Comment due Sunday, April 14
After last Friday’s magnitude 4.8 earthquake centered in New Jersey, I became curious about how linear algebra is applied in earthquake science. Most of what I found online was not quite relevant for us: the algebra often referred to is not linear (for example, when using triangulation to locate an earthquake’s epicenter, a system of quadratic equations is used… even ChatGPT got confused about this when I asked it for help).
However, I did find one topic that appears to use linear algebra: Geiger’s method for locating the hypocenter of an earthquake. We commonly refer to the epicenter of an earthquake as its location, but the epicenter is a point on the surface of the earth; an earthquake actually originates at some depth below the epicenter at its hypocenter. So, while the epicenter requires only two spacial coordinates, the hypocenter requires three spacial coordinates.
I am not an expert, but I have found a few resources that should help us understand Geiger’s method for locating the hypocenter of an earthquake:
- This assignment from an earth sciences class at the Saint Louis University Earthquake Center (this looks like a draft, watch out for typos!),
- This article (if you have seen partial derivatives in your differential equations or Calculus II class, this might make more sense to you),
- This article (the section on Geiger’s method looks similar to the section on Geiger’s method in the previous article).
For this assignment, you do not have to read any of these resources thoroughly (unless you want to). Read just the section on Geiger’s method in one of these resources (or another one if you find one) and try to understand:
- what the variables represent,
- where the equations in the linear system and/or matrices come from,
- what the solutions of the linear system and/or matrix equations represent,
- the steps in the procedure (this is an iterative method, which means it’s probably applied more than once).
By the way, some of these resources mention finding the “least squares solution” of a matrix equation; you can read more about that procedure here.
Don’t worry about understanding everything completely or getting everything right. Comment on this post with your ideas about items 1-4 above. If you’re not confident about one or more of them, no problem, you can just say so. Read your classmates’ comments and comment if you agree or disagree with any of their ideas. This should be a conversation among you and your classmates to try to understand this application of linear algebra, even if that understanding is just at a surface level at first.
Geiger’s method works by solving a system of equations based on data from seismometers. It estimates the three-dimensional coordinates of the earthquake’s origin point and the origin time. The linear system comes from the equations modeling the relationship between the observed arrival times of seismic waves and the distances from seismometers to the hypocenter. Solving this system helps determine the location and time of the earthquake. It uses iteration to refine the estimation with each cycle.
Geiner’s method is used to find the source of the earthquake. As earthquakes has a starting point and it spreads linearly around the radius, we can use linear algebra to find the hypocenter and arrival time. Using Taylor polynomials we can write the arrival time function from hypocenter. To guess the best arrival time we write iteration of the function that self corrects after each iteration. We can put the iteration in a matrix and solve the equation.
1.- Geiger’s Method variables represent the coordinates of the earthquake’s hypocenter by spatial coorditantes (X, Y, Z) and the origin time (T). On top of that you have coordinates for stations(x_j, y_j, o), arrival time(t_j), travel time (TT).
2.- The equations come from the study of the physics of wave propagation, They represent the travel time of the seismic waves from the hypocenter to seismic stations.
3.- The solutions represent estimates of the earthquake’s hypocenter and origin time by minimizing the difference between observed travel times and the ones being calculated.
4.- You take an initial guess for hypocenter and the origin time. You have to then calculate the travel time, which would adjust the hypocenter coordinates and origin time with the goal of minimizing the difference between the observed and calculated travel times. This is performed iteratively until the solution converges.