Delta Dec = C1 + C2*sinH + C3*cosH + C4*(sinD*cosH - tanL*cosD) + C5*Q*(sinL - sinD*cosZ)/cosD
(1)
Proposed Changes to the Pointing Model for the
140-ft Telescope
Ronald J. Maddalena
August 24, 1992
After investigating the 140-ft pointing model, and developing
new fitting software, I was able to determine that our existing pointing model
is probably in error and that my proposed model fits pointing data better. Some
suggestions are given as to how to further investigate the pointing problems of
the 140-ft and as to what observers should do to ensure good pointing of the
telescope.
The pointing model used by the 140-ft control system has gone through
numerous alterations over the years -- the history of these changes can be
ascertained by going through old reports and memos (Pauliny-Toth, 1969; Herrero,
1972; Gordon et al., 1973; von Hoerner, 1975; von Hoerner, 1976a, 1976b, 1977).
The currently used pointing model is almost identical to that proposed by von
Hoerner (1976b, 1977) and which I will label the von Hoerner model.
Recently, pointing problems with the 140-ft have nudged me to reinvestigate the
pointing model and the algorithms we use to fit data to the model. Although I am
still investigating the pointing problems, the new algorithm I have recently
developed is better suited to investigating pointing problems and is,
statistically, more correct. Using the new algorithm, I have determined that
some terms in the old pointing models need not exist while others terms which
have been either ignored or were unknown should be added.
I will first briefly describe the various versions of the von Hoerner model that
have been used since 1977 (section 2) and possible problems with those models
(section 3). Next, I will present my proposed model and the fitting algorithm I
have used that makes experimenting with pointing models simple (section 4). In
section 5, I report the results of the new fitting model and compare them with
thoseproduced by the old models. Finally, in section 6, I discuss
theimplications of my research; I also give suggestions on how to
furtherinvestigate the existing pointing problems and how observers can
ensuregood pointing of the 140-ft.
In the following description of the pointing models for the 140-ft, I use the
following definitions:
H, HA = Hour Angle
D, Dec = Declination
L = Latitude
Z = Zenith distance
The von Hoerner model (1976b, 1977) is based on how known
physical errors or features in the telescope can produce predictable pointing
error. It is, therefore, a theoretical model and not an empirical one. The Dec
and Ha pointing models he suggested are:
Delta Dec = C1 + C2*sinH + C3*cosH + C4*(sinD*cosH - tanL*cosD) + C5*Q*(sinL -
sinD*cosZ)/cosD (1)
Delta H = C6 + C7*sinD + C8*cosD + C9*sinH + C10*sinD*sinH +
C11*cosD*sinH + C5*Q*cosL*sinH + C2*sinD*cosH (2)
The factor Q is a weather dependent term which has the definition:
Q = K / [ cosZ + 0.00175*tan(Z-2.5) ]
where K = 0.354 P/T - 0.0585*Pw/T + 1701*Pw/(T**2); P = atmospheric pressure; T
= atmospheric. temperature; and Pw = water vapor pressure
He ignored three terms with physical causes, two in Dec (Q1*sinD
and Q2*cosD) and one in HA (Q3*cosD*cosH) for which the coefficients, he
ascertained, were probably close to zero. The various terms have the physical
causes outlined in Table 1.
Note that two coefficients, C2 and C5, are common to both the Dec and HA models;
their definition is a major difference between his model and the one I propose.
The coefficients C1, C6, and C8 should change with each receiver
installation since they depend upon the repeatability in which receivers are
mounted to the telescope. The changes in the threecoefficients are hopefully
corrected for by the process of finding thethree global pointing corrections we
call the 'PVLS' (see "ComputerAssisted Observing with the 140-ft", pp
12-13 for details about PVLS). Usually, the operator or Friend-of-the-telescope,
after every receiver installation, determines values for the PVLS by pointing on
three to four sources; the determined values are then used to alter the values
of C1, C6, and C8 that are stored in the Honeywell H316 computer.
