6_Inversion_with_any_forward_operator.ipynb

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    "<div>\n",
    "    <div style=\"float:left;width:50%;text-align:center\">\n",
    "        <h1>pyGIMLi Tutorial @ Transform 2021</h1>\n",
    "        <div>\n",
    "<table style=\"border-collapse:collapse;border-color:#ccc;border-spacing:0\" class=\"tg\"><thead><tr><th style=\"background-color:#f0f0f0;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;font-weight:bold;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\"></th><th style=\"background-color:#f0f0f0;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;font-weight:bold;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\">Info</th></tr></thead><tbody><tr><td style=\"background-color:#f9f9f9;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;font-weight:bold;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\">When</td><td style=\"background-color:#f9f9f9;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\">Live: Monday, April 19 • 8:00 - 10:00 UTC</td></tr><tr><td style=\"background-color:#fff;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;font-weight:bold;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\">Slack (Q&amp;A)</td><td style=\"background-color:#fff;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;text-align:left;text-decoration:underline;vertical-align:top;word-break:normal\"><a href=\"https://softwareunderground.org/\">Software Underground</a> channel <a href=\"https://swung.slack.com/archives/C01T5V5S9EV\">#t21-mon-pygimli</a></td></tr><tr><td style=\"background-color:#f9f9f9;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;font-weight:bold;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\">Recording</td><td style=\"background-color:#f9f9f9;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;text-align:left;text-decoration:underline;vertical-align:top;word-break:normal\"><a href=\"https://www.youtube.com/watch?v=w3pu0H3dXe8&t=6068\">https://www.youtube.com/watch?v=w3pu0H3dXe8&t=6068</a></td></tr><tr><td style=\"background-color:#fff;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;font-weight:bold;overflow:hidden;padding:10px 5px;text-align:left;vertical-align:middle;word-break:normal\">pyGIMLi documentation</td><td style=\"background-color:#fff;border-color:#ccc;border-style:solid;border-width:1px;color:#333;font-family:Arial, sans-serif;font-size:14px;overflow:hidden;padding:10px 5px;text-align:left;text-decoration:underline;vertical-align:top;word-break:normal\"><a href=\"https://www.pygimli.org/documentation.html\">https://www.pygimli.org/documentation.html</a></td></tr></tbody></table>\n",
    "        </div>\n",
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    "    <div style=\"float:right;width:30%\">\n",
    "        <img src=\"https://www.pygimli.org/_images/pg_logo.png\" alt=\"pyGIMLi logo\">\n",
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    "# Inversion with custom forward operators\n",
    "### Objectives\n",
    "1. Show how to use an own forward operator along with the pyGIMLi inversion\n",
    "2. Generate a synthetic model for drone-base CSEM\n",
    "3. Invert the synthetic data with pyGIMLi\n",
    "\n",
    "We use a modelling tool called `empymod` from Dieter Werthmüller.\n",
    "The repository can be found on https://emsig.github.io/\n",
    "\n",
    "`empymod` provides electromagnetic modelling for 1D VTI media with arbitrary transmitters and sensors in 3D space. For installation we call:\n",
    "```\n",
    "conda install -c conda-forge empymod\n",
    "```\n",
    "or run `pip install empymod` in the console or in a cell here."
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.style.use(\"seaborn-notebook\")\n",
    "%config InlineBackend.figure_format='svg'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pygimli as pg\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.colors as clr\n",
    "import empymod"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we define some settings that are to be used. We assume an electric dipole transmitter perpendicular to the x-z place of length `txLen` and magnetic receivers in the air (e.g. on a drone) along a profile line. Date are obtained in frequency domain for a number of frequencies."
