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Added installation functionality and renamed the package
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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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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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# Calculate the density of the dry air components, using the partial pressure of the dry air, i.e. p_dry = p_atm - p_water
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_rho_a = self.rho(p=p-RH*self.svp(T), T=T, RH=0, xc=xc) # Dry air component for the actual conditions.
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_rho_axs = self.rho(p=101325, T=288.15, RH=0, xc=xc) # Dry air component at standard conditions.
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if RH == 0:
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# nan is returned for _rho_w if we put in a (partial) pressure of 0
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_rho_w = 0
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else:
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# Partial pressure of water vapour is p_water = h*svp
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_rho_w = self.rho(p=RH*self.svp(T), T=T, RH=RH, xc=xc) # Moist air component for the actual conditions.
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_rho_ws = self.rho(p=1333, T=293.15, RH=1333/self.svp(293.15), xc=xc) # Moist air component at standard conditions.
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_Ltotal = ( _rho_a / _rho_axs ) * _La + ( _rho_w / _rho_ws ) * _Lw
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# print 'Dry air component = ', (_rho_a/_rho_axs)*(_n_a-1)*1e8
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# print 'Wet component = ', ( _rho_w / _rho_ws ) * (_n_w - 1)*1e8
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# print 'Sum of components = ', (_rho_a/_rho_axs)*(_n_a-1)*1e8+( _rho_w / _rho_ws ) * (_n_w - 1)*1e8
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return np.sqrt( (1 + 2 * _Ltotal) / (1 - _Ltotal) )
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def dn_tph(self, l, T, p, RH, xc, dl=None, dT=None, dP=None, dRH=None, dCO2=None):
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'''
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Calculates the uncertainty in the refractive index, following the Ciddor model,
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by using fully propagated error analysis.
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This involves almost endless use of the chain rule and evaluation of both the moist
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air and dry air components. Note: dry air does not contain any water, i.e. H=0, x_w=0, etc.
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Parameters:
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l - Wavelength in micron (0.4 - 2.5 micron)
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T - Temperature in Kelvin
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p - Pressure in Pascals
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RH - Relative humidity (no units, don't use percentage but value 0<x<1)
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xc - CO2 density in ppm
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dl - Uncertainty in the blue wavelength
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dT - Uncertainty in the temperature, default: 0.2 K
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dP - Uncertainty in the pressure, default: 20 Pa
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dRH - Uncertainty in the relative humidity, default: 0.02
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dCO2- Uncertainty in the CO2 density, default: 20
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'''
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if dP == None:
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dP = self.dP
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if dT == None:
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dT = self.dT
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if dRH == None:
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dRH = self.dH
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if dCO2 == None:
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dCO2 = self.dCO2
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# constants
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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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_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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_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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_Mw = 0.018015 # kg/mol
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_R = 8.314510 # J mol^-1 K^-1, gas constant
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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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_aa = 1.00062
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_bb = 3.14e-8 # Pa^-1
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_cc = 5.6e-7 # deg C^-2
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_RH_a = 0
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_RH_w = RH
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# Calculated values
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# _rho_obs = self.rho(p, T, RH, xc)
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# _ref = refraction(lat, h, p, _rho_obs)
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# _alpha = _ref.rc * np.sin(np.deg2rad(z)) / (_ref.rc + _ref.H)
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_p_a = p - RH*self.svp(T)
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_p_w = RH*self.svp(T)
