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green_func.py
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green_func.py
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"""
Lamb's Problem, Second type.
Refer to Johnson(1974)： https://academic.oup.com/gji/article/37/1/99/678320
:copyright:
Nanqiao Du ([email protected]), April 2019
:license:
GNU General Public License, Version 3
(http://www.gnu.org/copyleft/gpl.html)
"""
import numpy as np
import matplotlib.pyplot as plt
from numpy import sqrt,pi,sin,cos
from numpy.polynomial.legendre import leggauss
def gaussint(func,a,b,param,args,nseg = 20):
'''
integrate over (a,b), with gauss-legendre quadrature
Parameters:
func : integrand function
a,b : integral interval
param: gauss legendre nodes and weights for x in range (-1,1)
args : additional parameters for func
nseg : divide integral interval into nseg segments
Returns:
s : s=\\int_{a}^{b}func(x)dx
'''
x0,w=param
n=len(x0)
x=(x0*(b-a)+b+a)/2
s=0.0
y=np.linspace(a,b,nseg)
s = 0.0
for i in range(len(y)-1):
x=(x0*(y[i+1]-y[i])+y[i]+y[i+1])/2
s=s+sum([func(x[_i],args)*w[_i] for _i in range(n)])*(y[i+1]-y[i])/2
#s=sum([func(x[i],args)*w[i] for i in range(n)])*(b-a)/2
return s
def greenmatrix(p,q,phi,a,b):
'''
Construct green matrix M,N for P and S wave,
and sigma,etaa,etab for calculation
Parameters:
p,q,theta,phi: see Johnson(1979)
a,b : alpha,beta, P and S wave velocity
Returns:
M,N,sigma,etaa,etab
'''
etaa=(0.0j+1/a**2+p**2-q**2)**0.5
etab=(0.0j+1/b**2+p**2-q**2)**0.5
gamma=etab**2+p**2-q**2
sigma=gamma**2+4*etab*etaa*(q**2-p**2)
M=np.zeros((3,3),dtype = np.complex)
M[0,0]=2*etab*((p**2+q**2)*cos(phi)**2-p**2)
M[0,1]=2*etab*(p**2+q**2)*sin(phi)*cos(phi)
M[0,2]=2*q*etab*etaa*cos(phi)
M[1,0]=M[0,1]*1.0
M[1,1]=2*etab*((p**2+q**2)*sin(phi)**2-p**2)
M[1,2]=2*q*etaa*etab*sin(phi)
M[2,0]=q*gamma*cos(phi)
M[2,1]=q*gamma*sin(phi)
M[2,2]=etaa*gamma
N=np.zeros((3,3),dtype=np.complex)
N[0,0]=1/etab*(etab**2*gamma-(gamma-4*etaa*etab)*((p**2+q**2)*sin(phi)**2-p**2))
N[0,1]=1/etab*(q**2+p**2)*(gamma-4*etaa*etab)*sin(phi)*cos(phi)
N[0,2]=-q*gamma*cos(phi)
N[1,0]=N[0,1]*1.0
N[1,1]=1/etab*(etab**2*gamma-(gamma-4*etaa*etab)*((p**2+q**2)*cos(phi)**2-p**2))
N[1,2]=-q*gamma*sin(phi)
N[2,0]=-2*q*etaa*etab*cos(phi)
N[2,1]=-2*q*etaa*etab*sin(phi)
N[2,2]=2*etaa*(q**2-p**2)
return M,N,sigma,etaa,etab
def inta(p,args):
'''
Construct integrand for P wave, on the Cagniard path
'''
theta,phi,r,a,b,rho,t=args
s=np.zeros((3,3))
if t>r*sqrt(1/a**2+p**2):
mu=rho*b**2
q=-t/r*sin(theta)+1j*(sqrt((t/r)**2-1/a**2-p**2)*cos(theta))
M,_,sigma,etaa,_=greenmatrix(p,q,phi,a,b)
sp=etaa/sigma/sqrt((t/r)**2-1/a**2-p**2)*M
s=sp.real/(pi**2*mu*r)
return s
def intb(p,args):
'''
Construct integrand for S wave, on the second part of Cagniard path
'''
theta,phi,r,a,b,rho,t=args
s=np.zeros((3,3))
if t>r*sqrt(1/b**2+p**2):
mu=rho*b**2
q=-t/r*sin(theta)+1j*(sqrt((t/r)**2-1/b**2-p**2)*cos(theta))
_,N,sigma,_,etab=greenmatrix(p,q,phi,a,b)
sp=etab/sigma/sqrt((t/r)**2-1/b**2-p**2)*N
s=sp.real/(pi**2*mu*r)
return s
