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Add error propagation to the models
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Joost van den Born committed
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@@ -0,0 +1,2 @@
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.DS_Store
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@@ -13143,7 +13143,7 @@ div#notebook {
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<span class="n">l2</span> <span class="o">=</span> <span class="mf">1.78</span> <span class="c1"># micron</span>
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<span class="c1"># Initializing dispersion model</span>
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<span class="n">at</span> <span class="o">=</span> <span class="n">Ciddor1996</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">T</span><span class="o">=</span><span class="n">T</span><span class="p">,</span> <span class="n">H</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">xc</span><span class="o">=</span><span class="n">xc</span><span class="p">)</span>
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<span class="n">at</span> <span class="o">=</span> <span class="n">Observatory</span><span class="p">()</span>
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<span class="c1"># Calculating indices of refraction for l1 and l2</span>
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<span class="n">n1</span> <span class="o">=</span> <span class="n">at</span><span class="o">.</span><span class="n">n_tph</span><span class="p">(</span><span class="n">l</span><span class="o">=</span><span class="n">l1</span><span class="p">,</span> <span class="n">T</span><span class="o">=</span><span class="n">T</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">RH</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">xc</span><span class="o">=</span><span class="n">xc</span><span class="p">)</span>
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@@ -13171,8 +13171,9 @@ div#notebook {
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Density of the atmosphere (following CIPM-81/91 equations)</span>
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<span class="n">rho</span> <span class="o">=</span> <span class="n">at</span><span class="o">.</span><span class="n">rho</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">T</span><span class="o">=</span><span class="n">T</span><span class="p">,</span> <span class="n">RH</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">xc</span><span class="o">=</span><span class="n">xc</span><span class="p">)</span>
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<span class="c1"># Initializing refraction model</span>
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<span class="n">ref</span> <span class="o">=</span> <span class="n">refraction</span><span class="p">(</span><span class="n">lat</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>
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<span class="c1"># Initializing refraction model and setting the reduced height</span>
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<span class="n">ref</span> <span class="o">=</span> <span class="n">refraction</span><span class="p">(</span><span class="n">lat</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
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<span class="n">ref</span><span class="o">.</span><span class="n">setReducedHeight</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>
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</pre></div>
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</div>
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@@ -13225,20 +13226,32 @@ div#notebook {
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<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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</div><div class="inner_cell">
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<div class="text_cell_render border-box-sizing rendered_html">
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<p>Dispersion is simply the refraction differential between two wavelengths. This is calculated with this module by</p>
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<p>Some error propagation is included in the module as well. It takes is able to calculate the uncertainty in the refractive index and uses that to propagate it into the refraction formule for the cassini model. Note: The Cassini model is the only refraction model for which error propagation is possible, in this module.</p>
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<p>In this example we assume the following uncertainties:</p>
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<ul>
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<li>Wavelength: 1 nm</li>
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<li>Temperature: 0.2 K</li>
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<li>Pressure: 30 Pa</li>
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<li>Relative humidity: 0.03 (3%)</li>
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<li>CO2 density: 25 ppm</li>
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<li>Observed zenith angle: 1 arc second</li>
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</ul>
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</div>
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</div>
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [4]:</div>
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<div class="prompt input_prompt">In [17]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Initializing dispersion object</span>
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<span class="n">disp</span> <span class="o">=</span> <span class="n">dispersion</span><span class="p">(</span><span class="n">lat</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>
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<span class="n">dispersion_obs</span> <span class="o">=</span> <span class="n">disp</span><span class="o">.</span><span class="n">cassini</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">zenith</span><span class="o">=</span><span class="n">zenith_obs</span><span class="p">)</span><span class="o">*</span><span class="mf">3.6e6</span>
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<span class="nb">print</span> <span class="p">(</span><span class="s1">'The dispersion is </span><span class="si">%.03f</span><span class="s1"> milli arc seconds'</span> <span class="o">%</span><span class="p">(</span><span class="n">dispersion_obs</span><span class="p">))</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Calculating the uncertainty in the refractive index</span>
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<span class="n">n1</span> <span class="o">=</span> <span class="n">at</span><span class="o">.</span><span class="n">n_tph</span><span class="p">(</span><span class="n">l</span><span class="o">=</span><span class="n">l1</span><span class="p">,</span> <span class="n">T</span><span class="o">=</span><span class="n">T</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">RH</span><span class="o">=</span><span class="n">H</span><span class="p">,</span> <span class="n">xc</span><span class="o">=</span><span class="n">xc</span><span class="p">)</span>
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<span class="n">dn1</span> <span class="o">=</span> <span class="n">at</span><span class="o">.</span><span class="n">dn_tph</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">xc</span><span class="p">,</span> <span class="n">dl</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">dT</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">dP</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">dRH</span><span class="o">=</span><span class="mf">0.03</span><span class="p">,</span> <span class="n">dCO2</span><span class="o">=</span><span class="mi">25</span><span class="p">)</span>
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<span class="c1"># Calculating the uncertainty in the atmospheric refraction</span>
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<span class="n">refError</span> <span class="o">=</span> <span class="n">ref</span><span class="o">.</span><span class="n">cassiniError</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">dn1</span><span class="p">,</span> <span class="n">zenith_obs</span><span class="p">,</span> <span class="n">dz</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="mi">3600</span><span class="p">)</span><span class="o">*</span><span class="mf">3.6e3</span>
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<span class="nb">print</span> <span class="p">(</span><span class="s1">'The refraction is: </span><span class="si">%.3f</span><span class="s1"> +/- </span><span class="si">%.3f</span><span class="s1"> arc seconds'</span> <span class="o">%</span><span class="p">(</span><span class="n">refraction_obs</span><span class="p">,</span> <span class="n">refError</span><span class="p">))</span>
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</pre></div>
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</div>
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@@ -13255,7 +13268,57 @@ div#notebook {
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<div class="output_subarea output_stream output_stdout output_text">
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<pre>The dispersion is 31.019 milli arc seconds
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<pre>The refraction is: 40.697 +/- 0.035 arc seconds
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</pre>
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</div>
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</div>
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</div>
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</div>
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</div>
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<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
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</div><div class="inner_cell">
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<div class="text_cell_render border-box-sizing rendered_html">
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<p>Dispersion is simply the refraction differential between two wavelengths. This is calculated with a dispersion object. Again, we can also calculate the uncertainty in the dispersion based on the measurement uncertainties. We take the same values as above.</p>
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<p>Note that, since the unertainties in the refractive index at two wavelengths observated at the same time are correlated, we cannot simply use n1, n2, dn1 and dn2 as an input but require the whole set of conditions and uncertainties.</p>
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</div>
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</div>
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="input">
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<div class="prompt input_prompt">In [23]:</div>
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<div class="inner_cell">
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<div class="input_area">
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Initializing dispersion object</span>
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<span class="n">disp</span> <span class="o">=</span> <span class="n">dispersion</span><span class="p">(</span><span class="n">lat</span><span class="p">,</span> <span class="n">h</span><span class="p">)</span>
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<span class="n">disp</span><span class="o">.</span><span class="n">setReducedHeight</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">rho</span><span class="p">)</span>
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<span class="n">dispersion_obs</span> <span class="o">=</span> <span class="n">disp</span><span class="o">.</span><span class="n">cassini</span><span class="p">(</span><span class="n">n1</span><span class="p">,</span> <span class="n">n2</span><span class="p">,</span> <span class="n">zenith_obs</span><span class="p">)</span><span class="o">*</span><span class="mf">3.6e6</span>
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<span class="c1"># Calculating the uncertainty in the atmospheric dispersion</span>
