Morning, everyone.
Thank you very much for the invitation to be here today.
I’m going to be talking about a recent project on Bay Area
site effects from kappa and site spectral parameters.
And this work is mainly from UO undergrad Elias King and
graduate students Tara Nye and Alexis Klimasewski
with collaborators for various parts of it being Annemarie Baltay,
Joe Fletcher, and Jack Boatwright.
What is kappa?
Kappa is the decay of high – at high frequencies of
the acceleration spectrum, which you can see in this
cartoon here as the orange line, the decay being the teal line.
It could also be fit with this form, which shows the acceleration spectrum
as a function of a-naught and some exponential decay with kappa.
And this is usually fit after some frequency, f-e, which is above f-max,
that you can see labeled on this diagram.
Why do we care about kappa, and what does it have to do with site effects?
We think of kappa as likely as an effect of site conditions.
And it’s been used in estimating ground motions since the ’90s.
It’s also been found to correlate with PGA as of late.
That can be seen in the top right here in this figure, showing kappa
as a function of PGA for sites in New Zealand where you can see that
higher values of kappa tend to correlate with lower values of PGA.
And this makes sense because kappa is a decay at high frequencies,
so faster decay at the high frequencies seems to make sense that produces
lower PGA, as we think of PGA as coming from this range.
Klimasewski et al. 2019 found the same thing.
Here, the axes are flipped. Kappa is on the X axis, and the
Y axis is plotting a ground motion model site residual for predicting PGA.
And what we find is that higher values of kappa tend to correlate with lower
site residuals, meaning correlate with lower PGA than expected.
And recently, ground motion models have started to include kappa
as a site term, such as Laurendeau et al. 2013.
And there are other applications. So, for example, these could be used
in nonergodic spatially varying models, one of which is shown on the right,
where each panel shows the map variability of coefficients for
a ground motion model from Landwehr et al. 2016.
Kappa could be included as an additional constraint to these.
They’re also often – it’s often also featured in simulations as a site effect.
And so these simulations could directly include
measured values of kappa for the region.
However, we don’t actually have any measured values
or computed values of kappa in the Bay Area.
And it could be useful for a lot of things, so, as I mentioned,
to adapt the existing ground motion models or for future spatially
varying ground motion models. But, because it’s also a site condition
in the shallow surface and has a form similar to t-star and seismic attenuation,
it could perhaps be used to inform updates to community velocity models
in the shallow subsurface and simulations.
And it could also be used as a constraint in a variety of
different seismological studies in the area.
It’s typically computed in a few different ways.
The first, from Anderson and Hough, was to fit this A-naught
times e to the minus pi of kappa form to the acceleration spectra
of any single recording past some frequency.
But it was determined that these values changed as a function
of distance away form a site. And so another way of doing this
that’s common is computing kappa for a number of different records and then
representing kappa as a function of a kappa at the particular site,
which is considered to be a site kappa, as well as how that varies with distance
away from the site, which is more representative of the path attenuation.
And then some have also simultaneously solved for source and
site, such as Anderson and Humphreys. And you can see that here where this
is the original form from Anderson and Hough in natural log space.
And this is the source that’s being solved for.
We’ve taken a slightly different approach.
This is showing Klimasewski et al. 2019, where we first take a data set
of a number of different records from any earthquake, i, and station, j,
and we decompose that record into its contributions from the event
and the site for all given events and sites.
That’s in the cartoon here on the right, where the record is in blue,
and any particular event in the data set here is represented in green,
and the site is in red. When we run this inversion,
the resulting event and site spectra that come out are unconstrained,
as there is a degree of freedom. And Andrews 1986, who started
this method – or, who used this method commonly,
constrained the inversion by taking a reference site and dividing
all the other sites and multiplying events by that reference value.
This, however, yields one of the sites ineffective.
And so, to preserve all of them, what we have done is constrain
the inversion with a theoretical Brune spectra.
So we’ve found an event in the data set that looks to have the
shape closest to a Brune spectra – and these are the velocity spectra,
remember, in this cartoon.
And we then compute what its theoretical spectra would be for
a stress drop of 5 megapascals. The constraint function is then the
difference between these two spectra. So between the unconstrained single –
that single event and its theoretical Brune spectra.
This constraint, which you can see in the arrows here, is then applied
to all other events and sites as a multiplication or division factor.
And, because it doesn’t change the shape of the spectra,
it really only shifts them up or down.
And then finally, we obtain kappa by fitting this traditional form
to each site’s respective acceleration spectra,
and that’s how we – how we get it for each site.
We applied this in southern California when developing the method, so you
can see some examples from that here, where here, the two horizontal
components of the waveform, the record spectra with their
uncertainties in the spectral computation process with mtspec,
the event and site spectra, and you can see that the event
and site multiplied back so that we can get some level of misfit.
We have uncertainty in many different levels in this process,
so we propagate them all through, and that allows us to obtain
an uncertainty of kappa in the end.
And we were fortunate in that we did this is an area that
many other studies have focused on. So many of our sites had kappa values
computed from several different studies.
We compared our values to those studies, and we found that they’re
actually pretty similar in the region.
So we take this as a good sign that the method is robust, at least here.
And lastly, one of the benefits of this method is that we do preserve
the entire site spectra. So, in addition to obtaining kappa,
we can obtain spectral levels, or the average of the spectra in
given frequency bins, and these can be really useful to compare
to different intensity measures. Kappa, of course, is a function of
decay at high frequencies, but because we are preserving the spectra for all of
these sites, would could, for example, determine if these lower frequencies
correlate with different intensity measures.
