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What does it mean that the earthquake occurred at a depth of 0 km? How can an earthquake have a negative depth? What is the geoid, and what does it have to do with earthquake depth?

An earthquake cannot physically occur at a depth of 0 km or -1km (above the surface of the earth). In order for an earthquake to occur, two blocks of crust must slip past one another, and it is impossible for this to happen at or above the surface of the earth. So why do we report that the earthquake occurred at a depth of 0 km or event as a negative depth sometimes?

First of all, the depth of an earthquake is usually the most difficult part of the location to nail down with great accuracy.  Since most earthquakes are deep within the crust, an error of +/- 1 or 2 km is irrelevant; in other words, it is a small error when the depth is something like 13 km.  If the earthquake depth is relatively shallow, however, it becomes more of an issue.  A negative depth can sometimes be an artifact of the poor resolution for a shallow event.

For quarry blasts that are recorded by the seismic network, the depth is fixed at 0 km since we can never determine a precise depth for these, but we know they are very close to the surface.

Sometimes because of the density of the seismic network and the close proximity of the seismic stations to an earthquake epicenter, we are able to determine a very precise depth.  When the earthquake depth is very shallow, it can be reported as a negative depth.  This is where the geoid comes in; our earthquake depths are calculated using the geoid as a reference (or as the o km baseline) rather than the actual surface of the earth, which can vary.  A virtual representation of the geoid looks rather like a misshapen rock instead of a perfect sphere or ellipse.  The reference geoid that we currently use is called WGS84.