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Restructured the code
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@@ -1,2 +1,3 @@
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.DS_Store
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__pychache__/
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@@ -10,16 +10,21 @@ Author info: *Joost van den Born*, *born@astron.nl*
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*Below you can find a set of models describing the atmospheric refraction and dispersion.*
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**Contents:**
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- atmosphericModel.py : Python model that can be used to calculate the atmospheric refraction and dispersion
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- refractionModel.py : Python script that compares the output of the different rerfraction models.
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- atmosphericModel : Python model that can be used to calculate the atmospheric refraction and dispersion, the main content of this repository.
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- refractionModel.py : Python script that compares the output of the different refraction models.
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Can be used as an example on how to use atmosphericModel.py
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- 1976USSA.txt : Table with temperature, pressure, density and more as a function of altitude.
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Values derived from the 1976 U.S. Standard Atmosphere.
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- manual.pdf : Manual for using atmosphericModel.py
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- atmospheric refraction example: jupyter notebook example
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- [Tech note]...models.pdf : Tech note describing the models.
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**On the contents of the atmosphericModel package**
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The package contains the following files:
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- 1976USSA.txt : Data file of the 1976 US Standard Atmosphere.
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- ciddorModel.py : Contains the `Observatory()` class to calculate the refractive index of atmospheric air, using the Ciddor (1996) model.
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- dispersionModels.py : Contains various atmospheric models that can be used to calculate the atmospheric dispersion
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- examples.py : Contains several quick functions, e.g. to quickly calculate the refractive index.
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- refractionModels.py : Contains various atmospheric models that can be used to calculate the atmospheric refraction
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- refractivityModels.py : Contains various functions from literature to calculate the refractive index. Some examples of works included are Ciddor (1996), Barrell & Sears (1939), SLALIB, Owens (1967) and Edlen (1966).
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**On the dispersion model:**
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@@ -13286,6 +13286,57 @@ div#notebook {
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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>There are also some quick calculation functions defined in the <code>examples.py</code> file, if you just want to know the refractive index, the refraction or the dispersion at standard atmospheric conditions (<code>conditions='STANDARD'</code>) or at Cerro Armazones (<code>conditions='CERRO_ARMAZONES'</code>).</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 [6]:</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"># Quick refractive index</span>
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<span class="n">nq</span> <span class="o">=</span> <span class="n">quick_refractive_index</span><span class="p">(</span><span class="n">l</span><span class="o">=</span><span class="mf">1.49</span><span class="p">,</span> <span class="n">conditions</span><span class="o">=</span><span class="s1">'STANDARD'</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="s1">'Quick refractive index: </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">nq</span><span class="p">))</span>
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<span class="c1"># Quick refraction</span>
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<span class="n">rq</span> <span class="o">=</span> <span class="n">quick_refraction</span><span class="p">(</span><span class="n">l1</span><span class="p">,</span> <span class="n">zenith_obs</span><span class="p">,</span> <span class="n">conditions</span><span class="o">=</span><span class="s1">'CERRO_ARMAZONES'</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">'Quick refraction: </span><span class="si">{:.3f}</span><span class="s1"> arcsec'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">rq</span><span class="p">))</span>
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<span class="c1"># Quick dispersion</span>
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<span class="n">dq</span> <span class="o">=</span> <span class="n">quick_dispersion</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">zenith_obs</span><span class="p">,</span> <span class="n">conditions</span><span class="o">=</span><span class="s1">'CERRO_ARMAZONES'</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">'Quick dispersion: </span><span class="si">{:.3f}</span><span class="s1"> mas'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">dq</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>Quick refractive index: 1.0002733131243147
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Quick refraction: 40.697 arcsec
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Quick dispersion: 31.019 mas
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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>
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</div>
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@@ -159,6 +159,42 @@
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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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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"There are also some quick calculation functions defined in the `examples.py` file, if you just want to know the refractive index, the refraction or the dispersion at standard atmospheric conditions (`conditions='STANDARD'`) or at Cerro Armazones (`conditions='CERRO_ARMAZONES'`)."
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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": 6,
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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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"Quick refractive index: 1.0002733131243147\n",
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"Quick refraction: 40.697 arcsec\n",
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"Quick dispersion: 31.019 mas\n"
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]
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}
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],
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"source": [
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"# Quick refractive index\n",
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"nq = quick_refractive_index(l=1.49, conditions='STANDARD')\n",
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"print('Quick refractive index: {}'.format(nq))\n",
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"\n",
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"# Quick refraction\n",
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"rq = quick_refraction(l1, zenith_obs, conditions='CERRO_ARMAZONES') * 3.6e3\n",
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"print('Quick refraction: {:.3f} arcsec'.format(rq))\n",
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"\n",
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"# Quick dispersion\n",
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"dq = quick_dispersion(l1, l2, zenith_obs, conditions='CERRO_ARMAZONES') * 3.6e6\n",
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"print('Quick dispersion: {:.3f} mas'.format(dq))\n"
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]
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}
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],
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"metadata": {
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-1025
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@@ -0,0 +1,10 @@
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#!/usr/bin/env Python3
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import os
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from .ciddorModel import *
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from .dispersionModels import *
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from .refractionModels import *
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from .refractionModels import *
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from .examples import *
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from .refractivityModels import *
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#!/usr/bin/env Python3
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import numpy as np
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from scipy import integrate
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'''
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@ Author: Joost van den Born
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@ Contact: born@astron.nl
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@ Description: This script offers a complete model of the atmospheric refraction
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and dispersion. It uses the Ciddor1996 paper to calculate the
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index of refraction for a wide range of conditions, with corresponding
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error propagation if needed.
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The actual refraction is calculated by any of the following models:
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- Plane parallel atmosphere
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- Cassini spherical homogeneous atmosphere
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- Other methods described in Corbard et al., 2019.
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'''
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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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Formula that calculate the refractive index for standard atmospheric
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conditions, i.e. 15 C, 101,325 Pa, 0% RH, 450 ppm CO2.
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It returns n_as, not (n_as - 1) * 1e8!
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'''
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_s = 1 / l
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_k0 = 238.0185 # um^-2
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_k1 = 5792105 # um^-2
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_k2 = 57.362 # um^-2
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_k3 = 167917 # um^-2
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return (_k1 / (_k0 - _s**2) + _k3 / (_k2 - _s**2)) / 1e8 + 1
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def n_axs(self, l, xc):
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'''
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Formula that calculate the refractive index for standard atmospheric
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conditions, i.e. 15 C, 101,325 Pa, 0% RH, but with xc ppm of CO2.