The C4 and C9 coefficients will depend on whether the telescope is used at
prime or Cassegrain focus and whether at Cassegrain focus the lateral focus
mechanism is in or out of use. We, thus, should expect three different set of
values for C4 and C9 which depend upon the optical configuration of the
telescope.
In the early 1980's, Harry Payne added the term
C12*cosH*cosZ.
to the HA part of the von Hoerner model. There is no known
physical reason for the term but Harry added it since he said it 'looked' like
the term reduced the residuals of the fit.
When I looked closely in 1987 at pointing data, plus after studying the von
Hoerner memos, I noticed that Harry's C12 term was closely related to the Q3
term originally ignored by von Hoerner. When I replaced Harry's C12 term with
C12*cosH*cosD
the residuals from the fit were similarly reduced. The new term, unlike the
Payne term, does have a physical cause (Q3 term, Table 1).
Thus, between the early 1980's and today, the pointing model has had 12 terms
(11 of the von Hoerner terms plus two different versions of the 12th term).
For the rest of this report, my usage of the term 'von Hoerner model' means his
original eleven term model modified by the addition of my 12th term.
The coefficients C1 through C12 have traditionally been found by empirically
fitting pointing data to the model, although theory does predict the values for
some of the coefficients.
Let me define two quantities:
HA(predicted) = HA(catalog) + delta HA
Dec(predicted) = Dec(catalog) + delta Dec
where 'catalog' means the known coordinates of a source and where delta HA and
Dec are from the pointing model (eqs. 1 and 2). Some of the algorithms that have
been developed perform a non-linear, least-squares fit for the coefficients C1
through C12 of:
sqrt ( { HA(observed)-HA(predicted) }2 + {
Dec(observed)-Dec(predicted)}2 ) . (3)
The coefficients found by the fit should minimize the distance
between where the telescope should be pointed and where it does point. Because
coefficients C2 and C5 are in both the delta HA and delta Dec equations [and,
thus, in HA(predicted) and Dec(predicted)], one cannot minimize
HA(observed)-HA(predicted) and Dec(observed)-Dec(predicted) separately.
We can assume that the measured quantities HA(observed)-HA(predicted) and
Dec(observed)-Dec(predicted), have errors which have a Gaussian distribution.
However, in equation 3, we fit the quadratic sum of these two quantities and the
net result is that we are not fitting an observed quantity that has errors which
have a Gaussian distribution. Thus, we should have been using the more-general
maximum likelihood method of fitting as opposed to the least-squares method
whose basic assumptions we are violating. The coefficients that have resulted
from using these non-linear, least-squares methods probably have had reasonable
values but the formal errors for the coefficients cannot be trusted. One cannot
ascertain whether the found coefficients, statistically speaking, have their
most likely values.
Other algorithms that have been developed (e.g., Harry Payne's POINT140
program) perform a linear, least-squares fit of:
HA(observed)-HA(predicted) + Dec(observed)-Dec(predicted)
Again, because of the common C2 and C5 coefficients, one cannot fit
HA(observed)-HA(predicted) and Dec(observed)-Dec(predicted) separately. The
least-squares method can be used here since the fitted quantities should have a
Gaussian error distribution. However, the coefficients found will be such as to
minimize the sum of the HA and Dec components of the pointing error instead of
the magnitude of the pointing error. The algorithm do not minimize what is
usually defined as the total pointing error but minimize a related quantity.
The existing algorithms and computer programs, if we ignore their statistical
blunders, were very hard to modify in case one wanted to add or delete terms
from the pointing model. While this cannot be considered a flaw, it would have
been nice if they the were designed so as to make experimenting or playing with
the pointing model easy.
The von Hoerner model assumes that there are no other causes of pointing
problems besides those listed in the Herrero memo of 1972. Although this is a
very reasonable policy to take, its one that I think we must violate in order to
improve the pointing model. As shown in sections 5 and 6, I think I have found
additional terms to the pointing equation that do not have a presently-known
physical cause.
The work described here was an effort to overcome these deficiencies in how we
determine coefficients and terms in the pointing model.
