{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "**Objective:** \n", "\n", "In this tutorial we will create a simple magnetic problem from scratch using the SimPEG framework.\n", "\n", "We are using the integral form of the magnetostatic problem. In the absence of free-currents or changing magnetic field, magnetic material can give rise to a secondary magnetic field according to:\n", "\n", "$$\\vec b = \\frac{\\mu_0}{4\\pi} \\int_{V} \\vec M \\cdot \\nabla \\nabla \\left(\\frac{1}{r}\\right) \\; dV $$\n", "\n", "Where $\\mu_0$ is the magnetic permealitity of free-space, $\\vec M$ is the magnetization per unit volume and $r$ defines the distance between the observed field $\\vec b$ and the magnetized object. Assuming a purely induced response, the strenght of magnetization can be written as:\n", "\n", "$$ \\vec M = \\mu_0 \\kappa \\vec H_0 $$\n", "\n", "where $\\vec H$ is an external inducing magnetic field, and $\\kappa$ the magnetic susceptibility of matter.\n", "As derived by Sharma 1966, the integral can be evaluated for rectangular prisms such that:\n", "\n", "$$ \\vec b(P) = \\mathbf{T} \\cdot \\vec H_0 \\; \\kappa $$\n", "\n", "Where the tensor matrix $\\bf{T}$ relates the three components of magnetization $\\vec M$ to the components of the field $\\vec b$:\n", "\n", "$$\\mathbf{T} =\n", "\t \\begin{pmatrix}\n", " \t\tT_{xx} & T_{xy} & T_{xz} \\\\\n", "\t\tT_{yx} & T_{yy} & T_{yz} \\\\\n", "\t\tT_{zx} & T_{zy} & T_{zz} \n", "\t\\end{pmatrix} $$\n", " \n", "In general, we discretize the earth into a collection of cells, each contributing to the magnetic data such that:\n", "\n", "$$\\vec b(P) = \\sum_{j=1}^{nc} \\mathbf{T}_j \\cdot \\vec H_0 \\; \\kappa_j$$\n", "\n", "giving rise to a linear problem.\n", "\n", "The remaining of this notebook goes through all the important components of a 3D magnetic experiment. From mesh creation, topography, data and inverse problem. \n", "\n", "Enjoy.\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Using matplotlib backend: nbAgg\n", "Populating the interactive namespace from numpy and matplotlib\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "WARNING: pylab import has clobbered these variables: ['linalg']\n", "`%matplotlib` prevents importing * from pylab and numpy\n" ] } ], "source": [ "%matplotlib notebook\n", "%pylab\n", "from SimPEG import *\n", "import simpegPF as PF\n", "from simpegPF import BaseMag as MAG\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "C:\\Users\\dominiquef.MIRAGEOSCIENCE\\Documents\\GIT\\SimPEG\\simpeg\\SimPEG\\Maps.py:533: FutureWarning: `ActiveCells` is deprecated and will be removed in future versions. Use `InjectActiveCells` instead\n", " FutureWarning)\n" ] } ], "source": [ "# First we need to define the direction of the inducing field\n", "# As a simple case, we pick a vertical inducing field of magnitude 50,000nT. \n", "# From old convention, field orientation is given as an azimuth from North \n", "# (positive clockwise) and dip from the horizontal (positive downward).\n", "H0 = (50000.,90.,0.)\n", "\n", "\n", "# Create a mesh\n", "dx = 5.