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It returns n_axs, not (n_axs - 1)
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'''
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_n_as = self.n_as(l)
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return ( (_n_as - 1) * (1 + 0.534e-6 * (xc - 450) ) ) + 1
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def n_ws(self, l):
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'''
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Formula that calculates the refractivity of air for standard moist conditions
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as defined by B&S (1939), i.e. 20 C, 1333 Pa
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Returns n_ws, not (n_ws - 1) * 1e8
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'''
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_s = 1 / l
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_cf = 1.022
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_w0 = 295.235 # um^-2
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_w1 = 2.6422 # um^-2
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_w2 = -0.032380 # um^-4
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_w3 = 0.004028 # um^-6
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return (_cf * (_w0 + _w1 * _s**2 + _w2 * _s**4 + _w3 * _s**6) ) / 1e8 + 1
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def f(self, p, T):
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'''
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Enhancement factor of water vapour in air
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t in degrees celsius
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p in Pascal (Pa)
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'''
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_t = T - 273.15
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_a = 1.00062
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_b = 3.14e-8 # Pa^-1
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_c = 5.6e-7 # deg C^-2
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return _a + _b * p + _c * _t**2
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def svp(self, T):
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'''
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Saturation vapour pressure in Pa.
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'''
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_A = 1.2378847e-5 # K^-2
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_B = -1.9121316e-2 # K^-2
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_C = 33.93711047
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_D = -6.3431645e3 # K^-2
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return np.exp(_A*T**2 + _B*T + _C + _D/T)
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def svp2(self, T):
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'''
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Calculation of svp from IAPWS formula, based on documentation by NIST toolbox.
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'''
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K1 = 1.16705214528e+03
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K2 = -7.24213167032e+05
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K3 = -1.70738469401e+01
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K4 = 1.20208247025e+04
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K5 = -3.23255503223e+06
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K6 = 1.49151086135e+01
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K7 = -4.82326573616e+03
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K8 = 4.05113405421e+05
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K9 = -2.38555575678e-01
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K10 = 6.50175348448e+02
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_W = T + K9 / (T - K10)
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_A = _W**2 + K1*_W + K2
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_B = K3*_W**2 + K4*_W + K5
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_C = K6*_W**2 + K7*_W + K8
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_X = -_B + np.sqrt(_B**2 - 4*_A*_C)
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return (2*_C / _X)**4 * 1e6
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def xw(self, p, T, RH):
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'''
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Molar fraction of water vapour in moist air.
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'''
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_f = self.f(p, T)
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_svp = self.svp(T)
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return _f * RH * _svp / p
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def Z(self, p, T, RH, xw=''):
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'''
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Compressibility of moist air
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'''
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_a0 = 1.58123e-6 # K Pa^-1
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_a1 = -2.9331e-8 # Pa^-1
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_a2 = 1.1043e-10 # K^-1 Pa^-1
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_b0 = 5.707e-6 # K Pa^-1
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_b1 = -2.051e-8 # Pa^-1
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_c0 = 1.9898e-4 # K Pa^-1
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_c1 = -2.376e-6 # Pa^-1
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_d = 1.83e-11 # K^2 Pa^-2
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_e = -0.765e-8 # K^2 Pa^-2
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_t = T - 273.15
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if xw == '':
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_xw = self.xw(p, T, RH)
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else:
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_xw = xw
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Z = 1 - (p/T) * (_a0 + _a1*_t + _a2*_t**2 + (_b0 + _b1*_t)*_xw + (_c0 + _c1*_t)*_xw**2) + (p/T)**2 * (_d + _e*_xw**2)
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return Z
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def rho(self, p, T, RH, xc, xw=''):
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'''
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Calculates the density of air for the given parameters.
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T in Kelvin
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p in Pascal
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RH between 0 and 1
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xc in ppm
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'''
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_Ma = 1e-3 * (28.9635 + 12.011e-6 * (xc - 400) ) # Molar mass in kg/mol of dry air containing xc ppm of CO2
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_Mw = 0.018015 # kg/mol
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_R = 8.314510 # J mol^-1 K^-1, gas constant
|
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if xw == '':
|
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_xw = self.xw(p, T, RH)
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else:
|
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_xw = xw
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_Z = self.Z(p, T, RH, xw=_xw)
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return ( ( p * _Ma ) / (_Z * _R * T) ) * ( 1 - _xw * (1 - _Mw/_Ma))
|
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|
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|
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def n_tph(self, l, T, p, RH, xc):
|
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'''
|
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Calculates the refraction of the atmosphere for any conditions.
|
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'''
|
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_n_a = self.n_axs(l, xc)
|
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_n_w = self.n_ws(l)
|
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_La = (_n_a**2 - 1) / (_n_a**2 + 2)
|
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_Lw = (_n_w**2 - 1) / (_n_w**2 + 2)
|
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|
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# Calculate the density of the dry air components, using the partial pressure of the dry air, i.e. p_dry = p_atm - p_water
|
||||
_rho_a = self.rho(p=p-RH*self.svp(T), T=T, RH=0, xc=xc) # Dry air component for the actual conditions.
|
||||
_rho_axs = self.rho(p=101325, T=288.15, RH=0, xc=xc) # Dry air component at standard conditions.
|
||||
|
||||
if RH == 0:
|
||||
# nan is returned for _rho_w if we put in a (partial) pressure of 0
|
||||
_rho_w = 0
|
||||
else:
|
||||
# Partial pressure of water vapour is p_water = h*svp
|
||||
_rho_w = self.rho(p=RH*self.svp(T), T=T, RH=RH, xc=xc) # Moist air component for the actual conditions.
|
||||
_rho_ws = self.rho(p=1333, T=293.15, RH=1333/self.svp(293.15), xc=xc) # Moist air component at standard conditions.
|
||||
|
||||
_Ltotal = ( _rho_a / _rho_axs ) * _La + ( _rho_w / _rho_ws ) * _Lw
|
||||
|
||||
# print 'Dry air component = ', (_rho_a/_rho_axs)*(_n_a-1)*1e8
|
||||
# print 'Wet component = ', ( _rho_w / _rho_ws ) * (_n_w - 1)*1e8
|
||||
# print 'Sum of components = ', (_rho_a/_rho_axs)*(_n_a-1)*1e8+( _rho_w / _rho_ws ) * (_n_w - 1)*1e8
|
||||
return np.sqrt( (1 + 2 * _Ltotal) / (1 - _Ltotal) )
|
||||
|
||||
|
||||
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
|
||||
@@ -0,0 +1,448 @@
|
||||
#!/usr/bin/env Python3
|
||||
import numpy as np
|
||||
from scipy import integrate
|
||||
from .ciddorModel import Observatory
|
||||
'''
|
||||
@ Author: Joost van den Born
|
||||
@ Contact: born@astron.nl
|
||||
@ Description: This script offers a complete model of the atmospheric refraction
|
||||
and dispersion. It uses the Ciddor1996 paper to calculate the
|
||||
index of refraction for a wide range of conditions, with corresponding
|
||||
error propagation if needed.
