WEBVTT Kind: captions Language: en-US 00:00:00.130 --> 00:00:02.780 … everyone for coming to this week’s Earthquake Science seminar. 00:00:02.780 --> 00:00:06.140 I have a few announcements before we get started. 00:00:06.140 --> 00:00:12.340 Next week, we will officially be moving this seminar to Moffett Field. 00:00:12.340 --> 00:00:18.560 So Austin Elliott’s seminar will be in Building 19. 00:00:18.560 --> 00:00:21.499 The conference room is still to be determined. 00:00:21.499 --> 00:00:25.390 So we will have probably an announcement by email 00:00:25.390 --> 00:00:29.089 about where that location will be. 00:00:29.089 --> 00:00:32.960 As you probably know, tomorrow is the 30th anniversary of the Loma Prieta 00:00:32.960 --> 00:00:36.600 earthquake, and there are plenty of events going on related to that, 00:00:36.600 --> 00:00:41.280 including a governor’s press conference at 11:00 a.m. tomorrow and a few 00:00:41.280 --> 00:00:45.180 nighttime events in San Francisco at the Exploratorium and the 00:00:45.180 --> 00:00:52.170 California Academy of Sciences, featuring USGS speakers. 00:00:52.170 --> 00:00:54.520 So come out if you can. 00:00:56.340 --> 00:00:59.200 And, with that, it’s my pleasure to introduce 00:00:59.200 --> 00:01:02.180 this week’s speaker, Nico Kuehn. 00:01:02.190 --> 00:01:06.960 He got his Ph.D. at the University of Potsdam and was there for 00:01:06.960 --> 00:01:13.700 a few years as a research scientist. He then came to the Pacific Earthquake 00:01:13.700 --> 00:01:18.450 Engineering Research Center – PEER – at Berkeley as a postdoc and is now 00:01:18.450 --> 00:01:25.080 at the UCLA Garrick Risk Institute as a research scientist. 00:01:25.080 --> 00:01:33.520 And today he’ll be talking about the use of nonergodic GMPEs in PSHA. 00:01:35.260 --> 00:01:37.899 - Thank you, Grace. Can everybody hear me okay? 00:01:37.899 --> 00:01:41.479 All right. Perfect. And thank you for having me. 00:01:41.479 --> 00:01:45.859 So I’ll talk about – mainly I’ll talk about nonergodic ground motion 00:01:45.859 --> 00:01:49.149 prediction equations, or I probably should have called that 00:01:49.149 --> 00:01:53.600 ground motion models, but pretty much the same thing. 00:01:53.600 --> 00:01:59.579 And then a little bit about the use in PSHA and how that’s going to change. 00:02:00.220 --> 00:02:07.840 Oh, this one. So the reason why we’re doing this is summarized in this plot. 00:02:07.840 --> 00:02:09.920 I should probably use this. 00:02:09.920 --> 00:02:12.250 So this is a plot from Norm Abrahamson’s paper 00:02:12.250 --> 00:02:18.450 from 2019 showing a comparison of an ergodic seismic hazard analysis 00:02:18.450 --> 00:02:21.500 and a nonergodic PSHA for two sites – 00:02:21.500 --> 00:02:25.720 one in San Jose and one in northeastern California. 00:02:25.720 --> 00:02:30.390 And the ergodic hazard curves are shown in blue, 00:02:30.390 --> 00:02:33.890 and the nonergodic hazard curves are shown in red. 00:02:33.890 --> 00:02:36.330 And nonergodic is pretty much the standard the way we’re 00:02:36.330 --> 00:02:41.600 doing it right now pretty routinely. And the thing is, for nonergodic, 00:02:41.600 --> 00:02:47.140 things change quite a bit. In San Jose, on the left, they stay – 00:02:47.150 --> 00:02:51.360 at least the mean hazard stays pretty much the same. 00:02:51.360 --> 00:02:55.020 It’s very similar in northeastern California. 00:02:55.020 --> 00:02:59.280 And so I’m going to talk about how did we do – so, how did we 00:02:59.280 --> 00:03:03.090 come up with these hazard curves, and why are these so different. 00:03:03.090 --> 00:03:05.260 You probably know all this, and it’s not – it’s been known 00:03:05.260 --> 00:03:08.820 for a while that this is going to change hazard quite a bit. 00:03:08.820 --> 00:03:14.290 The main thing to note is, on the left, San Jose, Bay Area, there is some 00:03:14.290 --> 00:03:17.260 data around. That means we change the hazard quite a bit. 00:03:17.260 --> 00:03:20.440 Northeastern California, not really that much data. 00:03:20.440 --> 00:03:23.260 That means the mean hazard stays the same. 00:03:23.260 --> 00:03:27.900 But we increase the range of the hazard curve distribution. 00:03:27.900 --> 00:03:31.300 So we got much wider uncertainty in the hazard curves. 00:03:31.300 --> 00:03:34.320 And that’s pretty much all there is to say about this. 00:03:34.320 --> 00:03:37.220 And we’ll just go through how got it. 00:03:37.220 --> 00:03:42.510 So basically, the way Norm did this is he combined two methods – one from 00:03:42.510 --> 00:03:50.650 Landwehr et al. 2016, and one from the one we did from 2019 for path effects. 00:03:50.650 --> 00:03:55.260 And I’ll just probably mainly talk about the Landwehr et al. paper 00:03:55.260 --> 00:03:59.080 and the methodology about that because I think that’s the way we are 00:03:59.080 --> 00:04:03.840 currently envisioning going forward with nonergodic GMPEs. 00:04:03.900 --> 00:04:08.800 There are some pitfalls in doing that, and I’ll also briefly talk about this. 00:04:08.800 --> 00:04:14.100 This is also – both Norm and Niels Landwehr in Potsdam, 00:04:14.100 --> 00:04:16.440 so they’re really instrumental in this. 00:04:16.440 --> 00:04:22.790 And I’m mainly showing this – a lot of their work as well. 00:04:22.790 --> 00:04:28.050 So I assume you’re all familiar with seismic hazard. 00:04:28.050 --> 00:04:33.090 If not, this is just a brief refresher. We’re trying to estimate the 00:04:33.090 --> 00:04:36.030 rate of exceedance for some ground motion parameter, 00:04:36.030 --> 00:04:40.810 like PGA or response spectrum or something. 00:04:40.810 --> 00:04:44.290 The rate of exceedance for a certain ground motion level A, 00:04:44.290 --> 00:04:47.840 and that’s just by integrating over the ground motion distribution 00:04:47.840 --> 00:04:51.380 and integrating all possible magnitudes and distances, and then 00:04:51.380 --> 00:04:55.300 summing that up over all possible sources. That’s just basically it. 00:04:55.310 --> 00:04:58.490 And the key ingredient for that is the ground motion distribution. 00:04:58.490 --> 00:05:01.660 So what we need to know is, given an earthquake scenario, 00:05:01.660 --> 00:05:05.910 given a magnitude, the distance, and maybe some other parameters, 00:05:05.910 --> 00:05:09.500 what kind of ground motions can we expect? 00:05:09.500 --> 00:05:13.100 And that is typically modeled as a log normal distribution, 00:05:13.100 --> 00:05:17.220 which has a certain median and a certain standard deviation, sigma. 00:05:17.220 --> 00:05:21.840 And the median is a function of magnitude, distance, and so on. 00:05:21.840 --> 00:05:27.510 Vs30 and event type and depth are some other things. 00:05:27.510 --> 00:05:31.480 And often – and then the standard deviation can be a function of those 00:05:31.480 --> 00:05:34.840 as well, but it’s also – it’s the sum of multiple terms. 00:05:34.840 --> 00:05:39.990 And depending a bit on what we kind of think is – this is noise, this is not noise 00:05:39.990 --> 00:05:44.590 that we can model, we’re kind of moving some parts from here to there. 00:05:44.590 --> 00:05:48.660 And I’ll talk a little bit how that’s going to go. 00:05:48.660 --> 00:05:54.460 In a simple example, just to clarify, or to introduce – to illustrate what a 00:05:54.460 --> 00:06:00.720 nonergodic hazard is, imagine we have 20 sites in California, or just 20 sites, 00:06:00.720 --> 00:06:06.680 and each of them is only affected by a magnitude 6 in 20 kilometers’ distance. 00:06:06.680 --> 00:06:11.710 That just makes simulation a lot easier. So I simulated some simple example. 00:06:11.710 --> 00:06:16.830 And these events are super frequent, so at each site, we have 100 recordings. 00:06:16.830 --> 00:06:20.380 So the way we would do this right now is, we – well, the way it’s been – 00:06:20.380 --> 00:06:25.520 it’s done right now is to get a ground motion model for this. 00:06:27.120 --> 00:06:30.091 So, if we want to estimate the probability of exceedance, we need 00:06:30.091 --> 00:06:33.230 to have a ground motion model. For this case, it’s pretty simple. 00:06:33.230 --> 00:06:38.580 We’ll just look at the data, calculate the mean and the – or, the median 00:06:38.580 --> 00:06:42.680 and the standard deviation, and then we can have some estimate of our 00:06:42.680 --> 00:06:44.139 ground motion distribution. 00:06:44.140 --> 00:06:48.720 So this is all the data that we have – 2,000 data points. 00:06:48.720 --> 00:06:52.860 Log PGA. And that’s just the histogram of the event types, so that’s a nice 00:06:52.860 --> 00:06:56.640 normal distribution, so that would be log normal distribution PGA. 00:06:56.650 --> 00:07:01.300 So what we can do is we can just go ahead and look at – oh, this is – 00:07:01.300 --> 00:07:04.370 this is the mean, and then we get some estimate of the standard deviation, 00:07:04.370 --> 00:07:10.840 and then, if we want to know what’s the probability that the PGA is larger 00:07:10.840 --> 00:07:13.931 than some certain value, we just go around here, 00:07:13.940 --> 00:07:19.580 and look it up or calculate the CDF or 1 minus the CDF. 00:07:19.580 --> 00:07:22.620 This is all pretty nice. We can make some statistic tests. 00:07:22.620 --> 00:07:27.199 Oh, this looks – this is a probability plot, so this is really a normal 00:07:27.199 --> 00:07:30.310 distribution, so we can do all these kinds of things. 00:07:30.310 --> 00:07:33.699 And then we go ahead, and then we can calculate the empirical mean and 00:07:33.699 --> 00:07:38.430 standard deviation and then just calculate the probability of occurrence. 00:07:38.430 --> 00:07:44.120 So this is the exceedance probability that Y – that PGA would be larger than 00:07:44.120 --> 00:07:48.220 some ground motion value, and then this is the hazard curve. 00:07:48.220 --> 00:07:52.930 And then we can also look at the standard error of the mean and say – 00:07:52.930 --> 00:07:57.020 and get some uncertainty around that mean, put that on the hazard curve. 00:07:57.020 --> 00:08:00.710 You probably can’t really see that here, but there are some – there are some 00:08:00.710 --> 00:08:02.740 dashed lines around that, which would be, like, 00:08:02.740 --> 00:08:06.520 some uncertainty around this mean hazard curve. 00:08:06.520 --> 00:08:09.020 The problem with this is that this is actually not correct. 00:08:09.020 --> 00:08:13.900 Because, if we look at the sites individually, we see clear 00:08:13.900 --> 00:08:16.790 differences in their distribution. This is just one example. 00:08:16.790 --> 00:08:20.780 These are two sites. We know we have a lot less recordings per site. 00:08:20.780 --> 00:08:24.890 But we also see that the means of those two are really, really different. 00:08:24.890 --> 00:08:29.130 So that means, if we just aggregate those two sites together, calculate an 00:08:29.130 --> 00:08:33.560 empirical mean, we’re going to make – we’re going to be fine, like, on average, 00:08:33.560 --> 00:08:37.580 for those two sites. But individually, we’re going to be wrong. 00:08:37.580 --> 00:08:43.560 And if we do this for all those 20 sites – so we can actually just go ahead 00:08:43.560 --> 00:08:48.470 here and say this is the mean for Site 1. This is the mean for Site 2. 00:08:48.470 --> 00:08:50.930 And then we go through all those 20 sites. 00:08:50.930 --> 00:08:53.900 And then we get these individual hazard curves. 00:08:53.900 --> 00:08:56.460 I also plotted the mean of those hazard curves. 00:08:56.470 --> 00:09:01.190 So the mean of those hazard curves is actually the same as the mean 00:09:01.190 --> 00:09:04.450 that is calculated when we aggregate the data. 00:09:04.450 --> 00:09:08.500 What we see is, we see a really, really big spread along those 20 sites. 00:09:08.500 --> 00:09:11.740 I mean, this is simulated data, but this is also something 00:09:11.740 --> 00:09:14.970 that you will see in practice. 