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<span class="n">dispersion_unc</span> <span class="o">=</span> <span class="n">disp</span><span class="o">.</span><span class="n">cassiniError</span><span class="p">(</span><span class="n">zenith_obs</span><span class="p">,</span> <span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">,</span> <span class="n">T</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">H</span><span class="p">,</span> <span class="n">xc</span><span class="p">,</span>
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<span class="n">dl1</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">dl2</span><span class="o">=</span><span class="mf">0.001</span><span class="p">,</span> <span class="n">dT</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">dP</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">dRH</span><span class="o">=</span><span class="mf">0.03</span><span class="p">,</span> <span class="n">dCO2</span><span class="o">=</span><span class="mi">25</span><span class="p">,</span> <span class="n">dz</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="mi">3600</span><span class="p">)</span> <span class="o">*</span> <span class="mf">3.6e6</span>
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<span class="nb">print</span> <span class="p">(</span><span class="s1">'The dispersion is </span><span class="si">%.03f</span><span class="s1"> +/- </span><span class="si">%.03f</span><span class="s1"> milli arc seconds'</span> <span class="o">%</span><span class="p">(</span><span class="n">dispersion_obs</span><span class="p">,</span> <span class="n">dispersion_unc</span><span class="p">))</span>
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</pre></div>
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</div>
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</div>
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</div>
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<div class="output_wrapper">
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<div class="output">
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<div class="output_area">
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<div class="prompt"></div>
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<div class="output_subarea output_stream output_stdout output_text">
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<pre>The dispersion is 31.019 +/- 0.164 milli arc seconds
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</pre>
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</div>
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</div>
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@@ -31,12 +31,11 @@
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"l2 = 1.78 # micron\n",
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"\n",
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"# Initializing dispersion model\n",
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"at = Ciddor1996(p=P, T=T, H=H, xc=xc)\n",
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"at = Observatory()\n",
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"\n",
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"# Calculating indices of refraction for l1 and l2\n",
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"n1 = at.n_tph(l=l1, T=T, p=P, RH=H, xc=xc)\n",
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"n2 = at.n_tph(l=l2, T=T, p=P, RH=H, xc=xc)\n",
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"\n"
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"n2 = at.n_tph(l=l2, T=T, p=P, RH=H, xc=xc)\n"
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]
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},
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{
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@@ -56,8 +55,9 @@
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"# Density of the atmosphere (following CIPM-81/91 equations)\n",
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"rho = at.rho(p=P, T=T, RH=H, xc=xc)\n",
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"\n",
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"# Initializing refraction model\n",
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"ref = refraction(lat, h, P, rho)"
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"# Initializing refraction model and setting the reduced height\n",
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"ref = refraction(lat, h)\n",
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"ref.setReducedHeight(P, rho)"
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]
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},
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{
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@@ -91,27 +91,74 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Dispersion is simply the refraction differential between two wavelengths. This is calculated with this module by"
|
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"Some error propagation is included in the module as well. It takes is able to calculate the uncertainty in the refractive index and uses that to propagate it into the refraction formule for the cassini model. Note: The Cassini model is the only refraction model for which error propagation is possible, in this module.\n",
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"\n",
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"In this example we assume the following uncertainties:\n",
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"* Wavelength: 1 nm\n",
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"* Temperature: 0.2 K\n",
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"* Pressure: 30 Pa\n",
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"* Relative humidity: 0.03 (3%)\n",
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"* CO2 density: 25 ppm\n",
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"* Observed zenith angle: 1 arc second"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"The dispersion is 31.019 milli arc seconds\n"
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"The refraction is: 40.697 +/- 0.035 arc seconds\n"
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]
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}
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],
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"source": [
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"# Calculating the uncertainty in the refractive index\n",
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"n1 = at.n_tph(l=l1, T=T, p=P, RH=H, xc=xc)\n",
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"dn1 = at.dn_tph(l1, T, P, H, xc, dl=0.001, dT=0.2, dP=30, dRH=0.03, dCO2=25)\n",
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"\n",
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"# Calculating the uncertainty in the atmospheric refraction\n",
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"refError = ref.cassiniError(n1, dn1, zenith_obs, dz=1/3600)*3.6e3\n",
|
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"print ('The refraction is: %.3f +/- %.3f arc seconds' %(refraction_obs, refError))"
|
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
|
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"source": [
|
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"Dispersion is simply the refraction differential between two wavelengths. This is calculated with a dispersion object. Again, we can also calculate the uncertainty in the dispersion based on the measurement uncertainties. We take the same values as above.\n",
|
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"\n",
|
||||
"Note that, since the unertainties in the refractive index at two wavelengths observated at the same time are correlated, we cannot simply use n1, n2, dn1 and dn2 as an input but require the whole set of conditions and uncertainties."
|
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]
|
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},
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{
|
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"cell_type": "code",
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"execution_count": 23,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
|
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"The dispersion is 31.019 +/- 0.164 milli arc seconds\n"
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]
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}
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],
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"source": [
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"# Initializing dispersion object\n",
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"disp = dispersion(lat, h, P, rho)\n",
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"dispersion_obs = disp.cassini(n1, n2, zenith=zenith_obs)*3.6e6\n",
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"print ('The dispersion is %.03f milli arc seconds' %(dispersion_obs))"
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"disp = dispersion(lat, h)\n",
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"disp.setReducedHeight(P, rho)\n",
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"\n",
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"dispersion_obs = disp.cassini(n1, n2, zenith_obs)*3.6e6\n",
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"\n",
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"# Calculating the uncertainty in the atmospheric dispersion\n",
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"dispersion_unc = disp.cassiniError(zenith_obs, l1, l2, T, P, H, xc, \n",
|
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" dl1=0.001, dl2=0.001, dT=0.2, dP=30, dRH=0.03, dCO2=25, dz=1/3600) * 3.6e6\n",
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"\n",
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"print ('The dispersion is %.03f +/- %.03f milli arc seconds' %(dispersion_obs, dispersion_unc))"
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]
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}
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],
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+497
-34
@@ -17,14 +17,13 @@ from scipy import integrate
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'''
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class Ciddor1996:
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def __init__(self, p=101325, T=288.15, H=0, xc=450):
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self.p = p # Pressure in Pa
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self.T = T # Temperature in Kelvin
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self.H = H # Relative humidity on a scale from 0 to 1
|
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self.xc = xc # ppm of CO2
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pass
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class Observatory:
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def __init__(self, dT=0.2, dP=20, dH=0.02, dCO2=20, dl=0.001):
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self.dT = dT # in K
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self.dP = dP # in Pa
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self.dH = dH # relative humidity between 0 and 1
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self.dCO2 = dCO2 # in ppm
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self.dl = dl # in micron
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def n_as(self, l):
|
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'''
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@@ -191,15 +190,216 @@ class Ciddor1996:
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||||
return np.sqrt( (1 + 2 * _Ltotal) / (1 - _Ltotal) )
|
||||
|
||||
|
||||
class error(Ciddor1996):
|
||||
def __init__(self, p=101325, T=288.15, H=0, xc=450, dT=0.2, dP=20, dH=0.02, dCO2=20, dl=0.001):
|
||||
Ciddor1996.__init__(self, p, T, H)
|
||||
self.dT = 0#dT # in K
|
||||
self.dP = 0#dP # in Pa
|
||||
self.dH = 0#dH # relative humidity between 0 and 1
|
||||
self.dCO2 = 0#dCO2 # in ppm
|
||||
self.dl = 0#dl # in micron
|
||||
def dn_tph(self, l, T, p, RH, xc, dl=None, dT=None, dP=None, dRH=None, dCO2=None):
|
||||
'''
|
||||
Calculates the uncertainty in the refractive index, following the Ciddor model,
|
||||
by using fully propagated error analysis.
|
||||
This involves almost endless use of the chain rule and evaluation of both the moist
|
||||
air and dry air components. Note: dry air does not contain any water, i.e. H=0, x_w=0, etc.
