updates to analytics and a 'test' in the notebook

- need to change this into a real test!

notebooks/Test FDEM Mag Dipole Wholespace.ipynb

@@ -0,0 +1,558 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:0c740be2795f45fed7e9988b98793c1d5c028cb904190367f5ac4b4fa72b1bc6"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "heading",
+     "level": 1,
+     "metadata": {},
+     "source": [
+      "FDEM Inversion for Conductivity"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Forward simulation and inversion for a block (fracture) on a whole space"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from SimPEG import *\n",
+      "import simpegEM as EM\n",
+      "from pymatsolver import MumpsSolver\n",
+      "from scipy.constants import mu_0"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 15
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Set up Mesh"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "cs = 5\n",
+      "ncore = 35\n",
+      "ncore_x = 15\n",
+      "pad = 8\n",
+      "padfactor = 1.4\n",
+      "hx = [(cs,pad,-padfactor),(cs,ncore_x),(cs,pad,padfactor)]\n",
+      "h = [(cs,pad,-padfactor),(cs,ncore),(cs,pad,padfactor)]\n",
+      "mesh = Mesh.TensorMesh([hx, h, h],'CCC')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sigwh = 1e-2 # conductivity of the wholespace background\n",
+      "sigma = 1e-2*np.ones(prod(mesh.nC))  # whole space\n",
+      "sigmaB = 1e-2*np.ones(prod(mesh.nC)) # with block \n",
+      "sigmaB[(np.abs(mesh.gridCC[:,0]) < 10.) & (np.abs(mesh.gridCC[:,1]) < 25.) & (np.abs(mesh.gridCC[:,2]) < 25.)] = 1\n",
+      "mesh.plotSlice(sigmaB,grid=True) # todo: fix axes\n",
+      "mesh.plotImage(sigmaB)           # "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stderr",
+       "text": [
+        "/opt/local/Library/Frameworks/Python.framework/Versions/2.7/lib/python2.7/site-packages/matplotlib/lines.py:503: RuntimeWarning: invalid value encountered in greater_equal\n",
+        "  return np.alltrue(x[1:] - x[0:-1] >= 0)\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 3,
+       "text": [
+        "<matplotlib.collections.QuadMesh at 0x10a2c6790>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x109c147d0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x109c14750>"
+       ]
+      }
+     ],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Comparing against analytic"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Start by comparing the solution for the whole-space model to what we expect for a dipole in a whole-space"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = np.linspace(-50,50,10)\n",
+      "XYZ = Utils.ndgrid(np.r_[0],np.r_[50],x)\n",
+      "\n",
+      "Rx0 = EM.FDEM.RxFDEM(XYZ, 'bzr')\n",
+      "Rx1 = EM.FDEM.RxFDEM(XYZ, 'bzi')\n",
+      "\n",
+      "txList = []\n",
+      "freq = 500.\n",
+      "Tx = EM.FDEM.TxFDEM(np.r_[0.,0.,0.], 'VMD',freq,[Rx0,Rx1])\n",
+      "txList.append(Tx)\n",
+      "\n",
+      "survey = EM.FDEM.SurveyFDEM(txList)\n",
+      "mapping = Maps.ExpMap(mesh)\n",
+      "prb = EM.FDEM.ProblemFDEM_b(mesh, mapping=mapping)\n",
+      "prb.pair(survey)\n",
+      "prb.Solver = MumpsSolver"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# SimPEG soln\n",
+      "fTest = prb.fields(np.log(sigma))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Bx, By, Bz = EM.Analytics.FDEM.AnalyticMagDipoleWholeSpace(Rx0.locs,Tx.loc,sigwh,freq, m = 1.0, orientation='Z')\n",
+      "Bx0, By0, Bz0 = EM.Analytics.FDEM.AnalyticMagDipoleWholeSpace(Rx0.locs,Tx.loc,sigwh,0.0, m = 1.0, orientation='Z')\n",
+      "BxA = Bx - Bx0\n",
+      "ByA = By - By0\n",
+      "BzA = Bz - Bz0\n",
+      "# XYZ2 = Utils.ndgrid(np.r_[0],np.r_[100],x)\n",
+      "# Bx, By, Bz = EM.Analytics.FDEM.AnalyticMagDipoleWholeSpace(np.r_[0,100,0],Tx.loc,sigwh,100., m = 1.0, orientation='Z')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 61
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plot(x, np.c_[BzA.real/data[Tx,Rx0]])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 66,
+       "text": [
+        "[<matplotlib.lines.Line2D at 0x114c64f90>]"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x111fd8650>"
+       ]
+      }
+     ],
+     "prompt_number": 66
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plot(x, np.c_[np.arctan2(Bz.imag,Bz.real)*180./np.pi, np.arctan2(data[Tx,Rx1],data[Tx,Rx0])*180./np.pi])\n",
+      "legend(('ana','data'))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 53,
+       "text": [
+        "<matplotlib.legend.Legend at 0x110a13a90>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x110a13e50>"
+       ]
+      }
+     ],
+     "prompt_number": 53
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "data = survey.projectFields(fTest)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 9
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "plot(x, np.c_[data[Tx,Rx0], Bz.real])\n",
+      "legend(('data','Ana'))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 19,
+       "text": [
+        "<matplotlib.legend.Legend at 0x10a9688d0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x10a968950>"
+       ]