I have changed the von Hoerner model in a very subtle way by making one
simplifying assumptions and one hypothesis.
First, I assume that the value for the refraction coefficient, C5, in the von
Hoerner model is known -- not as unreasonable assumption since all previous fits
of pointing data to the von Hoerner model suggest a value of 1.02 +/- 0.02
arcmin regardless of weather conditions, observing frequency, time of year, etc.
Theory also predicts a coefficient close to this value (Herrero, 1972). My
assumption, if not completely correct, at most introduces a few arcsec error
when pointing very close to the horizon.
My hypothesis is that there may be a term in the Dec equation that has a sinH
dependence that comes from a cause other then polar axis misalignment. Or,
equivalently, I could hypothesized that there is a term in the HA equation that
has a sinD*cosH dependence that comes from a cause other then polar axis
misalignment. If either of these assumptions is true, then the C2 coefficient in
the Dec equation should not have the same value as the C2 coefficient term in
the HA equation. In essence, I am replacing one term in the von Hoerner model, a
hypothesis I test in section 5 by looking at the results of fitting data to the
new model.
When I make these assumptions, I can modify the von Hoerner pointing model (eqs.
1 and 2) by making C5 a constant and by replacing the C2 coefficient in delta HA
with a new C13 coefficient. The new pointing equations can be written as:
Delta Dec = C1 + C2*sinH + C3*cosH + C4*(sinD*cosH - tanL*cosD)
+ 1.02*Q*(sinL - sinD*cosZ)/cosD (4)
Delta H = C6 + C7*sinD + C8*cosD + C9*sinH + C11*cosD*sinH +
C10*sinD*sinH + 1.02*Q*cosL*sinH + C12*cosD*cosH + C13*sinD*cosH (5)
Now, there are no coefficients which are common between the two equations and I no longer need to add quadratically the Dec and HA pointing errors (eq. 3). Instead, I can perform two linear multi-regressional least-squares fit of the two equations to pointing data. The two equations I want to minimize are simply:
HA(observed) - HA(predicted) (6)
and
Dec(observed) - Dec(predicted) .
(7)
Because of the form of the new fitting equations, I can legitimately use the
least-squares technique and believe in the formal errors in the coefficients
produced by the fit.
The least-squares fitting algorithm I used is that described by Press et al.
(1986). The advantage of their algorithm is that by changing a parameter
provided to the algorithm, I can change which terms are to be fit and which are
to remain constant. I wrapped the Press et al. algorithm in a program that makes
it extremely easy for the user to experiment with what the algorithm should fit
or hold constant. In addition, I freely added other terms to the pointing model
in order to look for any unknown but important terms that von Hoerner may have
missed. The agility with which my program can add or remove terms to be fitted
is the major factor which allowed me to do the investigating I needed to do.
The full versions of equations 4 and 5 that my program can handle are:
Delta Dec = D1 + D2*sinH + D3*cosH + D4*(sinD*cosH - tanL*cosD)
+ D5*sinD + D6*cosD + D7*cosD*sinH + D8*cosD*cosH + D9*sinD*sinH + D10*sinD*cosH
+ D11*sin2D + D12*cos2D + 1.02*Q*(sinL - sinD*cosZ)/cosD (8)
Delta HA = D13 + D14*sinD + D15*cosD + D16*sinH + D17*cosH + D18*cosD*sinH +
D19*cosD*cosH + D20*sinD*sinH + D21*sinD*cosH + D22*sin2H + D23*cos2H (9)
(Instructions are provided with the program on how to add additional terms.)
Note that I have used D's to designate my coefficients from the von Hoerner
model even though some of the coefficients are equivalent to those in the von
Hoerner model. Table 2 shows how my D coefficients map
into to the von Hoerner C coefficients:
I can now use equation 6 and 8 to fit for the D1-D12 coefficients and
equation 7 and 9 for the D13-D23 coefficients.
