# Space–Time Earthquake Prediction: The Error Diagrams

- 145 Downloads
- 16 Citations

## Abstract

The quality of earthquake prediction is usually characterized by a two-dimensional diagram *n* versus *τ*, where *n* is the rate of failures-to-predict and *τ* is a characteristic of space–time alarm. Unlike the time prediction case, the quantity *τ* is not defined uniquely. We start from the case in which *τ* is a vector with components related to the local alarm times and find a simple structure of the space–time diagram in terms of local time diagrams. This key result is used to analyze the usual 2-d error sets {*n*, *τ* _{ w }} in which *τ* _{ w } is a weighted mean of the *τ* components and *w* is the weight vector. We suggest a simple algorithm to find the (*n*, *τ* _{ w }) representation of all random guess strategies, the set *D*, and prove that there exists the unique case of *w* when *D* degenerates to the diagonal *n* + *τ* _{ w } = 1. We find also a confidence zone of *D* on the (*n*, *τ* _{ w }) plane when the local target rates are known roughly. These facts are important for correct interpretation of (*n*, *τ* _{ w }) diagrams when we discuss the prediction capability of the data or prediction methods.

## Keywords

Prediction earthquake dynamics statistical seismology## Notes

### Acknowledgements

This work was supported by the Russian Foundation for Basic Research through grant 08-05-00215. I thank D.L. Turcotte for useful discussions, which have stimulated the writing of the present paper.

## References

- Bolshev, L. N. and Smirnov, V. N.
*Tables of Mathematical Statistics*(Nauka, Moscow 1983).Google Scholar - Harte, D., and Vere-Jones, D. (2005),
*The entropy score and its uses in earthquake forecasting*, Pure Appl. Geophys.*162*, 1229–1253.Google Scholar - Hanssen, A. W., and Kuiper, W. J. A. (1965),
*On the relationship between the frequency of rain and various meteorological parameters*, Modedeelingen en Verhandelingen, Royal Notherlands Meteorological Institute, 81.Google Scholar - Jolliffe, I. T. and Stephenson, D. B. (eds.),
*Forecast Verification: A Practitioner’s Guide in Atmospheric Science*(John Wiley & Sons, Hoboken 2003).Google Scholar - Kagan, Y. Y. (2007),
*On earthquake predictability measurement: information score and error diagram*, Pure Appl. Geophys.*164*, 1947–1962.Google Scholar - Keilis-Borok, V. I. and Soloviev, A. A. (eds.),
*Nonlinear Dynamics of the Lithosphere and Earthquake Prediction*(Springer-Verlag, Berlin-Heidelberg 2003).Google Scholar - Kossobokov, V. G. (2005),
*Earthquake prediction: principles, implementation*, Perspect. Comput. Seismol.*36*-1, 3–175, (GEOS, Moscow).Google Scholar - Marzocchi, W., Sandri, L., and Boschi, E. (2003),
*On the validation of earthquake-forecasting models: The case of pattern recognition algorithms*, Bull. Seismol. Soc. Am.*93*, 5, 1994–2004.Google Scholar - Molchan, G. M. (1990),
*Strategies in strong earthquake prediction*, Phys. Earth Planet. Inter.*61*(1–2), 84–98.Google Scholar - Molchan, G. M. (1991),
*Structure of optimal strategies of earthquake prediction*, Tectonophysics*193*, 267–276.Google Scholar - Molchan, G. M. (1997),
*Earthquake prediction as a decision making problem*, Pure Appl. Geophys.*149*, 233–247.Google Scholar - Molchan, G. M.,
*Earthquake prediction strategies: A theoretical analysis*. In*Nonlinear Dynamics of the Lithosphere and Earthquake Prediction*(eds. Keilis-Borok, V.I. and Soloviev, A.A.) (Springer-Verlag, Berlin-Heidelberg 2003), pp. 209–237.Google Scholar - Molchan, G. M., and Kagan, Y. Y. (1992),
*Earthquake prediction and its optimization*, J. Geophys. Res.*97*, 4823–4838.Google Scholar - Molchan, G. M. and Keilis-Borok, V. I. (2008),
*Earthquake prediction: Probabilistic aspect*, Geophys. J. Int.*173*, 1012–1017.Google Scholar - Shcherbakov, R., Turcotte, D. L., Holliday, J. R., Tiampo, K. F., and Rundle, J. B. (2007),
*A Method for forecasting the locations of future large earthquakes: An analysis and verification*, AGU, Fall meeting 2007, abstract #S31D-03.Google Scholar - Shen, Z.-K., Jackson, D. D., and Kagan, Y. Y. (2007),
*Implications of geodetic strain rate for future earthquakes, with a five-year forecast of M 5 Earthquakes in Southern California*, Seismol. Res. Lett.*78*(1), 116–120.Google Scholar - Swets, J. A. (1973),
*The relative operating characteristic in psychology*, Science*182*, 4116, 990–1000.Google Scholar - Tiampo, K. F., Rundle, J. B., McGinnis, S., Gross, S., and Klein, W. (2002),
*Mean field threshold systems and phase dynamics: An application to earthquake fault systems*, Europhys. Lett.*60*(3), 481–487.Google Scholar - Zechar, J.D. and Jordan, Th., H. (2008),
*Testing alarm-based earthquake predictions*, Geophys. J. Int.*172*, 715–724.Google Scholar