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AtmosphericModel/AstroAtmosphere/refractivityModels.py
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#!/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, RH=0, f=None):
'''
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.
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pascal
RH : float
Relative humidity (0=<H=<1)
f : float
Water vapour pressure in Pascal
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
'''
if f is None:
f = RH * svp(T)
f_mmHg = hPa2mmHg(f/100)
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) * f_mmHg / (1 + 0.003661*T)
n = R/1e6 + 1
return n
def Ciddor(l, CO2=450):
'''
From Ciddor, 1996
Equations 1 and 2
At default conditions only. For more flexibility use
Observatory.n_tph()
Parameters
----------
l : float
Wavelength in microns
CO2 : float
CO2 density in parts per million
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
'''
s = 1 / l
R = 5792105 / (238.0185 - s**2) + 167917 / (57.362 - s**2)
if CO2 != 450:
R = R * (1 + 0.534e-6 * (CO2 - 450))
n = R/1e8 + 1
return n
def Edlen1953(l, T=288.15, p=101325, RH=0, f=None):
'''
From Edlen, 1953
Equation 2
At standard conditions only.
Following Filippenko (1982), eqs. 2 & 3 we also include
the modification proposed by Barrell in 1951.
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pascal
RH : float
Relative humidity (0=<H=<1)
f : float
Water vapor pressure in Pascal
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
'''
s = 1 / l
R = 6432.8 + 2949810 / (146 - s**2) + 25540/(41-s**2)
n_standard = R/1e8 + 1
pHg = hPa2mmHg(p/100)
Tc = K2C(T)
if f is None:
f = RH * svp(T)
f_mmHg = hPa2mmHg(f/100)
# Changes in temperature and pressure, from Barrell (1951)
n_tp = R/1e8 * (pHg * (1 + (1.049 - 0.0157*Tc) * 1e-6 * pHg ) ) / ( 720.883 * (1 + 0.003661 * Tc)) + 1
# Correction for water vapour pressure
n_svp = (n_tp - 1) * 1e6 - ( 0.0624 - 0.000680 * s**2 ) / (1 + 0.003661 * Tc) * f_mmHg
n = n_svp/1e6 + 1
return n
def Edlen1966(l, T=288.15, p=101325, RH=0, CO2=300, f=None):
'''
From Edlen, 1966
Equations 1, 12, 17, 22
At standard conditions only
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pascal
RH : float
Relative humidity (0=<H=<1)
CO2 : float
CO2 density in parts per million
f : float
Water vapor pressure in Pascal
Returns
-------
n : float
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
# Influence of temperature and pressure, eq. 12
Tc = K2C(T)
Ptorr = Pa2Torr(p)
R_tp = ( Ptorr * R/1e8 ) / 720.775 * (1 + Ptorr * (0.817 - 0.0133 * Tc) * 1e-6) / ( 1 + 0.0036610 * Tc)
# Allow for a CO2 fraction, eq. 17, but we include T and P dependence.
# This is okay from "Asthe refractivity of CO, is about 50% higher than
# that of air this CO,-content increases the refractivity over that of
# CO2-free air by approximately 0.5 x 0.0003 x (n-1)s" and "As the
# density factor of CO, is not much different from that of air this ratio r
# can be taken as independent of t and p ."
R_x = R_tp * (1 + 0.540 * (CO2/1e6 - 0.0003))
n_x = R_x + 1
# Correction for water vapour, eq. 22
if f is None:
f = RH * svp(T)
ftorr = Pa2Torr(f)
n = n_x - ftorr * (5.7224 - 0.0457 * s**2) * 1e-8
return n
def Owens(l, T=288.15, p=101325, RH=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
Parameters
----------
l : float
wavelength in micron
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pa
RH : float
Relative humdity (0<=H<=1)
Returns
-------
n : float
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)) - RH * (5.722 - 0.0457*s**2)
n = R/1e8 + 1
return n
def BonschPotulski(l, T=288.15, p=101325, RH=0, CO2=400, f=None):
'''
From Bonsch & Potulski, 1998
Equations 6, 7, 8 and 9a
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pa
RH : float
Relative humidity (0<=H<=1)
CO2 : float
Particles per million of CO2
f : float
Water vapor pressure in Pascal
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
'''
x = CO2 / 1e6
Tc = K2C(T)
s = 1 / l
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 * Tc)*p) / (1 + 0.0036610 * Tc)
if f is None:
f = RH * svp(T)
n_tp = Rtp + 1
n = n_tp - f * (3.8020 - 0.0384 * s**2) * 1e-10
return n
def BirchDowns(l, T=288.15, p=101325, RH=0, f=None):
'''
From Birch & Downs, 1993 and 1994 (correction)
Equations 1, 2 and 3
Only at standard conditions
Parameters
----------
l : float
wavelength in micron
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pa
RH : float
Relative humidity (0<=H<=1)
f : float
Water vapor pressure in Pascal
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
'''
Tc = K2C(T)
s = 1 / l
Rs = 8342.54 + 2406147 / (130 - s**2) + 15998 / (38.9-s**2)
n = Rs/1e8 + 1
R_tp = (p * Rs/1e8) / 96095.43 * (1 + 1e-8 * (0.601 - 0.00972 * Tc) * p) / (1 + 0.0036610 * Tc)
n_tp = R_tp + 1
if f is None:
f = RH * svp(T)
n = n_tp - f * (3.7345 - 0.0401 * s**2) * 1e-10
return n
def PeckReeder(l):
'''
From Peck & Reeder, 1972
Equation 3
Only at standard conditions
Parameters
----------
l : float
wavelength in micron
Returns
-------
n : float
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 HohenkerkAndSinclair(l, T=288.15, p=101325, RH=0, h=0, lat=0):
'''
Calcalation of the refractive index following Hohenkerk & Sinclair
(1985), which is the same one as used in SLALIB.