\n", "\n", "hxind = [(dx,5,-1.3), (dx, 20), (dx,5,1.3)]\n", "hyind = [(dx,5,-1.3), (dx, 20), (dx,5,1.3)]\n", "hzind = [(dx,5,-1.3),(5, 10)]\n", "\n", "mesh = Mesh.TensorMesh([hxind, hyind, hzind], 'CCC')\n", "\n", "# Get index of the center\n", "midx = int(mesh.nCx/2)\n", "midy = int(mesh.nCy/2)\n", "\n", "# Lets create a simple Gaussian topo and set the active cells\n", "[xx,yy] = np.meshgrid(mesh.vectorNx,mesh.vectorNy)\n", "zz = -np.exp( ( xx**2 + yy**2 )/ 75**2 ) + mesh.vectorNz[-1]\n", "\n", "topo = np.c_[mkvc(xx),mkvc(yy),mkvc(zz)] # We would usually load a topofile\n", "\n", "actv = PF.Magnetics.getActiveTopo(mesh,topo,'N') # Go from topo to actv cells\n", "\n", "#nC = mesh.nC \n", "#actv = np.asarray(range(mesh.nC))\n", "\n", "# Create active map to go from reduce space to full\n", "actvMap = Maps.ActiveCells(mesh, actv, -100)\n", "nC = len(actv)\n", "\n", "# Create and array of observation points\n", "xr = np.linspace(-20., 20., 20)\n", "yr = np.linspace(-20., 20., 20)\n", "X, Y = np.meshgrid(xr, yr)\n", "\n", "# Let just put the observation above the topo\n", "Z = -np.exp( ( X**2 + Y**2 )/ 75**2 ) + mesh.vectorNz[-1] + 5. \n", "\n", "# Create a MAGsurvey\n", "rxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]\n", "rxLoc = MAG.RxObs(rxLoc)\n", "srcField = MAG.SrcField([rxLoc],H0)\n", "survey = MAG.LinearSurvey(srcField) \n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have all our spatial components, we can create our linear system. For a single location and single component of the data, the system would look like this:\n", "\n", "$$ b_x =\n", "\t\\begin{bmatrix}\n", "\tT_{xx}^1 &... &T_{xx}^{nc} & T_{xy}^1 & ... & T_{xy}^{nc} & T_{xz}^1 & ... & T_{xz}^{nc}\\\\\n", "\t \\end{bmatrix}\n", "\t \\begin{bmatrix}\n", "\t\t\\mathbf{M}_x \\\\ \\mathbf{M}_y \\\\ \\mathbf{M}_z\n", "\t\\end{bmatrix} \\\\ $$\n", "\n", "where each of $T_{xx},\\;T_{xy},\\;T_{xz}$ are [nc x 1] long. For the $y$ and $z$ component, we need the two other rows of the tensor $\\mathbf{T}$.\n", "In our simple induced case, the magnetization direction $\\mathbf{M_x,\\;M_y\\;,Mz}$ are known and assumed to be constant everywhere, so we can reduce the size of the system such that: \n", "\n", "$$ \\vec{\\mathbf{d}}_{\\text{pred}} = (\\mathbf{T\\cdot M})\\; \\kappa$$\n", "\n", "\n", "\n", "In most geophysical surveys, we are not collecting all three components, but rather the magnitude of the field, or $Total\\;Magnetic\\;Intensity$ (TMI) data.\n", "Because the inducing field is really large, we will assume that the anomalous fields are parallel to $H_0$:\n", "\n", "$$ d^{TMI} = \\hat H_0 \\cdot \\vec d$$\n", "\n", "We then end up with a much smaller system:\n", "\n", "$$ d^{TMI} = \\mathbf{F\\; \\kappa}$$\n", "\n", "where $\\mathbf{F} \\in \\mathbb{R}^{nd \\times nc}$ is our $forward$ operator." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Begin calculation of forward operator: ind\n", "Done 0.0 %\n", "Done 10.0 %\n", "Done 20.0 %\n", "Done 30.0 %\n", "Done 40.0 %\n", "Done 50.0 %\n", "Done 60.0 %\n", "Done 70.0 %\n", "Done 80.0 %\n", "Done 90.0 %\n", "Done 100% ...forward operator completed!!