Mary Howerton

Math 308A

 

Earthquake Location

 

Abstract

 

Earthquake location is an important application of linear algebra which involves calculating the transpose and the inverse of matrices, and using matrix multiplication and multiplication by a vector. Using these concepts, one can locate the hypocenter (location and beginning) of an earthquake in all three spatial dimensions, as well as in time.

 

Background and Explanation

 

Locating earthquakes is important to seismologists for a variety of reasons. In particular, knowing the location of past earthquakes can help predict where earthquakes are likely to strike in the future. The exact location of the beginning of an earthquake in three spatial dimensions and in time is called the hypocenter. In order to find the hypocenter, you need to have the locations of several stations in three spatial dimensions and the time at which the earthquake reached those stations. With this information, you can find the exact location of the hypocenter.

 

First, given stations 1, 2, 3…n with coordinates (x, y, z) and arrival times t1, t2, t3 …tn, you can use the formulas t = d/v  and d = [(x-x0)2 + (y-y0)2 + (z-z0)2 ]1/2 to get the formula:

 

tc = [(x-x0)2 + (y-y0)2 + (z-z0)2 ]1/2 / v

 

Now, this is obviously a complicated and non-linear equation, which would not be very practical to use. Therefore, it can be linearized by doing a Taylor expansion, giving the following formula:

 

t0 = tc + t/m Dm

 

where m is your model, or in other words, your previous guess at the location of the earthquake, and

 

t/x = {(x- x0) / (v*[(x-x0)2 + (y-y0)2 + (z-z0)2 ]1/2)} Dx

 

The next step in precisely locating the hypocenter is finding the difference between the observed and calculated travel times, providing an estimate of the error in the model. This quantity is called a residual, which is represented by a vector r.

 

r = t0 -t c = t/m Dm

 

Since there are several stations with data, there are also several equations, meaning that you can form a system of equations.

 

r =       | (t0 -t c )1 |    =  | (t1/x) Dx1 + (t1/y) Dy1 + (t1/z) Dz1 + Dt1| (From Station 1)

| (t0 -t c )2 |    =  | (t2/x) Dx2 + (t2/y) Dy2 + (t2/z) Dz2 + Dt2| (From Station 2)

| (t0 -t c )3 |    =  | (t3/x) Dx3 + (t3/y) Dy3 + (t3/z) Dz3 + Dt3| (From Station 3)

| (t0 -t c )4 |    =  | (t4/x) Dx4 + (t4/y) Dy4 + (t4/z) Dz4 + Dt4| (From Station 4)

 

 

Now define a matrix G such that G Dm = G r.

 

G =      | (t1/x) (t1/y)            (t1/z)             1 |

            | (t2/x) (t2/y)            (t2/z)             1 |

            | (t3/x) (t3/y)            (t3/z)             1 |

            | (t4/x) (t4/y)            (t4/z)             1 |

 

Next, left multiply G Dm = G r by GT to get

 

(GTG) Dm = GTr

 

and then left multiply again by (GTG)-1.

 

(GTG)-1 (GTG) Dm = (GTG)-1GTr

 

Since (GTG)-1 (GTG) = I , you get

 

Dm = (GTG)-1GTr

 

Finally, you add Dm to your original model to get a new model, which is a set of coordinates for the hypocenter that is more accurate.

 

m0 + Dm = m

 

From this, you should have a set of coordinates (x, y, z, t) for the hypocenter of the earthquake. Calculate the residual once again, r = t0 -t c , in order to gauge the accuracy of the new model. The whole process should be repeated until ||r|| falls within a specified value, usually 0.1 seconds.

 

Example

 

For an example, I decided to take some data from the Nisqually Earthquake that Seattle experienced on February 28, 2001. It was a magnitude 6.8 and received a lot of attention, so finding data for it was actually possible. Distances are in kilometers, and times are in seconds. For the original model, I used Seattle as the hypocenter, and 6 km/s as an original velocity, as well as using a flat Earth.