|
||||
The actual refraction is calculated by any of the following models:
|
||||
- Plane parallel atmosphere
|
||||
- Cassini spherical homogeneous atmosphere
|
||||
- Other methods described in Corbard et al., 2019.
|
||||
'''
|
||||
|
||||
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.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.
|
||||
'''
|
||||
_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)
|
||||
|
||||
|
||||
|
||||
def cassini(self, n1, n2, zenith, reduced_height=None):
|
||||
'''
|
||||
Refraction of spherical atmosphere, derived from geometry using Sine law
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_R1 = np.arcsin(n1 * self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H)) - np.arcsin(self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H) )
|
||||
_R2 = np.arcsin(n2 * self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H)) - np.arcsin(self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H) )
|
||||
return np.degrees(_R1) - np.degrees(_R2)
|
||||
|
||||
|
||||
|
||||
def tan5(self, n1, n2, zenith, reduced_height=None):
|
||||
'''
|
||||
Same as oriani's theorem, but including a higher order term.
|
||||
Found in Corbard et al., 2019.
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_rList = []
|
||||
for _n in [n1, n2]:
|
||||
_a = _n - 1
|
||||
_b = H / self.rc
|
||||
_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3 + 3 * _a * (_b - _a/2)**2 * np.tan( np.deg2rad(zenith) )**5
|
||||
_rList.append(np.degrees(_R))
|
||||
return _rList[0] - _rList[1]
|
||||
|
||||
|
||||
|
||||
def oriani(self, n1, n2, zenith, reduced_height=None):
|
||||
'''
|
||||
Classic refraction formula with two tan(z) terms. This does not assume any
|
||||
information about the structure of the atmosphere. Found with Oriani's theorem.
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_rList = []
|
||||
for _n in [n1, n2]:
|
||||
_a = _n - 1
|
||||
_b = H / self.rc
|
||||
_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3
|
||||
_rList.append(np.degrees(_R))
|
||||
return _rList[0] - _rList[1]
|
||||
|
||||
|
||||
|
||||
def corbard(self, n1, n2, zenith, reduced_height=None):
|
||||
'''
|
||||
Corbard et al., 2019, mention an additional formula based on the error function.
|
||||
Oriani's theorem can be derived from this equation by 'keeping only the three first
|
||||
terms of its asymptotic expansion'.
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_rList = []
|
||||
for _n in [n1, n2]:
|
||||
_a = _n - 1
|
||||
_b = H / self.rc
|
||||
_R = _a * ( (2 - _a) / (np.sqrt( 2*_b - _a )) ) * np.sin( np.deg2rad(zenith) ) * self.psi( (np.cos(np.deg2rad(zenith))) / np.sqrt(2*_b - _a) )
|
||||
_rList.append(np.degrees(_R))
|
||||
return _rList[0] - _rList[1]
|
||||
|
||||
|
||||
|
||||
def refractionIntegral(self, n1, n2, z0, R0=None, useStandardAtmosphere=True, heightData=None, rhoData=None):
|
||||
'''
|
||||
Using data that gives temperature, pressure and density as a function of height,
|
||||
the gladstone-dale relation is invoked to calculate the path of a light ray
|
||||
along the atmosphere. This is allows us to also include atmospheric activity in
|
||||
different atmosphere layers.
|
||||
Recommended use with the US Standard Atmosphere from 1976.
|
||||
USSA1976 data generated from code supplied by http://www.pdas.com/atmos.html
|
||||
The method described in eq. 3 in Auer & Standish (2000) is used for the calculation.
|
||||
'''
|
||||
if not R0:
|
||||
R0 = self.rc
|
||||
if useStandardAtmosphere:
|
||||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt('1976USSA.txt', unpack=True, skip_header=6)
|
||||
else:
|
||||
_Dens = rhoData
|
||||
_h = heightData
|
||||
|
||||
|
||||
_rList = []
|
||||
z0 = np.deg2rad(z0)
|
||||
|
||||
# Integration of the refraction integral over the atmosphere layers.
|
||||
for _n in [n1, n2]:
|
||||
_Dens = np.asarray(_Dens)
|
||||
_h = np.asarray(_h)
|
||||
_R = R0 + _h * 1000
|
||||
|
||||
# Gladstone-Dale relation between refractive index and density
|
||||
_n_h = 1 + (_n - 1) / _Dens[0] * _Dens
|
||||
|
||||
# Calculation of the zenith angle for given height and refractive index, from the refractive invariant
|
||||
_z = np.arcsin( (_n * R0) / (_n_h * _R) * np.sin(z0) )
|
||||
|
||||
# Calculation of log space derivative of n and r
|
||||
_lnn = np.log(_n_h)
|
||||
_lnr = np.log(_R)
|
||||
_dlnn = np.asarray([_lnn[i+1] - _lnn[i] for i in range(len(_lnn)-1)])
|
||||
_dlnr = np.asarray([_lnr[i+1] - _lnr[i] for i in range(len(_lnr)-1)])
|
||||
_dz = np.asarray([(_z[i] - _z[i+1]) for i in range(len(_z)-1)])
|
||||
|
||||
# Calculation of the integrand in eq. 3 in Auer & Standish
|
||||
_Integrand = -1*(_dlnn/_dlnr) / (1 + _dlnn/_dlnr) * _dz
|
||||
_rList.append(np.degrees(np.sum(_Integrand)))
|
||||
return _rList[0] - _rList[1]
|
||||
|
||||
|
||||
|
||||
def psi(self, x):
|
||||
_f = lambda a: np.exp(-a**2)
|
||||
_int = integrate.quad(_f, x, np.inf)
|
||||
return np.exp(x**2) * _int[0]
|
||||
|
||||
|
||||
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
|
||||
ellipsoidality of the earth.
|
||||
'''
|
||||
_A = 6378.137 # km
|
||||
_B = 6356.752 # km
|
||||
_phi = np.deg2rad(self.lat)
|
||||
|
||||
_rc0 = (_A*_B)**2 / (_A**2 * np.cos(_phi)**2 + _B**2 * np.sin(_phi)**2)**(3/2)
|
||||
return _rc0 + self.h/1000
|
||||
|
||||
|
||||
def set_H(self, P, rho):
|
||||
'''
|
||||
Calculates the reduced height of the atmosphere, assuming ideal gas law.
