{ "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" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 1, "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": [ "C:\\Users\\dominiquef.MIRAGEOSCIENCE\\AppData\\Local\\Continuum\\Anaconda\\lib\\site-packages\\IPython\\kernel\\__init__.py:13: ShimWarning: The `IPython.kernel` package has been deprecated. You should import from ipykernel or jupyter_client instead.\n", " \"You should import from ipykernel or jupyter_client instead.\", ShimWarning)\n" ] } ], "source": [ "%matplotlib notebook\n", "%pylab" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Efficiency Warning: Interpolation will be slow, use setup.py!\n", "\n", " python setup.py build_ext --inplace\n", " \n" ] } ], "source": [ "from SimPEG import *\n", "import simpegPF as PF" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [], "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 = np.array(([90.,0.,50000.]))\n", "\n", "# Assume all induced so the magnetization M is also in the same direction\n", "M = np.array([90,0])\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", "# Assume flat topo for now, so all cells are active\n", "nC = mesh.nC \n", "actv = np.ones(nC)\n", "\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", "Z = np.ones(X.size)*(mesh.vectorNz[-1]+dx) # Let just put the observation flat\n", "\n", "rxLoc = np.c_[Utils.mkvc(X.T), Utils.mkvc(Y.T), Utils.mkvc(Z.T)]\n" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "59.390075000000003" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Z.max()" ] }, { "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 looks 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": 61, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Begin calculation of forward operator: tmi\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": [ "# First, convert the magnetization direction to Cartesian\n", "mi = np.ones(mesh.nC) * M[0]\n", "md = np.ones(mesh.nC) * M[1]\n", "M_xyz = PF.Magnetics.dipazm_2_xyz( mi , md ) # Ouputs an nc x 3 array\n", "\n", "# Create the forward model operator\n", "F = PF.Magnetics.Intrgl_Fwr_Op(mesh,H0,M_xyz,rxLoc,actv,'tmi')" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Begin calculation of distance weighting for R= 3.0\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% ...distance weighting completed!!\n", "\n" ] }, { "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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manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; 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": [ "# 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", "\n", "# Create a few models\n", "figure()\n", "ax = subplot(211)\n", "mesh.plotSlice(model, ax = ax, normal = 'Y', ind=midx, grid=True)\n", "title('A simple block model.')\n", "xlabel('x');ylabel('z')\n", "plt.gca().set_aspect('equal', adjustable='box')\n", "\n", "# We can now generate data\n", "data = F.dot(model) #: this is matrix multiplication!!\n", "subplot(212)\n", "imshow(data.reshape(X.shape), extent=[xr.min(), xr.max(), yr.min(), yr.max()])\n", "title('Predicted data.')\n", "plt.gca().set_aspect('equal', adjustable='box')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": false }, "outputs": [], "source": [ "class LinearSurvey(Survey.BaseSurvey):\n", " def projectFields(self, u):\n", " return u\n", " \n", " @property\n", " def nD(self):\n", " return self.prob.G.shape[1]\n", "\n", "class LinearProblem(Problem.BaseProblem):\n", " \n", " surveyPair = LinearSurvey\n", "\n", " def __init__(self, mesh, G, **kwargs):\n", " Problem.BaseProblem.__init__(self, mesh, **kwargs)\n", " self.G = G\n", "\n", " def fields(self, m):\n", " return self.G.dot(m)\n", "\n", " def Jvec(self, m, v, u=None):\n", " return self.G.dot(v)\n", "\n", " def Jtvec(self, m, v, u=None):\n", " return self.G.T.dot(v)\n" ] }, { "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": 97, "metadata": { "collapsed": false }, "outputs": [], "source": [ "prob = LinearProblem(mesh, F)\n", "prob.solverOpts['accuracyTol'] = 1e-4\n", "survey = LinearSurvey()\n", "survey.pair(prob)\n", "#survey.makeSyntheticData(data, std=0.01)\n", "survey.dobs=data\n", "survey.mtrue = model\n", "\n", "\n", "reg = Regularization.Tikhonov(mesh, mapping=wrMap)\n", "dmis = DataMisfit.l2_DataMisfit(survey)\n", "dmis.Wd = np.ones(len(data))*1.\n", "opt = Optimization.ProjectedGNCG(maxIter=6,lower=0.,upper=1.)\n", "# opt = Optimization.InexactGaussNewton(maxIter=6)\n", "invProb = InvProblem.BaseInvProblem(dmis, reg, opt, beta = 1e+4)\n", "beta = Directives.BetaSchedule()\n", "#betaest = Directives.BetaEstimate_ByEig()\n", "target = Directives.TargetMisfit()\n", "inv = Inversion.BaseInversion(invProb, directiveList=[beta, target])\n", "reg.alpha_s =0.0025\n", "m0 = np.ones_like(survey.mtrue)*1e-4" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Explore the documentation to see what other parameters you can tweak in the different elements of the inversion, but let's check how well we recovered the model just by using the default parameters:" ] }, { "cell_type": "code", "execution_count": 98, "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 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 5.58e+04 2.22e-07 5.58e+04 8.41e+01 0 \n", "------------------------- STOP! -------------------------\n", "1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 5.5847e+03\n", "1 : |xc-x_last| = 4.1645e-02 <= tolX*(1+|x0|) = 1.0116e-01\n", "0 : |proj(x-g)-x| = 8.4094e+01 <= tolG = 1.0000e-01\n", "0 : |proj(x-g)-x| = 8.4094e+01 <= 1e3*eps = 1.0000e-02\n", "0 : maxIter = 6 <= iter = 1\n", "------------------------- DONE! -------------------------\n" ] } ], "source": [ "mrec = inv.run(m0)" ] }, { "cell_type": "code", "execution_count": 99, "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. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " this.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width);\n", " canvas.attr('height', height);\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('