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Merge branch 'Examples' into feat/sparse-regularization
Conflicts: SimPEG/Examples/__init__.py
This commit is contained in:
D Fournier
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from SimPEG import *
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# import simpegDCIP as DC
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from SimPEG import DCIP as DC
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import scipy.interpolate as interpolation
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import matplotlib.pyplot as plt
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import time
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import re
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def run(loc=np.c_[[-50.,0.,-50.],[50.,0.,-50.]], sig=np.r_[1e-2,1e-1,1e-3], radi=np.r_[25.,25.], param = np.r_[30.,30.,5], stype = 'dpdp', plotIt=True):
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"""
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DC Forward Simulation
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=====================
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Forward model conductive spheres in a half-space and plot a pseudo-section
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Created on Mon Feb 01 19:28:06 2016
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@fourndo
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"""
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def getIndicesSphere(center,radius,ccMesh):
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"""
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Creates a vector containing the sphere indices in the cell centers mesh.
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Returns a tuple
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The sphere is defined by the points
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p0, describe the position of the center of the cell
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r, describe the radius of the sphere.
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ccMesh represents the cell-centered mesh
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The points p0 must live in the the same dimensional space as the mesh.
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"""
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# Validation: mesh and point (p0) live in the same dimensional space
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dimMesh = np.size(ccMesh[0,:])
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assert len(center) == dimMesh, "Dimension mismatch. len(p0) != dimMesh"
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if dimMesh == 1:
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# Define the reference points
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ind = np.abs(center[0] - ccMesh[:,0]) < radius
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elif dimMesh == 2:
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# Define the reference points
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ind = np.sqrt( ( center[0] - ccMesh[:,0] )**2 + ( center[1] - ccMesh[:,1] )**2 ) < radius
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elif dimMesh == 3:
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# Define the points
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ind = np.sqrt( ( center[0] - ccMesh[:,0] )**2 + ( center[1] - ccMesh[:,1] )**2 + ( center[2] - ccMesh[:,2] )**2 ) < radius
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# Return a tuple
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return ind
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# First we need to create a mesh and a model.
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# This is our mesh
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dx = 5.
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hxind = [(dx,15,-1.3), (dx, 75), (dx,15,1.3)]
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hyind = [(dx,15,-1.3), (dx, 10), (dx,15,1.3)]
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hzind = [(dx,15,-1.3),(dx, 15)]
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mesh = Mesh.TensorMesh([hxind, hyind, hzind], 'CCN')
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# Set background conductivity
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model = np.ones(mesh.nC) * sig[0]
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# First anomaly
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ind = getIndicesSphere(loc[:,0],radi[0],mesh.gridCC)
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model[ind] = sig[1]
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# Second anomaly
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ind = getIndicesSphere(loc[:,1],radi[1],mesh.gridCC)
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model[ind] = sig[2]
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# Get index of the center
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indy = int(mesh.nCy/2)
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# Plot the model for reference
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# Define core mesh extent
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xlim = 200
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zlim = 125
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# Specify the survey type: "pdp" | "dpdp"
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# Then specify the end points of the survey. Let's keep it simple for now and survey above the anomalies, top of the mesh
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ends = [(-175,0),(175,0)]
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ends = np.c_[np.asarray(ends),np.ones(2).T*mesh.vectorNz[-1]]
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# Snap the endpoints to the grid. Easier to create 2D section.
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indx = Utils.closestPoints(mesh, ends )
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locs = np.c_[mesh.gridCC[indx,0],mesh.gridCC[indx,1],np.ones(2).T*mesh.vectorNz[-1]]
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# We will handle the geometry of the survey for you and create all the combination of tx-rx along line
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[Tx, Rx] = DC.gen_DCIPsurvey(locs, mesh, stype, param[0], param[1], param[2])
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# Define some global geometry
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dl_len = np.sqrt( np.sum((locs[0,:] - locs[1,:])**2) )
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dl_x = ( Tx[-1][0,1] - Tx[0][0,0] ) / dl_len
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dl_y = ( Tx[-1][1,1] - Tx[0][1,0] ) / dl_len
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azm = np.arctan(dl_y/dl_x)
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#Set boundary conditions
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mesh.setCellGradBC('neumann')
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# Define the differential operators needed for the DC problem
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Div = mesh.faceDiv
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Grad = mesh.cellGrad
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Msig = Utils.sdiag(1./(mesh.aveF2CC.T*(1./model)))
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A = Div*Msig*Grad
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# Change one corner to deal with nullspace
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A[0,0] = 1
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A = sp.csc_matrix(A)
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# We will solve the system iteratively, so a pre-conditioner is helpful
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# This is simply a Jacobi preconditioner (inverse of the main diagonal)
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dA = A.diagonal()
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P = sp.spdiags(1/dA,0,A.shape[0],A.shape[0])
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# Now we can solve the system for all the transmitters
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# We want to store the data
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data = []
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# There is probably a more elegant way to do this, but we can just for-loop through the transmitters
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for ii in range(len(Tx)):
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start_time = time.time() # Let's time the calculations
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#print("Transmitter %i / %i\r" % (ii+1,len(Tx)))
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# Select dipole locations for receiver
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rxloc_M = np.asarray(Rx[ii][:,0:3])
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rxloc_N = np.asarray(Rx[ii][:,3:])
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# For usual cases "dpdp" or "gradient"
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if not re.match(stype,'pdp'):
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inds = Utils.closestPoints(mesh, np.asarray(Tx[ii]).T )
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RHS = mesh.getInterpolationMat(np.asarray(Tx[ii]).T, 'CC').T*( [-1,1] / mesh.vol[inds] )
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else:
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# Create an "inifinity" pole
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tx = np.squeeze(Tx[ii][:,0:1])
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tinf = tx + np.array([dl_x,dl_y,0])*dl_len*2
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inds = Utils.closestPoints(mesh, np.c_[tx,tinf].T)
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RHS = mesh.getInterpolationMat(np.asarray(Tx[ii]).T, 'CC').T*( [-1] / mesh.vol[inds] )
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# Iterative Solve
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Ainvb = sp.linalg.bicgstab(P*A,P*RHS, tol=1e-5)
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# We now have the potential everywhere
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phi = mkvc(Ainvb[0])
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# Solve for phi on pole locations
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P1 = mesh.getInterpolationMat(rxloc_M, 'CC')
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P2 = mesh.getInterpolationMat(rxloc_N, 'CC')
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# Compute the potential difference
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dtemp = (P1*phi - P2*phi)*np.pi
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data.append( dtemp )
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print '\rTransmitter {0} of {1} -> Time:{2} sec'.format(ii,len(Tx),time.time()- start_time),
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print 'Transmitter {0} of {1}'.format(ii,len(Tx))
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print 'Forward completed'
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# Let's just convert the 3D format into 2D (distance along line) and plot
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[Tx2d, Rx2d] = DC.convertObs_DC3D_to_2D(Tx,Rx)
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# Here is an example for the first tx-rx array
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if plotIt:
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fig = plt.figure()
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ax = plt.subplot(2,1,1, aspect='equal')
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mesh.plotSlice(np.log10(model), ax =ax, normal = 'Y', ind = indy,grid=True)
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ax.set_title('E-W section at '+str(mesh.vectorCCy[indy])+' m')
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plt.gca().set_aspect('equal', adjustable='box')
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plt.scatter(Tx[0][0,:],Tx[0][2,:],s=40,c='g', marker='v')
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plt.scatter(Rx[0][:,0::3],Rx[0][:,2::3],s=40,c='y')
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plt.xlim([-xlim,xlim])
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plt.ylim([-zlim,mesh.vectorNz[-1]+dx])
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ax = plt.subplot(2,1,2, aspect='equal')
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# Plot the location of the spheres for reference
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circle1=plt.Circle((loc[0,0]-Tx[0][0,0],loc[2,0]),radi[0],color='w',fill=False, lw=3)
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circle2=plt.Circle((loc[0,1]-Tx[0][0,0],loc[2,1]),radi[1],color='k',fill=False, lw=3)
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ax.add_artist(circle1)
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ax.add_artist(circle2)
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# Add the speudo section
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DC.plot_pseudoSection(Tx2d,Rx2d,data,mesh.vectorNz[-1],stype)
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plt.scatter(Tx2d[0][:],Tx[0][2,:],s=40,c='g', marker='v')
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plt.scatter(Rx2d[0][:],Rx[0][:,2::3],s=40,c='y')
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plt.plot(np.r_[Tx2d[0][0],Rx2d[-1][-1,-1]],np.ones(2)*mesh.vectorNz[-1], color='k')
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plt.ylim([-zlim,mesh.vectorNz[-1]+dx])
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plt.show()
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return fig, ax
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if __name__ == '__main__':
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run()
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from SimPEG import *
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from SimPEG import EM
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from pymatsolver import MumpsSolver
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from scipy.constants import mu_0
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def run(plotIt=True):
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"""
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EM: FDEM: Effects of susceptibility
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===================================
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When airborne freqeuncy domain EM (AFEM) survey is flown over
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the earth including significantly susceptible bodies (magnetite-rich rocks),
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negative data is often observed in the real part of the lowest frequency
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(e.g. Dighem system 900 Hz). This phenomenon mostly based upon magnetization
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occurs due to a susceptible body when the magnetic field is applied.
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To clarify what is happening in the earth when we are exciting the earth with
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a loop source in the frequency domain we run three forward modelling:
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- F[:math:`\sigma`, :math:`\mu`]: Anomalous conductivity and susceptibility
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- F[:math:`\sigma`, :math:`\mu_0`]: Anomalous conductivity
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- F[:math:`\sigma_{air}`, :math:`\mu_0`]: primary field
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We plot vector magnetic fields in the earth. For secondary fields we provide
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F[:math:`\sigma`, :math:`\mu`]-F[:math:`\sigma`, :math:`\mu_0`]. Following
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figure show both real and parts.
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"""
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# Generate Cylindrical mesh
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cs, ncx, ncz, npad = 5, 25, 24, 20.
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hx = [(cs,ncx), (cs,npad,1.3)]
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hz = [(cs,npad,-1.3), (cs,ncz), (cs,npad,1.3)]
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mesh = Mesh.CylMesh([hx,1,hz], '00C')
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sighalf = 1e-3
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sigma = np.ones(mesh.nC)*1e-8
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sigmahomo = sigma.copy()
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mu = np.ones(mesh.nC)*mu_0
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sigma[mesh.gridCC[:,-1]<0.] = sighalf
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blkind = np.logical_and(mesh.gridCC[:,0]<30., (mesh.gridCC[:,2]<0)&(mesh.gridCC[:,2]>-150)&(mesh.gridCC[:,2]<-50))
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sigma[blkind] = 1e-1
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mu[blkind] = mu_0*1.1
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offset = 0.
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frequency = np.r_[10., 100., 1000.]
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rx0 = EM.FDEM.Rx(np.array([[8., 0., 30.]]), 'bzr')
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rx1 = EM.FDEM.Rx(np.array([[8., 0., 30.]]), 'bzi')
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srcLists = []
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nfreq = frequency.size
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for ifreq in range(nfreq):
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src = EM.FDEM.Src.CircularLoop([rx0, rx1], frequency[ifreq], np.array([[0., 0., 30.]]), radius=5.)
