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SUBROUTINE slRFCQ ( TDK, PMB, RH, WL, REFA, REFB )
*+
* - - - - - - -
* R F C Q
* - - - - - - -
*
* Determine the constants A and B in the atmospheric refraction
* model dZ = A tan Z + B tan**3 Z. This is a fast alternative
* to the slRFCO routine - see notes.
*
* Z is the "observed" zenith distance (i.e. affected by refraction)
* and dZ is what to add to Z to give the "topocentric" (i.e. in vacuo)
* zenith distance.
*
* Given:
* TDK d ambient temperature at the observer (K)
* PMB d pressure at the observer (millibar)
* RH d relative humidity at the observer (range 0-1)
* WL d effective wavelength of the source (micrometre)
*
* Returned:
* REFA d tan Z coefficient (radian)
* REFB d tan**3 Z coefficient (radian)
*
* The radio refraction is chosen by specifying WL > 100 micrometres.
*
* Notes:
*
* 1 The model is an approximation, for moderate zenith distances,
* to the predictions of the slRFRO routine. The approximation
* is maintained across a range of conditions, and applies to
* both optical/IR and radio.
*
* 2 The algorithm is a fast alternative to the slRFCO routine.
* The latter calls the slRFRO routine itself: this involves
* integrations through a model atmosphere, and is costly in
* processor time. However, the model which is produced is precisely
* correct for two zenith distance (45 degrees and about 76 degrees)
* and at other zenith distances is limited in accuracy only by the
* A tan Z + B tan**3 Z formulation itself. The present routine
* is not as accurate, though it satisfies most practical
* requirements.
*
* 3 The model omits the effects of (i) height above sea level (apart
* from the reduced pressure itself), (ii) latitude (i.e. the
* flattening of the Earth) and (iii) variations in tropospheric
* lapse rate.
*
* The model was tested using the following range of conditions:
*
* lapse rates 0.0055, 0.0065, 0.0075 K/metre
* latitudes 0, 25, 50, 75 degrees
* heights 0, 2500, 5000 metres ASL
* pressures mean for height -10% to +5% in steps of 5%
* temperatures -10 deg to +20 deg with respect to 280 deg at SL
* relative humidity 0, 0.5, 1
* wavelengths 0.4, 0.6, ... 2 micron, + radio
* zenith distances 15, 45, 75 degrees
*
* The accuracy with respect to direct use of the slRFRO routine
* was as follows:
*
* worst RMS
*
* optical/IR 62 mas 8 mas
* radio 319 mas 49 mas
*
* For this particular set of conditions:
*
* lapse rate 0.0065 K/metre
* latitude 50 degrees
* sea level
* pressure 1005 mb
* temperature 280.15 K
* humidity 80%
* wavelength 5740 Angstroms
*
* the results were as follows:
*
* ZD slRFRO slRFCQ Saastamoinen
*
* 10 10.27 10.27 10.27
* 20 21.19 21.20 21.19
* 30 33.61 33.61 33.60
* 40 48.82 48.83 48.81
* 45 58.16 58.18 58.16
* 50 69.28 69.30 69.27
* 55 82.97 82.99 82.95
* 60 100.51 100.54 100.50
* 65 124.23 124.26 124.20
* 70 158.63 158.68 158.61
* 72 177.32 177.37 177.31
* 74 200.35 200.38 200.32
* 76 229.45 229.43 229.42
* 78 267.44 267.29 267.41
* 80 319.13 318.55 319.10
*
* deg arcsec arcsec arcsec
*
* The values for Saastamoinen's formula (which includes terms
* up to tan^5) are taken from Hohenkerk and Sinclair (1985).
*
* The results from the much slower but more accurate slRFCO
* routine have not been included in the tabulation as they are
* identical to those in the slRFRO column to the 0.01 arcsec
* resolution used.
*
* 4 Outlandish input parameters are silently limited to mathematically
* safe values. Zero pressure is permissible, and causes zeroes to
* be returned.
