SimPEG Play - DC Resistivity Inversion

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Working to incorporate DC resistivity inversions in the new SimPEG framework.

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<div class="highlight"><pre><span class="kn">from</span> <span class="nn">SimPEG</span> <span class="kn">import</span> <span class="o">*</span>
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Warning: mumps solver not available.

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<div class="highlight"><pre><span class="c"># Create the mesh</span>
<span class="n">h1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
<span class="n">h2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">21</span><span class="p">)</span>
<span class="n">M</span> <span class="o">=</span> <span class="n">Mesh</span><span class="o">.</span><span class="n">TensorMesh</span><span class="p">([</span><span class="n">h1</span><span class="p">,</span><span class="n">h2</span><span class="p">])</span>

<span class="c"># Create some parameters for the model</span>
<span class="n">sig1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mf">0.001</span><span class="p">)</span>
<span class="n">sig2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>

<span class="c"># Create a synthetic model from a block in a half-space</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">Model</span><span class="o">.</span><span class="n">LogModel</span><span class="p">(</span><span class="n">M</span><span class="p">)</span>
<span class="n">mSynth</span> <span class="o">=</span> <span class="n">Utils</span><span class="o">.</span><span class="n">ModelBuilder</span><span class="o">.</span><span class="n">defineBlockConductivity</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">gridCC</span><span class="p">,[</span><span class="mi">50</span><span class="p">,</span> <span class="mi">10</span><span class="p">],[</span><span class="mi">80</span><span class="p">,</span> <span class="mi">16</span><span class="p">],[</span><span class="n">sig1</span><span class="p">,</span> <span class="n">sig2</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">plotImage</span><span class="p">(</span><span class="n">mSynth</span><span class="p">))</span>
<span class="n">Tx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">10.5</span><span class="p">,</span><span class="mi">90</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="n">Rx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">9.5</span><span class="p">,</span><span class="mi">90</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
<span class="n">yLoc</span> <span class="o">=</span> <span class="mf">13.5</span>
<span class="n">plot</span><span class="p">(</span><span class="n">Rx</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">Rx</span><span class="p">)</span><span class="o">*</span><span class="n">yLoc</span><span class="p">,</span><span class="s">&#39;r.&#39;</span><span class="p">,</span><span class="n">ms</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">Tx</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">Tx</span><span class="p">)</span><span class="o">*</span><span class="n">yLoc</span><span class="p">,</span><span class="s">&#39;c.&#39;</span><span class="p">,</span><span class="n">ms</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
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[&lt;matplotlib.lines.Line2D at 0x10f9e1510&gt;]
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<div class="highlight"><pre><span class="n">inds</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Rx</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
    <span class="k">for</span> <span class="n">jj</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Rx</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="n">ii</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">inds</span><span class="o">.</span><span class="n">append</span><span class="p">([</span> <span class="n">ii</span><span class="p">,</span> <span class="n">ii</span> <span class="o">+</span> <span class="n">jj</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">])</span>
<span class="n">posElectrodes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">Rx</span><span class="p">[</span><span class="n">ind</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span><span class="n">yLoc</span><span class="p">]</span> <span class="k">for</span> <span class="n">ind</span> <span class="ow">in</span> <span class="n">inds</span><span class="p">])</span>
<span class="n">negElectrodes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">Rx</span><span class="p">[</span><span class="n">ind</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span><span class="n">yLoc</span><span class="p">]</span> <span class="k">for</span> <span class="n">ind</span> <span class="ow">in</span> <span class="n">inds</span><span class="p">])</span>

<span class="n">Ppos</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">getInterpolationMat</span><span class="p">(</span><span class="n">posElectrodes</span><span class="p">,</span> <span class="s">&#39;CC&#39;</span><span class="p">)</span>
<span class="n">Pneg</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">getInterpolationMat</span><span class="p">(</span><span class="n">negElectrodes</span><span class="p">,</span> <span class="s">&#39;CC&#39;</span><span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">Ppos</span> <span class="o">-</span> <span class="n">Pneg</span>

