WEBVTT Kind: captions Language: en 00:00:01.689 --> 00:00:04.859 Hello. Good morning, everyone. Welcome to this week’s 00:00:04.859 --> 00:00:09.339 Earthquake Science Center seminar. Before we get started, next week, 00:00:09.339 --> 00:00:13.469 we’ll be doing a – SSA preview talks. Andrea, Brad, and Annemarie 00:00:13.469 --> 00:00:16.410 will be giving previews of what they’ll be presenting 00:00:16.410 --> 00:00:19.080 down in Miami the following week. 00:00:19.080 --> 00:00:21.360 And then a reminder that, during SSA week, which is 00:00:21.360 --> 00:00:27.100 two Wednesdays from now, we will not be having a regular Wednesday seminar. 00:00:27.100 --> 00:00:31.760 So today we’re joined by Roger Borcherdt, USGS emeritus 00:00:31.760 --> 00:00:35.270 and former Shimizu and consulting professor at Stanford. 00:00:35.270 --> 00:00:40.290 I won’t go through quite the impressive biography for Roger, 00:00:40.290 --> 00:00:43.899 but some highlights include he’s a Bolt Medal award winner. 00:00:43.899 --> 00:00:46.929 He has received the Presidential Distinguished Service Award as well as 00:00:46.929 --> 00:00:53.260 a DOI Merit – Meritorious Service Award. I’m struggling today. 00:00:53.900 --> 00:00:58.559 Roger is probably most famous for his work on Vs30 and site characterization. 00:00:58.559 --> 00:01:02.710 And this morning, he’ll be talking about seismic implications of advances 00:01:02.710 --> 00:01:06.860 on anelastic wave theory and general viscoelastic wave theory. 00:01:06.860 --> 00:01:11.280 Roger? Now if you just unmute your mic, you’re good to go. 00:01:16.200 --> 00:01:17.880 - Thank you, Alex. 00:01:19.250 --> 00:01:22.280 I have to say, it’s been a pleasure working 00:01:22.280 --> 00:01:24.590 at the Survey for all of these years. 00:01:24.590 --> 00:01:32.080 And it’s a great place to work on a lot of different and exciting problems. 00:01:32.080 --> 00:01:35.860 And in that context, I’d like to start off with a couple comments. 00:01:35.860 --> 00:01:40.360 First of all, I’d like to give a bit of credit where credit’s due. 00:01:40.360 --> 00:01:45.410 And that credit really goes to Steve Hickman for his appreciation of science, 00:01:45.410 --> 00:01:48.780 which he exhibits in so many different ways. 00:01:50.240 --> 00:01:54.880 After retiring, I was invited to give a presentation 00:01:54.880 --> 00:01:58.200 on advances in anelastic wave theory. 00:01:59.560 --> 00:02:06.000 And Steve enthusiastically encouraged me to do that. 00:02:06.000 --> 00:02:12.620 And the workshop was being held in the Czech Republic, and it was organized by 00:02:12.620 --> 00:02:20.940 Professor Pšenčík and Professor – who’s a colleague of Professor Červený. 00:02:20.940 --> 00:02:28.880 And they were very interested in trying to extend 00:02:28.880 --> 00:02:34.440 anelasticity theory to ray theory. 00:02:34.440 --> 00:02:40.960 And Červený had just published this big treatise on elastic ray theory. 00:02:40.960 --> 00:02:45.470 And so after long discussions, why they – or I became convinced 00:02:45.470 --> 00:02:49.640 that we could try to do that from the point of view of first principles. 00:02:49.640 --> 00:02:54.740 And that initiated a journey [chuckles] I couldn’t have imagined. 00:02:54.740 --> 00:02:58.570 And Steve really gets credit for supporting me in attending 00:02:58.570 --> 00:03:01.960 that workshop and interacting with those guys. 00:03:02.860 --> 00:03:05.920 The other thing I wanted to say was, is that one of the comments that was 00:03:05.920 --> 00:03:14.910 made at the workshop was that the anelastic results were suggesting 00:03:14.910 --> 00:03:18.340 we were kind of on the doorstep of a new era in seismology 00:03:18.340 --> 00:03:21.520 in the sense that the anelastic wave theory 00:03:21.520 --> 00:03:24.130 is suggesting a number of new characteristics for 00:03:24.130 --> 00:03:28.700 seismic wave fields that are not predicted by elastic solutions. 00:03:28.700 --> 00:03:34.680 And so, in that context, there’s kind of a wealth of problems to be worked on. 00:03:35.620 --> 00:03:41.460 And going forward on this is really going to – I’m retired – 00:03:41.470 --> 00:03:44.900 it’s really going to have to be up to the younger generation. 00:03:44.900 --> 00:03:49.020 And so, with that said, I’ve tried to focus my talk 00:03:49.020 --> 00:03:53.180 in that direction and try to give a brief overview of – 00:03:53.180 --> 00:03:58.160 first of all, I’ll give a brief overview from a historical perspective. 00:03:58.160 --> 00:04:04.430 And then talk about the implications of the anelastic solution on some of 00:04:04.430 --> 00:04:08.260 the fundamental characteristics of body and surface waves. 00:04:08.260 --> 00:04:13.220 And then talk about these recent developments in viscoelastic ray theory 00:04:13.220 --> 00:04:17.900 and what their implications are with respect to seismic waves. 00:04:18.769 --> 00:04:21.420 So the first – if we start from a historical point of view, 00:04:21.430 --> 00:04:25.550 first from the elastic point of view, of course, the first thing we have – 00:04:25.550 --> 00:04:31.070 from an elastic wave theory point of view is the constitutive law, and the 00:04:31.070 --> 00:04:36.940 elastic constitutive law simply says that stress is proportional to strain or that 00:04:36.940 --> 00:04:44.620 an instantaneous applied stress will result in a proportional instantaneous stress. 00:04:45.800 --> 00:04:49.560 And – whoops, excuse me. I hit the wrong button. 00:04:50.640 --> 00:04:54.260 But, based on this simple constitutive law, then there’s 00:04:54.270 --> 00:04:58.030 a number of basic problems that have been solved. 00:04:58.030 --> 00:05:03.220 And these are just – history perspective is really taken from Ben-Menahem’s 00:05:03.220 --> 00:05:08.860 article on the history of seismology in BSSA in 1995. 00:05:09.490 --> 00:05:15.280 But the constitutive law results in being able to come up with 00:05:15.290 --> 00:05:20.020 an elastic equation of motion with the real parameters for the bulk 00:05:20.020 --> 00:05:26.850 and shear modulus, elastic scaler Helmholtz equation, solutions of which 00:05:26.850 --> 00:05:31.720 describe P and S waves and their solutions of the equation of motion. 00:05:31.720 --> 00:05:36.570 And it also indicates that the only type of body waves 00:05:36.570 --> 00:05:40.100 we can have are homogeneous waves. 00:05:40.100 --> 00:05:44.139 And homogeneous waves are waves in which surfaces of 00:05:44.140 --> 00:05:48.360 constant phase are parallel to surfaces of constant amplitude. 00:05:48.360 --> 00:05:52.320 Or another way to say that is, is that the amplitudes do not vary 00:05:52.320 --> 00:05:55.200 along the surface of constant phase. 00:05:56.120 --> 00:06:00.140 And so we have solutions – the basic solutions of the reflection-refraction 00:06:00.140 --> 00:06:05.440 problems, solutions for critically refracted waves or head waves. 00:06:05.449 --> 00:06:10.300 Solutions for surface waves – Rayleigh and Love waves. 00:06:10.300 --> 00:06:14.949 And the response of a stock of layered media to incident homogeneous waves. 00:06:14.949 --> 00:06:19.780 What we know now today as kind of the Thompson-Haskell formulation. 00:06:20.280 --> 00:06:25.160 And then this is – these solutions have been utilized to develop solutions 00:06:25.170 --> 00:06:31.050 for the forward ray-tracing problem. And also for the inverse solutions 00:06:31.050 --> 00:06:34.530 for the ray-tracing problem where we try to use travel time data 00:06:34.530 --> 00:06:38.560 to infer intrinsic wave speeds. And these are based on solutions 00:06:38.560 --> 00:06:42.080 of the Abel integral transform for spherical media. 00:06:42.080 --> 00:06:45.160 This was the first solution of the inverse problem 00:06:45.160 --> 00:06:48.660 as initially developed by Herglotz and Wiechert. 00:06:49.380 --> 00:06:55.919 And then finally, the way we generally treat anelasticity in seismology 00:06:55.919 --> 00:07:00.400 is we use a one-dimensional anelastic damping approximation 00:07:00.400 --> 00:07:06.039 for an elastic homogeneous wave. And what that assumes is that 00:07:06.039 --> 00:07:10.410 the attenuation is in the direction of propagation. 00:07:10.410 --> 00:07:14.169 And the low-loss approximation, which we are all familiar with, 00:07:14.169 --> 00:07:19.380 says that the attenuation coefficient is approximately equal to circular 00:07:19.380 --> 00:07:26.550 frequency divided by 2 times the wave speed times Q, or the quality factor. 00:07:26.550 --> 00:07:32.000 Well, these basic fundamental solutions provide the theoretical basis 00:07:32.000 --> 00:07:35.960 for many of the seismic wave propagations in seismology, 00:07:35.960 --> 00:07:39.100 exploration geophysics, and engineering seismology. 00:07:39.100 --> 00:07:41.180 A number of examples are shown here. 00:07:41.180 --> 00:07:44.800 These are just – this is a very partial incomplete list. 00:07:44.800 --> 00:07:47.470 I won’t go through it, but it gives you an idea 00:07:47.470 --> 00:07:51.500 of how these basic solutions are utilized. 00:07:51.500 --> 00:07:55.759 And they’re well-documented and developed in – and their applications – 00:07:55.760 --> 00:08:00.350 in a number of different textbooks, as indicated here. 00:08:01.860 --> 00:08:08.380 Now, if we – but during the same time that a lot of these efforts have been 00:08:08.380 --> 00:08:12.210 applied and developed, there’s another constitutive law 00:08:12.210 --> 00:08:18.420 that was developed and kind of came into its own in the early ’60s. 00:08:18.420 --> 00:08:23.880 And it was first proposed by Boltzmann back in 1874, 00:08:23.880 --> 00:08:31.130 but has to do with viscoelastic behavior, which the constitutive law 00:08:31.130 --> 00:08:37.190 for that characterizes elastic and anelastic material behavior. 