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| /* Implementation of the formulae of Okada, 1985, "Surface deformation
due to shear and tensile faults in a half-space", Bulletin of the
Seismological Society of America, 75:4, 1135-1154, */
/* formulae (25)-(30) */
static void rectangular_source (const double U[3], double cosd, double sind,
double mulambda, double d,
double psi, double eta, double q,
double u[3])
{
double R = sqrt (psi*psi + eta*eta + q*q);
double X = sqrt (psi*psi + q*q);
double dtilde = eta*sind - q*cosd;
double ytilde = eta*cosd + q*sind;
double atanp = fabs (q) > 1e-6 ? atan (psi*eta/(q*R)) : 0.;
mulambda = mulambda/(1. + mulambda);
double logReta = R + eta > 1e-6 ? log (R + eta) : - log (R - eta);
double Reta = fabs (R + eta) > 1e-6 ? R + eta : 1e30;
double I1, I2, I3, I4, I5;
if (fabs (cosd) > 1e-6) {
/* formula (28) */
I5 = fabs (psi) < 1e-6 ? 0. :
mulambda*2./cosd*atan ((eta*(X + q*cosd) +
X*(R + X)*sind)/(psi*(R + X)*cosd));
I4 = mulambda/cosd*(log (R + dtilde) - sind*logReta);
I3 = mulambda*(1./cosd*ytilde/(R + dtilde) - logReta) + sind/cosd*I4;
I2 = mulambda*(- logReta) - I3;
I1 = mulambda*(-1./cosd*psi/(R + dtilde)) - sind/cosd*I5;
}
else {
/* formula (29) */
double R1 = R + dtilde;
I1 = - mulambda/2.*psi*q/(R1*R1);
I3 = mulambda/2.*(eta/R1 + ytilde*q/(R1*R1) - logReta);
I2 = mulambda*(- logReta) - I3;
I4 = - mulambda*q/R1;
I5 = - mulambda*psi*sind/R1;
}
/* strike-slip, formula (25) */
if (U[0] != 0.) {
double U1pi = U[0]/(2.*M_PI);
u[0] -= U1pi*(psi*q/(R*Reta) + atanp + I1*sind);
u[1] -= U1pi*(ytilde*q/(R*Reta) + q*cosd/Reta + I2*sind);
u[2] -= U1pi*(dtilde*q/(R*Reta) + q*sind/Reta + I4*sind);
}
/* dip-slip, formula (26) */
if (U[1] != 0.) {
double U2pi = U[1]/(2.*M_PI);
u[0] -= U2pi*(q/R - I3*sind*cosd);
u[1] -= U2pi*(ytilde*q/(R*(R + psi)) + cosd*atanp - I1*sind*cosd);
u[2] -= U2pi*(dtilde*q/(R*(R + psi)) + sind*atanp - I5*sind*cosd);
}
/* tensile, formula (27) */
if (U[2] != 0.) {
double U3pi = U[2]/(2.*M_PI);
u[0] += U3pi*(q*q/(R*Reta) - I3*sind*sind);
u[1] += U3pi*(-dtilde*q/(R*(R + psi)) -
sind*(psi*q/(R*Reta) - atanp) - I1*sind*sind);
u[2] += U3pi*(ytilde*q/(R*(R + psi)) +
cosd*(psi*q/(R*Reta) - atanp) - I5*sind*sind);
}
}
/* formula (24) */
static void okada_rectangular_source (const double U[3],
double L, double W, double d,
double delta, double mulambda,
double x, double y,
double u[3])
{
double cosd = cos (delta), sind = sin (delta);
double p = y*cosd + d*sind;
double q = y*sind - d*cosd;
u[0] = u[1] = u[2] = 0.;
rectangular_source (U, cosd, sind, mulambda, d,
x, p, q,
u);
rectangular_source (U, cosd, sind, mulambda, d,
x - L, p - W, q,
u);
double u1[3] = {0., 0., 0.};
rectangular_source (U, cosd, sind, mulambda, d,
x, p - W, q,
u1);
rectangular_source (U, cosd, sind, mulambda, d,
x - L, p, q,
u1);
u[0] -= u1[0];
u[1] -= u1[1];
u[2] -= u1[2];
}
static double dtheta (double theta1, double theta2)
{
double d = theta1 - theta2;
if (d > 180.) d -= 360.;
if (d < -180.) d += 360.;
return d;
}
struct Okada {
scalar d;
double x, y, depth;
double strike, dip, rake;
double mu, lambda;
double length, width, vU[3], U;
double R;
int (* iterate) (void);
bool flat;
};
void okada (struct Okada p)
{
// default settings
if (p.mu == 0.) p.mu = 1.;
if (p.lambda == 0.) p.lambda = 1.;
if (p.R == 0.) p.R = 6371220.; /* Earth radius (metres) */
double dtr = pi/180.;
if (p.rake != nodata) {
p.vU[0] = p.U*cos (p.rake*dtr);
p.vU[1] = p.U*sin (p.rake*dtr);
}
double sina = sin ((90. - p.strike)*dtr);
double cosa = cos ((90. - p.strike)*dtr);
double sind = sin (p.dip*dtr);
/* depth of the bottom edge */
double depth = sind > 0. ? p.depth + p.width*sind : p.depth;
/* origin to the centroid */
double x0 = p.length/2., y0 = p.width/2.*cos (p.dip*dtr);
foreach() {
if ( p.flat ) {
x -= p.x;
y -= p.y;
}
else {
x = p.R*cos(y*dtr)*dtheta(x, p.x)*dtr;
y = p.R*dtheta(y, p.y)*dtr;
}
double x1 = cosa*x + sina*y;
double y1 = - sina*x + cosa*y;
double oka[3];
okada_rectangular_source (p.vU, p.length, p.width, depth,
p.dip*dtr,
p.mu/p.lambda,
x0 + x1, y0 + y1,
oka);
val(p.d,0,0) = oka[2];
}
}
void fault (struct Okada p)
{
scalar hold[];
// save the initial water depth
scalar_clone (hold, h);
foreach()
hold[] = h[];
boundary ({hold});
p.d = h;
do {
okada (p);
// h[] now contains the Okada vertical displacement
foreach() {
// deformation is added to hold[] (water depth) only in wet areas
h[] = hold[] > dry ? max (0., hold[] + h[]) : hold[];
eta[] = zb[] + h[];
}
} while (p.iterate && p.iterate());
}
|
