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gmpe_as_2008_mod_wCD.m
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gmpe_as_2008_mod_wCD.m
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% coded by Yoshifumi Yamamoto, 5/6/10
% Stanford University
% yama4423@stanford.edu
% based on AS1997 by Jack W. Baker, 5/5/05
% Stanford University
% bakerjw@stanford.edu
%
% updated 2011/01/19
% Hanging Wall term
% from the Errata for “AS NGA” model (http://peer.berkeley.edu/products/abrahamson-silva_nga_report_files/AS08_NGA_errata.pdf)
% This version includes constant displacement effect
% updated 2010/05/20
% updated 2009/05/05
%
% Summary of the Abrahamson & Silva Ground-Motion Relations
% Norman Abrahamson, and Walter Silva
% Earthquake Spectra, Volume 24, No.1, pages 67-97, February 2008
%
% This script has been modified to correct an error based on the openSHA
% about constant displacement model(equation 22) and standard
% deviation(equation 24 and 26).
function [Sa, tsigma, period1, pga_rock, sigma, tau] = gmpe_as_2008_mod_wCD(M, Vs30, T, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, FVS30)
% Initialize struct V = constants
V = struct('period',0,'lin',0,...
'a1',0,'a2',0,'a3',0,'a4',0,'a5',0,'a8',0,'a10',0,'a12',0,'a13',0,'a14',0,'a15',0,'a16',0,'a18',0,...
'b',0,'c',0,'c1',0,'c2',0,'c4',0,'n',0,'s1',0,'s2',0,'s3',0,'s4',0,...
'ro',0,'v1',0,'e2',0,'a22',0,'Vs30s',0);
% Get info for rock conditions
Vrock = get_abrahamson_silva_constants(1,1100,FVS30,V); % Update constants for rock
pga_rock = exp(calc_val(M, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, 0, 1100, Vrock));
Vupdated = get_abrahamson_silva_constants(1,Vs30,FVS30,V);
[~, ~, ~, pga_sigmaB, pga_tauB] = abrahamson_silva_sigma(M, pga_rock, Vs30,0, 0, Vupdated);
% Get data for Td
Td = 10^( -1.25 + 0.3*M );
[SaTd] = AS_2008_nga_sub(M, 1100, Td, Rrup, Rjb, Rx, dip, Ztor, 0, W, FRV, FNM, FAS, FHW, FVS30, 0, 0, 1, V, pga_rock, pga_sigmaB, pga_tauB);
% Actual GMPM
[Sa, tsigma, period1, pga_rock, sigma, tau] = AS_2008_nga_sub(M, Vs30, T, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, FVS30, Td, SaTd, 0, V, pga_rock, pga_sigmaB, pga_tauB);
function [Sa, tsigma, period1, pga_rock, sigma, tau] = AS_2008_nga_sub(M, Vs30, T, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, FVS30, Td, SaTd, irock, V, pga_rock, pga_sigmaB, pga_tauB)
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INPUT
%
% M = moment magnitude
% T = period of vibration
% Use 1000 for output the array of Sa with period
% Rrup = closest distance to fault rupture (rupture distance)(km)
% Rjb = Joyner-Boore distance(km)
% Rx = Horizontal distance (km) from top edge of rupture
% dip = dip angle of fault in degree
% Ztor = Depth-to-top of rupture (km)
% Z10 = Depth to Vs=1.0km/s at the site(m)
% W = Down-dip rupture width (km)