00:09:14.970 --> 00:09:20.220 So that means, if we just aggregate all those data from those 20 sites, calculate 00:09:20.220 --> 00:09:23.810 the hazard curve, we will be, on average, correct over all those 20 sites. 00:09:23.810 --> 00:09:27.260 But, at each site, we will be wrong. 00:09:27.260 --> 00:09:31.310 And then this is just – so basically what that means is the probability 00:09:31.310 --> 00:09:34.510 of exceedance becomes a lot less certain. 00:09:34.510 --> 00:09:39.810 So this is showing the mean ergodic and the mean ergodic plus/minus the sigma 00:09:39.810 --> 00:09:46.180 calculated from the standard error of the mean of the aggregated data. 00:09:46.180 --> 00:09:55.430 And those are the plus/minus 1 sigma for – I call this nonergodic hazard here. 00:09:55.430 --> 00:09:59.580 So this would be the – this would be the five and – 00:09:59.580 --> 00:10:03.760 or, plus/minus sigma of those 20 hazard curves. 00:10:03.770 --> 00:10:08.010 So the thing why this becomes important is, if we just now – 00:10:08.010 --> 00:10:12.810 we go to a new site, and I say, this new site is also affected by 00:10:12.810 --> 00:10:17.800 just magnitude 6 and 20 kilometers, but we don’t have any data. 00:10:17.800 --> 00:10:22.350 So if we want to calculate the hazard for that – or, the probability of 00:10:22.350 --> 00:10:26.310 exceedance for that new site, we don’t really know what it’s going to be. 00:10:26.310 --> 00:10:33.020 So we can just aggregate all those data, and then we get the black line. 00:10:33.020 --> 00:10:38.280 But, if we just do that, we severely underestimate the uncertainty. 00:10:38.290 --> 00:10:42.330 because, from what we see, it’s probable that there will be 00:10:42.330 --> 00:10:47.430 some really strong site effect or really strong effect at that particular site. 00:10:47.430 --> 00:10:49.310 So it could be somewhere around here. 00:10:49.310 --> 00:10:53.110 It could be any of those 20 curves, or it could be somewhere in between. 00:10:53.110 --> 00:10:56.610 So that means we should – even if we don’t know anything 00:10:56.610 --> 00:11:00.220 about that particular site, we need to take into account that 00:11:00.220 --> 00:11:05.100 there will probably be strong effects that we don’t model. 00:11:05.100 --> 00:11:11.080 Otherwise, if we then collect data at that new site, and it – then the 00:11:11.080 --> 00:11:14.010 hazard curve will shift. So it will – it will be very different from 00:11:14.010 --> 00:11:17.170 the black line. And then people will complain, and people complain 00:11:17.170 --> 00:11:20.890 all the time – why does the hazard change? What happened? 00:11:20.890 --> 00:11:24.720 Well, the hazard changed, but we were really, really uncertain before. 00:11:24.720 --> 00:11:28.649 So now we’ve got more data, but now we just move it a little bit 00:11:28.649 --> 00:11:32.310 around in this uncertainty range. And that’s just really, really important 00:11:32.310 --> 00:11:36.170 to know that we shouldn’t just look at the mean hazard. 00:11:36.170 --> 00:11:40.060 We need to capture the – well, we try to capture – the uncertainties 00:11:40.060 --> 00:11:44.740 are captured in this if we just move it into the standard deviation. 00:11:44.740 --> 00:11:52.519 But that’s just – that won’t make it – make it less easy to then go, once we 00:11:52.520 --> 00:11:58.600 collect data, do something else and try to incorporate the site-specific hazards. 00:11:58.600 --> 00:12:03.060 So basically, what happens is, if we just aggregate those data from the 20 sites, 00:12:03.060 --> 00:12:06.740 we get larger variability in our ground motion distribution. 00:12:06.740 --> 00:12:08.860 But we get really, really small uncertainty. 00:12:08.860 --> 00:12:13.480 We’ve got 2,000 records, and it’s easy to just get 00:12:13.480 --> 00:12:17.540 a good estimate of the mean from 2,000 records. 00:12:17.540 --> 00:12:22.760 It does – it ignores the differences between the distributions at each site. 00:12:22.780 --> 00:12:28.620 On average, over all 20 sites, we’ll be correct, but at each site, we’ll be wrong. 00:12:30.640 --> 00:12:34.410 The individual hazard curves and the individual site can be very different. 00:12:34.410 --> 00:12:39.740 And they will be very different. And it is important to recognize that 00:12:39.740 --> 00:12:46.570 and also to communicate that to people that the hazard curves – because we – 00:12:46.570 --> 00:12:50.269 even if we don’t have any data, they will be very, very different 00:12:50.269 --> 00:12:55.360 from what – from the mean hazard. It’s very, very uncertain. 00:12:55.360 --> 00:13:01.470 All this coming together is that, if we just aggregate all the data together, 00:13:01.470 --> 00:13:04.720 that means we’re kind of making the ergodic assumption, which means 00:13:04.720 --> 00:13:09.910 that the – that we think the ground motion distribution of all sites is 00:13:09.910 --> 00:13:15.660 the same as the ground motion distribution at one site over space. 00:13:15.660 --> 00:13:17.040 And that’s wrong. 00:13:17.040 --> 00:13:21.900 And people have been – have known that for a long time that that’s wrong. 00:13:21.900 --> 00:13:26.720 But it’s very easy, and it’s very convenient, to make these assumptions 00:13:26.720 --> 00:13:30.700 because it makes the statistics very easy, and you can – you can aggregate data 00:13:30.700 --> 00:13:34.649 from many different sources. And you don’t have to go to 00:13:34.649 --> 00:13:38.100 some complicated model to take that into account. 00:13:38.100 --> 00:13:42.670 Because it was very easy to do that in the simulated example because it’s just 00:13:42.670 --> 00:13:46.959 one magnitude and distance scenario. But once you aggregate data from 00:13:46.959 --> 00:13:49.839 more sources and over many different magnitudes, 00:13:49.839 --> 00:13:53.230 things just become quite messy. 00:13:53.230 --> 00:14:00.269 So just a little bit of math – math is nice – to see how that works. 00:14:00.269 --> 00:14:05.530 So, in this – in the ergodic example, by aggregating, we calculated the mean 00:14:05.530 --> 00:14:09.820 and standard deviation from all the – from the aggregate data. 00:14:09.820 --> 00:14:15.850 For the nonergodic exceedance probability at one site, we just take 00:14:15.850 --> 00:14:21.100 the site at that – the median at that particular site and the standard deviation 00:14:21.100 --> 00:14:26.320 of the distribution at that particular site. And then we can – we can do this. 00:14:26.320 --> 00:14:32.580 We can usually write this – instead of saying, oh, we’ll just take the median 00:14:32.580 --> 00:14:37.670 at that particular site, we can also just take the average of all sites 00:14:37.670 --> 00:14:42.000 and then just have an adjustment. So it’s either – at that site, it’s either 00:14:42.000 --> 00:14:46.360 larger than the – than the average or it’s smaller than the average. 00:14:46.360 --> 00:14:50.590 And what we typically – or, what we should see if we just go site-specific, 00:14:50.590 --> 00:14:52.660 we have a smaller value of the aleatory – 00:14:52.660 --> 00:14:57.480 I mean, that must happen because otherwise, if we aggregate data, 00:14:57.480 --> 00:15:00.350 it’s always going to be larger. 00:15:00.350 --> 00:15:05.330 So what we have is, each site has an individual median value. 00:15:05.330 --> 00:15:10.920 And we typically try to write this as some adjustment to some 00:15:10.920 --> 00:15:15.700 aggregated value, and it usually also has smaller variability 00:15:15.700 --> 00:15:19.120 than the aggregated distribution of the sites. 00:15:19.930 --> 00:15:22.120 So people have known that for a long time, 00:15:22.120 --> 00:15:27.600 and a lot of work has been done in trying to relax the ergodic assumptions. 00:15:28.240 --> 00:15:32.680 So basically, I think, if you go, like, 20 years ago, 00:15:32.680 --> 00:15:36.870 data was quite sparse, and data just had to be aggregated. 00:15:36.870 --> 00:15:41.660 And then people started doing global models, so that means – 00:15:41.660 --> 00:15:45.140 and that’s not 20 years ago, but for example, NGA-West1, 00:15:45.140 --> 00:15:48.520 they did – they had global models. And they had data from, let’s say, 00:15:48.520 --> 00:15:52.560 Japan in there, California, Europe, Taiwan. 00:15:52.570 --> 00:15:55.769 And they just make one particular model. 00:15:55.769 --> 00:16:00.940 And then – I’m actually – I’m one slide ahead, actually. 00:16:00.940 --> 00:16:09.910 So what I actually wanted to say with this slide is, from an ergodic – 00:16:09.910 --> 00:16:12.579 moving from an ergodic to nonergodic is actually – 00:16:12.579 --> 00:16:16.930 we just take an ergodic and we make some adjustments. 00:16:16.930 --> 00:16:20.139 And those adjustments need to become location-specific, right? 00:16:20.139 --> 00:16:24.200 Like, in the other – in the simple example, there were adjustments 00:16:24.200 --> 00:16:29.450 at each particular site. And – but in the real world, 00:16:29.450 --> 00:16:33.410 we just make them location-specific. So that means we have – if we have 00:16:33.410 --> 00:16:37.490 an earthquake in a particular location, we maybe add something to the median 00:16:37.490 --> 00:16:41.410 or subtract something from the median. 00:16:41.410 --> 00:16:45.600 At a particular site, we have site effects, so we add that. 00:16:45.600 --> 00:16:49.850 And then there might be path effects that are not modeled in an ergodic model, 00:16:49.850 --> 00:16:56.150 so we also have to account for that. So, throughout this talk, I will 00:16:56.150 --> 00:17:01.670 use t for location, which was used in the Landwehr paper, 00:17:01.670 --> 00:17:07.260 and I just decided to be consistent with this. 00:17:07.260 --> 00:17:14.520 So this should be – this should be there at every particular – at every particular 00:17:14.520 --> 00:17:19.410 scenario that we want to calculate. If we have a source and we have a site, 00:17:19.410 --> 00:17:24.690 that – and for, let’s say, a magnitude 6 in 20 kilometers or in 100 kilometers, that 00:17:24.690 --> 00:17:28.840 should be different if we’re in northern California or in southern California. 00:17:28.840 --> 00:17:34.390 Or that should be different even if the – if we have one source, and the path 00:17:34.390 --> 00:17:38.299 goes to the east or the path goes to the – to the west. 00:17:38.300 --> 00:17:42.540 Because the geology will be different, and there will be different path effects. 00:17:42.540 --> 00:17:48.500 Sometimes these things will be strong, and sometimes these will be less strong. 00:17:49.120 --> 00:17:53.610 Since this is something, if we go to, like, nonergodic models, 00:17:53.610 --> 00:17:58.360 this is something that we – so this is an unmodeled effect before. 00:17:58.360 --> 00:18:01.890 So now we’re modeling it. That means we should have a smaller value 00:18:01.890 --> 00:18:08.120 of the noise. So we have a smaller value of the aleatory variability. 00:18:09.620 --> 00:18:13.720 One big problem is that data is sparse. Even though we have thousands of 00:18:13.720 --> 00:18:20.000 data points in California, and even more in Japan, it’s still quite sparse 00:18:20.000 --> 00:18:26.500 to get a really good grip of those, and especially at all possible locations 00:18:26.500 --> 00:18:29.700 that we want to look at. So that means we have to kind of – 00:18:29.700 --> 00:18:33.360 we still have to aggregate the data somehow. 00:18:33.360 --> 00:18:38.720 And what’s really, really important is – for hazard is that 00:18:38.730 --> 00:18:43.140 we need to take into account the adjustment in those things. 00:18:43.140 --> 00:18:47.720 We can estimate the site effect from, like, five records at that site. 00:18:47.720 --> 00:18:51.550 And we can estimate it from 100 records at that site. 00:18:51.550 --> 00:18:54.800 If we have 100 records, we’ll be more certain about that site effect. 00:18:54.800 --> 00:18:57.440 And from five records, we will be less certain. 00:18:57.440 --> 00:19:01.270 So that needs to be reflected in the application. 