+      }
+     ],
+     "prompt_number": 19
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fTest"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 48,
+       "text": [
+        "<simpegEM.FDEM.SurveyFDEM.FieldsFDEM at 0x122bed68>"
+       ]
+      }
+     ],
+     "prompt_number": 48
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = np.linspace(-50,50,10)\n",
+      "XYZ = Utils.ndgrid(np.r_[0],np.r_[50],x)\n",
+      "Rx0 = EM.FDEM.RxFDEM(XYZ, 'bzr')\n",
+      "Rx1 = EM.FDEM.RxFDEM(XYZ, 'bzi')\n",
+      "txList = []\n",
+      "for z in np.linspace(-50,50,10):\n",
+      "    for freq in [300.,600.,900.]:\n",
+      "        Tx = EM.FDEM.TxFDEM(np.r_[0.,-50.,z], 'VMD', freq, [Rx0,Rx1])\n",
+      "        txList.append(Tx)\n",
+      "survey = EM.FDEM.SurveyFDEM(txList)\n",
+      "mapping = Maps.ExpMap(mesh)\n",
+      "prb = EM.FDEM.ProblemFDEM_b(mesh, mapping=mapping)\n",
+      "prb.pair(survey)\n",
+      "prb.Solver = MumpsSolver"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 16
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fB = prb.fields(np.log(sigmaB))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 41
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "dB = survey.projectFields(fB)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "f300 = dB[survey.getTransmitters(300.)[-1],Rx0]\n",
+      "f600 = dB[survey.getTransmitters(600.)[-1],Rx0]\n",
+      "f900 = dB[survey.getTransmitters(900.)[-1],Rx0]\n",
+      "\n",
+      "semilogy(x, np.abs(np.c_[f300,f600,f900]))\n",
+      "legend(('3','6','9'))\n",
+      "# plot(x, np.c_[dpred[Tx0,Rx0], dB[Tx0,Rx0]])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 71,
+       "text": [
+        "<matplotlib.legend.Legend at 0x10da95950>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x10da95710>"
+       ]
+      }
+     ],
+     "prompt_number": 71
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "survey.dobs = survey.dpred(None, u=fB)\n",
+      "survey.std = 0.01"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 80
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "reg = Regularization.Tikhonov(mesh)\n",
+      "dmis = DataMisfit.l2_DataMisfit(survey)\n",
+      "opt = Optimization.InexactGaussNewton(maxIter=3)\n",
+      "invProb = InvProblem.BaseInvProblem(dmis, reg, opt)\n",
+      "beta = Directives.BetaSchedule()\n",
+      "betaest = Directives.BetaEstimate_ByEig()\n",
+      "inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest])\n",
+      "m0 = np.ones_like(sigmaB)*1e-2"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 81
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "inv.run(np.log(m0))"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "SimPEG.InvProblem will set Regularization.mref to m0.\n",
+        "SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.\n",
+        "                    ***Done using same solver as the problem***\n",
+        "SimPEG.l2_DataMisfit is creating default weightings for Wd."
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "============================ Inexact Gauss Newton ============================"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   \n",
+        "-----------------------------------------------------------------------------\n",
+        "   0  4.46e+03  6.92e+04  0.00e+00  6.92e+04    2.31e+04      0              "
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "   1  4.46e+03  6.51e+03  2.09e+00  1.58e+04    3.71e+02      0              "
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "   2  4.46e+03  6.15e+03  2.16e+00  1.58e+04    2.17e+01      0              "
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "   3  5.57e+02  6.13e+03  2.16e+00  7.33e+03    3.68e+03      0   Skip BFGS  "
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "------------------------- STOP! -------------------------\n",
+        "0 : |fc-fOld| = 8.4323e+03 <= tolF*(1+|f0|) = 6.9198e+03\n",
+        "1 : |xc-x_last| = 4.2277e-02 <= tolX*(1+|x0|) = 8.0690e+01\n",
+        "0 : |proj(x-g)-x|    = 3.6813e+03 <= tolG          = 1.0000e-01\n",
+        "0 : |proj(x-g)-x|    = 3.6813e+03 <= 1e3*eps       = 1.0000e-02\n",
+        "1 : maxIter   =       3    <= iter          =      3\n",
+        "------------------------- DONE! -------------------------\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 82,
+       "text": [
+        "array([-4.62771738, -4.62789076, -4.62806113, ..., -4.62159749,\n",
+        "       -4.62166636, -4.62176178])"
+       ]
+      }
+     ],
+     "prompt_number": 82
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "mopt = _"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 83
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "colorbar(mesh.plotSlice(mapping*mopt)[0])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 90,
+       "text": [
+        "<matplotlib.colorbar.Colorbar instance at 0x1141e51b8>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x11473f510>"
+       ]
+      }
+     ],
+     "prompt_number": 90
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 85
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}

