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GudniRos
109 lines
4.4 KiB
Python
109 lines
4.4 KiB
Python
# Analytic solution of EM fields due to a plane wave
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import numpy as np, SimPEG as simpeg
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from scipy.constants import mu_0, epsilon_0 as eps_0
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def getEHfields(m1d,sigma,freq,zd,scaleUD=True):
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'''Analytic solution for MT 1D layered earth. Returns E and H fields.
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:param SimPEG.mesh, object m1d: Mesh object with the 1D spatial information.
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:param numpy.array, vector sigma: Physical property of conductivity corresponding with the mesh.
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:param float, freq: Frequency to calculate data at.
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:param numpy array, vector zd: location to calculate EH fields at
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:param bollean, scaleUD: scales the output to be 1 at the top, increases numeracal stability.
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Assumes a halfspace with the same conductive as the last cell below.
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'''
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# Note add an error check for the mesh and sigma are the same size.
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# Constants: Assume constant
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mu = mu_0*np.ones((m1d.nC+1))
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eps = eps_0*np.ones((m1d.nC+1))
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# Angular freq
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w = 2*np.pi*freq
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# Add the halfspace value to the property
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sig = np.concatenate((np.array([sigma[0]]),sigma))
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# Calculate the wave number
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k = np.sqrt(eps*mu*w**2-1j*mu*sig*w)
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# Initiate the propagation matrix, in the order down up.
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UDp = np.zeros((2,m1d.nC+1),dtype=complex)
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UDp[1,0] = 1. # Set the wave amplitude as 1 into the half-space at the bottom of the mesh
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# Loop over all the layers, starting at the bottom layer
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for lnr, h in enumerate(m1d.hx): # lnr-number of layer, h-thickness of the layer
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# Calculate
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yp1 = k[lnr]/(w*mu[lnr]) # Admittance of the layer below the current layer
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zp = (w*mu[lnr+1])/k[lnr+1] # Impedance in the current layer
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# Build the propagation matrix
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# Convert fields to down/up going components in layer below current layer
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Pj1 = np.array([[1,1],[yp1,-yp1]])
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# Convert fields to down/up going components in current layer
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Pjinv = 1./2*np.array([[1,zp],[1,-zp]])
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# Propagate down and up components through the current layer
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elamh = np.array([[np.exp(-1j*k[lnr+1]*h),0],[0,np.exp(1j*k[lnr+1]*h)]])
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# The down and up component in current layer.
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UDp[:,lnr+1] = elamh.dot(Pjinv.dot(Pj1)).dot(UDp[:,lnr])
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if scaleUD:
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UDp[:,lnr+1::-1] = UDp[:,lnr+1::-1]/UDp[1,lnr+1]
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# Calculate the fields
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Ed = np.empty((zd.size,),dtype=complex)
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Eu = np.empty((zd.size,),dtype=complex)
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Hd = np.empty((zd.size,),dtype=complex)
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Hu = np.empty((zd.size,),dtype=complex)
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# Loop over the layers and calculate the fields
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# In the halfspace below the mesh
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dup = m1d.vectorNx[0]
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dind = dup >= zd
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Ed[dind] = UDp[1,0]*np.exp(-1j*k[0]*(dup-zd[dind]))
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Eu[dind] = UDp[0,0]*np.exp(1j*k[0]*(dup-zd[dind]))
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Hd[dind] = (k[0]/(w*mu[0]))*UDp[1,0]*np.exp(-1j*k[0]*(dup-zd[dind]))
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Hu[dind] = -(k[0]/(w*mu[0]))*UDp[0,0]*np.exp(1j*k[0]*(dup-zd[dind]))
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for ki,mui,epsi,dlow,dup,Up,Dp in zip(k[1::],mu[1::],eps[1::],m1d.vectorNx[:-1],m1d.vectorNx[1::],UDp[0,1::],UDp[1,1::]):
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dind = np.logical_and(dup >= zd, zd > dlow)
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Ed[dind] = Dp*np.exp(-1j*ki*(dup-zd[dind]))
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Eu[dind] = Up*np.exp(1j*ki*(dup-zd[dind]))
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Hd[dind] = (ki/(w*mui))*Dp*np.exp(-1j*ki*(dup-zd[dind]))
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Hu[dind] = -(ki/(w*mui))*Up*np.exp(1j*ki*(dup-zd[dind]))
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# Return return the fields
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return Ed, Eu, Hd, Hu
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def getImpedance(m1d,sigma,freq):
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"""Analytic solution for MT 1D layered earth. Returns the impedance at the surface.
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:param SimPEG.mesh, object m1d: Mesh object with the 1D spatial information.
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:param numpy.array, vector sigma: Physical property corresponding with the mesh.
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:param numpy.array, vector freq: Frequencies to calculate data at.
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"""
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# Initiate the impedances
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Z1d = np.empty(len(freq) , dtype='complex')
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h = m1d.hx #vectorNx[:-1]
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# Start the process
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for nrFr, fr in enumerate(freq):
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om = 2*np.pi*fr
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Zall = np.empty(len(h)+1,dtype='complex')
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# Calculate the impedance for the bottom layer
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Zall[0] = (mu_0*om)/np.sqrt(mu_0*eps_0*(om)**2 - 1j*mu_0*sigma[0]*om)
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for nr,hi in enumerate(h):
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# Calculate the wave number
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# print nr,sigma[nr]
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k = np.sqrt(mu_0*eps_0*om**2 - 1j*mu_0*sigma[nr]*om)
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Z = (mu_0*om)/k
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Zall[nr+1] = Z *((Zall[nr] + Z*np.tanh(1j*k*hi))/(Z + Zall[nr]*np.tanh(1j*k*hi)))
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#pdb.set_trace()
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Z1d[nrFr] = Zall[-1]
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return Z1d
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