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576 lines
18 KiB
Python
576 lines
18 KiB
Python
#!/usr/bin/env Python3
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import numpy as np
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from scipy import integrate
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import os
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from .ciddorModel import Observatory
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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 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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- Mathar's Barometric exponential model
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- Other methods described in Corbard et al., 2019.
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'''
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class refraction(Observatory):
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'''
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Class that sets the observer location.
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These location attributes are then used to calculate the
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appropriate atmospheric refraction.
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See also:
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Observatory()
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dispersion()
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...
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Attributes
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-----------
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lat : float
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Latitude of the observer in degrees
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h : float
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Altitude above mean sea level of the observer in meters
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Other Observatory() attributes
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Methods
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-------
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setReducedHeight(p, rho)
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Sets the reduced atmospheric height.
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planeParallel(n1, n2, zenith)
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Returns the atmospheric refraction assuming a plane parallel
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atmosphere
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cassini(n1, n2, zenith, reduced_height=None)
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Returns the atmospheric refraction asusming Cassini's homegeneous
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atmosphere model
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tan5(n1, n2, zenith, reduced_height=None)
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Returns the atmospheric refraction following Oriani's theorem,
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including a fifth order term in the expansion.
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oriani(n1, n2, zenith, reduced_height=None)
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Returns the atmospheric refraction following the classic version
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of Oriani's theorem.
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corbard(n1, n2, zenith, reduced_height=None)
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Returns the atmospheric refraction following the method described
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in Corbard et al., 2019.
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refractionIntegral(n1, n2, z0, R0=None, useStandardAtmosphere=True, heightData=None, rhoData=None)
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Returns the atmospheric refraction following the refraction integral.
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psi(x)
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Submethod for corbard()
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set_rc()
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Sets the local radius of curvature of the Earth at the observer.
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set_H(p, rho)
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Sets the reduced height for the conditions at the observer, assuming
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an adiabatic atmosphere.
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H_isotherm(T)
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Sets the reduced height for the conditions at the observer, assuming
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an isothermal atmosphere.
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cassiniError(z, l1, l2, T, p, RH, xc, dl1=0, dl2=0, dT=0.2, dP=20,
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dRH=0.02, dCO2=20, dz=0, lat=None, h=None)
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Returns the uncertainty in the atmospheric refraction, assuming
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Cassini's homegeneous atmosphere model and the given parameters.
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Other Observatory() methods
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'''
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def __init__(self, lat, h):
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Observatory.__init__(self)
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self.h = h # height above sea level in meters
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self.lat = lat # Latitude of the observer in degrees
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self.rc = self.set_rc() * 1000 # Radius of curvature of the earth at the observer
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# self.H2 = self.H_isotherm(T) # Reduced height of isotherm atmosphere
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def cassini(self, n1, zenith, reduced_height=None):
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'''
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Refraction of spherical atmosphere, derived from geometry using Sine law
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Parameters
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----------
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n1 : float
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Refractive index at the wavelength of interest
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zenith : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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reduced_height : float (optional)
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Reduced height of the atmosphere. Default value is calculated
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from the object attributes.
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Returns
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-------
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R : float
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Atmospheric refraction in degrees
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'''
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if reduced_height:
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H = reduced_height
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else:
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H = self.H
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_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) )
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return np.degrees(_R1)
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def cassiniError(self, n, dn, zenith, dz=0, reduced_height=None):
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'''
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Calculates the uncertainty in the refraction angle in degrees.
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See Observatory().n_tph() and Observatory().dn_tph() on how to
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calculate the refractive index and it's uncertainty.
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Parameters
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----------
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n : float
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Refractive index of air
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dn : float
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Uncertainty in the refractive index of air
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zenith : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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dz : float (optional)
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Uncertainty in the zenith angle, in degrees.
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reduced_height : float (optional)
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Reduced height of the atmosphere. Default value is calculated
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from the object attributes.
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Returns
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-------
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dR : float
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Uncertainty in the atmospheric refraction in degrees.
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'''
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if reduced_height:
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_H = reduced_height
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else:
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_H = self.H
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_zenith = np.deg2rad(zenith)
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_dz = np.deg2rad(dz)
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_R = self.rc
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_dRdn = (_R * np.sin(_zenith)) / ((_R + _H) * np.sqrt(1 - ((n**2 * _R**2 * np.sin(_zenith)**2)/(_R + _H)**2)))
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_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) )
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_dR = np.sqrt( (_dRdn * dn)**2 + (_dRdz * _dz)**2)
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return np.degrees(_dR)
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def corbard(self, n1, zenith, reduced_height=None):
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'''
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Corbard et al., 2019, mention an additional formula based on the error function.