simpegEM/Analytics/FDEM.py

@@ -1,7 +1,9 @@
+from __future__ import division
 import numpy as np
 from scipy.constants import mu_0, pi
 from scipy.special import erf
 import matplotlib.pyplot as plt
+from SimPEG import Utils
 
 def hzAnalyticDipoleF(r, freq, sigma, secondary=True):
     """
@@ -12,7 +14,7 @@ def hzAnalyticDipoleF(r, freq, sigma, secondary=True):
         import matplotlib.pyplot as plt
         import simpegEM as EM
         freq = np.logspace(-1, 6, 61)
-        test = EM.Utils.Ana.FEM.hzAnalyticDipoleF(100, freq, 0.001, secondary=False)
+        test = EM.Analytics.FDEM.hzAnalyticDipoleF(100, freq, 0.001, secondary=False)
         plt.loglog(freq, abs(test.real))
         plt.loglog(freq, abs(test.imag))
         plt.title('Response at $r$=100m')
@@ -36,7 +38,7 @@ def hzAnalyticDipoleF(r, freq, sigma, secondary=True):
 
     return hz
 
-def AnalyticMagDipoleWholeSpace(x,y,z,sig,f,xs=0.,ys=0.,zs=0.,m=1.,orientation='X'):
+def AnalyticMagDipoleWholeSpace(XYZ, txLoc, sig, f, m=1., orientation='X'):
     """
     Analytical solution for a dipole in a whole-space.
 
@@ -47,14 +49,30 @@ def AnalyticMagDipoleWholeSpace(x,y,z,sig,f,xs=0.,ys=0.,zs=0.,m=1.,orientation='
         - add E-fields
         - handle multiple frequencies
         - add divide by zero safety
+
+
+    .. plot::
+
+        import simpegEM as EM
+        import matplotlib.pyplot as plt
+        freqs = np.logspace(-2,5,100)
+        Bx, By, Bz = EM.Analytics.FDEM.AnalyticMagDipoleWholeSpace([0,100,0], [0,0,0], 1e-2, freqs, m=1, orientation='Z')
+        plt.loglog(freqs, np.abs(Bz.real)/mu_0, 'b')
+        plt.loglog(freqs, np.abs(Bz.imag)/mu_0, 'r')
+        plt.legend(('real','imag'))
+        plt.show()
+
+
     """
 
-    dx = x-xs
-    dy = y-ys
-    dz = z-zs
+    XYZ = Utils.asArray_N_x_Dim(XYZ, 3)
+
+    dx = XYZ[:,0]-txLoc[0]
+    dy = XYZ[:,1]-txLoc[1]
+    dz = XYZ[:,2]-txLoc[2]
 
     r  = np.sqrt( dx**2. + dy**2. + dz**2.)
-    k  = np.sqrt(-1j*2.*np.pi*f*mu_0*sig)
+    k  = np.sqrt( -1j*2.*np.pi*f*mu_0*sig )
     kr = k*r
 
     front = m / (4.*pi * r**3.) * np.exp(-1j*kr)