\n", "\n" ] } ], "source": [ "# We can now create a susceptibility model and generate data\n", "# Lets start with a simple block in half-space\n", "model = np.zeros((mesh.nCx,mesh.nCy,mesh.nCz))\n", "model[(midx-2):(midx+2),(midy-2):(midy+2),-6:-2] = 0.01\n", "model = mkvc(model)\n", "model = model[actv]\n", "\n", "# Create active map to go from reduce set to full\n", "actvMap = Maps.InjectActiveCells(mesh, actv, -100)\n", "\n", "# Creat reduced identity map\n", "idenMap = Maps.IdentityMap(nP = len(actv))\n", "\n", "# Create the forward model operator\n", "prob = PF.Magnetics.MagneticIntegral(mesh, mapping = idenMap, actInd = actv)\n", "\n", "# Pair the survey and problem\n", "survey.pair(prob)\n", "\n", "# Compute linear forward operator and compute some data\n", "d = prob.fields(model)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Create distance weights from our linera forward operator\n", "wr = np.sum(prob.G**2.,axis=0)**0.5\n", "wr = ( wr/np.max(wr) )\n", "\n", "\n", "wr_FULL = actvMap * wr\n", "plt.figure()\n", "ax = subplot()\n", "mesh.plotSlice(wr_FULL, ax = ax, normal = 'Y', ind=midx, grid=True, clim = (0, wr.max()))\n", "title('Distance weighting')\n", "xlabel('x');ylabel('z')\n", "plt.gca().set_aspect('equal', adjustable='box')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Once we have our problem, we can use the inversion tools in SimPEG to run our inversion:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#survey.makeSyntheticData(data, std=0.01)\n", "survey.dobs=data\n", "survey.std = np.ones(len(data))\n", "survey.mtrue = model\n", "\n", "# Create a regularization\n", "reg = Regularization.Simple(mesh, indActive = actv, mapping = idenMap)\n", "reg.wght = wr\n", "\n", "# Directive to cool the trade-off parameter beta\n", "beta = Directives.BetaSchedule(coolingFactor=2, coolingRate=1)\n", "beta_in = 1e+4\n", "\n", "# Create a jacobi pre-conditioner\n", "diagA = np.sum(prob.G**2.,axis=0) + beta_in*(reg.W.T*reg.W).diagonal()*wr\n", "PC = Utils.sdiag(diagA**-1.)\n", "\n", "# Define the misfit function\n", "dmis = DataMisfit.l2_DataMisfit(survey)\n", "dmis.Wd = 1./survey.std\n", "\n", "\n", "# Solver for the optimization problem\n", "opt = Optimization.ProjectedGNCG(maxIter=10, lower=0.,upper=1., maxIterCG= 30, tolCG = 1e-3)\n", "opt.approxHinv = PC\n", "\n", "# Put everything together\n", "invProb = InvProblem.BaseInvProblem(dmis, reg, opt, beta = beta_in)\n", "target = Directives.TargetMisfit()\n", "\n", "inv = Inversion.BaseInversion(invProb, directiveList=[beta,target])\n", "m0 = np.ones(mesh.nC)*1e-4\n", "m0 = m0[actv]" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "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 and solverOpts as the problem***\n", "=============================== Projected GNCG ===============================\n", " # beta phi_d phi_m f |proj(x-g)-x| LS Comment \n", "-----------------------------------------------------------------------------\n", " 0 1.00e+04 6.70e+04 0.00e+00 6.70e+04 8.30e+01 0 \n", " 1 5.00e+03 2.94e+03 4.20e-02 3.15e+03 6.89e+01 0 \n", " 2 2.50e+03 3.61e+02 6.08e-02 5.13e+02 7.04e+01 0 \n", " 3 1.25e+03 2.58e+02 5.84e-02 3.31e+02 8.36e+01 0 Skip BFGS \n", "------------------------- STOP! -------------------------\n", "1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 6.6998e+03\n", "1 : |xc-x_last| = 8.5475e-03 <= tolX*(1+|x0|) = 1.0111e-01\n", "0 : |proj(x-g)-x| = 8.3598e+01 <= tolG = 1.0000e-01\n", "0 : |proj(x-g)-x| = 8.3598e+01 <= 1e3*eps = 1.0000e-02\n", "0 : maxIter = 10 <= iter = 4\n", "------------------------- DONE! -------------------------\n" ] } ], "source": [ "mrec = inv.run(m0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Inversion has converged. We can plot sections through the model." ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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