00:08:37.190 --> 00:08:41.860 Any material behavior that’s linear, it basically says that stress 00:08:41.860 --> 00:08:45.209 can be expressed as a convolution interval with strain 00:08:45.209 --> 00:08:50.480 or the kernel for the convolution interval – so relaxation function. 00:08:50.480 --> 00:08:55.270 And one of the things that impeded the development of this was it took some 00:08:55.270 --> 00:08:58.920 developments in the theory of linear functionals before this became a 00:08:58.920 --> 00:09:04.620 mathematically tractable problem. But one of the things about this is 00:09:04.620 --> 00:09:09.810 Bland wrote a book in 1960 that indicated that all linear material 00:09:09.810 --> 00:09:14.540 behavior could be characterized as some combination of 00:09:14.540 --> 00:09:19.660 springs and dashpots in series or parallel. 00:09:19.660 --> 00:09:24.140 And some examples are shown here where, for the dashpot, 00:09:24.140 --> 00:09:26.800 stress is proportional to strain rate. 00:09:26.800 --> 00:09:32.290 And for the spring, of course, stress is proportional to strain. 00:09:33.480 --> 00:09:38.540 Well, if we turn to isotropic materials, then it turns out we need only 00:09:38.540 --> 00:09:42.360 two relaxation functions – one in characterized behavior in bulk, 00:09:42.360 --> 00:09:44.920 one in shear. And the Fourier transforms 00:09:44.920 --> 00:09:51.580 of these are related to the bulk and shear modulus as indicated here. 00:09:51.580 --> 00:09:56.460 Now, it’s also possible to characterize material behavior from the point of view 00:09:56.460 --> 00:10:03.300 of the wave speeds and the reciprocal Q’s for homogeneous S and P waves, 00:10:03.300 --> 00:10:08.100 and that’s the way we do it now. And it turns out there’s a number 00:10:08.100 --> 00:10:12.250 of conveniences mathematically taking that approach. 00:10:12.250 --> 00:10:16.430 And also, it’s very intuitive because measurements 00:10:16.430 --> 00:10:23.200 of the wave speeds and Q exist for many materials. 00:10:23.200 --> 00:10:26.750 So that’s kind of where we’re starting from now with respect 00:10:26.750 --> 00:10:32.430 to viscoelasticity is this much more general constitutive law. 00:10:32.430 --> 00:10:41.300 And then, in 1960, Hunter, who did a review on viscoelasticity, 00:10:41.300 --> 00:10:45.720 indicated that the application of the general theory of viscoelasticity, 00:10:45.730 --> 00:10:49.990 more than one-dimensional wave propagation was incomplete. 00:10:49.990 --> 00:10:53.500 And by more than one-dimensional, he meant its application to 00:10:53.500 --> 00:10:57.960 two- and three-dimensional waves. And you’ll see why in just a moment. 00:10:59.020 --> 00:11:04.860 So now the situation is – but there’s been a lot of progress since 1960. 00:11:04.860 --> 00:11:07.260 And a lot of these problems are now solved. 00:11:07.260 --> 00:11:11.399 We now have an equation of motion with complex bulk and shear moduli. 00:11:11.399 --> 00:11:17.290 We have a vector Helmholtz equation that describes P and 00:11:17.290 --> 00:11:21.560 two types of S waves. And physical characteristics for P 00:11:21.560 --> 00:11:26.400 and these two types of body waves have been thoroughly developed now. 00:11:26.400 --> 00:11:28.920 We have low-loss approximations, 00:11:28.920 --> 00:11:32.240 closed-form expressions for their characteristics. 00:11:32.240 --> 00:11:36.279 Expressions for volumetric strain for body and surface waves. 00:11:36.279 --> 00:11:41.510 And we have the welded boundary reflection-refraction problem solved now 00:11:41.510 --> 00:11:46.040 for incident inhomogeneous waves as well as homogeneous waves. 00:11:46.040 --> 00:11:48.900 A description of viscoelastic head waves. 00:11:49.960 --> 00:11:56.760 And solutions for the Rayleigh- and Love-type surface waves, the 00:11:56.779 --> 00:12:01.510 Thompson-Haskell formulation, and then there’s a number of other problems here. 00:12:01.510 --> 00:12:05.120 Line source near welded boundary solved by Buchen. 00:12:05.120 --> 00:12:11.660 Synthetic seismogram – SII synthetic seismograms by Krebes. 00:12:11.670 --> 00:12:16.060 Then characteristics of body waves and anisotropic 00:12:16.060 --> 00:12:20.750 viscoelastic media by Professor Pšenčík and Červený. 00:12:20.750 --> 00:12:27.370 We have some more recent stuff on amplitude versus offset 00:12:27.370 --> 00:12:31.209 exploration analyses that have just been developed. 00:12:31.209 --> 00:12:35.350 And then I’ll talk about, a little later, the developments with respect to 00:12:35.350 --> 00:12:39.580 general viscoelastic ray theory as far as forward and inverse problems. 00:12:39.580 --> 00:12:43.390 So that’s – the intent is from – point of view of the overall perspective 00:12:43.390 --> 00:12:47.750 is kind of where we are with respect to solving the elastic problems and then 00:12:47.750 --> 00:12:55.430 the solutions for viscoelastic problems. And what we find is that, now that 00:12:55.430 --> 00:13:00.040 these problems are solved, that each of these solutions – 00:13:00.040 --> 00:13:05.740 general solutions imply that the characteristics of the body waves from 00:13:05.740 --> 00:13:11.280 the anelastic point of view are distinct or unique from those for elastic. 00:13:11.280 --> 00:13:14.959 This is not to mean that one’s not a limit of the other one, but the general 00:13:14.959 --> 00:13:22.570 solutions for anelastic media can be in the limit if we let the intrinsic material 00:13:22.570 --> 00:13:29.020 absorption decrease to zero, we end up with the special case elastic results. 00:13:29.020 --> 00:13:35.980 Now, to kind of intuitively try to tell you about this, I’m going to try to 00:13:35.980 --> 00:13:42.101 do it without equations and try to do it from the point of view of just what the 00:13:42.101 --> 00:13:46.180 intuitive behavior of these waves are. So if it’s not – what I’m saying is 00:13:46.180 --> 00:13:51.120 not clear, why, feel free to interrupt, and we can talk about it some more. 00:13:51.120 --> 00:13:54.582 But to see basically why we get a distinction between the elastic 00:13:54.582 --> 00:14:01.149 and anelastic, we need just simply to consider the simple problem 00:14:01.149 --> 00:14:06.220 of an incident homogeneous wave incident on a boundary, 00:14:06.220 --> 00:14:10.139 and then consider what the refracted wave looks like. 00:14:10.139 --> 00:14:15.440 In this case, I’m showing it first for a homogeneous wave – and this is 00:14:15.440 --> 00:14:20.570 a P wave with particle motions in the direction of propagation. 00:14:20.570 --> 00:14:24.860 For elastic media – and the situation in elastic media is, is that, 00:14:24.860 --> 00:14:29.940 since there’s no damping on either – in the material on either side, 00:14:29.940 --> 00:14:35.450 the energy that propagates along this left-most boundary basically 00:14:35.450 --> 00:14:41.850 will be the same amplitude on this refracted wave front as will be the 00:14:41.850 --> 00:14:48.760 energy that propagates along this path. And so, basically, if it starts off 00:14:48.760 --> 00:14:52.520 as a homogeneous wave, it will be a homogeneous wave. 00:14:52.520 --> 00:14:57.480 Also, since it is a homogeneous wave, it’s going to be propagating at the wave 00:14:57.480 --> 00:15:01.560 speed of the intrinsic – or, the intrinsic wave speed of the material. 00:15:01.560 --> 00:15:06.600 And this is, you know, the way we – this is our common understanding of waves. 00:15:06.600 --> 00:15:11.180 Also, what’s going to happen is, is that the direction of maximum energy flux, 00:15:11.180 --> 00:15:14.220 or the energy flux that’s carried by that wave is going to 00:15:14.220 --> 00:15:17.880 be parallel to the direction of phase propagation. 00:15:19.280 --> 00:15:23.340 And that’s also true for the refracted wave as well. 00:15:23.340 --> 00:15:28.580 In this case, it would be a refracted SI wave 00:15:28.580 --> 00:15:30.760 whose particle motions would be perpendicular 00:15:30.779 --> 00:15:35.120 to the direction of phase propagation. Now, if we go to the anelastic case, 00:15:35.120 --> 00:15:38.110 then we see we have quite a different situation. 00:15:38.110 --> 00:15:45.180 And that comes about because, if we have damping in this incident material, 00:15:45.180 --> 00:15:49.740 then the amplitude at this point on the wave front, when it reaches 00:15:49.740 --> 00:15:58.070 the boundary, is going to be larger than the amplitude that goes along this path. 00:15:58.070 --> 00:15:59.779 Because there’s going to be additional damping 00:15:59.779 --> 00:16:02.380 and it’s traveled an additional distance. 00:16:02.380 --> 00:16:06.360 So the amplitudes of the refracted rate at this point are going to be 00:16:06.360 --> 00:16:11.860 less than they will be at this point, assuming that the amount of 00:16:11.860 --> 00:16:16.650 attenuation in the rock is much less. And, of course, this can vary. 00:16:16.650 --> 00:16:21.260 Well, so that means that the amplitudes vary along this surface of constant phase. 00:16:21.260 --> 00:16:25.700 That means there’s a strain gradient along this surface of constant phase. 00:16:25.700 --> 00:16:30.700 And that gives rise to elliptical particle motions for this refracted wave. 00:16:31.800 --> 00:16:38.700 Well, also what transpires is that, because the wave is now what we 00:16:38.709 --> 00:16:46.390 call an inhomogeneous wave, basically, its wave speed is less than 00:16:46.390 --> 00:16:50.410 that of a homogeneous wave. That is, if this wave were homogeneous, 00:16:50.410 --> 00:16:56.529 and the amplitudes are the same, then it would be propagating more quickly. 