Volcano Seismic Network

What is modelled

An earthquake sends out a faster P wave and a slower S wave. The delay between their arrivals at a station grows in proportion to the distance, so one delay gives one distance. With several stations the distances define circles (spheres in three dimensions) whose intersection is the source. Picking errors blur the intersection, and the blur depends strongly on the network geometry.

Equations

• Delay tS − tP = r (1/VS − 1/VP) with r = √(Δx2 + Δy2 + Δz2).
• Least squares by Levenberg-Marquardt on the three unknown coordinates.
• Error prediction C = σ2 (JTJ)−1 with J = (1/VS − 1/VP) ∂r/∂x.
• Poisson ratio ν = (k2 − 2) / (2 (k2 − 1)) with k = VP/VS.

Checks run in the page

• The delay converted back to distance, and the Poisson ratio from two routes.
• The noise-free fit recovering the true position.
• The scatter of 80 noisy fits against the covariance prediction.
• The linear growth of the delay with distance.

Honest simplifications

The ground is a uniform half-space with straight rays, whereas a volcano has strong velocity contrasts, topography and a magma body that bends and delays waves; the stations lie at one height; picking errors are independent Gaussian numbers with one standard deviation; and only S minus P delays are used, not absolute arrival times.

• Press Three stations only: the delays fix the position, but the error is large.
• Press Event outside the ring and compare the predicted error with the first preset.
• Raise the picking error and watch the yellow cloud grow.