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anisotropic_potential.py
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import numpy as np
from numpy import fft
from scipy.interpolate import RectBivariateSpline as Interpolator
class Halfspace:
def __init__(self, sig1, sig2, sig3, alpha, beta, gamma):
"""
sig1, sig2, sig3, alpha, beta, gamma
"""
self._sig1 = sig1
self._sig2 = sig2
self._sig3 = sig3
sina = np.sin(alpha)
cosa = np.cos(alpha)
sinb = np.sin(beta)
cosb = np.cos(beta)
sing = np.sin(gamma)
cosg = np.cos(gamma)
sxx = (sig1*cosa**2*cosb**2
+ sig2*(sina*cosg-sinb*sing*cosa)**2
+ sig3*(sina*sing+sinb*cosa*cosg)**2)
sxy = (sig1*sina*cosa*cosb**2
- sig2*(sina*cosg-sinb*sing*cosa)*(sina*sinb*sing+cosa*cosg)
+ sig3*(sina*sing+sinb*cosa*cosg)*(sina*sinb*cosg-sing*cosa))
sxz = (-sig1*sinb*cosa
- sig2*(sina*cosg-sinb*sing*cosa)*sing
+ sig3*(sina*sing+sinb*cosa*cosg)*cosg)*cosb
syy = (sig1*sina**2*cosb**2
+ sig2*(sina*sinb*sing+cosa*cosg)**2
+ sig3*(sina*sinb*cosg-sing*cosa)**2)
syz = (-sig1*sina*sinb
+ sig2*(sina*sinb*sing+cosa*cosg)*sing
+ sig3*(sina*sinb*cosg-sing*cosa)*cosg)*cosb
szz = (sig1*sinb**2
+ sig2*sing**2*cosb**2
+ sig3*cosb**2*cosg**2)
self._sxx = sxx
self._sxy = sxy
self._sxz = sxz
self._syy = syy
self._syz = syz
self._szz = szz
def phi(self, x, y, z=None):
"""Potential at x, y, z due to unit current source at the origin."""
x = np.array(x)
shape = x.shape
x = x.reshape(-1)
y = np.array(y).reshape(-1)
if z is None:
z = np.zeros_like(x)
else:
z = np.array(z).reshape(-1)
assert(len(x) == len(y) and len(y) == len(z))
sig = np.array([[self._sxx, self._sxy, self._sxz],
[self._sxy, self._syy, self._syz],
[self._sxz, self._syz, self._szz]])
rho = np.linalg.inv(sig)
scale = 1/(2*np.pi*np.sqrt(self._sig1*self._sig2*self._sig3))
xyzRxyz = np.sqrt(rho[0, 0]*x*x + rho[1, 1]*y*y + rho[2, 2]*z*z +
2*(rho[0, 1]*x*y + rho[0, 2]*x*z + rho[1, 2]*y*z))
phi = scale/xyzRxyz
phi = phi.reshape(shape)
return phi
def voltage(self, A, B, M, N, current=1):
"""
Voltage difference at M & N, due to current sources at A and B.
A : Tuple of numpy arrays for x and y position of current source
B : Tuple of numpy arrays for x and y position of current sink
M : Tuple of numpy arrays for x and y position of positive electrode
N : Tuple of numpy arrays for x and y position of negative electrode
"""
MAx = M[0]-A[0]
MAy = M[1]-A[1]
NAx = N[0]-A[0]
NAy = N[1]-A[1]
MBx = M[0]-B[0]
MBy = M[1]-B[1]
NBx = N[0]-B[0]
NBy = N[1]-B[1]
MA_potential = self.phi(MAy, MAx)
MB_potential = self.phi(MBy, MBx)
NA_potential = self.phi(NAy, NAx)
NB_potential = self.phi(NBy, NBx)
volts = MA_potential-NA_potential-MB_potential+NB_potential
return volts*current
class LayeredHalfspace:
def __init__(self, z, sig1, sig2, sig3, alpha, beta, gamma):
"""
z, sig1, sig2, sig3, alpha, beta, gamma.
Bam!
