Plate Tectonics and People

Raypaths Through the Earth

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The structure of the Earth makes tracing raypaths more complicated than the simple layered models we considered in the problem set. For one thing, the Earth is round, not flat. This means that all waves that start at the surface return to the surface, whether they encounter velocity changes along their path or not. Also, the concentric layers of material that make up the Earth aren't necessarily homogeneous. This means that sometimes waves are transmitted faster in certain directions than others. Let's discuss how raypaths travel through a homogeneous spherical body and then add complexity to see how seismic waves travel through a more "Earth-like" structure of concentric shells. Then we'll check out some actual seismic data to see how well the Earth mimics our simple model.

If the Earth were a Homogeneous Sphere

The animation below shows a cross-section through a homogeneous sphere. In this case, an earthquake that occurs somewhere on its surface would send seismic waves out in all directions and these waves would travel straight through to the other side.

Explanation of ray paths through a homogeneous sphere.
Click here for transcript.

This is a cross section through a homogeneous sphere. Let us just imagine that we have an earthquake that happened somewhere on the surface of the sphere over here. The wavefronts will travel outward from the source sort of like this, you know, the way that noise travels in all directions from a noise source. Or, if you toss a rock in a pond, the waves travel outward in concentric circles from the source. These are pictures of what the wavefronts look like. The ray paths are perpendicular to the wavefronts. We draw them like this. In a homogeneous sphere, all the ray paths would just travel straight through the sphere to the other side.

Hopefully you guessed that there was a point to all those refraction calculations you just did. What was that point? Well, the Earth is not a homogeneous sphere. It is composed of concentric shells of material and each one transmits seismic waves at a different velocity.

If the Earth had two layers over a homogeneous center

Let's consider a model like the sketch below, which is just a couple of spherical layers over a homogeneous center. This one is a little more Earth-like than the previous model we considered. When the incident ray strikes the boundary between the two layers, it will be refracted. It is refracted again at the next boundary, and then it happily travels along until it strikes the underside of that same boundary again and makes its way back up to the surface. In the animation below, note the shape of the ray paths through this model as opposed to the last model.

Animation of raypaths through two layers over a homogeneous sphere
Click here for the transcript.

I've drawn here two layers over a homogeneous middle part. Each of these layers has a transmitting velocity. Let us say that each one of them is slower than the one below. Now let's say we've got an earthquake up here. One of the ray paths comes down like this. It hits this layer, and since the lower layer is a little faster, (you remember this, right) it is going to refract away from the normal, so it's going to go like that. Then it's going to do that same thing again because the third layer is even a little bit faster. Now it hits the underside of the layer and refracts back toward the normal and up. This can be generalized to any ray path that comes from this event. Here is another one. It even works if you have some ray paths that do not ever get as deep as that bottom layer. Here is an example of some ray paths that go through this layered media. Remember the difference between what this looks like and what the ray paths looked like in the earlier sketch where the Earth was completely homogeneous.

Earth model in which transmitting velocity increases smoothly with depth

Now what if you imagined a sphere composed of an infinite number of concentric layers, and let's say that each layer transmits seismic waves just a little faster than the layer above. This model in which velocity increases with depth is even more Earth-like than the previous model. At each local boundary, the ray still has to obey Snell's law, but since the layers are infinitely thin, the path ends up being a smooth curve. To see some of these curved ray paths, check out the animation below.

Description of the curved ray paths
Click here for video transcript.

This model is even more Earth-like than the previous two. There are two big differences here. This model does not have a core in it, and also you need to have a little bit of imagination here because what I am hoping to have you think about is that you have got this gradation from dark to light that is smooth. I know you can see rings here but pretend they aren't there. what this model would show is if the Earth only had a mantle and it got successively denser and denser and denser from the outside to the inside, the waves would be transmitted faster and faster and faster. The effect of this would be to make the ray paths curved because at each layer's boundary the waves would still have to obey Snell's Law and refract, but if each layer is infinitely thin then mathematically that ends up being a curve. So let me just draw some sample rays. Here is an earthquake right here and so ray paths would curve like this or like that. Or like this. Anything that went straight down would just head through the middle but that is the only path that would be the same as the analogous path in a homogeneous Earth. Every other path curves. You might want to look back at the previous sketches to see the difference between these ray paths and the ray paths that went through a homogeneous Earth, or even the one that just had a couple of layers at the top.