|
||||
|
||||
Parameters:
|
||||
l - Wavelength in micron (0.4 - 2.5 micron)
|
||||
T - Temperature in Kelvin
|
||||
p - Pressure in Pascals
|
||||
RH - Relative humidity (no units, don't use percentage but value 0<x<1)
|
||||
xc - CO2 density in ppm
|
||||
dl - Uncertainty in the blue wavelength
|
||||
dT - Uncertainty in the temperature, default: 0.2 K
|
||||
dP - Uncertainty in the pressure, default: 20 Pa
|
||||
dRH - Uncertainty in the relative humidity, default: 0.02
|
||||
dCO2- Uncertainty in the CO2 density, default: 20
|
||||
'''
|
||||
|
||||
if dP == None:
|
||||
dP = self.dP
|
||||
if dT == None:
|
||||
dT = self.dT
|
||||
if dRH == None:
|
||||
dRH = self.dH
|
||||
if dCO2 == None:
|
||||
dCO2 = self.dCO2
|
||||
|
||||
# constants
|
||||
_cf = 1.022
|
||||
_w0 = 295.235 # um^-2
|
||||
_w1 = 2.6422 # um^-2
|
||||
_w2 = -0.032380 # um^-4
|
||||
_w3 = 0.004028 # um^-6
|
||||
|
||||
_k0 = 238.0185 # um^-2
|
||||
_k1 = 5792105 # um^-2
|
||||
_k2 = 57.362 # um^-2
|
||||
_k3 = 167917 # um^-2
|
||||
|
||||
_A = 1.2378847e-5 # K^-2
|
||||
_B = -1.9121316e-2 # K^-2
|
||||
_C = 33.93711047
|
||||
_D = -6.3431645e3 # K^-2
|
||||
|
||||
_Mw = 0.018015 # kg/mol
|
||||
_R = 8.314510 # J mol^-1 K^-1, gas constant
|
||||
|
||||
_a0 = 1.58123e-6 # K Pa^-1
|
||||
_a1 = -2.9331e-8 # Pa^-1
|
||||
_a2 = 1.1043e-10 # K^-1 Pa^-1
|
||||
_b0 = 5.707e-6 # K Pa^-1
|
||||
_b1 = -2.051e-8 # Pa^-1
|
||||
_c0 = 1.9898e-4 # K Pa^-1
|
||||
_c1 = -2.376e-6 # Pa^-1
|
||||
_d = 1.83e-11 # K^2 Pa^-2
|
||||
_e = -0.765e-8 # K^2 Pa^-2
|
||||
_t = T - 273.15
|
||||
|
||||
_aa = 1.00062
|
||||
_bb = 3.14e-8 # Pa^-1
|
||||
_cc = 5.6e-7 # deg C^-2
|
||||
|
||||
_RH_a = 0
|
||||
_RH_w = RH
|
||||
|
||||
# Calculated values
|
||||
# _rho_obs = self.rho(p, T, RH, xc)
|
||||
# _ref = refraction(lat, h, p, _rho_obs)
|
||||
# _alpha = _ref.rc * np.sin(np.deg2rad(z)) / (_ref.rc + _ref.H)
|
||||
_p_a = p - RH*self.svp(T)
|
||||
_p_w = RH*self.svp(T)
|
||||
_x_w_a = self.xw(_p_a, T, _RH_a)
|
||||
_x_w_w = self.f(_p_w, T)
|
||||
_Z_a = self.Z(p=_p_a, T=T, RH=_RH_a)
|
||||
_Z_w = self.Z(p=_p_w, T=T, RH=_RH_w, xw=_x_w_w)
|
||||
_Ma = 1e-3 * (28.9635 + 12.011e-6 * (xc - 400) ) # Molar mass in kg/mol of dry air containing xc ppm of CO2
|
||||
_p_sv = self.svp(T)
|
||||
_rho_a = self.rho(p=p-RH*self.svp(T), T=T, RH=0, xc=xc)
|
||||
_rho_axs = self.rho(p=101325, T=288.15, RH=0, xc=xc)
|
||||
if RH == 0.0:
|
||||
# nan is returned for _rho_w if we put in a (partial) pressure of 0
|
||||
_rho_w = 0
|
||||
else:
|
||||
_rho_w = self.rho(p=RH*self.svp(T), T=T, RH=RH, xc=xc)
|
||||
_rho_ws = self.rho(p=1333, T=293.15, RH=1333/self.svp(293.15), xc=xc)
|
||||
_n_a1 = self.n_axs(l, xc)
|
||||
_n_w1 = self.n_ws(l)
|
||||
|
||||
_L1_a = (_n_a1**2 - 1) / (_n_a1**2 + 2) # Dry air component of L
|
||||
_L1_w = (_n_w1**2 - 1) / (_n_w1**2 + 2)
|
||||
_L1 = ( _rho_a / _rho_axs ) * _L1_a + ( _rho_w / _rho_ws ) * _L1_w # Overall L
|
||||
_n1 = np.sqrt( (1 + 2 * _L1) / (1 - _L1) )
|
||||
|
||||
# Partial derivatives for derivation to l1
|
||||
_dn_wdl = _cf * 1e-8 * (-1 * _w1 * 2/l**3 - _w2 * 4/l**5 - _w3 * 6/l**7)
|
||||
_dLdn_w1 = _rho_w / _rho_ws * 6 * _n_w1 / (_n_w1**2 + 2)**2
|
||||
_dn_adl = 1e-8 * (1 + 0.534e-6 * (xc - 450)) * ( - (2 * _k1 * l ) / ((_k0 * l**2 - 1)**2) - (2 * _k3 * l) / ((_k2 * l**2 - 1)**2) )
|
||||
_dLdn_a1 = _rho_a / _rho_axs * 6 * _n_a1 / (_n_a1**2 + 2)**2
|
||||
_dndL1 = ( ( (1 + 2*_L1) / (1 - _L1)**2 ) + (2 / (1-_L1)) ) / (2 * np.sqrt( (1 + 2*_L1) / (1 - _L1) ) )
|
||||
_dndl = _dndL1 * ( _dLdn_a1 * _dn_adl + _dLdn_w1 * _dn_wdl)
|
||||
|
||||
# # Partial derivatives for derivation to l2
|
||||
|
||||
|
||||
# Partial derivatives for derivation to T
|
||||
# Dry air parts:
|
||||
_drho_adp_a = _Ma / (_Z_a * _R * T) * (1 - _x_w_a * (1 - _Mw / _Ma))
|
||||
_dp_adp_sv = - RH
|
||||
_dp_svdT = (2*_A*T + _B - _D/(T**2)) * np.exp(_A * T**2 + _B * T + _C + _D/T )
|
||||
_drho_adTZ = - _p_a * _Ma / (_R * ( _Z_a * T)**2) * (1 - _x_w_a * (1 - _Mw / _Ma))
|
||||
_dTZdT_a = _p_a / T * (_a2 * _t**2 + _a1 * _t + _a0 + (_b1*_t + _b0) * _x_w_a + (_c0 + _c1*_t) * _x_w_a**2 ) - _p_a * (_a1 + 2*_a2*_t + _b1*_x_w_a + _c1*_x_w_a**2) - 2*(_p_a/T)**2 * (_d + _e * _x_w_a**2) + _Z_a
|
||||
_dTZdxw_a = 2*_e*_p_a**2 / T * _x_w_a - _p_a * (_b1*_t + _b0 + 2 * (_c0 + _c1*_t) * _x_w_a)
|
||||
_dxw_adf = _RH_a * _p_sv / _p_a
|
||||
_dfdT = 2*_cc*_t
|
||||
_dxw_adp_sv = self.f(_p_a, T) * _RH_a / _p_a
|
||||
_drho_adxw_a = 0 # as rho_a has not water content, i.e. rho = p*M/ZRT
|
||||
|
||||
|
||||
# Moist air parts:
|
||||
_drho_wdp_w = _Ma / (_Z_w * _R * T) * (1 - _x_w_w * (1 - _Mw / _Ma))
|
||||
_dp_wdp_sv = RH
|
||||
_dp_svdT = (2*_A*T + _B - _D/(T**2)) * np.exp(_A * T**2 + _B * T + _C + _D/T )
|
||||
_drho_wdTZ = - _p_w * _Ma / (_R * (_Z_w * T)**2) * (1 - _x_w_w * (1 - _Mw / _Ma))