00:16:56.529 --> 00:17:02.220 Same thing is true for energy absorption. The amount of intrinsic – 00:17:02.220 --> 00:17:07.560 or, the amount of absorption associated with this wave – 00:17:07.569 --> 00:17:11.619 the fractional energy loss, which is proportional to the reciprocal Q, 00:17:11.620 --> 00:17:14.740 will be greater than for a homogeneous wave. 00:17:15.540 --> 00:17:18.420 And actually, it happens from a – this is from a theoretical point of view 00:17:18.429 --> 00:17:23.220 is that what you can see is, if this wave were normally incident – 00:17:23.220 --> 00:17:27.189 so these two distances were the same – even though there were damping, 00:17:27.189 --> 00:17:30.980 then the refracted wave would be homogeneous. 00:17:30.980 --> 00:17:34.800 So what that means is these new physical characteristics that we’re 00:17:34.800 --> 00:17:40.480 talking about for the wave basically are dependent on angle of incident. 00:17:40.480 --> 00:17:45.500 Now, this is a phenomenon we don’t, of course, encounter in elastic media. 00:17:47.340 --> 00:17:52.559 And that means that what happens is that, since this wave speed depends on 00:17:52.559 --> 00:17:59.529 angle of incidence, the actual theoretical speed that a wave will – 00:17:59.529 --> 00:18:04.809 refracted wave will, say, go through this point in the medium, 00:18:04.809 --> 00:18:10.280 is going to depend on angle of incidence. 00:18:10.280 --> 00:18:14.600 And so the wave speed of the wave in this material is not unique, 00:18:14.600 --> 00:18:19.259 is another way to say it. And this – these same comments 00:18:19.260 --> 00:18:24.080 that I’ve just made also apply to the refracted S wave. 00:18:26.580 --> 00:18:31.500 And before I leave this slide, I just want to say that this is kind of 00:18:31.509 --> 00:18:35.559 the basic fundamental concept that gives rise to the different 00:18:35.559 --> 00:18:40.770 characteristics of anelastic waves versus elastic waves that we’re going to 00:18:40.770 --> 00:18:44.100 see be repeated over and over again. 00:18:46.140 --> 00:18:51.159 In fact, what you can show also is that the – if you look at the solution 00:18:51.159 --> 00:18:53.039 in the Helmholtz equation – the vector Helmholtz equation, 00:18:53.039 --> 00:18:58.100 you readily show that basically the only type of inhomogeneous wave 00:18:58.100 --> 00:19:01.539 that can propagate in elastic media 00:19:01.539 --> 00:19:05.570 can’t propagate in anelastic media, and vice versa. 00:19:05.570 --> 00:19:09.090 The other thing which you can show quickly is, in layered media, 00:19:09.090 --> 00:19:14.940 the predominant type of anelastic wave is a wave that’s inhomogeneous – 00:19:14.940 --> 00:19:18.560 has these amplitudes varying on a surface of constant phase. 00:19:18.560 --> 00:19:23.380 And in elastic media, that kind of wave can’t propagate. 00:19:23.380 --> 00:19:27.180 Only the homogeneous wave can propagate. 00:19:27.180 --> 00:19:31.780 Okay, so quickly, the Helmholtz equation will quickly show you that the 00:19:31.789 --> 00:19:37.409 particle motions have to be elliptical for both P and type I S waves with tilt 00:19:37.409 --> 00:19:43.029 of the particle motion ellipse parallel – perpendicular – well, the tilt particle 00:19:43.029 --> 00:19:46.680 motion ellipse inclined with respect to the direction of propagation for the 00:19:46.680 --> 00:19:52.420 P wave and inclined with respect to the perpendicular direction in the SI wave. 00:19:52.429 --> 00:19:56.799 Also show that, for type II S waves, 00:19:56.799 --> 00:20:00.609 the particle motion is linear and perpendicular to the direction 00:20:00.609 --> 00:20:03.600 of propagation for all degrees of inhomogeneity. 00:20:03.600 --> 00:20:06.619 So there’s really fundamentally two different types of P waves. 00:20:06.620 --> 00:20:11.400 One’s elliptical particle motions, and one with linear. 00:20:11.400 --> 00:20:16.640 And you can also look quickly at the wave speeds for the inhomogeneous 00:20:16.640 --> 00:20:20.840 waves and find that they’re less than that for a homogeneous wave. 00:20:20.840 --> 00:20:24.039 The attenuation coefficients maximum are greater than that 00:20:24.040 --> 00:20:30.049 for a homogeneous wave. And here’s curves computed to show that. 00:20:30.700 --> 00:20:35.680 And what you see is, is that the wave speeds and attenuation coefficients 00:20:35.680 --> 00:20:40.400 for inhomogeneous anelastic body waves are significantly less and larger, 00:20:40.400 --> 00:20:43.940 respectively, than those for homogeneous waves 00:20:43.940 --> 00:20:47.100 for sufficiently large values of inhomogeneity. 00:20:47.100 --> 00:20:50.210 And that’s what’s plotted here is the inhomogeneity of the wave, 00:20:50.210 --> 00:20:53.940 or the angle between the direction of propagation and attenuation. 00:20:53.940 --> 00:20:59.900 And you see that they’ll always differ if the inhomogeneity gets large enough. 00:20:59.909 --> 00:21:05.739 But on the other hand, it also shows that, for smaller degrees of inhomogeneity, 00:21:05.739 --> 00:21:10.419 that basically these wave speeds and attenuation coefficients approach 00:21:10.419 --> 00:21:15.039 those for a homogeneous wave. In other words, approach those for 00:21:15.040 --> 00:21:19.440 wave fields that we’re much more used to dealing with in elastic media. 00:21:19.440 --> 00:21:23.220 And so this gives you some insight that there’s going to be problems 00:21:23.220 --> 00:21:27.559 where this anelastic theory is important for certain seismic problems. 00:21:27.560 --> 00:21:32.629 And in other cases, it’s going to be sufficient to just use elasticity theory. 00:21:34.260 --> 00:21:38.080 And of course, when we look at energy flux, we find that, you know, 00:21:38.080 --> 00:21:42.669 the mean energy flux, kinetic energy densities, potential energy densities – 00:21:42.669 --> 00:21:45.099 everything for the inhomogeneous waves 00:21:45.100 --> 00:21:48.320 is greater than it is for the homogeneous waves. 00:21:49.580 --> 00:21:52.619 And same thing for fractional energy losses. 00:21:52.619 --> 00:21:58.539 The fractional energy loss for – which is proportional to reciprocal Q – basically 00:21:58.539 --> 00:22:03.570 for an inhomogeneous wave is greater than it is for a homogeneous wave. 00:22:03.570 --> 00:22:09.519 And we also find that, when we look at the fractional – or, the reciprocal Q’s 00:22:09.519 --> 00:22:14.600 for type I and type II S waves that they’re basically different. 00:22:14.600 --> 00:22:19.240 And the one with elliptical particle motions, the fractional energy loss are – 00:22:19.240 --> 00:22:24.049 which are proportional to reciprocal Q, basically is greater than it is for 00:22:24.049 --> 00:22:33.399 the linear S wave, or the wave with the type II S wave. 00:22:34.360 --> 00:22:39.159 Okay, now I want to give you an example that kind of confirms and 00:22:39.159 --> 00:22:43.869 is probably one of the best examples we have that confirms these theoretical 00:22:43.869 --> 00:22:48.669 predictions that have been derived. And this is a problem that has to do 00:22:48.669 --> 00:22:56.279 with having a source – an acoustic source at 00:22:56.279 --> 00:23:01.009 a water-stainless steel boundary and basically shining an acoustic beam 00:23:01.009 --> 00:23:05.179 on the stainless steel and then recording the reflected wave. 00:23:05.179 --> 00:23:09.890 Now, if this is an elastic – if we used an elastic model, then we would predict 00:23:09.890 --> 00:23:16.259 that, when we got to the critical angle, that basically an incident homogeneous 00:23:16.259 --> 00:23:23.029 acoustic beam would be refracted as an SI wave and a P wave. 00:23:23.029 --> 00:23:27.520 But these inhomogeneous elastic waves would propagate parallel to 00:23:27.520 --> 00:23:31.610 the boundary and attenuate perpendicular to the boundary. 00:23:31.610 --> 00:23:37.190 And what we would predict for a normalized reflected amplitude is this, 00:23:37.190 --> 00:23:41.299 is that, when we got to the critical angle for the SI wave, 00:23:41.299 --> 00:23:45.409 which is about 30 degrees, we would get total internal reflection. 00:23:45.409 --> 00:23:51.899 That is, all of the energy from the acoustic beam is reflected back in the water. 00:23:52.600 --> 00:23:57.159 Now, if we go to anelastic, then we get a different picture. 00:23:57.159 --> 00:24:03.840 Because now what happens is that the anelastic theory predicts that there’s 00:24:03.840 --> 00:24:08.450 going to be refracted P and S waves with elliptical particle motions that 00:24:08.450 --> 00:24:11.940 are going to propagate away from this boundary for all angles of incidence. 00:24:11.940 --> 00:24:15.059 It’s never quite going to get to being parallel 00:24:15.059 --> 00:24:17.719 and propagating parallel to the boundary. 00:24:17.719 --> 00:24:22.109 And of course, in anelastic media, this direction of propagation can 00:24:22.109 --> 00:24:26.499 never become perpendicular to the direction of attenuation. 00:24:26.499 --> 00:24:30.919 So we get energy that propagates away for all angles of incidence. 00:24:30.919 --> 00:24:33.669 And the reflection curve looks like this. 00:24:33.669 --> 00:24:38.799 It predicts that, at the head wave critical angle for the SI wave, 00:24:38.800 --> 00:24:43.300 there’s a significant amount of energy that goes across the boundary. 