Yan Y. Kagan Dept. Earth and Space Sciences, UCLA, Los Angeles, CA , Testing spatial distribution forecast of point processes with application to seismology

CMT catalog: Shallow earthquakes,

The table below and accompanying plots are calculated on 2007/ 4/19 at midnight Los Angeles time. The last earthquake with scalar seismic moment M>=10^17.7 Nm (Mw>=5.8) entered in the catalog occurred in the region 0.0 > LAT. > -60.0, > LONG. > on 2007/ 4/16 at latitude and longitude , Mw = 6.42 ____________________________________________________________________ LONG-TERM FORECAST | SHORT-TERM Probability Focal mechanism | Probability Probability M>5.8 T-axis P-axis M>5.8 ratio eq/day*km^2 Pl Az Pl Az eq/day*km^2 Time- Longitude | | | Rotation Time- dependent/ | Latitude | | | angle dependent independent v v v v degree ……………………………………………………………………………………………………… E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 ………………………………………………………………………………………………………

NW Pacific -- Here we demonstrate forecast effectiveness: displayed earthquakes (108 events) occurred after smoothed seismicity forecast had been calculated.

SW Pacific -- Here we demonstrate forecast effectiveness: displayed earthquakes (170 events) occurred after smoothed seismicity forecast had been calculated.

1080 events (10 times the actual number of events in ) have been simulated, using the forecast density. Some points overlap; in low activity area we specified some low-level density to avoid surprises.

Long-term Forecast Efficiency Evaluation We simulate synthetic catalogs using smoothed seismicity map. Likelihood function for simulated catalogs and for real earthquakes in the time period of forecast is computed. If the `real earthquakes’ likelihood value is within 2.5— 97.5% of synthetic distribution, the forecast is considered successful. Kagan, Y. Y., and D. D. Jackson, Probabilistic forecasting of earthquakes, Geophys. J. Int., 143,

Cumulative likelihood score distribution Questions: 1. How fast approaches Gaussian; 2. Standard error behavior; 3. Influence of cell size

Earthquake potential in and around China: Estimated from past earthquakes, Yufang Rong and David D. Jackson, GRL, 29(16), 2002.

Kossobokov, Testing earthquake prediction methods: ``The West Pacific short-term forecast of earthquakes with magnitude MwHRV \ge 5.8", Tectonophysics, 413(1-2), See also Kagan & Jackson, TECTO, 2006, pp

Testing alarm-based earthquake predictions J. Douglas Zechar and Thomas H. Jordan Geophys. J. Int. (2008) 172, 715 – 724 Results of Molchan trajectory/area skill score analysis for PI (squares) and NSHM (triangles) relative to the RI reference model. Top panel shows complete Molchan trajectories for both predictions and bottom panel shows corresponding area skill score curve. Each plot also shows the α = 1, 5 and 10 per cent critical boundaries.

Calculation of Information Score from EQ density table (log-likelihood) Cell No Long Lat EQ density Cell Probab. EQ Gain EQ size

Error diagram (ED), NW Pacific

Error diagram, SW Pacific

Calculation of Information Score from EQ density table (log-likelihood)

Calculation of Information Score from EQ density table (log-likelihood), I= Several other log-likelihood scores can be calculated: (1)Gain for cells where earthquakes occurred (I=2.3625); (2) Gain for actual EQ location (I=2.4014); (3) Gain for simulated events (I=2.3609); (4) Info score for EQ numbers in Error Diagram (I=3.0922) – Kagan, PAGEOPH, These scores converge if cells are becoming small, or the EQ number increases.

Distribution of log-likelihood score Blue -- SW Red -- NW

Two segments ED

Two segments ED, dependence on slope of the first segment Magenta – standard error; Blue –coefficient of skewness; Green – coefficient of kurtosis.

World seismicity: 1990 – 2000 (PDE)

Error diagram with CMT and PDE concentrations Magenta – CMT ; Cyan – PDE ; Red – forecast based on CMT ; Blue – EQs 04-06; Green – simulation; Black – Poisson rate.

Error diagram with forecast used as template (one simulation) Magenta – CMT ; Cyan – PDE ; Red – forecast based on CMT ; Blue – EQs 04-06; Green – simulation; Black – Poisson rate.

Error diagram with forecast used as template (10 simulations) Magenta – CMT ; Cyan – PDE ; Red – forecast based on CMT ; Blue – EQs 04-06; Green – simulations; Black – Poisson rate.

Molchan, G., and V. Keilis-Borok, Earthquake Prediction: Probabilistic Aspect, 21 pages, Typical examples of \phi (strategy) that are used at the research stage are max(n, \tau) and sum (n + \tau).

END Thank you

Abstract We propose a new method to test the effectiveness of spatial point process forecast based on a log-likelihood score for a predicted point density and the information gain for events that actually occurred in the test period. The method largely avoids simulation use and allows to calculate the information score for each event as well as the standard error of each forecast. As the number of predicted events increases, the score distribution approaches the Gaussian law. The degree of its similarity to the Gaussian distribution can be measured by the computed coefficients of skewness and kurtosis. To display the forecasted point density and the point events we use an event concentration diagram or a variant of the Error Diagram (ED ).

Abstract (cont.) We demonstrate the application of the method by using our long-term forecast of seismicity in two west Pacific regions. We compare the Error Diagram (ED) for these regions with simplified diagrams based on two-segment approximations. Since in these regions earthquakes are concentrated in narrow subduction belts, the use of the forecast density as a template for the ED is a more convenient display technique. We also show, using simulated event occurrence, that some proposed criteria for measuring forecast effectiveness at EDs, would be strongly biased for small number of events.

Frequency plot for likelihood score distribution

Distribution of log-likelihood score Blue -- SW Red -- NW

Two segments ED