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AtmosphericModel/AstroAtmosphere/dispersionModels.py
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#!/usr/bin/env Python3
import numpy as np
from scipy import integrate
import os
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):
'''
Class that sets the observer location.
These location attributes are then used to calculate the
appropriate atmospheric dispersion.
See also:
Observatory()
refraction()
...
Attributes
-----------
lat : float
Latitude of the observer in degrees
h : float
Altitude above mean sea level of the observer in meters
Other Observatory() attributes
Methods
-------
setReducedHeight(p, rho)
Sets the reduced atmospheric height.
planeParallel(n1, n2, zenith)
Returns the atmospheric dispersion assuming a plane parallel
atmosphere
cassini(n1, n2, zenith, reduced_height=None)
Returns the atmospheric dispersion asusming Cassini's homegeneous
atmosphere model
tan5(n1, n2, zenith, reduced_height=None)
Returns the atmospheric dispersion following Oriani's theorem,
including a fifth order term in the expansion.
oriani(n1, n2, zenith, reduced_height=None)
Returns the atmospheric dispersion following the classic version
of Oriani's theorem.
corbard(n1, n2, zenith, reduced_height=None)
Returns the atmospheric dispersion following the method described
in Corbard et al., 2019.
refractionIntegral(n1, n2, z0, R0=None, useStandardAtmosphere=True, heightData=None, rhoData=None)
Returns the atmospheric dispersion following the refraction integral.
psi(x)
Submethod for corbard()
set_rc()
Sets the local radius of curvature of the Earth at the observer.
set_H(p, rho)
Sets the reduced height for the conditions at the observer, assuming
an adiabatic atmosphere.
H_isotherm(T)
Sets the reduced height for the conditions at the observer, assuming
an isothermal atmosphere.
cassiniError(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)
Returns the uncertainty in the atmospheric dispersion, assuming
Cassini's homegeneous atmosphere model and the given parameters.
Other Observatory() methods
'''
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):
'''
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 planeParallel(self, n1, n2, zenith):
'''
Refraction based on a flat atmosphere, easily derived from Snell's law.
Parameters
----------
n1 : float
Refractive index at the shortest wavelength of interest
n2 : float
Refractive index at the longest wavelength of interest
zenith : float
Angle of observation in degrees.
zenith = 0 means the observation is directly overhead.
Returns
-------
dR : float
Atmospheric dispersion in degrees
'''
_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
Parameters
----------
n1 : float
Refractive index at the shortest wavelength of interest
n2 : float
Refractive index at the longest 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
-------
dR : float
Atmospheric dispersion in degrees
'''
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.
Parameters
----------
n1 : float
Refractive index at the shortest wavelength of interest
n2 : float
Refractive index at the longest 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
-------
dR : float
Atmospheric dispersion in degrees
'''
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.
Parameters
----------
n1 : float
Refractive index at the shortest wavelength of interest
n2 : float
Refractive index at the longest 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
-------
dR : float
Atmospheric dispersion in degrees
'''
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'.
Parameters
----------
n1 : float
Refractive index at the shortest wavelength of interest
n2 : float
Refractive index at the longest 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
-------
dR : float
Atmospheric dispersion in degrees
'''
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.
Parameters
----------
n1 : float
Refractive index at the shortest wavelength of interest
n2 : float
Refractive index at the longest wavelength of interest
z0 : float
Angle of observation in degrees.
zenith = 0 means the observation is directly overhead.
R0 : float (optional)
Local radius of curvature of the Earth in km.
useStandardAtmosphere : bool
Flag to use the included USSA1976 data
heightData : 1D Array
If "useStandardAtmosphere" is False, the function will ask
for alternative atmospheric data to integrate over. This
gives the altitudinal points
rhoData : 1D Array
If useStandardAtmosphere is False, the function will ask
for alternative atmospheric data to integrate over. This
gives the atmospheric density at the points defined by
"heightData".
Returns
-------
dR : float
Atmospheric dispersion in degrees
'''
if not R0:
R0 = self.rc
if useStandardAtmosphere:
_datafileLocation = os.path.join(os.path.dirname(__file__), 'data/1976USSA.txt')
_h, _, _, _, _Temp, _Press, _Dens,_,_,_ = np.genfromtxt(_datafileLocation, 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):
'''
Submethod for corbard(). Described explicitly in Corbard et al.,2019.
Parameters
----------
x : float
Input of the function
Returns
-------
y : float
Output of the function
'''
_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.
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 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 H_isotherm(self, T):
'''
Calculates the reduced height of the atmosphere,
assuming an isotherm distribution.
Parameters
----------
T : float
Atmospheric temperature in Kelvin
Returns
-------
H : float
Reduced height of the atmosphere in meters
'''
_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, from my results this should give a small or even
negligible difference.
Parameters
----------
z : float
Angle of observation in degrees.
zenith = 0 means the observation is directly overhead.
l1 : float
Wavelength of light in microns, at the shortest wavelength
of interest
l2 : float
Wavelength of light in microns, at the longest wavelength
of interest
T : float
Atmospheric temperature in Kelvin
p : float
Atmospheric pressure in Pascal
RH : float
Atmospheric relative humidity (0<=RH<=1)
xc : float
CO2 density in parts per million
dl1 : float
Uncertainty in the wavelength in microns, at the shortest
wavelength of interest
dl2 : float
Uncertainty in the wavelength in microns, at the longest
wavelength of interest
dT : float (default: 0.2)
Uncertainty in the atmospheric temperature in Kelvin
dP : float (default: 20)
Uncertainty in the atmospheric pressure in Pascal
dRH : float (default: 0.02)
Uncertainty in the atmospheric relative humidity
dCO2: float (default: 20)
Uncertainty in the atmospheric CO2 density in parts per million
dz : float (default: 0)
Uncertainty in the zenith angle, in degrees.
lat : float (default: None)
If entered, overrides the latitude attribute of the class.
h : float (default=None)
If entered, overrides the altitude attribute of the class.
Returns
-------
dRR : float
Uncertainty in the atmospheric dispersion in degrees.
'''
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)
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