1. Single station location Multiple station location
Computer implementation
Based on: Havskov and Ottemoller, Routine data processing in Earthquake seismology, Springer, 2010.

2. Location: Relative location methods
Lin et al., 2007

3. Location: Intro.
Earthquake location is defined by the earthquake hypocenter (x0,y0,z0) and the origin time t0.
The epicenter is the projection of the hypocenter on the Earth’s surface (x0,y0).
The epicentral distance, D, is the distance from the epicenter to the station (x,y,z) along the surface of the Earth. For local earthquake, where flat Earth is assumed, it is simply:
Similarly for local earthquakes, the hypocentral distance is:

4. Location: Intro.
The azimuth is calculated using the epicenter and the station location.
The back azimuth may also be calculated, but it can also be directly observed on three component stations.
The centroid location represents the average time and location for the entire event, and is obtained by moment tensor inversion of long period waves.

5. Location: Single station
In general, it is preferable to use as many as possible arrival times from different stations and phases. It is, however, also possible to locate an earthquake using a single 3-component station.

6. Location: Single station
There is an ambiguity of 1800, so the polarity must also be used in order to get the correct back-azimuth.

7. Location: Single station
After having obtained the direction of the seismic source, we can now obtain the distance to the source from the difference between the P and S arrivals.
Thus, the hypocentral distance is given by:
This result is only valid for hypocentral distances of up to km. Beyond these distances the Pn and Sn arrive first.

8. Location: Single station
In the absence of local travel-time curves, one can use a Vp to Vs ratio that is suitable for a Poisson solid:
and use that to derive simple rule of thumbs as follows:
Normal crust:
Old crust:
With both azimuth and distance, the epicenter can be obtained by measuring the distance along the azimuth of approach.
Finally, knowing the distance, we can calculate the travel time and thereby get the origin time.

9. Location: Multiple station
With back azimuth available from at least two stations, epicenter and origin time may be calculated.
In order to get the depth, at least one P and S reading must also be available.

10. Location: Multiple station
With readings from at least 3 stations, a simple manual location can be from circle drawing, with the center of each station at the station location and the radii equal to the epicentral distances calculated from the S-P times
The circles drawn around the stations with radius that is set according to ts-tp will normally not be crossing at a single point, but rather “overshoot”.

11. Location: Multiple station
With several stations available from a local earthquake, the origin time can be determined by a very simple technique called the Wadati diagram.
One can write:
After eliminating the p-wave travel time,
The S-P time is related to the origin time and the Vp to Vs ratio according to:
Note that it is possible to get the both the origin time and Vp/Vs without prior knowledge of of the crustal structure.

12. Location: Computer implementation
The calculated arrival time at station I can be written as a sum of the origin time and a calculated travel time given a velocity model and hypocenter location:
Because this equation has 4 unknowns, at least 4 arrival time observations are needed.
With more than 4 arrival times, the problem is over-determined.
A solution is found such that the difference between the observed and calculated arrival times is minimized

13. Location: Computer implementation
Note that the travel time is a non-linear function of the model parameters – not even for the simplest case where the waves travel through a medium of uniform wave speed:
Because the travel time does not scale linearly with either of the spatial coordinates, the problem must be solved numerically.

14. Location: Computer implementation
Grid search
Since it is inexpensive to calculate travel times to any point in a model, a very simple method is to perform a systematic grid search over all possible locations.
At each such point we obtain n arrival time residuals, ri, as follows:
with the only unknown being the origin time that can be obtained in an average sense using all the observations.

15. Location: Computer implementation
Grid search
The most common approach is to choose the solution which minimizes the root of the average residual squared:

16. Location: Computer implementation
Grid search
The simplest way to get an indications of the The location uncertainty is contour the RMS of the residuals. The final product of this procedure looks like this:
Location by a grid search is up to 1000 times slower than location by iterative methods.

17. Location: Computer implementation
Location by an iterative approach
These methods are based on linearizing the problem.
Step 1: Make a guess of the hypocenter and the origin time. In the simplest case, the event is near or within a seismic network, in which case the guessed hypocenter should be near the station to which the ground shaking arrived first.

18. Location: Computer implementation
Location by an iterative approach
Step 2: Linearize the problem. By so doing, it is assumed that the guessed hypocenter is located close enough to its true location, so that travel time residual are a linear function of the 3 spatial coordinates. The residual we seek to minimize may be written as:
Or in a matrix form:
Except for the simplest situation of constant velocity, the derivatives cannot be obtained analytically and must be obtained through interpolation of travel time tables.