|
||||
'''
|
||||
_phi = np.deg2rad(self.lat)
|
||||
_c1 = 5.2790414e-3
|
||||
_c2 = 2.32718e-5
|
||||
_c3 = 1.262e-7
|
||||
_c4 = 7e-10
|
||||
_g0 = 9.780327 # m/s^2
|
||||
_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
|
||||
_g = _g0_local - (3.0877e-6 - 4.3e-9 * np.sin(_phi)**2) * self.h + 7.2e-13 * self.h**2
|
||||
return P / (rho * _g)
|
||||
|
||||
def H_isotherm(self, T):
|
||||
'''
|
||||
Calculates the reduced height of the atmosphere, assuming an isotherm distribution.
|
||||
'''
|
||||
_kb = 1.38064852e-23 # kg m2 s-2 K-1
|
||||
_m = 4.809651698e-26 # kg (weight per molecule of air, assuming dry air)
|
||||
|
||||
_phi = np.deg2rad(self.lat)
|
||||
_c1 = 5.2790414e-3
|
||||
_c2 = 2.32718e-5
|
||||
_c3 = 1.262e-7
|
||||
_c4 = 7e-10
|
||||
_g0 = 9.780327 # m/s^2
|
||||
_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
|
||||
|
||||
return (_kb*T) / (_m * _g0_local)
|
||||
|
||||
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)
|
||||
@@ -0,0 +1,106 @@
|
||||
#!/usr/bin/env Python3
|
||||
from .ciddorModel import Observatory
|
||||
from .refractionModels import refraction
|
||||
from .dispersionModels import dispersion
|
||||
'''
|
||||
Author: Joost van den Born
|
||||
Some functions for quick testing.
|
||||
'''
|
||||
|
||||
|
||||
def quick_refractive_index(l, conditions='STANDARD'):
|
||||
if conditions == 'STANDARD':
|
||||
# Parameters at Cerro Armazones
|
||||
T = 288.15 # k
|
||||
P = 101325 # Pa
|
||||
H = 0.0
|
||||
xc = 450 # ppm
|
||||
lat = 0 # degrees
|
||||
h = 0 # m
|
||||
|
||||
|
||||
elif conditions == 'CERRO_ARMAZONES':
|
||||
# Parameters at Cerro Armazones
|
||||
T = 279.65 # k
|
||||
P = 71200 # Pa
|
||||
H = 0.22
|
||||
xc = 450 # ppm
|
||||
lat = -24.5983 # degrees
|
||||
h = 3064 # m
|
||||
|
||||
|
||||
# Initializing dispersion model
|
||||
at = Observatory()
|
||||
|
||||
# Calculating indices of refraction for l
|
||||
n = at.n_tph(l=l, T=T, p=P, RH=H, xc=xc)
|
||||
return n
|
||||
|
||||
def quick_refraction(l, z, conditions='STANDARD'):
|
||||
if conditions == 'STANDARD':
|
||||
# Parameters at Cerro Armazones
|
||||
T = 288.15 # k
|
||||
P = 101325 # Pa
|
||||
H = 0.0
|
||||
xc = 450 # ppm
|
||||
lat = 0 # degrees
|
||||
h = 0 # m
|
||||
|
||||
|
||||
elif conditions == 'CERRO_ARMAZONES':
|
||||
# Parameters at Cerro Armazones
|
||||
T = 279.65 # k
|
||||
P = 71200 # Pa
|
||||
H = 0.22
|
||||
xc = 450 # ppm
|
||||
lat = -24.5983 # degrees
|
||||
h = 3064 # m
|
||||
|
||||
# Initializing dispersion model
|
||||
at = Observatory()
|
||||
|
||||
# Calculating indices of refraction for l
|
||||
n = at.n_tph(l=l, T=T, p=P, RH=H, xc=xc)
|
||||
|
||||
# Density of the atmosphere (following CIPM-81/91 equations)
|
||||
rho = at.rho(p=P, T=T, RH=H, xc=xc)
|
||||
|
||||
# Initializing refraction model and setting the reduced height
|
||||
ref = refraction(lat, h)
|
||||
ref.setReducedHeight(P, rho)
|
||||
return ref.cassini(n, z)
|
||||
|
||||
def quick_dispersion(l1, l2, z, conditions='STANDARD'):
|
||||
if conditions == 'STANDARD':
|
||||
# Parameters at Cerro Armazones
|
||||
T = 288.15 # k
|
||||
P = 101325 # Pa
|
||||
H = 0.0
|
||||
xc = 450 # ppm
|
||||
lat = 0 # degrees
|
||||
h = 0 # m
|
||||
|
||||
|
||||
elif conditions == 'CERRO_ARMAZONES':
|
||||
# Parameters at Cerro Armazones
|
||||
T = 279.65 # k
|
||||
P = 71200 # Pa
|
||||
H = 0.22
|
||||
xc = 450 # ppm
|
||||
lat = -24.5983 # degrees
|
||||
h = 3064 # m
|
||||
|
||||
# Initializing dispersion model
|
||||
at = Observatory()
|
||||
|
||||
# Calculating indices of refraction for l1 and l2
|
||||
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)
|
||||
|
||||
# Density of the atmosphere (following CIPM-81/91 equations)
|
||||
rho = at.rho(p=P, T=T, RH=H, xc=xc)
|
||||
|
||||
# Initializing refraction model and setting the reduced height
|
||||
disp = dispersion(lat, h)
|
||||
disp.setReducedHeight(P, rho)
|
||||
return disp.cassini(n1, n2, z)
|
||||
@@ -0,0 +1,218 @@
|
||||
#!/usr/bin/env Python3
|
||||
import numpy as np
|
||||
from scipy import integrate
|
||||
from .ciddorModel import Observatory
|
||||
'''
|
||||
@ Author: Joost van den Born
|
||||
@ Contact: born@astron.nl
|
||||
@ Description: This script offers a complete model of the atmospheric refraction
|
||||
and dispersion. It uses the Ciddor1996 paper to calculate the
|
||||
index of refraction for a wide range of conditions, with corresponding
|
||||
error propagation if needed.
|
||||
The actual refraction is calculated by any of the following models:
|
||||
- Plane parallel atmosphere
|
||||
- Cassini spherical homogeneous atmosphere
|
||||
- Other methods described in Corbard et al., 2019.
|
||||
'''
|
||||
|
||||
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.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.
|
||||
'''
|
||||
_R = np.arcsin(n * np.sin( np.deg2rad(zenith) )) - np.deg2rad(zenith)
|
||||
return np.degrees(_R)
|
||||
|
||||
|
||||
|
||||
def cassini(self, n1, zenith, reduced_height=None):
|
||||
'''
|
||||
Refraction of spherical atmosphere, derived from geometry using Sine law
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_R1 = np.arcsin(n1 * self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H)) - np.arcsin(self.rc * np.sin(np.deg2rad(zenith)) / (self.rc + H) )
|
||||
return np.degrees(_R1)
|
||||
|
||||
|
||||
|
||||
def tan5(self, n1, zenith, reduced_height=None):
|
||||
'''
|
||||
Same as oriani's theorem, but including a higher order term.
|
||||
Found in Corbard et al., 2019.