|
||||
_dTZdT_w = _p_w / T * (_a2 * _t**2 + _a1 * _t + _a0 + (_b1*_t + _b0) * _x_w_w + (_c0 + _c1*_t) * _x_w_w**2 ) - _p_w * (_a1 + 2*_a2*_t + _b1*_x_w_w + _c1*_x_w_w**2) - 2*(_p_w/T)**2 * (_d + _e * _x_w_w**2) + _Z_w
|
||||
_dTZdxw_w = 2*_e*_p_w**2 / T * _x_w_w - _p_w * (_b1*_t + _b0 + 2 * (_c0 + _c1*_t) * _x_w_w)
|
||||
_dxw_wdf = 1
|
||||
_dfdT = 2*_cc*_t
|
||||
_dxw_wdp_sv = self.f(_p_w, T) / _p_sv
|
||||
_drho_wdxw_w = _p_w * _Ma / (_Z_w * _R * T) * (_Mw / _Ma - 1)
|
||||
|
||||
_tA = _drho_adp_a * _dp_adp_sv * _dp_svdT + _drho_adTZ * (_dTZdT_a + _dTZdxw_a * (_dxw_adf * _dfdT + _dxw_adp_sv * _dp_svdT) ) + _drho_adxw_a * ( _dxw_adf * _dfdT + _dxw_adp_sv * _dp_svdT )
|
||||
_tB = _drho_wdp_w * _dp_wdp_sv * _dp_svdT + _drho_wdTZ * (_dTZdT_w + _dTZdxw_w * (_dxw_wdf * _dfdT + _dxw_wdp_sv * _dp_svdT) ) + _drho_wdxw_w * ( _dxw_wdf * _dfdT + _dxw_wdp_sv * _dp_svdT )
|
||||
_dL1drho_a = _L1_a / _rho_axs
|
||||
_dL1drho_w = _L1_w / _rho_ws
|
||||
|
||||
_dndT = _tA * (_dndL1 * _dL1drho_a) + _tB * (_dndL1 * _dL1drho_w)
|
||||
|
||||
# Partial derivatives for derivation to p
|
||||
# Dry air parts:
|
||||
_drho_adZ = - _p_a * _Ma / ( _Z_a**2 * _R * T) * (1 - _x_w_a * (1 - _Mw / _Ma))
|
||||
_dZdp_a = - 1 / T * (_a0 + _a1 * _t + _a2 * _t**2 + (_b0 + _b1*_t)*_x_w_a + (_c0 + _c1*_t)*_x_w_a**2 ) + 2*_p_a/(T**2) * (_d + _e * _x_w_a**2)
|
||||
_dfdp = _bb
|
||||
_dxw_adp_a = - self.f(_p_a, T) * _RH_a * _p_sv / (_p_a)**2
|
||||
|
||||
# Moist air parts
|
||||
_dZdp_w = - 1 / T * (_a0 + _a1 * _t + _a2 * _t**2 + (_b0 + _b1*_t)*_x_w_w + (_c0 + _c1*_t)*_x_w_w**2 ) + 2*_p_w/(T**2) * (_d + _e * _x_w_w**2)
|
||||
# Not necessary, since dpwdp=0
|
||||
|
||||
_pA = _drho_adp_a + _drho_adZ * _dZdp_a + _drho_adxw_a * (_dxw_adf * _dfdp + _dxw_adp_a)
|
||||
_dndP = _pA * (_dndL1 * _dL1drho_a)
|
||||
|
||||
|
||||
# Partial derivatives for derivation to H
|
||||
# Dry air parts:
|
||||
_dp_adH = - _p_sv
|
||||
_dZdxw_a = (_p_a / T)**2 * 2 * _e * _x_w_a - (_p_a / T) * ( (_b0 + _b1 * _t) + 2 * (_c0 + _c1 * _t) * _x_w_a )
|
||||
_dxw_adH = self.f(_p_a, T) * _p_sv / _p_a
|
||||
|
||||
# Moist air parts:
|
||||
_dp_wdH = _p_sv
|
||||
_dZdxw_w = (_p_w / T)**2 * 2 * _e * _x_w_w - (_p_w / T) * ( (_b0 + _b1 * _t) + 2 * (_c0 + _c1 * _t) * _x_w_w )
|
||||
_dxw_wdH = 0
|
||||
_dxw_wdp_w = 0
|
||||
_drho_wdZ = - _p_w * _Ma / ( _Z_w**2 * _R * T) * (1 - _x_w_w * (1 - _Mw / _Ma))
|
||||
|
||||
_hA = _drho_adp_a * _dp_adH + _drho_adZ * ( _dZdp_a * _dp_adH + _dZdxw_a * (_dxw_adH + _dxw_adp_a * _dp_adH + _dxw_adf * _dfdp * _dp_adH) ) + _drho_adxw_a * (_dxw_adH + _dxw_adp_a * _dp_adH + _dxw_adf * _dfdp * _dp_adH)
|
||||
_hB = _drho_wdp_w * _dp_wdH + _drho_wdZ * ( _dZdp_w * _dp_wdH + _dZdxw_w * ( _bb * _p_sv ) ) + _drho_wdxw_w * ( _bb * _p_sv )
|
||||
_dndRH = _hA * (_dndL1 * _dL1drho_a) + _hB * (_dndL1 * _dL1drho_w)
|
||||
|
||||
|
||||
# Partial derivatives for derivation to xc
|
||||
# Dry air parts:
|
||||
_drho_adMa = _p_a * (1 - _x_w_a) / (_Z_a * _R * T)
|
||||
_dL1drho_axs = - (_rho_a * _L1_a)/(_rho_axs**2)
|
||||
_drho_axsdMa = _drho_adMa
|
||||
_dL1dL_a = _rho_a / _rho_axs
|
||||
_dL_adn_a1 = 6 * _n_a1 / (_n_a1**2 + 2)**2
|
||||
|
||||
# Moist air parts:
|
||||
_drho_wdMa = _p_w * (1 - _x_w_w) / (_Z_w * _R * T)
|
||||
_dL1drho_ws = - (_rho_w * _L1_w)/(_rho_ws**2)
|
||||
_drho_wsdMa = _drho_wdMa
|
||||
|
||||
# Other parts:
|
||||
_dMadx_c = 12.011e-9
|
||||
_dn_adx_c1 = (self.n_as(l) - 1) * 0.534e-6
|
||||
|
||||
_dL1dx_c = _dL1drho_a * _drho_adMa * _dMadx_c + _dL1drho_w * _drho_wdMa * _dMadx_c + _dL1drho_axs * _drho_axsdMa * _dMadx_c + _dL1drho_ws * _drho_wsdMa * _dMadx_c + _dL1dL_a * _dL_adn_a1 * _dn_adx_c1
|
||||
_dndCO2 = _dndL1 * _dL1dx_c
|
||||
|
||||
# Final error:
|
||||
_nError = np.sqrt( (_dndl * dl)**2 + (_dndT * dT)**2 + (_dndP * dP)**2 + (_dndRH * dRH)**2 + (_dndCO2 * dCO2)**2 )
|
||||
|
||||
return _nError
|
||||
|
||||
###
|
||||
### THIS ERROR CLASS IS OUTDATED
|
||||
### Use Observatory.dn_tph(), dispersion.cassiniError() or refraction.cassiniError() instead.
|
||||
###
|
||||
class error(Observatory):
|
||||
def __init__(self, dT=0.2, dP=20, dH=0.02, dCO2=20, dl=0.001):
|
||||
Observatory.__init__(self)
|
||||
self.dT = dT # in K
|
||||
self.dP = dP # in Pa
|
||||
self.dH = dH # relative humidity between 0 and 1
|
||||
self.dCO2 = dCO2 # in ppm
|
||||
self.dl = dl # in micron
|
||||
print('THIS ERROR CLASS IS OUTDATED\nUse Observatory.dn_tph(), dispersion.cassiniError() or refraction.cassiniError() instead.')
|
||||
|
||||
def dn_as(self, l, dl=None):
|
||||
'''
|
||||
@@ -384,6 +584,7 @@ class error(Ciddor1996):
|
||||
|
||||
def dn_tph(self, l, T, p, RH, xc, dl=None, dT=None, dP=None, dRH=None, dCO2=None):
|
||||
'''
|
||||
[DEPRECATED]
|
||||
Calculates the uncertainty in the refraction of the atmosphere for any conditions.