And that’s shown here from the study where we compared it to
a PGA ground motion model residual. And you can see that the lower
frequencies do not correlate, however the spectral level
at the higher frequencies does seem to correlate.
In the Bay Area, where we’re doing this, we were using data
from 30 broadband stations on three networks.
You can see that on the map over here where the stations are triangles
that are colored by the number of events that they record.
And we have almost 4,000 earthquakes of magnitude 2.5 and greater.
And those are the circles on this map colored by the number of stations
that record them to show the robustness of the data set.
We split it into several different data sets based on signal-to-noise
ratio cutoffs. So, for example, if we leave it at a signal clearance ratio of 3,
we have about 20,000 records or more. And, if we cut it – if we only
keep events that have a signal-to-noise ratio of 5 or greater,
then we still have nearly 15,000 records.
One thing that we do in this process that we didn’t do in the original study
is look at how different parameters affect the inversion,
and do they have a large impact on the kappa values.
So we look at not just signal-to-noise ratio, but we also look at the number
of bins that we use in the frequency inversion.
The spectral decomposition is done in each frequency bin.
And so, by performing it on, say, 150 bins as opposed to 75, we will
get a spectra that is less smooth and has more features to it – or variability.
And then finally, in the original study, we used a record length of 60 seconds
after the S wave arrival. And so, in this one,
we tried a more traditional value at 15.
On the bottom here, you can see the results for these six different
runs for three different stations. For example, the purple line is
for Run 6, which is 15 seconds, 150 bins, SNR 5.
And you can see that it is less smooth than some of the other –
the other runs or some of the other parameters.
And it does show more features. And so, in the end, we use this
run as our preferred, or final, model to make sure we have the best quality
data set and we are using these – using this S wave cut length of 15.
And then lastly, for some of the results, here’s our map of kappa values.
The warm colors are higher kappa values,
and the more green colors are lower kappa values.
And there’s a lot of variability in the area, but there are some
features that pop out. So, for example, we see relatively high values in
the Sacramento-San Joaquin River Delta over here.
But we see slightly lower values towards the coast.
Down here by the San Andreas Fault, we see fairly low values of kappa
in the Gabilan Range west of the San Andreas Fault.
And we see higher values at a station just to its east,
suggesting that perhaps this fault is creating sufficient difference
in geology from west to east that is reflected in the site conditions.
One of the caveats, or a downside here,
is that our kappa values are a little too high.
A typical reference kappa value is somewhere around 0.04 seconds
for a hard rock site, whereas, soft rock is generally considered
to be something like 0.02 seconds. But what you can see here is that
our kappa values are a little higher than that, so more like 0.06, even 0.08.
And we think that this is possibly because there’s
a path component included in there.
Our spectral decomposition process does not directly invert for the path.
It just inverts for the event and the site.
And this data set does have some long distances in it –
source-to-site distances, so we think what may be happening
is the path attenuation is being wrapped into the site term.
And the way that we are going to try and fix this,
or see if this makes a difference, is by correcting each record
by some average Q value before inverting.
We have models of seismic attenuation here.
This one on the right is showing Q-p from Eberhart-Phillips 2016.
But we also have Q-s for the region. So, for every record in our data set,
we will find the average Q value and then divide it out from the record
at the same time as correcting for distance for geometric spreading.
Lastly, one thing to show is that many studies have looked for
correlations between Vs30 and kappa. And this is perhaps useful in going back
and forth between ground motion models if you have a measured value
of one but not the other. And so we look at that for our data set, where
kappa is plotted on the X axis for every site, and its Vs30 is on the Y axis.
We don’t see a significant correlation here that’s statistically significant.
In fact, none at all, seemingly.
And this could be for a couple reasons.
It could be that our kappa values are not representative of the site
because we need to remove the path attenuation first.
It could be that perhaps the Vs30 values are – some of these are
proxy values and not measured. And the measured values may
lead to a better correlation. Or it could just be that this correlation
is not strong here, and it is just not a good correlation.
And other studies have seen a very weak correlation
but nothing particularly strong in the past.
And where to from here? What are some different things
that we are planning on doing or could do with this?
Well, first, correcting for Q is one big thing, and that is nearly complete.
Another thing to do here is to compare the observed ground motion
model residuals – or, observe ground motion model residuals
in the region to the kappa values.
Meaning, for the kappa values that we – that we get, how do those
compare to recorded ground motions at those stations?
Does kappa actually seem to correlate with lower PGA?
What about other intensity measures,
such as spectral accelerations or even Fourier amplitude spectra?
And lastly, for the spectral levels that we get from the same-site spectra,
how do those compare to, say, the lower frequencies
of recorded ground motion?
And then last, one of the goals for this region is to obtain
a spatially interpolated map of kappa, similar to what’s shown here
on the right from Van Houtte et al. 2018 in New Zealand.
This model contains not just kappa – the variability of kappa in the region,
but it also shows what the standard deviation is in each location.
With our stations and our values, we aim to create the same kind of
[inaudible] map, however, at the moment, there aren’t
quite enough stations or there isn’t quite enough density to
really come up with a robust map. And so we would like to include
more strong motion stations in our inversion so that we
have a little more data to obtain this.
That’s all that I have. Thanks for your attention,
and I’m looking forward to questions at the end.