James’s Musings

thoughts, photography, and geeky stuff
from an unrelentingly curious Silicon Valley entrepreneur

What’s Shaking?

by James G. Beldock on October 30, 2007

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:

Map of Earthquake Location

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.

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