
And we have the order statistic for two events
if we look at the bigger of the two – rank 2.
And so basically, it shows us that it’s more likely we’re going to
pick higher numbers. So what is this x that goes from zero to 1?
What it work – what it works out is the x that goes from zero to 1
is the cumulative probability distribution here from zero to 1.
So, if I’m more likely to pick large numbers of cumulative probability,
that means I’m more likely to pick large ground motions.
Because this curve monotonically increases.
So we’re likely to pick stuff out at this end, which corresponds to
the larger range of ground motions. Here’s how we do that.
First of all, probability density is not particularly useful in this case.
I want to do cumulative probability. These are the same curves.
Except for now, the one on the right is the cumulative sum
of the one on the left.
And so just keep that in mind.
It’s the same curve we showed before, but now it’s a cumulative curve.
So we have the cumulative probability, and now we’re labeling this first axis –
this is the cumulative probability of one event.
And this is the cumulative probability here of two events,
taking the larger one, so it’s n 2, rank 2.
And we can use this to transform our probability of ground motion
for one event to the probability of ground motions for two events.
Here’s how we do it. We take some input probability
for one event – say 0.5, which very conveniently goes
very nicely, for two events, will turn into a probability of 0.25.
So nice round numbers.
We first locate the probability of 0.5 on this curve for one event.
So we get the ground motion here. It’s probably around 0.2 g’s.
And we drop it down from 0.5 to 0.25.
So this is – following this blue line here becomes this arrow here.
And we can then do that for a whole bunch of input probabilities.
And then, if we connect those arrows, we get a new curve.
And so this curve here – the dashed line that connects the heads of the
blue arrows is the probability of ground motion being less than x
for two events if two events exactly happen.
And, if you want to look at the math, it’s down here.
It’s the probability of the ground motion of n events is the –
is the beta statistic of the probability of ground motion of one event,
and then given the appropriate rank and [inaudible] parameters based on
rank and number, which actually are behind
one of Teams’ little windows for me, so I can’t quite see.
So we can actually then do this for, say, 1 through 5 events.
And so here we have the exceedance curves.
Now I’m back to plotting log-log exceedance.
So ground motion seismologists feel more comfortable,
and we’re used to looking at it this way.
And we see that – you know, here’s the ground motion input
for one event, and the dashed lines go for 2, 3, 4, and 5 events.
And so, the more events we have, the higher the ground motions go.
This curve steps out to the right to higher ground motions.
And that’s what we would expect.
More earthquakes, more likely we have some large ground motion.
Okay. So now we want to compute hazard – compute hazard
from probability n and the probability of the ground motions
given different number of events. I can do the simplest possible thing.
I’m going to say that the probability of a different number of events is a constant.
And so, from zero to 5 events, it’s 1 over 6.
So any of these choices – any of these black curves on this plot
have an equal probability of happening, plus the probability of zero events.
And then we can just do the sum, from zero to 5.
The probability, which is a constant, times all these curves.
So basically, we’re taking some sort of average of these curves plus
the probability step-down here of no event.
And we’ve now created a non-Poissonian PSHA.
And we have a hazard curve that includes non-Poisson statistics.
And part of my point of putting in a constant probability here is,
I don’t have to put in a negative binomial.
I can put in anything I want. I can make even numbers of
earthquakes possible and odd numbers of earthquakes impossible.
Doesn’t matter. We can do any –
we can do any arbitrary earthquake behavior.
Okay. So remember the caldera? Back to it.
Do we have all the ingredients we need?
We have a distribution of the number of events.
We have the method, yes, but we don’t have a probability
of exceedance for one event. I’m going to actually just do
an empirical ground motion curve here.
We had 40 events studied by Rekoske et al.
This is spectral acceleration period here versus spectral acceleration in % g.
And the black lines are each of the 40 events.
And the red line, by the way, is the magnitude 6.9.
So you can see it was as strong as the magnitude 5s at short periods, but the
magnitude 5s were stronger at longer periods at HVO, or station UWE.
And we see that the peak ground motion was really at half a second.
So I’m actually going to use that.
If we plot this up as an exceedance probability, the empirical curves,
we can then fit log normals to them.
