- Introduction
The aim of here is to produce accurate formulae for determining refraction
as a function of local atmospheric conditions at the 4.1km
altitude of JCMT.
It was found during this study that the approximation of the type given by
Allen, for instance, of the form
R = A * tan(z) + B * tan**3(z)
(1)
is most suitable, and sufficiently accurate to large zenith distances. The
challenge was then to express A and B in terms of local
temperature, T, pressure,
p, and humidity, h, so that real-time calculations of refraction
can be made, and thus keep the JCMT pointing accuracy to acceptable levels.
The approach is to
- develop a model of the atmosphere for the calculation of refraction
under varying conditions
- test its sensitivity to various profiles of T,p and h
- compare the results with previous determinations
of refraction, and
- proceed to determine the values of A and B in (1) above so as to
reduce the calculation from the lengthy integrations through the
atmosphere required from the model to a single formula with real-time
application.
- The atmospheric model
The method of Allen(AQ3,1973,p124) is used to calculate refractive index.
The Earth's atmosphere is considered to be radially symmetric.
The basic variations of temperature, density and pressure with altitude are
taken to be the American Standard Atmosphere (ASA) as given by Allen(p121).
The local (Hawaiian) atmosphere has been studied by Takahasi (U.of H., Hilo).
His data reveals the LHA to be 14 degs warmer than the ASA at all altitudes
up to 15 km. Seasonal variations of 1-2 degrees are seen on top of this.
The model features the ASA temperature profile with the facility for adding
a constant at all altitudes, and the ASA pressure profile with the facility
of applying a constant multiplicative factor at all altitudes.
From Takahashi's results T=4degC is adopted as the reference temperature
for the summit of Mauna Kea in the formulations below. Local experience
prompted the adoption of 20% as the reference humidity level.
Takahashi's data were also examined for data on the local humidity structure.
A feature of his data is that the air is essentially quite dry above 11km.
Otherwise many profiles were seen, but it was found during the analysis that
refraction in the radially symmetric case, as in the plane case, depends
essentially only upon the refractive indices of the top and bottom layers,
and thus happily lends itself to formulation based on local conditions only.
The mathematical model for calculating the refraction maintains a radially
symmetric atmosphere and resolves the atmosphere according to the formula :
height-resolution = [ height * cos (Z) ] / [10 + Z**2/300 ]
where Z is the zenith distance in degrees
The height-resolution increases with height (altitude) and decreases with
increasing zenith distance. Resolutions at low altitudes are much in excess
of the 1km resolution of the ASA, but this was felt necessary in order to
achieve the highest possible accuracy in the derived refraction ( 0.1" ).
Fig.1 shows one interface between two levels in the atmosphere at which
refraction is assumed to take place. The refraction is calculated in the usual
way once the refractive indices of the two layers have been determined, and
the process repeated at the next level. The total refraction is then the
sum of these components.
It is clear at this stage that except at zenith distances close to
90o
refraction depends essentially on the humidity at the observer's locale
and not upon the atmospheric profile of the humidity.
- Checks on the results
A check on the programming was made by calculating the optical refractions as
given by AQ3,p125. Prevailing conditions are quoted as 760mmHg pressure,10C
temperature and 4mmHg water vapour pressure, which is equivalent to a relative
humidity of 88%. A model using a uniform 90% humidity to 11km was used.
The results were essentially as given by Allen, with differences of less than
1 arc second for elevations above 20 degrees, increasing to 2" at 3 degrees.
A comparison with the AQ3 results is given in
Table 1.
( Note: The formulation differs from that quoted by Allen for the contribution
due to the water : the term quoted, supposedly dependent upon the water vapour
pressure, f, is quoted without an f-dependence. The term quoted by Allen was
multiplied by f before being used in these calculations.)
A comparison of the millimeter refractions with those of Ulich(1981) is shown
in Table 2.
It is seen that except under extreme atmospheric conditions the
agreement with Ulich is very good.
With confidence that the model is yielding accurate refractions we proceed to
search for a simple formulation that will reproduce the results of the model
integrations.
- A simple formula - the optical (0.55micron) case
Results in the optical (0.55micron wavelength) showed that refraction could
indeed be expressed as in (1). The value of A in (1) is essentially the
refraction at 45deg elevation, and its variation with each of the parameters
T,p, and h was determined with the other two parameters fixed at nominal
values. It is found that for a wavelength of 0.55 microns and an observatory
height of 4.1 km : then :
a) allowing T alone to vary ( i.e. p = 0 %, h = 20% ) gives
A = 35.893 - 0.138(T-4) + 0.000432(T-4)**2
B = -0.0359 + 0.000127(T-4) + C(Z)
where T is the temperature in deg C
Z is the Zenith distance
and C(Z) = 0 for Z < 80deg
= 0.0002(Z-80)(Z-79) for 80deg < Z < 85deg
b) allowing p alone to vary (i.e. T = 4degC , h = 20%) gives
A = 35.893 + 0.359p
B = -0.0357 - 0.00034p + C(Z)
where p is the percentage variation in pressure from the ASA
and C(Z) is as above
c) allowing h alone to vary (i.e. T = 4degC , p = 0 ) gives
A = 35.893 - 0.000667(h-20)
B = -0.0360 + C(Z)
where h is the % relative humidity at the observers altitude.
These results suggest that, to a first approximation, in (1),
A = 35.893 - (h-20)/1500 + 0.359p -0.135(T-4) + 0.000432(T-4)**2
and B = -0.0359 + 0.000127(T-4) - 0.00034p + C(Z).