Now, armed with our model of a sphere in which velocity increases with depth, we should be ready to plop seismometers down all over the world, measure the P and S wave arrival times from earthquakes, and confirm our idea about how wave speed varies in the Earth. In fact, seismologists did do this, and when they did, they realized that at some distances away from earthquakes, they didn't get any P or S arrivals at the times they should have arrived. They realized that there had to be a major boundary inside the Earth where material properties changed drastically, thus altering the wave paths enough to create a "shadow zone" where there are no direct arrivals from P and S waves. This is how the Earth's core was discovered. Seismologists knew that seismic rays took a curved path through the Earth. As the distance between the source and the receiver increased, the "turning point" (the deepest point along the wave's path) got a little deeper. At the farthest distance direct body waves were recorded, the turning point corresponding to that raypath must have been the depth of the mystery boundary, because a ray that tries to turn at a deeper point will run into that boundary, get refracted, and follow a different path. The depth of the boundary between the mantle and the outer core was found to be about 2900 km. This corresponds to a shadow zone for direct arrivals of P and S waves beginning at about 104° from the source. The same method was used to discover the inner core of the Earth as well.

A Seismic Record Section from the 2004 Sumatra-Andaman Earthquake

Let's check out some actual seismic data to see if we can distinguish all the features of the raypaths in the Earth-like models we considered. Below is a record section from the 2004 26 December Sumatra-Andaman earthquake. Creating a record section means plotting a suite of seismograms that are arranged in order by their distance from the earthquake. In this record section, the closest station is CHTO, at about 16 degrees away, and the farthest station is PAYG, at about 173 degrees away (Can't get too much farther than that because 180 degrees would be exactly on the other side of the Earth!). Each seismogram has some colored bars on it. These colored bars are arrival time picks for various body waves. Watch the two screencasts below the figure to see me sketch the paths the seismic waves took through the Earth to produce each of these arrivals.

Contact the instructor if you have difficulty viewing this image
Record section of the 2004 Sumatra-Andaman earthquake. Data from the Incorporated Research Institutions for Seismology. The x-axis is time and the y-axis is distance in degrees. Each station's name appears on the y-axis at the appropriate distance. The dark blue wiggly lines are the seismograms. The grey icons that look like one-handed clocks in the right margin show the azimuth, the direction from the earthquake to the station. The colored vertical bars are marks where an automatic phase picker has predicted the arrivals of various body waves. For more details see the animations below.
Enter image credit here

Seismogram Arrivals Explained!

First I'll single out station SBA and draw a cross-sectional sketch of the Earth that shows how each arriving wave gets from the earthquake to SBA.

There is also a transcript of the screencast demonstrating arrivals at SBA.