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_a = n1 - 1
|
||||
_b = H / self.rc
|
||||
_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3 + 3 * _a * (_b - _a/2)**2 * np.tan( np.deg2rad(zenith) )**5
|
||||
return np.degrees(_R)
|
||||
|
||||
|
||||
|
||||
def oriani(self, n1, zenith, reduced_height=None):
|
||||
'''
|
||||
Classic refraction formula with two tan(z) terms. This does not assume any
|
||||
information about the structure of the atmosphere. Found with Oriani's theorem.
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_a = n1 - 1
|
||||
_b = H / self.rc
|
||||
_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3
|
||||
return np.degrees(_R)
|
||||
|
||||
|
||||
|
||||
def corbard(self, n1, zenith, reduced_height=None):
|
||||
'''
|
||||
Corbard et al., 2019, mention an additional formula based on the error function.
|
||||
Oriani's theorem can be derived from this equation by 'keeping only the three first
|
||||
terms of its asymptotic expansion'.
|
||||
'''
|
||||
if reduced_height:
|
||||
H = reduced_height
|
||||
else:
|
||||
H = self.H
|
||||
|
||||
_a = n1 - 1
|
||||
_b = H / self.rc
|
||||
_R = _a * ( (2 - _a) / (np.sqrt( 2*_b - _a )) ) * np.sin( np.deg2rad(zenith) ) * self.psi( (np.cos(np.deg2rad(zenith))) / np.sqrt(2*_b - _a) )
|
||||
return np.degrees(_R)
|
||||
|
||||
|
||||
|
||||
def refractionIntegral(self, n0, z0, R0=None, useStandardAtmosphere=True, heightData=None, rhoData=None):
|
||||
'''
|
||||
Using data that gives temperature, pressure and density as a function of height,
|
||||
the gladstone-dale relation is invoked to calculate the path of a light ray
|
||||
along the atmosphere. This is allows us to also include atmospheric activity in
|
||||
different atmosphere layers.
|
||||
Recommended use with the US Standard Atmosphere from 1976.
|
||||
USSA1976 data generated from code supplied by http://www.pdas.com/atmos.html
|
||||
'''
|
||||
if not R0:
|
||||
R0 = self.rc
|
||||
if useStandardAtmosphere:
|
||||
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt('1976USSA.txt', unpack=True, skip_header=6)
|
||||
else:
|
||||
_Dens = rhoData
|
||||
_h = heightData
|
||||
|
||||
z0 = np.deg2rad(z0)
|
||||
_Dens = np.asarray(_Dens)
|
||||
_h = np.asarray(_h)
|
||||
_R = R0 + _h * 1000
|
||||
|
||||
# Gladstone-Dale relation between refractive index and density
|
||||
_n_h = 1 + (n0 - 1) / _Dens[0] * _Dens
|
||||
|
||||
# Calculation of the zenith angle for given height and refractive index, from the refractive invariant
|
||||
_z = np.arcsin( (n0 * R0) / (_n_h * _R) * np.sin(z0) )
|
||||
|
||||
# Calculation of log space derivative of n and r
|
||||
_lnn = np.log(_n_h)
|
||||
_lnr = np.log(_R)
|
||||
_dlnn = np.asarray([_lnn[i+1] - _lnn[i] for i in range(len(_lnn)-1)])
|
||||
_dlnr = np.asarray([_lnr[i+1] - _lnr[i] for i in range(len(_lnr)-1)])
|
||||
_dz = np.asarray([(_z[i] - _z[i+1]) for i in range(len(_z)-1)])
|
||||
|
||||
# Calculation of the integrand in eq. 3 in Auer & Standish
|
||||
_Integrand = -1*(_dlnn/_dlnr) / (1 + _dlnn/_dlnr) * _dz
|
||||
|
||||
|
||||
return np.degrees(np.sum(_Integrand))
|
||||
|
||||
|
||||
|
||||
def psi(self, x):
|
||||
_f = lambda a: np.exp(-a**2)
|
||||
_int = integrate.quad(_f, x, np.inf)
|
||||
return np.exp(x**2) * _int[0]
|
||||
|
||||
|
||||
|
||||
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
|
||||
ellipsoidality of the earth.
|
||||
'''
|
||||
_A = 6378.137 # km
|
||||
_B = 6356.752 # km
|
||||
_phi = np.deg2rad(self.lat)
|
||||
|
||||
_rc0 = (_A*_B)**2 / (_A**2 * np.cos(_phi)**2 + _B**2 * np.sin(_phi)**2)**(3/2)
|
||||
return _rc0 + self.h/1000
|
||||
|
||||
|
||||
|
||||
def set_H(self, P, rho):
|
||||
'''
|
||||
Calculates the reduced height of the atmosphere, assuming ideal gas law.
|
||||
'''
|
||||
_phi = np.deg2rad(self.lat)
|
||||
_c1 = 5.2790414e-3
|
||||
_c2 = 2.32718e-5
|
||||
_c3 = 1.262e-7
|
||||
_c4 = 7e-10
|
||||
_g0 = 9.780327 # m/s^2
|
||||
_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
|
||||
_g = _g0_local - (3.0877e-6 - 4.3e-9 * np.sin(_phi)**2) * self.h + 7.2e-13 * self.h**2
|
||||
return P / (rho * _g)
|
||||
|
||||
def H_isotherm(self, T):
|
||||
'''
|
||||
Calculates the reduced height of the atmosphere, assuming an isotherm distribution.
|
||||
'''
|
||||
_kb = 1.38064852e-23 # kg m2 s-2 K-1
|
||||
_m = 4.809651698e-26 # kg (weight per molecule of air, assuming dry air)
|
||||
|
||||
_phi = np.deg2rad(self.lat)
|
||||
_c1 = 5.2790414e-3
|
||||
_c2 = 2.32718e-5
|
||||
_c3 = 1.262e-7
|
||||
_c4 = 7e-10
|
||||
_g0 = 9.780327 # m/s^2
|
||||
_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
|
||||
|
||||
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)
|
||||
@@ -0,0 +1,349 @@
|
||||
#!/usr/bin/env Python3
|
||||
import numpy as np
|
||||
|
||||
|
||||
'''
|
||||
@ Author: Joost van den Born
|
||||
@ Contact: born@astron.nl
|
||||
@ Description: A comparison of various atmospheric refraction models.
|
||||
Also these are used to calculate the difference in
|
||||
atmospheric dispersion, given a set of conditions.
|
||||
'''
|
||||
|
||||
def BarrellAndSears(l, T=288.15, p=101325, H=0):
|
||||
'''
|
||||
From Barrell & Sears, 1939
|
||||
Equation 7.7, page 52
|
||||
For atmospheric air (i.e. CO2 = 300 ppm)
|
||||
This equation is applicable for the ranges T=10-30C and p=720-800 mmHg.