 

 

 

 

Station

x

y

z

t

Ottawa, Canada

3527

-250

1

383

Golden, CO

1382

-865

1

205

Anchorage, AK

-1676

1540

1

279

Mexico City

1806

-3300

1

396

 

 

 

G =      | 0.166             -0.0118 1.89E-7          1 |

            | 0.141             -0.0884 1.02E-4          1 |

            | -0.123   0.113               7.32E-5          1 |

            | 0.0800 -0.146              4.43E-5          1 |

 

GT =    | 0.166             0.141               -0.123              0.0800            |

            | -0.0118 -0.0884 0.113               -0.146             |

            | 1.89E-7            1.02E-4          7.32E-5          4.43E-5|

            | 1                    1                      1                      1            |

 

GTG     =            | 0.0387 -0.0400 8.95E-6          0.264            |

                        | -0.0400 0.0420             -7.22E-6            -0.133            |          

                        | 8.95E-6            -7.22E-6            1.77E-8          2.20E-4|

                        | 0.264             -0.133              2.20E-4          4            |

 

(GTG)-1 =            | -99.5              -82.4                -9.76E4 9.19            |

                        | -82.4              -41.7                -8.13E4 8.53            |

                        | -9.76E4 -8.13E4 8.29E7             -822            |

                        | 9.19               8.53                 -822                 -281            |

 

(GTG)-1 GT =            | -6.37              -7.51                4.97                 8.94            |

                        | -4.67              -7.69                8.00                 4.42            |

                        | -1.6E4   1.06E3             8.06E3             6.91E3            |

                        | -280               -281                 -281                 -282            |

 

r0 =      | 383            |

            | 205            |

            | 279            |

            | 396            |

 

Dm = (GTG)-1 GT r0 = | -2440             -1540               1904                3420            |

                                    | -957               -1580               1640                906            |

                                    | -4.46E6 2.96E5             2.25E6             1.93E6            |

                                    | -1.11E5 -1.11E5 -1.11E5 -1.12E5|

 

 

 

 

r = | t0 – tc | =            | -3040            |

                        | -701            |

                        | -1.93E6|

                        | -1.12E5|

 

Since ||r|| is clearly not less than 0.1, we have to add our new model onto our old model and repeat the process.

 

m0 + Dm = m

 

m =      | (-2440 + 3527)            (-1540 + -250)                        (1904 + 1)      -3040            |

            | (-957 + 1382)            (-1580 + -865)                        (1640 + 1)      -701            |

            | (-4.46E6 + -1676)            (2.96E5 + 1540)            (2.25E6 + 1)   -1.93E6|

            | (-1.11E5 + 1806)   (-1.11E5 + -3300)   (-1.11E5 + 1)            -1.12E5|

 

m =      | 1090              -1790               1905                -3040            |

            | 425                -2450               1641                -701            |

            | -4.46E6 2.98E5             2.25E6             -1.93E6|

            | -1.10E5 -1.14E5 -1.11E5 -1.12E5|

 

After a few iterations, you come up with ||r|| £ 0.1, at which point you can locate the hypocenter relative to the original guess, which was 1 kilometer below the city of Seattle.

 

Bibliography

 

Espanshade Jr., Edward B. (Editor) Goode’s World Atlas. 19th Ed. Rand McNally, 1995.

 

IRIS Special Report – Nisqually, WA 2/2001.            http://www.iris.washington.edu/NEWS/news20010228110948.htm

 

Jones, Josh. (University of Washington Department of Earth and Space Sciences,

Geophysics) Interview.

 

Lay, David C. Linear Algebra and Its Applications. 2nd Ed. Addison-Wesley, 2000.

 

Shearer, Peter M. Introduction to Seismology. Cambridge, 1999.

 

Evaluation

 

I enjoyed the project because it allowed me to actually see the applications of linear algebra to topics that interest me. I spent several hours researching the topic, and several more hours putting all of the information together; however, I think that time was a lot more helpful than just sitting and doing exercises from the book. I’ll most likely get to use the information I learned from this project in the future. Seismology is one of the fields that I might decide to go into later in my life.