Complex Earthquake Sequences Algorithm
The Virtual Seismologist (VS) method is an algorithm developed for seismic early warning systems using a Bayesian approach (Cua, G., and Heaton, T., 2007). The VS method makes use of ratios of available ground motion amplitudes (as early as 3 seconds after the first P wave arrival to a station) for estimating magnitude. We are trying to answer the question: what if we thought we had only one earthquake, but “suddenly” we realized that we have two (foreshock/mainshock pair)? Another way of putting the question would be: how can we distinguish between a foreshock and a mainshock in an ongoing earthquake? How would VS work on it? And how could we improve it?
Envelopes of channels of ground motion such as acceleration, velocity and displacement (horizontal and vertical components) are input to the VS method. For this work, we combine (numerically add) different magnitude earthquakes that occurred in close proximity in space to simulate foreshock/mainshock pairs. Because information must be obtained in real time, the ground motion envelopes should be monitored for deviation from the envelopes expected from a single event. If there is a foreshock/mainshock pair then the predicted envelope from the proposed foreshock location and magnitude will significantly diverge from the recorded data. Once the perturbation is greater than a specified threshold, the system will realize that something is not right.
In order to discover the relative size and timing of a foreshock/mainshock pair, we propose to use P/S discriminants (whether a wave is a P- or an S- wave) (Cua 2004, Caltech Ph.D.) introduced for detection of the phases. To determine if a phase is a P- or as an S- wave, ratios of vertical to horizontal ground motions are used with the help of linear discriminant analysis. It is imperative to be able to distinguish between the phases because P- and S- waves have different envelope attenuation relationships. Once we determine what the station is detecting, we will proceed with predicting new envelopes with magnitudes computed using the ratios of the ground motions recorded after the detection of the greater perturbation. Once the deviation between the initial predicted envelope and the observed exceeds the threshold, our proposed algorithm commences:
We combine the predicted envelopes in a “square root of sum of squares” sense, so we subtract the initial predicted envelope from the observed envelope:
New Observed Envelope = √( (Observed Envelope)^2 – (Initial Predicted Envelope)^2 )
Then using the initial values of the “New Observed Envelope” from the previous step, we predict a new magnitude using the equation given in Cua 2005:
M = -0.59log(acc) + 1.51log(disp) + 8.94
Note that the VS method needs a distance (R) and a magnitude (M) to predict envelopes. For the distance input, we assume the mainshock took place around the same location as the foreshock so we use the R predicted in the foreshock part of the problem. Then with the M estimation from the equation above and R are used to come up with a new predicted envelope for the rest of the earthquake. Then, we keep monitoring the perturbation between the observed and the predicted envelopes for the rest of the record.
Thomas Heaton, Maren Böse, Gokcan Karakus