00:19:01.270 --> 00:19:03.380 Right. 00:19:03.380 --> 00:19:08.840 So people have known this for a long time. 00:19:08.840 --> 00:19:19.740 And they try to include that in, like, recent models over the last few years. 00:19:19.740 --> 00:19:24.170 And one way is to make it – to make some model broadly 00:19:24.170 --> 00:19:28.900 location-specific is to put in some region-specific effects. 00:19:29.600 --> 00:19:34.640 So, for example, like in NGA, NGA-West1 was global. 00:19:34.640 --> 00:19:41.580 NGA-West2 had adjustment coefficients for Japan, California, and Taiwan and – 00:19:41.580 --> 00:19:44.570 I don’t know – I think – don’t think they had adjustments for Europe. 00:19:44.570 --> 00:19:49.740 But you can basically do – you have the same sort of model structure, 00:19:49.740 --> 00:19:56.600 but then you say, okay, if I’m in Japan, I change my Vs30 scaling a bit because 00:19:56.600 --> 00:19:59.660 the site effects in Japan are slightly different. 00:19:59.660 --> 00:20:04.159 If I’m in Taiwan, I also change it a little bit. 00:20:04.159 --> 00:20:07.990 So what that typically looks like, if you have some function for the 00:20:07.990 --> 00:20:12.370 median, is that you have some function of magnitude, distance, and so on. 00:20:12.370 --> 00:20:16.230 And then you have some, let’s say, global coefficient. And then you have 00:20:16.230 --> 00:20:20.140 some adjustment coefficient that depends on the particular region. 00:20:20.140 --> 00:20:24.360 And people have been doing that for a long time, and then this also 00:20:24.360 --> 00:20:26.470 should have some sort of uncertainty. 00:20:26.470 --> 00:20:32.269 So we could usually model that as a random effect or something like that. 00:20:32.269 --> 00:20:35.870 Another way is – and that’s, I think, probably what a lot of people know 00:20:35.870 --> 00:20:43.710 is single-station sigma. That means, instead of just adding 00:20:43.710 --> 00:20:49.790 or just aggregating data from all sites, we’ll look at each site individually. 00:20:49.790 --> 00:20:53.910 And so even sites that have the same Vs30 value might have – 00:20:53.910 --> 00:20:58.100 might lead to different average predictions. 00:20:58.100 --> 00:21:04.299 So then we have some – just some sort of site constant that is – 00:21:04.299 --> 00:21:08.880 that S term – that will be specific to each particular site. 00:21:08.880 --> 00:21:12.650 And then also this actually takes a good chunk of the 00:21:12.650 --> 00:21:17.020 aleatory variability out of the sigma. 00:21:19.040 --> 00:21:23.800 A simple example for this. If you just go further and further and 00:21:23.800 --> 00:21:30.570 just to show how that would work, is example regionalization of California. 00:21:30.570 --> 00:21:35.730 So what this shows is event terms from the ASK14 model, 00:21:35.730 --> 00:21:38.750 so that’s one of the NGA-West2 models. 00:21:38.750 --> 00:21:44.640 And so that basically is, like, an average shift for each event. 00:21:44.640 --> 00:21:49.120 And that’s plotted at each hypocenter location. 00:21:49.120 --> 00:21:55.140 And what we can see – I don’t know if you can see this, but on average, 00:21:55.150 --> 00:21:59.980 we have more bluish colors up here and more reddish colors down there. 00:21:59.980 --> 00:22:06.490 And that means that, on average, we might have slightly lower predictions in 00:22:06.490 --> 00:22:11.860 northern California and slightly larger predictions in southern California. 00:22:11.860 --> 00:22:17.680 There’s probably a lot of other things mapped into these event terms if we 00:22:17.690 --> 00:22:24.580 only have event terms, but right now, this is just a simple example. 00:22:24.580 --> 00:22:29.980 So one way to just try to put that into a model is to have some 00:22:29.980 --> 00:22:32.870 very, very simple regionalization. That means we have a different 00:22:32.870 --> 00:22:36.700 constant for northern California and southern California. 00:22:36.700 --> 00:22:42.240 That means I just put it as an example. I say if the latitude of the hypocenter 00:22:42.240 --> 00:22:49.690 is larger than 36.5, it gets a different constant than if the latitude – 00:22:49.690 --> 00:22:53.760 if the latitude is – so it’s different if it’s in northern California 00:22:53.760 --> 00:22:55.630 and in southern California. 00:22:55.630 --> 00:23:00.270 And if you do this, then this actually shows on the map. 00:23:00.270 --> 00:23:03.560 It shows the c-1 and c-2 value. Then c-1 is, on average, 00:23:03.560 --> 00:23:08.610 a little bit negative. And c-2 is just slightly positive. 00:23:08.610 --> 00:23:13.419 And then, on average, it works out that, if you would redo these event terms 00:23:13.419 --> 00:23:17.799 using the regionalized models, you don’t really see the strength anymore. 00:23:17.799 --> 00:23:21.740 That means we’ve kind of taken care of that regional trend 00:23:21.740 --> 00:23:25.059 that we saw in those event terms. 00:23:25.059 --> 00:23:30.610 One big problem with this, or one problem with this, is, if we look back 00:23:30.610 --> 00:23:34.540 at this, we have mainly data here, and we have mainly data there. 00:23:34.540 --> 00:23:39.280 So that means this whole thing is kind of driven by data that’s here, 00:23:39.280 --> 00:23:43.620 and it’s driven by data that’s here. But we sort of apply it everywhere – 00:23:43.620 --> 00:23:47.890 we would probably apply it everywhere in California. 00:23:47.890 --> 00:23:53.720 So that means, are we really certain that something – if an event happened 00:23:53.720 --> 00:23:58.360 somewhere around here, that would be lower on average? 00:23:58.360 --> 00:24:02.150 Because it’s, like, mainly driven from data in the Bay Area. 00:24:02.150 --> 00:24:05.300 And the other thing is, 36.5 is pretty arbitrary. 00:24:05.300 --> 00:24:09.710 Why is it not 36.4? Why is not 36-something? 00:24:09.710 --> 00:24:15.820 We can maybe optimize for that. But that’s not really – so this is kind of – 00:24:15.820 --> 00:24:19.720 it’s a simple model. It works. It’s statistically – the difference 00:24:19.720 --> 00:24:24.440 is statistically significant. It leads to a slightly smaller aleatory 00:24:24.440 --> 00:24:28.630 variability, so that makes us happy. But it’s not really a good model, 00:24:28.630 --> 00:24:33.460 and it’s not really – I just drew a line on the map. 00:24:33.460 --> 00:24:37.520 Why did I not drew this like this? Why – well, this was easier. 00:24:37.520 --> 00:24:41.560 But, so it’s not really – it’s not really a good model. 00:24:41.560 --> 00:24:45.300 The other thing is, okay, to take care of that, maybe we can make this 00:24:45.309 --> 00:24:51.720 thing maybe dependent on geology. Or maybe we can – if we don’t want to 00:24:51.720 --> 00:24:55.850 apply this everywhere in California, we just have one region here, and we 00:24:55.850 --> 00:24:59.470 have more regions somewhere else. But then this becomes – 00:24:59.470 --> 00:25:03.669 we make the region smaller. That should give us a little bit better 00:25:03.669 --> 00:25:09.190 handle on those regional – on the regionalization. 00:25:09.190 --> 00:25:15.049 But that’s not really that great. So basically – and that was the reason 00:25:15.049 --> 00:25:21.660 why we came up with this varying coefficient model is that – I mean, 00:25:21.660 --> 00:25:24.429 defining those regional boundaries is kind of arbitrary. 00:25:24.429 --> 00:25:29.520 You can make it sort of based on geology, but it’s also – it’s usually just, 00:25:29.520 --> 00:25:32.520 we draw a line on the map based on where we have data 00:25:32.530 --> 00:25:34.600 and where we don’t really have data. 00:25:34.600 --> 00:25:39.880 And then the other thing is, there’s – I think there’s probably a trend to make 00:25:39.880 --> 00:25:43.799 it – if we have more and more data, we can make the region smaller and smaller. 00:25:43.799 --> 00:25:47.520 But if we make the region smaller and smaller and smaller and smaller, 00:25:47.520 --> 00:25:51.040 that means, at some point, the regions become a point. 00:25:51.040 --> 00:25:55.549 That means we get something like a continuous regionalization. 00:25:55.549 --> 00:26:00.500 And that is basically the idea behind the varying coefficient 00:26:00.500 --> 00:26:07.039 model of Landwehr et al. So that is a model where coefficients 00:26:07.040 --> 00:26:10.340 are a continuous function of location. 00:26:10.900 --> 00:26:16.060 Location is a different topic. What does location really mean? 00:26:16.060 --> 00:26:20.880 But that’s the way it is. And I think Niels was – 00:26:20.880 --> 00:26:26.890 Niels Landwehr was a postdoc or so in the machine learning department 00:26:26.890 --> 00:26:31.360 in Potsdam, and I knew him from there. And then he sent me an email saying, 00:26:31.360 --> 00:26:36.280 oh, we’ve got this cool spatial model. We think you’ve got earthquake data. 00:26:36.280 --> 00:26:39.500 That’s spatial data. Let’s try it out on that. 00:26:39.500 --> 00:26:43.179 And so that’s where you came, and then we brought Norm in. 00:26:43.179 --> 00:26:48.950 And that’s how we do this. So just a very, very simple example. 00:26:48.950 --> 00:26:54.740 Typically, we would have, let’s say, a ground motion model – 00:26:54.740 --> 00:26:57.220 not a very good one. Arguably, it would look 00:26:57.230 --> 00:26:59.870 maybe something like this. We have some constants. 00:26:59.870 --> 00:27:05.650 We have some beta coefficients, and then we have some magnitude scaling. 00:27:05.650 --> 00:27:11.700 We have a geometric spreading term. We have an anelastic attenuation term. 00:27:11.700 --> 00:27:16.860 And then the only thing that we did in the varying coefficient model 00:27:16.860 --> 00:27:20.510 is make some of those coefficients dependent on location. 00:27:21.280 --> 00:27:26.299 So that means, in this case, t-e is the coordinate of the event, 00:27:26.299 --> 00:27:30.279 and t-s is the coordinate of the station. And it means we have some 00:27:30.279 --> 00:27:32.940 coefficients – let’s say, like, 00:27:32.940 --> 00:27:37.279 we have constants here, and we make those constants 00:27:37.279 --> 00:27:40.630 dependent on earthquake location and site location. 00:27:40.630 --> 00:27:45.280 That means we have a different constant at each point in California, 00:27:45.280 --> 00:27:48.340 depending on whether the – if there’s an earthquake. 00:27:48.340 --> 00:27:52.710 We also make the – we can make the geometrical spreading. 00:27:52.710 --> 00:27:58.420 So in the Landwehr paper, we made that event-specific – dependent. 00:27:58.420 --> 00:28:01.770 And then we have the anelastic attenuation, 00:28:01.770 --> 00:28:06.309 which should depend on both event location and site location. 00:28:06.309 --> 00:28:10.910 Because it’s a path-specific term, right? So it depends, if I have an earthquake 00:28:10.910 --> 00:28:16.090 here, and an earthquake at the site to the west, and the site to the north, 00:28:16.090 --> 00:28:19.180 that should – the path is different, so I should have a different 00:28:19.180 --> 00:28:23.330 anelastic attenuation. We didn’t really do that in the original 00:28:23.330 --> 00:28:27.340 paper, but this is definitely something where things can be improved. 00:28:27.340 --> 00:28:31.679 So basically, what this does is, we take some function that is 00:28:31.679 --> 00:28:34.460 a function of magnitude, distance, and some other parameters, 00:28:34.460 --> 00:28:39.920 and we make some of the coefficients a function of event and station location. 00:28:39.920 --> 00:28:43.610 Which immediately raises the question, which ones? 00:28:43.610 --> 00:28:47.870 And so, for that, we sort of said we probably cannot do this for 00:28:47.870 --> 00:28:50.299 the magnitude scaling because we don’t really have that many 00:28:50.299 --> 00:28:54.310 large-magnitude events. So we don’t really want to do that. 00:28:54.310 --> 00:29:00.620 We think the constants should be fine. We think – I didn’t put it up here, 00:29:00.620 --> 00:29:07.381 but there’s a difference in Vs30 scaling, probably between northern – it might 00:29:07.381 --> 00:29:12.200 be between northern – so we make that one location-dependent. 