% FRV = Flag for reverse faulting earthquakes
% = 1 for reverse and reverse/oblique earthquakes defined
% by rake angles between 30 and 150 degrees
% = 0 otherwise
%
% FNM = Flag for normal faulting earthquakes
% = 1 for normal earthquakes defined by rake angles
% between -60 and -120 degrees
% = 0 otherwise
%
% FAS = Flag for aftershocks
% = 1 for aftershocks
% = 0 for mainshocks, foreshocks, and swarms
%
% FHW = 1 for Hanging Wall sites
% = 0 otherwise
%
% FVS30 = 1 for estimated Vs30
% = 0 for measured Vs30
%
% ircok = 1 rock
% = 0 not rock
%
% OUTPUT
%
% Sa = median spectral acceleration prediction
% tsigma = logarithmic standard deviation of spectral acceleration
% prediction FOR AN ARBITRARY OR AVERAGE COMPONENT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% for the given period T, get the index for the constants
period = [ 0, -1, 0.01, 0.02, 0.03, 0.04, 0.05, 0.075, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 7.5, 10];
nT=length(T); % Length of input vector of vibration periods T
iflg=0;
if (nT==1 && T==1000); % If input period is scalar and if input period is 1000 sec
iflg=1;
nperi=length(period);
Sa=zeros(1,nperi-2);
tsigma=zeros(1,nperi-2);
period1=period(3:end);
for index=3:1:nperi;
% get constants for the given index value
V = get_abrahamson_silva_constants(index,Vs30,FVS30,V);
if period(index)<=Td || SaTd==0;
Sa(index-2) = exp(calc_val(M, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, pga_rock, Vs30, V) + (1-irock)* f_10(Z10, Vs30, V));
else
Sa(index-2) = exp(calc_val2(SaTd,Td, period(index), Z10, pga_rock, Vs30, V));
end;
[tsigma(index-2), sigma, tau, sigmaB, tauB] = abrahamson_silva_sigma(M, pga_rock, Vs30, pga_sigmaB, pga_tauB, V);
end;
end;
if(iflg==0);
Sa=zeros(1,nT);
tsigma=zeros(1,nT);
sigma = zeros(1,nT);
tau = zeros(1,nT);
period1=T;
for it=1:1:nT;
Teach=T(it); % Current value of period within input period vector
if Teach>period(end); Teach = period(end); end;
% interpolate between periods if neccesary
if all(Teach ~= period) % If current period not one of those considered in GMPM; use logical indexing
T_low = max(period(period<Teach)); % Avoid using find()
T_hi = min(period(period>Teach));
% Reduce calls to AS_2008_nga_sub
T_sub = [T_low T_hi];
[sa_sub, tsigma_sub, ~, ~, sigma_sub, tau_sub] = AS_2008_nga_sub(M, Vs30, T_sub, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, FVS30, Td, SaTd, irock, V, pga_rock, pga_sigmaB, pga_tauB);
sa_low = sa_sub(1,1); sa_hi = sa_sub(1,2);
tsigma_low = tsigma_sub(1,1); tsigma_hi = tsigma_sub(1,2);
sigma_low = sigma_sub(1,1); sigma_hi = sigma_sub(1,2);
tau_low = tau_sub(1,1); tau_hi = tau_sub(1,2);
x = [log(T_low) log(T_hi)];
Y_sa = [log(sa_low) log(sa_hi)];
Y_tsigma = [tsigma_low tsigma_hi];
Y_sigma = [sigma_low sigma_hi];
Y_tau = [tau_low tau_hi];
Sa(it) = exp(interp1(x,Y_sa,log(Teach)));