<span class="n">inds</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">ii</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Tx</span><span class="o">.</span><span class="n">size</span><span class="p">):</span>
    <span class="k">for</span> <span class="n">jj</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">Tx</span><span class="o">.</span><span class="n">size</span> <span class="o">-</span> <span class="n">ii</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">inds</span><span class="o">.</span><span class="n">append</span><span class="p">([</span> <span class="n">ii</span><span class="p">,</span> <span class="n">ii</span> <span class="o">+</span> <span class="n">jj</span> <span class="o">+</span> <span class="mi">1</span> <span class="p">])</span>
<span class="n">posElectrodes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">Tx</span><span class="p">[</span><span class="n">ind</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span><span class="n">yLoc</span><span class="p">]</span> <span class="k">for</span> <span class="n">ind</span> <span class="ow">in</span> <span class="n">inds</span><span class="p">])</span>
<span class="n">negElectrodes</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">Tx</span><span class="p">[</span><span class="n">ind</span><span class="p">[</span><span class="mi">1</span><span class="p">]],</span><span class="n">yLoc</span><span class="p">]</span> <span class="k">for</span> <span class="n">ind</span> <span class="ow">in</span> <span class="n">inds</span><span class="p">])</span>

<span class="n">Qpos</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">getInterpolationMat</span><span class="p">(</span><span class="n">posElectrodes</span><span class="p">,</span> <span class="s">&#39;CC&#39;</span><span class="p">)</span>
<span class="n">Qneg</span> <span class="o">=</span> <span class="n">M</span><span class="o">.</span><span class="n">getInterpolationMat</span><span class="p">(</span><span class="n">negElectrodes</span><span class="p">,</span> <span class="s">&#39;CC&#39;</span><span class="p">)</span>
<span class="n">Q</span> <span class="o">=</span> <span class="p">(</span><span class="n">Qpos</span> <span class="o">-</span> <span class="n">Qneg</span><span class="p">)</span><span class="o">.</span><span class="n">T</span><span class="o">.</span><span class="n">toarray</span><span class="p">()</span>
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In&nbsp;[4]:
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<div class="highlight"><pre><span class="n">prob</span> <span class="o">=</span> <span class="n">Examples</span><span class="o">.</span><span class="n">DC</span><span class="o">.</span><span class="n">DCProblem</span><span class="p">(</span><span class="n">M</span><span class="p">,</span> <span class="n">model</span><span class="p">)</span>
<span class="c"># Create some data</span>
<span class="n">data</span> <span class="o">=</span> <span class="n">prob</span><span class="o">.</span><span class="n">createSyntheticData</span><span class="p">(</span><span class="n">mSynth</span><span class="p">,</span> <span class="n">std</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">P</span><span class="o">=</span><span class="n">P</span><span class="p">,</span> <span class="n">RHS</span><span class="o">=</span><span class="n">Q</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">prob</span><span class="o">.</span><span class="n">field</span><span class="p">(</span><span class="n">mSynth</span><span class="p">)</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">data</span><span class="o">.</span><span class="n">reshapeFields</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">M</span><span class="o">.</span><span class="n">plotImage</span><span class="p">(</span><span class="n">u</span><span class="p">[:,</span><span class="mi">2</span><span class="p">])</span>
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&lt;matplotlib.collections.QuadMesh at 0x10fc77310&gt;
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<div class="highlight"><pre><span class="c"># Now set up the prob to do some minimization</span>
<span class="c"># prob.dobs = dobs</span>
<span class="c"># prob.std = dobs*0 + 0.05</span>
<span class="n">m0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">nC</span><span class="p">)</span> <span class="o">+</span> <span class="n">sig2</span>