This equation is based on the Barrell & Sears equation, but with
slightly different values. The refractive index values here seem
to be a bit higher than the original Barrel and Sears values.
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pa
RH : float
fractional humidity (0=<H=<1)
h : float
Altitude of the observer (default=0 m)
lat : float
Latitude of the observer (default=-24)
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
'''
# Some constants
alpha = 0.0065
delta = 18.36
R = 8314.36
Md = 28.966
Mw = 18.016
# Pressure in millibars
P0 = p/100
# Local acceleration due to gravity
g_local = 9.784 * (1 - 0.0026 * np.cos(2*np.deg2rad(lat))) - 0.00000028*h
gamma = g_local * Md / (R * alpha)
# Partial pressure of water vapour at observer
Pw = RH * (T/247.1)**18.36
# Refractivity * 1e6 at reference conditions
A = (287.604 + 1.6288*l**-2 + 0.0136*l**-4) * 273.15/1013.25
# A = (287.5522 + 1.6288*l**-2 + 0.0136*l**-4) * 273.15/1013.25
P = (P0 + (1 - Mw/Md) * gamma/(delta - gamma) * Pw )\
- (1 - Mw/Md) * gamma / (delta-gamma) * Pw
# Refractive index at given conditions
n = 1 + 1e-6 * (A * P - 11.2684*Pw)/T
return n
def slalib(l, T=288.15, p=101325, RH=0, h=0, lat=-24, tlr=0.0065, eps=1e-10):
'''
NOTE: This function still works for now, but is considered DEPRECATED.
Use BarrellAndSears() or HohenkerkAndSinclair() instead.
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.
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pa
RH : float
Relative humidity (0=<H=<1)
h : float
Altitude of the observer (default=0 m)
lat : float
Latitude of the observer (default=-24)
tlr : float
temperature lapse rate (default=0.0065)
eps : float
precision at which the calculation is aborted (default=1e-10)
Returns
-------
n : float
Refractive index of atmospheric air at the given conditions
Notes
-----
1. I don't understand why this function works. It calculates the
refraction and somehow returns the refractive index. Perhaps
this is a very particular set of input parameters that make it
work. It does closely follow BarrellAndSears() for standard
conditions.
'''
try:
from pyslalib import slalib
except ImportError as e:
print('pySLALIB package not found! Try installing it first.\nReturning n=1.')
return 1
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, RH, 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, RH=0):
'''
Refractivity model by Mathar (2007), valid above 1.4 microns
Code taken from
https://github.com/polyanskiy/refractiveindex.info-scripts/blob/master/scripts/Mathar%202007%20-%20Air%201.3-2.5um.py
(better known from refractiveindex.info). The code was subsequently
minimally modified to make it suitable for this package.
Parameters
----------
l : float
Wavelength in microns
T : float
Temperature in Kelvin
p : float
Atmospheric pressure in Pa
H : float
fractional humidity (0=<H=<1)
Returns
-------
n : float
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))
elif 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
########################################################################
elif 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
########################################################################
elif 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
########################################################################
elif 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
########################################################################
elif 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 = RH*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
# Other necessary conversion or calculations
def K2C(K):
'''
Kelvin to Celsius conversion
Parameters
----------
K : float
Temperature in Kelvin
Returns
-------
C : float
Temperature in Celsius
'''
return K - 273.15
def hPa2mmHg(hPa):
'''
hPa to mmHg conversion
Parameters
----------
hPa : float
Atmospheric pressure in hPa (hecto Pascal)
Returns
-------
mmHg : float
Atmospheric pressure in mmHg
'''
return 0.7500616827 * hPa
def Pa2Torr(Pa):
'''
Pa to Torr conversion
Parameters
----------
Pa : float
Atmospheric pressure in Pa (Pascal)
Returns
-------
Torr : float
Atmospheric pressure in Torr
'''
return Pa * 760/101325
def svp(T):
'''
Saturation vapour pressure in Pa.
Parameters
----------
T : float
Atmospheric temperature in Kelvin
Returns
-------
svp : float
Saturation vapour pressure in Pa
'''
_A = 1.2378847e-5 # K^-2
_B = -1.9121316e-2 # K^-2
_C = 33.93711047
_D = -6.3431645e3 # K^-2
return np.exp(_A*T**2 + _B*T + _C + _D/T)
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