tsigma(it) = interp1(x,Y_tsigma,log(Teach));
sigma(it) = interp1(x,Y_sigma,log(Teach));
tau(it) = interp1(x,Y_tau,log(Teach));
else
index = find(abs((period - Teach)) < 0.0001); % Identify the period
% get constants for the given index value
V = get_abrahamson_silva_constants(index,Vs30,FVS30,V);
if period(index)<=Td || SaTd==0;
Sa(it) = exp(calc_val(M, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, pga_rock, Vs30, V) + (1-irock)* f_10(Z10, Vs30, V));
else
Sa(it) = exp(calc_val2(SaTd,Td, period(index), Z10, pga_rock, Vs30, V));
end;
[tsigma(it), sigma(it), tau(it), sigmaB, tauB] = abrahamson_silva_sigma(M, pga_rock, Vs30, pga_sigmaB, pga_tauB, V);
end;
end;
end;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% local functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [f1] = f_1(M, R, V)
% value of f1
if (M <= V.c1)
f1 = V.a1 + V.a4 * (M - V.c1) + V.a8 * (8.5 - M)^2 + (V.a2 + V.a3 * (M - V.c1)) * log(R);
else
f1 = V.a1 + V.a5 * (M - V.c1) + V.a8 * (8.5 - M)^2 + (V.a2 + V.a3 * (M - V.c1)) * log(R);
end
function [f4] = f_4(Rjb, Rx, dip, Ztor, M, W, V)
% value of f_4
if Rjb<30
T1=1-Rjb/30;
else
T1=0;
end;
W1=W*cos(dip);
if Rx<=W1;
T2=0.5+Rx/(2*W1);
elseif Rx>W1 || dip==90;
T2=1;
end;
if Rx>=Ztor;
T3=1;
else
T3=Rx/Ztor;
end;
if M<=6;
T4=0;
elseif M<7;
T4=M-6;
else
T4=1;
end;
% from paper AS08
% if dip>=70;
% T5=1-(dip-70)/20;
% else
% T5=1;
% end;
% from the Errata for “AS NGA” model (http://peer.berkeley.edu/products/abrahamson-silva_nga_report_files/AS08_NGA_errata.pdf)
if dip>=30;
T5=1-(dip-30)/60;
else
T5=1;
end;
f4 = V.a14*T1*T2*T3*T4*T5;
function [f5] = f_5(pga_rock, Vs30, V)
% value of f_5
if Vs30 < V.lin;
f5 = V.a10 * log(V.Vs30s/V.lin) - V.b*log(pga_rock+V.c) + V.b*log(pga_rock+V.c*((V.Vs30s/V.lin)^V.n));
else
f5 = (V.a10 + V.b*V.n) * log(V.Vs30s/V.lin);
end;
function [f6] = f_6(Ztor, V)
% value of f_6
if Ztor<10
f6=V.a16*Ztor/10;
else
f6=V.a16;
end;
function [f8] = f_8(Rrup, M, V)
% value of f_8
if M<5.5;
T6=1;
elseif M<=6.5;
T6=0.5*(6.5-M)+0.5;
else
T6=0.5;
end;
if Rrup<100
f8=0;
else
f8=V.a18*(Rrup-100)*T6;
end;
function [f10] = f_10(Z10, Vs30, V)
% value of f_10
if Vs30<180;
Z10h=exp(6.745);
elseif Vs30<=500;
Z10h=exp(6.745-1.35*log(Vs30/180));
else
Z10h=exp(5.394-4.48*log(Vs30/500));
end;
a211=(V.a10+V.b*V.n)*log(V.Vs30s/min(V.v1,1000));
a212=log((Z10+V.c2)/(Z10h+V.c2));
if Vs30>=1000;
a21=0;
elseif a211+V.e2*a212<0;
a21=-a211/a212;
else
a21=V.e2;
end;
f10=a21*a212;
if Z10>=200
f10=f10+V.a22*log(Z10/200);
end;
function [X] = calc_val(M, Rrup, Rjb, Rx, dip, Ztor, Z10, W, FRV, FNM, FAS, FHW, pga_rock, Vs30, constants)
% calculate predicted value
R = sqrt(Rrup^2 + constants.c4^2);
X = f_1(M, R, constants) + constants.a12*FRV + constants.a13*FNM + constants.a15*FAS ...
+ f_5(pga_rock, Vs30, constants) + FHW*f_4(Rjb, Rx, dip, Ztor, M, W, constants) + f_6(Ztor, constants) ...
