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      SUBROUTINE slREFV (VU, REFA, REFB, VR)

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      R E F V
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  Adjust an unrefracted Cartesian vector to include the effect of
  atmospheric refraction, using the simple A tan Z + B tan**3 Z
  model.

  Given:
    VU    dp    unrefracted position of the source (Az/El 3-vector)
    REFA  dp    tan Z coefficient (radian)
    REFB  dp    tan**3 Z coefficient (radian)

  Returned:
    VR    dp    refracted position of the source (Az/El 3-vector)

  Notes:

  1  This routine applies the adjustment for refraction in the
     opposite sense to the usual one - it takes an unrefracted
     (in vacuo) position and produces an observed (refracted)
     position, whereas the A tan Z + B tan**3 Z model strictly
     applies to the case where an observed position is to have the
     refraction removed.  The unrefracted to refracted case is
     harder, and requires an inverted form of the text-book
     refraction models;  the algorithm used here is equivalent to
     one iteration of the Newton-Raphson method applied to the above
     formula.

  2  Though optimized for speed rather than precision, the present
     routine achieves consistency with the refracted-to-unrefracted
     A tan Z + B tan**3 Z model at better than 1 micro-arcsecond within
     30 degrees of the zenith and remains within 1 milliarcsecond to
     beyond ZD 70 degrees.  The inherent accuracy of the model is, of
     course, far worse than this - see the documentation for slRFCO
     for more information.

  3  At low elevations (below about 3 degrees) the refraction
     correction is held back to prevent arithmetic problems and
     wildly wrong results.  Over a wide range of observer heights
     and corresponding temperatures and pressures, the following
     levels of accuracy (arcsec) are achieved, relative to numerical
     integration through a model atmosphere:

              ZD    error

              80      0.4
              81      0.8
              82      1.6
              83      3
              84      7
              85     17
              86     45
              87    150
              88    340
              89    620
              90   1100
              91   1900         } relevant only to
              92   3200         } high-elevation sites

  4  See also the routine slREFZ, which performs the adjustment to
     the zenith distance rather than in Cartesian Az/El coordinates.
     The present routine is faster than slREFZ and, except very low down,
     is equally accurate for all practical purposes.  However, beyond
     about ZD 84 degrees slREFZ should be used, and for the utmost
     accuracy iterative use of slRFRO should be considered.

  P.T.Wallace   Starlink   26 December 1994

  Copyright (C) 1995 Rutherford Appleton Laboratory
  Copyright (C) 1995 Association of Universities for Research in Astronomy Inc.

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