Characteristics of Ground Motion

ATL Rule

2 : RMS of displacement ( y ) between two points with a distance L after a time T.

A is site-dependent;10-16~10-21 m/sec.

ATL at Esashi

L = 50m

T=1979.6 ~ 1994.4

A=(0.33±0.05)・ 10-18m/s

National Astronomical Observatory Mizusawa

a random walk(or Brownian) process

ATL at HERA6km long tunnel under the city

GM Measurements

GM Model1. 2 dimentional power spectrum

P(,k) = lim lim 1/T 1/L∬x(t,s)e-ite-iks dt ds

P(,k) = A/2/k2 (1-cos(kB/A/ 2)) + Di()Ui(,k)

where, Ui(,k)=2/√k2cut - k2, k2

cut = /vi, vi: phase velociy

Di()=ai/(1+[di(- i)/ i]4), fi=2i

2. Displacement2 = <x(t,s)2> = ∬ P (,k) d dk /(2)2

3. Relative Displacement

2 (T,L) = ∬ P (,k) 2(1-cosT)2(1-coskL) d dk /(2)2

T→∞ 　 L→∞ -T/2 -L/2

T/2 L/2

-∞ -∞

∞ ∞

-∞ -∞

∞ ∞

GM mode (2)l

2 (T,L) = ∫ (,L) 2(1-cosT) d /(2)

(,L) = ∫P(,k)2(1-coskL) dk/(2p)

　　　 = AL/ 2 if 0 < < o

= B/ 4 if o < < ∞

where, o = (B/AL)1/2

2 (T,L) = ATL at 0 < < o

(,L) = 2 P() { 1 - Re[N12(,L)]}= 2 P() R (,L)

where, Re(N12(,L)) is correlation;

e.g. N12(,L) = Jo( L/vi) for waves; vi kcut=

-∞

∞

Parameters of P(,k) in ILC-TRCParameter Units Model A Model B Model C

A of ATL m/s 1.0・ 10-19 0.5・ 10-18 1.0・ 10-17

B of ATL m2/s3 0.5・ 10-18 1.0・ 10-18 5.0・ 10-18

f1 Hz 0.001 0.001 0.14

a1 m2/Hz 1.0・ 10-9 1.0・ 10-9 1.0・ 10-11

d1 1 1 5

v1 m/s 3・ 103 vapp(f) vapp(f)

f2 Hz 0.2 0.2 2.5

a2 m2/Hz 3.5・ 10-13 3.5・ 10-9 1.0・ 10-15

d2 3.5 3.5 1.5

v2 m/s 3・ 103 vapp(f) vapp(f)

f3 Hz 5.0 4.5 50.0

a3 m2/Hz 1.0・ 10-21 2.5・ 10-20 1.0・ 10-19

d3 1.3 3.5 1.5

v3 m/s 3・ 103 vapp(f) vapp(f)

Vapp(f) = 450 + 1900 exp(-f/2) m/s

GM Model Specta

Give up region

GM Power Spectra and Models

GM Model at Tsukuba

a1,2,3= 1 ・ 10-8 , 3 ・ 10-16, 5 ・ 10-17 m2/Hz

N. Uchida, Tsukuba univ.

GL=0m

GL=-36m

scaled

fitted to GL-36m data