+ f_8(Rrup, M, constants);
% + f_8(Rrup, M, constants) + f_10(Z10, Vs30, constants);
function [X] = calc_val2(SaTd, Td, T, Z10, pga_rock, Vs30, constants)
% calculate predicted value
X = log((SaTd) * Td^2 / T^2) - f_5(pga_rock, 1100, constants) + f_5(pga_rock, Vs30, constants) + f_10(Z10, Vs30, constants);
function [constants] = get_abrahamson_silva_constants(index,Vs30,FVS30,constants)
% get relevant constants
% arrays with values by index
period = [ 0, -1, 0.01, 0.02, 0.03, 0.04, 0.05, 0.075, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 7.5, 10];
lin = [ 865.1, 400.0, 865.1, 865.1, 907.8, 994.5, 1053.5, 1085.7, 1032.5, 877.6, 748.2, 654.3, 587.1, 503.0, 456.6, 410.5, 400.0, 400.0, 400.0, 400.0, 400.0, 400.0, 400.0, 400.0];
b = [ -1.186, -1.955, -1.186, -1.219, -1.273, -1.308, -1.346, -1.471, -1.624, -1.931, -2.188, -2.381, -2.518, -2.657, -2.669, -2.401, -1.955, -1.025, -0.299, 0.0, 0.0, 0.0, 0.0, 0.0];
a1 = [ 0.804, 5.7578, 0.811, 0.855, 0.962, 1.037, 1.133, 1.375, 1.563, 1.716, 1.687, 1.646, 1.601, 1.511, 1.397, 1.137, 0.915, 0.510, 0.192, -0.280, -0.639, -0.936, -1.527, -1.993];
a2 = [-0.9679, -0.9046, -0.9679, -0.9774, -1.0024, -1.0289, -1.0508, -1.0810, -1.0833, -1.0357, -0.9700, -0.9202, -0.8974, -0.8677, -0.8475, -0.8206, -0.8088, -0.7995, -0.7960, -0.7960, -0.7960, -0.7960, -0.7960, -0.7960];
a8 = [-0.0372, -0.12, -0.0372, -0.0372, -0.0372, -0.0315, -0.0271, -0.0191, -0.0166, -0.0254, -0.0396, -0.0539, -0.0656, -0.0807, -0.0924, -0.1137, -0.1289, -0.1534, -0.1708, -0.1954, -0.2128, -0.2263, -0.2509, -0.2683];
a10 = [ 0.9445, 1.5390, 0.9445, 0.9834, 1.0471, 1.0884, 1.1333, 1.2808, 1.4613, 1.8071, 2.0773, 2.2794, 2.4201, 2.5510, 2.5395, 2.1493, 1.5705, 0.3991, -0.6072, -0.9600, -0.9600, -0.9208, -0.7700, -0.6630];
a12 = [ 0.0000, 0.0800, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0181, 0.0309, 0.0409, 0.0491, 0.0619, 0.0719, 0.0800, 0.0800, 0.0800, 0.0800, 0.0800, 0.0800, 0.0800, 0.0800, 0.0800];
a13 = [-0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600, -0.0600];
a14 = [ 1.0800, 0.7000, 1.0800, 1.0800, 1.1331, 1.1708, 1.2000, 1.2000, 1.2000, 1.1683, 1.1274, 1.0956, 1.0697, 1.0288, 0.9971, 0.9395, 0.8985, 0.8409, 0.8000, 0.4793, 0.2518, 0.0754, 0.0000, 0.0000];
a15 = [-0.3500, -0.3900, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3500, -0.3191, -0.2629, -0.2230, -0.1668, -0.1270, -0.0708, -0.0309, 0.0000, 0.0000, 0.0000];
a16 = [ 0.9000, 0.6300, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.9000, 0.8423, 0.7458, 0.5704, 0.4460, 0.2707, 0.1463, -0.0291, -0.1535, -0.2500, -0.2500, -0.2500];
a18 = [-0.0067, 0.0000, -0.0067, -0.0067, -0.0067, -0.0067, -0.0076, -0.0093, -0.0093, -0.0093, -0.0083, -0.0069, -0.0057, -0.0039, -0.0025, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 0.0000];
s1e = [ 0.590, 0.590, 0.590, 0.590, 0.605, 0.615, 0.623, 0.630, 0.630, 0.630, 0.630, 0.630, 0.630, 0.630, 0.630, 0.630, 0.630, 0.615, 0.604, 0.589, 0.578, 0.570, 0.611, 0.640];