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Oriani's theorem can be derived from this equation by 'keeping only the three first
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terms of its asymptotic expansion'.
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Parameters
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----------
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n1 : float
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Refractive index at the wavelength of interest
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zenith : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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reduced_height : float (optional)
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Reduced height of the atmosphere. Default value is calculated
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from the object attributes.
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Returns
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-------
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R : float
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Atmospheric refraction in degrees
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'''
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if reduced_height:
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H = reduced_height
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else:
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H = self.H
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_a = n1 - 1
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_b = H / self.rc
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_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) )
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return np.degrees(_R)
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def gammaCoefficients(self, n0, reduced_height=None):
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'''
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Submethod for matharExponential(). Calculates the gamma
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coefficients.
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Parameters
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----------
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n0 : float
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Refractive index at the wavelength of interest at the observer
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reduced_height : float (optional)
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Reduced height of the atmosphere. Default value is calculated
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from the object attributes.
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Returns
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-------
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g : array
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List of values, corresponding to the gamma coefficients.
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g[0] corresponds to g_1, g[1] to g_3, g[2] to g_5, and so on.
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'''
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if reduced_height:
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H = reduced_height
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else:
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H = self.H
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_chi0 = n0**2 - 1
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_Khat = H / self.rc
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_g1 = (1 - 5/24 * _Khat * _chi0**2 - 3/4 * _chi0 - 3/8 * _Khat**2 * _chi0 \
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- _Khat - 6 * _Khat**3 + 5/8 * _chi0**2 + 63/128 * _chi0**4 + 3/8 \
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* _Khat * _chi0 + 35/256 * _Khat * _chi0**3 + 2 * _Khat**2 + 5/36 \
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* _Khat**2 * _chi0**2 - 35/64 * _chi0**3 + 9/16 * _Khat**3 * _chi0 \
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- 120 * _Khat**5 + 24 * _Khat**4 - 231 / 512 * _chi0**5 - 63 / 640 \
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* _Khat * _chi0**4 - 35 / 512 * _Khat**2 * _chi0**3 - 5 / 36 * _Khat**3\
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* _chi0**2 - 9 / 8 * _Khat**4 * _chi0 )
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_g3 = (35/96 * _chi0**3 - 5/12 * _chi0**2 + 333 * _Khat**4 * _chi0 - 2400 \
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* _Khat**5 + 3605/4608 * _Khat**2 * _chi0**3 + 63/256 * _Khat * _chi0**4\
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- 819/16 * _Khat**3 * _chi0 - 35/18 * _Khat**2 * _chi0**2 - 35/96 \
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* _Khat * _chi0**3 + 69/8 * _Khat**2 * _chi0 + 5/8 * _Khat * _chi0**2 \
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- 3/2 * _Khat * _chi0 + 1355/216 * _Khat**3 * _chi0**2 + 10 * _Khat**2 \
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+ 336 * _Khat**4 - 2 * _Khat + _chi0/2 - 21/64 * _chi0**4 - 54 * _Khat**3 \
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+ 77/256 * _chi0**5 ) * 1/2
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_g5 = (-96 * _Khat**3 + 8 * _Khat**2 - 7/24 * _chi0**3 + 65/4 * _Khat**2 \
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* _chi0**2 + 35/48 * _Khat * _chi0**3 + 30 * _Khat**2 * _chi0 - 5/3 \
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* _Khat * _chi0**2 - 3 * _Khat * _chi0 + 20091/8 * _Khat**4 * _chi0 \
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- 32605/216 * _Khat**3 * _chi0**2 - 35/9 * _Khat**2 * _chi0**3 + 984 \
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* _Khat**4 + _chi0**2/3 - 10200 * _Khat**5 - 77/320 * _chi0**5 + 21/80 \
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* _chi0**4 - 1089/4 * _Khat**3 * _chi0 - 7/16 * _Khat * _chi0**4 ) * 3/8
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_g7 = (-17520 * _Khat**5 - 9/40 * _chi0**4 - 8755/12 * _Khat**3 * _chi0**2 \
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+ 1056 * _Khat**4 + _chi0**3/4 + 23517/4 * _Khat**4 * _chi0 - 11/3 \
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* _Khat * _chi0**2 + 21 * _Khat**2 * _chi0 - 7/4 * _Khat * _chi0**3 \
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+ 505/9 * _Khat**2 * _chi0**2 - 801/2 * _Khat**3 * _chi0 + 33/160 \
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* _chi0**5 - 48 * _Khat**3 + 63/80 * _Khat * _chi0**4 + 2345/96 \
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* _Khat**2 * _chi0**3) * 5/16
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_g9 = (_chi0**4 / 5 + 384 * _Khat**4 - 13440 * _Khat**5 + 5562 * _Khat**4 \
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* _chi0 + 3125/36 * _Khat**2 * _chi0**3 - 11/60 * _chi0**5 - 25/6 \
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* _Khat * _chi0**3 - 26950/27 * _Khat**3 * _chi0**2 + 340/9 * _Khat**2 \
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* _chi0**2 - 180 * _Khat**3 * _chi0 - 9/5 * _Khat * _chi0**4) * 35/128
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g = np.array([_g1, _g3, _g5, _g7, _g9])
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g*= n0*_chi0/2
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return g
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def H_isotherm(self, T):
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'''
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Calculates the reduced height of the atmosphere,
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assuming an isotherm distribution.
