Determining the location of earthquakes is one of the most important tasks of seismology, extremely important for informing the public about earthquakes that occured, and at the same time the first step for numerous scientific researches in seismology. When an earthquake occurs, part of the energy is released in the form of seismic waves that we feel on the surface as a shaking of the ground. We “catch” the movement of the ground with seismographs at seismic stations and we can use the obtained records of earthquakes to determine their location. An earthquake occurs at a point called the focal point or hypocenter, where two types of waves are generated: longitudinal or P-wave and transverse or S-wave (Figure 1). Both waves propagate through space, the P-wave at a higher speed than the S-wave. If the P-wave moves at speed *v _{P}* and reaches the seismograph at time

*t*, and the S-wave moves at speed

_{P}*v*and is recorded by the seismograph at time

_{S}*t*, then the distance of the focal point from the station can be calculated using the expression

_{S}From the earthquake record at one station, we know how far the epicentre is from it, but it can be anywhere below the surface of the ground at that distance. Therefore, to determine the location of the earthquake, we need its records on at least three stations. If we observe the simplest case when the focus of the earthquake is just below the surface, in the epicentre, then the location of the earthquake can be found using the triangulation method. Namely, the epicentre is at the point where three circles with centres in seismic stations touch. For example, if we calculate from the difference in the arrival time of S- and P-waves t_S-t_P that the epicentral distance for the HVAR station is 56 km, and for the CRONOS project stations HR01A and HR11A the epicentral distance is 103 km and 73 km, then the epicentre of the earthquake was not far from Turjaci in Dalmatinska zagora (Figure 2).

**Complications**

In a real situation, these three circles usually do not intersect at one point for several reasons. First, it is not always easy to determine the arrival times of seismic waves in records due to noise in the record, earthquake mechanism effects, and wave propagation effects. The measured wave arrival times carry with them a certain error. Second, the speeds of seismic waves in the ground are not the same everywhere. There are variations of velocities in space, especially with depth, that this simple calculation does not consider.

In addition, we want to calculate the depth of the focus and the time of the earthquake, not just the location of the epicentre. Also, for a more precise determination of the location of the earthquake, it is preferable to use as many seismic stations as possible, i.e. as many onset times of the earthquake waves as possible (Figure 3). This carries the complication that not all the measured onset times can be modelled with any single hotspot location. Therefore, determining the location of an earthquake is also an interesting mathematical problem that today is usually solved by computer methods. It’s an inverse problem, similar to when we hear a sound we don’t know where it’s coming from and try to guess where the speaker is that’s making it.

**Computer methods**

To determine the location of the epicentre and the depth of the earthquake and the time of the earthquake, it is necessary to know how to calculate the theoretical times of the earthquake waves arriving at the observed stations, that is, to have the best possible model of seismic wave velocities. Then the procedure of determining the location of the hotspot is relatively simple using iterative methods. It starts with an arbitrary location of the focal point and calculates the difference between the theoretical and measured arrival times. Then the location is slightly changed in such a way as to reduce the specified difference, i.e. so that the theoretical times are as close as possible to the observed times. By repeating the procedure, you get closer and closer to the real focus of the earthquake.

In the 21st century, additional progress has been made in the development of methods for calculating the location of earthquakes. In order to avoid unreliability related to poor knowledge of ground velocities, focal locations for multiple earthquakes are calculated simultaneously. The most famous of them is the method of double differences, where the relative location of two earthquakes is calculated. The procedure is extremely useful for swarms of earthquakes with close foci and often allows the geometry of smaller faults to be determined. For a larger number of earthquakes, probabilistic methods are suitable, in which a simpler, direct problem is solved. In recent years, machine learning methods have been used more and more for the entire process: from the identification of earthquakes in the records, through the determination of onset times to the calculation of the location.

*Prepared by Dr. Marija Mustać Brčić*