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srcLists.append(src)
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survey = EM.FDEM.Survey(srcLists)
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iMap = Maps.IdentityMap(nP=int(mesh.nC))
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# Use PhysPropMap
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maps = [('sigma', iMap), ('mu', iMap)]
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prob = EM.FDEM.Problem_b(mesh, mapping=maps)
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prob.Solver = MumpsSolver
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survey.pair(prob)
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m = np.r_[sigma, mu]
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survey0 = EM.FDEM.Survey(srcLists)
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prob0 = EM.FDEM.Problem_b(mesh, mapping=maps)
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prob0.Solver = MumpsSolver
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survey0.pair(prob0)
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m = np.r_[sigma, mu]
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m0 = np.r_[sigma, np.ones(mesh.nC)*mu_0]
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m00 = np.r_[np.ones(mesh.nC)*1e-8, np.ones(mesh.nC)*mu_0]
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# Anomalous conductivity and susceptibility
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F = prob.fields(m)
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# Only anomalous conductivity
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F0 = prob.fields(m0)
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# Primary field
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F00 = prob.fields(m00)
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if plotIt:
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import matplotlib.pyplot as plt
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def vizfields(ifreq=0, primsec="secondary",realimag="real"):
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titles = ["F[$\sigma$, $\mu$]", "F[$\sigma$, $\mu_0$]", "F[$\sigma$, $\mu$]-F[$\sigma$, $\mu_0$]"]
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actind = np.logical_and(mesh.gridCC[:,0]<200., (mesh.gridCC[:,2]>-400)&(mesh.gridCC[:,2]<200))
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if primsec=="secondary":
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bCCprim = (mesh.aveF2CCV*F00[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')
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bCC = (mesh.aveF2CCV*F[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')-bCCprim
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bCC0 = (mesh.aveF2CCV*F0[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')-bCCprim
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elif primsec=="primary":
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bCC = (mesh.aveF2CCV*F[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')
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bCC0 = (mesh.aveF2CCV*F0[:,'b'][:,ifreq]).reshape(mesh.nC, 2, order='F')
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XYZ = mesh.gridCC[actind,:]
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X = XYZ[:,0].reshape((31,43), order='F')
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Z = XYZ[:,2].reshape((31,43), order='F')
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bx = bCC[actind,0].reshape((31,43), order='F')
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bz = bCC[actind,1].reshape((31,43), order='F')
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bx0 = bCC0[actind,0].reshape((31,43), order='F')
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bz0 = bCC0[actind,1].reshape((31,43), order='F')
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bxsec = (bCC[actind,0]-bCC0[actind,0]).reshape((31,43), order='F')
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bzsec = (bCC[actind,1]-bCC0[actind,1]).reshape((31,43), order='F')
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absbreal = np.sqrt(bx.real**2+bz.real**2)
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absbimag = np.sqrt(bx.imag**2+bz.imag**2)
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absb0real = np.sqrt(bx0.real**2+bz0.real**2)
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absb0imag = np.sqrt(bx0.imag**2+bz0.imag**2)
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absbrealsec = np.sqrt(bxsec.real**2+bzsec.real**2)
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absbimagsec = np.sqrt(bxsec.imag**2+bzsec.imag**2)
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fig = plt.figure(figsize=(15,5))
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ax1 = plt.subplot(131)
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ax2 = plt.subplot(132)
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ax3 = plt.subplot(133)
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typefield="real"
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scale=20
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if realimag=="real":
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ax1.contourf(X, Z,np.log10(absbreal), 100)
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ax1.quiver(X, Z,bx.real/absbreal,bz.real/absbreal,scale=scale,width=0.005, alpha = 0.5)
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ax2.contourf(X, Z,np.log10(absb0real), 100)
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ax2.quiver(X, Z,bx0.real/absb0real,bz0.real/absb0real,scale=scale,width=0.005, alpha = 0.5)
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ax3.contourf(X, Z,np.log10(absbrealsec), 100)
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ax3.quiver(X, Z,bxsec.real/absbrealsec,bzsec.real/absbrealsec,scale=scale,width=0.005, alpha = 0.5)
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elif realimag=="imag":
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ax1.contourf(X, Z,np.log10(absbimag), 100)
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ax1.quiver(X, Z,bx.imag/absbimag,bz.imag/absbimag,scale=scale,width=0.005, alpha = 0.5)
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ax2.contourf(X, Z,np.log10(absb0imag), 100)
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ax2.quiver(X, Z,bx0.imag/absb0imag,bz0.imag/absb0imag,scale=scale,width=0.005, alpha = 0.5)
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ax3.contourf(X, Z,np.log10(absbimagsec), 100)
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ax3.quiver(X, Z,bxsec.imag/absbimagsec,bzsec.imag/absbimagsec,scale=scale,width=0.005, alpha = 0.5)
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ax = [ax1, ax2, ax3]
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ax3.text(30, 50, ("Frequency=%5.2f Hz")%(frequency[ifreq]), color="k", fontsize=18)
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ax2.text(30, 50, primsec, color="k", fontsize=18)
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ax1.text(30, 50, realimag, color="k", fontsize=18)
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for i, axtemp in enumerate(ax):
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axtemp.plot(np.r_[0, 29.75], np.r_[-50, -50], 'w', lw=3)
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axtemp.plot(np.r_[29.5, 29.5], np.r_[-50, -142.5], 'w', lw=3)
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axtemp.plot(np.r_[0, 29.5], np.r_[-142.5, -142.5], 'w', lw=3)
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axtemp.plot(np.r_[0, 100.], np.r_[0, 0], 'w', lw=3)
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axtemp.set_ylim(-200, 100.)
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axtemp.set_xlim(10, 100.)
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axtemp.set_title(titles[i])
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plt.show()
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return fig, ax
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fig1, ax1 = vizfields(1, primsec="primary", realimag="real")
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fig2, ax2 = vizfields(1, primsec="secondary", realimag="real")
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fig4, ax4 = vizfields(1, primsec="secondary", realimag="imag")
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||||
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||||
if __name__ == '__main__':
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run()
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||||
@@ -0,0 +1,90 @@
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from SimPEG import *
|
||||
|
||||
|
||||
def run(N=100, plotIt=True):
|
||||
"""
|
||||
Inversion: Linear Problem
|
||||
=========================
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||||
|
||||
Here we go over the basics of creating a linear problem and inversion.
|
||||
|
||||
"""
|
||||
|
||||
class LinearSurvey(Survey.BaseSurvey):
|
||||
def projectFields(self, u):
|
||||
return u
|
||||
|
||||
class LinearProblem(Problem.BaseProblem):
|
||||
|
||||
surveyPair = LinearSurvey
|
||||
|
||||
def __init__(self, mesh, G, **kwargs):
|
||||
Problem.BaseProblem.__init__(self, mesh, **kwargs)
|
||||
self.G = G
|
||||
|
||||
def fields(self, m, u=None):
|
||||
return self.G.dot(m)
|
||||
|
||||
def Jvec(self, m, v, u=None):
|
||||
return self.G.dot(v)
|
||||
|
||||
def Jtvec(self, m, v, u=None):
|
||||
return self.G.T.dot(v)
|
||||
|
||||
|
||||
np.random.seed(1)
|
||||
|
||||
mesh = Mesh.TensorMesh([N])
|
||||
|
||||
nk = 20
|
||||
jk = np.linspace(1.,20.,nk)
|
||||
p = -0.25
|
||||
q = 0.25
|
||||
|
||||
g = lambda k: np.exp(p*jk[k]*mesh.vectorCCx)*np.cos(2*np.pi*q*jk[k]*mesh.vectorCCx)
|
||||
|
||||
G = np.empty((nk, mesh.nC))
|
||||
|
||||
for i in range(nk):
|
||||
G[i,:] = g(i)
|
||||
|
||||
mtrue = np.zeros(mesh.nC)
|
||||
mtrue[mesh.vectorCCx > 0.3] = 1.
|
||||
mtrue[mesh.vectorCCx > 0.45] = -0.5
|
||||
mtrue[mesh.vectorCCx > 0.6] = 0
|
||||
|
||||
prob = LinearProblem(mesh, G)
|
||||
survey = LinearSurvey()
|
||||
survey.pair(prob)
|
||||
survey.makeSyntheticData(mtrue, std=0.01)
|
||||
|
||||
M = prob.mesh
|
||||
|
||||
reg = Regularization.Tikhonov(mesh)
|
||||
dmis = DataMisfit.l2_DataMisfit(survey)
|
||||
opt = Optimization.ProjectedGNCG(maxIter=20,lower=-1.,upper=1., maxIterCG= 20, tolCG = 1e-3)
|
||||
invProb = InvProblem.BaseInvProblem(dmis, reg, opt)
|
||||
beta = Directives.BetaSchedule()
|
||||
betaest = Directives.BetaEstimate_ByEig()
|
||||
inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest])
|
||||
m0 = np.zeros_like(survey.mtrue)
|
||||
|
||||
mrec = inv.run(m0)
|
||||
|
||||
if plotIt:
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
fig, axes = plt.subplots(1,2,figsize=(12*1.2,4*1.2))
|
||||
for i in range(prob.G.shape[0]):
|
||||
axes[0].plot(prob.G[i,:])
|
||||
axes[0].set_title('Columns of matrix G')
|
||||
|
||||
axes[1].plot(M.vectorCCx, survey.mtrue, 'b-')
|
||||
axes[1].plot(M.vectorCCx, mrec, 'r-')
|
||||
axes[1].legend(('True Model', 'Recovered Model'))
|
||||
plt.show()
|
||||
|
||||
return prob, survey, mesh, mrec
|
||||
|
||||
if __name__ == '__main__':
|
||||
run()
|
||||
@@ -0,0 +1,443 @@
|
||||
from scipy.constants import epsilon_0, mu_0
|
||||
import matplotlib.pyplot as plt
|
||||
import numpy as np
|
||||
from ipywidgets import *
|
||||
#from SimPEG.EM.Utils import k, omega
|
||||
|
||||
"""
|
||||
MT1D: n layered earth problem
|
||||
*****************************
|
||||
|
||||
Author: Thibaut Astic
|
||||
Contact: thast@eos.ubc.ca
|
||||
Date: January 2016
|
||||
|
||||
This code compute the analytic response of a n-layered Earth to a plane wave (Magneto-Tellurics).
|
||||
|
||||
We start by looking at Maxwell's equations in the electric
|
||||
field \\\(\\\mathbf{E}\\) and the magnetic flux
|
||||
\\\(\\\mathbf{H}\\) to write the wave equations
|
||||
\\(\\ \nabla ^2 \mathbf{E_x} + k^2 \mathbf{E_x} = 0 \\) &
|
||||
\\(\\ \nabla ^2 \mathbf{H_y} + k^2 \mathbf{H_y} = 0 \\)
|
||||
|
||||
Then solving the equations in each layer "j" between z_{j-1} and z_j in the form of
|
||||
\\(\\ E_{x,j} (z) = U_j e^{i k (z-z_{j-1})} + D_j e^{-i k (z-z_{j-1})} \\)
|
||||
\\(\\ H_{y,j} (z) = \frac{1}{Z_j} (D_j e^{-i k (z-z_{j-1})} - U_j e^{i k (z-z_{j-1})}) \\)
|
||||
|
||||
With U and D the Up and Down components of the E-field.