By fitting pointing data to to the pointing equations, and by using the new
program which allows the user to dynamically state which set of coefficients are
to be fitted and which are to be held constant, I was able to ascertain which of
the D coefficients have statistically relevant values. I used data for eight
pointing runs which went back to November 1990 and which were at frequencies
that ranged from 22 GHz down to 1.6 GHz.
My first attempt was to fit for D1-D3 and D5-D23. I didn't fit for D4 at this
point since it is a coefficient to a term that has a similar dependence as the
combination of the D6 and D10 terms (see eq. 8). That is, I wasn't sure whether
we should be using the D4 term or the more general combination of the D6 and D10
terms. If the ratio of the derived values of D6 to that of D10 is consistently
close to the value of -tanL, then I can assume that the D4 term is all we need
to fit and we shouldn't use the D6 and D10 terms. If the ratio is consistently
different from -tanL, then the D4 term should not be fitted for.
Finding which coefficients had significant values wasn't too difficult.
Basically, I performed a fit on each of the eight pointing runs and produced
eight sets of coefficients. By simple inspections, many coefficients were seen
to be non significant -- their values for every run were much smaller than their
standard deviations. I then redid the fits but specified that these coefficients
were to be held constant and have a value of zero. The results of the second fit
indicated that other terms were not significant, so I similarly eliminated them.
The values of Chi^2 (the rms average values of eq. 6 and 7) did not rise
significantly between the first and second fit which confirmed that the
eliminated terms probably were not needed. By the third or fourth iteration, it
was clear that all of the remaining terms were statistically significant. If I
eliminated any additional terms, Chi^2 would increase substantially but if I
reintroduced any previously eliminated terms, Chi^2 wouldn't decrease
substantially. The results of the first few fits are too voluminous to provide
here but the results of the last iteration are given in Table
3.
It was easy to determine that I should be fitting D4 and that the D6 and D10
terms are not needed. In addition, the values for the coefficients I found for
the D5, D7-D9 and D11-D12 terms are not statistically significant but the D1-D3
terms are significant. Thus, except for the definition of D2 (related to C2), I
agree completely with the Dec part of the von Hoerner model -- no terms like
those I tried need to be added to and no remaining terms should be removed from
the Dec equation.
The D13-D16 and D20-D21 terms are significant but the D17-D19 and D22-D23 terms
are not significant. Note that the eliminated D18 and D19 terms are a part of
the von Hoerner model which implies that we have had two terms in the von
Hoerner's HA model that probably are not needed.
If my hypothesis in section 3 were wrong, then the values of my D2 coefficient
should be statistically close to that found for D21, which is obviously not
true. Thus, my hypothesis seems to be valid.
My model, once I eliminate the insignificant terms becomes:
Delta Dec = D1 + D2*sinH + D3*cosH + D4*(sinD*cosH - tanL*cosD)
+ 1.02*Q*(sinL - sinD*cosZ)/cosD (10)
Delta HA = D13 + D14*sinD + D15*cosD + D16*sinH + D20*sinD*sinH
+ D21*sinD*cosH + 1.02*Q*cosL*sinH (11)
I then gave all insignificant coefficients a value of zero, held them fixed,
and fitted for D1-D4, D13-D16, and D20-D21. The results of the final fit are
given in Table 4.
In Table 5, I provide the HA and Dec rms errors
produced by the von Hoerner model using Harry Payne's POINT140 program as well
as the errors produced by my model and algorithm. In figure 1, I compare the
residuals for a subset of the data for the old and new models. The superiority
of the new model over the old is quickly apparent.
A careful investigation of Table 4 reveals certain
interesting results. I will step through the various coefficients one by one and
relate what I notice.
The D1, D13, and D15 coefficients (equivalent to C1, C6, and C8 in the von
Hoerner model) are supposed to vary from one receiver installation to another
(i.e., they are related to the PVLS) but, amazingly enough, the scatter in the
values for these coefficients suggest that we mount are receivers in a somewhat
consistent fashion.
