Leave it to we Californians to be utterly nonplussed by tonight’s “moderate earthquake.” The event in question, registering 5.6 on the Richter Scale, took place about 17 miles from my home this evening at 8:04pm:

Those of us who live near the Hayward Fault, which runs to the East of Silicon Valley, have known for a while that we are “overdue” for a quake (with apologies to the Wizard of Odds—yes, of Odds—for flirting with the Gambler’s Fallacy). [Aside: anyone remember that great '80s Bond film with the man-made Silicon Valley earthquake?]

When the quake hit, I was having dinner with two colleagues from work, one of whom had been around during the much more serious Loma Prieta quake in 1989. That one measured about 7.1 on the Richter Magnitude Scale. Most people know that the Richter scale is a logarithmic scale, but many don’t realize that the scale actually provides two different output values: the physical displacement at the fault location goes up by a factor of 10 (i.e., the log is to the base 10) for each single unit of increase, but the energy released goes up by a factor of 32 for each single unit increase. Thus the ‘89 Loma Prieta quake was approximately 181 times more energetic than today’s quake.

A truly astonishing amount of information was immediately available about today’s quake, thanks to automated processing and reporting of seismic events. For example, here’s a map of all the recent earthquakes in California from the USGS. (The large blue rectangle is, of course, the quake in question. Note the smaller aftershocks.)

The detailed report on the quake confirms that the sophistication of automated earthquake analysis is truly impressive. Want to know whether there’s a tsunami risk? Read the Tsunami Message from WCATWC. (No risk.) Or how many people in your neighborhood felt it? Read this map. (84 in Mountain View.) Or what the ground looks like above the epicenter? Check this out:

This is all particularly interesting to my colleagues and to me because the mathematics used to locate earthquakes (by seismic triangulation) are pretty much identical to the mathematics use at ShotSpotter to locate gunfire (by acoustic triangulation). Both techniques are fundamentally based on the difference in time of arrival of a transient signal (the seismic or acoustic wave) at sensors located in different places. Based on this difference in time of arrival (also known as TDOA, or time difference of arrival), a series of hyperbolae can be plotted, and the intersection of these hyperbolae will be the origin of the transient. Why hyperbolae? Hyperbolae are “curves of constant difference in distance” between two points, or foci. Wolfram has an excellent article explaining them. So, geometrically, if you have two different points F₁ and F₂, then there is a hyperbola is the set of all points whose distance from F₁ and F₂ always differs by a specific constant, which we can call k. In the diagram below, for example, the difference between the distances r1 and r2 will equal k, as will the difference between the distances r3 and r4.

The physical interpretation of this k is straight forward: it is the difference (in time) between when someone standing at F₁ and someone standing at F₂ would hear a noise which originated somewhere on the hyperbola.

If you have only two points (F₁ and F₂ ), then you have a single hyperbola. But if you have three points (F₁, F₂ and F₃ ), then of course you have three hyperbolae, reflecting the difference in time of arrival at the three points (F₁/F₂ , F₂/F₃ , and F₁/F₃ ). This diagram, from Suruj Dutta’s site explaining the technical underpinnings of location-based services, shows how cell phone triangulation works and gets it mostly right, although it only shows two of the three hyperbolae:

Moreover, the reflections and “echoes” caused by different geologic layers of the earth are quite similar to the reflections and echoes caused by the complex urban terrain in which ShotSpotter systems are deployed.