Time slice movies of Miyagi Earthquake

Bob Clapp and Jon Claerbout

See earlier results before low-pass filtering.

First we take the Miyagi earthquake and low-pass filter it. Now bin it onto a regular mesh. We have done some trace scaling.

Time slice movie Data(x,y,t)

The data will fill an outline of Japan with north to the right.

Changing numbers are seconds. P-waves are an expanding ring. S-waves are an expanding ring at about 2/3 the radius. The Rayleigh wave closely follows the S-wave and is stronger. Without the low-pass filtering you would see random numbers and not be able to recognize the expanding wavefronts. The noisy appearance of the original movie results from the large receiver separation compared to the wavelength. Temporal hi-cut filtering has lowered the visible wavelength.

Moved out data(x,y, tau=t*r0/r)

We stretch the time axes of the data so that the p-wave should arrive in approximately one burst all over Japan. A little while later, likewise should the S wave. A little while later, likewise should the Rayleigh wave. The stretching transformation, however, is very simple and only approximately achieves this goal. The time stretching transformation depends only on the radial distance of the receiver from the epicenter and approximately "stops" radial motion in the viewing surface.

We had a bug, the motion we saw was mostly lateral (perpendicular to the direction of propagation) which we interpreted as lateral velocity variation. We are not ready to interpret the new result.

Now let us look at some details of the math of the time stretching function. It will explain a few other features of the movie. People often time shift for the first arrival, but since there are three main arrivals of interest that would require three separate movies. Instead we do something like normal moveout (time-variable stretching). The time-distance curves for these waves are approximately (very approximately) straight lines thru the origin. This suggests a transformation from t to tau: tau = t*r_mid/r where r is radial distance of a receiver from the epicenter, and r_mid is any distance about half way across Japan. Seismograms at distance r_mid are unstretched. Closer ones are stretched while further ones compressed. At constant tau, t/r is constant. As the movie progresses, the linear stepout t/r progresses. Each movie frame is a frame of constant tau.

Arrival bursts are spread out a little in tau because event travel time curves have some curvature in t(r). If we had set the time origin perfectly, the first arrival would be simultaneously at r=0 and r_max. Mid distance arrivals come a little later in tau. So, you see the whole of Japan illuminated by the p-wave in a very short span of viewing time tau. Likewise for S and R. We also seem to see that the northern island Hokaido arrivals are unexpectedly late.

Future work

There's lots more to do on this earthquake data. We're looking for somebody else to do it because we've other things we need to do.