|
||||
|
||||
The iteration from one layer to another is ensure by:
|
||||
|
||||
\\(\\ \left(\begin{matrix} E_{x,j} \\ H_{y,j} \end{matrix} \right) =
|
||||
P_j T_j P^{-1}_J \left(\begin{matrix} E_{x,j+1} \\ H_{y,j+1} \end{matrix} \right) \\)
|
||||
|
||||
And the Boundary Condition is set for the E-field in the last layer, with no Up component (=0)
|
||||
and only a down component (=1 then normalized by the highest amplitude to ensure numeric stability)
|
||||
|
||||
The layer 0 is assumed to be the air layer.
|
||||
|
||||
"""
|
||||
|
||||
#Frequency conversion
|
||||
omega = lambda f: 2.*np.pi*f
|
||||
|
||||
#Evaluate k wavenumber
|
||||
k = lambda mu,sig,eps,f: np.sqrt(mu*mu_0*eps*epsilon_0*(2.*np.pi*f)**2.-1.j*mu*mu_0*sig*omega(f))
|
||||
|
||||
#Define a frquency range for a survey
|
||||
frange = lambda minfreq, maxfreq, step: np.logspace(minfreq,maxfreq,num = step, base = 10.)
|
||||
|
||||
#Functions to create random physical Perties for a n-layered earth
|
||||
thick = lambda minthick, maxthick, nlayer: np.append(np.array([1.2*10.**5]),
|
||||
np.ndarray.round(minthick + (maxthick-minthick)* np.random.rand(nlayer-1,1)
|
||||
,decimals =1))
|
||||
|
||||
sig = lambda minsig, maxsig, nlayer: np.append(np.array([0.]),
|
||||
np.ndarray.round(10.**minsig + (10.**maxsig-10.**minsig)* np.random.rand(nlayer,1)
|
||||
,decimals=3))
|
||||
|
||||
mu = lambda minmu, maxmu, nlayer: np.append(np.array([1.]),
|
||||
np.ndarray.round(minmu + (maxmu-minmu)* np.random.rand(nlayer,1)
|
||||
,decimals=1))
|
||||
|
||||
eps = lambda mineps, maxeps, nlayer: np.append(np.array([1.]),
|
||||
np.ndarray.round(mineps + (maxeps-mineps)* np.random.rand(nlayer,1)
|
||||
,decimals=1))
|
||||
|
||||
#Evaluate Impedance Z of a layer
|
||||
ImpZ = lambda f, mu, k: omega(f)*mu*mu_0/k
|
||||
|
||||
#Complex Cole-Cole Conductivity - EM utils
|
||||
PCC= lambda siginf,m,t,c,f: siginf*(1.-(m/(1.+(1j*omega(f)*t)**c)))
|
||||
|
||||
|
||||
#Converted thickness array into top of layer array
|
||||
def top(thick):
|
||||
topv= np.zeros(len(thick)+1)
|
||||
|
||||
topv[0]=-thick[0]
|
||||
|
||||
for i in range(1,len(topv),1):
|
||||
topv[i] = topv[i-1] + thick[i-1]
|
||||
|
||||
return topv
|
||||
|
||||
#Propagation Matrix and theirs inverses
|
||||
|
||||
#matrix T for transition of Up and Down components accross a layer
|
||||
T = lambda h,k: np.matrix([[np.exp(1j*k*h),0.],[0.,np.exp(-1j*k*h)]],dtype='complex_')
|
||||
|
||||
Tinv = lambda h,k: np.matrix([[np.exp(-1j*k*h),0.],[0.,np.exp(1j*k*h)]],dtype='complex_')
|
||||
|
||||
#transition of Up and Down components accross a layer
|
||||
UD_Z = lambda UD,z,zj,k : T((z-zj),k)*UD
|
||||
|
||||
|
||||
#matrix P relating Up and Down components with E and H fields
|
||||
P = lambda z: np.matrix([[1.,1,],[-1./z,1./z]],dtype='complex_')
|
||||
|
||||
Pinv = lambda z: np.matrix([[1.,-z],[1.,z]],dtype='complex_')/2.
|
||||
|
||||
|
||||
#Time Variation of E and H
|
||||
E_ZT = lambda U,D,f,t : np.exp(1j*omega(f)*t)*(U+D)
|
||||
H_ZT = lambda U,D,Z,f,t : (1./Z)*np.exp(1j*omega(f)*t)*(D-U)
|
||||
|
||||
#Plot the configuration of the problem
|
||||
def PlotConfiguration(thick,sig,eps,mu,ax,widthg,z):
|
||||
|
||||
topn = top(thick)
|
||||
widthn = np.arange(-widthg,widthg+widthg/10.,widthg/10.)
|
||||
|
||||
ax.set_ylim([z.min(),z.max()])
|
||||
ax.set_xlim([-widthg,widthg])
|
||||
|
||||
ax.set_ylabel("Depth (m)", fontsize=16.)
|
||||
ax.yaxis.tick_right()
|
||||
ax.yaxis.set_label_position("right")
|
||||
|
||||
#define filling for the different layers
|
||||
hatches=['/' , '+', 'x', '|' , '\\', '-' , 'o' , 'O' , '.' , '*' ]
|
||||
|
||||
#Write the physical properties of air
|
||||
ax.annotate(("Air, $\sigma$ =%1.0f mS/m")%(sig[0]*10**(3)),
|
||||
xy=(-widthg/2., -np.abs(z.max())/2.), xycoords='data',
|
||||
xytext=(-widthg/2., -np.abs(z.max())/2.), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.annotate(("$\epsilon_r$= %1i")%(eps[0]),
|
||||
xy=(-widthg/2., -np.abs(z.max())/3.), xycoords='data',
|
||||
xytext=(-widthg/2., -np.abs(z.max())/3.), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.annotate(("$\mu_r$= %1i")%(mu[0]),
|
||||
xy=(-widthg/2., -np.abs(z.max())/3.), xycoords='data',
|
||||
xytext=(0, -np.abs(z.max())/3.), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
#Write the physical properties of the differents layers up to the (n-1)-th and fill it with pattern
|
||||
for i in range(1,len(topn)-1,1):
|
||||
if topn[i] == topn[i+1]:
|
||||
pass
|
||||
else:
|
||||
ax.annotate(("$\sigma$ =%3.3f mS/m")%(sig[i]*10**(3)),
|
||||
xy=(0., (2.*topn[i]+topn[i+1])/3), xycoords='data',
|
||||
xytext=(0., (2.*topn[i]+topn[i+1])/3), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.annotate(("$\epsilon_r$= %1i")%(eps[i]),
|
||||
xy=(-widthg/1.1, (2.*topn[i]+topn[i+1])/3), xycoords='data',
|
||||
xytext=(-widthg/1.1, (2.*topn[i]+topn[i+1])/3), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.annotate(("$\mu_r$= %1.2f")%(mu[i]),
|
||||
xy=(-widthg/2., (2.*topn[i]+topn[i+1])/3), xycoords='data',
|
||||
xytext=(-widthg/2., (2.*topn[i]+topn[i+1])/3), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.plot(widthn,topn[i]*np.ones_like(widthn),color='black')
|
||||
ax.fill_between(widthn,topn[i],topn[i+1],alpha=0.3,color="none",edgecolor='black', hatch=hatches[(i-1)%10])
|
||||
|
||||
#Write the physical properties of the n-th layer and fill it with pattern
|
||||
ax.plot(widthn,topn[-1]*np.ones_like(widthn),color='black')
|
||||
ax.fill_between(widthn,topn[-1],z.max(),alpha=0.3,color="none",edgecolor='black', hatch=hatches[(len(topn)-2)%10])
|
||||
|
||||
ax.annotate(("$\sigma$ =%3.3f mS/m")%(sig[-1]*10**(3)),
|
||||
xy=(0., (2.*topn[-1]+z.max())/3), xycoords='data',
|
||||
xytext=(0., (2.*topn[-1]+z.max())/3), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.annotate(("$\epsilon_r$= %1i")%(eps[-1]),
|
||||
xy=(-widthg/1.1, (2.*topn[-1]+z.max())/3), xycoords='data',
|
||||
xytext=(-widthg/1.1, (2.*topn[-1]+z.max())/3), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
ax.annotate(("$\mu_r$= %1.2f")%(mu[-1]),
|
||||
xy=(-widthg/2., (2.*topn[-1]+z.max())/3), xycoords='data',
|
||||
xytext=(-widthg/2., (2.*topn[-1]+z.max())/3), textcoords='data',
|
||||
fontsize=14.)
|
||||
|
||||
#plot Trees!
|
||||
ax.annotate("",
|
||||
xy=(widthg/2., -1.*z.max()/5.), xycoords='data',
|
||||
xytext=(widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.2,head_length=1.2',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(widthg/2., -3./4.*z.max()/5.), xycoords='data',
|
||||
xytext=(widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.4,head_length=1.4',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(widthg/2., -1./2.*z.max()/5.), xycoords='data',
|
||||
xytext=(widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.6,head_length=1.6',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(1.2*widthg/2., -1.*z.max()/5.), xycoords='data',
|
||||
xytext=(1.2*widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.2,head_length=1.2',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(1.2*widthg/2., -3./4.*z.max()/5.), xycoords='data',
|
||||
xytext=(1.2*widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.4,head_length=1.4',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(1.2*widthg/2., -1./2.*z.max()/5.), xycoords='data',
|
||||
xytext=(1.2*widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.6,head_length=1.6',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(1.5*widthg/2., -1.*z.max()/5.), xycoords='data',
|
||||
xytext=(1.5*widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.2,head_length=1.2',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(1.5*widthg/2., -3./4.*z.max()/5.), xycoords='data',
|
||||
xytext=(1.5*widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.4,head_length=1.4',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
ax.annotate("",
|
||||
xy=(1.5*widthg/2., -1./2.*z.max()/5.), xycoords='data',
|
||||
xytext=(1.5*widthg/2., 0.), textcoords='data',
|
||||
arrowprops=dict(arrowstyle='->, head_width=1.6,head_length=1.6',color='green',linewidth=2.)
|
||||
)
|
||||
|
||||
|
||||
ax.invert_yaxis()
|
||||
|
||||
return ax
|
||||
|
||||
#Propagate Up and Down component for a certain frequency & evaluate E and H field
|
||||
|
||||
def Propagate(f,H,sig,chg,taux,c,mu,eps,n):
|
||||
|
||||
sigcm = np.zeros_like(sig,dtype='complex_')
|
||||
|
||||
for j in range(1,len(sig)):
|
||||
sigcm[j]=PCC(sig[j],chg[j],taux[j],c[j],f)
|
||||
|
||||
K = k(mu,sigcm,eps,f)
|
||||
Z = ImpZ(f,mu,K)
|
||||
|
||||
EH = np.matrix(np.zeros((2,n+1),dtype = 'complex_'),dtype = 'complex_')
|
||||
UD = np.matrix(np.zeros((2,n+1),dtype = 'complex_'),dtype = 'complex_')
|
||||
|
||||
UD[1,-1] = 1.
