Earthquake Recurrence Law and the Weibull Distribution

Synonyms

Earthquake; Weibull distribution

Introduction

Earthquakes are unpredictable given the difficulty of measuring the stresses and strains on a fault plane (Geller et al. 1997). The situation reflects the reality that recent catastrophic events went unpredicted, such as the 2008 Wenchuan earthquake in China and the 2011 Japan earthquake. Under the circumstances, a few alternatives for earthquake hazard mitigation are developed and practiced, including earthquake early warning, seismic hazard analysis, and earthquake statistics analysis (e.g., Hsiao et al. 2011; Chen et al. 2013; Wang et al. 2011, 2012a, b; Wu and Kanamori 2008).

With statistical models, earthquake probability can be best estimated accounting for the inevitable earthquake uncertainty in nature. One of the famous examples is the use of the Poisson model to estimate the earthquake probability with time. However, besides the support from mainly engineering judgments (earthquakes are rare), more tangible support was...

Appendix: VBA Scripts of the In-House Weibull Functions Calibrating Model Parameters

As mentioned, the two in-house functions are basically the same except for returning α or β. Therefore, the script for calculating α is only given in the following.

Public Function weibull_afa(mn, sd)

'**************************************

'This function to calculate afa of Weibull

'mn = mean value of X

'sd = standard deviation of X

'inc = increment of afa

'k0 = initial value of afa

'k and c = interim variables

'**************************************

inc = 0.005

k0 = 0.01

k = 0

Do

k = k + 1

afa1 = inc * k + k0

beta1 = mn / gamma_function(1 + 1 / afa1)

c1 = beta1 ^ 2 * (gamma_function(1 + 2 / afa1) - (gamma_function(1 + 1 / afa1)) ^ 2)

c1 = Abs(c1 - sd ^ 2)

afa2 = inc * (k + 1) + k0

beta2 = mn / gamma_function(1 + 1 / afa2)

c2 = beta2 ^ 2 * (gamma_function(1 + 2 / afa2) - (gamma_function(1 + 1 / afa2)) ^ 2)

c2 = Abs(c2 - sd ^ 2)

If c2 > c1 Then

Exit Do

End If

Loop

weibull_afa = afa1

End Function