It is empirically known that the estimation of the depth of an earthquake is less reliable than the estimation of the epicenter. For example, for my PhD thesis, I assumed that the uncertainty in depth was about three times as high as the uncertainty in horizontal location, as indicated by earlier work on earthquakes in Greece in the 1980s. It's good to know this, but why exactly does it happen?
It's all about the method routinely used to jointly estimate epicenters, depths and occurrence times. It was devised by Geiger (1910) and a description of it without the math goes like this: Start with an initial guess for the epicenter and the depth. The location of the station in which the seismic waves were first observed will do. Also, an initial depth of a few km will do. Now compare the theoretical seismic wave arrival times according to your initial guess with the actual, observed seismic wave arrival times. This will indicate, as explained later, how you should correct your initial guess. Repeat with your new guess. It is very unlikely that all your observations will be satisfied perfectly, but at some point all your observations will be satisfied equally and no improvement will be possible using this algorithm. At this point you have the best estimates for the epicenter and the depth.
Now, how do the corrections work? Let's say that at station X the P waves arrive 3s earlier than predicted by the initial guess. This means that the epicenter is closer to X than you thought on the seismic ray, and by knowing the speed of the P waves you can tell by how much. If at station Y the P waves arrive 2s later than predicted, then the epicenter is farther away from Y than you thought on the seismic ray and, again, you can estimate by how much. But why does this generally work fine with the two horizontal coordinates but not with depth?
One reason is that as all seismometers are on the surface of the Earth, for an earthquake occurring inside the area covered by the network, the epicenter will be constrained better than the depth, as there will be stations all around the epicenter but no stations bellow the earthquake's focus. Similarly, when locating an earthquake at a great distance from the network of seismometers, its location along the direction from the network to the epicenter is not constrained very well. Its location along the direction perpendicular to that, is not usually that bad.
It also turns out that because of the way seismic waves propagate in the Earth, a small change in the epicenter has a small effect on the arrival time, while a small change in depth has an even smaller effect on the arrival time. And as earthquake location works the other way round, a small error in arrival time, no matter what its source may be, has a great effect on the depth. In fact, at certain distances, you can't even tell if the correction in depth has to be positive or negative (See Lomnitz 2006). For an explanation of this in terms of travel time curves, have a look at the following figure (translated from Papazachos et al. 2005). The depth step is 100km, which is large for areas like Greece, where there are no earthquakes below 160km or so, but you get the idea. At large epicentral distances, the travel time curves converge. Only stations near the epicenter, which, remember, is still an unknown, can provide information on the depth of the earthquake.
Actually, there is a way to accurately determine the depth of an earthquake as long as it is not based on joint determination of epicenter, depth and occurrence time using arrival times. This is done with the Wadati method which uses pP waves, which are not routinely marked at seismological centers, so it has to happen at a later time. pP waves, are P waves that were reflected once at the surface of the Earth. The point of reflection is relatively close to the epicenter. This means that the difference in arrival times between pP and P changes slowly with distance but rapidly with depth. Tables with focal depth as a function of the pP-P time interval and the epicentral distance have been published and they are the most reliable way to estimate the focal depth.
One reason is that as all seismometers are on the surface of the Earth, for an earthquake occurring inside the area covered by the network, the epicenter will be constrained better than the depth, as there will be stations all around the epicenter but no stations bellow the earthquake's focus. Similarly, when locating an earthquake at a great distance from the network of seismometers, its location along the direction from the network to the epicenter is not constrained very well. Its location along the direction perpendicular to that, is not usually that bad.
It also turns out that because of the way seismic waves propagate in the Earth, a small change in the epicenter has a small effect on the arrival time, while a small change in depth has an even smaller effect on the arrival time. And as earthquake location works the other way round, a small error in arrival time, no matter what its source may be, has a great effect on the depth. In fact, at certain distances, you can't even tell if the correction in depth has to be positive or negative (See Lomnitz 2006). For an explanation of this in terms of travel time curves, have a look at the following figure (translated from Papazachos et al. 2005). The depth step is 100km, which is large for areas like Greece, where there are no earthquakes below 160km or so, but you get the idea. At large epicentral distances, the travel time curves converge. Only stations near the epicenter, which, remember, is still an unknown, can provide information on the depth of the earthquake.
Actually, there is a way to accurately determine the depth of an earthquake as long as it is not based on joint determination of epicenter, depth and occurrence time using arrival times. This is done with the Wadati method which uses pP waves, which are not routinely marked at seismological centers, so it has to happen at a later time. pP waves, are P waves that were reflected once at the surface of the Earth. The point of reflection is relatively close to the epicenter. This means that the difference in arrival times between pP and P changes slowly with distance but rapidly with depth. Tables with focal depth as a function of the pP-P time interval and the epicentral distance have been published and they are the most reliable way to estimate the focal depth.
References
Geiger, L.(1910). Herdbestimmung bei Erdbeben aus den Ankunftszeiten, K. Ges.Wiss. Goett., 4, 331–349.
Lomnitz, C. (2006). Three theorems of earthquake location, Bull. Seism. Soc. Am. 96, 306-312.
Papazachos, B., Karakaisis, G., Hatzidimitriou, P. (2005). Introduction to Seismology, Ziti Publications, Thessaloniki, Greece, 520pp. (In Greek).
Geiger, L.(1910). Herdbestimmung bei Erdbeben aus den Ankunftszeiten, K. Ges.Wiss. Goett., 4, 331–349.
Lomnitz, C. (2006). Three theorems of earthquake location, Bull. Seism. Soc. Am. 96, 306-312.
Papazachos, B., Karakaisis, G., Hatzidimitriou, P. (2005). Introduction to Seismology, Ziti Publications, Thessaloniki, Greece, 520pp. (In Greek).
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