|
||||
|
||||
for i in range(-2,-(n+2),-1):
|
||||
|
||||
UD[:,i] = Tinv(H[i+1],K[i])*Pinv(Z[i])*P(Z[i+1])*UD[:,i+1]
|
||||
UD = UD/((np.abs(UD[0,:]+UD[1,:])).max())
|
||||
|
||||
for j in range(0,n+1):
|
||||
EH[:,j] = np.matrix([[1.,1,],[-1./Z[j],1./Z[j]]])*UD[:,j]
|
||||
|
||||
return UD, EH, Z ,K
|
||||
|
||||
|
||||
#Evaluate the apparent resistivity and phase for a frequency range
|
||||
def appres(F,H,sig,chg,taux,c,mu,eps,n):
|
||||
|
||||
Res = np.zeros_like(F)
|
||||
Phase = np.zeros_like(F)
|
||||
App_ImpZ= np.zeros_like(F,dtype='complex_')
|
||||
|
||||
for i in range(0,len(F)):
|
||||
|
||||
UD,EH,Z ,K = Propagate(F[i],H,sig,chg,taux,c,mu,eps,n)
|
||||
|
||||
App_ImpZ[i] = EH[0,1]/EH[1,1]
|
||||
|
||||
Res[i] = np.abs(App_ImpZ[i])**2./(mu_0*omega(F[i]))
|
||||
Phase[i] = np.angle(App_ImpZ[i], deg = True)
|
||||
|
||||
return Res,Phase
|
||||
|
||||
#Evaluate Up, Down components, E and H field, for a frequency range,
|
||||
#a discretized depth range and a time range (use to calculate envelope)
|
||||
def calculateEHzt(F,H,sig,chg,taux,c,mu,eps,n,zsample,tsample):
|
||||
|
||||
topc = top(H)
|
||||
|
||||
layer = np.zeros(len(zsample),dtype=np.int)-1
|
||||
|
||||
Exzt = np.matrix(np.zeros((len(zsample),len(tsample)),dtype = 'complex_'),dtype = 'complex_')
|
||||
Hyzt = np.matrix(np.zeros((len(zsample),len(tsample)),dtype = 'complex_'),dtype = 'complex_')
|
||||
Uz = np.matrix(np.zeros((len(zsample),len(tsample)),dtype = 'complex_'),dtype = 'complex_')
|
||||
Dz = np.matrix(np.zeros((len(zsample),len(tsample)),dtype = 'complex_'),dtype = 'complex_')
|
||||
UDaux = np.matrix(np.zeros((2,len(zsample)),dtype = 'complex_'),dtype = 'complex_')
|
||||
|
||||
for i in range(0,n+1,1):
|
||||
layer = layer+(zsample>=topc[i])*1
|
||||
|
||||
for j in range(0,len(F)):
|
||||
|
||||
UD,EH,Z ,K = Propagate(F[j],H,sig,chg,taux,c,mu,eps,n)
|
||||
|
||||
for p in range(0,len(zsample)):
|
||||
|
||||
UDaux[:,p] = UD_Z(UD[:,layer[p]],zsample[p],topc[layer[p]],K[layer[p]])
|
||||
|
||||
for q in range(0,len(tsample)):
|
||||
|
||||
Exzt[p,q] = Exzt[p,q] + E_ZT(UDaux[0,p],UDaux[1,p],F[j],tsample[q])/len(F)
|
||||
Hyzt[p,q] = Hyzt[p,q] + H_ZT(UDaux[0,p],UDaux[1,p],Z[layer[p]],F[j],tsample[q])/len(F)
|
||||
Uz[p,q] = Uz[p,q] + UDaux[0,p]*np.exp(1j*omega(F[j])*tsample[q])/len(F)
|
||||
Dz[p,q] = Dz[p,q] + UDaux[1,p]*np.exp(1j*omega(F[j])*tsample[q])/len(F)
|
||||
|
||||
return Exzt,Hyzt,Uz,Dz,UDaux,layer
|
||||
|
||||
|
||||
#Function to Plot Apparent Resistivity and Phase
|
||||
def PlotAppRes(F,H,sig,chg,taux,c,mu,eps,n,fenvelope,PlotEnvelope):
|
||||
|
||||
Res, Phase = appres(F,H,sig,chg,taux,c,mu,eps,n)
|
||||
|
||||
fig,ax = plt.subplots(1,2,figsize=(16,10))
|
||||
|
||||
ax[0].scatter(Res,F,color='black')
|
||||
ax[0].set_xscale('Log')
|
||||
ax[0].set_yscale('Log')
|
||||
ax[0].set_xlim([10.**(np.log10(Res.min())-1.),10.**(np.log10(Res.max())+1.)])
|
||||
ax[0].set_ylim([F.min(),F.max()])
|
||||
ax[0].set_xlabel('Apparent Resistivity (Ohm*m)',fontsize=16.,color="black")
|
||||
ax[0].set_ylabel('Frequency (Hz)',fontsize=16.)
|
||||
ax[0].grid(which='major')
|
||||
|
||||
ax0 = ax[0].twiny()
|
||||
|
||||
ax0.set_xlim([0.,90.])
|
||||
ax0.set_ylim([F.min(),F.max()])
|
||||
ax0.scatter(Phase,F,color='purple')
|
||||
ax0.set_xlabel('Phase (Degrees)',fontsize=16.,color="purple")
|
||||
|
||||
zc=np.arange(-(H[1:].max()+10)*n,(H[1:].max()+10)*n,10.)
|
||||
|
||||
ax[0].tick_params(labelsize=16)
|
||||
ax[1].tick_params(labelsize=16)
|
||||
ax0.tick_params(labelsize=16)
|
||||
|
||||
if PlotEnvelope:
|
||||
|
||||
widthn=np.logspace(np.log10(Res.min())-1., np.log10(Res.max())+1., num=100, endpoint=True, base=10.0)
|
||||
fenvelope1n=np.ones(100)*fenvelope
|
||||
ax[0].plot(widthn,fenvelope1n,linestyle='dashed',color='black')
|
||||
|
||||
tc=np.arange(0.,1./fenvelope,0.01/(fenvelope))
|
||||
Exzt,Hyzt,Uz,Dz,UDaux,layer = calculateEHzt(np.array([fenvelope]),H,sig,chg,taux,c,mu,eps,n,zc,tc)
|
||||
|
||||
ax1=ax[1].twiny()
|
||||
|
||||
ax[1].tick_params(labelsize=16)
|
||||
ax1.tick_params(labelsize=16)
|
||||
|
||||
ax[1].set_xlabel('Amplitude Electric Field E (V/m)',color='blue',fontsize=16)
|
||||
|
||||
ax1.set_xlabel('Amplitude Magnetic Field H (A/m)',color='red',fontsize=16)
|
||||
|
||||
ax[1].fill_betweenx(zc,np.squeeze(np.asarray(np.real(Exzt.min(axis=1)))),
|
||||
np.squeeze(np.asarray(np.real(Exzt.max(axis=1)))),
|
||||
color='blue', alpha=0.1)
|
||||
|
||||
ax1.fill_betweenx(zc,np.squeeze(np.asarray(np.real(Hyzt.min(axis=1)))),
|
||||
np.squeeze(np.asarray(np.real(Hyzt.max(axis=1)))),
|
||||
color='red', alpha=0.1)
|
||||
|
||||
ax[1] = PlotConfiguration(H,sig,eps,mu,ax[1],(1.5*np.abs(Exzt).max()),zc)
|
||||
ax1.set_xlim([-1.5*np.abs(Hyzt).max(),1.5*np.abs(Hyzt).max()])
|
||||
ax1.set_xlim([-1.5*np.abs(Hyzt).max(),1.5*np.abs(Hyzt).max()])
|
||||
else:
|
||||
print 'No envelop (if True, might be slow)'
|
||||
ax[1] = PlotConfiguration(H,sig,eps,mu,ax[1],1.,zc)
|
||||
ax[1].get_xaxis().set_ticks([])
|
||||
|
||||
plt.show()
|
||||
|
||||
#Interactive MT for Notebook
|
||||
def PlotAppRes3LayersInteract(h1,h2,sigl1,sigl2,sigl3,mul1,mul2,mul3,epsl1,epsl2,epsl3,PlotEnvelope,F_Envelope):
|
||||
|
||||
frangn=frange(-5,5,100.)
|
||||
sig3= np.array([0.,0.001,0.1, 0.001])
|
||||
thick3 = np.array([120000.,50.,50.])
|
||||
eps3=np.array([1.,1.,1.,1])
|
||||
mu3=np.array([1.,1.,1.,1])
|
||||
chg3=np.array([0.,0.1,0.,0.2])
|
||||
chg3_0=np.array([0.,0.1,0.,0.])
|
||||
taux3=np.array([0.,0.1,0.,0.1])
|
||||
c3=np.array([1.,1.,1.,1.])
|
||||
|
||||
sig3[1]=sigl1
|
||||
sig3[1]=10.**sig3[1]
|
||||
sig3[2]=sigl2
|
||||
sig3[2]=10.**sig3[2]
|
||||
sig3[3]=sigl3
|
||||
sig3[3]=10.**sig3[3]
|
||||
mu3[1]=mul1
|
||||
mu3[2]=mul2
|
||||
mu3[3]=mul3
|
||||
eps3[1]=epsl1
|
||||
eps3[2]=epsl2
|
||||
eps3[3]=epsl3
|
||||
thick3[1]=h1
|
||||
thick3[2]=h2
|
||||
|
||||
PlotAppRes(frangn,thick3,sig3,chg3_0,taux3,c3,mu3,eps3,3,F_Envelope,PlotEnvelope)
|
||||
|
||||
|
||||
def run(n,plotIt=True):
|
||||
# something to make a plot
|
||||
|
||||
F = frange(-5.,5.,20)
|
||||
H = thick(50.,100.,n)
|
||||
sign = sig(-5.,0.,n)
|
||||
mun = mu(1.,2.,n)
|
||||
epsn = eps(1.,9.,n)
|
||||
chg = np.zeros_like(sign)
|
||||
taux = np.zeros_like(sign)
|
||||
c = np.zeros_like(sign)
|
||||
|
||||
Res, Phase = appres(F,H,sign,chg,taux,c,mun,epsn,n)
|
||||
|
||||
if plotIt:
|
||||
|
||||
PlotAppRes(F, H, sign, chg, taux, c, mun, epsn, n, fenvelope=1000., PlotEnvelope=True)
|
||||
|
||||
return Res, Phase
|
||||
|
||||
if __name__ == '__main__':