00:29:12.200 --> 00:29:16.220 So that’s a little bit of, like, judgment. What can we do? 00:29:16.220 --> 00:29:21.340 What do we think? Which kind of coefficients can we get to? 00:29:21.340 --> 00:29:25.320 The other thing is really, how should this function look like? 00:29:25.320 --> 00:29:27.960 So it’s a function of location. So that means, how should it 00:29:27.960 --> 00:29:31.080 go from northern California to southern California? 00:29:31.080 --> 00:29:33.980 Is it a sine function? Is it a polynomial? 00:29:33.980 --> 00:29:37.190 Is it exponential? So we have no idea. 00:29:37.190 --> 00:29:42.610 And that’s basically the main part of the varying coefficient model is 00:29:42.610 --> 00:29:47.490 that we assume that it is drawn from a Gaussian process prior. 00:29:47.490 --> 00:29:51.549 And that’s some very nice – that has some very nice properties that we can 00:29:51.549 --> 00:29:57.500 use and that we want to have here. So basically, in Bayesian 00:29:57.500 --> 00:30:02.520 nonparametrics, a Gaussian process prior is often used as 00:30:02.530 --> 00:30:06.559 a prior over functions. So that means, if you have – if you’re 00:30:06.559 --> 00:30:09.570 familiar with Bayesian, you often have – you have a prior distribution for 00:30:09.570 --> 00:30:12.860 some coefficient, and then you have a posterior distribution. 00:30:12.860 --> 00:30:18.020 Here, what we have is, we have a distribution over functions. 00:30:18.020 --> 00:30:23.049 And then, once we observe data, we get some sort of posterior distribution over 00:30:23.049 --> 00:30:28.419 functions, which makes – which allows us to quantify them a little bit. 00:30:28.419 --> 00:30:33.309 So this is one example – just one-dimensional example. 00:30:33.309 --> 00:30:39.050 So here that would be three different functions drawn from this Gaussian 00:30:39.050 --> 00:30:43.779 process prior in the – if we don’t really have any data at all. 00:30:43.779 --> 00:30:47.039 And then this is, like, the width of our prior. 00:30:47.039 --> 00:30:50.450 So we think, okay, this is plus/minus one standard deviation. 00:30:50.450 --> 00:30:55.650 So if I had, like, 1,000 realizations and looked at the standard deviation 00:30:55.650 --> 00:31:00.940 at every point, they would sort of mostly fall in this range around zero. 00:31:00.940 --> 00:31:08.140 Once we observe data, like here, here, and here, we get a posterior distribution. 00:31:08.140 --> 00:31:13.559 And, depending a bit on how we set it up, now we have – now we see that all 00:31:13.559 --> 00:31:18.410 the functions go through those points. Once we’re far away from those points, 00:31:18.410 --> 00:31:24.659 they kind of can still do whatever they want, but they all go through 00:31:24.659 --> 00:31:29.620 the data points that we observe. And that is really a nice property that we 00:31:29.620 --> 00:31:35.750 have – that we can use for nonergodic. If we’re close to some data, 00:31:35.750 --> 00:31:39.020 then we know something about that. Then we should know about 00:31:39.020 --> 00:31:41.880 something about how those coefficients should scale 00:31:41.880 --> 00:31:45.670 or should behave close to that data point. 00:31:45.670 --> 00:31:50.700 And whereas, if we go far away, they can still do whatever they want. 00:31:50.700 --> 00:31:54.920 And that is – and so this also shows the posterior uncertainty. 00:31:54.920 --> 00:31:58.330 And then, in this case, we also see, close to data, 00:31:58.330 --> 00:32:00.540 we have very, very small uncertainty. 00:32:00.540 --> 00:32:06.830 But the farther we go away from the data, we have really large uncertainty. 00:32:06.830 --> 00:32:10.850 And that is exactly what we want for nonergodic models. 00:32:10.850 --> 00:32:14.850 Because, if we have some data, then we can – then we think 00:32:14.850 --> 00:32:17.510 we know something. If there’s no data – if we go 00:32:17.510 --> 00:32:21.020 to northeastern California where we don’t have data, 00:32:21.020 --> 00:32:24.140 we don’t know anything. So the uncertainty needs to be large. 00:32:24.140 --> 00:32:28.020 So that means, in this case, we’re kind of maybe here. 00:32:28.020 --> 00:32:31.520 So we – like, really, really big uncertainty. 00:32:32.740 --> 00:32:37.840 This is how it looks like, and so I decided to put this in. 00:32:37.850 --> 00:32:41.460 Might look intimidating at first, but it really isn’t. 00:32:41.460 --> 00:32:46.060 So what we have is, we have some of those coefficients, like beta-i, like the 00:32:46.060 --> 00:32:51.800 constant, or anelastic continuation – is a function of location. 00:32:51.809 --> 00:32:54.480 And the way we write this, we put a Gaussian process 00:32:54.480 --> 00:32:58.320 prior over that particular coefficient. 00:32:58.320 --> 00:33:02.140 And then Gaussian process prior is defined by a mean function 00:33:02.140 --> 00:33:04.120 and a covariance function. 00:33:04.120 --> 00:33:06.340 And the covariance function is this. 00:33:06.340 --> 00:33:11.960 And we use these two – or, this covariance function 00:33:11.960 --> 00:33:18.770 for all the coefficients in our model. So what that basically means is that 00:33:18.770 --> 00:33:24.140 we have – for those coefficients that are not spatially varying, 00:33:24.140 --> 00:33:26.240 we put a constant in there. 00:33:26.240 --> 00:33:30.080 And, for those that are spatially varying, 00:33:30.080 --> 00:33:34.779 that actually depends on the distance between two points. 00:33:34.779 --> 00:33:36.500 That covariance matrix. 00:33:36.500 --> 00:33:38.510 And that is all function. 00:33:38.510 --> 00:33:42.350 That means, if these two points are really, really close together, 00:33:42.350 --> 00:33:45.850 then their correlation is pretty – is large. 00:33:45.850 --> 00:33:51.149 And then, if we go one step further and say, how was the data generated, well, 00:33:51.149 --> 00:33:54.240 the data is generated from a normal distribution. 00:33:54.240 --> 00:33:58.450 So all our records are generated from normal distribution, which has 00:33:58.450 --> 00:34:05.910 a mean zero. And all the modeling goes on in the covariance function. 00:34:05.910 --> 00:34:10.109 So we have the covariance function, which is defined by this covariant 00:34:10.109 --> 00:34:14.089 or this covariance matrix, which is defined by this covariance function. 00:34:14.089 --> 00:34:17.079 And then we also have some noise term on the diagonal. 00:34:17.079 --> 00:34:21.319 Because there will always be noise. We don’t really – what this actually 00:34:21.319 --> 00:34:27.179 means – so I don’t really want to go into this, but this constant here means 00:34:27.179 --> 00:34:32.679 that we have a linear model in the coefficients that do not vary spatially. 00:34:32.679 --> 00:34:36.450 And what this actually encodes is that the spatially varying coefficients 00:34:36.450 --> 00:34:42.319 are correlated. That means the closer to locations are, the stronger there is 00:34:42.319 --> 00:34:46.970 a correlation between the values of those coefficients. 00:34:46.970 --> 00:34:52.259 And that actually allows us to do some aggregation of data, and that – because, 00:34:52.260 --> 00:34:55.740 if you think about it, it’s a continuously varying model. 00:34:55.740 --> 00:34:59.059 We have infinitely many data locations. 00:34:59.059 --> 00:35:01.069 But we don’t really have the data to do that. 00:35:01.069 --> 00:35:04.680 So that means, by assuming – or, by modeling them as 00:35:04.680 --> 00:35:09.230 spatially correlated, or modeling them with the GP prior, we can 00:35:09.230 --> 00:35:13.700 actually aggregate some data that is sort of close together. 00:35:15.620 --> 00:35:20.160 How that actually looks like – looks like compared to a typical 00:35:20.160 --> 00:35:23.940 regression is that, say, in a typical regression, let’s say 00:35:23.940 --> 00:35:27.660 we observe Y-1 to Y-N. That’s our observed data. 00:35:27.660 --> 00:35:32.640 Let’s say log PGA at N different records. 00:35:32.650 --> 00:35:37.180 And then we typically have a normal distribution, and the mean is some 00:35:37.180 --> 00:35:40.359 function depending on magnitude, distance, and so on, 00:35:40.359 --> 00:35:45.369 and some coefficients. And then we just have the covariance 00:35:45.369 --> 00:35:49.920 matrix for that normal distribution. It’s just a diagonal matrix with – 00:35:49.920 --> 00:35:55.160 actually that should be sigma-squared – with the sigma-squared on the diagonal. 00:35:55.160 --> 00:36:02.979 And on the non-diagonal, the entries of that covariance matrix are zero. 00:36:02.979 --> 00:36:09.450 On the other hand, if we – in a typical GMPE, that looks pretty similar. 00:36:09.450 --> 00:36:15.540 The only thing is, now we replace sigma with phi-squared plus tau-squared. 00:36:15.540 --> 00:36:19.940 And then we have some off – we can have off-diagonal elements. 00:36:19.940 --> 00:36:24.160 And in a typical GMPE, that’s just accounting for event terms. 00:36:24.160 --> 00:36:30.000 That means, if we have – if the i-th and j-th record are from the same event, 00:36:30.009 --> 00:36:34.089 we have a tau-squared on the – in the covariance matrix. 00:36:34.089 --> 00:36:37.359 And then the only thing that we’re doing, we’re just making 00:36:37.359 --> 00:36:42.119 this one a lot more complicated. So now, in the varying coefficient 00:36:42.119 --> 00:36:48.380 GP model, we’re just moving everything to the covariance matrix – 00:36:48.380 --> 00:36:52.450 into the covariance matrix of the data. We still assume that all of our 00:36:52.450 --> 00:36:57.319 observed data is a multivariate normal distribution. 00:36:57.319 --> 00:37:00.720 But now we have – we assume that has a mean zero, and all the 00:37:00.720 --> 00:37:04.840 modeling goes on in our covariance matrix. 00:37:06.380 --> 00:37:10.340 There’s probably some sort of step in between, and I think that’s probably 00:37:10.349 --> 00:37:15.040 the way to go is to put some part back into the mean function. 00:37:15.040 --> 00:37:19.760 But still have some sort of complicated model in the covariance 00:37:19.760 --> 00:37:27.780 function which accounts for spatial correlation between some coefficients 00:37:27.789 --> 00:37:31.109 or some effects that we want to model. 00:37:31.109 --> 00:37:35.749 And then we still need something – and we can make the covariance 00:37:35.749 --> 00:37:38.469 matrix scale as complicated as we want. 00:37:38.469 --> 00:37:43.809 And, depending on how we model – how we set it up, we can model 00:37:43.809 --> 00:37:47.839 different things. But this is – this is really something that we need to think 00:37:47.840 --> 00:37:54.520 about, and the next models will not look exactly like the previous ones. 00:37:55.780 --> 00:37:58.809 So this is – again, this is just the model that we had, so we have 00:37:58.809 --> 00:38:06.229 some sort of coefficients that we make spatially correlated. 00:38:06.229 --> 00:38:12.640 And then we set it up that each of them is – has this sort of covariance function. 00:38:12.640 --> 00:38:16.280 And what actually turns out is that the parameters of the models that 00:38:16.280 --> 00:38:22.579 we estimate are only the parameters of the covariance functions. 00:38:22.579 --> 00:38:27.970 So we only need the variances and the length scales of our covariance 00:38:27.970 --> 00:38:31.760 functions to make predictions. That means we do not estimate 00:38:31.760 --> 00:38:36.150 the individual coefficients. We do not estimate beta-1, beta-2. 00:38:36.150 --> 00:38:39.599 We do not estimate beta-zero at each location. 00:38:39.600 --> 00:38:45.920 We don’t need that to do prediction, and so this model is a very – 00:38:45.920 --> 00:38:48.869 it’s geared to what’s prediction of new data points. 