Image: GeoLurking, rights reserved. Basic triangulation, how to find a spot or a station
First, my qualifications. None. I have been a fan of geophysical processes and phenomena for the last 35 years. More so if you count the time spent hanging out in the archeological part of the library.
Image: GeoLurking, rights reserved. Margin of error from one station within one standard deviation
When an earthquake occurs, the energy from that quake travels directly to the seismic station. Granted, there is more to it than that, and it actually takes a curved path through the earth depending on the density of the material (refraction), but for our purposes it’s a direct path.
When an earthquake appears in a seismic catalogue, you are given the latitude and longitude of the epicenter, and the depth. The addition of depth turns that position report into a hypocenter, because it locates the quake in three dimensions. In the more detailed phase portion of the reports, you can obtain the arrival times of the quakes. Take the difference in arrival times for each station with respect to the event time, and you have how long it took for the quake’s energy to reach the station. Knowing the distance from the station to the quake, and you can work out the speed of that seismic wave.
Image: GeoLurking, rights reserved. The P-wave
This presents a problem if you don’t realize what you are looking at. The distance from the station to the quake, is usually seen as surface distance if you use a map. This is not the actual path that the wave took. Remember, it takes a direct path. How do you find it?
Geologists, and geological organizations, think in terms of central angle when discussing seismic events. This is the angular distance as measured from the center of the earth that describes the arc length on the surface between two points. I’ve mentioned AK135 before, in it you can find the expected arrival times for the various phases based on our current model of the Earth’s structure. All of its data is listed by the central angle. It is used to assist in identifying what each individual squiggle in a seismic trace represents. (Based on the path that particular portion of the wave took)
Image: GeoLurking, rights reserved. The S-wave
For the simplistic technique employed here, AK135 and the more advanced concepts aren’t really needed… as long as we remember that this is a simplified approach, and prone to error.
Treating the orientation of the seismic station and the quake as being two points on a slice of a sphere, the problem reduces to being a series of calculations on a circle. You have the chord, which is point from the seismic station to a point equidistant from the quake on the other side of the quake. That chord will have a height which is the distance from it to the part directly below the epicenter. Calculate that and you can then find the parts of a right triangle when you determine the depth of the quake in relation to the midpoint of the chord. Find the hypotenuse and you have the direct patch distance to the quake. If your eyes have glassed over by now, don’t sweat it. It took me three days to get the spreadsheet formulas down. Sometimes you just have to get up and walk away for a while.
Image: GeoLurking, all rights reserved. P and S wave paths
Now that we have the direct path distance, we can calculate the actual P-wave and S-wave speeds. We can do something with that.
Before we do, I want to point out (yet again) that this IS NOT seismic tomography. That involves far more than our simple juggling of the geometry and data.
Image: GeoLurking, rights reserved. Preliminary image
P-waves are compression waves, much like sounds are a compression waves. The molecules move toward and away from the direction of travel. S-waves are transverse waves, they move side to side with respect to the direction of travel. The density of the medium that the waves travel through affects the speed of propagation, or how fast the waves moves. S-waves cannot travel through a pure liquid as S-waves.
When the rock density goes up, the difference in the speed betweed P and S waves grows smaller. With lower rock density, the speed difference grows larger.
Image: GeoLurking, rights reserved.
About the Plots
I’m not going to interpret what the plots mean. I’ll leave that to Carl and the others. But I will tell you what they are (it took me a few days to figure this one out). They are a plot of the speed field of quakes in a particular area. The quadratic surface interpolates the regions between the individual quakes, and shows you what a quake originating there should look like if one occurred there… based on the speed of the ones that did occur.
Like I said, it’s not tomography, but failing that, it’s about as good a representation of what is going down there that we can obtain as non geologists, based on available data.
Image: GeoLurking, rights reserved.
Since quakes occur over a period of time, these plots are a summary of what was observed over that period of time. In other words, they have very poor temporal resolution. Keep that in mind.
Other aspects of the plots show what would be expected from rising pockets of magma… compression of the overlying material, and a subsequent decrease in the speed difference of P and S waves.
Image: GeoLurking, rights reserved.
The unnerving part of these plots, and why I was apprehensive about releasing them, is that they dovetail well with anecdotal information about the dynamics in play.
Image: GeoLurking, rights reserved. The original image that started the discussion. Here we can see what might be a magma chamber, and the feeder tube that goes down into the deep. Location of the inferred magma chamber is below Tanganasoga volcano.
Again, my standard caveat: I am not a geologist and am not trained in that field. I could be very wrong in my methods. Take it with questioning view, which is sane way to look at it.
Afterword: Peer review
In science when you wish to publish something new, you send the paper in and then a review-board reads it through, and either it pass and becomes published (often after revisions), or not.
Since this is something that can affect a lot of peoples lives the author wanted the post checked. So, here is how the peer review was done.
First I picked a part the reasoning, and then I looked if I would get the same result. I also went through the physics of it to see that it checked out.
Then the paper was sent to an engineer in the oil industry that works with tomography on a daily basis. And he found it to be correct.
After that I started to think about what other ways one could “see” what the author had found. So I deduced that the quakes that are smaller then 2M are just hiding the real action. So I tricked KarenZ and Ursula into plotting that, and guess what, the same structures showed up again. For those who follow the comments in here, those plots are there to be viewed.
I guess there are more ways we could have checked it, but in the end, I wished to publish this before the volcano became demented with old age.