   ]
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  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# receiver setting\n",
    "rx = np.arange(50., 501., 50)\n",
    "ry = np.zeros(rx.size)\n",
    "# empymod arguments including transmitter\n",
    "txLen = 400\n",
    "inpdat = {'src': [0, 0, txLen/2, -txLen/2., 0.1, 0.1], 'strength': 1,  # ground\n",
    "           'mrec': True, 'rec': [rx, ry, -20, 0, 90],  # Hz air\n",
    "           # 'mrec': False, 'rec': [rx, ry, 0.1, 90, 0],  # Ey ground\n",
    "          'srcpts': 11, 'htarg': {'pts_per_dec': -1}, 'verb': 1}\n",
    "# frequencies\n",
    "freqs = [10., 20, 50, 100, 200, 500, 1000, 2000., 5000.]  # Hz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We define a function for calling empymod with vectors of resistivity and depth."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fwd(res, dep):\n",
    "    \"\"\"Call empymods function bipole with the above arguments.\"\"\"\n",
    "    assert len(res) == len(dep)\n",
    "    OUT = np.zeros((len(freqs), len(rx)), dtype=np.complex)\n",
    "    for i, f in enumerate(freqs):\n",
    "        OUT[i, :] = empymod.bipole(res=np.concatenate(([2e14], res)),\n",
    "                                   depth=dep, freqtime=f, **inpdat)\n",
    "\n",
    "    return OUT"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we create a functon to show the real and imaginary part of the data as image plots."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def showData(A, ax=None, sym=True):\n",
    "    \"\"\"Show data as two subplots for real and imaginary part.\"\"\"\n",
    "    if ax is None:\n",
    "        fig, ax = plt.subplots(1, 2)\n",
    "\n",
    "    if A.ndim == 1: # long vector\n",
    "        reim = np.reshape(A, (2, -1))\n",
    "        A = np.reshape(reim[0], (len(freqs), -1)) + \\\n",
    "            np.reshape(reim[1], (len(freqs), -1)) * 1j\n",
    " \n",
    "    norm = clr.NoNorm()\n",
    "    cmap = \"Spectral_r\"\n",
    "    if sym:\n",
    "        mi = np.min(np.abs(A))\n",
    "        ma = np.max(np.abs(A))\n",
    "        norm = clr.SymLogNorm(mi, vmin=-ma, vmax=ma, base=10)#, linscale=mi)\n",
    "        cmap = 'RdBu_r'\n",
    "\n",
    "    imr = ax[0].imshow(A.real, norm=norm, cmap=cmap)\n",
    "    plt.colorbar(imr, ax=ax[0], orientation=\"horizontal\")\n",
    "    imi = ax[1].imshow(A.imag, norm=norm, cmap=cmap)\n",
    "    plt.colorbar(imi, ax=ax[1], orientation=\"horizontal\")\n",
    "    if A.shape[1] == len(rx):\n",
    "        for a in ax:\n",
    "            a.xaxis.set_ticks(range(len(rx))[::3])\n",
    "            a.xaxis.set_ticklabels([str(int(r)) for r in rx][::3])\n",
    "            a.set_xlabel('x (m)')\n",
    "            a.yaxis.set_ticks([y for y in range(len(freqs))])\n",
    "            a.yaxis.set_ticklabels([str(int(f)) for f in freqs])\n",
    "            a.set_ylabel('f (Hz)')\n",
    "\n",
    "    ax[0].xaxis.set_ticks([])\n",
    "    ax[0].set_xlabel('')\n",
    "    ax[0].set_title(\"real part\")\n",
    "    ax[1].set_title(\"imaginary part\")\n",
    "    plt.tight_layout()\n",
    "    return ax"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now create a synthetic model of a good conductor embedded in a resistive halfspace."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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   "source": [
    "depth = [0, 100, 200]  # m\n",
    "res = [100, 10, 100]\n",
    "A0 = fwd(res, depth)\n",
    "showData(A0);"
   ]
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   "cell_type": "markdown",
   "metadata": {
    "tags": []
   },
   "source": [
    "To use the forward modelling in pyGIMLi, we create a class, derived from the pg modelling base class, and set the `response` and `createStartModel` functions, and init a 1D mesh for an Occam-type inversion. "
   ]
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  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "class myFwd(pg.Modelling):\n",
    "    def __init__(self, depth):\n",
    "        \"\"\"Initialize the model.\"\"\"\n",
    "        self.dep = depth\n",
    "        self.mesh1d = pg.meshtools.createMesh1D(len(self.dep))\n",
    "        super().__init__()\n",
    "        self.setMesh(self.mesh1d)\n",
    "    \n",
    "    def response(self, model):\n",
    "        \"\"\"Forward response.\"\"\"\n",
    "        A = fwd(model, self.dep)\n",
    "        Avec = A.ravel()\n",
    "        return np.hstack((Avec.real, Avec.imag))\n",
    "        \n",