def intb2(p,args):
'''
Construct integrand for S wave, on the branch cut
'''
theta,phi,r,a,b,rho,t=args
s=np.zeros((3,3))
if t<r*sqrt(1/b**2+p**2):
mu=rho*b**2
q=-t/r*sin(theta)+sqrt(-(t/r)**2+1/b**2+p**2)*cos(theta)
_,N,sigma,_,etab=greenmatrix(p,q,phi,a,b)
sp=etab/sigma/sqrt(-(t/r)**2+1/b**2+p**2)*N
s=sp.imag/(pi**2*mu*r)
return s
def green(theta,phi,r,a,b,rho,t,gauss_points):
'''
Compute g^H in time domain
Parameters:
theta,phi,r : source and receiver parameters
a,b,rho : media parameters
t : time range
Returns:
G : green's function, convolved with a Heaviside funciton,3*3 matrix
'''
nt=t.shape[0]
# allocate space
P=np.zeros((nt,3,3))
S=np.zeros((nt,3,3))
# Generate gauss-legendre points
param=leggauss(gauss_points)
# compute green's function
t2=r/a*sin(theta)+r*cos(theta)*sqrt(1/b**2-1/a**2)
for i in range(nt):
t0 = t[i]
print("computing green's function: %f%% "%((i + 1) / nt * 100))
pp=(t0/r)**2-(1/a)**2
pb=(t0/r)**2-(1/b)**2
p2=((t0/r-sqrt(1/b**2-1/a**2)*cos(theta))/sin(theta))**2-1/a**2
args=(theta,phi,r,a,b,rho,t0)
if pp>0:
P[i,:,:]=gaussint(inta,0,sqrt(pp),param,args)
if pb>0:
S[i,:,:]=gaussint(intb,0,sqrt(pb),param,args)
if sin(theta)>b/a:
if t0<r/b and t0>t2:
S[i,:,:] -= gaussint(intb2,0,sqrt(p2),param,args)
elif t0>r/b:
S[i,:,:] -= gaussint(intb2,sqrt(pb),sqrt(p2),param,args)
return P+S
def show_green(x,xs,r,a,b,theta,t,G):
'''
Show green's function g^H, only 11,13,22,31,33 components
'''
plt.figure(figsize=(6,6))
left=[0,2,1,0,2]
right=[0,0,1,2,2]
for i in range(5):
g=G[:,left[i],right[i]]
if i!=4:
plt.plot(t,g*1000+(5-i)*0.2,'k')
else:
plt.plot(t,g*1000+(5-i)*0.0,'k')
strtext='$g_{'+str(left[i]+1)+str(right[i]+1)+'}^{H}$'
strtitle='$\\mathbf{G}^{H}('
for j in range(3):
strtitle+=str(x[j])+','
strtitle+='t;'
for j in range(3):
strtitle+=str(xs[j])+','
strtitle+='0'+')$'
if i!=4:
plt.text(0.65,(5-i)*0.2,strtext,fontsize=16)
else:
plt.text(0.65,(5-i)*0.0,strtext,fontsize=16)
plt.title(strtitle)
plt.xlabel('t/s')
plt.yticks([])
plt.xlim(0.5,4)
plt.show()
def main():
# you could change parameters here
#=======================================================
# media parameters, here we don't convert it to SI units
x=[10.0,0,0] # receiver
xs=[0,0,2.0] # source , make sure xs[2] > 0
a=8.0 # vp
b=4.62 # vs
rho=3.3 # density
# time vector
dt=0.01 # time interval
t=np.arange(1,4+dt,dt)
# if xs[2] is small, please increase this number,
# or increase "nseg" in function `gaussint`
gauss_points=7
# end parameters
# ============================================================
# Utilize that the y component of source and field point are
# both 0, this simplify the calculation of theta and phi
#theta=np.arctan(x[0]/xs[2])
#phi=np.arctan2(xs[1],x[0])
r=sqrt(sum([(x[i]-xs[i])**2 for i in range(3)]))
r_plane = sqrt(sum([(x[i]-xs[i])**2 for i in range(2)]))
theta = np.arctan(r_plane / (-x[2] + xs[2]))
phi = np.arctan2(x[1] - xs[1],x[0] - xs[0])
# Compute and show g
G=green(theta,phi,r,a,b,rho,t,gauss_points)
show_green(x,xs,r,a,b,theta,t,G)
if __name__ == '__main__':
main()