The constant terms are the averages of their values derived above, and
are accurate to about 0.002 and 0.0002 respectively.
It is seen that humidity is relatively unimportant in 'optical' refraction.
- Non-linear terms and final errors
By examining the residuals ( formula minus model) at zenith distances of
45 degrees, it is possible to resolve ther non-linear terms arising from the
interactions of variations of T,p,h from their nominal values. The only non-
linear term of significance may be approximated by the following addition to
the term A in (1) :
- 0.0013p(T-4).
Note that this is independent of humidity.
The formula was then compared with the model calculations at a comprehensive
number of combinations of T,p,h and Z. It is found that, even at the most
extreme conditions considered, the errors are less than 1/3" for Z<75, and
<1/2" for Z<80. At atmospheric conditions closer to nominal the errors are
essentially neglible. Experience at Mauna Kea suggests that temperatures
outside the range -15 to 15C and pressure variations from nominal of greater
than 5% are extremely rare.
The error contours in the T-p plane for the case of
Z=85 are shown in Fig.2 for the humidity values of 20%, 50% and 80%.
It can be seen that only in extreme weather conditions will refraction
errors be larger than 1".
- Radio refraction
Refraction in the radio window is described by all sources (e.g. AQ3) as being
wavelength independent. The formulation of Allen is used, and a check on the
results is made against those for Kitt Peak at 2km altitude given by Ulich
(J. mm & IR ast. nn, nn. 198n), though Ulich's quoted barometric pressure at
Kitt Peak of 614 mbar is maintained in the present calculations at the ASA
value of nnn mbar.
In a similar fashion to the optical case the form of (1) is adopted and
a) allowing T alone to vary (i.e. p = 0 %, h = 20% ) gives
A = 36.798 - 0.0294(T-4) + 0.00329(T-4)**2
+ 0.000042(T-4)**3
B = -0.0356 + 0.0001(T-4) + C(Z)
where T is the temperature in deg C
Z is the Zenith distance
and C(Z) = 0 for Z < 80deg
= 0.0002(Z-80)(Z-79) for 80deg < Z < 85deg
b) allowing p alone to vary (i.e. T = 4degC , h = 20%) gives
A = 36.801 + 0.3527p
B = -0.0355 - 0.00030p + C(Z)
where p is the percentage variation in pressure from the ASA
and C(Z) is as above
c) allowing h alone to vary (i.e. T = 4degC , p = 0 ) gives
A = 36.800 + 0.0768(h-20)
B = -0.0357 - 0.00001(h-20) + C(Z)
where h is the % relative humidity .
This result is independent of the precise humidity profile thereabove.
- A simple formula
The results in section 6 suggest that, to a first approximation, in (1),
A = 36.800 + 0.0768(h-20) + 0.3527p -0.0294(T-4) +
0.00329(T-4)**2 + 0.000042(T-4)**3
and B = -0.0356 + 0.00010(T-4) - 0.00030p - 0.00001(h-20) + C(Z).
The constant terms are the averages of their values derived in section 3, and
are accurate to about 0.002 and 0.0002 respectively.
It is seen that humidity is much more important in 'radio' refraction,
and that the formulae for A and B are similar in form and size in both cases.
- Second order terms
The formula above was tested at an elevation of 45 degrees and revealed
second-order ('cross') terms in T and h and a small term in T and p.
These were evaluated as additional terms for A :
+ 0.00133(h-20)
+ 0.00490(h-20)(T-4) + 0.000140(h-20)(T-4)**2
+ 0.00000222(h-20)(T-4)**3 - 0.00125p(T-4)
(Note added 02 Jun 1995 : should not the first of these terms, and its
transcription below, be 0.00133(h-20)p ? )
- The errors
For Z<80 errors of greater than 1" occur only when the temperature drops
to -20C. We show in Fig.3 the error contours in T-p space for zenith distance
85 degrees, and for humidity values of 20%, 50% and 80%. Again, it is only
under the most extreme of conditions that the new formula diverges from an
integrated solution by more than 1".
- Summary and recommendations
The following formulae are suggested for determining the refraction at the
summit of Mauna Kea :
Z = Zenith Distance (o)
T = temperature (oC)
p = pressure variation (%)
h = relative humidity (%)
C(Z) = 0 for Z < 80deg
= 0.0002(Z-80)(Z-79) for 80deg < Z < 85deg
a. optical refraction
A = 35.893 - (h-20)/1500 + 0.359p -0.135(T-4) + 0.000432(T-4)**2
- 0.0013p(T-4).
B = -0.0359 + 0.000127(T-4) - 0.00034p + C(Z).
b. (sub)millimetre refraction
A = 36.800 + 0.0768(h-20) + 0.3527p - 0.0294(T-4) + 0.00329(T-4)**2
+ 0.000042(T-4)**3 + 0.00133(h-20)
+ 0.00490(h-20)(T-4) + 0.000140(h-20)(T-4)**2
+ 0.00000222(h-20)(T-4)**3 - 0.00125p(T-4)
B = -0.0356 + 0.00010(T-4) - 0.00030p - 0.00001(h-20) + C(Z).
The refraction is then given by
A tan(Z) + B tan3(Z)
These formulae are, of course, limited in application, and will not account
for local cloud structures or non-laminar air flows which destroy the radial
symmetry requirements of the model.
If it is required that neither optical nor millimeter refrcation errors
exceed 1" when the elevation is greater than 10 degrees then the errors of
measurement of temperature, pressure and relative humidity should not
exceed 2C, 0.5% and 2% respectively.
Iain Coulson
Original Version : Sep 1987
Latest Update : Aug 2001