00:24:43.820 --> 00:24:52.279 And this was realized earlier that there was something wrong with the elastic 00:24:52.279 --> 00:24:58.200 prediction because, when [inaudible], in his book for example, on ways of – 00:24:58.200 --> 00:25:04.919 layered media, he had empirical measurements of just this case and found 00:25:04.920 --> 00:25:10.929 that the empirical measurements did not track the predicted elastic curve. 00:25:12.540 --> 00:25:16.220 And you can see what the anelastic model predicts with respect to the 00:25:16.220 --> 00:25:20.479 amplitudes, that it predicts there is a peak in the amplitudes at this point and 00:25:20.479 --> 00:25:30.019 that also there’s a significant amplitude beyond that that is predicted to be 00:25:30.020 --> 00:25:35.160 transmitted away from the boundary or refracted away from the boundary. 00:25:35.160 --> 00:25:39.479 And laboratory measurements that were conducted back in 1970 00:25:39.479 --> 00:25:45.739 by Becker and Richardson do show that – the empirical 00:25:45.739 --> 00:25:50.039 measurements do show this dip in the reflection coefficient. 00:25:50.039 --> 00:25:54.890 And in fact, it matches theoretically predicted curves extremely well. 00:25:54.890 --> 00:25:59.710 I’ve shown other curves here predicted for other values of Q, 00:25:59.710 --> 00:26:04.340 but it comes about because we’re saying that the Q for the stainless steel at a 00:26:04.340 --> 00:26:08.999 frequency of 10 megahertz is about 79. That’s the one that fits best. 00:26:08.999 --> 00:26:14.769 But you can see that, if you go to more attenuation of the material, 00:26:14.769 --> 00:26:17.840 that basically you get to the point where essentially all of the energy goes across 00:26:17.840 --> 00:26:21.919 the boundary, and that’s a situation not predicted by elasticity theory. 00:26:21.919 --> 00:26:25.259 You also see a good correspondence between phase shifts that are 00:26:25.259 --> 00:26:30.619 predicted at two different frequencies for the stainless steel. 00:26:30.619 --> 00:26:35.989 And those two are in good agreement. In fact, this phenomenon has been 00:26:35.989 --> 00:26:41.759 used in the non-destructive testing of materials industry for a long time. 00:26:41.759 --> 00:26:46.359 And basically, now a commercial procedure for finding impurities in 00:26:46.359 --> 00:26:54.070 sheets of stainless steel as you shine an acoustic beam across it and attempt to 00:26:54.070 --> 00:26:58.239 find these impurities simply by looking at the reflections you get. 00:26:58.239 --> 00:27:04.619 Now, it also has applications in the marine seismic world and basically 00:27:04.619 --> 00:27:08.789 has been picked up and modeled now for that purpose. 00:27:08.789 --> 00:27:14.270 But you can have various configurations of ships either moving apart or 00:27:14.270 --> 00:27:19.879 in the same direction to scan over that particular range and angles 00:27:19.879 --> 00:27:28.539 of incidence in order to try to discern what the underlying material looks like. 00:27:28.540 --> 00:27:34.560 These are some reflection curves that we predicted some time ago, but for – 00:27:34.560 --> 00:27:40.740 like, for basalt with a velocity for a homogeneous shear wave 00:27:40.750 --> 00:27:45.749 of about 2.8 kilometers per second. These are the kinds of curves that 00:27:45.749 --> 00:27:50.720 you would get for various Q’s with a Q for 11 showing essentially 00:27:50.720 --> 00:27:52.940 all the energy goes across the boundary. 00:27:52.940 --> 00:27:56.119 And these are some of the phase shifts. But you can see that this kind of 00:27:56.120 --> 00:28:00.140 thing can definitely have applications in exploration. 00:28:00.140 --> 00:28:04.639 And, at the time, we were working with [inaudible], and this was seafloor 00:28:04.639 --> 00:28:09.179 spreading time, but we were proposing this as some way of perhaps mapping Q 00:28:09.180 --> 00:28:12.540 away from the Mid-Atlantic Ridge and trying to correlate that with age. 00:28:12.540 --> 00:28:18.260 But anyway, another case where this is kind of exciting from the point of view 00:28:18.269 --> 00:28:24.019 of what viscoelasticity or anelasticity is implying about seismic waves 00:28:24.019 --> 00:28:27.940 has to do with head waves. And a head wave, of course, is really 00:28:27.940 --> 00:28:33.489 the basis for refraction seismology. It’s often the first arrival 00:28:33.489 --> 00:28:38.200 on seismic record sections. An example is shown here. 00:28:39.100 --> 00:28:42.839 And if we were thinking about this from an elastic ray theory point of view, 00:28:42.839 --> 00:28:47.519 then basically, we’d think of a ray coming down, 00:28:47.519 --> 00:28:51.460 basically being critically refracted along this boundary, 00:28:51.460 --> 00:28:58.639 and basically then returning back to the surface at the same angle of incidence. 00:28:58.639 --> 00:29:03.090 Well, the conundrum is, when you try to do this from an elasticity point of view, 00:29:03.090 --> 00:29:09.139 is that it predicts no energy propagates along this boundary 00:29:09.140 --> 00:29:12.920 by this – or is carried by this refracted wave. 00:29:12.920 --> 00:29:18.600 However, when we go to the anelastic case, then we have a different situation. 00:29:18.609 --> 00:29:22.390 It does predict that this refracted wave carries energy. 00:29:22.390 --> 00:29:27.440 It predicts that it’s inhomogeneous and it has a significant point of 00:29:27.440 --> 00:29:32.379 energy parallel to the boundary. So the travel times that basically 00:29:32.379 --> 00:29:40.190 are so prevalent in refraction seismology basically are consistent with this 00:29:40.190 --> 00:29:43.879 wave actually carrying energy along the boundary if we use 00:29:43.879 --> 00:29:47.519 a plane-wave anelastic reflection model. 00:29:47.520 --> 00:29:52.020 And it’s consistent with what we observe. 00:29:52.820 --> 00:29:55.960 Now, a few more examples quickly. 00:29:55.970 --> 00:30:00.890 Thompson-Haskell formulation. Basically, if we consider an incident 00:30:00.890 --> 00:30:07.109 wave on a soft soil layer, then this is the response when we calculate. 00:30:07.109 --> 00:30:10.169 It’s normalized by the fundamental frequency 00:30:10.169 --> 00:30:12.929 and plotted as a function of angle incidence. 00:30:12.929 --> 00:30:15.710 If we consider the case of an inhomogeneous wave, 00:30:15.710 --> 00:30:20.200 then the anelastic wave theory predicts an increased response 00:30:20.200 --> 00:30:23.660 of the incident inhomogeneous wave near grazing incidence. 00:30:23.660 --> 00:30:27.460 And, of course, this has some implications for basin response. 00:30:29.040 --> 00:30:38.360 Also, with respect to the simple case of a soil layer over rock, if we were – 00:30:38.369 --> 00:30:42.119 since anelastic solutions that have been developed now are valid for 00:30:42.119 --> 00:30:47.399 any media with – regardless of the amount of damping, 00:30:47.399 --> 00:30:51.059 we don’t have to worry about low-loss approximations, and we can basically 00:30:51.059 --> 00:30:55.289 develop estimates of site response independent of the amount of damping. 00:30:55.289 --> 00:31:00.740 And these are a couple that were developed for code purposes. 00:31:03.100 --> 00:31:08.499 Rayleigh waves – this is a low-loss – grid for low-loss 00:31:08.499 --> 00:31:13.099 Rayleigh wave and one with moderate loss, propagating from left to right. 00:31:13.100 --> 00:31:16.880 And the anelastic theory indicates that the propagation and 00:31:16.880 --> 00:31:20.759 attenuation vectors for these waves are not parallel and perpendicular 00:31:20.759 --> 00:31:28.950 as they are for elastic media, but tilted. And then, of course, the anelastic theory 00:31:28.950 --> 00:31:35.159 predicts that there’s – tilt is a function of depth, and the amplitudes, 00:31:35.160 --> 00:31:39.420 vertical and horizontal, also depend on the intrinsic absorption. 00:31:40.840 --> 00:31:46.740 If we go to Love-type surface waves, then that problem also has been solved. 00:31:46.740 --> 00:31:51.620 Basically, that’s a question of finding pairs of real values for the velocity and 00:31:51.620 --> 00:31:56.279 the absorption coefficient that satisfy this viscoelastic period equation, 00:31:56.279 --> 00:31:59.690 which for elastic media, reduces down to the simple one 00:31:59.690 --> 00:32:04.259 that we’re all familiar with. But again, what we find is that the 00:32:04.259 --> 00:32:10.089 anelastic solution predicts normalized attenuation coefficients, wave speed, 00:32:10.089 --> 00:32:16.159 and dispersion that vary with the intrinsic absorption in the material. 00:32:16.160 --> 00:32:23.420 One more of – this is the last one, but this has to do with basically 00:32:23.420 --> 00:32:29.620 predictions of anelastic wave theory for displacements in volumetric strains. 00:32:29.630 --> 00:32:37.259 This is basically work that was initiated based on the excellent 00:32:37.259 --> 00:32:40.940 seismic observation locations that Malcolm Johnston 00:32:40.940 --> 00:32:44.590 and his crew set up with volumetric strain meters. 00:32:44.590 --> 00:32:49.039 This is an example of a recording from the now-famous Parkfield earthquake. 00:32:49.039 --> 00:32:54.129 Co-located measurement of strain and vertical displacement, 00:32:54.129 --> 00:32:57.350 and you can see that the volumetric strain meters, 00:32:57.350 --> 00:33:02.249 which record just dilation and not shear, basically respond to P waves, and the 00:33:02.249 --> 00:33:05.759 two traces track each other with the onset of the P wave quite well. 00:33:05.760 --> 00:33:10.280 But then we get to the S wave. The volumetric strain meter shouldn’t 00:33:10.280 --> 00:33:15.800 respond to the S wave, but what it does respond to is the reflected P wave. 00:33:15.800 --> 00:33:19.460 And you see that there is a significant response. 