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# refco

erfa.refco(phpa, tc, rh, wl)

  Determine the constants A and B in the atmospheric refraction model dZ = A tan Z + B tan^3 Z.

  Parameters
    phpa double array
    tc double array
    rh double array
    wl double array
  Returns
    refa double array
    refb double array

  Notes

  Wraps ERFA function `eraRefco`. The ERFA documentation is:

      - - - - - - - - -
       e r a R e f c o
      - - - - - - - - -

      Determine the constants A and B in the atmospheric refraction model
      dZ = A tan Z + B tan^3 Z.

      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:
        phpa   double    pressure at the observer (hPa = millibar)
        tc     double    ambient temperature at the observer (deg C)
        rh     double    relative humidity at the observer (range 0-1)
        wl     double    wavelength (micrometers)

      Returned:
        refa   double         tan Z coefficient (radians)
        refb   double         tan^3 Z coefficient (radians)

      Notes:

      1) The model balances speed and accuracy to give good results in
         applications where performance at low altitudes is not paramount.
         Performance is maintained across a range of conditions, and
         applies to both optical/IR and radio.

      2) 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), (iii) variations in tropospheric lapse
         rate and (iv) dispersive effects in the radio.

         The model was tested using the following range of conditions:

           lapse rates 0.0055, 0.0065, 0.0075 deg/meter
           latitudes 0, 25, 50, 75 degrees
           heights 0, 2500, 5000 meters 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 raytracing through a model
         atmosphere 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/meter
           latitude 50 degrees
           sea level
           pressure 1005 mb
           temperature 280.15 K
           humidity 80%
           wavelength 5740 Angstroms

         the results were as follows:

           ZD       raytrace     eraRefco   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).

      3) A wl value in the range 0-100 selects the optical/IR case and is
         wavelength in micrometers.  Any value outside this range selects
         the radio case.

      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
            Equations (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), Equation (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, wavelength.  The formulae for the two cases are
            developed from Hohenkerk & Sinclair (1985) and Rueger (2002).
            The IAG (1999) optical refractivity for dry air is used.

         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 Equation (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), Equation (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.

         IAG Resolutions adopted at the XXIIth General Assembly in
         Birmingham, 1999, Resolution 3.

         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.

      This revision:   2021 February 24

      Copyright (C) 2013-2021, NumFOCUS Foundation.
      Derived, with permission, from the SOFA library.  See notes at end of file.