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Parameters
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----------
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T : float
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Atmospheric temperature in Kelvin
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Returns
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-------
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H : float
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Reduced height of the atmosphere in meters
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'''
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_kb = 1.38064852e-23 # kg m2 s-2 K-1
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_m = 4.809651698e-26 # kg (weight per molecule of air, assuming dry air)
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_phi = np.deg2rad(self.lat)
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_c1 = 5.2790414e-3
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_c2 = 2.32718e-5
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_c3 = 1.262e-7
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_c4 = 7e-10
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_g0 = 9.780327 # m/s^2
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_g0_local = _g0 * (1 + _c1 * np.sin(_phi)**2 + _c2 * np.sin(_phi)**4 + _c3 * np.sin(_phi)**6 + _c4 * np.sin(_phi)**8)
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return (_kb*T) / (_m * _g0_local)
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def matharExponential(self, n1, zenith, reduced_height=None):
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'''
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Model on atmospheric refraction described by Richard Mathar in
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"A Barometric Exponential Model of the Atmosphere’s Refractive Index: ZenithAngles and Second Order Aberration in the Entrance Pupil" (2020)
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ArXiv id: 2004.11808 (https://arxiv.org/abs/2004.11808)
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The code in this function is adapted from eqs 3, 16, 17 and 18 in this
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work. It is an adaptation of the RefrExpo.cxx C++ file written by Mathar.
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This model assumes the refractive index, n, is related to the susceptibility,
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chi, following: n(r) = sqrt(1+chi(r)) (eq. 9).
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R. Mathar then assumes that the susceptibility is proportional to the
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air density and reduced height, following eq. 10:
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chi(r) = chi0 * exp(-(r-rho)/K).
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The refraction is described using a Taylor expansion; R in terms of
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the odd powers of the tangent of the observed zenith angle (eq. 3), not
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unlike Oriani's theorem. This requires coefficients g_i.
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Parameters
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----------
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n1 : float
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Refractive index at the wavelength of interest at the observer
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zenith : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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reduced_height : float (optional)
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Reduced height of the atmosphere. Default value is calculated
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from the object attributes.
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Returns
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-------
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R : float
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Atmospheric refraction in degrees
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'''
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if reduced_height:
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_H = reduced_height
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else:
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_H = self.H
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_zenith = np.deg2rad(zenith)
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_rc = self.rc # rho
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# Atmospheric refraction in radians
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_gamma = self.gammaCoefficients(n1, reduced_height=_H)
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_R = sum([_gamma[i]*np.tan(_zenith)**(2*i+1) for i in range(len(_gamma))])
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return np.degrees(_R)
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def oriani(self, n1, zenith, reduced_height=None):
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'''
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Classic refraction formula with two tan(z) terms. This does not
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assume any information about the structure of the atmosphere.
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Found with Oriani's theorem.
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Parameters
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----------
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n1 : float
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Refractive index at the wavelength of interest
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zenith : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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reduced_height : float (optional)
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Reduced height of the atmosphere. Default value is calculated
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from the object attributes.
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Returns
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-------
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R : float
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Atmospheric refraction in degrees
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'''
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if reduced_height:
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H = reduced_height
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else:
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H = self.H
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_a = n1 - 1
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_b = H / self.rc
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_R = _a * (1 - _b) * np.tan( np.deg2rad(zenith) ) - _a * (_b - _a/2) * np.tan( np.deg2rad(zenith) )**3
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return np.degrees(_R)
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def planeParallel(self, n1, zenith):
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'''
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Refraction based on a flat atmosphere, easily derived from Snell's law.