If I use all the data, I get the dashed lines.
The black is the observed, and the red is the fit.
It’s not a good fit to most of the data because there seem to be
two magnitude 5 earthquakes that are right down here that have extremely
low ground motions for some reason. So I’m going to rule those out –
toss them out, and then fit the rest of the data, so I actually
get a better fit to the bulk of the data.
And the red line will be my ground motion curve.
So now we have all the ingredients.
So we have cumulative probability of the number of events.
I’m using the magnitude 5 here.
And we have an exceedance probability curve, which is the solid red line.
The first thing I want to do is compute the rate of exceedance in 50 years.
And I do this for a particular reason. I don’t have a method for doing this
for non-Poisson, so just to be sure, I did 10 million simulations of
the next 50 years, drawing out of these probability distributions.
The black line is the theoretical Poisson.
And the red dashed line is the simulated non-Poisson.
And they’re exactly identical.
Basically, if you want to – if you work in rate of exceedance,
it completely hides non-Poisson effects.
So, to see non-Poisson effects, we have to work in probability.
Now we can compute the probability. Both the Poisson models showing here.
This is now probability of exceedance,
so different ground motions at half a second.
Spectral acceleration.
The black is the Poisson model. And this can be computed either
as I showed before or with order statistics.
And then the red line is a non-Poisson model.
Notice that one big difference is here at zero probability, the Poisson model
makes it very unlikely that there would be no earthquake in the next 50 years
in the – a caldera collapse event. Whereas, it’s actually quite likely –
it’s, I think, about 60% of the time, in the next 50 years, there would be
no magnitude 5s. And the dotted red lines
are plus or minus one sigma for the non-Poisson model.
The uncertainty in the Poisson model is much smaller.
And then, here we have 2% in 50 years, 10% in 50 years, and 50%.
And we can see, for one exceedance – the standard things we compute –
that the Poisson model does a pretty good job at 2%, really quite a good job
within uncertainties at 10%, but – so, for engineering purposes,
for building structures, it probably does pretty well,
even in this highly clustered case. But, for 50%, which is a standard that’s
close to those used for say, operating standards, keeping things running,
for infrastructure and tall buildings, there starts to be a very large
divergence because of the probability of zero events.
Of course, we can do more – we can do more exceedances.
And we can do that both with Poisson or with non-Poissonian order statistics.
So here we have the plot we had before, which was for one exceedance.
If we model the ground motion we would exceed twice in the
next 50 years, we can see that the pattern changes a bit.
Notice that, now, as we go to higher ground motions, the non-Poisson model
actually exceeds the ground motions from the Poisson model, and that’s
because it’s more likely to have large numbers of earthquakes.
We can go to – now here’s 2 again, and now it’s 3. The effect gets larger.
And we can even be almost silly about this.
We can go to 10 or 20 exceedances because, you know, we had
50 earthquakes magnitude 5 and greater in 2018.
And you can see that this gets – these differences get very large.
And that’s because, you know, for instance, for 2018, it says
the Poisson model basically tells us that 20 earthquakes can’t happen.
Or at least it’s – looks like it’s, you know, well under 1%.
So, after you compute multiple exceedances, the non-Poisson
behavior may become more important, and that could be important for
applications such as in the insurance industry.
So I want to finish now very quickly with how clustering
and anti-clustering affects PSHA with some simulations.
We’re going to measure clustering with the coefficient of variation, which is
just the mean by the variance of the number of events in the next 50 years.
These are simulated sequences of events.
The middle one here is actually a Poisson model.
So any clustering you see is randomness.
At the top, we have anti-cluster events.
This would be maybe due to elastic rebound.
And, at the bottom, we have highly clustered events –
may be due to aftershocks.
I used a negative binomial distribution to produce different
coefficients of variations. One is the Poisson model –
2, 5, and 10 are artificial ones. If there’s one earthquake per average
in 50 years – so a rare earthquake – slightly rare earthquake, we can see
a fairly large difference as we change coefficient of variation in the hazard.
If we get to 5 earthquakes – we multiply the underlying rate by 5 –
this becomes smaller. So lower rates and higher coefficients
of variability is the line here. The dotted line is 10, 5, and 2.