00:33:19.460 --> 00:33:24.830 Well, if we use elasticity to model this particular problem, then basically 00:33:24.830 --> 00:33:32.509 the incident S wave at, say, the critical angle, would be reflected as an S wave, 00:33:32.509 --> 00:33:36.500 but the corresponding P wave would be a wave 00:33:36.500 --> 00:33:41.259 that propagates parallel and doesn’t carry energy again. 00:33:41.259 --> 00:33:48.429 But if we go to the anelastic solution, then what we find is that there is 00:33:48.429 --> 00:33:51.219 phase propagation away from the boundary for all angles 00:33:51.219 --> 00:33:56.279 of incidence by the inhomogeneous wave, as indicated here. 00:33:56.279 --> 00:34:01.019 And then the peak of the response in the volumetric strain meter and 00:34:01.019 --> 00:34:05.179 the seismometer, which both occur after the critical angle, 00:34:05.179 --> 00:34:12.269 we see that the maximum reductions in phase and energy speeds are, like, 23%. 00:34:12.269 --> 00:34:17.361 The reductions at these maxima in terms of reciprocal – or, the increase 00:34:17.361 --> 00:34:22.660 in reciprocal Q, and in circularity, the particle motions are, like, 67%. 00:34:22.660 --> 00:34:26.720 So there’s some dramatic changes in the characteristics of the wave field 00:34:26.730 --> 00:34:31.059 that are predicted by anelasticity. They’re not predicted by the elasticity. 00:34:31.060 --> 00:34:37.740 Okay, with that, let’s turn to ray theory. First, I’d like to talk about 00:34:37.740 --> 00:34:42.840 kind of general viscoelastic ray theory from a first principles point of view 00:34:42.849 --> 00:34:45.940 with a single boundary reflection-refraction problem really 00:34:45.940 --> 00:34:51.269 being the basis of it and the implications of it for generalized Snell’s law. 00:34:51.269 --> 00:34:59.240 And then the introduction of viscoelastic phase parameters – ray parameters. 00:34:59.240 --> 00:35:02.330 A ray parameter for phase propagation as well as 00:35:02.330 --> 00:35:06.340 a ray parameter for amplitude attenuation. 00:35:06.340 --> 00:35:10.020 Okay, we have talked about the solutions of the forward ray-tracing 00:35:10.020 --> 00:35:15.580 problem for horizontal and spherical media with layers and with gradients. 00:35:16.840 --> 00:35:21.400 Then I’ll talk about some computation steps that readily 00:35:21.400 --> 00:35:25.170 admit themselves once the closed-form solutions have been 00:35:25.170 --> 00:35:29.480 obtained that can be used for general ray-tracing computer codes. 00:35:29.480 --> 00:35:32.160 And also make mention of Earth-flattening transformations 00:35:32.160 --> 00:35:36.460 and then show you a few numerical results. 00:35:36.460 --> 00:35:42.200 And then have to talk a little bit about the inverse problem for the anelastic case 00:35:42.210 --> 00:35:49.140 and the solution of the viscoelastic Herglotz-Wiechert integral solutions. 00:35:49.140 --> 00:35:53.440 And finally, discuss what some of the implications are. 00:35:53.440 --> 00:36:01.569 So first of all, again, trying to do this from kind of an intuitive point of view. 00:36:01.569 --> 00:36:07.670 If we think about a ray in elastic media, then we think in terms of a ray 00:36:07.670 --> 00:36:13.510 that’s depicted somewhat like this, but it’s a ray for which the waves 00:36:13.510 --> 00:36:16.860 are homogeneous, the direction of the energy flow is coincident 00:36:16.860 --> 00:36:22.280 with the direction of phase propagation, and the waves travel as homogeneous 00:36:22.280 --> 00:36:27.060 waves in each layer. So the wave speed that the ray travels through – 00:36:27.069 --> 00:36:33.500 say this second layer is the same as the intrinsic wave speed 00:36:33.500 --> 00:36:37.380 of the material and doesn’t depend on angle of incidence. 00:36:38.000 --> 00:36:41.460 Now, if we go to anelasticity, then the situation is somewhat different 00:36:41.460 --> 00:36:46.410 because now the physical characteristics of the anelastic inhomogeneous waves 00:36:46.410 --> 00:36:51.920 that are refracted vary with the inhomogeneity of the wave that’s 00:36:51.920 --> 00:36:56.280 induced by contrast in the intrinsic absorption at the boundaries. 00:36:57.380 --> 00:37:00.740 And, as a result, you end up with links 00:37:00.740 --> 00:37:04.300 to the ray paths and travel times that are different. 00:37:04.300 --> 00:37:06.460 I’m going to just say a few more words about this. 00:37:06.460 --> 00:37:13.200 If we think about even a homogeneous wave incident on this boundary, 00:37:13.200 --> 00:37:18.360 where we have a contrast in anelastic absorption, then the wave field 00:37:18.360 --> 00:37:22.480 that’s transmitted across this boundary is an inhomogeneous wave 00:37:22.480 --> 00:37:25.740 with direction of propagation in one direction and the direction 00:37:25.740 --> 00:37:28.980 of attenuation in a slightly different direction. 00:37:28.980 --> 00:37:32.670 And then it’s incident on – and that comes about because of that 00:37:32.670 --> 00:37:38.300 contrast in Q at the boundary – or the contrast in the anelastic absorption. 00:37:38.300 --> 00:37:42.970 And then the same thing happens here, and it becomes more inhomogeneous. 00:37:42.970 --> 00:37:46.650 And the fact that the wave is inhomogeneous means 00:37:46.650 --> 00:37:51.280 that it refracts at a different angle than the homogeneous wave. 00:37:51.280 --> 00:37:55.800 So that’s why this ray path starts taking on a different location. 00:37:55.800 --> 00:37:59.140 Now, this is exaggerated in the sense of – 00:37:59.140 --> 00:38:02.880 basically to illustrate what’s going on. 00:38:02.880 --> 00:38:08.559 But theoretically, this is basically what the anelastic wave theory predicts. 00:38:08.560 --> 00:38:13.860 And so our travel times and actually amplitude attenuation along an 00:38:13.860 --> 00:38:18.320 anelastic ray are different than those from an elastic ray. 00:38:19.300 --> 00:38:22.490 And it comes back to the solution of this simple problem. 00:38:22.490 --> 00:38:26.081 The single – of an incident inhomogeneous wave 00:38:26.081 --> 00:38:29.049 on a single boundary. This problem is solved 00:38:29.049 --> 00:38:32.550 simply by assuming solutions for the reflected and refracted 00:38:32.550 --> 00:38:38.059 and incident waves, applying boundary conditions, and then expressing the – 00:38:38.059 --> 00:38:42.420 and being able to show, then, that the characteristics of the reflected and 00:38:42.420 --> 00:38:47.580 refracted waves can be expressed in terms of those of the incident wave. 00:38:47.580 --> 00:38:50.720 And it implies that the complex wave number for 00:38:50.720 --> 00:38:55.220 each of these solutions is the same. And that gives rise to generalized 00:38:55.220 --> 00:39:01.300 Snell’s law, which says that – the real part of which says that the horizontal 00:39:01.300 --> 00:39:06.380 component of the propagation vector for each of the waves is the same. 00:39:06.380 --> 00:39:08.780 And the same thing – the imaginary part says 00:39:08.780 --> 00:39:13.140 the horizontal component attenuation vectors are the same. 00:39:14.380 --> 00:39:20.500 And in elastic parlance, if we take this real part and transform it in – 00:39:20.510 --> 00:39:23.289 or, write it in the form of wave speeds, 00:39:23.289 --> 00:39:30.200 then it says the horizontal slowness of each of the different solutions are equal. 00:39:30.200 --> 00:39:36.220 And – but there’s one significant difference. 00:39:36.230 --> 00:39:38.609 This appears much like it does in elastic theory, 00:39:38.609 --> 00:39:43.090 but this expression – this expression for the wave speed is the wave speed of 00:39:43.090 --> 00:39:47.589 an inhomogeneous wave and not the wave speed of a homogeneous wave. 00:39:47.589 --> 00:39:51.220 And so there’s this dependence on angle of incidence, 00:39:51.220 --> 00:39:54.280 and everything gets taken into account. 00:39:55.620 --> 00:39:57.260 So now, what’s that suggest? 00:39:57.260 --> 00:40:03.960 It suggests defining a phase ray parameter that’s constant along the ray. 00:40:03.960 --> 00:40:10.560 Also an amplitude attenuation ray parameter that’s constant along the ray. 00:40:12.760 --> 00:40:18.059 And then, generalized Snell’s law then says that, for an assumed incident – 00:40:18.059 --> 00:40:24.089 angles of incidence for this incident wave, that basically the ray parameters – 00:40:24.089 --> 00:40:27.950 and assumed material parameters, that the ray parameters are 00:40:27.950 --> 00:40:31.780 determined in terms of those assumed or given parameters. 00:40:33.140 --> 00:40:37.599 And that they are constant – they’re also constant along the ray path. 00:40:37.599 --> 00:40:39.890 There’s – the horizontal slowness is constant, 00:40:39.890 --> 00:40:43.100 and the horizontal attenuation is constant. 00:40:44.990 --> 00:40:49.240 So now, basically, that means that what we can do is, 00:40:49.250 --> 00:40:53.099 is basically we have a solution for the – for the ray-tracing problem 00:40:53.099 --> 00:40:57.940 in the sense that, by determination of the propagation – we can determine 00:40:57.940 --> 00:41:02.289 the propagation and attenuation vectors for the general wave 00:41:02.289 --> 00:41:04.950 in each layer in terms of the ray parameters 00:41:04.950 --> 00:41:09.420 and the given material parameters, and those expressions are shown here. 00:41:10.640 --> 00:41:16.720 And that says that the phase wave speed and direction of 00:41:16.720 --> 00:41:21.059 phase propagation is determined. It says that the maximum attenuation 00:41:21.060 --> 00:41:25.580 of the wave is determined. And its direction is determined. 