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seogi_macbook
119 lines
3.8 KiB
Python
119 lines
3.8 KiB
Python
import numpy as np
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from scipy.constants import mu_0, pi
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from scipy import special
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def DCAnalyticHalf(txloc, rxlocs, sigma, earth_type="wholespace"):
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"""
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Analytic solution for electric potential from a postive pole
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:param array txloc: a xyz location of A (+) electrode (np.r_[xa, ya, za])
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:param list rxlocs: xyz locations of M (+) and N (-) electrodes [M, N]
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e.g.
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rxlocs = [M, N]
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M: xyz locations of M (+) electrode (np.c_[xmlocs, ymlocs, zmlocs])
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N: xyz locations of N (-) electrode (np.c_[xnlocs, ynlocs, znlocs])
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:param float or complex sigma: values of conductivity
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:param string earth_type: values of conductivity ("wholsespace" or "halfspace")
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"""
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M = rxlocs[0]
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N = rxlocs[1]
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rM = np.sqrt( (M[:,0]-txloc[0])**2 + (M[:,1]-txloc[1])**2 + (M[:,2]-txloc[1])**2 )
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rN = np.sqrt( (N[:,0]-txloc[0])**2 + (N[:,1]-txloc[1])**2 + (N[:,2]-txloc[1])**2 )
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phiM = 1./(4*np.pi*rM*sigma)
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phiN = 1./(4*np.pi*rN*sigma)
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phi = phiM - phiN
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if earth_type == "halfspace":
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phi *= 2
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return phi
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deg2rad = lambda deg: deg/180.*np.pi
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rad2deg = lambda rad: rad*180./np.pi
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def DCAnalyticSphere(txloc, rxloc, xc, radius, sigma, sigma1, \
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field_type = "secondary", order=12, halfspace=False):
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# def DCSpherePointCurrent(txloc, rxloc, xc, radius, rho, rho1, \
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# field_type = "secondary", order=12):
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"""
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Parameters:
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:param array txloc: A (+) current electrode location (x,y,z)
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:param array xc: x center of depressed sphere
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:param array rxloc: M(+) electrode locations / (Nx3 array, # of electrodes)
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:param float radius: radius (float): radius of the sphere (m)
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:param float rho: resistivity of the background (ohm-m)
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:param float rho1: resistivity of the sphere
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:param string field_type: : "secondary", "total", "primary"
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(default="secondary")
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"secondary": secondary potential only due to sphere
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"primary": primary potential from the point source
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"total": "secondary"+"primary"
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:param float order: maximum order of Legendre polynomial (default=12)
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Written by Seogi Kang (skang@eos.ubc.ca)
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Ph.D. Candidate of University of British Columbia, Canada
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"""
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Pleg = []
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# Compute Legendre Polynomial
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for i in range(order):
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Pleg.append(special.legendre(i, monic=0))
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rho = 1./sigma
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rho1 = 1./sigma1
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# Center of the sphere should be aligned in txloc in y-direction
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yc = txloc[1]
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xyz = np.c_[rxloc[:,0]-xc, rxloc[:,1]-yc, rxloc[:,2]]
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r = np.sqrt( (xyz**2).sum(axis=1) )
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x0 = abs(txloc[0]-xc)
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costheta = xyz[:,0]/r * (txloc[0]-xc)/x0
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phi = np.zeros_like(r)
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R = (r**2+x0**2.-2.*r*x0*costheta)**0.5
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# primary potential in a whole space
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prim = rho*1./(4*np.pi*R)
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if field_type =="primary":
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return prim
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sphind = r < radius
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out = np.zeros_like(r)
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for n in range(order):
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An, Bn = AnBnfun(n, radius, x0, rho, rho1)
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dumout = An*r[~sphind]**(-n-1.)*Pleg[n](costheta[~sphind])
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out[~sphind] += dumout
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dumin = Bn*r[sphind]**(n)*Pleg[n](costheta[sphind])
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out[sphind] += dumin
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out[~sphind] += prim[~sphind]
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if halfspace:
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scale = 2
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else:
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scale = 1
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if field_type == "secondary":
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return scale*(out-prim)
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elif field_type == "total":
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return scale*out
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def AnBnfun(n, radius, x0, rho, rho1, I=1.):
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const = I*rho/(4*np.pi)
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bunmo = n*rho + (n+1)*rho1
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An = const * radius**(2*n+1) / x0 ** (n+1.) * n * \
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(rho1-rho) / bunmo
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Bn = const * 1. / x0 ** (n+1.) * (2*n+1) * (rho1) / bunmo
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return An, Bn
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