So lower rates and higher CV makes non-Poissonian more important.
And here, we show, for the same rate, rate of 1, we show one exceedance
and 3 exceedances. And we also see that number of
exceedances changes the impact. For 1 exceedance, the non-Poissonian
behavior was actually lower hazard. For 3 exceedances, it’s higher hazard.
Okay. So, in summary, so earthquake probabilities – we can determine them
by combining the probability of underlying events and the earthquake
response to a single underlying event. And that’s what we did at Kilauea.
Order of statistics gives us an analytic, computationally efficient way to
compute PSHA with arbitrary earthquake behavior.
So I think this fulfills Allin Cornell’s goal of having an analytic method.
Obviously, there’s some penalties, still.
This isn’t as fast as the Poisson statistics, but it’s so fast,
even on my laptop, that I can’t notice it running.
Notice that rate of exceedance, if that’s how we want to present PSHA,
will hide non-Poissonian behavior and – which increases with lower
rates and higher variability. Lower rates are equivalent to shorter
time periods, which are used in re-insurance applications.
For one exceedance, the Poisson model is actually pretty good.
So, for a lot of what we do, the Poisson model is fine, and that actually confirms
work by Marzocchi and Taroni, who basically inferred this
from a logical argument. But it’s not actually a good model
for higher probabilities that may be used for some applications.
For greater numbers of exceedances, the Poisson model is not always
a good approximation, even at lower probabilities.
And something we’re going to do in the future is start applying this
order of statistics approach to extend past work on aftershock PSHA,
which actually used non-stationary time-dependent Poisson models.
And we can actually include the true variability we see
in aftershock sequences. And I think this could give us a way
to get to a full national operational earthquake forecast in a way that
we can actually compute in real time. So, thank you. I’ll stop there.
[silence]
- Thank you, Andy, for that talk. If anyone has questions, please feel free
to type them in the chat, raise your hand, or you can simply dive in.
Turn your camera on, and microphone, and ask away.
[chime sound] So far, there are no questions in the chat.
So I will – I can kick off with a question going back to how you
might – how you would – you essentially describe a fairly
generic model for applying to these pretty complex – or, poorly understood
sequences of earthquakes around the caldera.
So how might you change this? Or, can you talk a little bit more about
the varying mechanisms and rates of the earthquakes themselves
in these different cases, with the caldera collapses, right –
some have absolutely none. Some have – the mechanism
seems to differ for why these are happening. And …
- Yeah. - … who would – how would you
actually implement those sorts of different [inaudible] ...
- Well, I think the – one way we could it is –
I mean, there seems to be, obviously,
some relationship between the size of the collapse and the
number of earthquakes in it. It’s not a simple enough relationship
for us to implement it at this point. We’ve just considered them
all one class. So, if we could – if we could get better at, say,
the size of collapses, that might give us a better forecast of the
size of future collapses and the number of earthquakes.
I think the other thing that’s not taken into account here is – that worried
me a lot – is to what degree did the 2018 eruption reset the system?
You know, were we – I mean, here we are, all the sudden,
after 20 years, redoing the Hawaii hazard model.
And we have this monstrous eruption with all these really interesting
earthquakes in it. And we have to immediately incorporate them.
Well, you know, did that collapse just make that – a future collapse
much less likely in the next 50 years? Or is the system in a situation
right now where it’s more likely to have another collapse?
And, you know, over the last month, actually, the bottom 200 meters of the
caldera – of that collapsed area have now filled in with lava again.
So it’s already refilling with lava.
So I really don’t think I can rule out future collapses in the next 50 years.
And I think an interesting thing is, when the – when the volcanologists
talk about hazard, they’re often talking about the response –
the hazard due to an eruption if it happens.
So very important things like, here’s where the lava would run.
You know, which you can try to forecast based on
where the vents are and where they – what the topography is.
That’s essentially our ground motion prediction equations.
They get very hesitant about actual probabilistic hazard
in the way we do it – at least, that’s been my experience talking to them.
And I think, you know, it’s a very complicated system.
And it’s probably less constant and time-independent than, say,
plate motions driving earthquakes on the San Andreas.
