To calculate the distribution of the seismic moment at a site, we assume that on a

Transcription

1 KAGAN : EARTHQUAKE SLIP DISTRIBUTION X - 1 Appendix A: Seismic Moment Distribution at a Site To calculate the distribution of the seismic moment at a site, we assume that on a fault zone of total length L the rate of earthquake with moment M 0 = N m (m = 7) or greater and rupture length L 0 = 37.5 km is equal to α 0 (see section 4.3). The complementary moment function is then = α 0L 0 L C 1 1 x 1/d φ(x)dx, M (A1) φ(x) is defined by equations (4), (5), (6), and (7) and C 1 1 is a normalizing coefficient. Similar to the moment-frequency relation, the function shows the number of fractures caused by an earthquake with moment M at a fault site. For the characteristic distribution, = α 0L 0 L ξd βd(m 0/M) ξ (M 0 /M xc ) ξ ] = 0 M > M xc, (A2) ξ = (βd 1)/d. Using equation 7 of Kagan 2002b], we can convert (A2) = 1 β ξd Ṁ 0 M β 0 M β 1 xc βd(m 0 /M) ξ (M 0 /M xc ) ξ ] (A3) Ṁ0 is the seismic moment rate on L 0.

2 X - 2 KAGAN : EARTHQUAKE SLIP DISTRIBUTION = α 0L 0 βη p L ξ (M 0 /M) ξ (M 0 /M xp ) ξ ] M xp > M M 0, = 0 M M xp, (A4) η p is defined by η p = M β xp M β xp M β 0, (A5) Alternately, = 1 β ξd Ṁ 0 M β 0 M β 1 xp (M 0 /M) ξ (M 0 /M xp ) ξ ] M xp > M M 0. (A6) If βd = 1, equations (A2) (A4) need to be modified. For the characteristic distribution, = α 0L 0 1 log(m L xc/m)] = 0 M > M xc, (A7) = α 0L 0 βη p L log(m xp /M) M xp > M M 0, = 0 M M xp, (A8)

3 KAGAN : EARTHQUAKE SLIP DISTRIBUTION X - 3 Appendix B: Slip Distribution The distribution density of the average slip is obtained from υ(m) (see equations (18) and (23)), as f(u) υ u d/(d 1)] u 1/(d 1). (B1) Inserting in (B1) appropriate expressions for seismic moment distribution and integrating to obtain a cumulative function, we obtain the following formula for the complementary function of displacement: Ψ(u) = C 1 f(x)dx. u (B2) To simplify the equations, we normalize the displacement by dividing them by u 0 = 1.87 m; u x is the slip corresponding to the maximum earthquake (normalized by dividing it by u 0 ). For the characteristic distribution, Ψ(u) = βd u ζ u ζ xc βd u ζ 0 u ζ xc, u xc > u u 0, Ψ(u) = 0 u u xc, (B3) ζ = βd 1 d 1 ; (B4) hence ζ 0 for d 1/β. For the truncated Pareto distribution of the seismic moment, Ψ(u) = u ζ u ζ xp u ζ 0 u ζ xp, u xp > u u 0, Ψ(u) = 0 u u xp. (B5)

4 X - 4 KAGAN : EARTHQUAKE SLIP DISTRIBUTION Ψ(u) = C 1 C g { Γ 1 βd 1, ( ) ] d/(d 1) u d + ( u ) (1 βd)/(d 1) exp ( u ) d/(d 1) ]}, (B6) Γ(a, x) is the incomplete gamma function Abramowitz and Stegun, 1972, p. 260] x P (x, a) = (Γ(a)) 1 e t t a 1 dt, a > 0, (B7) Γ(a) is the gamma function Abramowitz and Stegun, 1972, p. 260]. 0 C 1 ( = β d 1 βd ) ( u0 ) (1 βd)/(d 1) exp ( u0 ) d/(1 d) ], (B8) C g = C 1 Γ 1 βd 1, ( ) ] d/(1 d) u 0 d + βd. βd 1 (B9) If βd = 1, equations (B3), (B5), and (B6) need to be modified. For the characteristic distribution, Ψ(u) = d 1+log u xc log u d 1+log u xc log u 0, u xc > u u 0, Ψ(u) = 0 u u xc. (B10) Ψ(u) = log u xp log u log u xp log u 0, u xp > u u 0, Ψ(u) = 0 for u u xp. (B11) ( ) / ( ) u u0 Ψ(u) = E 1 E 1 u u 0, (B12) E 1 (z) is an exponential integral Abramowitz and Stegun, 1972, p. 228] E 1 (z) = t 1 e t dt. (B13) z We calculate this integral using the mathematica package Wolfram, 1999].

5 KAGAN : EARTHQUAKE SLIP DISTRIBUTION X - 5 Appendix C: Cumulative Slip Distribution at a Site The cumulative distribution of the total slip at a site can be expressed as follows: For the characteristic distribution of the seismic moment tensor, Φ(u) = C 2 u 1 ζ 1 u 1 ζ xc 1, u xc > u 1, Φ(u) = 1 u u xc, (C1) ζ = βd 1 d 1, (C2) see (B4) and C 2 = β 1 uζ 1 xc 1 βu ζ 1. (C3) xc For the truncated Pareto distribution of the seismic moment tensor, Φ(u) = u1 ζ 1 u 1 ζ xp 1, u xp > u 1, Φ(u) = 1 u u xp. (C4) see (B7) for the definition of the P function. Φ(u) = P 1 P 2 1 P 2, (C5) ( ( u ) ) d/(1 d) P 1 = P, 1 β, (C6) ( ( 1 ) ) d/(1 d) P 2 = P, 1 β, (C7)