David Brillinger and [Bruce Bolt] Objective Estimates of the Seismic Intensity of the 1755 Lisbon Earthquake David Brillinger and [Bruce Bolt] University of California, Berkeley

Introduction. Quantifying damage Intensity scales ordinal-valued Isoseismals Probabilistic approach Lisbon event n = 810

Methods. Grouped continuous model Latent r.v.  , cutpoints j  Y= II if  < II  Y= j if j-1 < < j if j = II,…,IX   = X if IX <  j increasing.

Objectives. 1. Automatic modelling method for ordinal-valued data Ordinal scale: “OK, good, very good, excellent” Can merge adjacent categories 2. Chance model for probabilistic risk assessment 3. Treat ordinal values as numerical?

Including explanatory X Spatial case ’X = (x,y),  smooth function extreme value distribution Prob{Y=j|(x,y)} = exp{-exp{ j-1+(x,y)}}-exp{-exp{j+(x,y)}} Generalized additive model

Results. Data: MSK intensity counts II II+ III III+ IV IV+ V V+ VI 5 11 27 128 114 104 48 111 VI+ VII VII+ VIII VIII+ IX IX+ X 59 84 27 29 17 13 11

Estimate of (x,y)

Estimate of j

- Assessment of fit. Prob{Y=j|Y>j-1,X} = 1 – exp{-exp{φj+ (x,y)}} Proportion of successes on y-axis vs. cells for lp: φj+ (x,y) on x-axis

Prob{Y=j|(x,y)} for j = X, VII, II

Dependence on distance.

Joyner-Boore type attenuation log(-log(1-Prob{Y=j|Y>j-1})) = j + d + log(d) d, distance

Difficulties for small d

Discussion and conclusions. Empirical relations for intensity probability as function of location Stochastic model – assessment, tests Limitations – other variables, e.g. regional and local geology Hypercenter error Uncertainties Other functional forms (small d’s)

REFERENCES. Martinez-Solares, J. M., Lopez-Arroyo, A. The great historical 1755 earthquake, effects and damages in Spain, J. Seismology 8, pp 275-294. 2004. Mendes-Victor, L., Baptista, M. A., Miranda, J. M., Miranda, P. M., Can Hydrodynamic Modelling of Tsunami Contribute to Seismic Risk Assessment? Phys. Chem. Earth 24, pp. 139-144. 1999. Brillinger, D. R. Earthquake risk and insurance, Environmetrics 4, pp. 1-21. 1993

Acknowledgements. J. M. Miranda J. M. Martinez-Solares The Organizers Beverley Bolt [Bruce Bolt] Errors are mine