"""
z = np.array(z)
self._z = z
self._h = z[1:]-z[:-1]
assert np.all(self._h >= 0)
n_layers = len(z)
self.n_layers = n_layers
sig1 = np.array(sig1)
sig2 = np.array(sig2)
sig3 = np.array(sig3)
alpha = np.array(alpha)
beta = np.array(beta)
gamma = np.array(gamma)
self._sig1 = sig1
self._sig2 = sig2
self._sig3 = sig3
self._alpha = alpha
self._beta = beta
self._gamma = gamma
assert len(sig1) == n_layers
assert len(sig2) == n_layers
assert len(sig3) == n_layers
assert len(alpha) == n_layers
assert len(beta) == n_layers
assert len(gamma) == n_layers
sina = np.sin(alpha)
cosa = np.cos(alpha)
sinb = np.sin(beta)
cosb = np.cos(beta)
sing = np.sin(gamma)
cosg = np.cos(gamma)
sxx = (sig1*cosa**2*cosb**2
+ sig2*(sina*cosg-sinb*sing*cosa)**2
+ sig3*(sina*sing+sinb*cosa*cosg)**2)
sxy = (sig1*sina*cosa*cosb**2
- sig2*(sina*cosg-sinb*sing*cosa)*(sina*sinb*sing+cosa*cosg)
+ sig3*(sina*sing+sinb*cosa*cosg)*(sina*sinb*cosg-sing*cosa))
sxz = (-sig1*sinb*cosa
- sig2*(sina*cosg-sinb*sing*cosa)*sing
+ sig3*(sina*sing+sinb*cosa*cosg)*cosg)*cosb
syy = (sig1*sina**2*cosb**2
+ sig2*(sina*sinb*sing+cosa*cosg)**2
+ sig3*(sina*sinb*cosg-sing*cosa)**2)
syz = (-sig1*sina*sinb
+ sig2*(sina*sinb*sing+cosa*cosg)*sing
+ sig3*(sina*sinb*cosg-sing*cosa)*cosg)*cosb
szz = (sig1*sinb**2
+ sig2*sing**2*cosb**2
+ sig3*cosb**2*cosg**2)
self._sxx = sxx
self._sxy = sxy
self._sxz = sxz
self._syy = syy
self._syz = syz
self._szz = szz
def phi_tilde(self, wx, wy):
"""
Calculates phi_tilde for unitary source at the surface.
wx, wy
"""
ix0 = np.where(wx == 0)[0]
iy0 = np.where(wy == 0)[0]
wx, wy = np.meshgrid(wx, wy)
wx[ix0, iy0] = 1.0
wy[ix0, iy0] = 1.0
sxx = self._sxx
sxy = self._sxy
sxz = self._sxz
syy = self._syy
syz = self._syz
szz = self._szz
h = self._h
N = self.n_layers
a = szz[-1]
b = sxz[-1]*wx+syz[-1]*wy
c = sxx[-1]*wx*wx+2*sxy[-1]*wx*wy+syy[-1]*wy*wy
al = np.sqrt(a*c-b*b)
Q = 1/(al)
for i in range(N-2, -1, -1):
a = szz[i]
b = sxz[i]*wx+syz[i]*wy
c = sxx[i]*wx*wx+2*sxy[i]*wx*wy+syy[i]*wy*wy
al = np.sqrt(a*c-b*b)
l = al/a
tanh = np.tanh(h[i]*l)
alQ = al*Q
Q = (alQ+tanh)/(al*(alQ*tanh+1))
Q[ix0, iy0] = 0
return Q/(2*np.pi)
def phi_tilde_J(self, wx, wy):
"""
Calculates phi_tilde for unitary source at the surface.