s2e = [ 0.470, 0.470, 0.470, 0.470, 0.478, 0.483, 0.488, 0.495, 0.501, 0.509, 0.514, 0.518, 0.522, 0.527, 0.532, 0.539, 0.545, 0.552, 0.558, 0.565, 0.570, 0.587, 0.618, 0.640];
s1m = [ 0.576, 0.576, 0.576, 0.576, 0.591, 0.602, 0.610, 0.617, 0.617, 0.616, 0.614, 0.612, 0.611, 0.608, 0.606, 0.602, 0.594, 0.566, 0.544, 0.527, 0.515, 0.510, 0.572, 0.612];
s2m = [ 0.453, 0.453, 0.453, 0.453, 0.461, 0.466, 0.471, 0.479, 0.485, 0.491, 0.495, 0.497, 0.499, 0.501, 0.504, 0.506, 0.503, 0.497, 0.491, 0.500, 0.505, 0.529, 0.579, 0.612];
s3 = [ 0.470, 0.420, 0.420, 0.420, 0.462, 0.492, 0.515, 0.550, 0.550, 0.550, 0.520, 0.497, 0.479, 0.449, 0.426, 0.385, 0.350, 0.350, 0.350, 0.350, 0.350, 0.350, 0.350, 0.350];
s4 = [ 0.300, 0.300, 0.300, 0.300, 0.305, 0.309, 0.312, 0.317, 0.321, 0.326, 0.329, 0.332, 0.335, 0.338, 0.341, 0.346, 0.350, 0.350, 0.350, 0.350, 0.350, 0.350, 0.350, 0.350];
ro = [ 1.000, 0.740, 1.000, 1.000, 0.991, 0.982, 0.973, 0.952, 0.929, 0.896, 0.874, 0.856, 0.841, 0.818, 0.783, 0.680, 0.607, 0.504, 0.431, 0.328, 0.255, 0.200, 0.200, 0.200];
c1=6.75;
c4=4.5;
a3=0.265;
a4=-0.231;
a5=-0.398;
n=1.18;
c=1.88;
c2=50;
constants.period = period(index);
constants.lin = lin(index);
constants.a1 = a1(index);
constants.a2 = a2(index);
constants.a3 = a3;
constants.a4 = a4;
constants.a5 = a5;
constants.a8 = a8(index);
constants.a10 = a10(index);
constants.a12 = a12(index);
constants.a13 = a13(index);
constants.a14 = a14(index);
constants.a15 = a15(index);
constants.a16 = a16(index);
constants.a18 = a18(index);
constants.b = b(index);
constants.c = c;
constants.c1 = c1;
constants.c2 = c2;
constants.c4 = c4;
constants.n = n;
if FVS30==1;
constants.s1 = s1e(index);
constants.s2 = s2e(index);
else
constants.s1 = s1m(index);
constants.s2 = s2m(index);
end;
constants.s3 = s3(index);
constants.s4 = s4(index);
constants.ro = ro(index);
T=period(index);
if index==2;
constants.v1 = 862;
elseif T<=0.5;
constants.v1 = 1500;
elseif T<=1;
constants.v1 = exp(8.0-0.795*log(T/0.21));
elseif T<2;
constants.v1 = exp(6.76-0.297*log(T));
else
constants.v1 = 700;
end;
if T<0.35 || Vs30>1000;
constants.e2=0;
elseif T<=2;
constants.e2=-0.25*log(Vs30/1000)*log(T/0.35);
else
constants.e2=-0.25*log(Vs30/1000)*log(2/0.35);
end;
if T<2;
constants.a22=0;
else
constants.a22=0.0625*(T-2);
end;
constants.Vs30s=min(Vs30,constants.v1);
function [tsigma, sigma, tau, sigmaB, tauB] = abrahamson_silva_sigma(M, pga_rock, Vs30, pga_sigmaB, pga_tauB, V)
% calculate the sigma
% use the published coefficients for the geometric mean
if M<5;
sigma0 = V.s1;
elseif M<=7;
sigma0 = V.s1 + (V.s2-V.s1)/2 * (M-5);
else
sigma0 = V.s2;
end;
sigmaAMP=0.3;
sigmaB=sqrt(sigma0^2-sigmaAMP^2);
if M<5;
tau0 = V.s3;
elseif M<=7;
tau0 = V.s3 + (V.s4-V.s3)/2 * (M-5);
else
tau0 = V.s4;
end;
tauB=tau0;
if Vs30>=V.lin;
term1 = 0;
else
% from openSHA
term1 = V.b * pga_rock * ( (-1/(pga_rock+V.c)) + (1/(pga_rock + V.c*((Vs30/V.lin)^V.n))) );
end;
% from openSHA
sigma = sqrt(sigma0^2 + term1^2 * pga_sigmaB^2 + 2*term1 * sigmaB*pga_sigmaB*V.ro);
tau = sqrt(tau0^2 + term1^2 * pga_tauB^2 + 2*term1 * tauB *pga_tauB *V.ro);
tsigma = sqrt(sigma^2 + tau^2);