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_x_w_a = self.xw(_p_a, T, _RH_a)
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_x_w_w = self.f(_p_w, T)
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_Z_a = self.Z(p=_p_a, T=T, RH=_RH_a)
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_Z_w = self.Z(p=_p_w, T=T, RH=_RH_w, xw=_x_w_w)
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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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_p_sv = self.svp(T)
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_rho_a = self.rho(p=p-RH*self.svp(T), T=T, RH=0, xc=xc)
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_rho_axs = self.rho(p=101325, T=288.15, RH=0, xc=xc)
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if RH == 0.0:
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# nan is returned for _rho_w if we put in a (partial) pressure of 0
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_rho_w = 0
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else:
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_rho_w = self.rho(p=RH*self.svp(T), T=T, RH=RH, xc=xc)
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_rho_ws = self.rho(p=1333, T=293.15, RH=1333/self.svp(293.15), xc=xc)
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_n_a1 = self.n_axs(l, xc)
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_n_w1 = self.n_ws(l)
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_L1_a = (_n_a1**2 - 1) / (_n_a1**2 + 2) # Dry air component of L
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_L1_w = (_n_w1**2 - 1) / (_n_w1**2 + 2)
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_L1 = ( _rho_a / _rho_axs ) * _L1_a + ( _rho_w / _rho_ws ) * _L1_w # Overall L
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_n1 = np.sqrt( (1 + 2 * _L1) / (1 - _L1) )
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# Partial derivatives for derivation to l1
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_dn_wdl = _cf * 1e-8 * (-1 * _w1 * 2/l**3 - _w2 * 4/l**5 - _w3 * 6/l**7)
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_dLdn_w1 = _rho_w / _rho_ws * 6 * _n_w1 / (_n_w1**2 + 2)**2
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_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) )
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_dLdn_a1 = _rho_a / _rho_axs * 6 * _n_a1 / (_n_a1**2 + 2)**2
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_dndL1 = ( ( (1 + 2*_L1) / (1 - _L1)**2 ) + (2 / (1-_L1)) ) / (2 * np.sqrt( (1 + 2*_L1) / (1 - _L1) ) )
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_dndl = _dndL1 * ( _dLdn_a1 * _dn_adl + _dLdn_w1 * _dn_wdl)
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# # Partial derivatives for derivation to l2
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# Partial derivatives for derivation to T
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# Dry air parts:
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_drho_adp_a = _Ma / (_Z_a * _R * T) * (1 - _x_w_a * (1 - _Mw / _Ma))
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_dp_adp_sv = - RH
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_dp_svdT = (2*_A*T + _B - _D/(T**2)) * np.exp(_A * T**2 + _B * T + _C + _D/T )
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_drho_adTZ = - _p_a * _Ma / (_R * ( _Z_a * T)**2) * (1 - _x_w_a * (1 - _Mw / _Ma))
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_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
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_dTZdxw_a = 2*_e*_p_a**2 / T * _x_w_a - _p_a * (_b1*_t + _b0 + 2 * (_c0 + _c1*_t) * _x_w_a)
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_dxw_adf = _RH_a * _p_sv / _p_a
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_dfdT = 2*_cc*_t
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_dxw_adp_sv = self.f(_p_a, T) * _RH_a / _p_a
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_drho_adxw_a = 0 # as rho_a has not water content, i.e. rho = p*M/ZRT
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# Moist air parts:
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_drho_wdp_w = _Ma / (_Z_w * _R * T) * (1 - _x_w_w * (1 - _Mw / _Ma))
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_dp_wdp_sv = RH
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_dp_svdT = (2*_A*T + _B - _D/(T**2)) * np.exp(_A * T**2 + _B * T + _C + _D/T )
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_drho_wdTZ = - _p_w * _Ma / (_R * (_Z_w * T)**2) * (1 - _x_w_w * (1 - _Mw / _Ma))
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_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
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_dTZdxw_w = 2*_e*_p_w**2 / T * _x_w_w - _p_w * (_b1*_t + _b0 + 2 * (_c0 + _c1*_t) * _x_w_w)
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_dxw_wdf = 1
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_dfdT = 2*_cc*_t
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_dxw_wdp_sv = self.f(_p_w, T) / _p_sv
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_drho_wdxw_w = _p_w * _Ma / (_Z_w * _R * T) * (_Mw / _Ma - 1)
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_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 )
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_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 )
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_dL1drho_a = _L1_a / _rho_axs
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_dL1drho_w = _L1_w / _rho_ws
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_dndT = _tA * (_dndL1 * _dL1drho_a) + _tB * (_dndL1 * _dL1drho_w)
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# Partial derivatives for derivation to p
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# Dry air parts:
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_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
|
||||
Reference in new issue
Block a user