The large standard deviations I found for the value of the coefficients for a
large number of data points suggest that our determination of PVLS after a
receiver installation with only three or four measurements cannot be very
accurate. It isn't clear to me whether our practices in determining PVLS
improves the pointing or whether we would be better off to simply set the PVLS
to zero and always use the same values for D1, D13, and D15.
A clue as to why we find large standard deviations comes from a glance at the
covariance matrix produced by the fit. The matrix indicates a strong correlation
between the D13 and D15 terms which suggests that one cannot determine very well
D13 and D15 separately but that D13+D15 should have a well determined value. I
suggest that we investigate whether our pointing would be better if we set P2 =
0, only fitted for P1, and applied a correction only to D13.
In addition, we should look into whether or not it is practical to take great
care in mounting receivers. Then, we could possibly find a set of values for the
coefficients that would be better than our error-prone determinations of PVLS.
More work is required to determine which of these options is best.
From Table 4, it is apparent that the value for the D2
coefficient jumps between two values -- approximately -0.5 and -1.2. The
standard deviations from the fits are so small that this must be interpreted as
a real jump. Historically, D2 (equivalent to C2 in the von Hoerner model) has
had a value of -0.5 so I believe the normal state of the telescope is when D2 =
-0.5 and that something is amiss with the telescope when D2 = -1.2.
The jump in pointing has plagued the telescope for at least two years and we
are not any closer in finding the culprit. It is noticed with the Cassegrain
system but may exist at prime focus since we do not carefully monitor pointing
at the lower frequencies. Pointing jumps do not seem to correlate with any
particular combination of lateral focus status, usage of the beam splitter, or
which Cassegrain receiver is in use. Sometimes, the simple act of remounting the
subreflector after a week hiatus eliminates the problem.
According to the von Hoerner model, D2 and D21 should have the same values
but, according to my results, they do not -- D21 doesn't statistically fluctuate
very much, it doesn't jump when D2 jumps, nor does it have a similar value to
D2. This suggests that the jump in pointing is not a change in the polar axis
alignment; something else must be producing a declination pointing error of
magnitude 0.7 arcmin which depends upon the sine of HA.
With the von Hoerner model, we could not adjust D2 to correct the declination
errors without making the HA errors worse. With my new model, we have the
freedom to alter D2 whenever we think we need to without compromising the HA
pointing. I suggest that we keep a record of the circumstances when D2
apparently needs to have its value changed. I also suggest that we give D2 a
default value of -0.5 and prepare instructions for the operator in case they
need to alter D2 to a value of -1.2.
The need to alter D2 should be easily apparent to the operator. The change only affects the Dec pointing and has a sinH dependence. That is, if the operator sees a difference of 1.4 arcmin in the Dec pointing between 6 hours east and west, for all sources at all declinations, then we should use a D2 value of -1.2. If they see an error in HA pointing, or the Dec pointing doesn't follow a sinH dependence with the proper magnitude, then they should not alter the value of D2.
D3 (equivalent to von Hoerner's C3) has very consistent values and apparently
is not a problem. It also has a value very close to that found historically. I
suggest we take and use an average value of D3.
D4 and D16 (equivalent to von Hoerner's C4 and C9) have consistent values
when the lateral focus mechanism is in operation or when at prime focus. The
values are also consistent with historical values. My current results do not
provide good enough values to use for Cassegrain focus operation without the
lateral focus mechanism.
I suggest that we only update the values of D4 and D16 for prime focus and
Cassegrain, lateral-focus-on observations and that we try to obtain better
values for the coefficients in the case of Cassegrain, lateral-focus-off
observations. In the meantime, we should continue to use previously-determined
values for D4 and D16 for Cassegrain, non-lateral focus observations.
The standard deviations for D14 (equivalent to von Hoerner's C7) are
constantly high and a look at the covariance matrix produced by the fit
indicates that the determinations of the D13, D14, and D15 coefficients are
correlated. The correlation is not as strong as that between D13 and D15
(discussed above), but it does mean that only a large number of data points will
improve our determination of D14.