<span class="n">reg</span> <span class="o">=</span> <span class="n">Regularization</span><span class="o">.</span><span class="n">Tikhonov</span><span class="p">(</span><span class="n">model</span><span class="p">)</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">Parameters</span><span class="o">.</span><span class="n">BetaEstimate</span><span class="p">(</span><span class="n">beta0_ratio</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">obj</span> <span class="o">=</span> <span class="n">ObjFunction</span><span class="o">.</span><span class="n">BaseObjFunction</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">reg</span><span class="p">,</span> <span class="n">beta</span><span class="o">=</span><span class="n">beta</span><span class="p">)</span>
<span class="n">opt</span> <span class="o">=</span> <span class="n">Optimization</span><span class="o">.</span><span class="n">InexactGaussNewton</span><span class="p">(</span><span class="n">maxIterLS</span><span class="o">=</span><span class="mi">20</span><span class="p">,</span> <span class="n">maxIter</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">maxIterCG</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">inv</span> <span class="o">=</span> <span class="n">Inversion</span><span class="o">.</span><span class="n">BaseInversion</span><span class="p">(</span><span class="n">obj</span><span class="p">,</span> <span class="n">opt</span><span class="p">)</span>

<span class="c"># Check Derivative</span>
<span class="n">derChk</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">m</span><span class="p">:</span> <span class="p">[</span><span class="n">obj</span><span class="o">.</span><span class="n">dataObj</span><span class="p">(</span><span class="n">m</span><span class="p">),</span> <span class="n">obj</span><span class="o">.</span><span class="n">dataObjDeriv</span><span class="p">(</span><span class="n">m</span><span class="p">)]</span>
<span class="c"># Tests.checkDerivative(derChk, mSynth)</span>
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<div class="highlight"><pre><span class="n">m</span> <span class="o">=</span> <span class="n">inv</span><span class="o">.</span><span class="n">run</span><span class="p">(</span><span class="n">m0</span><span class="p">)</span>
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Regularization has not set mref. SimPEG will set it to m0.
============================ Inexact Gauss Newton ============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment   
-----------------------------------------------------------------------------
   0  1.80e+02  1.25e+06  0.00e+00  1.25e+06    5.24e+03      0              
   1  1.80e+02  1.21e+06  6.15e+00  1.21e+06    6.17e+03      0              
   2  1.80e+02  1.14e+06  9.45e+01  1.16e+06    6.83e+03      0              
   3  1.80e+02  9.87e+05  1.53e+02  1.01e+06    1.69e+04      0              
   4  1.80e+02  8.33e+05  1.73e+02  8.64e+05    6.31e+04      2              
   5  1.80e+02  6.92e+05  2.17e+02  7.31e+05    6.27e+04      1              
   6  1.80e+02  5.64e+05  2.30e+02  6.05e+05    3.48e+04      0              
   7  1.80e+02  4.35e+05  2.70e+02  4.83e+05    1.48e+05      0              
   8  1.80e+02  3.55e+05  2.79e+02  4.05e+05    3.84e+04      0              
   9  1.80e+02  2.84e+05  2.82e+02  3.34e+05    5.42e+04      0              
  10  1.80e+02  2.01e+05  3.24e+02  2.60e+05    6.50e+04      0              
------------------------- STOP! -------------------------
1 : |fc-fOld| = 7.4684e+04 &lt;= tolF*(1+|f0|) = 1.2546e+05
0 : |xc-x_last| = 6.7408e+00 &lt;= tolX*(1+|x0|) = 1.0000e-01
0 : |proj(x-g)-x|    = 6.5036e+04 &lt;= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 6.5036e+04 &lt;= 1e3*eps       = 1.0000e-02
1 : maxIter   =      10    &lt;= iter          =     10
------------------------- DONE! -------------------------

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<div class="highlight"><pre><span class="n">plt</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">M</span><span class="o">.</span><span class="n">plotImage</span><span class="p">(</span><span class="n">m</span><span class="p">))</span>
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&lt;matplotlib.colorbar.Colorbar instance at 0x111e14368&gt;
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