|
||||
run(3)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@@ -1,10 +1,10 @@
|
||||
# Run this file to add imports.
|
||||
|
||||
##### AUTOIMPORTS #####
|
||||
import DC_Analytic_Dipole
|
||||
import DC_Forward_PseudoSection
|
||||
#import DC_PseudoSection_Simulation
|
||||
import EM_FDEM_1D_Inversion
|
||||
import EM_FDEM_Analytic_MagDipoleWholespace
|
||||
#import EM_FDEM_SusEffects
|
||||
import EM_TDEM_1D_Inversion
|
||||
import FLOW_Richards_1D_Celia1990
|
||||
import Forward_BasicDirectCurrent
|
||||
@@ -16,10 +16,8 @@ import Mesh_QuadTree_Creation
|
||||
import Mesh_QuadTree_FaceDiv
|
||||
import Mesh_QuadTree_HangingNodes
|
||||
import Mesh_Tensor_Creation
|
||||
import MT_1D_ForwardAndInversion
|
||||
import MT_3D_Foward
|
||||
|
||||
__examples__ = ["DC_Analytic_Dipole", "DC_Forward_PseudoSection", "EM_FDEM_1D_Inversion", "EM_FDEM_Analytic_MagDipoleWholespace", "EM_TDEM_1D_Inversion", "FLOW_Richards_1D_Celia1990", "Forward_BasicDirectCurrent", "Inversion_Linear", "Mesh_Basic_PlotImage", "Mesh_Basic_Types", "Mesh_Operators_CahnHilliard", "Mesh_QuadTree_Creation", "Mesh_QuadTree_FaceDiv", "Mesh_QuadTree_HangingNodes", "Mesh_Tensor_Creation", "MT_1D_ForwardAndInversion", "MT_3D_Foward"]
|
||||
__examples__ = ["DC_PseudoSection_Simulation", "EM_FDEM_1D_Inversion", "EM_FDEM_Analytic_MagDipoleWholespace", "EM_FDEM_SusEffects", "EM_TDEM_1D_Inversion", "FLOW_Richards_1D_Celia1990", "Forward_BasicDirectCurrent", "Inversion_Linear", "Mesh_Basic_PlotImage", "Mesh_Basic_Types", "Mesh_Operators_CahnHilliard", "Mesh_QuadTree_Creation", "Mesh_QuadTree_FaceDiv", "Mesh_QuadTree_HangingNodes", "Mesh_Tensor_Creation"]
|
||||
|
||||
##### AUTOIMPORTS #####
|
||||
|
||||
|
||||
@@ -0,0 +1,775 @@
|
||||
from scipy.constants import epsilon_0
|
||||
import matplotlib.pyplot as plt
|
||||
import matplotlib.colors as colors
|
||||
import numpy as np
|
||||
from SimPEG.Utils import ndgrid, mkvc
|
||||
|
||||
'''
|
||||
Authors: Thibaut Astic, Lindsey Heagy, Sanna Tyrvainen, Ronghua Peng
|
||||
Date: December 2015
|
||||
|
||||
This code defines function to resolve analytically the electrostatic sphere problem.
|
||||
We first define a problem configuration, with a conductive or resistive sphere in a
|
||||
wholespace background.
|
||||
We then calculate the potential, then the electric field, then the current density and
|
||||
finally the charges accumulation.
|
||||
|
||||
Several plotting functions are defined for data visualisation.
|
||||
|
||||
|
||||
'''
|
||||
|
||||
# Plot options
|
||||
ftsize_title = 18 #font size for titles
|
||||
ftsize_axis = 14 #font size for axis ticks
|
||||
ftsize_label = 14 #font size for axis labels
|
||||
|
||||
# Radius function, useful sigma ratio, and log scale converter
|
||||
r = lambda x,y,z: np.sqrt(x**2.+y**2.+z**2.)
|
||||
sigf = lambda sig0,sig1: (sig1-sig0)/(sig1+2.*sig0)
|
||||
|
||||
#tools to convert log conductivity in conductivity
|
||||
def conductivity_log_wrapper(log_sig0,log_sig1):
|
||||
sig0 = 10.**log_sig0
|
||||
sig1 = 10.**log_sig1
|
||||
|
||||
return sig0,sig1
|
||||
|
||||
# Examples
|
||||
#Plot the configuration. Label=False is used to generate a general case figure
|
||||
def get_Setup(XYZ,sig0,sig1,R,E0,ax,label,colorsphere):
|
||||
'''
|
||||
XYZ: ndgrid
|
||||
sig0: conductivity of the background
|
||||
sig1: conductivity of the sphere
|
||||
R: radius of the sphere
|
||||
E0: Amplitude of the uniform electrostatic field
|
||||
ax: ax where to plot the configuration
|
||||
label: True: plot real values, False: plot general case
|
||||
colorsphere: color of the sphere, format [x,x,x]
|
||||
'''
|
||||
|
||||
xplt = np.linspace(-R, R, num=100)
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
dx = xr[1]-xr[0]
|
||||
top = np.sqrt(R**2-xplt**2)
|
||||
bot = -np.sqrt(R**2-xplt**2)
|
||||
|
||||
if R != 0:
|
||||
ax.plot(xplt, top, xplt, bot, color=colorsphere,linewidth=1.5)
|
||||
ax.fill_between(xplt,bot,top,color=colorsphere,alpha=0.5 )
|
||||
ax.arrow(0.,0.,np.sqrt(2.)*R/2.,np.sqrt(2.)*R/2.,head_width=0.,head_length=0.)
|
||||
|
||||
if label:
|
||||
ax.annotate(("$\sigma_1$=%3.3f mS/m")%(sig1*10.**(3.)),
|
||||
xy=(0.,-R/2.), xycoords='data',
|
||||
xytext=(0.,-R/2.), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.annotate(("$\sigma_0$= %3.3f mS/m")%(sig0*10.**(3.)),
|
||||
xy=(0.,-1.5*R), xycoords='data',
|
||||
xytext=(0.,-1.5*R), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.annotate(('$\mathbf{E_0} = %1i \mathbf{\hat{x}}$ V/m')%(E0),
|
||||
xy=(xr.min()+np.abs(xr.max()-xr.min())/20.,0), xycoords='data',
|
||||
xytext=(xr.min()+np.abs(xr.max()-xr.min())/20.,0), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.annotate(('$R$ = %1i m')%(R),
|
||||
xy=(R/4.+(xr[1]-xr[0]),R/4.), xycoords='data',
|
||||
xytext=(R/4.+(xr[1]-xr[0]),R/4.), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
|
||||
else:
|
||||
ax.set_xticklabels([])
|
||||
ax.set_yticklabels([])
|
||||
ax.text(-1.,-np.sqrt(R)/2.-10.,'$\sigma_1$',fontsize=14)
|
||||
ax.text(-0.05,-R-10,'$\sigma_0$',fontsize=14)
|
||||
ax.annotate(('$\mathbf{E_0} = E_0 \mathbf{\hat{x}}$ V/m'),
|
||||
xy=(xr.min()+np.abs(xr.max()-xr.min())/20.,0), xycoords='data',
|
||||
xytext=(xr.min()+np.abs(xr.max()-xr.min())/20.,0), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.annotate(('$R$'),
|
||||
xy=(R/4.+(xr[1]-xr[0]),R/4.), xycoords='data',
|
||||
xytext=(R/4.+(xr[1]-xr[0]),R/4.), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.set_xlabel('x',fontsize=12)
|
||||
ax.set_ylabel('y',fontsize=12)
|
||||
|
||||
else:
|
||||
if label:
|
||||
ax.annotate(("$\sigma_0$= %3.3f mS/m")%(sig0*10.**(3.)),
|
||||
xy=(0.,-1.5*R), xycoords='data',
|
||||
xytext=(0.,-1.5*R), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.annotate(('$\mathbf{E_0} = %1i \mathbf{\hat{x}}$ V/m')%(E0),
|
||||
xy=(xr.min()+np.abs(xr.max()-xr.min())/20.,0), xycoords='data',
|
||||
xytext=(xr.min()+np.abs(xr.max()-xr.min())/20.,0), textcoords='data',
|
||||
fontsize=14.)
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
|
||||
else:
|
||||
ax.set_xticklabels([])
|
||||
ax.set_yticklabels([])
|
||||
ax.text(-0.05,-10,'$\sigma_0$',fontsize=14)
|
||||
ax.text(xr.min()+np.abs(xr.max()-xr.min())/20., 0, '$\mathbf{E_0} = E_0 \mathbf{\hat{x}}$ V/m', fontsize=14)
|
||||
ax.set_xlabel('x',fontsize=12)
|
||||
ax.set_ylabel('y',fontsize=12)
|
||||
|
||||
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
[ax.arrow(xr.min(),_,np.abs(xr.max()-xr.min())/20.,0.,head_width=5.,head_length=2.,color='k') for _ in np.linspace(yr.min(),yr.max(),num=10)]
|
||||
ax.patch.set_facecolor([0.4,0.7,0.4])
|
||||
ax.patch.set_alpha(0.2)
|
||||
|
||||
ax.set_aspect('equal')
|
||||
|
||||
|
||||
|
||||
return ax
|
||||
|
||||
def get_Conductivity(XYZ,sig0,sig1,R):
|
||||
'''
|
||||
Define the conductivity for each point of the space
|
||||
'''
|
||||
x,y,z = XYZ[:,0],XYZ[:,1],XYZ[:,2]
|
||||
r_view=r(x,y,z)
|
||||
|
||||
ind0= (r_view>R)
|
||||
ind1= (r_view<=R)
|
||||
|
||||
assert (ind0 + ind1).all(), 'Some indicies not included'
|
||||
|
||||
Sigma = np.zeros_like(x)
|
||||
|
||||
Sigma[ind0] = sig0
|
||||
Sigma[ind1] = sig1
|
||||
|
||||
return Sigma
|
||||
|
||||
|
||||
def get_Potential(XYZ,sig0,sig1,R,E0):
|
||||
|
||||
'''
|
||||
Function that returns the total, the primary and the secondary potentials, assumes an x-oriented inducing field and that the sphere is at the origin
|
||||
:input: grid, outer sigma, inner sigma, radius of the sphere, strength of the electric field
|
||||
'''
|
||||
|
||||
x,y,z = XYZ[:,0],XYZ[:,1],XYZ[:,2]
|
||||
|
||||
sig_cur = sigf(sig0,sig1)
|
||||
|
||||
r_cur = r(x,y,z) # current radius
|
||||
|
||||
ind0 = (r_cur > R)
|
||||
ind1 = (r_cur <= R)
|
||||
|
||||
assert (ind0 + ind1).all(), 'Some indicies not included'
|
||||
|
||||
Vt = np.zeros_like(x)
|
||||
Vp = np.zeros_like(x)
|
||||
Vs = np.zeros_like(x)
|
||||
|
||||
Vt[ind0] = -E0*x[ind0]*(1.-sig_cur*R**3./r_cur[ind0]**3.) # total potential outside the sphere
|
||||
Vt[ind1] = -E0*x[ind1]*3.*sig0/(sig1+2.*sig0) # inside the sphere
|
||||
|
||||
|
||||
Vp = - E0*x # primary potential
|
||||
|
||||
Vs = Vt - Vp # secondary potential
|
||||
|
||||
return Vt,Vp,Vs
|
||||
|
||||
#plot the primary potential on ax
|
||||
def Plot_Primary_Potential(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Vt,Vp,Vs = get_Potential(XYZ,sig0,sig1,R,E0)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
Pplot = ax.pcolor(xr,yr,Vp.reshape(xr.size,yr.size))
