# The Algorithm M8

The Algorithm M8 was designed by retroactive analysis of the seismicity preceding the greatest (M8+) earthquakes worldwide, hence its name. It is based on a simple physical scheme of prediction, which can be briefly described as follows:

1. Prediction is aimed at earthquakes of magnitude M0 and above. We consider different values of M0 with a step 0.5. Overlapping circles with the diameter D(M0) scan the seismic territory. Within each circle the sequence of earthquakes is considered with aftershocks removed {ti, mi, hi, bi(e)}, i = 1, 2 … Here ti is the origin time, ti <= t i + 1; mi is the magnitude, hi is focal depth, and bi(e) is the number of aftershocks during the first e days. The sequence is normalized by the lower magnitude cutoff Mmin(С), С being the standard value of the average annual number of earthquakes in the sequence. The magnitude scale we use should reflect the size of the earthquake sources. Accordingly, MS usually is taken for larger magnitudes while mb is used for smaller ones. For many catalogs, using the maximal reported magnitude could set this up, (we do so in the case of the NEIC GHDB).

2. Several running averages are computed for this sequence in the sliding time windows (t – s, t) and magnitude range M0>=Mi >=Mmin(С). They depict different measures of intensity in earthquake flow, its deviation from the long-term trend, and clustering of earthquakes. These averages include: N(t), the number of main shocks; L(t), the deviation of N(t) from the long-term trend, L(t) = N(t) – Ncum(t-s)x(t-t0)/(t-s-t0), Ncum(t) being the cumulative number of main shocks with M >= Mmin(С) from the beginning of the sequence t0 to t; Z(t), linear concentration of the main shocks estimated as the ratio of the average diameter of the source, l, to the average distance, r, between them; and B(t) = max{bi}, the maximal number of aftershocks (a measure of earthquake clustering). The earthquake sequence {i} is considered in the time window (t – s’, t) and in the magnitude range (M0 – p, M0 – q). Each of the functions N, L, Z is calculated for С = 20 and С = 10. As a result, the earthquake sequence is given a robust averaged description by seven functions: N, L, Z (twice each), and B.

3. “Very large” values are identified for each function using the condition that they exceed Q percentiles (i.e., they are higher than Q% of the encountered values).

4. An alarm or a TIP, “time of increased probability”, is declared for five years, when at least six out of seven functions, including B, become “very large” within a narrow time window (t – u, t). To stabilize prediction, this condition is required for two consecutive moments, t and t+0.5 years.

The following standard values of parameters indicated above are prefixed in the algorithm M8: D(M0)={exp(M0- 5.6)+1}0 in degrees of meridian (this is 384 km, 560 km, 854 km and 1333 km for M0 = 6.5, 7.0, 7.5 and 8 respectively), s = 6 years, s‘ = 1 year, g = .5, p = 2, q = .2, u = 3 years, and Q = 75% for B and 90% for the other six functions. The running averages are defined in a robust way, so that a reasonable variation of parameters does not affect the predictions.

The stabilised M8 (M8S) algorithm

Supplementary Materials