wx, wy
"""
ix0 = np.where(wx == 0)[0]
iy0 = np.where(wy == 0)[0]
wx, wy = np.meshgrid(wx, wy)
wx[ix0, iy0] = 1.0
wy[ix0, iy0] = 1.0
sxx = self._sxx
sxy = self._sxy
sxz = self._sxz
syy = self._syy
syz = self._syz
szz = self._szz
h = self._h
N = self.n_layers
a = szz[:, None, None]
b = sxz[:, None, None] * wx + syz[:, None, None] * wy
c = sxx[:, None, None] * wx * wx + 2 * sxy[:, None, None] * wx * wy + syy[:, None, None] * wy * wy
al = np.sqrt(a * c - b * b)
l = (al/a)[:-1]
tanh = np.tanh(h[:, None, None] * l)
Qs = np.empty((N, *wx.shape))
Qs[-1] = 1/al[-1]
for i in range(N-2, -1, -1):
top = al[i] * Qs[i+1] + tanh[i]
bot = al[i] * ( al[i] * Qs[i+1] * tanh[i] + 1)
Qs[i] = top / bot
Qs[0, ix0, iy0] = 0
d_a = np.empty((N, *wx.shape))
d_b = np.empty((N, *wx.shape))
d_c = np.empty((N, *wx.shape))
d_h = np.empty((N-1, *wx.shape))
gout = 1.0
gQ = 1.0/(2*np.pi)*gout
for i in range(N-1):
top = al[i] * Qs[i+1] + tanh[i]
bot = al[i] * ( al[i] * Qs[i+1] * tanh[i] + 1)
gtop = 1.0/bot * gQ
gbot = -Qs[i] / bot * gQ
gal = Qs[i+1] * gtop + (2 * al[i] * Qs[i+1] * tanh[i] + 1)*gbot
gtanh = 1.0 * gtop + al[i] * al[i] * Qs[i+1] * gbot
gQ = al[i] * gtop + al[i] * al[i] * tanh[i] * gbot
# tanh = tanh(h * l)
front = (1 - tanh[i] * tanh[i])
d_h[i] = front * l[i] * gtanh
gl = front * h[i] * gtanh
# l = al/a
gal += gl / a[i]
d_a[i] = -gl * al[i] / (a[i] * a[i])
# al = np.sqrt(a * c - b * b)
d_a[i] += c[i] / (2 * al[i]) * gal
d_b[i] = - b[i] / al[i] * gal
d_c[i] = a[i] / (2 * al[i]) * gal
# Qs[-1] = 1/al[-1]
gal = -1/(al[-1] * al[-1]) * gQ
d_a[-1] = c[-1] / (2 * al[-1]) * gal
d_b[-1] = - b[-1] / al[-1] * gal
d_c[-1] = a[-1] / (2 * al[-1]) * gal
d_h[:, ix0, iy0] = 0.0
d_a[:, ix0, iy0] = 0.0
d_b[:, ix0, iy0] = 0.0
d_c[:, ix0, iy0] = 0.0
d_szz = d_a
d_sxz = wx * d_b
d_syz = wy * d_b
d_sxx = wx * wx * d_c
d_sxy = 2 * wx * wy * d_c
d_syy = wy * wy * d_c
return d_sxx, d_sxy, d_sxz, d_syy, d_syz, d_szz, d_h
def J_params(self, Jsxx, Jsxy, Jsxz, Jsyy, Jsyz, Jszz, transpose=False):
s1 = self._sig1
s2 = self._sig2
s3 = self._sig3
cosa = np.cos(self._alpha)
cosb = np.cos(self._beta)
cosg = np.cos(self._gamma)
sina = np.sin(self._alpha)
sinb = np.sin(self._beta)
sing = np.sin(self._gamma)
if transpose:
Jsxx = Jsxx.T
Jsxy = Jsxy.T
Jsxz = Jsxz.T
Jsyy = Jsyy.T
Jsyz = Jsyz.T
Jszz = Jszz.T
# DsxxD_params
Js1 = Jsxx*(cosa**2*cosb**2)
Js2 = Jsxx*(sina*cosg - sinb*sing*cosa)**2
Js3 = Jsxx*(sina*sing + sinb*cosa*cosg)**2
Ja = Jsxx*(2*(-s1*sina*cosa*cosb**2
+ s2*(sina*cosg - sinb*sing*cosa)*(sina*sinb*sing + cosa*cosg)
+ s3*(sina*sing + sinb*cosa*cosg)*(sing*cosa - sina*sinb*cosg)))
Jb = Jsxx*(2*(-s1*sinb*cosa**2*cosb
- s2*(sina*cosg - sinb*sing*cosa)*sing*cosa*cosb
+ s3*(sina*sing + sinb*cosa*cosg)*cosa*cosb*cosg))