00:38:48.869 --> 00:38:53.499 To predict new data – to make new predictions, the only thing 00:38:53.500 --> 00:39:00.420 you need to know are the parameters of the covariance function. 00:39:01.020 --> 00:39:05.520 So a quick word about prediction. 00:39:05.529 --> 00:39:11.819 If you know your multivariate normal distribution, then it’s pretty easy to do. 00:39:11.820 --> 00:39:15.820 We just condition – if we want to make a prediction, add a new, let’s say, 00:39:15.820 --> 00:39:19.480 magnitude-distance scenario at a new location, we just 00:39:19.489 --> 00:39:24.319 condition on observed data. And that basically means we just write it out. 00:39:24.319 --> 00:39:29.989 We have our observed data, and then we have our new data point that 00:39:29.989 --> 00:39:36.369 we want to predict. That’s just – again, that’s a normal distribution. And it has 00:39:36.369 --> 00:39:41.359 this covariance. So this is just the covariance of the observed data. 00:39:41.359 --> 00:39:44.410 We can calculate that. We know all of that. 00:39:44.410 --> 00:39:49.900 This is the covariance between the new data point and the observed data. 00:39:49.900 --> 00:39:53.960 Then only depends on location and magnitude, distance, and so on. 00:39:53.960 --> 00:39:55.970 So that’s pretty easy to do. 00:39:55.970 --> 00:40:01.269 And that’s the covariance of the new data point with itself. 00:40:01.269 --> 00:40:08.329 And then it turns out that, if you do that, then you can calculate the mean 00:40:08.329 --> 00:40:12.949 and the variance of the – of Y-star. That means the mean of the variance 00:40:12.949 --> 00:40:17.410 of the ground motion distribution at that new set of predictor variables, 00:40:17.410 --> 00:40:24.329 which is – I think it’s – the mean is just this times – so this covariance times 00:40:24.329 --> 00:40:31.449 this covariance times the observed data. And if you want to know the variance 00:40:31.449 --> 00:40:36.460 of this one, then it’s just this minus this times this times this. 00:40:36.460 --> 00:40:41.059 That’s pretty – I mean, it’s very easy to calculate. 00:40:41.059 --> 00:40:45.440 It can take a while to compute, depending on how large these 00:40:45.440 --> 00:40:50.840 matrices are, but that’s all there is to it – there is to it. 00:40:51.000 --> 00:40:57.180 So then, results. So this is just taken from Landwehr et al. 00:40:57.199 --> 00:41:01.670 I think there’s a lot of room for improvement for that particular model. 00:41:01.670 --> 00:41:05.520 So what we see – what we see here is, on the left is the – are the median 00:41:05.520 --> 00:41:11.540 predictions for a magnitude 6 and a distance of 20 kilometers 00:41:11.549 --> 00:41:17.329 all over California. And I think, just for simplicity, 00:41:17.329 --> 00:41:22.099 we set the station location the same as the earthquake location, 00:41:22.099 --> 00:41:25.849 which is not really realistic, but it made it easy to plot. 00:41:25.849 --> 00:41:29.779 And on the right we have the associated epistemic uncertainty. 00:41:29.779 --> 00:41:33.839 So that is just calculated by conditioning on the data and then 00:41:33.839 --> 00:41:36.979 just moving through all of California and then calculating 00:41:36.979 --> 00:41:39.309 a prediction at each point. 00:41:39.309 --> 00:41:45.579 What we see is, we see – there is quite a very – quite a large variation 00:41:45.579 --> 00:41:50.059 in predictions across California. We have some lower predictions 00:41:50.059 --> 00:41:52.499 in northern California. We have some larger predictions 00:41:52.499 --> 00:41:56.731 in southern California. But we also see that, once we get to 00:41:56.731 --> 00:42:00.589 northeast [inaudible] on the regions where we don’t really have a lot of data, 00:42:00.589 --> 00:42:05.489 we sort of go back to – so we don’t really see those big effects anymore. 00:42:05.489 --> 00:42:08.180 And that’s because we don’t have data there. 00:42:08.180 --> 00:42:12.599 That means we don’t – we cannot really constrain different predictions. 00:42:12.599 --> 00:42:15.259 So we just revert back to some sort of mean. 00:42:15.260 --> 00:42:20.280 We also see this in the associated uncertainty. 00:42:21.020 --> 00:42:26.510 That is small here and here. So that’s small everywhere where we have data. 00:42:26.510 --> 00:42:30.089 And it’s large everywhere else where we don’t have data. 00:42:30.089 --> 00:42:35.059 And that kind of coincides nicely with this because, well, on average, 00:42:35.059 --> 00:42:40.569 our adjustment – our adjustments to a median prediction will be zero there. 00:42:40.569 --> 00:42:43.529 But that doesn’t mean that those adjustments don’t exist. 00:42:43.529 --> 00:42:45.999 We just don’t know how large they are. 00:42:45.999 --> 00:42:50.680 That means we need to put these – we need to account that by having 00:42:50.680 --> 00:42:55.069 a very, very large epistemic uncertainty. 00:42:55.069 --> 00:43:00.280 We also have coefficients. And I said that we don’t estimate 00:43:00.280 --> 00:43:04.739 coefficients, but the way these coefficients – we can still calculate 00:43:04.740 --> 00:43:09.260 them, and the way these coefficients are calculated is by actually just 00:43:09.260 --> 00:43:13.880 generating synthetic data at all – at different locations 00:43:13.880 --> 00:43:16.619 and then fitting a linear model to that. 00:43:16.619 --> 00:43:20.700 And that gives us – so, because the predictions are different, 00:43:20.700 --> 00:43:24.119 that will give us a different set of coefficients [inaudible]. 00:43:24.119 --> 00:43:25.959 So, on the left, we have a constant. 00:43:25.959 --> 00:43:29.459 And, on the right, we have the geometrical spreading. 00:43:29.459 --> 00:43:34.819 And, as you can see, these things are correlated. 00:43:34.819 --> 00:43:40.420 Annemarie makes that point all the time that these things go together. 00:43:40.420 --> 00:43:43.660 And that is true. So we can’t just say, oh, yeah, this is – 00:43:43.660 --> 00:43:47.000 we just – oh, we have this constant. We just apply this. 00:43:47.000 --> 00:43:51.940 No. At the same time, we – if we make the constant larger, we have to make the 00:43:51.941 --> 00:43:55.989 geometrical spreading a little bit smaller. Those two things go together. 00:43:55.989 --> 00:43:58.580 These things go together all the time. 00:43:58.580 --> 00:44:05.140 So, yeah, I mean, that’s just – but the way we then use this is 00:44:05.140 --> 00:44:09.440 we use these coefficients, calculated from synthetic data or 00:44:09.460 --> 00:44:15.580 from model predictions, we use these coefficients later in the hazard run. 00:44:16.180 --> 00:44:19.780 The one thing that’s not really well done 00:44:19.780 --> 00:44:23.900 in the VCM is the anelastic attenuation. 00:44:24.780 --> 00:44:29.979 Initially, we had the anelastic attenuation as a function of event 00:44:29.979 --> 00:44:34.650 location, which means it's isotropic in all directions, which is certainly 00:44:34.650 --> 00:44:41.430 not correct. We had written a paper. It was pretty much finished. 00:44:41.430 --> 00:44:46.009 And then I think Frank Scherbaum, my old adviser, said, why don’t you 00:44:46.009 --> 00:44:49.239 just use the midpoint between earthquake and station location? 00:44:49.239 --> 00:44:52.459 That actually is a better thing to do instead of making it – 00:44:52.459 --> 00:44:55.509 but it was way too late to actually change that. 00:44:55.509 --> 00:44:59.529 But that would be a better way to do it. But ideally, it should depend on 00:44:59.529 --> 00:45:03.160 the event and station location. That means if we have an event 00:45:03.160 --> 00:45:08.920 here, and – I don’t know. So, if we have an event and a station, 00:45:08.920 --> 00:45:11.680 and if we have the same event and the station is somewhere else, 00:45:11.680 --> 00:45:15.459 we should get a different value for that coefficient. 00:45:15.459 --> 00:45:20.209 The problem is that that makes this a little bit – you can still do it, 00:45:20.209 --> 00:45:23.969 but it makes it, like, four-dimensional. 00:45:23.969 --> 00:45:25.979 Because you have four different locations. 00:45:25.979 --> 00:45:30.039 And it’s still not really great. There are still some other problems. 00:45:30.039 --> 00:45:33.339 And – because – and then – so we decided not to actually 00:45:33.339 --> 00:45:37.140 do this for the hazard runs. And because we think, overall, 00:45:37.140 --> 00:45:42.390 the anelastic attenuation should actually depend on the path traveled. 00:45:42.390 --> 00:45:48.210 So what we then did, we used a model that was introduced by 00:45:48.210 --> 00:45:52.930 Dawood and Rodriguez-Marek. And we used a cell-specific attenuation 00:45:52.930 --> 00:45:56.470 model. That’s very, very simple to do. It’s very simple to calculate. 00:45:56.470 --> 00:46:02.180 Basically, what the [inaudible] of this is the – or, this is the event location. 00:46:02.180 --> 00:46:05.150 That’s the station. 00:46:05.150 --> 00:46:08.259 They just divide the whole space where they are 00:46:08.259 --> 00:46:10.979 into different cells of a certain size. 00:46:10.979 --> 00:46:15.150 And then you just calculate the path traveled in each of those cells. 00:46:15.150 --> 00:46:20.969 And then each cell gets a different attenuation coefficient. 00:46:20.969 --> 00:46:25.269 And then you just sum the attenuation coefficient for that cell 00:46:25.269 --> 00:46:27.859 times the path traveled in that cell. 00:46:27.859 --> 00:46:33.559 It’s very easy to do. It’s very easy to calculate those distances. 00:46:33.559 --> 00:46:38.630 And it’s not really that complicated to get the – get those coefficients. 00:46:38.630 --> 00:46:42.890 In an ergodic model, this would just be beta-a times R. 00:46:42.890 --> 00:46:46.009 So this would just be one coefficient everywhere. 00:46:46.009 --> 00:46:50.069 And if you go from here to there, it’s the same attenuation if you go from here to 00:46:50.069 --> 00:46:55.220 here. It’s always the same attenuation, which is certainly not correct. 00:46:56.040 --> 00:46:59.819 And then what we did is we extended the cell-specific model because 00:46:59.819 --> 00:47:05.619 one thing that we really wanted to have is uncertainty in those cell-specific 00:47:05.619 --> 00:47:11.150 coefficients. Because that’s really, really important to carry that through. 00:47:11.150 --> 00:47:15.890 So they didn’t do that in the original paper. 00:47:15.890 --> 00:47:21.190 It was also – it was done for Japan and subduction data, so we needed to 00:47:21.190 --> 00:47:24.720 have it for California. So we made the whole thing – 00:47:24.720 --> 00:47:28.880 we modeled everything as a Bayesian model, and we modeled the cell-specific 00:47:28.890 --> 00:47:33.599 coefficients as random effects, which is – which allows us to nicely 00:47:33.599 --> 00:47:37.880 keep track of the associated uncertainty. So basically, what we have – each of 00:47:37.880 --> 00:47:44.189 those cell-specific coefficients is assumed to be normally distributed 00:47:44.189 --> 00:47:48.630 with a certain mean. That would be, like, an average California attenuation 00:47:48.630 --> 00:47:53.099 and a certain standard deviation. And that is, like, the range of 00:47:53.099 --> 00:47:56.670 possible attenuation coefficients across California. 00:47:56.670 --> 00:48:05.900 We also make it truncated so that we don’t get – that we don’t get 00:48:05.900 --> 00:48:11.800 coefficients that are larger than zero. What we see here is, on the left is all 00:48:11.809 --> 00:48:17.959 the paths, and so that’s the stations in red and the events in blue and all the 00:48:17.959 --> 00:48:24.609 paths in the ASK14 data sets for the – for the NGA-West2 model. 