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Parameters
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----------
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n1 : float
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Refractive index at the wavelength of interest
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zenith : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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Returns
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-------
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R : float
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Atmospheric refraction in degrees
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'''
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_R = np.arcsin(n * np.sin( np.deg2rad(zenith) )) - np.deg2rad(zenith)
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return np.degrees(_R)
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def psi(self, x):
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'''
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Submethod for corbard(). Described explicitly in Corbard et al.,2019.
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Parameters
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----------
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x : float
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Input of the function
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Returns
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-------
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y : float
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Output of the function
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'''
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_f = lambda a: np.exp(-a**2)
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_int = integrate.quad(_f, x, np.inf)
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return np.exp(x**2) * _int[0]
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def refractionIntegral(self, n1, z0, R0=None, useStandardAtmosphere=True,
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heightData=None, rhoData=None):
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'''
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Using data that gives temperature, pressure and density as a
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function of height, the gladstone-dale relation is invoked to
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calculate the path of a light ray along the atmosphere.
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This is allows us to also include atmospheric activity in
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different atmosphere layers.
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Recommended use with the US Standard Atmosphere from 1976.
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USSA1976 data generated from code supplied by http://www.pdas.com/atmos.html
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The method described in eq. 3 in Auer & Standish (2000) is used for the
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calculation.
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Parameters
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----------
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n1 : float
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Refractive index at the wavelength of interest
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z0 : float
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Angle of observation in degrees.
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zenith = 0 means the observation is directly overhead.
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R0 : float (optional)
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Local radius of curvature of the Earth in km.
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useStandardAtmosphere : bool
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Flag to use the included USSA1976 data
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heightData : 1D Array
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If "useStandardAtmosphere" is False, the function will ask
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for alternative atmospheric data to integrate over. This
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gives the altitudinal points
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rhoData : 1D Array
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If useStandardAtmosphere is False, the function will ask
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for alternative atmospheric data to integrate over. This
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gives the atmospheric density at the points defined by
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"heightData".
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Returns
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-------
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R : float
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Atmospheric refraction in degrees
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'''
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if not R0:
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R0 = self.rc
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if useStandardAtmosphere:
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_datafileLocation = os.path.join(os.path.dirname(__file__), 'data/1976USSA.txt')
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_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt(_datafileLocation, unpack=True, skip_header=6)
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else:
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_Dens = rhoData
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_h = heightData
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z0 = np.deg2rad(z0)
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_Dens = np.asarray(_Dens)
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_h = np.asarray(_h)
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_R = R0 + _h * 1000
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# Gladstone-Dale relation between refractive index and density
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_n_h = 1 + (n0 - 1) / _Dens[0] * _Dens
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# Calculation of the zenith angle for given height and refractive index, from the refractive invariant
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_z = np.arcsin( (n0 * R0) / (_n_h * _R) * np.sin(z0) )
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# 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 setReducedHeight(self, p, rho):
|
||
'''
|
||
Sets the reduced atmospheric height.
|
||
|
||
Parameters
|
||
----------
|
||
p : float
|
||
Atmospheric pressure in Pascal
|
||
rho : float
|
||
Density of the atmospheric air at the observer
|
||
'''
|
||
self.H = self.set_H(p, rho) # Reduced height of the atmosphere assuming ideal gas law
|
||
|
||
|
||
def set_H(self, p, rho):
|
||
'''
|
||
Calculates the reduced height of the atmosphere, assuming ideal gas law.
|
||
|
||
Parameters
|
||
----------
|
||
p : float
|
||
Atmospheric pressure in Pascal
|
||
rho : float
|
||
Density of the atmospheric air at the observer
|
||
|
||
Returns
|
||
-------
|
||
H : float
|
||
Reduced height of the atmosphere in meters
|
||
'''
|
||
_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 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.
|
||
|
||
Returns
|
||
-------
|
||
rc : float
|
||
Local radius of curvature of the Earth in kilometers.
|
||
'''
|
||
_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 tan5(self, n1, zenith, reduced_height=None):
|
||
'''
|
||
Same as oriani's theorem, but including a higher order term.
|
||
Found in Corbard et al., 2019.
|
||
|
||
Parameters
|
||
----------
|
||
n1 : float
|
||
Refractive index at the wavelength of interest
|
||
zenith : float
|
||
Angle of observation in degrees.
|
||
zenith = 0 means the observation is directly overhead.
|
||
reduced_height : float (optional)
|
||
Reduced height of the atmosphere. Default value is calculated
|
||
from the object attributes.
|
||
|
||
Returns
|
||
-------
|
||
R : float
|
||
Atmospheric refraction in degrees
|
||
'''
|
||
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)
|
||
|