Jg = Jsxx*(2*(s2*(sina*sing + sinb*cosa*cosg)*(sinb*sing*cosa - sina*cosg)
+ s3*(sina*sing + sinb*cosa*cosg)*(sina*cosg - sinb*sing*cosa)))
# DsxyD_params
Js1 += Jsxy*(sina*cosa*cosb**2)
Js2 += Jsxy*(-(sina*cosg-sinb*sing*cosa)*(sina*sinb*sing+cosa*cosg))
Js3 += Jsxy*((sina*sing+sinb*cosa*cosg)*(sina*sinb*cosg-sing*cosa))
Ja += Jsxy*(-s1*sina**2*cosb**2
+ s1*cosa**2*cosb**2
- s2*(sinb*sing*cosa - sina*cosg)*(sina*cosg - sinb*sing*cosa)
- s2*(sina*sinb*sing + cosa*cosg)**2
+ s3*(sina*sing + sinb*cosa*cosg)**2
+ s3*(sing*cosa - sina*sinb*cosg)*(sina*sinb*cosg - sing*cosa))
Jb += Jsxy*(-2*s1*sina*sinb*cosa*cosb
- s2*(sina*cosg - sinb*sing*cosa)*sina*sing*cosb
+ s2*(sina*sinb*sing + cosa*cosg)*sing*cosa*cosb
+ s3*(sina*sing + sinb*cosa*cosg)*sina*cosb*cosg
+ s3*(sina*sinb*cosg - sing*cosa)*cosa*cosb*cosg)
Jg += Jsxy*(s2*(sina*sing + sinb*cosa*cosg)*(sina*sinb*sing + cosa*cosg)
- s2*(sina*cosg - sinb*sing*cosa)*(sina*sinb*cosg - sing*cosa)
- s3*(sina*sing + sinb*cosa*cosg)*(sina*sinb*sing + cosa*cosg)
+ s3*(sina*cosg - sinb*sing*cosa)*(sina*sinb*cosg - sing*cosa))
# DsxzD_params
Js1 += Jsxz*(-sinb*cosa*cosb)
Js2 += Jsxz*((sinb*sing*cosa - sina*cosg)*sing*cosb)
Js3 += Jsxz*((sina*sing + sinb*cosa*cosg)*cosb*cosg)
Ja += Jsxz*((s1*sina*sinb
- s2*(sina*sinb*sing + cosa*cosg)*sing
+ s3*(-sina*sinb*cosg + sing*cosa)*cosg)*cosb)
Jb += Jsxz*(((s1*sinb*cosa
+ s2*(sina*cosg - sinb*sing*cosa)*sing
- s3*(sina*sing + sinb*cosa*cosg)*cosg)*sinb
+ (-s1*cosa*cosb
+ s2*sing**2*cosa*cosb
+ s3*cosa*cosb*cosg**2)*cosb))
Jg += Jsxz*((-s2*(-sina*sing - sinb*cosa*cosg)*sing
- s2*(sina*cosg - sinb*sing*cosa)*cosg
- s3*(sina*sing + sinb*cosa*cosg)*sing
+ s3*(sina*cosg - sinb*sing*cosa)*cosg)*cosb)
# DsyyD_params
Js1 += Jsyy*(sina**2*cosb**2)
Js2 += Jsyy*((sina*sinb*sing + cosa*cosg)**2)
Js3 += Jsyy*((sina*sinb*cosg - sing*cosa)**2)
Ja += Jsyy*(2*(s1*sina*cosa*cosb**2
+ s2*(sinb*sing*cosa - sina*cosg)*(sina*sinb*sing + cosa*cosg)
+ s3*(sina*sing + sinb*cosa*cosg)*(sina*sinb*cosg - sing*cosa)))
Jb += Jsyy*(2*(-s1*sina**2*sinb*cosb
+ s2*(sina*sinb*sing + cosa*cosg)*sina*sing*cosb
+ s3*(sina*sinb*cosg - sing*cosa)*sina*cosb*cosg))
Jg += Jsyy*(2*(s2*(sina*sinb*sing + cosa*cosg)*(sina*sinb*cosg - sing*cosa)
- s3*(sina*sinb*sing + cosa*cosg)*(sina*sinb*cosg - sing*cosa)))
# DsyzD_params
Js1 += Jsyz*(-sina*sinb*cosb)
Js2 += Jsyz*((sina*sinb*sing + cosa*cosg)*sing*cosb)
Js3 += Jsyz*((sina*sinb*cosg - sing*cosa)*cosb*cosg)
Ja += Jsyz*((- s1*sinb*cosa
+ s2*(sinb*sing*cosa - sina*cosg)*sing
+ s3*(sina*sing + sinb*cosa*cosg)*cosg)*cosb)
Jb += Jsyz*((-(-s1*sina*sinb
+ s2*(sina*sinb*sing + cosa*cosg)*sing
+ s3*(sina*sinb*cosg - sing*cosa)*cosg)*sinb
+ (-s1*sina*cosb
+ s2*sina*sing**2*cosb
+ s3*sina*cosb*cosg**2)*cosb))
Jg += Jsyz*((s2*(sina*sinb*sing + cosa*cosg)*cosg