|
||||
'''
|
||||
if not dP:
|
||||
@@ -443,24 +644,27 @@ class error(Ciddor1996):
|
||||
_dnLL = np.sqrt( ( (3*np.sqrt(1 - _Ltotal )) / ( 2 * (1 - _Ltotal)**2 * np.sqrt(1 + 2*_Ltotal) ) * _dLtotal )**2 )
|
||||
|
||||
return (_nLL, _dnLL)
|
||||
###
|
||||
### END OF OUTDATED ERROR CLASS
|
||||
###
|
||||
|
||||
|
||||
class dispersion:
|
||||
def __init__(self, lat, h, P, rho):
|
||||
|
||||
class dispersion(Observatory):
|
||||
def __init__(self, lat, h):
|
||||
Observatory.__init__(self)
|
||||
self.h = h # height above sea level in meters
|
||||
self.lat = lat # Latitude of the observer in degrees
|
||||
self.rc = self.rc() * 1000 # Radius of curvature of the earth at the observer
|
||||
self.H = self.H(P, rho) # Reduced height of the atmosphere assuming ideal gas law
|
||||
# self.H2 = self.H_isotherm(T) # Reduced height of isotherm atmosphere
|
||||
self.rc = self.set_rc() * 1000 # Radius of curvature of the earth at the observer
|
||||
|
||||
def setReducedHeight(self, P, rho):
|
||||
self.H = self.set_H(P, rho) # Reduced height of the atmosphere assuming ideal gas law
|
||||
|
||||
|
||||
def planeParallel(self, n1, n2, zenith):
|
||||
'''
|
||||
Refraction based on a flat atmosphere, easily derived from Snell's law.
|
||||
'''
|
||||
# _theta1 = np.degrees( np.arcsin( np.sin( np.deg2rad(zenith) / n1) ) )
|
||||
# _theta2 = np.degrees( np.arcsin( np.sin( np.deg2rad(zenith) / n2) ) )
|
||||
_R1 = np.arcsin(n1 * np.sin( np.deg2rad(zenith) )) - np.deg2rad(zenith)
|
||||
_R2 = np.arcsin(n2 * np.sin( np.deg2rad(zenith) )) - np.deg2rad(zenith)
|
||||
return np.degrees(_R1) - np.degrees(_R2)
|
||||
@@ -555,7 +759,7 @@ class dispersion:
|
||||
if not R0:
|
||||
R0 = self.rc
|
||||
if useStandardAtmosphere:
|
||||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt('1976USSA.txt', unpack=True, skip_header=6)
|
||||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt('./US1976 Standard Atmosphere/si1py_modified.txt', unpack=True, skip_header=6)
|
||||
else:
|
||||
_Dens = rhoData
|
||||
_h = heightData
|
||||
@@ -589,7 +793,7 @@ class dispersion:
|
||||
return np.exp(x**2) * _int[0]
|
||||
|
||||
|
||||
def rc(self):
|
||||
def set_rc(self):
|
||||
'''
|
||||
Returns the radius of the curvature at the observer in km,
|
||||
taking into account the latitude of the observer and the corresponding
|
||||
@@ -603,7 +807,7 @@ class dispersion:
|
||||
return _rc0 + self.h/1000
|
||||
|
||||
|
||||
def H(self, P, rho):
|
||||
def set_H(self, P, rho):
|
||||
'''
|
||||
Calculates the reduced height of the atmosphere, assuming ideal gas law.
|
||||
'''
|
||||
@@ -634,21 +838,262 @@ class dispersion:
|
||||
|
||||
return (_kb*T) / (_m * _g0_local)
|
||||
|
||||
class refraction:
|
||||
def __init__(self, lat, h, P, rho):
|
||||
def cassiniError(self, z, l1, l2, T, p, RH, xc, dl1=0, dl2=0, dT=0.2, dP=20, dRH=0.02, dCO2=20, dz=0, lat=None, h=None):
|
||||
'''
|
||||
Calculates the uncertainty in the atmospheric dispersion, following the Cassini model,
|
||||
by using fully propagated error analysis.
|
||||
This involves almost endless use of the chain rule and evaluation of both the moist
|
||||
air and dry air components. Note: dry air does not contain any water, i.e. H=0, x_w=0, etc.
|
||||
|
||||
Not taken into account is the errror propagation in the scale height of the atmosphere or
|
||||
the zenith error, from my results this should give a small or even negligible difference.
|
||||
|
||||
Parameters:
|
||||
z - Zenith angle in degrees
|
||||
l1 - Bluest wavelength in micron (0.4 - 2.5 micron)
|
||||
l2 - Reddest wavelength in micron (0.4 - 2.5 micron)
|
||||
T - Temperature in Kelvin
|
||||
p - Pressure in Pascals
|
||||
RH - Relative humidity (no units, don't use percentage but value 0<x<1)
|
||||
xc - CO2 density in ppm
|
||||
dl1 - Uncertainty in the blue wavelength
|
||||
dl2 - Uncertainty in the red wavelength
|
||||
dT - Uncertainty in the temperature, default: 0.2 K
|
||||
dP - Uncertainty in the pressure, default: 20 Pa
|
||||
dRH - Uncertainty in the relative humidity, default: 0.02
|
||||
dCO2- Uncertainty in the CO2 density, default: 20
|
||||
lat - [Optional] Latitude of the observatory in degrees
|
||||
h - [Optional] Height in m of the observatory
|
||||
'''
|
||||
if lat == None:
|
||||
lat = self.lat
|
||||
else:
|
||||
self.lat = lat
|
||||
if h == None:
|
||||
h = self.h
|
||||
else:
|
||||
self.h = h
|
||||
|
||||
# constants
|
||||
_cf = 1.022
|
||||
_w0 = 295.235 # um^-2
|
||||
_w1 = 2.6422 # um^-2
|
||||
_w2 = -0.032380 # um^-4
|
||||
_w3 = 0.004028 # um^-6
|
||||
|
||||
_k0 = 238.0185 # um^-2
|
||||
_k1 = 5792105 # um^-2
|
||||
_k2 = 57.362 # um^-2
|
||||
_k3 = 167917 # um^-2
|
||||
|
||||
_A = 1.2378847e-5 # K^-2
|
||||
_B = -1.9121316e-2 # K^-2
|
||||
_C = 33.93711047
|
||||
_D = -6.3431645e3 # K^-2
|
||||
|
||||
_Mw = 0.018015 # kg/mol
|
||||
_R = 8.314510 # J mol^-1 K^-1, gas constant
|
||||
|
||||
_a0 = 1.58123e-6 # K Pa^-1
|
||||
_a1 = -2.9331e-8 # Pa^-1
|
||||
_a2 = 1.1043e-10 # K^-1 Pa^-1
|
||||
_b0 = 5.707e-6 # K Pa^-1
|
||||
_b1 = -2.051e-8 # Pa^-1
|
||||
_c0 = 1.9898e-4 # K Pa^-1
|
||||
_c1 = -2.376e-6 # Pa^-1
|
||||
_d = 1.83e-11 # K^2 Pa^-2