00:41:26.760 --> 00:41:29.460 And now, if we have a solution of the forward ray-tracing problem, 00:41:29.460 --> 00:41:33.839 then we can turn and go to a solution – then we can write down expressions for 00:41:33.840 --> 00:41:40.000 the horizontal distance of a given ray, travel time, and its slope. 00:41:40.000 --> 00:41:43.390 And we can do the same thing for amplitude attenuation. 00:41:43.390 --> 00:41:49.869 So now we have an expression for the travel time and amplitude 00:41:49.869 --> 00:41:55.500 attenuation of rays that takes into account the inhomogeneity of the wave 00:41:55.500 --> 00:42:02.420 fields along their entire – and changing inhomogeneity along the entire ray path. 00:42:03.660 --> 00:42:06.300 Now, I’m not going to go through the details, 00:42:06.309 --> 00:42:11.779 but we’ve solved the same problem now for media with – 00:42:11.779 --> 00:42:16.119 this is a half-space with vertical material gradients. 00:42:16.120 --> 00:42:21.020 And we have similar kind of expressions that we’ve derived for this problem. 00:42:21.920 --> 00:42:24.920 And the same thing for amplitude attenuation. 00:42:26.580 --> 00:42:32.780 We also have solutions for the classic problem of spherical media with layers. 00:42:35.140 --> 00:42:38.900 And these are their expressions. We won’t go through them. 00:42:38.900 --> 00:42:42.200 And finally, for kind of the classical problem in seismology, 00:42:42.200 --> 00:42:47.880 which was the problem of spherical media with continuous 00:42:47.880 --> 00:42:52.359 gradients – radial gradients. And the solutions for that are here. 00:42:52.359 --> 00:42:58.070 And this, to me, is kind of – it’s pretty exciting because now we have exact, 00:42:58.070 --> 00:43:02.500 closed-form theoretical solutions for travel times and amplitude 00:43:02.500 --> 00:43:08.000 attenuation that take into account inhomogeneity of the wave fields. 00:43:09.420 --> 00:43:13.740 And what these closed-form solutions then readily admit, 00:43:13.750 --> 00:43:17.289 one can just quickly write down what should be the steps for 00:43:17.289 --> 00:43:21.079 developing general computer codes to do this. 00:43:21.079 --> 00:43:24.980 And this is where this focus – I keep looking at these younger guys in the 00:43:24.980 --> 00:43:28.970 audience, but this is where this focus on the younger generation comes in. 00:43:28.970 --> 00:43:32.490 But it’s pretty straightforward now to do this. 00:43:32.490 --> 00:43:37.150 And so basically what the problem is, is basically – I’m showing it here for 00:43:37.150 --> 00:43:42.840 horizontal and spherical homogeneous isotopic linear viscoelastic layers. 00:43:42.840 --> 00:43:45.760 But the first part of the problem is the given parameters are the 00:43:45.769 --> 00:43:49.890 material parameters in both cases. And then the angles of emergence 00:43:49.890 --> 00:43:53.270 or angles of incidence at the pre-surface for the wave. 00:43:53.270 --> 00:43:58.410 And then the thickness of the layers if it’s horizontal, or the radii of the 00:43:58.410 --> 00:44:02.500 spherical layers, and the frequency. And then, from that, we can write 00:44:02.500 --> 00:44:06.540 the wave speed and the maximum attenuation coefficients down. 00:44:06.540 --> 00:44:08.940 We can write down the expression for the wave parameters 00:44:08.940 --> 00:44:12.609 for the ray – for each ray and their constant. 00:44:12.609 --> 00:44:15.309 Then what you can do is go on to the next step and write down 00:44:15.309 --> 00:44:18.059 the propagation and attenuation vectors for each. 00:44:18.059 --> 00:44:21.920 And then, basically everything is determined from that point on, 00:44:21.920 --> 00:44:25.849 and you can do a lot of other characteristics of the rays. 00:44:25.849 --> 00:44:29.340 But you can write down the travel times and the amplitude 00:44:29.340 --> 00:44:33.609 attenuation for the rays. And, of course, this kind of a format, 00:44:33.609 --> 00:44:37.880 or this mapping, if you like, provides a direct correspondence 00:44:37.880 --> 00:44:42.620 between parameters for the spherical problem and the horizontal problem. 00:44:42.620 --> 00:44:48.700 And in the past, that has been, you know, used for, like, 00:44:48.710 --> 00:44:52.109 the Earth-flattening transformations. If you solved your problem with spherical 00:44:52.109 --> 00:44:56.779 media, you could turn then and use the Earth-flattening transformations or 00:44:56.780 --> 00:45:00.780 these correspondences to have a solution for the horizontal media. 00:45:00.780 --> 00:45:02.460 And since we already have these solutions, 00:45:02.460 --> 00:45:05.210 that’s not going to be quite so important here, but it could be 00:45:05.210 --> 00:45:09.900 useful from the point of view of efficiency in the general computer codes. 00:45:10.700 --> 00:45:18.460 So let me turn now to a few numerical considerations. 00:45:18.460 --> 00:45:21.540 And the first thing I want to show is where we’ve applied 00:45:21.550 --> 00:45:25.940 some of these formulas. First thing I want to look at is just the 00:45:25.940 --> 00:45:30.150 simplest problem possible, and that is, what do the travel time and amplitude 00:45:30.150 --> 00:45:38.000 attenuation curves look like for homogeneous waves in a single layer? 00:45:38.000 --> 00:45:42.440 And so this is the travel time curve. 00:45:42.440 --> 00:45:45.970 And if the incident wave is homogeneous, then the transmitted 00:45:45.970 --> 00:45:51.519 waves will be – reflected waves will be homogeneous. 00:45:51.519 --> 00:45:55.359 And this is the amplitude attenuation curves 00:45:55.359 --> 00:45:57.930 for the direct, head, and reflected waves. 00:45:57.930 --> 00:46:00.569 And these have been calculated now for media 00:46:00.569 --> 00:46:04.690 with a lot of absorption in Q’s of 2 and 3. 00:46:04.690 --> 00:46:08.359 But they’re very similar to what we would see from elastic – 00:46:08.359 --> 00:46:11.579 if we predicted these curves elastically in terms of shape. 00:46:11.579 --> 00:46:15.019 And that’s because these – I’ve assumed the incident wave is homogeneous, 00:46:15.020 --> 00:46:19.540 the reflected, then, is homogeneous. And the direct wave is homogeneous. 00:46:20.290 --> 00:46:27.780 So now if we – but one of the questions we can – maybe let me back up here. 00:46:27.780 --> 00:46:33.540 One of the things to really ask ourselves is, what’s the next – 00:46:33.540 --> 00:46:35.800 what are the waves that are refracted that are going to be 00:46:35.800 --> 00:46:40.220 refracted as inhomogeneous waves imply? 00:46:41.640 --> 00:46:45.740 And if we first of all consider this from an elastic point of view, 00:46:45.740 --> 00:46:53.140 then basically elastic waves are refracted up to the head wave 00:46:53.140 --> 00:46:57.789 critical angle, but then they’re not refracted beyond that. 00:46:57.789 --> 00:47:05.210 And so the travel time curve for waves reflected from this second layer that 00:47:05.210 --> 00:47:10.559 would be predicted by the elastic model, the travel time curve looks like this. 00:47:10.560 --> 00:47:17.080 That is, as this wave gets closer to being refracted, the head wave angle travels 00:47:17.080 --> 00:47:23.100 a long distance before it gets reflected. And so you can see this travel time 00:47:23.120 --> 00:47:28.920 essentially approaches infinity as we approach the critical angle. 00:47:30.160 --> 00:47:35.119 Now, if we go to the anelastic case, then the situation we find is 00:47:35.119 --> 00:47:39.809 somewhat different. Because this refracted wave – 00:47:39.809 --> 00:47:44.020 there is refracted wave energy beyond this head wave critical angle. 00:47:44.020 --> 00:47:48.500 And the travel – and then it can reflect from the second layer. 00:47:48.500 --> 00:47:52.060 And so this would be the elastic prediction. 00:47:52.880 --> 00:48:00.140 But this is the prediction predicted by the anelastic theory. 00:48:00.150 --> 00:48:06.880 And this is a prediction for two layers in the Earth’s mantle where the Q now, 00:48:06.880 --> 00:48:12.660 I’m saying is – we’re modeling it as 50 in one layer and 60 in the other. 00:48:12.660 --> 00:48:16.600 So there’s not much of a change in the Q, but nevertheless, 00:48:16.600 --> 00:48:20.660 we end up with this additional travel time curve. 00:48:22.140 --> 00:48:26.540 Now, if we go to a higher Q, or a lower intrinsic absorption 00:48:26.540 --> 00:48:31.480 in this material, or, if you like, a larger contrast in intrinsic absorption 00:48:31.480 --> 00:48:37.920 at this boundary, then what we find is yet another travel time curve. 00:48:37.920 --> 00:48:41.880 And what we see is, is that the travel time curves that we’re getting 00:48:41.880 --> 00:48:47.329 from these second reflections is, they’re strongly dependent on the 00:48:47.329 --> 00:48:53.529 contrast in absorption at the boundary. And that’s kind of an exciting result 00:48:53.529 --> 00:49:00.039 in the sense that, first of all, anelasticity theory is predicting 00:49:00.039 --> 00:49:04.970 that there might be some other reflections that we haven’t been 00:49:04.970 --> 00:49:08.920 looking for based on elasticity theory. And, of course, that’s going to 00:49:08.920 --> 00:49:12.849 take some modeling and so on and so on to see if they’re really there. 00:49:12.849 --> 00:49:15.960 But this is beginning to guess they might be there. 00:49:17.120 --> 00:49:21.140 And the second thing is, is that we’re using travel times to 00:49:21.140 --> 00:49:25.859 say something about the contrast and absorption at that boundary. 00:49:25.859 --> 00:49:30.269 So it’s a way possibly of looking into seeing if we can 00:49:30.269 --> 00:49:34.980 find out something about the anelasticity in the upper mantle 00:49:34.980 --> 00:49:38.400 based on looking simply at travel time curves. 