Epidemic Type Earthquake Sequence (ETES) model  Seismicity rate = "background" + "aftershocks":  Magnitude distribution: uniform G.R. law with b=1 (Fig. 1a and Fig. 2)  Aftershocks: Omori law and increase of aftershock # with magnitude (Fig. 1)  Normalized spatial distribution of aftershocks: (1) “small” m <5.5 EQS (Figs. 3 and 4) - power -law - Gaussian with  Spatial distribution of aftershocks: (2) “large” m ≥ 5.5 EQS (Fig. 11) We use early aftershocks to predict the location of future ones:` with K(r) power law or Gaussian kernel Application for EQ forecasting  Forecasts: probability of occurrence of an EQ in each range of: - space: grid of 0.05 o x 0.05 o - time: updated each day for the next 24 hours - magnitude: forecasts for m ≥2 (or larger), magnitude bin of 0.1  Model optimization, testing and comparison - Log Likelihood: - non-stationary spatially variable Poisson process : predicted rate in bin i t,i x,i y,i m n : observed number of events - estimation of , k, p,  0 and f d. c is fixed to 5 minutes. - retrospective tests with data, maximization of LL - probability gain Observations: properties of triggered seismicity A/ Magnitude distribution and number of triggered earthquakes as a function of the ‘mainshock’ magnitudeB/ Spatial distribution of aftershocks, triggering distance as a function of the mainshock magnitude Comparison of short-term and long-term earthquake forecast models for Southern California Agnès Helmstetter 1, Yan Y. Kagan 2 and David D. Jackson 2 1 Lamont-Doherty Earth Observatory, Columbia University, New York 2 Department of Earth and Space Sciences, University of California Los Angeles Abstract We consider the problem of forecasting earthquakes on two different time scales: years, and days. We evaluate some published forecast models on these time scales, and suggest improvements in forecast models on both time scales. For time scales of several years, we modify the “smoothed seismicity” method of Kagan and Jackson [1994] by smoothing the locations of magnitude 2 and larger earthquakes. Kagan and Jackson used only magnitude 5 and larger. The new long-term model outperforms the best known published models in a retrospective test on magnitude 5 and larger, primarily because it has much higher spatial resolution. We have also developed a model to estimate daily earthquake probabilities in southern California, using the Epidemic Type Earthquake Sequence model [Kagan and Knopoff, 1987; Ogata, 1988; Kagan and Jackson 2000]. The forecasted seismicity rate is the sum of a constant external loading and of the aftershocks of all past earthquakes. The background rate is estimated by smoothing past seismicity. Each earthquake triggers aftershocks with a rate that increases exponentially with its magnitude and decreases with time following Omori's law. We use an isotropic kernel to model the spatial distribution of aftershocks for small (M≤5.5) mainshocks, and a smoothing of the location of early aftershocks for larger mainshocks. The model also assumes that all earthquake magnitudes follow the Gutenberg-Richter law with a uniform b-value. We use a maximum likelihood method to estimate the model parameters and test the short- term and long-term forecasts. A retrospective test using a daily update of the forecasts between 1985/1/1 and 2004/3/10 shows that the short-term model increases the average probability of an earthquake occurrence by a factor 11.5 compared to the long-term time-independent forecast. See forecasts and draft at Time-independent models: construction, testing and comparison Comparison with other long-term models: Likelihood results: Time-dependent forecasts using the ETES model: results Fig. 5. Declustered [Reasenberg, JGR 1985] ANSS catalog using m ≥3 and m ≥2 earthquakes. Fig. 6. Density of seismicity  (r) obtained by smoothing EQ locations using an isotropic adaptative power-law kernel K(r) ~1/(r 2 +d 2 ) 1.5 with d(r)=d o /√  (r) Fig. 7. EQ density for Kagan and Jackson [1994] model, obtained by smoothing m ≥ 5.5 earthquakes with an anisotropic power-law kernel K(r,  )~cos 2  /r. White circles are m ≥ 5 [ ] earthquakes. Fig. 8. Density of m ≥ 5 EQS for Frankel et al. [1997] model, obtained by smoothing m ≥3 earthquakes and adding characteristic earthquakes on known faults (m ≥ 6). White circles are m ≥ 5 [ ] earthquakes. Conclusions  Including small m ≥ 2 EQS improves long-term and short-term forecasts  Most ~70% earthquakes are “aftershocks” of m ≥2 EQ  Automatic, fast method to predict the spatial distribution of aftershocks - better than Coulomb static stress maps?  