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_title('Primary Potential',fontsize=ftsize_title)
|
||||
cb = plt.colorbar(Pplot,ax=ax)
|
||||
cb.set_label(label= 'Potential ($V$)',size=ftsize_label)
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_aspect('equal')
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
|
||||
return ax
|
||||
|
||||
#plot the total potential on ax
|
||||
def Plot_Total_Potential(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Vt,Vp,Vs = get_Potential(XYZ,sig0,sig1,R,E0)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
|
||||
Pplot = ax.pcolor(xr,yr,Vt.reshape(xr.size,yr.size))
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_title('Total Potential',fontsize=ftsize_title)
|
||||
cb = plt.colorbar(Pplot,ax=ax)
|
||||
cb.set_label(label= 'Potential ($V$)',size=ftsize_label)
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_aspect('equal')
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
|
||||
return ax
|
||||
|
||||
#plot the secondary potential on ax
|
||||
def Plot_Secondary_Potential(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Vt,Vp,Vs = get_Potential(XYZ,sig0,sig1,R,E0)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
Pplot = ax.pcolor(xr,yr,Vs.reshape(xr.size,yr.size))
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_title('Secondary Potential',fontsize=ftsize_title)
|
||||
cb = plt.colorbar(Pplot,ax=ax)
|
||||
cb.set_label(label= 'Potential ($V$)',size=ftsize_label)
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_aspect('equal')
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
|
||||
return ax
|
||||
|
||||
|
||||
def get_ElectricField(XYZ,sig0,sig1,R,E0):
|
||||
'''
|
||||
Function that returns the total, the primary and the secondary electric fields,
|
||||
input: grid, outer sigma, inner sigma, radius of the sphere, strength of the electric field
|
||||
'''
|
||||
|
||||
x,y,z= XYZ[:,0], XYZ[:,1], XYZ[:,2]
|
||||
|
||||
r_cur=r(x,y,z) # current radius
|
||||
|
||||
ind0= (r_cur>R)
|
||||
ind1= (r_cur<=R)
|
||||
|
||||
assert (ind0 + ind1).all(), 'Some indicies not included'
|
||||
|
||||
Ep = np.zeros(shape=(len(x),3))
|
||||
Ep[:,0] = E0
|
||||
|
||||
Et = np.zeros(shape=(len(x),3))
|
||||
|
||||
Et[ind0,0] = E0 + E0*R**3./(r_cur[ind0]**5.)*sigf(sig0,sig1)*(2.*x[ind0]**2.-y[ind0]**2.-z[ind0]**2.);
|
||||
Et[ind0,1] = E0*R**3./(r_cur[ind0]**5.)*3.*x[ind0]*y[ind0]*sigf(sig0,sig1);
|
||||
Et[ind0,2] = E0*R**3./(r_cur[ind0]**5.)*3.*x[ind0]*z[ind0]*sigf(sig0,sig1);
|
||||
|
||||
Et[ind1,0] = 3.*sig0/(sig1+2.*sig0)*E0;
|
||||
Et[ind1,1] = 0.;
|
||||
Et[ind1,2] = 0.;
|
||||
|
||||
Es = Et - Ep
|
||||
|
||||
return Et, Ep, Es
|
||||
|
||||
#plot the total electric field on ax
|
||||
def Plot_Total_ElectricField(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Et, Ep, Es = get_ElectricField(XYZ,sig0,sig1,R,E0)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
EtXr = Et[:,0].reshape(xr.size, yr.size)
|
||||
EtYr = Et[:,1].reshape(xr.size, yr.size)
|
||||
EtAmp = np.sqrt(Et[:,0]**2+Et[:,1]**2 + Et[:,2]**2).reshape(xr.size, yr.size)
|
||||
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_aspect('equal')
|
||||
|
||||
Eplot = ax.pcolor(xr,yr,EtAmp)
|
||||
cb = plt.colorbar(Eplot,ax=ax)
|
||||
cb.set_label(label= 'Amplitude ($V/m$)',size=ftsize_label) #weight='bold')
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.streamplot(xr,yr,EtXr,EtYr,color='gray',linewidth=2.,density=0.75)#angles='xy',scale_units='xy',scale=0.05)
|
||||
ax.set_title('Total Field',fontsize=ftsize_title)
|
||||
|
||||
|
||||
return ax
|
||||
|
||||
#plot the secondary electric field on ax
|
||||
def Plot_Secondary_ElectricField(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Et, Ep, Es = get_ElectricField(XYZ,sig0,sig1,R,E0)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
EsXr = Es[:,0].reshape(xr.size, yr.size)
|
||||
EsYr = Es[:,1].reshape(xr.size, yr.size)
|
||||
EsAmp = np.sqrt(Es[:,0]**2+Es[:,1]**2+Es[:,2]**2).reshape(xr.size, yr.size)
|
||||
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize = ftsize_label)
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_aspect('equal')
|
||||
|
||||
Eplot = ax.pcolor(xr,yr,EsAmp)
|
||||
cb = plt.colorbar(Eplot,ax=ax)
|
||||
cb.set_label(label= 'Amplitude ($V/m$)',size=ftsize_label) #weight='bold')
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.streamplot(xr,yr,EsXr,EsYr,color='gray',linewidth=2.,density=0.75)#,angles='xy',scale_units='xy',scale=0.05)
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_title('Secondary Field',fontsize=ftsize_title)
|
||||
|
||||
return ax
|
||||
|
||||
|
||||
def get_Current(XYZ,sig0,sig1,R,Et,Ep,Es):
|
||||
'''
|
||||
Function that returns the total, the primary and the secondary current densities,
|
||||
:input: grid, outer sigma, inner sigma, radius of the sphere, total, the primary and the seconadry electric fields,
|
||||
'''
|
||||
|
||||
x,y,z= XYZ[:,0], XYZ[:,1], XYZ[:,2]
|
||||
|
||||
r_cur=r(x,y,z)
|
||||
|
||||
ind0= (r_cur>R)
|
||||
ind1= (r_cur<=R)
|
||||
|
||||
assert (ind0 + ind1).all(), 'Some indicies not included'
|
||||
|
||||
Jt = np.zeros(shape=(len(x),3))
|
||||
J0 = np.zeros(shape=(len(x),3))
|
||||
Js = np.zeros(shape=(len(x),3))
|
||||
|
||||
|
||||
Jp = sig0*Ep
|
||||
|
||||
Jt[ind0,:] = sig0*Et[ind0,:]
|
||||
Jt[ind1,:] = sig1*Et[ind1,:]
|
||||
|
||||
Js[ind0,:] = sig0*(Et[ind0,:]-Ep[ind0,:])
|
||||
Js[ind1,:] = sig1*Et[ind1,:]-sig0*Ep[ind1,:]
|
||||
|
||||
return Jt,Jp,Js
|
||||
|
||||
#plot the total currents density on ax
|
||||
def Plot_Total_Currents(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Et,Ep,Es = get_ElectricField(XYZ,sig0,sig1,R,E0)
|
||||
Jt,Jp,Js = get_Current(XYZ,sig0,sig1,R,Et,Ep,Es)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
JtXr = Jt[:,0].reshape(xr.size, yr.size)
|
||||
JtYr = Jt[:,1].reshape(xr.size, yr.size)
|
||||
JtAmp = np.sqrt(Jt[:,0]**2+Jt[:,1]**2+Jt[:,2]**2).reshape(xr.size, yr.size)
|
||||
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize=ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize=ftsize_label)
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_aspect('equal')
|
||||
|
||||
Jplot = ax.pcolor(xr,yr,JtAmp.reshape(xr.size,yr.size))
|
||||
cb = plt.colorbar(Jplot,ax=ax)
|
||||
cb.set_label(label= 'Current Density ($A/m^2$)',size=ftsize_label) #weight='bold')
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.streamplot(xr,yr,JtXr,JtYr,color='gray',linewidth=2.,density=0.75)#,angles='xy',scale_units='xy',scale=1)
|
||||
ax.set_title('Total Current Density',fontsize=ftsize_title)
|
||||
|
||||
return ax
|
||||
|
||||
|
||||
#plot the secondary currents density on ax
|
||||
def Plot_Secondary_Currents(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
Et,Ep,Es = get_ElectricField(XYZ,sig0,sig1,R,E0)
|
||||
Jt,Jp,Js = get_Current(XYZ,sig0,sig1,R,Et,Ep,Es)
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
JsXr = Js[:,0].reshape(xr.size, yr.size)
|
||||
JsYr = Js[:,1].reshape(xr.size, yr.size)
|
||||
JsAmp = np.sqrt(Js[:,1]**2+Js[:,0]**2+Jt[:,2]**2).reshape(xr.size,yr.size)
|
||||
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize=ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize=ftsize_label)
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_aspect('equal')
|
||||
|
||||
Jplot = ax.pcolor(xr,yr,JsAmp.reshape(xr.size,yr.size))
|
||||
cb = plt.colorbar(Jplot,ax=ax)
|
||||
cb.set_label(label= 'Current Density ($A/m^2$)',size=ftsize_label) #weight='bold')
|
||||
cb.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.streamplot(xr,yr,JsXr,JsYr,color='gray',linewidth=2.,density=0.75)#,angles='xy',scale_units='xy',scale=1)
|
||||
ax.set_title('Secondary Current Density',fontsize=ftsize_title)
|
||||
|
||||
return ax
|
||||
|
||||
|
||||
def get_ChargesDensity(XYZ,sig0,sig1,R,Et,Ep):
|
||||
'''
|
||||
Function that returns the charges accumulation at the background/sphere interface,
|
||||
:input: grid, outer sigma, inner sigma, radius of the sphere, total and the primary electric fields,
|
||||
'''
|
||||
|
||||
x,y,z= XYZ[:,0], XYZ[:,1], XYZ[:,2]
|
||||
|
||||
dx = x[1]-x[0]
|
||||
|
||||
r_cur=r(x,y,z)
|
||||
|
||||
ind0 = (r_cur > R)
|
||||
ind1 = (r_cur < R)
|
||||
ind2 = ((r_cur < (R+dx/2)) & (r_cur > (R-dx/2)) )
|
||||
|
||||
assert (ind0 + ind1 + ind2).all(), 'Some indicies not included'
|
||||
|
||||
rho = np.zeros_like(x)
|
||||
|
||||
rho[ind0] = 0
|
||||
rho[ind1] = 0
|
||||
rho[ind2] = epsilon_0*3.*Ep[ind2,0]*sigf(sig0,sig1)*x[ind2]/(np.sqrt(x[ind2]**2.+y[ind2]**2.))