00:49:39.740 --> 00:49:43.660 Now, if we – and just to look at this in a case where we have 00:49:43.660 --> 00:49:49.700 lots of absorption now, I’ve redone this thing for, say, soft soil and stiff soil 00:49:49.700 --> 00:49:53.550 with really low Q’s or large amounts of intrinsic absorption, 00:49:53.550 --> 00:49:59.280 and the first boundary is a contrast between a Q of 2 and 2-1/2. 00:49:59.280 --> 00:50:02.440 And we get this travel time curve. 00:50:03.720 --> 00:50:08.560 And if we boost it up to 3, then we get this one. 00:50:08.560 --> 00:50:13.079 You can see, again, the contrast in absorption is playing a role here. 00:50:13.079 --> 00:50:17.410 And boost it up to 10, which is kind of at the boundary between low-loss and 00:50:17.410 --> 00:50:22.660 non-low-loss, which you can see we’re getting a different curve as well. 00:50:22.660 --> 00:50:28.180 And so basically, again, we’re seeing that it’s this contrast in absorption that’s 00:50:28.180 --> 00:50:36.380 important with respect to giving rise to the different travel time curves. 00:50:36.380 --> 00:50:42.490 And so with that said, now I’d like to quickly turn to the inverse problem. 00:50:42.490 --> 00:50:46.440 That’s the problem of trying to utilize the travel times – 00:50:46.440 --> 00:50:49.460 measured travel times and amplitude attenuation 00:50:49.460 --> 00:50:53.530 to infer the intrinsic material parameters for the media. 00:50:53.530 --> 00:50:56.630 And of course, with anelasticity, it’s a little more complicated 00:50:56.630 --> 00:51:00.690 in the sense that now we’re going to try to infer intrinsic reciprocal Q’s 00:51:00.690 --> 00:51:05.509 for homogeneous waves, if you like, for the material. 00:51:05.509 --> 00:51:09.779 And so question is, just how far can you go with this? 00:51:09.779 --> 00:51:11.950 And that is basically, we take a travel time curve, 00:51:11.950 --> 00:51:18.440 make a measurement off of it for a particular distance and slope. 00:51:18.440 --> 00:51:22.420 And same thing on the amplitude curves, and then see what we can do with them. 00:51:22.420 --> 00:51:26.850 But the first approach that I think is going to prove quite useful is 00:51:26.850 --> 00:51:31.890 some of the trial-and-error methods, just like we do it in elasticity theory. 00:51:31.890 --> 00:51:36.480 And that’s used – trial-and-error anelastic material models and then 00:51:36.480 --> 00:51:41.079 the general ray tracing computer codes that these guys are going to write 00:51:41.079 --> 00:51:43.499 to match the computed and observed travel time 00:51:43.499 --> 00:51:47.660 and amplitude curves for inhomogeneous waves. 00:51:47.660 --> 00:51:55.240 And I think this has got a lot of promise because the general solutions provide 00:51:55.240 --> 00:52:01.359 the solutions for elastic and anelastic. And so if we can – once we get these 00:52:01.359 --> 00:52:07.069 codes set up, we basically just have to set reciprocal material Q’s to zero, 00:52:07.069 --> 00:52:11.289 and we’ve got the elastic results. And then we just have to put in the 00:52:11.289 --> 00:52:16.380 trial model for the anelastic part to start trying to adjust it to see where things 00:52:16.380 --> 00:52:20.560 really – you know, where this contrast in absorption really starts to kick in. 00:52:21.660 --> 00:52:28.180 But to talk about kind of a rigorous solution to this problem, first of all, 00:52:28.180 --> 00:52:32.299 I’m just going to do it for one of these, but basically, for spherical media with 00:52:32.299 --> 00:52:37.269 radial material gradients, the idea is – now is to make measurements of the 00:52:37.269 --> 00:52:40.920 travel time and amplitude, as might be measured for this 00:52:40.920 --> 00:52:46.339 particular ray at a particular location to infer the anelastic material parameter 00:52:46.340 --> 00:52:50.500 for intrinsic wave speed and intrinsic Q. 00:52:51.000 --> 00:52:55.500 And so the first step in this is to measure the angular distance and travel time 00:52:55.520 --> 00:53:00.190 at this point. And then to measure the slope on the travel time curve. 00:53:01.000 --> 00:53:06.160 Then to use that to infer the ray parameter for this ray that’s got a turning 00:53:06.160 --> 00:53:11.440 point of the depth radius of r-sub-p. That’s where this subscript comes from. 00:53:11.440 --> 00:53:19.220 And remembering now that this thing’s inhomogeneous and not homogeneous. 00:53:19.220 --> 00:53:23.470 And then to infer the radius at this particular turning point 00:53:23.470 --> 00:53:27.180 using the solution – the Herglotz-Wiechert integral. 00:53:27.180 --> 00:53:32.660 And this integral basically has been now extended to viscoelasticity. 00:53:32.660 --> 00:53:35.520 It was first solved for elasticity. 00:53:35.520 --> 00:53:39.640 And the steps for the solution to this, I’ll say, are pretty similar to 00:53:39.640 --> 00:53:43.349 what they are for elastic media. It’s just that you have to be conscious 00:53:43.349 --> 00:53:46.600 of the fact that we’re dealing with inhomogeneous waves 00:53:46.600 --> 00:53:50.000 instead of homogeneous waves. And this ray parameter here now is a ray 00:53:50.000 --> 00:53:54.380 parameter for an inhomogeneous wave and not one for a homogeneous wave. 00:53:54.380 --> 00:54:00.109 But with that said, then if we got the radius and the ray parameter, 00:54:00.109 --> 00:54:03.990 then we’ve got the intrinsic wave speed – or, not the intrinsic wave speed. 00:54:03.990 --> 00:54:08.809 We’ve got the wave speed of this inhomogeneous wave at this point. 00:54:08.809 --> 00:54:12.000 And we’ve got the propagation vector for it at this point. 00:54:12.000 --> 00:54:16.950 Now, the next step is, we’ve got to do the same thing for the attenuation vector. 00:54:16.950 --> 00:54:19.079 So we measure angular distance and amplitude 00:54:19.080 --> 00:54:23.440 off the amplitude attenuation curve and its slope. 00:54:24.430 --> 00:54:28.900 We get the ray parameter for attenuation. 00:54:28.900 --> 00:54:34.260 Which is normalized – it’s the slope plus the – normalized with the amplitude. 00:54:35.340 --> 00:54:39.500 We get the attenuation then in the direction of phase propagation. 00:54:40.180 --> 00:54:44.340 And now we still need to be able to get back to the 00:54:44.340 --> 00:54:46.980 actual intrinsic material characteristics, though, 00:54:46.980 --> 00:54:50.740 because these are characteristics of the inhomogeneous wave. 00:54:50.740 --> 00:54:54.829 That comes from the solution of that vector Helmholtz equation again, 00:54:54.829 --> 00:55:01.019 which relates the magnitudes of the propagation of the attenuation vectors 00:55:01.019 --> 00:55:07.989 back to the imaginary real part of the squares of the complex wave numbers. 00:55:07.989 --> 00:55:12.029 And so we use those measured values to get those. 00:55:12.029 --> 00:55:14.710 And then the intrinsic material absorption 00:55:14.710 --> 00:55:19.320 is given by the ratio of the imaginary part to the real part. 00:55:19.320 --> 00:55:24.000 And the wave speed is given by this expression here, which is – 00:55:24.009 --> 00:55:29.200 utilizes the expression for the intrinsic absorption and the real part. 00:55:29.200 --> 00:55:32.569 So we have now a solution for the inverse problem – 00:55:32.569 --> 00:55:36.420 a rigorous solution for the inverse. And in practice, it’s going to have a lot 00:55:36.420 --> 00:55:42.700 more complications because this is dependent on how – the quality of the 00:55:42.700 --> 00:55:46.860 amplitude-attenuation curve and the travel time curves and so on and so on. 00:55:48.140 --> 00:55:52.340 And so then the final step is, we need to repeat that for all the waves. 00:55:52.340 --> 00:55:56.800 So this provides, then, a solution for the – for the inverse problem. 00:55:57.720 --> 00:56:05.760 So in summary, first of all, from a viscoelastic point of view, 00:56:05.769 --> 00:56:09.010 the fundamental problems for seismology are now solved 00:56:09.010 --> 00:56:12.749 for viscoelastic media with the solutions being valid 00:56:12.749 --> 00:56:17.960 for any media with a linear response – elastic or anelastic. 00:56:17.960 --> 00:56:23.559 And each of the anelastic solutions – because of this inhomogeneous 00:56:23.559 --> 00:56:27.130 business – implies characteristics for anelastic seismic waves 00:56:27.130 --> 00:56:32.239 that are not implied by elastic solutions. Not that every elastic solution we have 00:56:32.239 --> 00:56:36.670 now is going to be invalid or not useful. It’s just that there’s going to be certain 00:56:36.670 --> 00:56:41.580 situations where the anelastic solution is going to provide new insight. 00:56:42.000 --> 00:56:44.620 And the solutions that are now available – 00:56:44.630 --> 00:56:47.309 I’m not going to go through these. We’ve already been through them. 00:56:47.309 --> 00:56:52.540 But various problems are now available for viscoelastic media are these. 00:56:53.140 --> 00:56:57.599 And what we find from the viscoelastic ray theory as developed from 00:56:57.599 --> 00:57:05.950 first principles is that the solutions of the forward ray-tracing problems 00:57:05.950 --> 00:57:10.999 now exist for horizontal and spherical media with layers and gradients 00:57:10.999 --> 00:57:15.190 that accounts for the anelastic wave field inhomogeneity 00:57:15.190 --> 00:57:18.280 induced by variations in material absorption. 