This is a record section from the 2004 Sumatra Andaman earthquake. A record section just means that I have taken a bunch of seismograms and arranged them in order of their distance from the earthquake. So, on the x-axis here is time. On the y-axis is distance in degrees. Each of these seismograms is plotted with the name of its station and its distance away from the earthquake. What I'm going to do now is just focus on one of these stations. This station, SBA, here. I'm going to show you just by sketching that path that each of these arriving waves took to get from the earthquake to this station. These colored bars here are arrival time picks for various waves. This was probably done by an automatic picker that knows about how long it takes for each type of wave to get from one point to another on Earth. If I just make this little sketch right here of the Earth. I'm leaving out the crust here, but basically, this is a cross-section. Here's the mantle, and here's the outer core, and here's the inner core. And let us just pretend that my earthquake happened right up here at the top. We can pick any spot I guess. I'm going to choose this station SBA because it is handily almost exactly 90 degrees away and that is easy for me to freehand. So we'll draw a little house. That's our seismometer. The first arriving wave that gets from the earthquake to the seismometer is this one that is marked in green right here. It is the direct P wave. The path that the direct P wave takes through the mantle is kind of like that. Notice how it curves. It does not follow a straight line path. That is because seismic wave speed increases with depth. The next arriving wave is this arrival marked in red. That is PP. What PP does is it goes through the mantle and bounces once between the earthquake and the station, and then it continues on its way to the station. Each of these waves is a P wave in the mantle so we call it PP. The next arriving wave; well actually this orange one and this yellow one come practically on top of each other. The orange one is the direct S wave and the direct S wave follows the exact same path as the direct P wave only shear waves are slower so that is why it takes longer than the P wave to get from the earthquake to the station. The yellow one is a little more interesting. It is an S wave that bounces off the core mantle boundary and then gets to the station. So it is S and then that bounce point is called little c and then it is S again so the entire wave is called ScS. The next arriving wave is this pink one. The pink one is SS which follows the same path as PP only shear waves are slower so it takes it longer to get to the station. Those are all the waves marked here. We have this big high amplitude this coming in later on and those are actually surface waves. They travel along a path like this. Takes them longer to get there because the crust doesn't transmit waves as fast.

body wave paths from earthquake to station 90 degrees away
Body wave paths to SBA, a station about 90 degrees from the earthquake
E. Richardson

Now...Station PAYG is trickier! No direct P wave! No direct S wave! In fact, by looking at seismic records like this one, seismologists figured out that the Earth must have a core made of significantly different material than the mantle. Based on the "shadow zone," the distance range over which no direct mantle body waves are observed, seismologists also figured out the size of the core. The fact that no S waves could make it through the core showed scientists that at least part of it had to be a liquid. Watch my sketch of how this works for the arrivals at station PAYG.

There is also a transcript of the screencast of the sketched paths to PAYG.

Let us look at the same record section, but a different station. This time I am going to look at station PAYG, which is almost 180 degrees away from the earthquake. I have drawn it down here. What you will notice right away is that the number of arrivals is fewer. There is no direct P wave and there is no direct S wave. The reason for that is that the core gets in the way. Remember at the earlier station that P waves and S waves take a kind of curving path through the mantle that I'm showing you right now with the mouse. If you are farther away than about 104 degrees then these waves will bounce off the core or they will be refracted within the core as P waves and so you will not get a direct arrival. This is how the core was discovered. What are the arrivals that are coming in at this station? The first one is this one here and that one is labeled PKiKP. What that wave does is, it is a P wave in the mantle, it comes down here, it gets refracted in the outer core, and the inner core, and back through the outer core, and back out through the mantle to this station. The path is called a P wave when it is in the mantle, and then a P wave in the outer core is called K. I do not know why that is. In the inner core it is called i. And then it goes back out as K and P. The whole thing together is called PKiKP. The next arriving wave is PP. That is our old friend. We know how that works. It goes through the mantle and it bounces and it goes to the station. Each of these paths is a P wave in the mantle. It is called PP. The last arriving wave is SS. It follows the same path as PP except shear waves are slower so it takes it longer to get there.

body wave arrivals at station PAYG which is almost 180 degrees from eq
Body wave paths at PAYG, a station about 180 degrees from the earthquake
E. Richardson

Want to make your own record section of a recent earthquake?

Thanks to the good folks at IRIS, the Incorporated Research Institutions for Seismology, it is not too hard for anyone to do. I recommend letting your students play around with this!

  1. Go to Wilber3, where you can request seismic data from recent earthquakes.

  2. The default page has a map with recent (last month or so) earthquakes on it. The map is interactive, so you can draw a box to zoom in, and there are also some dialog boxes you can type into in order to narrow down the number of events in the list by date, location and magnitude.

  3. The map locations are clickable colored circles. Clicking on one of them will highlight that event in the list on the same page.

  4. After you've clicked on your selection in the list, the map will refresh and now show you all the stations that recorded data from your selected earthquake.

  5. At this point you can click a button that says "Show Record Section"