+ s2*(sina*sinb*cosg - sing*cosa)*sing
- s3*(sina*sinb*sing + cosa*cosg)*cosg
- s3*(sina*sinb*cosg - sing*cosa)*sing)*cosb)
#DszzD_params
Js1 += Jszz*(sinb**2)
Js2 += Jszz*(sing**2*cosb**2)
Js3 += Jszz*(cosb**2*cosg**2)
# Ja += np.zeros_like(Jszz.T)
Jb += Jszz*(-2*(- s1*sinb*cosb
+ s2*sinb*sing**2*cosb
+ s3*sinb*cosb*cosg**2))
Jg += Jszz*(2*(s2*sing*cosb**2*cosg
- s3*sing*cosb**2*cosg))
if transpose:
Js1 = Js1.T
Js2 = Js2.T
Js3 = Js3.T
Ja = Ja.T
Jb = Jb.T
Jg = Jg.T
return Js1, Js2, Js3, Ja, Jb, Jg
def phi(self, nx=128, x_width=128, ny=None, y_width=None, return_grid=False):
if(ny is None):
ny = nx
if(y_width is None):
y_width = x_width
dx = x_width/(nx-1)
dy = y_width/(ny-1)
wx = 2*np.pi*(fft.fftfreq(nx, dx))
wy = 2*np.pi*(fft.fftfreq(ny, dy))
dwx = wx[1]-wx[0]
dwy = wy[1]-wy[0]
phi_tilde = self.phi_tilde(wx, wy)
scale = len(wx)*len(wy)*dwx*dwy/(2*np.pi)
phi = fft.fftshift(fft.irfft2(phi_tilde, s=phi_tilde.shape))*scale
if return_grid:
x = np.linspace(-x_width/2, x_width/2, nx)
y = np.linspace(-y_width/2, y_width/2, ny)
return phi, x, y
return phi
def phi_ds(self, nx=128, x_width=128, ny=None, y_width=None, return_grid=False):
if(ny is None):
ny = nx
if(y_width is None):
y_width = x_width
dx = x_width/(nx-1)
dy = y_width/(ny-1)
wx = 2*np.pi*(fft.fftfreq(nx, dx))
wy = 2*np.pi*(fft.fftfreq(ny, dy))
dwx = wx[1]-wx[0]
dwy = wy[1]-wy[0]
Js = self.phi_tilde_J(wx, wy)
scale = len(wx)*len(wy)*dwx*dwy/(2*np.pi)
Jsxx = fft.fftshift(fft.irfft2(Js[0], s=Js[0].shape[1:]), axes=(-2,-1))*scale
Jsxy = fft.fftshift(fft.irfft2(Js[1], s=Js[1].shape[1:]), axes=(-2,-1))*scale
Jsxz = fft.fftshift(fft.irfft2(Js[2], s=Js[2].shape[1:]), axes=(-2,-1))*scale
Jsyy = fft.fftshift(fft.irfft2(Js[3], s=Js[3].shape[1:]), axes=(-2,-1))*scale
Jsyz = fft.fftshift(fft.irfft2(Js[4], s=Js[4].shape[1:]), axes=(-2,-1))*scale
Jszz = fft.fftshift(fft.irfft2(Js[5], s=Js[5].shape[1:]), axes=(-2,-1))*scale
if return_grid:
x = np.linspace(-x_width/2, x_width/2, nx)
y = np.linspace(-y_width/2, y_width/2, ny)
return Jsxx, Jsxy, Jsxz, Jsyy, Jsyz, Jszz, x, y
return Jsxx, Jsxy, Jsxz, Jsyy, Jsyz, Jszz
def phi_dh(self, nx=128, x_width=128, ny=None, y_width=None, return_grid=False):
if(ny is None):
ny = nx
if(y_width is None):
y_width = x_width
dx = x_width/(nx-1)
dy = y_width/(ny-1)
wx = 2*np.pi*(fft.fftfreq(nx, dx))
wy = 2*np.pi*(fft.fftfreq(ny, dy))
dwx = wx[1]-wx[0]
dwy = wy[1]-wy[0]
Jh = self.phi_tilde_J(wx, wy)[-1]
scale = len(wx)*len(wy)*dwx*dwy/(2*np.pi)
Jh = fft.fftshift(fft.irfft2(Jh, s=Jh.shape[1:]), axes=(-2, -1))*scale
if return_grid:
x = np.linspace(-x_width/2, x_width/2, nx)
y = np.linspace(-y_width/2, y_width/2, ny)
return Jh, x, y
return Jh
def voltage(self, A, B, M, N, current=1):
"""