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
T - Temperature in Kelvin
|
||||
p - Atmospheric pressure in Pa
|
||||
H - fractional humidity (0=<H=<1)
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
T = K2C(T)
|
||||
P = hPA2mmHg(p/100)
|
||||
R = (0.378125 + 0.0021414 / l**2 + 0.00001793 / l**4) * (P*(1 + (1.049 - 0.0157*T) * P * 1e-6)/(1 + 0.003661*T)) - (0.0624 - 0.000680 / l**2) * H / (1 + 0.003661*T)
|
||||
n = R/1e6 + 1
|
||||
return n
|
||||
|
||||
def Ciddor(l):
|
||||
'''
|
||||
From Ciddor, 1996
|
||||
Equations 1
|
||||
At default conditions only. For more flexibility use
|
||||
Observatory.n_tph()
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
s = 1 / l
|
||||
R = 5792105 / (238.0185 - s**2) + 167917 / (57.362 - s**2)
|
||||
n = R/1e8 + 1
|
||||
return n
|
||||
|
||||
def Edlen1953(l):
|
||||
'''
|
||||
From Edlen, 1953
|
||||
Equation 2
|
||||
|
||||
At standard conditions only
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
s = 1 / l
|
||||
R = 6432.8 + 2949810 / (146 - s**2) + 25540/(41-s**2)
|
||||
n = R/1e8 + 1
|
||||
return n
|
||||
|
||||
def Edlen1966(l):
|
||||
'''
|
||||
From Edlen, 1966
|
||||
Equation 1
|
||||
|
||||
At standard conditions only
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
s = 1 / l
|
||||
R = 8342.13 + 2406030 / (130 - s**2) + 15997 / (38.9 - s**2)
|
||||
n = R/1e8 + 1
|
||||
return n
|
||||
|
||||
def Owens(l, T=288.15, p=101325, H=0):
|
||||
'''
|
||||
From Owens, 1967
|
||||
Equation 32
|
||||
Note that the pressure P and H are in torricelli (Torr),
|
||||
however, this is almost precisely 1 Torr = 1 mmHg
|
||||
|
||||
Input:
|
||||
l - wavelength in micron
|
||||
T - Temperature in Kelvin
|
||||
p - Atmospheric pressure in Pa
|
||||
H - Humdity (0<=H<=1)
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
P = hPA2mmHg(p/100)
|
||||
T = K2C(T)
|
||||
s = 1 / l
|
||||
R = 8342.13 + 2406030 / (130 - s**2) + 15997 / (38.9 - s**2) * (P/720.775) * ((1 + P * (0.817 - 0.0133) * 1e-6)/(1 + 0.0036610*T)) - H * (5.722 - 0.0457*s**2)
|
||||
n = R/1e8 + 1
|
||||
return n
|
||||
|
||||
def BonschPotulski(l, T=288.15, p=101325, CO2=450):
|
||||
'''
|
||||
From Bonsch & Potulski, 1998
|
||||
Equations 6, 7, 8
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
T - Temperature in Kelvin
|
||||
p - Atmospheric pressure in Pa
|
||||
CO2 - Particles per million of CO2
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
x = CO2 / 1e6
|
||||
T = K2C(T)
|
||||
s = 1 / l
|
||||
# Rs = 8092.33 + 2333983 / (130 - s**2) + 15518 / (38.9 - s**2)
|
||||
Rs = 8091.37 + 2333983 / (130 - s**2) + 15518 / (38.9 - s**2)
|
||||
Rs = Rs / 1e8
|
||||
Rx = Rs * (1 + 0.5327 * (x - 0.0004))
|
||||
Rtp = Rx * p / 93214.60 * (1 + 1e-8 * (0.5953 - 0.009876 * T)*p) / (1 + 0.0036610 * T)
|
||||
n = Rtp + 1
|
||||
return n
|
||||
|
||||
def BirchDowns(l):
|
||||
'''
|
||||
From Birch & Downs, 1993 and 1994 (correction)
|
||||
Equations 1 and 2
|
||||
|
||||
Only at standard conditions
|
||||
|
||||
Input:
|
||||
l - wavelength in micron
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
s = 1 / l
|
||||
R = 8342.54 + 2406147 / (130 - s**2) + 15998 / (38.9-s**2)
|
||||
n = R/1e8 + 1
|
||||
return n
|
||||
|
||||
def PeckReeder(l):
|
||||
'''
|
||||
From Peck & Reeder, 1972
|
||||
Equation 3
|
||||
|
||||
Only at standard conditions
|
||||
|
||||
Input:
|
||||
l - wavelength in micron
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
s = 1 / l
|
||||
R = 8060.51 + 2480990 / (132.274 - s**2) + 17455.7 / (39.32957 - s**2)
|
||||
n = R/1e8 + 1
|
||||
return n
|
||||
|
||||
|
||||
def slalib(l, T=288.15, p=101325, H=0, h=0, lat=-24, tlr=0.0065, eps=1e-10):
|
||||
'''
|
||||
SLALIB uses the same atmospheric model as ZEMAX
|
||||
based on the works of Hohenkerk & Sinclair and Seidelmann.
|
||||
They take the B&S refraction formula as a basis.
|
||||
Requires the pySLALIB package to work.