So I think the more we – I think my biggest uncertainty in there is probably
really – you know, these really big collapses, what happens after them?
Because I really think we need to think about forecasting
not just any 50-year period, but the next 50 years.
- Do you think also – with the earthquakes themselves, I mean you
showed these plots where you can take the average ground motion, right, but
the ground motion varies substantially, even within this 2018 sequence.
You know, the frequency content distribution.
Do you suspect that that varies between different collapse events?
And how might that change what you’re …
- Oh, from one collapse event to another?
- Yeah. - It’s an interesting question.
There’s been some questions about, say, resonance in the magma
chambers and if the magma chamber was very different, and that’s a factor.
Sure, it could. But I think we have enough sequences where these events
seem to have similar – I mean, [inaudible]
and they also think they’re similar mechanisms.
I don’t actually know about the spectral accelerations there.
I think there are likely to be, you know, similar sorts of things.
They’re – you know, in – also the behavior we saw in the Galapagos is –
seems like these were earthquakes at the beginning of each –
of a down-dropping, you know, the way they had ramping-up
earthquakes then a sudden cutoff.
So I’m guessing the source mechanism is similar and non-tectonic.
And so maybe the spectral will be similar.
All right. Brian Shiro asks – he says thanks – thanks for
an interesting analysis. What do you think of the Butler 2020
paper on a reverse aftershock description for the collapsed sequences?
- Yeah. So I have looked at that paper a little bit.
And there’s also a paper by a student from Davis who
I think also looked at accelerating release time into them.
I think what’s fascinating about these earthquakes is that you have
some sort of failure process that’s accelerating, then the magnitude 5s
are some sort of release of pressure, and down-dropping, and shutoff.
I think they’re fascinating. I don’t think they affect the hazard very much.
Because those small earthquakes aren’t producing that much ground motion.
We can leave them out.
It did mean, for instance, that we didn’t try putting an ETAS model on these.
Because the ETAS model [chuckles] would be
the magnitude 5 happens, then everything happens after it.
So I think it’s fascinating.
I’m not sure it affects what I would do here, how those papers turn out.
- Thanks for those thoughts. So Morgan Page asks – well, points out
that she’s also interested in your thoughts on just skipping over
magnitude entirely and having just a probability distribution for
ground motion. Is that something we should consider elsewhere also?
- Well, in essence, we – well, okay.
Here, we can do it basically – I mean, I still took a – I had to take
a class of earthquakes 5 and above. I could have taken 4 and above,
except for, I don’t know that they computed the spectral accelerations
for all the 4s and above. So, if we have the ground motions
for a class of earthquakes, yes, we can just use an empirical model.
You know, the problem is, we have very few recordings of large earthquakes.
And so we wind up using GMPEs and models so we can, you know,
extend them around the world. And then this gets to – for instance,
I think, you know, the paper that Kevin Milner et al. just published
where they used a earthquake simulator to produce possible earthquakes and
then computed ground motions due to those earthquakes using CyberShake.
So, you know, to get a large suite of earthquakes for southern California,
they had to resort to simulations. That’s one approach to doing that.
Yeah, it was sort of nice, in this case, that, you know, for this very specific
case of the station UWE, which is close to HVO,
I can just use the ground motions. I do need to pick a class of earthquakes,
in some sense, that my data set of ground motions represents.
And so here I used magnitude 5 and above.
- Well, it was nice for you because you didn’t really have a magnitude
distribution. [laughs] It was well-defined for
your earthquakes. [laughs] - Well, actually, I mean,
there is a magnitude – it actually – it’s almost a normal distribution
centered on magnitude 5.3. I could put that in.
I mean, the way we combine magnitudes in GMPEs is done
essentially as an empirical distribution too, so ...
- Yeah.
- But, yeah, I liked skipping that step. [laughs]
- All right. Thanks for this interesting talk and the thoughtful answers, Andy.
I’ll – we can turn on our microphones and give a brief round of applause
for Andy, and then we’ll finish the recording here.
[applause]
- Thank you. - Thank you.
- Thanks, Andy.
- Thanks for the talk. So if anyone wants to stay around, we’ll –
the meeting will stay open and you can chat a little bit more.
I’ve got to take off, but thank you all.