Voltage measured at M and N, due to current sources at A and B.
A : Tuple of numpy arrays for x and y position of current source
B : Tuple of numpy arrays for x and y position of current sink
M : Tuple of numpy arrays for x and y position of positive electrode
N : Tuple of numpy arrays for x and y position of negative electrode
"""
MAx = M[0]-A[0]
MAy = M[1]-A[1]
NAx = N[0]-A[0]
NAy = N[1]-A[1]
MBx = M[0]-B[0]
MBy = M[1]-B[1]
NBx = N[0]-B[0]
NBy = N[1]-B[1]
MA_potential = np.zeros_like(MAx)
MB_potential = np.zeros_like(MAx)
NA_potential = np.zeros_like(MAx)
NB_potential = np.zeros_like(MAx)
"""
x_width = max(np.abs(MAx).max(), np.abs(NAx).max(),
np.abs(MBx).max(), np.abs(NBx).max())
y_width = max(np.abs(MAy).max(), np.abs(NAy).max(),
np.abs(MBy).max(), np.abs(NBy).max())
width = max(x_width, y_width)*4
phi, x, y = self.phi(nx=4096, x_width=width, return_grid=True)
interp = Interpolator(x, y, phi)
MA_potential = interp(MAx, MAy, grid=False)
NA_potential = interp(NAx, NAy, grid=False)
MB_potential = interp(MBx, MBy, grid=False)
NB_potential = interp(NBx, NBy, grid=False)
"""
n_obs = len(M[0])
for i in range(n_obs):
x_width = max(np.abs(MAx[i]), np.abs(NAx[i]),
np.abs(MBx[i]), np.abs(NBx[i]))
y_width = max(np.abs(MAy[i]), np.abs(NAy[i]),
np.abs(MBy[i]), np.abs(NBy[i]))
width = max(x_width, y_width)*5
phi, x, y = self.phi(nx=128, x_width=width, return_grid=True)
wsMA = sinc_weights(MAx[i], MAy[i], x, y)
wsNA = sinc_weights(NAx[i], NAy[i], x, y)
wsMB = sinc_weights(MBx[i], MBy[i], x, y)
wsNB = sinc_weights(NBx[i], NBy[i], x, y)
MA_potential[i] = np.sum(wsMA*phi)
NA_potential[i] = np.sum(wsNA*phi)
MB_potential[i] = np.sum(wsMB*phi)
NB_potential[i] = np.sum(wsNB*phi)
"""
ixMA, iyMA, wsMA = get_weights(MAx[i], MAy[i], x, y)
ixNA, iyNA, wsNA = get_weights(NAx[i], NAy[i], x, y)
ixMB, iyMB, wsMB = get_weights(MBx[i], MBy[i], x, y)
ixNB, iyNB, wsNB = get_weights(NBx[i], NBy[i], x, y)
# interp = Interpolator(x, y, phi)
MA_potential[i] = wsMA.dot(phi[ixMA, iyMA])
NA_potential[i] = wsNA.dot(phi[ixNA, iyNA])
MB_potential[i] = wsMB.dot(phi[ixNA, iyNA])
NB_potential[i] = wsNB.dot(phi[ixNB, iyNB])
interp = Interpolator(x, y, phi)
MA_potential[i] = interp(MAx[i], MAy[i], grid=False)
NA_potential[i] = interp(NAx[i], NAy[i], grid=False)
MB_potential[i] = interp(MBx[i], MBy[i], grid=False)
NB_potential[i] = interp(NBx[i], NBy[i], grid=False)
"""
volts = MA_potential-NA_potential-MB_potential+NB_potential
return volts*current
def voltage_ds(self, A, B, M, N, current=1):
"""
Voltage measured at M and N, due to current sources at A and B.
A : Tuple of numpy arrays for x and y position of current source
B : Tuple of numpy arrays for x and y position of current sink
M : Tuple of numpy arrays for x and y position of positive electrode