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
T - Temperature in Kelvin
|
||||
p - Atmospheric pressure in Pa
|
||||
H - fractional humidity (0=<H=<1)
|
||||
h - Altitude of the observer (default=0 m)
|
||||
lat - Latitude of the observer (default=-24)
|
||||
tlr - temperature lapse rate (default=0.0065)
|
||||
eps - precision at which the calculation is aborted (default=1e-10)
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
from pyslalib import slalib
|
||||
zObs = np.deg2rad(45.06563)
|
||||
p = p/100
|
||||
R = []
|
||||
if type(l) == float:
|
||||
l = [l]
|
||||
for wl in l:
|
||||
R.append(slalib.sla_refro(zObs, h, T, p, H, wl, lat, tlr, eps))
|
||||
if len(l) == 1:
|
||||
return R[0]+1
|
||||
else:
|
||||
return np.asarray(R)+1
|
||||
|
||||
|
||||
def Mathar(l, T=288.15, p=101325, H=0):
|
||||
'''
|
||||
Refractivity model by Mathar (2007), valid above 1.4 microns
|
||||
Code taken and modified from:
|
||||
https://github.com/polyanskiy/refractiveindex.info-scripts/blob/master/scripts/Mathar%202007%20-%20Air%201.3-2.5um.py
|
||||
|
||||
Input:
|
||||
l - Wavelength in microns
|
||||
T - Temperature in Kelvin
|
||||
p - Atmospheric pressure in Pa
|
||||
H - fractional humidity (0=<H=<1)
|
||||
|
||||
Returns:
|
||||
n - Refractive index of atmospheric air at the given conditions
|
||||
'''
|
||||
if l < 1.3 or l > 24:
|
||||
print('Using the Mathar model at this wavelength (l={:.3f}) is not recommended!'.format(l))
|
||||
|
||||
if l <= 2.65:
|
||||
########################################################################
|
||||
# model parameters for wavelengths between 1.3 and 2.5 microns
|
||||
cref = [ 0.200192e-3, 0.113474e-9, -0.424595e-14, 0.100957e-16, -0.293315e-20, 0.307228e-24] # cm^j
|
||||
cT = [ 0.588625e-1, -0.385766e-7, 0.888019e-10, -0.567650e-13, 0.166615e-16, -0.174845e-20] # cm^j · K
|
||||
cTT = [-3.01513, 0.406167e-3, -0.514544e-6, 0.343161e-9, -0.101189e-12, 0.106749e-16] # cm^j · K^2
|
||||
cH = [-0.103945e-7, 0.136858e-11, -0.171039e-14, 0.112908e-17, -0.329925e-21, 0.344747e-25] # cm^j · %^-1
|
||||
cHH = [ 0.573256e-12, 0.186367e-16, -0.228150e-19, 0.150947e-22, -0.441214e-26, 0.461209e-30] # cm^j · %^-2
|
||||
cp = [ 0.267085e-8, 0.135941e-14, 0.135295e-18, 0.818218e-23, -0.222957e-26, 0.249964e-30] # cm^j · Pa^-1
|
||||
cpp = [ 0.609186e-17, 0.519024e-23, -0.419477e-27, 0.434120e-30, -0.122445e-33, 0.134816e-37] # cm^j · Pa^-2
|
||||
cTH = [ 0.497859e-4, -0.661752e-8, 0.832034e-11, -0.551793e-14, 0.161899e-17, -0.169901e-21] # cm^j · K · %^-1
|
||||
cTp = [ 0.779176e-6, 0.396499e-12, 0.395114e-16, 0.233587e-20, -0.636441e-24, 0.716868e-28] # cm^j · K · Pa^-1
|
||||
cHp = [-0.206567e-15, 0.106141e-20, -0.149982e-23, 0.984046e-27, -0.288266e-30, 0.299105e-34] # cm^j · %^-1 · Pa^-1
|
||||
|
||||
sref = 1e4/2.25 # cm^−1, reference wavenumber
|
||||
Tref = 273.15+17.5 # K, reference temperature
|
||||
pref = 75000 # Pa, reference pressure
|
||||
Href = 10 #%, reference humdity in percent
|
||||
########################################################################
|
||||
|
||||
if l > 2.65 and l <= 4.275:
|
||||
########################################################################
|
||||
# model parameters for wavelengths between 2.8 and 4.2 microns
|
||||
cref = [ 0.200049e-3, 0.145221e-9, 0.250951e-12, -0.745834e-15, -0.161432e-17, 0.352780e-20] # cm^j
|
||||
cT = [ 0.588432e-1, -0.825182e-7, 0.137982e-9, 0.352420e-13, -0.730651e-15, -0.167911e-18] # cm^j · K
|
||||
cTT = [-3.13579, 0.694124e-3, -0.500604e-6, -0.116668e-8, 0.209644e-11, 0.591037e-14] # cm^j · K^2
|
||||
cH = [-0.108142e-7, 0.230102e-11, -0.154652e-14, -0.323014e-17, 0.630616e-20, 0.173880e-22] # cm^j · %^-1
|
||||
cHH = [ 0.586812e-12, 0.312198e-16, -0.197792e-19, -0.461945e-22, 0.788398e-25, 0.245580e-27] # cm^j · %^-2
|
||||
cp = [ 0.266900e-8, 0.168162e-14, 0.353075e-17, -0.963455e-20, -0.223079e-22, 0.453166e-25] # cm^j · Pa^-1
|
||||
cpp = [ 0.608860e-17, 0.461560e-22, 0.184282e-24, -0.524471e-27, -0.121299e-29, 0.246512e-32] # cm^j · Pa^-2
|
||||
cTH = [ 0.517962e-4, -0.112149e-7, 0.776507e-11, 0.172569e-13, -0.320582e-16, -0.899435e-19] # cm^j · K · %^-1
|
||||
cTp = [ 0.778638e-6, 0.446396e-12, 0.784600e-15, -0.195151e-17, -0.542083e-20, 0.103530e-22] # cm^j · K · Pa^-1
|
||||
cHp = [-0.217243e-15, 0.104747e-20, -0.523689e-23, 0.817386e-26, 0.309913e-28, -0.363491e-31] # cm^j · %^-1 · Pa^-1
|
||||
|
||||
sref = 1e4/3.4 # cm^−1, reference wavenumber
|
||||
Tref = 273.15+17.5 # K, reference temperature
|
||||
pref = 75000 # Pa, reference pressure
|
||||
Href = 10 #%, reference humdity in percent
|
||||
########################################################################
|
||||
|
||||
if l > 4.275 and l <= 6.35:
|
||||
########################################################################
|
||||
# model parameters for wavelengths between 4.35 and 5.2 microns
|
||||
cref = [ 0.200020e-3, 0.275346e-9, 0.325702e-12, -0.693603e-14, 0.285610e-17, 0.338758e-18] # cm^j
|
||||
cT = [ 0.590035e-1, -0.375764e-6, 0.134585e-9, 0.124316e-11, 0.508510e-13, -0.189245e-15] # cm^j · K
|
||||
