Locating an Earthquake

What is modelled

An earthquake sends out a fast P wave and a slower S wave from the same point and the same moment. A station does not need to know when the quake happened: the delay between the two arrivals grows in proportion to the distance, so one S-P time gives one distance. Three or more distances fix the epicentre where the circles meet. Each pick has a small timing error, which becomes a distance error of the pick error divided by (1/Vs − 1/Vp), and the network geometry decides how that error becomes a long thin ellipse or a round one. The Wadati diagram plots S-P against the P arrival time and gives Vp/Vs as its slope and the origin time as its zero crossing.

Equations

• S-P time tS−P = r (1/Vs − 1/Vp), with r = √(d2 + h2) for epicentral distance d and depth h.
• Gauss-Newton least squares on ri = |x − si| with Jacobian rows (x − si) / ri.
• Covariance σr2 (JT J)−1, σr = σt / (1/Vs − 1/Vp); the 95 % ellipse has semi-axes 2.45 times the standard deviations.
• Wadati line tS − tP = (Vp/Vs − 1)(tP − t0).

Checks run in the page

• Exact S-P times returned to the true epicentre by the solver.
• Thousands of noisy locations against the linearised error ellipse.
• Halving the pick error against the expected fourfold drop in squared error.
• Wadati slope and origin time averaged over noisy sets.

Honest simplifications

The Earth is a uniform half-space with straight rays and constant wave speeds, the focal depth is known, picking errors are independent and Gaussian, and wrongly identified phases, site effects and the layered structure of a real volcano are left out.

• Press Stations on one side and watch the ellipse stretch.
• Press Sloppy picks and compare the location error.
• Press Event outside the network to see the error grow outside the stations.