N : Tuple of numpy arrays for x and y position of negative electrode
"""
MAx = M[0]-A[0]
MAy = M[1]-A[1]
NAx = N[0]-A[0]
NAy = N[1]-A[1]
MBx = M[0]-B[0]
MBy = M[1]-B[1]
NBx = N[0]-B[0]
NBy = N[1]-B[1]
n_obs = len(M[0])
n_layers = self.n_layers
Jxx = np.empty((n_obs, n_layers))
Jxy = np.empty(Jxx.shape)
Jxz = np.empty(Jxx.shape)
Jyy = np.empty(Jxx.shape)
Jyz = np.empty(Jxx.shape)
Jzz = np.empty(Jxx.shape)
for i in range(n_obs):
x_width = max(np.abs(MAx[i]), np.abs(NAx[i]),
np.abs(MBx[i]), np.abs(NBx[i]))
y_width = max(np.abs(MAy[i]), np.abs(NAy[i]),
np.abs(MBy[i]), np.abs(NBy[i]))
width = max(x_width, y_width)*5
out = self.phi_ds(nx=128, x_width=width, return_grid=True)
Jsxx, Jsxy, Jsxz, Jsyy, Jsyz, Jszz, x, y = out
wsMA = sinc_weights(MAx[i], MAy[i], x, y)
wsNA = sinc_weights(NAx[i], NAy[i], x, y)
wsMB = sinc_weights(MBx[i], MBy[i], x, y)
wsNB = sinc_weights(NBx[i], NBy[i], x, y)
MA_potential = np.sum(Jsxx*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jsxx*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jsxx*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jsxx*wsNB, axis=(-2, -1))
Jxx[i] = MA_potential-NA_potential-MB_potential+NB_potential
MA_potential = np.sum(Jsxy*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jsxy*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jsxy*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jsxy*wsNB, axis=(-2, -1))
Jxy[i] = MA_potential-NA_potential-MB_potential+NB_potential
MA_potential = np.sum(Jsxz*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jsxz*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jsxz*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jsxz*wsNB, axis=(-2, -1))
Jxz[i] = MA_potential-NA_potential-MB_potential+NB_potential
MA_potential = np.sum(Jsyy*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jsyy*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jsyy*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jsyy*wsNB, axis=(-2, -1))
Jyy[i] = MA_potential-NA_potential-MB_potential+NB_potential
MA_potential = np.sum(Jsyz*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jsyz*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jsyz*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jsyz*wsNB, axis=(-2, -1))
Jyz[i] = MA_potential-NA_potential-MB_potential+NB_potential
MA_potential = np.sum(Jszz*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jszz*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jszz*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jszz*wsNB, axis=(-2, -1))
Jzz[i] = MA_potential-NA_potential-MB_potential+NB_potential
return Jxx, Jxy, Jxz, Jyy, Jyz, Jzz
def voltage_dh(self, A, B, M, N, current=1):
"""
Voltage measured at M and N, due to current sources at A and B.