00:48:24.609 --> 00:48:29.319 And on the right we see how many paths are traveled through each cell. 00:48:29.319 --> 00:48:33.650 We have lots of paths here and not really that many 00:48:33.650 --> 00:48:36.760 in northeastern California, and so on. 00:48:37.300 --> 00:48:42.760 And then, this is how that looks like. On the left, we have the average 00:48:42.769 --> 00:48:47.239 difference to the mean. So that would be beta minus mu-beta. 00:48:47.239 --> 00:48:52.640 So that’s – so we see, where we actually do have data, we do change the mean. 00:48:52.640 --> 00:48:56.580 So we do see differences to, let’s say, the average coefficient. 00:48:56.580 --> 00:49:01.759 Whereas, where we don’t have data, we don’t really – it all just falls back 00:49:01.759 --> 00:49:06.150 towards the mean. So the adjustment is pretty much zero there. 00:49:06.150 --> 00:49:09.640 And that corresponds, and that always goes together. 00:49:09.640 --> 00:49:14.650 We have strong systematic effects. It means we have – but we also are 00:49:14.650 --> 00:49:17.200 more certain about that because we have data there. 00:49:17.200 --> 00:49:20.109 Whereas, if we don’t have data, then we have really, really 00:49:20.109 --> 00:49:24.340 large uncertainty in those coefficients. 00:49:24.340 --> 00:49:26.619 What you can see here is – I don’t know if you can, 00:49:26.619 --> 00:49:29.779 but there’s a – these two little symbols. 00:49:29.779 --> 00:49:35.489 So we then calculated predictions for paths from here to here and the 00:49:35.489 --> 00:49:39.859 path from here to here. So one path through cells where 00:49:39.859 --> 00:49:43.440 we have lots of uncertainty, and one path through cells where 00:49:43.440 --> 00:49:48.099 we have very little uncertainty because there’s a lot of data there. 00:49:48.099 --> 00:49:51.829 They’re all – they have this – you know, same length, so they are – 00:49:51.829 --> 00:49:56.460 I think it’s 100 kilometers. And these are the predictions of those. 00:49:56.460 --> 00:50:01.900 So on the left, we have the predictions for the paths that goes through here. 00:50:01.900 --> 00:50:04.859 And on the right, we have the path that goes 00:50:04.859 --> 00:50:07.439 here somewhere in southern California. 00:50:07.439 --> 00:50:15.989 And what we can see is – so this is the posterior distribution of the predictions. 00:50:15.989 --> 00:50:20.099 So that means we get out of the model, we get some posterior samples. 00:50:20.099 --> 00:50:24.729 We get, like, 1,000 different samples of coefficients that cover the distribution 00:50:24.729 --> 00:50:28.130 and the uncertainty in each cell. And then we calculate for each of 00:50:28.130 --> 00:50:31.140 those cells, and we calculate the median prediction. 00:50:31.140 --> 00:50:36.299 And that – and that gives us an estimate of the epistemic uncertainty 00:50:36.299 --> 00:50:40.559 of our predictions just based on that particular path. 00:50:40.559 --> 00:50:46.010 And that is very, very wide on the left, and it’s very narrow on the right. 00:50:46.010 --> 00:50:49.189 But it’s also very different from the mean. 00:50:49.189 --> 00:50:54.529 What we also have here is – so this one is calculated using the average 00:50:54.529 --> 00:51:00.000 coefficient. So that’s a prediction using this new beta. 00:51:00.000 --> 00:51:05.100 And so that’s the same in both cases because it’s 100 kilometers. 00:51:05.109 --> 00:51:14.210 And then this is the prediction using the ASK14 anelastic attenuation coefficient. 00:51:14.210 --> 00:51:18.739 That one is different from this one, so even though we kind of use – 00:51:18.739 --> 00:51:23.119 we use all the other coefficients from ASK14, so that’s what we have 00:51:23.120 --> 00:51:29.780 is an adjustment – an anelastic adjustment to the ASK14 model. 00:51:29.780 --> 00:51:35.740 But we actually – our average coefficient for California is different 00:51:35.740 --> 00:51:42.220 from the one that they have. And the reason for that is that the 00:51:42.220 --> 00:51:45.320 ASK14, that’s an ergodic model. That means it’s aggregated – it’s 00:51:45.320 --> 00:51:50.520 aggregating all data from California. And that means we have – theirs has 00:51:50.529 --> 00:51:54.719 more data from southern California than it does from northern California. 00:51:54.719 --> 00:52:01.119 And that means the ASK14 anelastic attenuation coefficient is sort of 00:52:01.119 --> 00:52:05.749 biased toward southern California. And if there are strong differences 00:52:05.749 --> 00:52:11.260 between southern California and northern California, then it won’t really 00:52:11.260 --> 00:52:14.650 work that well in northern California. On the other hand, if you model 00:52:14.650 --> 00:52:20.499 everything as a random effect, that gets taken care of. 00:52:20.499 --> 00:52:23.720 And what we have – we have an average over all those cells. 00:52:23.720 --> 00:52:25.440 And then the cells are different. 00:52:25.440 --> 00:52:32.210 So that’s why we get a slightly different average coefficient of California. 00:52:32.210 --> 00:52:36.829 If we do – if we weight the anelastic attenuation coefficient of each cell 00:52:36.829 --> 00:52:41.719 by the number of cells and then calculate the weighted average, 00:52:41.719 --> 00:52:48.859 we get the same – the same value, pretty much, as ASK14 out. 00:52:48.859 --> 00:52:58.329 And then, basically, the only thing that was done in Norm’s paper for 2019 was 00:52:58.329 --> 00:53:02.609 just combining the varying coefficient model and the cell-specific attenuation 00:53:02.609 --> 00:53:07.729 model to calculate site-specific nonergodic seismic hazard 00:53:07.729 --> 00:53:10.279 for three different sites in PSHA. 00:53:10.279 --> 00:53:14.920 So I say the only thing. Took two years to write that paper. 00:53:14.920 --> 00:53:21.600 So – and to get all the calculations right. But it’s really not that – theoretically, 00:53:21.600 --> 00:53:25.820 it’s not that complicated. So what we took from the varying 00:53:25.820 --> 00:53:30.360 coefficient model is we have an event location-specific constant 00:53:30.369 --> 00:53:34.880 and geometrical spreading. We have a site location-specific 00:53:34.880 --> 00:53:41.299 constant and the Vs30 scaling adjustment coefficient that is – 00:53:41.299 --> 00:53:45.209 that comes out of Vs30. And we also take the epistemic 00:53:45.209 --> 00:53:51.079 uncertainty from Landwehr to – and we apply that to the – 00:53:51.080 --> 00:53:55.900 to the constant. That was just – that’s not really well handled. 00:53:55.900 --> 00:54:01.580 Because it’s really complicated to actually get the uncertainty of the – 00:54:01.599 --> 00:54:08.380 of individual coefficients. So we just put all the uncertainty into the constant. 00:54:08.380 --> 00:54:12.069 And then, from the cell-specific model, we just took – we took the cell-specific 00:54:12.069 --> 00:54:17.150 attenuation coefficients, and we estimated different ones. 00:54:17.150 --> 00:54:22.540 Because there will be some tradeoff with the geometrical spreading coefficient. 00:54:22.540 --> 00:54:25.940 So we took – we actually calculated five different sets of cell-specific 00:54:25.949 --> 00:54:31.349 attenuation coefficients to account for uncertainty in the geometrical spreading 00:54:31.349 --> 00:54:35.660 of the base model. And then the base model is just using, like, 00:54:35.660 --> 00:54:41.359 an aggregate model from NGA-West2. So we took the five NGA-West2 model, 00:54:41.360 --> 00:54:47.580 averaged them, sort of, and then accounted for there within – 00:54:47.580 --> 00:54:50.671 accounted for the between model uncertainty to get some 00:54:50.680 --> 00:54:54.029 sort of epistemic uncertainty in the background model. 00:54:54.900 --> 00:55:00.180 And now, for site-specific hazard, that is pretty – and that’s all you need, 00:55:00.189 --> 00:55:05.529 and that’s pretty easy, then, to put that into hazard calculations. 00:55:05.529 --> 00:55:11.349 So what we did is, if we have a site – let’s say the site in San Jose. 00:55:11.349 --> 00:55:14.780 We just put a grid – I don’t know – I think it was, like, 00:55:14.780 --> 00:55:19.839 200 by 200 kilometers around that site and have lots of grid points. 00:55:19.839 --> 00:55:25.890 And for each grid point, we calculate – we find the geometrical spreading 00:55:25.890 --> 00:55:30.900 coefficient from the VCM. The VCM has points all over California. 00:55:30.900 --> 00:55:35.019 So we find that particular value for one particular grid point. 00:55:35.019 --> 00:55:40.440 And then, for those same grid points around the site, we randomly sample 00:55:40.440 --> 00:55:46.769 100 different constants from the distribution that was implied by 00:55:46.769 --> 00:55:50.160 the varying coefficient model. So varying – it gives us – for each 00:55:50.160 --> 00:55:56.029 of those grid points, it gives us the constant adjustment and an uncertainty. 00:55:56.029 --> 00:56:01.479 So we randomly sample that uncertainty 100 times so that we have some – 00:56:01.480 --> 00:56:07.440 so that we carry that uncertainty through to the end hazard curves. 00:56:07.440 --> 00:56:13.220 And so that basically means we have, like, 100 different branches on the 00:56:13.230 --> 00:56:19.279 logic tree, each with a set of coefficients for all of our grid points. 00:56:19.279 --> 00:56:23.959 And then, for the site location, we just find the Vs30 scaling coefficient 00:56:23.959 --> 00:56:29.559 from the VCM, apply that there, and we also have a – like, a site constant from 00:56:29.559 --> 00:56:35.390 the VCM, together with an uncertainty, and we randomly sample 00:56:35.390 --> 00:56:38.760 100 site [inaudible]. So we randomly sample 100 site 00:56:38.760 --> 00:56:43.000 constants for that particular site and also put that on the logic tree. 00:56:43.000 --> 00:56:47.529 That means we have, again, 100 different – 100 different 00:56:47.529 --> 00:56:50.560 sample site terms that sample the uncertainty and the 00:56:50.560 --> 00:56:53.839 site adjustments for that particular site. 00:56:53.839 --> 00:56:57.450 And then, for all the grid points around the site, we also – we calculate 00:56:57.450 --> 00:57:04.720 the distance – the cell-specific distances to the site, calculate – and then calculate 00:57:04.720 --> 00:57:08.980 the anelastic attenuation term together with the cell-specific constant. 00:57:08.980 --> 00:57:13.060 And then that means, for each grid point that we have – that we have, let’s say, 00:57:13.060 --> 00:57:19.819 200 by 200, every 5 kilometers – I don’t know how much that is 00:57:19.819 --> 00:57:23.079 right now, but then, for each of those grid points, we have an adjustment. 00:57:23.079 --> 00:57:29.420 And that – and then, if we – if we want to calculate the median prediction for 00:57:29.420 --> 00:57:36.189 a source, we just – for one particular source, we find the grid point that is 00:57:36.189 --> 00:57:41.549 closest to the source, and then just add the adjustments that we had previously 00:57:41.549 --> 00:57:45.680 calculated to the median prediction there. 00:57:45.680 --> 00:57:53.039 So that’s pretty easy for a site-specific. If you want to repeat that for many 00:57:53.040 --> 00:57:56.940 different sites, that’s going to be more complicated. 00:57:57.829 --> 00:58:01.700 So that’s basically just – it’s actually not from the Landwehr paper, 00:58:01.710 --> 00:58:10.019 but this is kind of what this shows. So what we see here is, we see – 00:58:10.019 --> 00:58:18.219 this is actually the data that we have. And these are sort of like the constants – 00:58:18.220 --> 00:58:23.100 the constants that we estimate for each source location. 00:58:23.100 --> 00:58:27.390 So for each location, we estimate, oh, this is actually the constant that we 00:58:27.390 --> 00:58:30.660 need to apply based on the model that we have. 00:58:30.660 --> 00:58:36.949 And then we go through this grid here and calculate the mean of those 00:58:36.949 --> 00:58:41.969 constants at all those grid points, conditioned on what we observe 00:58:41.969 --> 00:58:45.219 at the – at the data points that we have. 00:58:45.219 --> 00:58:49.710 And then, what we see is here. So here we have negative values. 