ETES improves our long-term model by a factor ~11 in average probability per EQ  fluctuations of aftershocks productivity not explained by our model - due to errors in magnitude? - introduce sequence specific parameters K,p,b as in STEP [Gerstenberger et al., 2004]?  Future work: introduce an upper cut-off in P(m) deduced from largest fault length? References  Frankel, A. et al. (1997), Seismic hazard maps for California, Nevada, and Western Arizona/Utah, U. S. Geological Survey Open-File Report  Gerstenberger, M., S. Wiemer, and L. Jones (2004) Real-time forecasts of tomorrow's earthquakes in California: a new mapping tool, U.S. Geological survey, Open file report  Helmstetter, A., Kagan, Y.Y. and D. D. Jackson (2005), Importance of small earthquakes for stress transfers and earthquake triggering, J. Geophys. Res. 110, B05S08, /2004JB  Kagan, Y.Y. and L. Knopoff (1987) Statistical short- term earthquake prediction, Science 236,  Kagan, Y.Y. and D. D. Jackson (1994), Long-term probabilistic forecasting of earthquakes, J. Geophys. Res. 99, 13,685-13,700.  Ogata, Y. (1988) Statistical models for earthquake occurrence and residual analysis for point processes, J. Amer. Statist. Assoc. 83,  Shearer, P., E. Hauksson, G. Lin and D. Kilb, Comprehensive waveform cross-correlation of southern California seismograms: Part 2. Event locations obtained using cluster analysis. Eos Trans. AGU, 84(46), Fall Meet. Suppl., Abstract S21D-0326, 2003 Acknowledgments: Fig. 11. Density of aftershocks estimated by smoothing the location of early (dt<2 hours) aftershocks (white circles) for Landers (left) and Hector-Mine (right) mainshocks. Black dots are aftershocks with 2 hours<dt<1 year. Fig. 13. (a) Observed (black) and predicted (pink) number of m ≥ 2 earthquakes per day in southern California. Blue line is the long- term rate  l.t. =10. The model parameters are  =0.8 (fixed, non constrained by maximizing LL), k=0.45, p=1.18 (Omori),  0 =2.81 (background rate) and f d =0.41 (width of the Gaussian kernel f(r)). (b) Probability gain per EQ G for the ETES model compared with the long-term model. The average gain is G=11. Fig. 12. (top) Forecasts for April 10 th. Average number of m ≥2 EQS predicted for the next day, and for each cell, using the ETES model, including background events (Fig. 6) and aftershocks. (bottom): ratio of the ETES and long-term seismicity rates. The predicted number of events for this day was =8.0, the number of observed events (black circles) was n=6. Fig. 9. Magnitude versus time for a few days in the ANSS catalog. A significant fraction of EQS that will occur in the next day [t p t p +T] may be triggered by EQS that will occur in the next day (t p <t<t p +T). Implication: using the “true” model under- estimates future rate, we thus use ETES with “effective” optimized parameters Fig. 10. Aftershocks magnitude versus time after (a) Joshua-Tree m=6.1, (b) San-Simeon m=6.5, and (c) Landers m=7.3.The solid line is an estimation of the completeness magnitude m c (m,t)=m log 10 (t). We use this relation to correct the forecasts for the effect of unreported small early aftershocks. Modeldata used to build the modelsdata used to test the modelsLLGain This work (Fig. 6)m≥ , m≥ m≥ (N=56) K.&J. [1994] (Fig. 7)m≥ m≥ (N=56)-463 This work (Fig. 6)m≥ , m≥ m≥ (N=12) Frankel et al. [1997] (Fig. 8)m≥ , m≥ m≥ (N=12)  (M<4)  (M>4) Mainshocks M=7-7.5 aftershocks foreshocks  (M<4)  (M>4) Mainshock M=2-2.5  =1.05 b=1 M=7 M=2 Omori p=0.9 Mainshocks M=7-7.5 Mainshock M=2-2.5 b=1 Fig. 1 (a) Average rate (M,t) of m≥2 earthquakes as a function of the time t after a triggering earthquake of magnitude (b) Aftershock productivity K(M) as a function of M (circles) and cumulative magnitude distribution P(m) (crosses). Fig. 2. Cumulative magnitude distribution of triggered earthquakes for different values of the mainshock magnitude M. All curves follow the Gutenberg-Richter law (black line) with b=1 independently of M, i.e., the magnitude of a triggered earthquake is not constrained by M. Fig. 3. Distribution of distances between foreshocks and mainshocks (dashed lines, 1 day before the mainshock) and mainshock-aftershocks pairs (solid lines, 1 day after), using the relocated catalog for Southern California by Shearer et al. [2003], with  z and  h <  (M). Fig. 4. Distance r b (M) (diamonds) at which the rate of events within 1 day after a mainshock of magnitude M is equal to the rate of activity of the day before. The average (blue) and median (circles) triggering distance, estimated for all aftershocks with r<r b (M), is very close to the mainshock rupture length (black line) L(M)= 0.01x10 0.5M corresponding to a constant stress drop  =3MPa Fall Meeting 2004 Paper Number: S23A-0287