*
* 5 The algorithm draws on several sources, as follows:
*
* a) The formula for the saturation vapour pressure of water as
* a function of temperature and temperature is taken from
* expressions A4.5-A4.7 of Gill (1982).
*
* b) The formula for the water vapour pressure, given the
* saturation pressure and the relative humidity, is from
* Crane (1976), expression 2.5.5.
*
* c) The refractivity of air is a function of temperature,
* total pressure, water-vapour pressure and, in the case
* of optical/IR but not radio, wavelength. The formulae
* for the two cases are developed from Hohenkerk & Sinclair
* (1985) and Rueger (2002).
*
* The above three items are as used in the slRFRO routine.
*
* d) The formula for beta, the ratio of the scale height of the
* atmosphere to the geocentric distance of the observer, is
* an adaption of expression 9 from Stone (1996). The
* adaptations, arrived at empirically, consist of (i) a
* small adjustment to the coefficient and (ii) a humidity
* term for the radio case only.
*
* e) The formulae for the refraction constants as a function of
* n-1 and beta are from Green (1987), expression 4.31.
*
* References:
*
* Crane, R.K., Meeks, M.L. (ed), "Refraction Effects in the Neutral
* Atmosphere", Methods of Experimental Physics: Astrophysics 12B,
* Academic Press, 1976.
*
* Gill, Adrian E., "Atmosphere-Ocean Dynamics", Academic Press, 1982.
*
* Green, R.M., "Spherical Astronomy", Cambridge University Press, 1987.
*
* Hohenkerk, C.Y., & Sinclair, A.T., NAO Technical Note No. 63, 1985.
*
* Rueger, J.M., "Refractive Index Formulae for Electronic Distance
* Measurement with Radio and Millimetre Waves", in Unisurv Report
* S-68, School of Surveying and Spatial Information Systems,
* University of New South Wales, Sydney, Australia, 2002.
*
* Stone, Ronald C., P.A.S.P. 108 1051-1058, 1996.
*
* Last revision: 2 December 2005
*
* Copyright P.T.Wallace. All rights reserved.
*
* License:
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program (see SLA_CONDITIONS); if not, write to the
* Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
* Boston, MA 02110-1301 USA
*
* Copyright (C) 1995 Association of Universities for Research in Astronomy Inc.
*-
IMPLICIT NONE
DOUBLE PRECISION TDK,PMB,RH,WL,REFA,REFB
LOGICAL OPTIC
DOUBLE PRECISION T,P,R,W,TDC,PS,PW,WLSQ,GAMMA,BETA
* Decide whether optical/IR or radio case: switch at 100 microns.
OPTIC = WL.LE.100D0
* Restrict parameters to safe values.
T = MIN(MAX(TDK,100D0),500D0)
P = MIN(MAX(PMB,0D0),10000D0)
R = MIN(MAX(RH,0D0),1D0)
W = MIN(MAX(WL,0.1D0),1D6)
* Water vapour pressure at the observer.
IF (P.GT.0D0) THEN
TDC = T-273.15D0
PS = 10D0**((0.7859D0+0.03477D0*TDC)/(1D0+0.00412D0*TDC))*
: (1D0+P*(4.5D-6+6D-10*TDC*TDC))
PW = R*PS/(1D0-(1D0-R)*PS/P)
ELSE
PW = 0D0
END IF
* Refractive index minus 1 at the observer.
IF (OPTIC) THEN
WLSQ = W*W
GAMMA = ((77.53484D-6+(4.39108D-7+3.666D-9/WLSQ)/WLSQ)*P
: -11.2684D-6*PW)/T
ELSE
GAMMA = (77.6890D-6*P-(6.3938D-6-0.375463D0/T)*PW)/T
END IF
* Formula for beta adapted from Stone, with empirical adjustments.
BETA=4.4474D-6*T
IF (.NOT.OPTIC) BETA=BETA-0.0074D0*PW*BETA
* Refraction constants from Green.
REFA = GAMMA*(1D0-BETA)
REFB = -GAMMA*(BETA-GAMMA/2D0)
END
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