cTT = [-4.09830, 0.250037e-2, 0.275187e-6, -0.653398e-8, -0.310589e-9, 0.127747e-11] # cm^j · K^2
|
||||
cH = [-0.140463e-7, 0.839350e-11, -0.190929e-14, -0.121399e-16, -0.898863e-18, 0.364662e-20] # cm^j · %^-1
|
||||
cHH = [ 0.543605e-12, 0.112802e-15, -0.229979e-19, -0.191450e-21, -0.120352e-22, 0.500955e-25] # cm^j · %^-2
|
||||
cp = [ 0.266898e-8, 0.273629e-14, 0.463466e-17, -0.916894e-23, 0.136685e-21, 0.413687e-23] # cm^j · Pa^-1
|
||||
cpp = [ 0.610706e-17, 0.116620e-21, 0.244736e-24, -0.497682e-26, 0.742024e-29, 0.224625e-30] # cm^j · Pa^-2
|
||||
cTH = [ 0.674488e-4, -0.406775e-7, 0.289063e-11, 0.819898e-13, 0.468386e-14, -0.191182e-16] # cm^j · K · %^-1
|
||||
cTp = [ 0.778627e-6, 0.593296e-12, 0.145042e-14, 0.489815e-17, 0.327941e-19, 0.128020e-21] # cm^j · K · Pa^-1
|
||||
cHp = [-0.211676e-15, 0.487921e-20, -0.682545e-23, 0.942802e-25, -0.946422e-27, -0.153682e-29] # cm^j · %^-1 · Pa^-1
|
||||
σref = 1e4/4.8 # cm^−1, reference wavenumber
|
||||
Tref = 273.15+17.5 # K, reference temperature
|
||||
pref = 75000 # Pa, reference pressure
|
||||
Href = 10 #%, reference humdity in percent
|
||||
########################################################################
|
||||
|
||||
if l > 6.35 and l <= 15.05:
|
||||
########################################################################
|
||||
# model parameters for wavelengths between 7.5 and 14.1 microns
|
||||
cref = [ 0.199885e-3, 0.344739e-9, -0.273714e-12, 0.393383e-15, -0.569488e-17, 0.164556e-19] # cm^j
|
||||
cT = [ 0.593900e-1, -0.172226e-5, 0.237654e-8, -0.381812e-11, 0.305050e-14, -0.157464e-16] # cm^j · K
|
||||
cTT = [-6.50355, 0.103830e-1, -0.139464e-4, 0.220077e-7, -0.272412e-10, 0.126364e-12] # cm^j · K^2
|
||||
cH = [-0.221938e-7, 0.347377e-10, -0.465991e-13, 0.735848e-16, -0.897119e-19, 0.380817e-21] # cm^j · %^-1
|
||||
cHH = [ 0.393524e-12, 0.464083e-15, -0.621764e-18, 0.981126e-21, -0.121384e-23, 0.515111e-26] # cm^j · %^-2
|
||||
cp = [ 0.266809e-8, 0.695247e-15, 0.159070e-17, -0.303451e-20, -0.661489e-22, 0.178226e-24] # cm^j · Pa^-1
|
||||
cpp = [ 0.610508e-17, 0.227694e-22, 0.786323e-25, -0.174448e-27, -0.359791e-29, 0.978307e-32] # cm^j · Pa^-2
|
||||
cTH = [ 0.106776e-3, -0.168516e-6, 0.226201e-9, -0.356457e-12, 0.437980e-15, -0.194545e-17] # cm^j · K · %^-1
|
||||
cTp = [ 0.77368e-6, 0.216404e-12, 0.581805e-15, -0.189618e-17, -0.198869e-19, 0.589381e-22] # cm^j · K · Pa^-1
|
||||
cHp = [-0.206365e-15, 0.300234e-19, -0.426519e-22, 0.684306e-25, -0.467320e-29, 0.126117e-30] # cm^j · %^-1 · Pa^-1
|
||||
σref = 1e4/10.1 # cm^−1, reference wavenumber
|
||||
Tref = 273.15+17.5 # K, reference temperature
|
||||
pref = 75000 # Pa, reference pressure
|
||||
Href = 10 #%, reference humdity in percent
|
||||
########################################################################
|
||||
|
||||
|
||||
if l > 15.05:
|
||||
########################################################################
|
||||
# model parameters for wavelengths between 16 and 24 microns
|
||||
cref = [ 0.199436e-3, 0.299123e-8, -0.214862e-10, 0.143338e-12, 0.122398e-14, -0.114628e-16] # cm^j
|
||||
# something seems to be wrong with cT...
|
||||
cT = [ 0.621723e-1, -0.177074e-4, 0.152213e-6, -0.954584-9, -0.996706e-11, 0.921476e-13] # cm^j · K
|
||||
cTT = [-23.2409, 0.108557, -0.102439e-2, 0.634072e-5, 0.762517e-7, -0.675587e-9 ] # cm^j · K^2
|
||||
cH = [-0.772707e-7, 0.347237e-9, -0.272675e-11, 0.170858e-13, 0.156889e-15, -0.150004e-17] # cm^j · %^-1
|
||||
cHH = [-0.326604e-12, 0.463606e-14, -0.364272e-16, 0.228756e-18, 0.209502e-20, -0.200547e-22] # cm^j · %^-2
|
||||
cp = [ 0.266827e-8, 0.120788e-14, 0.522646e-17, 0.783027e-19, 0.753235e-21, -0.228819e-24] # cm^j · Pa^-1
|
||||
cpp = [ 0.613675e-17, 0.585494e-22, 0.286055e-24, 0.425193e-26, 0.413455e-28, -0.812941e-32] # cm^j · Pa^-2
|
||||
cTH = [ 0.375974e-3, -0.171849e-5, 0.146704e-7, -0.917231e-10, -0.955922e-12, 0.880502e-14] # cm^j · K · %^-1
|
||||
cTp = [ 0.778436e-6, 0.461840e-12, 0.306229e-14, -0.623183e-16, -0.161119e-18, 0.800756e-20] # cm^j · K · Pa^-1
|
||||
cHp = [-0.272614e-15, 0.304662e-18, -0.239590e-20, 0.149285e-22, 0.136086e-24, -0.130999e-26] # cm^j · %^-1 · Pa^-1
|
||||
σref = 1e4/20 # cm^−1, reference wavenumber
|
||||
Tref = 273.15+17.5 # K, reference temperature
|
||||
pref = 75000 # Pa, reference pressure
|
||||
Href = 10 #%, reference humdity in percent
|
||||
########################################################################
|
||||
|
||||
s = 1e4/l # cm^-1
|
||||
n = 1
|
||||
H = H*100
|
||||
|
||||
for j in range(0, 6):
|
||||
n += ( cref[j] + cT[j]*(1/T-1/Tref) + cTT[j]*(1/T-1/Tref)**2
|
||||
+ cH[j]*(H-Href) + cHH[j]*(H-Href)**2
|
||||
+ cp[j]*(p-pref) + cpp[j]*(p-pref)**2
|
||||
+ cTH[j]*(1/T-1/Tref)*(H-Href)
|
||||
+ cTp[j]*(1/T-1/Tref)*(p-pref)
|
||||
+ cHp[j]*(H-Href)*(p-pref) ) * (s-sref)**j
|
||||
return n
|
||||
|
||||
|
||||
def K2C(K):
|
||||
'''
|
||||
Kelvin to Celsius conversion
|
||||
'''
|
||||
return K - 273.15
|
||||
|
||||
def hPA2mmHg(hPa):
|
||||
'''
|
||||
hPa to mmHg conversion
|
||||
'''
|
||||
return 0.7500616827 * hPa
|
||||
BIN
Binary file not shown.
Reference in new issue
Block a user