A : Tuple of numpy arrays for x and y position of current source
B : Tuple of numpy arrays for x and y position of current sink
M : Tuple of numpy arrays for x and y position of positive electrode
N : Tuple of numpy arrays for x and y position of negative electrode
"""
MAx = M[0]-A[0]
MAy = M[1]-A[1]
NAx = N[0]-A[0]
NAy = N[1]-A[1]
MBx = M[0]-B[0]
MBy = M[1]-B[1]
NBx = N[0]-B[0]
NBy = N[1]-B[1]
n_obs = len(M[0])
n_layers = self.n_layers
Jh = np.empty((n_obs, n_layers-1))
for i in range(n_obs):
x_width = max(np.abs(MAx[i]), np.abs(NAx[i]),
np.abs(MBx[i]), np.abs(NBx[i]))
y_width = max(np.abs(MAy[i]), np.abs(NAy[i]),
np.abs(MBy[i]), np.abs(NBy[i]))
width = max(x_width, y_width)*5
Jhi, x, y = self.phi_dh(nx=128, x_width=width, return_grid=True)
wsMA = sinc_weights(MAx[i], MAy[i], x, y)
wsNA = sinc_weights(NAx[i], NAy[i], x, y)
wsMB = sinc_weights(MBx[i], MBy[i], x, y)
wsNB = sinc_weights(NBx[i], NBy[i], x, y)
MA_potential = np.sum(Jhi*wsMA, axis=(-2, -1))
NA_potential = np.sum(Jhi*wsNA, axis=(-2, -1))
MB_potential = np.sum(Jhi*wsMB, axis=(-2, -1))
NB_potential = np.sum(Jhi*wsNB, axis=(-2, -1))
Jh[i] = MA_potential-NA_potential-MB_potential+NB_potential
return Jh
def sinc_weights(xi, yi, x, y):
dx = x[1]-x[0]
dy = y[1]-y[0]
wsx = np.sinc((xi-x)/dx)
wsy = np.sinc((yi-y)/dy)
return np.outer(wsx, wsy)
def get_weights(xi, yi, x, y):
# indices and weights for 2d bilinear interpolation.
ix = np.searchsorted(x, xi)
iy = np.searchsorted(y, yi)
hx = (x[ix]-xi)/(x[ix]-x[ix-1])
hy = (y[iy]-yi)/(y[iy]-y[iy-1])
ixs = [ix-1, ix-1, ix, ix]
iys = [iy-1, iy, iy-1, iy]
ws = np.array([hx*hy, hx*(1-hy), (1-hx)*hy, (1-hx)*(1-hy)])
return ixs, iys, ws
def apparent_conductivity(volts, A, B, M, N, current=1):
"""
Calculates the apparent conductivity for a given array configuration.
Bam!
"""
R = (1/np.sqrt((M[0]-A[0])**2+(M[1]-A[1])**2)
- 1/np.sqrt((M[0]-B[0])**2+(M[1]-B[1])**2)
- 1/np.sqrt((N[0]-A[0])**2+(N[1]-A[1])**2)
+ 1/np.sqrt((N[0]-B[0])**2+(N[1]-B[1])**2))
return current*R/(volts*2*np.pi)