|
||||
_e = -0.765e-8 # K^2 Pa^-2
|
||||
_t = T - 273.15
|
||||
|
||||
_aa = 1.00062
|
||||
_bb = 3.14e-8 # Pa^-1
|
||||
_cc = 5.6e-7 # deg C^-2
|
||||
|
||||
_RH_a = 0
|
||||
_RH_w = RH
|
||||
|
||||
# Calculated values
|
||||
_rho_obs = self.rho(p, T, RH, xc)
|
||||
self.rc = self.set_rc() * 1000
|
||||
self.H = self.set_H(p, _rho_obs)
|
||||
_alpha = self.rc * np.sin(np.deg2rad(z)) / (self.rc + self.H)
|
||||
_p_a = p - RH*self.svp(T)
|
||||
_p_w = RH*self.svp(T)
|
||||
_x_w_a = self.xw(_p_a, T, _RH_a)
|
||||
_x_w_w = self.f(_p_w, T)
|
||||
_Z_a = self.Z(p=_p_a, T=T, RH=_RH_a)
|
||||
_Z_w = self.Z(p=_p_w, T=T, RH=_RH_w, xw=_x_w_w)
|
||||
_Ma = 1e-3 * (28.9635 + 12.011e-6 * (xc - 400) ) # Molar mass in kg/mol of dry air containing xc ppm of CO2
|
||||
_p_sv = self.svp(T)
|
||||
_rho_a = self.rho(p=p-RH*self.svp(T), T=T, RH=0, xc=xc)
|
||||
_rho_axs = self.rho(p=101325, T=288.15, RH=0, xc=xc)
|
||||
if RH == 0.0:
|
||||
# nan is returned for _rho_w if we put in a (partial) pressure of 0
|
||||
_rho_w = 0
|
||||
else:
|
||||
_rho_w = self.rho(p=RH*self.svp(T), T=T, RH=RH, xc=xc)
|
||||
_rho_ws = self.rho(p=1333, T=293.15, RH=1333/self.svp(293.15), xc=xc)
|
||||
_n_a1 = self.n_axs(l1, xc)
|
||||
_n_w1 = self.n_ws(l1)
|
||||
_n_a2 = self.n_axs(l2, xc)
|
||||
_n_w2 = self.n_ws(l2)
|
||||
|
||||
_L1_a = (_n_a1**2 - 1) / (_n_a1**2 + 2) # Dry air component of L
|
||||
_L1_w = (_n_w1**2 - 1) / (_n_w1**2 + 2)
|
||||
_L1 = ( _rho_a / _rho_axs ) * _L1_a + ( _rho_w / _rho_ws ) * _L1_w # Overall L
|
||||
_L2_a = (_n_a2**2 - 1) / (_n_a2**2 + 2) # Dry air component of L
|
||||
_L2_w = (_n_w2**2 - 1) / (_n_w2**2 + 2)
|
||||
_L2 = ( _rho_a / _rho_axs ) * _L2_a + ( _rho_w / _rho_ws ) * _L2_w # Overall L
|
||||
|
||||
_n1 = np.sqrt( (1 + 2 * _L1) / (1 - _L1) )
|
||||
_n2 = np.sqrt( (1 + 2 * _L2) / (1 - _L2) )
|
||||
|
||||
# Partial derivatives for derivation to l1
|
||||
_dn_wdl1 = _cf * 1e-8 * (-1 * _w1 * 2/l1**3 - _w2 * 4/l1**5 - _w3 * 6/l1**7)
|
||||
_dLdn_w1 = _rho_w / _rho_ws * 6 * _n_w1 / (_n_w1**2 + 2)**2
|
||||
_dn_adl1 = 1e-8 * (1 + 0.534e-6 * (xc - 450)) * ( - (2 * _k1 * l1 ) / ((_k0 * l1**2 - 1)**2) - (2 * _k3 * l1) / ((_k2 * l1**2 - 1)**2) )
|
||||
_dLdn_a1 = _rho_a / _rho_axs * 6 * _n_a1 / (_n_a1**2 + 2)**2
|
||||
_dndL1 = ( ( (1 + 2*_L1) / (1 - _L1)**2 ) + (2 / (1-_L1)) ) / (2 * np.sqrt( (1 + 2*_L1) / (1 - _L1) ) )
|
||||
_dddn1 = _alpha / np.sqrt( 1 - _alpha**2 * _n1**2 )
|
||||
_dddl1 = _dddn1 * _dndL1 * ( _dLdn_a1 * _dn_adl1 + _dLdn_w1 * _dn_wdl1)
|
||||
|
||||
# Partial derivatives for derivation to l2
|
||||
_dn_wdl2 = _cf * 1e-8 * (-1 * _w1 * 2/l2**3 - _w2 * 4/l2**5 - _w3 * 6/l2**7)
|
||||
_dLdn_w2 = _rho_w / _rho_ws * 6 * _n_w2 / (_n_w2**2 + 2)**2
|
||||
_dn_adl2 = 1e-8 * (1 + 0.534e-6 * (xc - 450)) * ( - (2 * _k1 * l2 ) / ((_k0 * l2**2 - 1)**2) - (2 * _k3 * l2) / ((_k2 * l2**2 - 1)**2) )
|
||||
_dLdn_a2 = _rho_a / _rho_axs * 6 * _n_a2 / (_n_a2**2 + 2)**2
|
||||
_dndL2 = ( ( (1 + 2*_L2) / (1 - _L2)**2 ) + (2 / (1-_L2)) ) / (2 * np.sqrt( (1 + 2*_L2) / (1 - _L2) ) )
|
||||
_dddn2 = _alpha / np.sqrt( 1 - _alpha**2 * _n2**2 )
|
||||
_dddl2 = _dddn2 * _dndL2 * ( _dLdn_a2 * _dn_adl2 + _dLdn_w2 * _dn_wdl2)
|
||||
|
||||
|
||||
# Partial derivatives for derivation to T
|
||||
# Dry air parts:
|
||||
_drho_adp_a = _Ma / (_Z_a * _R * T) * (1 - _x_w_a * (1 - _Mw / _Ma))
|
||||
_dp_adp_sv = - RH
|
||||
_dp_svdT = (2*_A*T + _B - _D/(T**2)) * np.exp(_A * T**2 + _B * T + _C + _D/T )
|
||||
_drho_adTZ = - _p_a * _Ma / (_R * ( _Z_a * T)**2) * (1 - _x_w_a * (1 - _Mw / _Ma))
|
||||
_dTZdT_a = _p_a / T * (_a2 * _t**2 + _a1 * _t + _a0 + (_b1*_t + _b0) * _x_w_a + (_c0 + _c1*_t) * _x_w_a**2 ) - _p_a * (_a1 + 2*_a2*_t + _b1*_x_w_a + _c1*_x_w_a**2) - 2*(_p_a/T)**2 * (_d + _e * _x_w_a**2) + _Z_a
|
||||
_dTZdxw_a = 2*_e*_p_a**2 / T * _x_w_a - _p_a * (_b1*_t + _b0 + 2 * (_c0 + _c1*_t) * _x_w_a)
|
||||
_dxw_adf = _RH_a * _p_sv / _p_a
|
||||
_dfdT = 2*_cc*_t
|
||||
_dxw_adp_sv = self.f(_p_a, T) * _RH_a / _p_a
|
||||
_drho_adxw_a = 0 # as rho_a has not water content, i.e. rho = p*M/ZRT
|
||||
|
||||
|
||||
# Moist air parts:
|
||||
_drho_wdp_w = _Ma / (_Z_w * _R * T) * (1 - _x_w_w * (1 - _Mw / _Ma))
|
||||
_dp_wdp_sv = RH
|
||||
_dp_svdT = (2*_A*T + _B - _D/(T**2)) * np.exp(_A * T**2 + _B * T + _C + _D/T )
|
||||
_drho_wdTZ = - _p_w * _Ma / (_R * (_Z_w * T)**2) * (1 - _x_w_w * (1 - _Mw / _Ma))
|
||||
_dTZdT_w = _p_w / T * (_a2 * _t**2 + _a1 * _t + _a0 + (_b1*_t + _b0) * _x_w_w + (_c0 + _c1*_t) * _x_w_w**2 ) - _p_w * (_a1 + 2*_a2*_t + _b1*_x_w_w + _c1*_x_w_w**2) - 2*(_p_w/T)**2 * (_d + _e * _x_w_w**2) + _Z_w
|
||||
_dTZdxw_w = 2*_e*_p_w**2 / T * _x_w_w - _p_w * (_b1*_t + _b0 + 2 * (_c0 + _c1*_t) * _x_w_w)
|
||||
_dxw_wdf = 1 # _RH_w * _p_sv / _p_w
|
||||
_dfdT = 2*_cc*_t
|
||||
_dxw_wdp_sv = self.f(_p_w, T) / _p_sv # * _RH_w / _p_w
|
||||
_drho_wdxw_w = _p_w * _Ma / (_Z_w * _R * T) * (_Mw / _Ma - 1)
|
||||
|
||||
_tA = _drho_adp_a * _dp_adp_sv * _dp_svdT + _drho_adTZ * (_dTZdT_a + _dTZdxw_a * (_dxw_adf * _dfdT + _dxw_adp_sv * _dp_svdT) ) + _drho_adxw_a * ( _dxw_adf * _dfdT + _dxw_adp_sv * _dp_svdT )
|
||||