00:58:49.710 --> 00:58:55.009 And that means all the points close to those negative constants 00:58:55.009 --> 00:58:59.309 that we have from the model also get a negative adjustment – 00:58:59.309 --> 00:59:01.700 a negative average adjustment. 00:59:01.700 --> 00:59:08.140 And then same here. We get a slightly positive or more zero adjustment. 00:59:08.140 --> 00:59:11.839 And then, the farther away we go from the data, the adjustment terms 00:59:11.839 --> 00:59:16.249 become zero. And on the right, we see the standard deviation of those. 00:59:16.249 --> 00:59:21.140 And that’s the standard deviation of those points that 00:59:21.140 --> 00:59:22.749 go together with that mean. 00:59:22.749 --> 00:59:28.089 And, again, it’s small close to the data and large very far of the data. 00:59:28.089 --> 00:59:31.849 And then what we do is we just – for each of those grid points, 00:59:31.849 --> 00:59:36.469 we randomly sample from this distribution implied by this mean 00:59:36.469 --> 00:59:41.509 and the standard deviation 100 times. And that gives us – if we have a source 00:59:41.509 --> 00:59:47.799 somewhere around here, we just find the closest point and add that prediction at 00:59:47.800 --> 00:59:52.680 that adjustment to the median prediction for that particular source. 00:59:54.060 --> 01:00:01.060 These are the sites that we use. So one is somewhere south of San Jose. 01:00:01.060 --> 01:00:07.960 I think that was mainly chosen because it’s somewhere in a spot 01:00:07.979 --> 01:00:12.439 where we actually have data and where the uncertainty is low. 01:00:12.439 --> 01:00:15.819 And this is chosen because that’s where we don’t have data 01:00:15.819 --> 01:00:18.249 and the uncertainty is large. 01:00:18.249 --> 01:00:25.469 And then this is a site west of San Luis Obispo pretty close to the – 01:00:25.469 --> 01:00:29.520 close to the water. And you can – I asked Norm if 01:00:29.520 --> 01:00:33.420 we should be more specific in the paper, and he said no, 01:00:33.420 --> 01:00:37.329 but you can probably guess which site that is. 01:00:37.329 --> 01:00:41.849 And this just basically shows the epistemic uncertainty associated with 01:00:41.849 --> 01:00:46.009 median predictions, again from the Landwehr paper. 01:00:46.009 --> 01:00:50.329 And then, this is how we end up with these hazard curves. 01:00:50.329 --> 01:00:57.509 So this is calculated hazard using NGA-West2 as is. 01:00:57.509 --> 01:01:03.829 And this is calculated hazard using the nonergodic adjustment terms 01:01:03.829 --> 01:01:07.660 from Landwehr and the cell-specific models. 01:01:07.660 --> 01:01:12.080 So, on the left for San Jose. On the right for northeastern California. 01:01:12.080 --> 01:01:17.630 Here, we don’t really have data. That means, on average, our 01:01:17.630 --> 01:01:21.170 adjustment coefficients will be zero, but it will have very large uncertainty. 01:01:21.170 --> 01:01:24.640 And that means, if we – if they have very large uncertainty, 01:01:24.640 --> 01:01:30.499 and we sample from that 100 times, each different run will be very different. 01:01:30.499 --> 01:01:34.089 But if we average them, we go back to the mean hazard. 01:01:34.089 --> 01:01:37.630 But the fractiles just become really, really large. 01:01:37.630 --> 01:01:43.820 On the left, we actually do have data. That means we actually have the 01:01:43.820 --> 01:01:48.820 adjustment coefficients that are non-zero and that drive the median 01:01:48.820 --> 01:01:53.960 predictions away from their ergodic values implied by NGA-West2. 01:01:53.969 --> 01:01:59.969 And that means we actually change the mean hazard. 01:01:59.969 --> 01:02:03.240 On the other hand, since we have data, we’re also 01:02:03.240 --> 01:02:05.430 a little bit more certain about those values. 01:02:05.430 --> 01:02:10.380 That means we decrease the fractiles compared to the [inaudible]. 01:02:10.380 --> 01:02:18.309 So we see some larger differences at the very low exceedance rates. 01:02:18.309 --> 01:02:22.950 Because there’s some contributions from far-away sources where we 01:02:22.950 --> 01:02:27.900 actually do have a little bit of data for northeastern California. 01:02:29.089 --> 01:02:35.419 The last thing I want to mention is that we can do these things in 01:02:35.419 --> 01:02:39.930 California because there’s a lot of data in California. 01:02:39.930 --> 01:02:46.040 So the – so systematic effects are pretty strong in California. 01:02:46.040 --> 01:02:52.919 And the thing is, if we go to some regions where we don’t have data, that 01:02:52.920 --> 01:02:57.880 doesn’t mean that these non-systematic, nonergodic effects do not exist. 01:02:57.880 --> 01:03:00.199 It just means that we cannot estimate them. 01:03:00.199 --> 01:03:03.739 But that doesn’t mean that you can’t really run nonergodic hazard there. 01:03:03.739 --> 01:03:07.419 It’s just more complicated. You just – if data is sparse, 01:03:07.419 --> 01:03:09.759 you just have to make more assumptions. 01:03:09.759 --> 01:03:17.239 So one thing that we just did, and we need to get back to that, is – 01:03:17.240 --> 01:03:20.160 so that’s an example from Georgia. 01:03:20.160 --> 01:03:27.619 And so I think we had maybe, like, 50 records from Georgia. 01:03:27.619 --> 01:03:31.519 And then we tried to calculate nonergodic adjustment terms. 01:03:31.519 --> 01:03:35.709 So this is – on the left is nonergodic adjustment terms relative to 01:03:35.709 --> 01:03:41.300 this particular site here. And that is, I think, where they have a 01:03:41.300 --> 01:03:46.440 dam. They don’t really have data there, and they don’t have stations there. 01:03:46.440 --> 01:03:52.560 But this dam is really, really important for the electricity supply in Georgia. 01:03:52.560 --> 01:03:56.920 So, if something happened to that dam, it would be pretty bad, and they 01:03:56.920 --> 01:04:02.840 would have to buy electricity from somewhere else. 01:04:02.840 --> 01:04:06.840 So probably from neighboring countries. 01:04:06.849 --> 01:04:11.420 So they’re – my understanding is, they’re a little bit concerned, 01:04:11.420 --> 01:04:15.799 and they wanted to, like, look at that. And then Norm wanted to go and say, 01:04:15.799 --> 01:04:20.519 look, you actually can do this. So we – so they have some pocket of 01:04:20.519 --> 01:04:25.569 data here and some data down there. And we actually calculated 01:04:25.569 --> 01:04:30.580 nonergodic adjustment terms relative to that particular site. 01:04:30.580 --> 01:04:35.060 And we also calculated nonergodic adjustment terms relative to a site 01:04:35.061 --> 01:04:39.469 that sits somewhere in that pocket where we have data. 01:04:39.469 --> 01:04:42.360 So I don’t know if you can read those numbers. 01:04:44.029 --> 01:04:47.959 The contours sort of make sense. So here we are – that’s where 01:04:47.960 --> 01:04:51.900 we don’t really know. The adjustments are pretty constant. 01:04:51.900 --> 01:04:55.880 Whereas, they get a little bit stronger towards the points where we have 01:04:55.880 --> 01:04:59.789 data because there we have data. That means we might – but, since we 01:04:59.789 --> 01:05:07.179 only have these, like, I don’t know, 40 data points, the average adjustments 01:05:07.179 --> 01:05:09.750 are extremely small. 01:05:09.750 --> 01:05:13.039 The same thing here. That shows – oh, yeah. 01:05:13.039 --> 01:05:15.079 That actually shows the median adjustment. 01:05:15.079 --> 01:05:18.919 So I didn’t put in the variances. We barely see a dent. 01:05:18.919 --> 01:05:23.989 So you – the variances get a little bit smaller where you have data, 01:05:23.989 --> 01:05:28.569 but nothing like in California where we go from 0.5 to 0.2 or something. 01:05:28.569 --> 01:05:33.049 Here, we go from 0.55 to 0.54 or something like that. 01:05:33.049 --> 01:05:39.089 So the data that we have is not really enough to actually change this. 01:05:39.089 --> 01:05:42.099 But we can still do this. 01:05:42.099 --> 01:05:47.589 So we just have to make some sort of assumptions how these 01:05:47.589 --> 01:05:53.010 nonergodic effects will be – will be distributed. 01:05:53.010 --> 01:05:57.650 So what we did here is we just assumed that the covariant structure 01:05:57.650 --> 01:06:02.960 that we have for California is transferable to Georgia. 01:06:02.960 --> 01:06:06.459 That means we take the same length scales and the same variances of the 01:06:06.459 --> 01:06:10.969 nonergodic effects and apply them in Georgia. 01:06:10.969 --> 01:06:14.540 Whether that is correct or not, that’s probably … 01:06:16.540 --> 01:06:19.680 It’s probably not totally correct, but it’s at least a start. 01:06:19.690 --> 01:06:24.229 And then basically, if you just do that, your hazard will probably look like this. 01:06:24.229 --> 01:06:27.519 So that means the nonergodic hazard will not be different from the ergodic 01:06:27.519 --> 01:06:31.140 hazard because we don’t know what the adjustments will be. 01:06:31.140 --> 01:06:34.319 We just have really, really wide uncertainties. 01:06:34.319 --> 01:06:37.799 These wide uncertainties will be a lot more realistic than 01:06:37.800 --> 01:06:41.000 uncertainties from an ergodic model. 01:06:44.800 --> 01:06:54.219 Things to be careful about or things to – let’s say issues or things to look into 01:06:54.219 --> 01:07:01.579 going forward is – there are some issues or – yeah, there are 01:07:01.579 --> 01:07:04.329 some issues with the varying coefficient model. 01:07:04.329 --> 01:07:09.130 And some of them are just computational, in that is, if you 01:07:09.130 --> 01:07:16.410 remember, like, the full VCM models everything in the covariance matrix. 01:07:16.410 --> 01:07:24.200 That means we have a covariance matrix that has non-zero entries everywhere. 01:07:24.200 --> 01:07:27.589 And that means, if we have 12,000 records, we have a covariance 01:07:27.589 --> 01:07:34.309 matrix of 12,000 by 12,000. That will not – that will crash your 01:07:34.309 --> 01:07:40.650 computer if you try to invert that matrix. So that’s not something you can just do. 01:07:40.650 --> 01:07:44.429 So the question – so what we have to do is we have to make some 01:07:44.429 --> 01:07:48.809 approximations, which means, how good are these approximations 01:07:48.809 --> 01:07:55.920 really to capture – to capture all the stuff that we want to capture, 01:07:55.920 --> 01:08:03.289 especially if data is maybe not – is maybe not regularly spaced. 01:08:03.289 --> 01:08:07.019 Often when people do, oh, I’ve got this new approximation. 01:08:07.019 --> 01:08:11.579 It works really well on this simulated data set, which is – 01:08:11.579 --> 01:08:15.160 has points everywhere. That’s not real data, so that’s 01:08:15.160 --> 01:08:20.880 not really how we have. We have classes of data here, classes of data there. 01:08:20.880 --> 01:08:23.839 So the question is, how good are these approximations, 01:08:23.839 --> 01:08:26.259 and can they really capture what we want to have? 01:08:26.259 --> 01:08:28.969 There are tradeoffs between some parameters. 01:08:28.969 --> 01:08:32.711 So you can often make the variance larger and maybe the length scale 01:08:32.711 --> 01:08:37.529 a little bit smaller and get very similar predictions. 01:08:37.529 --> 01:08:45.230 So that is something that probably needs some sort of input. 01:08:45.230 --> 01:08:50.779 And then there’s a question – some parameters, I don’t really – 01:08:50.779 --> 01:08:54.420 are not really well-identifiable, so that means you can just 01:08:54.420 --> 01:08:58.450 swap them out and get the same results. 01:08:58.450 --> 01:09:05.360 So that’s just computational stuff, and that’s some – we’re looking into that. 01:09:05.360 --> 01:09:10.480 There is also modeling stuff. It is kind of – how should we really 01:09:10.489 --> 01:09:13.549 model the covariance function? Like, right now, we just use the 01:09:13.549 --> 01:09:17.520 exponential covariance function. You can do some other – you can 01:09:17.520 --> 01:09:22.730 do a squared exponential. You can do something other. 