19. Location: Computer implementation
Location by an iterative approach
Step 3: Solve for the 4 unknowns. Usually the best solution is obtained using the least-squares criterion, i.e. through solution of:
Step 4: Apply correction. The previous solution is corrected, and the new solution is used as a trial solution for the next iteration.
Steps 1 trough 4 until a pre-specified break point is reached.
It may happen that the solution converges towards a local minima, and unless the residuals are very large this would be hard to detect.

20. Location: Computer implementation
Location by an iterative approach
If one or several back azimuth observations are available, they can be used together with the arrival time observations, and this equation:
may be jointly inverted with:
Note that the residual of the previous equation is in terms of time (i.e. seconds), whereas that of the new equation is in terms of angles (i.e., degrees). Thus, in order to solve the two equations jointly, the degrees must be converted to seconds.

21. Location: Computer implementation
Location by an iterative approach
In the location program HYPOCENTER (Lienert and Havskov, 1995), a 10 degrees back azimuth is set equivalent to a 1S travel time residual. The effect of setting a 20 degrees back azimuth equivalent to a 1S residual is to lower the weight of the back azimuth data in the inversion.
In the location program HYPOSAT (Schweitzer, 2001), the uncertainties in the back azimuth observations are used to weight the individual equations that go into the inversion, in a manner that less certain data contribute less to the final result.

22. Location: Computer implementation
Example of location in a homogeneous model
The travel time equation is:
And the partial derivatives, assuming the station is at the surface, are:
and:

23. Location: Computer implementation
Example of location in a homogeneous model
In this simple example, because the data is noise-free, two iterations are enough to recover the exact location.

24. Location: Relative location methods
Note of warning
When a good station distribution is not available, location errors can be quite large. For example, the distance to events occurring outside of a seismic array, since there is a large trade-off between position and origin time.

25. Location: Relative location methods
Note of warning
Another problem is the trade-of between depth and origin time that occurs when stations at close range are not available.
If a fault separates the crust into two blocks of different velocities, events occurring on the fault will tend to be mislocated off the fault and into the faster block.

26. Location: Relative location methods
Master Event Technique (MET)
In many cases, the relative location between events within a localized region can be determined with much greater accuracy than the absolute location. This is because the velocity variation outside the source region, will have nearly the same effect on all of earthquakes within that region.
If we designate one of the events as a master event, we can compute arrival times of other events relative to the master event time:
The relative location, Dm, is given by the best-fitting solution to:
where the solution is valid provided that Dm is small enough, so that the linear approximation holds.

27. Location: Relative location methods
Master Event Technique (MET)
The relative location between events within a certain region can often be made with a much greater accuracy than the absolute location of any of the events
The events in the local area are relocated relative to one particularly well-located event, which is then called the “master event”.
The MET can be used only when the distance to the station is much larger than the distance between the events, since it is assumed that the travel time residual cause by the velocity structure outside the local area are the same for all event-station pairs.

28. Location: Relative location methods
Master Event Technique (MET)
Two earthquakes recorded by a local network of 3 stations.
The two events have nearly identical signals at stations a few hundreds of kilometers apart, which is another evidence that they occurred at the same location and have similar focal mechanism.

29. Location: Relative location methods
Master Event Technique (MET)
The actual implementation of the MET follows these steps:
Step 1. Select the subset of stations and phases that are common among the events to be used. Use clear phases whose waveforms are similar to that of the master event at a given station. Get relative times – ideally by cross-correlation.
Step 2. Locate the master event at the fixed location with the selected phases and stations only and observe the residuals at the stations. It is assumed that these residuals exactly reflect the differences between the actual velocity variations and the velocity model employed by the inversion.

30. Location: Relative location methods
Master Event Technique (MET)
Step 3. Add the residuals to the readings for all events to be located relative to the master event.
Step 4. Relocate all events including the master event without any fixing. A good test of checking that residuals have been added correctly is to check the location and RMS of the master event. The RMS should be 0 and the location the same as before. In practice there will always be some round off errors so the master event moves a bit.

31. Location: Relative location methods
Joint Hypocenter Determination (JHD)
In the MET, it was assumed that true structure dependent residuals could be obtained absolutely correct from the master event, however other errors could be present in the readings for the master event. Instead, in the JHD, the most precise station residuals is determined using the whole data set.

32. Location: Relative location methods
Joint Hypocenter Determination (JHD)

33. Location: Relative location methods
Double difference earthquake location

34. Location: Relative location methods
Double difference earthquake location

35. Location: Relative location methods
Double difference earthquake location