_tB = _drho_wdp_w * _dp_wdp_sv * _dp_svdT + _drho_wdTZ * (_dTZdT_w + _dTZdxw_w * (_dxw_wdf * _dfdT + _dxw_wdp_sv * _dp_svdT) ) + _drho_wdxw_w * ( _dxw_wdf * _dfdT + _dxw_wdp_sv * _dp_svdT )
|
||||
_dL1drho_a = _L1_a / _rho_axs
|
||||
_dL2drho_a = _L2_a / _rho_axs
|
||||
_dL1drho_w = _L1_w / _rho_ws
|
||||
_dL2drho_w = _L2_w / _rho_ws
|
||||
|
||||
_dddT = _tA * (_dddn1 * _dndL1 * _dL1drho_a - _dddn2 * _dndL2 * _dL2drho_a) + _tB * (_dddn1 * _dndL1 * _dL1drho_w - _dddn2 * _dndL2 * _dL2drho_w)
|
||||
|
||||
# Partial derivatives for derivation to p
|
||||
# Dry air parts:
|
||||
_drho_adZ = - _p_a * _Ma / ( _Z_a**2 * _R * T) * (1 - _x_w_a * (1 - _Mw / _Ma))
|
||||
_dZdp_a = - 1 / T * (_a0 + _a1 * _t + _a2 * _t**2 + (_b0 + _b1*_t)*_x_w_a + (_c0 + _c1*_t)*_x_w_a**2 ) + 2*_p_a/(T**2) * (_d + _e * _x_w_a**2)
|
||||
_dfdp = _bb
|
||||
_dxw_adp_a = - self.f(_p_a, T) * _RH_a * _p_sv / (_p_a)**2
|
||||
|
||||
# Moist air parts
|
||||
_dZdp_w = - 1 / T * (_a0 + _a1 * _t + _a2 * _t**2 + (_b0 + _b1*_t)*_x_w_w + (_c0 + _c1*_t)*_x_w_w**2 ) + 2*_p_w/(T**2) * (_d + _e * _x_w_w**2)
|
||||
# Not necessary, since dpwdp=0
|
||||
|
||||
_pA = _drho_adp_a + _drho_adZ * _dZdp_a + _drho_adxw_a * (_dxw_adf * _dfdp + _dxw_adp_a)
|
||||
_dddP = _pA * (_dddn1 * _dndL1 * _dL1drho_a - _dddn2 * _dndL2 * _dL2drho_a)
|
||||
|
||||
|
||||
# Partial derivatives for derivation to H
|
||||
# Dry air parts:
|
||||
_dp_adH = - _p_sv
|
||||
_dZdxw_a = (_p_a / T)**2 * 2 * _e * _x_w_a - (_p_a / T) * ( (_b0 + _b1 * _t) + 2 * (_c0 + _c1 * _t) * _x_w_a )
|
||||
_dxw_adH = self.f(_p_a, T) * _p_sv / _p_a
|
||||
|
||||
# Moist air parts:
|
||||
_dp_wdH = _p_sv
|
||||
_dZdxw_w = (_p_w / T)**2 * 2 * _e * _x_w_w - (_p_w / T) * ( (_b0 + _b1 * _t) + 2 * (_c0 + _c1 * _t) * _x_w_w )
|
||||
_dxw_wdH = 0
|
||||
_dxw_wdp_w = 0
|
||||
_drho_wdZ = - _p_w * _Ma / ( _Z_w**2 * _R * T) * (1 - _x_w_w * (1 - _Mw / _Ma))
|
||||
|
||||
_hA = _drho_adp_a * _dp_adH + _drho_adZ * ( _dZdp_a * _dp_adH + _dZdxw_a * (_dxw_adH + _dxw_adp_a * _dp_adH + _dxw_adf * _dfdp * _dp_adH) ) + _drho_adxw_a * (_dxw_adH + _dxw_adp_a * _dp_adH + _dxw_adf * _dfdp * _dp_adH)
|
||||
_hB = _drho_wdp_w * _dp_wdH + _drho_wdZ * ( _dZdp_w * _dp_wdH + _dZdxw_w * ( _bb * _p_sv ) ) + _drho_wdxw_w * ( _bb * _p_sv )
|
||||
_dddRH = _hA * (_dddn1 * _dndL1 * _dL1drho_a - _dddn2 * _dndL2 * _dL2drho_a) + _hB * (_dddn1 * _dndL1 * _dL1drho_w - _dddn2 * _dndL2 * _dL2drho_w)
|
||||
|
||||
|
||||
# Partial derivatives for derivation to xc
|
||||
# Dry air parts:
|
||||
_drho_adMa = _p_a * (1 - _x_w_a) / (_Z_a * _R * T)
|
||||
_dL1drho_axs = - (_rho_a * _L1_a)/(_rho_axs**2)
|
||||
_dL2drho_axs = - (_rho_a * _L2_a)/(_rho_axs**2)
|
||||
_drho_axsdMa = _drho_adMa
|
||||
_dL1dL_a = _rho_a / _rho_axs
|
||||
_dL_adn_a1 = 6 * _n_a1 / (_n_a1**2 + 2)**2
|
||||
_dL2dL_a = _dL1dL_a
|
||||
_dL_adn_a2 = 6 * _n_a2 / (_n_a2**2 + 2)**2
|
||||
|
||||
# Moist air parts:
|
||||
_drho_wdMa = _p_w * (1 - _x_w_w) / (_Z_w * _R * T)
|
||||
_dL1drho_ws = - (_rho_w * _L1_w)/(_rho_ws**2)
|
||||
_dL2drho_ws = - (_rho_w * _L2_w)/(_rho_ws**2)
|
||||
_drho_wsdMa = _drho_wdMa
|
||||
|
||||
# Other parts:
|
||||
_dMadx_c = 12.011e-9
|
||||
_dn_adx_c1 = (self.n_as(l1) - 1) * 0.534e-6
|
||||
_dn_adx_c2 = (self.n_as(l2) - 1) * 0.534e-6
|
||||
|
||||
_dL1dx_c = _dL1drho_a * _drho_adMa * _dMadx_c + _dL1drho_w * _drho_wdMa * _dMadx_c + _dL1drho_axs * _drho_axsdMa * _dMadx_c + _dL1drho_ws * _drho_wsdMa * _dMadx_c + _dL1dL_a * _dL_adn_a1 * _dn_adx_c1
|
||||
_dL2dx_c = _dL2drho_a * _drho_adMa * _dMadx_c + _dL2drho_w * _drho_wdMa * _dMadx_c + _dL2drho_axs * _drho_axsdMa * _dMadx_c + _dL2drho_ws * _drho_wsdMa * _dMadx_c + _dL2dL_a * _dL_adn_a2 * _dn_adx_c2
|
||||
_dddCO2 = _dddn1 * _dndL1 * _dL1dx_c - _dddn2 * _dndL2 * _dL2dx_c
|
||||
|
||||
|
||||
# Partial derivatives for derivation to z
|
||||
_zenith = np.deg2rad(z)
|
||||
_dz = np.deg2rad(dz)
|
||||
_R = self.rc
|
||||
_H = self.H
|
||||
_dddz = (_R * np.cos(_zenith)) / (_R + _H) * ( _n1 / np.sqrt(1-(_n1**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2) - _n2 / np.sqrt(1-(_n2**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2) )
|
||||
|
||||
# Final error:
|
||||
_dispersionError = np.sqrt( (_dddl1 * dl1)**2 + (_dddl2 * dl2)**2 + (_dddT * dT)**2 + (_dddP * dP)**2 + (_dddRH * dRH)**2 + (_dddCO2 * dCO2)**2 + (_dddz * _dz)**2)
|
||||
|
||||
return np.degrees(_dispersionError)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
class refraction(Observatory):
|
||||
def __init__(self, lat, h):
|
||||
Observatory.__init__(self)
|
||||
self.h = h # height above sea level in meters
|
||||
self.lat = lat # Latitude of the observer in degrees
|
||||
self.rc = self.rc() * 1000 # Radius of curvature of the earth at the observer
|
||||
self.H = self.H(P, rho) # Reduced height of the atmosphere assuming ideal gas law
|
||||
self.rc = self.set_rc() * 1000 # Radius of curvature of the earth at the observer
|
||||
# self.H2 = self.H_isotherm(T) # Reduced height of isotherm atmosphere
|
||||
|
||||
def setReducedHeight(self, P, rho):
|
||||
self.H = self.set_H(P, rho) # Reduced height of the atmosphere assuming ideal gas law
|
||||
|
||||
|
||||
def planeParallel(self, n, zenith):
|
||||
'''
|
||||
Refraction based on a flat atmosphere, easily derived from Snell's law.