01:09:22.730 --> 01:09:26.730 You can do [inaudible] or whatever. 01:09:26.730 --> 01:09:31.340 So does that have an input – an influence on the results? 01:09:31.340 --> 01:09:34.680 [loud static] 01:09:34.680 --> 01:09:35.900 [static stops] 01:09:35.900 --> 01:09:39.660 I’m sorry. [laughter] 01:09:39.660 --> 01:09:45.160 Okay. The other question is, so right now – so we moved everything 01:09:45.170 --> 01:09:48.349 to the covariance function. But should we maybe move 01:09:48.349 --> 01:09:52.190 some stuff back to the mean. So if we have a mean function, 01:09:52.190 --> 01:09:58.400 that might be a little bit easier to put some constraints on some of 01:09:58.400 --> 01:10:01.980 the scaling of the model. Because it’s harder – it’s harder 01:10:01.980 --> 01:10:05.980 to understand the covariance function a little bit and what 01:10:05.980 --> 01:10:08.530 that implies for the covariance. 01:10:08.530 --> 01:10:14.030 And then the other – the big thing is that people always come back is, 01:10:14.030 --> 01:10:17.230 is it physical? That’s a very empirical model right now. 01:10:17.230 --> 01:10:22.840 So we just – the data – we showed – I mean, we checked that we 01:10:22.840 --> 01:10:28.889 don’t do, like, crazy things. But it’s harder to check the model. 01:10:28.889 --> 01:10:36.179 Because you cannot just look at scaling at all possible locations in California. 01:10:36.179 --> 01:10:39.199 So that’s just not humanly possible. 01:10:39.200 --> 01:10:41.380 So … 01:10:43.280 --> 01:10:47.559 And it’s a very empirical model. I mean, we do check that it predicts 01:10:47.559 --> 01:10:51.270 better than the corresponding ergodic model. 01:10:51.270 --> 01:10:54.280 And we looked at the residuals at Napa, and they looked good, 01:10:54.280 --> 01:10:58.400 even though Napa was not in the data set. 01:10:58.400 --> 01:11:02.630 So they looked better than the typical NGA model. 01:11:02.630 --> 01:11:08.440 But how can we incorporate physical constraints in those adjustments? 01:11:08.440 --> 01:11:16.130 That’s a very open question, and I – yeah, that’s – we don’t really know. 01:11:16.130 --> 01:11:22.280 And then the other question is – I mean, I kind of touched – is, is it really – 01:11:22.280 --> 01:11:25.820 so right now, we just have the same length scale for northern California 01:11:25.820 --> 01:11:28.699 and southern California. Is that really true, or should it 01:11:28.699 --> 01:11:33.400 be something like non-stationary? That means that we can have the – 01:11:33.400 --> 01:11:36.531 that the covariance function can actually change – or, the length scale 01:11:36.531 --> 01:11:41.219 can somehow change depending on where we are. 01:11:41.220 --> 01:11:48.880 That just is possible. It just makes things very, very – 01:11:48.880 --> 01:11:52.960 it just makes the modeling more complex and then also the estimation 01:11:52.960 --> 01:11:55.280 of the coefficients more complex. 01:11:55.280 --> 01:12:01.949 And then, in the end, is how to make the varying coefficient model and 01:12:01.949 --> 01:12:09.840 how to make it easier applicable to seismic hazard instead of going 01:12:09.840 --> 01:12:15.020 actually through a logic tree of 100 or 200 sampled realizations. 01:12:16.000 --> 01:12:20.699 So basically, just as a summary, ergodic seismic hazard 01:12:20.700 --> 01:12:22.960 ignores systematic effects. 01:12:22.960 --> 01:12:28.980 It aggregates data, which is good because it’s statistically very nice, 01:12:28.980 --> 01:12:35.080 and you can do nice models with it, but it ignores the systematic source effects. 01:12:35.080 --> 01:12:39.160 And ignoring that will get actually just biased hazard curves. 01:12:39.170 --> 01:12:43.330 So those hazard curves will be, on average, correct over California, 01:12:43.330 --> 01:12:47.239 but they will not be correct at any individual site. 01:12:47.239 --> 01:12:52.699 So – and also, because we aggregate data, we get a really good handle on 01:12:52.700 --> 01:12:57.080 our coefficients, which means we get two small uncertainty estimates. 01:12:58.280 --> 01:13:02.460 The variant coefficient model and the Gaussian process is a very nice model 01:13:02.460 --> 01:13:08.060 to obtain those estimates and the uncertainty because it automatically – 01:13:08.060 --> 01:13:12.620 it only leads to deviations where we have data, and it leads to zero 01:13:12.630 --> 01:13:14.960 deviations where we don’t have data. 01:13:14.960 --> 01:13:19.160 But, at the same time, increases the uncertainty on [inaudible]. 01:13:19.160 --> 01:13:27.100 Site-specific PSHA is conceptually straightforward. 01:13:27.110 --> 01:13:32.230 We just calculate adjustments for a bunch of – adjustments for a bunch of 01:13:32.230 --> 01:13:38.130 grid points around the site and then just look wherever we are in terms of source 01:13:38.130 --> 01:13:44.340 and add that to the median predictions. And if you – if we do that, we get 01:13:44.340 --> 01:13:47.340 hazard curves that deviate strongly 01:13:47.340 --> 01:13:52.290 from ergodic ones and – if there is data. But there still are lots of things to 01:13:52.290 --> 01:13:59.270 do and lots of nonergodic models to estimate for – both for 01:13:59.270 --> 01:14:03.840 California and for lots of other regions as well. 01:14:03.840 --> 01:14:06.440 And that’s it. Thanks. 01:14:06.440 --> 01:14:11.040 [Applause] 01:14:11.060 --> 01:14:16.420 - I think we have time for a quick question or two if anyone has one. 01:14:19.600 --> 01:14:21.760 - Hey, Nico. Thanks. That was really nice. 01:14:21.760 --> 01:14:25.990 So I think you just sort of started talking about it a little bit at the end, but maybe 01:14:25.990 --> 01:14:30.719 not right now, but in the future, when these models become common practice, 01:14:30.719 --> 01:14:36.679 how do you envision people like me trying to plot them or people 01:14:36.679 --> 01:14:42.090 using them for other applications? Will you – like, you were talking about, 01:14:42.090 --> 01:14:46.030 should you have a median model that accompanies them, or – and maybe 01:14:46.030 --> 01:14:50.929 that gets in a little bit to the PSHA. Right, so you were saying, it’s relatively 01:14:50.929 --> 01:14:54.139 straightforward if you do all these steps. - When – yes. 01:14:54.139 --> 01:14:58.030 - But if you’re – you know, you want to just know, what is your ground motion 01:14:58.030 --> 01:15:02.619 at a location for a magnitude 6 at 20 kilometers, how can I – 01:15:02.620 --> 01:15:05.160 or, how do you envision that happening? 01:15:05.160 --> 01:15:09.320 - I actually envision that happening by being more thorough with 01:15:09.320 --> 01:15:13.889 documentation and making code available. 01:15:13.889 --> 01:15:16.679 And not just saying, look, this is the model. 01:15:16.679 --> 01:15:21.820 This is a – this is a paper written by a computer scientist. 01:15:21.820 --> 01:15:28.580 So that just needs to happen. So I think that’s – I think – 01:15:28.580 --> 01:15:32.940 like, we have weekly – or, biweekly calls about that. 01:15:32.940 --> 01:15:37.820 And Norm is always – explain this in plain English. 01:15:37.820 --> 01:15:45.070 It’s just hard for me, but I’m not a native speaker, so – but, no, it’s – 01:15:45.070 --> 01:15:52.260 I mean, it’s just – it’s [inaudible]. So I think, yeah, that will – to make 01:15:52.260 --> 01:15:57.440 the model applicable, there needs to be a lot more documentation 01:15:57.440 --> 01:16:03.670 and code to be made available. Which I think will happen. 01:16:03.670 --> 01:16:12.579 So that – and it needs to be a little bit maybe better explained that, if you – 01:16:12.579 --> 01:16:19.780 that people can still make their own calculations if you just know 01:16:19.780 --> 01:16:21.960 the variances and length scale parameters. 01:16:21.960 --> 01:16:23.940 - Yeah. I mean, it’s not so different, I guess, right now. 01:16:23.949 --> 01:16:25.250 - It’s not. - You still need to make 01:16:25.250 --> 01:16:26.650 some assumptions … - Right. 01:16:26.650 --> 01:16:28.959 - On your generic fault parameters or something. 01:16:28.959 --> 01:16:31.219 - Yes. Right. - But, yeah, it’s not – 01:16:31.219 --> 01:16:34.459 currently it’s not clearly explained. - No, I agree, so … 01:16:34.460 --> 01:16:37.580 - And this will take even more … - Right. 01:16:37.580 --> 01:16:43.770 So I think we’ll – like, the next model, we’ll try to explain that a little bit better. 01:16:43.770 --> 01:16:48.420 And also, I think, make code available. - Oh, yeah. Or not opensource. 01:16:48.420 --> 01:16:50.239 A closed code that … - Yes. [inaudible] 01:16:50.239 --> 01:16:52.960 - … you can’t mess with, but you just input your location 01:16:52.960 --> 01:16:56.400 and what you want to know. - Yeah. Maybe something like that. 01:16:56.400 --> 01:16:57.400 - Great. 01:17:02.560 --> 01:17:09.340 - I have a quick question about the cell-specific attenuation model. 01:17:09.340 --> 01:17:13.500 How do you – do you treat spatial correlation when you develop that? 01:17:13.500 --> 01:17:19.560 Or just let the data kind of determine … - No. I just let the data determine it. 01:17:22.650 --> 01:17:28.120 So you can see that a little bit. If you look at the posterior distributions 01:17:28.139 --> 01:17:31.570 and then plot them for one cell that’s, like, neighboring, you can 01:17:31.570 --> 01:17:35.409 see some sort of correlation. Not everywhere. 01:17:35.409 --> 01:17:43.110 So I try to do that, and so I tried to look – I estimated – it’s called 01:17:43.110 --> 01:17:45.940 a conditional auto-regressive model. 01:17:45.940 --> 01:17:51.980 So basically, there, you just assume that neighboring cells are correlated. 01:17:51.980 --> 01:17:54.800 Only neighboring cells. So that makes the matrix that 01:17:54.800 --> 01:17:59.780 [inaudible], and it makes it somewhat efficient. 01:17:59.780 --> 01:18:03.680 And things change. Things change a little bit. 01:18:03.690 --> 01:18:08.180 They mainly change for cells where you don’t have a lot of data, 01:18:08.180 --> 01:18:12.260 but which are close to cells where you do have a lot of data. 01:18:12.260 --> 01:18:17.280 Now, on the one hand, you can say, sure, that should be the case. 01:18:17.280 --> 01:18:20.679 And, I mean, that’s the assumption that’s made in the varying coefficient 01:18:20.679 --> 01:18:26.059 model, right, that things that are close together – on the other hand, attenuation 01:18:26.059 --> 01:18:29.539 can actually change pretty quickly. Like, if you go from – especially if you 01:18:29.539 --> 01:18:31.940 have a fault, you can actually have a very different attenuation 01:18:31.940 --> 01:18:38.610 between two different faults. So I don’t think the data is enough to 01:18:38.610 --> 01:18:42.889 actually – I mean, you can do that, and then you can look at some, like, 01:18:42.889 --> 01:18:47.539 cross-validation and say which one predicts the data better than the other. 01:18:47.539 --> 01:18:53.309 I don’t think it’s enough to actually distinguish between those two models. 01:18:53.309 --> 01:19:03.200 So I don’t know how best to do this. So I would probably prefer not to do it. 01:19:03.200 --> 01:19:08.840 When it is so sort of implied, then in the model, it comes out of the model. 01:19:08.840 --> 01:19:15.550 But basically it’s – the only thing is, if there’s no paths through a cell, 01:19:15.550 --> 01:19:19.199 but there’s a lot of paths in the neighboring cells, do you assume 01:19:19.199 --> 01:19:25.570 it’s the same correlation? And that’s probably true for some parts and 01:19:25.570 --> 01:19:30.139 not true for other parts. So – I mean, that’s always – ideally, 01:19:30.140 --> 01:19:34.280 you would have something that takes care – but we need a lot more 01:19:34.280 --> 01:19:38.500 data to actually distinguish between those things. 01:19:39.520 --> 01:19:42.520 - Yep. Okay. Thank you. 01:19:43.800 --> 01:19:45.280 - We lost everyone. 01:19:45.280 --> 01:19:48.760 - Yeah, I think everyone had meetings to get to.