|
||||
|
||||
return rho
|
||||
|
||||
#Plot charges density on ax
|
||||
def Plot_ChargesDensity(XYZ,sig0,sig1,R,E0,ax):
|
||||
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
xcirc = xr[np.abs(xr) <= R]
|
||||
|
||||
Et, Ep, Es = get_ElectricField(XYZ,sig0,sig1,R,E0)
|
||||
rho = get_ChargesDensity(XYZ,sig0,sig1,R,Et,Ep)
|
||||
|
||||
ax.set_xlim([xr.min(),xr.max()])
|
||||
ax.set_ylim([yr.min(),yr.max()])
|
||||
ax.set_aspect('equal')
|
||||
Cplot = ax.pcolor(xr,yr,rho.reshape(xr.size, yr.size))
|
||||
cb1 = plt.colorbar(Cplot,ax=ax)
|
||||
cb1.set_label(label= 'Charge Density ($C/m^2$)',size=ftsize_label) #weight='bold')
|
||||
cb1.ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.plot(xcirc,np.sqrt(R**2-xcirc**2),'--k',xcirc,-np.sqrt(R**2-xcirc**2),'--k')
|
||||
ax.set_ylabel('Y coordinate ($m$)',fontsize=ftsize_label)
|
||||
ax.set_xlabel('X coordinate ($m$)',fontsize=ftsize_label)
|
||||
ax.tick_params(labelsize=ftsize_axis)
|
||||
ax.set_title('Charges Density', fontsize=ftsize_title)
|
||||
|
||||
return ax
|
||||
|
||||
def MN_Potential_total(sig0,sig1,R,E0,start,end,nbmp,mn):
|
||||
|
||||
'''
|
||||
Function that return array of midpoints electrodes, electrodes positions,
|
||||
potentials differences for total and secondary potentials fields, unormalized and
|
||||
normalized to electrodes distances.
|
||||
sig0: background conductivity
|
||||
sig1: sphere conductivity
|
||||
R: Sphere's radius
|
||||
E0: uniform E field value
|
||||
start: start point for the profile start.shape = (2,)
|
||||
end: end point for the profile end.shape = (2,)
|
||||
nbmp: number of dipoles
|
||||
mn: Space between the M and N electrodes
|
||||
'''
|
||||
|
||||
#D: total distance from start to end
|
||||
D = np.sqrt((start[0]-end[0])**2.+(start[1]-end[1])**2.)
|
||||
|
||||
#MP: dipoles'midpoint positions (x,y)
|
||||
MP = np.zeros(shape=(nbmp,2))
|
||||
MP[:,0] = np.linspace(start[0],end[0],nbmp)
|
||||
MP[:,1] = np.linspace(start[1],end[1],nbmp)
|
||||
|
||||
#Dipoles'Electrodes positions around each midpoints
|
||||
EL = np.zeros(shape=(2*nbmp,2))
|
||||
for n in range(0,len(EL),2):
|
||||
EL[n,0] = MP[n/2,0] - ((end[0]-start[0])/D)*mn/2.
|
||||
EL[n+1,0] = MP[n/2,0] + ((end[0]-start[0])/D)*mn/2.
|
||||
EL[n,1] = MP[n/2,1] - ((end[1]-start[1])/D)*mn/2.
|
||||
EL[n+1,1] = MP[n/2,1] + ((end[1]-start[1])/D)*mn/2.
|
||||
|
||||
VtEL = np.zeros(2*nbmp) #Total Potential (Vt-) at each electrode (-EL)
|
||||
VsEL = np.zeros(2*nbmp) #Secondary Potential (Vt-) at each electrode (-EL)
|
||||
dVtMP = np.zeros(nbmp) #Diffence (d-) of Total Potential (Vt-) at each dipole (-MP)
|
||||
dVtMPn = np.zeros(nbmp) #Diffence (d-) of Total Potential (Vt-) at each dipole (-MP) normalized for the mn spacing (n)
|
||||
dVsMP = np.zeros(nbmp) #Diffence (d-) of Secondaty Potential (Vt-) at each dipole (-MP)
|
||||
dVsMPn = np.zeros(nbmp) #Diffence (d-) of Secondary Potential (Vt-) at each dipole (-MP) normalized for the mn spacing (n)
|
||||
dVpMP = np.zeros(nbmp) #Diffence (d-) of Primary Potential (Vt-) at each dipole (-MP)
|
||||
dVpMPn = np.zeros(nbmp) #Diffence (d-) of Primary Potential (Vt-) at each dipole (-MP) normalized for the mn spacing (n)
|
||||
|
||||
#Computing VtEL
|
||||
for m in range(0,2*nbmp):
|
||||
if (r(EL[m,0],EL[m,1],0) > R):
|
||||
VtEL[m] = -E0*EL[m,0]*(1.-sigf(sig0,sig1)*R**3./r(EL[m,0],EL[m,1],0)**3.)
|
||||
else:
|
||||
VtEL[m] = -E0*EL[m,0]*3.*sig0/(sig1+2.*sig0)
|
||||
|
||||
#Computing VsEL
|
||||
VsEL = VtEL + E0*EL[:,0]
|
||||
|
||||
#Computing dVtMP, dVsMP
|
||||
for p in range(0,nbmp):
|
||||
dVtMP[p] = VtEL[2*p]-VtEL[2*p+1]
|
||||
dVtMPn[p] = dVtMP[p]/mn
|
||||
dVsMP[p] = VsEL[2*p]-VsEL[2*p+1]
|
||||
dVsMPn[p] = dVsMP[p]/mn
|
||||
|
||||
return MP,EL,dVtMP,dVtMPn,dVsMP,dVsMPn
|
||||
|
||||
#Compare the DC response of two configurations
|
||||
def two_configurations_comparison(XYZ,sig0,sig1,sig2,R0,R1,E0,xstart,ystart,xend,yend,nb_dipole,electrode_spacing,PlotOpt):#,linearcolor):
|
||||
|
||||
#Define the mesh
|
||||
xr,yr,zr = np.unique(XYZ[:,0]),np.unique(XYZ[:,1]),np.unique(XYZ[:,2])
|
||||
|
||||
#Defining the Profile
|
||||
start = np.array([xstart,ystart])
|
||||
end = np.array([xend,yend])
|
||||
|
||||
#Calculating the data from the defined survey line for Configuration 0 and 1
|
||||
MP0,EL0,VtdMP0,VtdMPn0,VsdMP0,VsdMPn0 = MN_Potential_total(sig0,sig1,R0,E0,start,end,nb_dipole,electrode_spacing)
|
||||
MP1,EL1,VtdMP1,VtdMPn1,VsdMP1,VsdMPn1 = MN_Potential_total(sig0,sig2,R1,E0,start,end,nb_dipole,electrode_spacing)
|
||||
|
||||
|
||||
# Initializing the figure
|
||||
fig = plt.figure(figsize=(20,20))
|
||||
ax0 = plt.subplot2grid((20,12), (0, 0),colspan=6,rowspan=6)
|
||||
ax1 = plt.subplot2grid((20,12), (0, 6),colspan=6,rowspan=6)
|
||||
ax2 = plt.subplot2grid((20,12), (16, 2), colspan=9,rowspan=4)
|
||||
ax3 = plt.subplot2grid((20,12), (8, 0),colspan=6,rowspan=6)
|
||||
ax4 = plt.subplot2grid((20,12), (8, 6),colspan=6,rowspan=6)
|
||||
|
||||
#Plotting the Configuration 0
|
||||
ax0 = get_Setup(XYZ,sig0,sig1,R0,E0,ax0,True,[0.6,0.1,0.1])
|
||||
|
||||
#Plotting the Configuration 1
|
||||
ax1 = get_Setup(XYZ,sig0,sig2,R1,E0,ax1,True,[0.1,0.1,0.6])
|
||||
|
||||
#Plotting the Data (Legends)
|
||||
ax2.set_title('Potential Differences',fontsize=ftsize_title)
|
||||
ax2.set_ylabel('Potential difference ($V$)',fontsize=ftsize_label)
|
||||
ax2.set_xlabel('Distance from start point ($m$)',fontsize=ftsize_label)
|
||||
ax2.tick_params(labelsize=ftsize_axis)
|
||||
ax2.grid()
|
||||
|
||||
if PlotOpt == 'Total':
|
||||
ax3= Plot_Total_Potential(XYZ,sig0,sig1,R0,E0,ax3)
|
||||
ax4= Plot_Total_Potential(XYZ,sig0,sig2,R1,E0,ax4)
|
||||
|
||||
#Plot the Data (from Configuration 0)
|
||||
gphy0 = ax2.plot(np.sqrt((MP0[0,0]-MP0[:,0])**2+(MP0[:,1]-MP0[0,1])**2),VtdMP0
|
||||
,marker='o',color='blue',linewidth=3.,label ='Left Model Response' )
|
||||
|
||||
#Plot the Data (from Configuration 1)
|
||||
gphy1 = ax2.plot(np.sqrt((MP1[0,0]-MP1[:,0])**2+(MP1[:,1]-MP1[0,1])**2),VtdMP1
|
||||
,marker='o',color='red',linewidth=2.,label ='Right Model Response' )
|
||||
ax2.legend(('Left Model Response','Right Model Response'),loc=4)
|
||||
|
||||
elif PlotOpt == 'Secondary':
|
||||
#plot the secondary potentials
|
||||
ax3= Plot_Secondary_Potential(XYZ,sig0,sig1,R0,E0,ax3)
|
||||
ax4= Plot_Secondary_Potential(XYZ,sig0,sig2,R1,E0,ax4)
|
||||
|
||||
#Plot the data(from configuration 0)
|
||||
gphy0 = ax2.plot(np.sqrt((MP0[0,0]-MP0[:,0])**2+(MP0[:,1]-MP0[0,1])**2),VsdMP0,color='blue'
|
||||
,marker='o',linewidth=3.,label ='Left Model Response' )
|
||||
|
||||
|
||||
#Plot the Data (from Configuration 1)
|
||||
gphy1 = ax2.plot(np.sqrt((MP1[0,0]-MP1[:,0])**2+(MP1[:,1]-MP1[0,1])**2),VsdMP1
|
||||
,marker='o',color='red',linewidth=2.,label ='Right Model Response' )
|
||||
ax2.legend(('Left Model Response','Right Model Response'),loc=4 )
|
||||
|
||||
else:
|
||||
print('What dont you get? Total or Secondary?')