    "    def createStartModel(self):\n",
    "        return pg.Vector(len(self.dep)) * 100"
   ]
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   "cell_type": "markdown",
   "metadata": {
    "tags": []
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   "source": [
    "We organize the real and imaginary part and add relative error to the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "A0vec = A0.ravel()\n",
    "data = np.hstack((A0vec.real, A0vec.imag))\n",
    "# error = np.abs(data) * 0.01 + 0.00001\n",
    "relativeError = np.ones_like(data) * 0.01\n",
    "data *= (np.random.randn(len(data)) * relativeError + 1.0)"
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   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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   "source": [
    "depth_fixed = np.linspace(0., 300., 21)\n",
    "fop = myFwd(depth_fixed)\n",
    "resistivity = np.ones_like(depth_fixed) * 100\n",
    "response = fop.response(resistivity)\n",
    "showData(response);"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We set up the pyGIMLi inversion usng the forward operator created by `myFwd` (fop) using `setForwardOperator`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "inv = pg.Inversion()\n",
    "inv.setForwardOperator(fop)\n",
    "transModel = pg.trans.TransLog(1) # > 1 Ohmm\n",
    "inv.transModel = transModel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "28/04/21 - 17:37:49 - pyGIMLi - \u001b[0;32;49mINFO\u001b[0m - Startmodel set from given value. 100.0\n",
      "28/04/21 - 17:37:49 - pyGIMLi - \u001b[0;32;49mINFO\u001b[0m - Starting inversion.\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "fop: <__main__.myFwd object at 0x7fe2f8720680>\n",
      "Data transformation: <pgcore._pygimli_.RTransLin object at 0x7fe308308630>\n",
      "Model transformation: <pgcore._pygimli_.RTransLog object at 0x7fe2f8720270>\n",
      "min/max (data): -4.7e-04/0.0027\n",
      "min/max (error): 1.00%/1.00%\n",
      "min/max (start model): 100/100\n",
      "--------------------------------------------------------------------------------\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 2 ... chi² = 89.5 (dPhi = 78.51%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 3 ... chi² = 77.91 (dPhi = 13.29%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 4 ... chi² = 68.32 (dPhi = 12.54%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 5 ... chi² = 55.57 (dPhi = 18.84%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 6 ... chi² = 27.14 (dPhi = 50.45%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 7 ... chi² = 12.73 (dPhi = 53.28%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 8 ... chi² = 6.97 (dPhi = 46.14%) lam: 20.0\n",
      "--------------------------------------------------------------------------------\n",
      "inv.iter 9 ... chi² = 1.0 (dPhi = 82.39%) lam: 20.0\n",
      "\n",
      "\n",
      "################################################################################\n",
      "#                  Abort criterion reached: chi² <= 1 (1.00)                   #\n",
      "################################################################################\n"
     ]
    }
   ],
   "source": [
    "model = inv.run(data, relativeError, startModel=100, verbose=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot the model using a dedicated model plotter along with the synthetic model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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      "text/plain": [
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     },
     "metadata": {
      "needs_background": "light"
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     "output_type": "display_data"
    }
   ],
   "source": [
    "from pygimli.viewer.mpl import drawModel1D\n",
    "fig, ax = plt.subplots()\n",
    "drawModel1D(ax, np.diff(depth), res, color=\"red\", label=\"synthetic\")\n",
    "drawModel1D(ax, np.diff(depth_fixed), model, color=\"blue\", label=\"inverted\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "# Plot the model response (to be compared to the data)\n",
    "showData(response);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
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      "text/plain": [
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   ],
   "source": [
    "# Plot the misfit\n",
    "showData(inv.response-data);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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