00:57:18.280 --> 00:57:21.722 And this is the key thing, I think, is that we now have solutions 00:57:21.722 --> 00:57:27.730 for forward ray-tracing problems that account for the contrast in intrinsic 00:57:27.730 --> 00:57:32.759 absorption at boundaries or the gradients in intrinsic absorption with depth. 00:57:32.759 --> 00:57:35.410 And we also have a solution of the inverse problems 00:57:35.410 --> 00:57:39.730 to basically infer anelastic material absorption 00:57:39.730 --> 00:57:44.220 and wave speed from empirical measurements, as we just saw. 00:57:45.380 --> 00:57:51.839 And so the – just to summarize then, these recent advances imply that 00:57:51.839 --> 00:57:57.369 the anelastic seismic rays are distinct from elastic rays with 00:57:57.369 --> 00:58:02.410 measurable distances in travel time – or, ray path locations, distances, travel 00:58:02.410 --> 00:58:08.940 times, and amplitude attenuation in both the near-surface, crust, and mantle. 00:58:08.940 --> 00:58:13.200 That is basically for moderate-loss materials 00:58:13.200 --> 00:58:16.340 as well as materials with low loss. 00:58:16.340 --> 00:58:24.000 And the – and these closed-form solutions suggest definite steps 00:58:24.000 --> 00:58:27.569 for computer codes to be developed to accurately model 00:58:27.569 --> 00:58:34.359 what these implications are. And basically the final thing is 00:58:34.359 --> 00:58:41.809 is that these new insights suggest opportunities, I think, for interpretation 00:58:41.809 --> 00:58:45.540 of seismic waves to infer anelastic material characteristics of the Earth’s 00:58:45.540 --> 00:58:48.740 interior that just haven’t been there before. 00:58:48.740 --> 00:58:54.780 And I think it would really be exciting to basically get these codes working and to, 00:58:54.780 --> 00:59:00.359 say, recalculate the travel times and amplitudes for all of these rays 00:59:00.359 --> 00:59:03.510 in the Earth’s interior and then see what happens 00:59:03.510 --> 00:59:11.360 when we start perturbing those with anelastic material characteristics. 00:59:11.360 --> 00:59:14.920 So with that, I’ll stop. Thank you for your time. 00:59:14.920 --> 00:59:21.000 [Applause] 00:59:25.220 --> 00:59:27.300 - Questions, or if anyone wants to join the call to 00:59:27.300 --> 00:59:30.360 hire more youth, you’re welcome to do that as well. 00:59:31.440 --> 00:59:34.040 There’s only a couple of us millennials here. 00:59:36.080 --> 00:59:40.340 [Silence] 00:59:40.980 --> 00:59:42.980 - So early on, you said the material velocity 00:59:42.980 --> 00:59:46.180 was dependent on the angle of incidence? 00:59:46.180 --> 00:59:47.320 - Yeah. 00:59:47.320 --> 00:59:50.839 - Okay. How much – how much does that velocity change? 00:59:51.640 --> 00:59:56.620 - That’s – that goes back to that plot I showed where – I can – 00:59:56.620 --> 00:59:59.120 we can go back there if you like. 01:00:03.360 --> 01:00:07.300 It depends on the amount of inhomogeneity of the wave. 01:00:07.320 --> 01:00:15.100 And it turns out that there are certain situations … 01:00:17.620 --> 01:00:20.060 Let’s see here. Where is it? 01:00:23.020 --> 01:00:25.080 Okay, this plot. 01:00:25.820 --> 01:00:30.180 Okay, so it turns out, if the wave field’s really inhomogeneous – that is, 01:00:30.180 --> 01:00:34.839 the direction of propagation really differs from the direction of maximum 01:00:34.840 --> 01:00:39.620 attenuation, which – and when I say it gets really large, it’s getting up, like … 01:00:39.620 --> 01:00:41.660 - Which is dependent on Q. - What’s that? 01:00:41.660 --> 01:00:43.499 - Which is dependent on Q, right? 01:00:43.499 --> 01:00:45.450 - And it depends on Q. See, each one of these curves 01:00:45.450 --> 01:00:49.220 is for a different intrinsic absorption in the material. 01:00:49.220 --> 01:00:55.700 But it says that, basically, you can get to a large enough angle of 01:00:55.700 --> 01:01:00.260 incidence – or, a large enough degree of inhomogeneity for which there is 01:01:00.260 --> 01:01:03.760 a significant difference. But, on the other hand, on the other 01:01:03.760 --> 01:01:10.569 end of this plot, where the degrees of inhomogeneity are not so large, 01:01:10.569 --> 01:01:14.420 then the differences between the wave speed for the homogeneous 01:01:14.420 --> 01:01:17.289 and inhomogeneous wave will not be very large. 01:01:17.289 --> 01:01:22.630 That’s why these examples I’m showing you that are showing these significant 01:01:22.630 --> 01:01:28.680 differences, a lot of the cases are for wide angles of incidence or where – 01:01:28.680 --> 01:01:32.049 situations where a degree of inhomogeneity 01:01:32.049 --> 01:01:34.809 of the wave gets quite large. 01:01:34.809 --> 01:01:40.860 - So what – sorry, a sub-question is, so should we – should this be easy to see? 01:01:42.460 --> 01:01:44.960 - Ah … - And, you know, observe? 01:01:45.720 --> 01:01:50.800 - Well, it is easy to see in that case of the water-stainless steel. 01:01:50.800 --> 01:01:53.220 I mean, it’s a commercial procedure. - Well, but in the Earth, 01:01:53.220 --> 01:01:56.380 we’ve got contrast of, you know, 1 kilometer per second 01:01:56.380 --> 01:01:59.260 to 3 kilometers per second or something. 01:02:00.140 --> 01:02:04.420 - Well, I’m – in wide-angle reflection data, some of this stuff 01:02:04.430 --> 01:02:08.049 should be easy to see. In other situations, 01:02:08.049 --> 01:02:13.499 if you’re looking at more vertical reflection, you probably won’t see it. 01:02:14.380 --> 01:02:18.980 Of course, that doesn’t mean that the anelastic wave speeds are exactly 01:02:18.980 --> 01:02:23.289 the same as the elastic wave speeds. But it does mean that this – 01:02:23.289 --> 01:02:28.080 going from homogeneous phenomena to inhomogeneous phenomena, 01:02:28.080 --> 01:02:32.609 that’s a situation where you’re going to see the most – the most dramatic effects 01:02:32.609 --> 01:02:36.500 are going to be in those cases where the inhomogeneity is large. 01:02:39.200 --> 01:02:45.400 [Silence] 01:02:45.400 --> 01:02:46.940 Rufus? 01:02:49.160 --> 01:02:52.440 [Silence] 01:02:53.360 --> 01:02:55.700 - An interesting thing you brought up was that 01:02:55.700 --> 01:03:00.420 you might be able to infer attenuation from travel times. 01:03:00.920 --> 01:03:06.600 But how do you – how do you get the velocity? 01:03:06.600 --> 01:03:10.220 I mean, it seems like there’s a kind of a chicken-and-egg thing? 01:03:10.239 --> 01:03:13.250 - There’s what? - A chicken-and-egg, where you have to 01:03:13.250 --> 01:03:16.299 know the velocity in order to get the attenuation, or vice versa? 01:03:16.300 --> 01:03:19.049 Did I – or did I miss something? 01:03:19.500 --> 01:03:23.420 I mean, if you know the travel time – if the travel time can give you some 01:03:23.420 --> 01:03:31.520 indication of attenuation, don’t you have to know the velocities to begin with? 01:03:31.520 --> 01:03:36.430 - I had thought – yeah, you do. Basically, it’s, again, this – might be 01:03:36.430 --> 01:03:40.280 this trial-and-error thing where you first of all start off with an elastic model. 01:03:40.280 --> 01:03:43.000 You know, and you – and you’ve got your velocities. 01:03:43.000 --> 01:03:47.229 And then, what I’m – and then – and this phenomenon that I was 01:03:47.229 --> 01:03:50.509 talking about there was, you’re going to try to look for 01:03:50.509 --> 01:03:54.180 another arrival that’s basically from that second layer. 01:03:55.440 --> 01:03:59.700 And if you see these other arrivals, and you have your velocity structure, 01:03:59.700 --> 01:04:04.180 then you would try to model it, depending on what the Q contrast was, 01:04:04.180 --> 01:04:10.480 to see, you know, how best to match that travel time curve. 01:04:11.840 --> 01:04:16.400 But, yeah, it’s a bit of a – you got – you know, always pulling ourselves up 01:04:16.400 --> 01:04:19.700 by our boots – you know, bootstraps or whatever. 01:04:22.020 --> 01:04:26.040 [Silence] 01:04:26.680 --> 01:04:30.180 - Sorry. I don’t know much about this. When you use the expression 01:04:30.180 --> 01:04:37.109 “inhomogeneity of the medium,” what does that mean exactly? 01:04:37.109 --> 01:04:39.940 At a – at a – I don’t know a microscopic level, maybe. 01:04:39.940 --> 01:04:44.580 What is – what does inhomogeneity of the medium mean? 01:04:44.580 --> 01:04:46.519 What is that? - Well, if you use the term 01:04:46.519 --> 01:04:51.749 “inhomogeneous” with respect to media from seismology point of view, 01:04:51.749 --> 01:05:00.390 we often think of that as layering, or we often think of it as changes 01:05:00.390 --> 01:05:05.380 in material properties that seismic wave fields would reflect from, 01:05:05.380 --> 01:05:10.039 or we would see reflections from. - Okay. It’s talking about properties. 01:05:10.040 --> 01:05:12.880 - Of – if you used it in the context of media. 01:05:12.880 --> 01:05:17.300 If you use it in the context of the wave, I use the word 01:05:17.319 --> 01:05:20.380 “inhomogeneous” a lot with respect to the waves. 01:05:20.380 --> 01:05:23.540 And that means that the surfaces at constant amplitude are not parallel. 01:05:23.540 --> 01:05:25.760 - No. I’m talking about the medium. The medium. 01:05:25.760 --> 01:05:27.400 - The surface is constant phase. Yeah. 01:05:27.400 --> 01:05:34.520 - Okay, so it’s – it doesn’t make any assumption about what the 01:05:34.520 --> 01:05:40.220 medium is, but it’s just characteristics. Characteristics. Okay. 01:05:41.580 --> 01:05:45.420 - Yeah. It’s different than from a geological perspective, probably. 01:05:48.080 --> 01:05:50.980 - Okay, well, thank you all. And thanks, Roger. 01:05:50.980 --> 01:05:53.940 And we’ll see you next week for our SSA preview. 01:05:54.440 --> 01:05:58.760 [Applause] 01:06:00.800 --> 01:06:13.680 [Silence]