|
||||
'''
|
||||
# _theta1 = np.degrees( np.arcsin( np.sin( np.deg2rad(zenith) / n) ) )
|
||||
_R = np.arcsin(n * np.sin( np.deg2rad(zenith) )) - np.deg2rad(zenith)
|
||||
return np.degrees(_R)
|
||||
|
||||
@@ -732,7 +1177,7 @@ class refraction:
|
||||
if not R0:
|
||||
R0 = self.rc
|
||||
if useStandardAtmosphere:
|
||||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt('1976USSA.txt', unpack=True, skip_header=6)
|
||||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt('./US1976 Standard Atmosphere/si1py_modified.txt', unpack=True, skip_header=6)
|
||||
else:
|
||||
_Dens = rhoData
|
||||
_h = heightData
|
||||
@@ -760,7 +1205,7 @@ class refraction:
|
||||
|
||||
|
||||
|
||||
def rc(self):
|
||||
def set_rc(self):
|
||||
'''
|
||||
Returns the radius of the curvature at the observer in km,
|
||||
taking into account the latitude of the observer and the corresponding
|
||||
@@ -775,7 +1220,7 @@ class refraction:
|
||||
|
||||
|
||||
|
||||
def H(self, P, rho):
|
||||
def set_H(self, P, rho):
|
||||
'''
|
||||
Calculates the reduced height of the atmosphere, assuming ideal gas law.
|
||||
'''
|
||||
@@ -806,3 +1251,21 @@ class refraction:
|
||||
|
||||
return (_kb*T) / (_m * _g0_local)
|
||||
|
||||
def cassiniError(self, n, dn, zenith, dz=0, reduced_height=None):
|
||||
'''
|
||||
Calculates the uncertainty in the refraction angle in degrees.
|
||||
'''
|
||||
if reduced_height:
|
||||
_H = reduced_height
|
||||
else:
|
||||
_H = self.H
|
||||
_zenith = np.deg2rad(zenith)
|
||||
_dz = np.deg2rad(dz)
|
||||
_R = self.rc
|
||||
_dRdn = (_R * np.sin(_zenith)) / ((_R + _H) * np.sqrt(1 - ((n**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2)))
|
||||
|
||||
_dRdz = (_R * np.cos(_zenith)) / (_R + _H) * ( n / np.sqrt(1-(n**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2) - 1 / np.sqrt(1-(_R**2 * np.sin(_zenith)**2)/(_R + _H)**2) )
|
||||
|
||||
_dR = np.sqrt( (_dRdn * dn)**2 + (_dRdz * _dz)**2)
|
||||
|
||||
return np.degrees(_dR)
|
||||
+16
-5
@@ -23,6 +23,11 @@ import pandas as pd
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
plt.rc('font', family='serif')
|
||||
# plt.rc('xtick', labelsize='x-small')
|
||||
# plt.rc('ytick', labelsize='x-small')
|
||||
plt.rc('axes', labelsize='large')
|
||||
|
||||
# Conditions at Cerro Armazones
|
||||
# T = 279.65 # k
|
||||
# P = 71200 # Pa
|
||||
@@ -39,7 +44,7 @@ if __name__ == '__main__':
|
||||
zenith = 30 # degrees
|
||||
|
||||
# Calculate the index of refraction
|
||||
at = Ciddor1996(p=P, T=T, H=H, xc=xc)
|
||||
at = Observatory()
|
||||
n1 = at.n_tph(l=l1, T=T, p=P, RH=H, xc=xc)
|
||||
n2 = at.n_tph(l=l2, T=T, p=P, RH=H, xc=xc)
|
||||
rho = at.rho(p=P, T=T, RH=H, xc=xc)
|
||||
@@ -49,7 +54,8 @@ if __name__ == '__main__':
|
||||
tableCompare = True
|
||||
if dispersionAnalysis:
|
||||
# Initiate the refraction class
|
||||
ref = dispersion(lat, h, P, rho)
|
||||
ref = dispersion(lat, h)
|
||||
ref.setReducedHeight(P, rho)
|
||||
|
||||
# Compare the refraction models to check if the output is okay
|
||||
print ('Plane - parallel atmosphere dispersion: %s mas' %(ref.planeParallel(n1, n2, zenith) * 3.6e6))
|
||||
@@ -92,7 +98,7 @@ if __name__ == '__main__':
|
||||
|
||||
|
||||
# Plotting it all, with vertical lines where the variation between the models exceed 1, 0.1, 0.01 and 0.001 mas.
|
||||
fig = plt.figure(figsize=(11,6))
|
||||
fig = plt.figure(figsize=(9,5))
|
||||
ax = fig.add_subplot(111)
|
||||
ax.set_xlim([30,85])
|
||||
ax.set_xlabel('Zenith (degrees)')
|
||||
@@ -119,6 +125,8 @@ if __name__ == '__main__':
|
||||
ax.plot(zenithRange, orianiList_isotherm, 'y--', label='$Atan(z) - Btan^3(z)$ (isotherm)')
|
||||
ax.plot(zenithRange, corbardList_isotherm, 'r--', label='Error function (isotherm)')
|
||||
|
||||
|
||||
ax.tick_params(axis ='both', direction='in')
|
||||
ax.legend()
|
||||
plt.tight_layout()
|
||||
plt.show()
|
||||
@@ -128,6 +136,7 @@ if __name__ == '__main__':
|
||||
# Initiate the refraction class
|
||||
ref = refraction(lat, h, P, rho)
|
||||
|
||||
|
||||
# Compare the refraction models to check if the output is okay
|
||||
print ('Plane - parallel atmosphere refraction: %s arcsec' %(ref.planeParallel(n1, zenith) * 3.6e3))
|
||||
print ('Cassini refraction: %s arcsec' %(ref.cassini(n1, zenith) * 3.6e3))
|
||||
@@ -154,7 +163,7 @@ if __name__ == '__main__':
|
||||
corbardList_isotherm = [ref.corbard(n1, z, reduced_height=ref.H_isotherm(T)) * 3.6e3 for z in zenithRange]
|
||||
|
||||
# Plotting it all, with vertical lines where the variation between the models exceed 1, 0.1, 0.01 and 0.001 arcsec.
|
||||
fig = plt.figure(figsize=(11,6))
|
||||
fig = plt.figure(figsize=(9,5))
|
||||
ax = fig.add_subplot(111)
|
||||
ax.set_xlim([80,88])
|
||||
ax.set_xlabel('Zenith (degrees)')
|
||||
@@ -173,6 +182,7 @@ if __name__ == '__main__':
|
||||
ax.plot(zenithRange, orianiList_isotherm, 'y--', label='$Atan(z) - Btan^3(z)$ (isotherm)')
|
||||
ax.plot(zenithRange, corbardList_isotherm, 'r--', label='Error function (isotherm)')
|
||||
|
||||
ax.tick_params(axis ='both', direction='in')
|
||||
ax.legend()
|
||||
plt.tight_layout()
|
||||
plt.show()
|
||||
@@ -182,7 +192,8 @@ if __name__ == '__main__':
|
||||
refTableData = []
|
||||
dispTableData = []
|
||||
ref = refraction(lat, h, P, rho)
|
||||
disp = dispersion(lat, h, P, rho)
|
||||
disp = dispersion(lat, h)
|
||||
disp.setReducedHeight(P, rho)
|
||||
zenithRange = [10, 30, 50, 60, 70, 80, 85]
|
||||
headerList = np.array(['z', 'Plane-parallel', 'Cassini', 'tan3', 'tan5', 'Error function', 'Full integration'])
|
||||
refTableData.append(headerList)
|
||||
|
||||
Reference in new issue
Block a user