*Earth Planets Space,* Vol. 63 (No. 3), pp. 275-287, 2011

doi:10.5047/eps.2010.08.007

## Conventional N-, L-, and R-tests of earthquake forecasting models without simulated catalogs

Masajiro Imoto^{1}, David A. Rhoades^{2}, Hiroyuki Fujiwara^{1}, and Naoko Yamamoto^{1}

^{1}National Research Institute for Earth Science and Disaster Prevention, 3-1 Ten-nodai, Tsukuba-shi, Ibaraki-ken 305-0006, Japan

^{2}GNS Science, 1 Fairway Drive, Avalon, Lower Hutt 5010, P.O. Box 30-368, Lower Hutt, 5040 New
Zealand

(Received January 12, 2010; Revised July 27, 2010; Accepted August 24, 2010; Online published March 4, 2011)

**Abstract: **
We propose a new procedure for testing the expected number (N-test), log likelihood (L-test), and log likelihood-ratio (R-test) of seismicity models. In these tests, scores obtained from observed earthquakes are compared with distributions of scores estimated from earthquakes expected from the models under test. We introduce a method to estimate the test score distributions analytically where uncertainties in magnitude and hypocentral parameters are involved. The analytical formulas used to estimate expected values and standard deviations of the test scores for earthquakes conforming to the test models were derived in earlier published studies, which allowed calculation of normal approximations by which to test score distributions. Using these two methods simultaneously, we can perform N-, L-, and R-tests for seismicity models without using any simulated catalogs. As a case study, the proposed procedure was applied to two seismicity models for Kanto, central Japan. To compare our procedure with the current one based on the Monte Carlo method, we randomly generated sets of 10,000 earthquake catalogs of two kinds: one set conforming to the model under test, and the other derived from the observed catalog allowing for uncertainties in magnitude and hypocentral parameters. The distributions of L-scores obtained from both sets are in good agreement with those obtained by the proposed procedure. This comparison suggests that the analytical approach presented here could be useful for conducting the N-, L-, and R-tests in a conventional way.

**Key words: **
Seismicity models, probabilistic prediction, CSEP project, N-, L-, and R-scores, uncertainties in earthquake source parameters, simulated catalogs.

Corresponding author E-mail: imoto@bosai.go.jp

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