|
||||
|
||||
#Legends
|
||||
ax3.plot(MP0[:,0],MP0[:,1],color='gray')
|
||||
Dip_Midpoint0 = ax3.scatter(MP0[:,0],MP0[:,1],color='black')
|
||||
Electrodes0 = ax3.scatter(EL0[:,0],EL0[:,1],color='red')
|
||||
ax3.legend([Dip_Midpoint0,Electrodes0], ["Dipole Midpoint", "Electrodes"],scatterpoints=1)
|
||||
|
||||
ax4.plot(MP1[:,0],MP1[:,1],color='gray')
|
||||
Dip_Midpoint1 = ax4.scatter(MP1[:,0],MP1[:,1],color='black')
|
||||
Electrodes1 = ax4.scatter(EL1[:,0],EL1[:,1],color='red')
|
||||
ax4.legend([Dip_Midpoint1,Electrodes1], ["Dipole Midpoint", "Electrodes"],scatterpoints=1)
|
||||
|
||||
return fig
|
||||
|
||||
#Function to visualise and compare any two meaningful plots for the sphere in a uniform backgound with an unifom Electric Field
|
||||
def interact_conductiveSphere(R,log_sig0,log_sig1,Figure1a,Figure1b,Figure2a,Figure2b):
|
||||
|
||||
sig0,sig1 = conductivity_log_wrapper(log_sig0,log_sig1)
|
||||
E0 = 1. # inducing field strength in V/m
|
||||
n = 100 #level of discretisation
|
||||
xr = np.linspace(-200., 200., n) # X-axis discretization
|
||||
yr = xr.copy() # Y-axis discretization
|
||||
zr = np.r_[0] # identical to saying `zr = np.array([0])`
|
||||
XYZ = ndgrid(xr,yr,zr) # Space Definition
|
||||
|
||||
fig, ax = plt.subplots(1,2,figsize=(18,6))
|
||||
|
||||
#Setup figure 1 with options Configuration, Total or Secondary,
|
||||
#then Potential, ElectricField, Current Density or Charges Density
|
||||
if Figure1a == 'Configuration':
|
||||
ax[0] = get_Setup(XYZ,sig0,sig1,R,E0,ax[0],True,[0.1,0.1,0.6])
|
||||
|
||||
elif Figure1a == 'Total':
|
||||
|
||||
if Figure1b == 'Potential':
|
||||
ax[0] = Plot_Total_Potential(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1b == 'ElectricField':
|
||||
ax[0] = Plot_Total_ElectricField(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1b == 'CurrentDensity':
|
||||
ax[0] = Plot_Total_Currents(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1b == 'ChargesDensity':
|
||||
ax[0] = Plot_ChargesDensity(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1a == 'Secondary':
|
||||
|
||||
if Figure1b == 'Potential':
|
||||
ax[0] = Plot_Secondary_Potential(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1b == 'ElectricField':
|
||||
ax[0] = Plot_Secondary_ElectricField(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1b == 'CurrentDensity':
|
||||
ax[0] = Plot_Secondary_Currents(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
elif Figure1b == 'ChargesDensity':
|
||||
ax[0] = Plot_ChargesDensity(XYZ,sig0,sig1,R,E0,ax[0])
|
||||
|
||||
|
||||
if Figure1a== 'Configuration':
|
||||
ax[1] = Plot_Primary_Potential(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
print 'While figure1 is plotting Configuration, figure2 plots the primary field'
|
||||
|
||||
elif Figure2a == 'Total':
|
||||
if Figure2b == 'Potential':
|
||||
ax[1] = Plot_Total_Potential(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
elif Figure2b == 'ElectricField':
|
||||
ax[1] = Plot_Total_ElectricField(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
elif Figure2b == 'CurrentDensity':
|
||||
ax[1]=Plot_Total_Currents(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
elif Figure2b == 'ChargesDensity':
|
||||
ax[1] = Plot_ChargesDensity(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
|
||||
elif Figure2a == 'Secondary':
|
||||
if Figure2b == 'Potential':
|
||||
ax[1] = Plot_Secondary_Potential(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
elif Figure2b == 'ElectricField':
|
||||
ax[1] = Plot_Secondary_ElectricField(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
elif Figure2b == 'CurrentDensity':
|
||||
ax[1] = Plot_Secondary_Currents(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
elif Figure2b == 'ChargesDensity':
|
||||
ax[1] = Plot_ChargesDensity(XYZ,sig0,sig1,R,E0,ax[1])
|
||||
|
||||
plt.tight_layout(True)
|
||||
plt.show()
|
||||
|
||||
#Interactive Visualisation of the responses of two configurations to a (pseudo) DC resistivity survey
|
||||
def interactive_two_configurations_comparison(log_sig0,log_sig1,log_sig2,R0,R1,xstart,ystart,xend,yend,dipole_number,electrode_spacing,matching_spheres_example):
|
||||
|
||||
sig0,sig1 = conductivity_log_wrapper(log_sig0,log_sig1)
|
||||
sig2 = 10.**log_sig2
|
||||
E0 = 1. # inducing field strength in V/m
|
||||
n = 100 #level of discretisation
|
||||
xr = np.linspace(-200., 200., n) # X-axis discretization
|
||||
yr = xr.copy() # Y-axis discretization
|
||||
zr = np.r_[0] # identical to saying `zr = np.array([0])`
|
||||
XYZ = ndgrid(xr,yr,zr) # Space Definition
|
||||
PlotOpt = 'Total'
|
||||
|
||||
if matching_spheres_example:
|
||||
sig0 = 10.**(-3)
|
||||
sig1 = 10.**(-2)
|
||||
sig2 = 1.310344828 * 10**(-3)
|
||||
R0 = 20.
|
||||
R1 = 40.
|
||||
|
||||
two_configurations_comparison(XYZ,sig0,sig1,sig2,R0,R1,E0,xstart,ystart,xend,yend,dipole_number,electrode_spacing,PlotOpt)
|
||||
|
||||
else:
|
||||
two_configurations_comparison(XYZ,sig0,sig1,sig2,R0,R1,E0,xstart,ystart,xend,yend,dipole_number,electrode_spacing,PlotOpt)
|
||||
|
||||
plt.tight_layout(True)
|
||||
plt.show()
|
||||
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
sig0 = -3. # conductivity of the wholespace
|
||||
sig1 = -1. # conductivity of the sphere
|
||||
sig0, sig1 = conductivity_log_wrapper(sig0,sig1)
|
||||
R = 50. # radius of the sphere
|
||||
E0 = 1. # inducing field strength
|
||||
n = 100 #level of discretisation
|
||||
xr = np.linspace(-2.*R, 2.*R, n) # X-axis discretization
|
||||
yr = xr.copy() # Y-axis discretization
|
||||
zr = np.r_[0] # identical to saying `zr = np.array([0])`
|
||||
XYZ = ndgrid(xr,yr,zr) # Space Definition
|
||||
|
||||
fig, ax = plt.subplots(2,5,figsize=(50,10))
|
||||
ax[0,0] = get_Setup(XYZ,sig0,sig1,R,E0,ax[0,0],True,[0.6,0.1,0.1])
|
||||
ax[1,0] = Plot_Primary_Potential(XYZ,sig0,sig1,R,E0,ax[1,0])
|
||||
ax[0,1] = Plot_Total_Potential(XYZ,sig0,sig1,R,E0,ax[0,1])
|
||||
ax[1,1] = Plot_Secondary_Potential(XYZ,sig0,sig1,R,E0,ax[1,1])
|
||||
ax[0,2] = Plot_Total_ElectricField(XYZ,sig0,sig1,R,E0,ax[0,2])
|
||||
ax[1,2] = Plot_Secondary_ElectricField(XYZ,sig0,sig1,R,E0,ax[1,2])
|
||||
ax[0,3] = Plot_Total_Currents(XYZ,sig0,sig1,R,E0,ax[0,3])
|
||||
ax[1,3] = Plot_Secondary_Currents(XYZ,sig0,sig1,R,E0,ax[1,3])
|
||||
ax[0,4] = Plot_Primary_Potential(XYZ,sig0,sig1,R,E0,ax[0,4])
|
||||
ax[1,4] = Plot_ChargesDensity(XYZ,sig0,sig1,R,E0,ax[1,4])
|
||||
|
||||
|
||||
plt.show()
|
||||
|
||||
@@ -990,4 +990,18 @@ class ProjectedGNCG(BFGS, Minimize, Remember):
|
||||
cgFlag = 1
|
||||
# End CG Iterations
|
||||
|
||||
# Take a gradient step on the active cells if exist
|
||||
if temp != self.xc.size:
|
||||
|
||||
rhs_a = (Active) * -self.g
|
||||
|
||||
dm_i = max( abs( delx ) )
|
||||
dm_a = max( abs(rhs_a) )
|
||||
|
||||
delx = delx + rhs_a * dm_i / dm_a /10.
|
||||
|
||||
# Only keep gradients going in the right direction on the active set
|
||||
indx = ((self.xc<=self.lower) & (delx < 0)) | ((self.xc>=self.upper) & (delx > 0))
|
||||
delx[indx] = 0.
|
||||
|
||||
return delx
|
||||
|
||||
@@ -646,7 +646,7 @@ class Sparse(Simple):
|
||||
eps = 1e-1
|
||||
curModel = None # use a model to compute the weights
|
||||
gamma = 1.
|
||||
p = 0.
|
||||
norms = [0., .2, 2., 2., 1.]
|
||||
qx = 2.
|
||||
qy = 2.
|
||||
qz = 2.
|
||||
@@ -666,7 +666,7 @@ class Sparse(Simple):
|
||||
|
||||
else:
|
||||
f_m = self.curModel - self.reg.mref
|
||||
self.rs = self.R(f_m , self.p)
|
||||
self.rs = self.R(f_m , self.norms[0])
|
||||
#print "Min rs: " + str(np.max(self.rs)) + "Max rs: " + str(np.min(self.rs))
|
||||
self.Rs = Utils.sdiag( self.rs )
|
||||
|
||||
|
||||
@@ -0,0 +1,31 @@
|
||||
.. _examples_DC_PseudoSection_Simulation:
|
||||
|
||||
.. --------------------------------- ..
|
||||
.. ..
|
||||
.. THIS FILE IS AUTO GENEREATED ..
|
||||
.. ..
|
||||
.. SimPEG/Examples/__init__.py ..
|
||||
.. ..
|
||||
.. --------------------------------- ..
|
||||
|
||||
|
||||
|
||||
DC Forward Simulation
|
||||
=====================
|
||||
|
||||
Forward model conductive spheres in a half-space and plot a pseudo-section
|
||||
|
||||
Created on Mon Feb 01 19:28:06 2016
|
||||
|
||||
@fourndo
|
||||
|
||||
|
||||
|
||||
.. plot::
|
||||
|
||||
from SimPEG import Examples
|
||||
Examples.DC_PseudoSection_Simulation.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/DC_PseudoSection_Simulation.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -0,0 +1,41 @@
|
||||
.. _examples_EM_FDEM_SusEffects:
|
||||
|
||||
.. --------------------------------- ..
|
||||
.. ..
|
||||
.. THIS FILE IS AUTO GENEREATED ..
|
||||
.. ..
|
||||
.. SimPEG/Examples/__init__.py ..
|
||||
.. ..
|
||||
.. --------------------------------- ..
|
||||
|
||||
|
||||
EM: FDEM: Effects of susceptibility
|
||||
===================================
|
||||
|
||||
When airborne freqeuncy domain EM (AFEM) survey is flown over
|
||||
the earth including significantly susceptible bodies (magnetite-rich rocks),
|
||||
negative data is often observed in the real part of the lowest frequency
|
||||
(e.g. Dighem system 900 Hz). This phenomenon mostly based upon magnetization
|
||||
occurs due to a susceptible body when the magnetic field is applied.
|
||||
|
||||
To clarify what is happening in the earth when we are exciting the earth with
|
||||
a loop source in the frequency domain we run three forward modelling:
|
||||
|
||||
- F[:math:`\sigma`, :math:`\mu`]: Anomalous conductivity and susceptibility
|
||||
- F[:math:`\sigma`, :math:`\mu_0`]: Anomalous conductivity
|
||||
- F[:math:`\sigma_{air}`, :math:`\mu_0`]: primary field
|
||||
|
||||
We plot vector magnetic fields in the earth. For secondary fields we provide
|
||||
F[:math:`\sigma`, :math:`\mu`]-F[:math:`\sigma`, :math:`\mu_0`]. Following
|
||||
figure show both real and parts.
|
||||
|
||||
|
||||
|
||||
.. plot::
|
||||
|
||||
from SimPEG import Examples
|
||||
Examples.EM_FDEM_SusEffects.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/EM_FDEM_SusEffects.py
|
||||
:language: python
|
||||
:linenos:
|
||||
Reference in New Issue
Block a user