The determined values of D14 are very close to those reported historically
for the 140-ft.
The D20 term is equivalent to von Hoerner's C10 and has values which are very
close to those reported historically. There is some correlation between D20 and
D16, hence the large standard deviations.
The D21 term has no equivalent in the von Hoerner model but, as described in
section 4, was originally the same as von Hoerner's C2. In reality, it is a new
term in the equation for which no complete physical cause is known, although
Polar axis orientation probably plays some role. There is some correlation
between D21 and D14, hence the large standard deviations.
My best estimates of the values for the various coefficients were determined
by a weighted average of the values in Table 4 and are
reported in Table 6.
In order to ascertain what would be the pointing errors if we use the
suggested coefficients, I performed one more fit specifying that only D1, D12,
and D14 were to be fitted (i.e, the PVLS terms which should differ between
pointing experiments) and I held the rest constant with the appropriate values
from Table 6. I am trying to mimic what an average
observer should expect if: (1) we used my model and the coefficients in Table
6; (2) if we do a good job in determining the PVLS; and (3) if no other
pointing is done (i.e., the observer doesn't ascertain local pointing
corrections and points blindly).
The rms after the fit (Table 7) suggests that, except
for D2, the pointing does not change much over the course of a few years. The
changes are significant enough that our use of average values for the
coefficients in the model will only provide good but not excellent pointing.
Since it is impractical to perform pointing runs every few weeks, this is the
best that can be done with a static pointing model.
Observers who blindly point the 140-ft (assuming that PVLS have been
determined) can expect their pointing to be in error occasionally by more than
20 arcsec rms! I recommend that if observers want more accurate pointing they
should perform measurements of local pointing corrections. When observing a
source as it rises and sets, observers should also occasionally repeat these
local pointing measurement every one to two hour so as to ensure that, baring a
problem in D2, the pointing will be better than 15 arcsec rms.
The following points summarize the results of this report:
Gordon, M.A., Huang, S., Cate, O., Kellerman, K.I., and Vance, B.,
"On-line Pointing Correction of the 140-ft Telescope", Telescope
Operations Division Report, No. 11, 1973.
Herrero, V., "140-Ft Telescope Pointing Corrections", 1972.
Pauliny-Toth, I., "Pointing Corrections for the 140-ft. under computer
control", 1969.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T.,
"Numerical Recipes: The Art of Scientific Computing", Cambridge
University Press, 1986, pp. 509-515.
von Hoerner, S., "140-Ft Pointing Errors and Possible Corrections",
Electronics Division Internal Report, No. 164, 1975.
von Hoerner, S., "Refraction Correction for the 140-Ft Pointing",
Engineering Division Internal Report, No. 101, 1976a.
von Hoerner, S., "140-Ft Pointing Program, and Thermal Shielding of Shaft
and Yoke", Engineering Division Internal Report, No. 102, 1976b.
von Hoerner, S., "140-Ft Pointing Errors after Thermal Shielding",
Engineering Division Internal Report, No. 107, 1977.
Figure 1: The residuals (in arcsec) for the June 1992, 22 GHz pointing run using
the von Hoerner model and Harry Payne's POINT140 program. Each plot depicts the
residuals for the indicated range of declinations. The top row of plots show the
HA residuals and the bottom row the Dec.
Figure 2: The same as Figure 1 but using the new model and algorithm.
The ignored terms correspond to:
| Q1, Q2 | Dec Encoder eccentricity |
| Q3 | HA Encoder eccentricity |
Note: Values for D1 through D20 are in arcmin. The formal standard deviations
of the fitted coefficients is given in parenthesis. The average HA and Dec
residuals after the fit are given as well as the number of data points that went
into the fit. Whether the receiver was a Cassegrain or prime focus, as well as
whether the lateral focus tracking was on, is also indicated.
Note: The D13 through D21 coefficients are identical to those in Table 3
since the HA equations was not altered between the fits.
HA and Dec rms are defined as the rms average values of
HA(observed)-HA(predicted) and Dec(observed)-Dec(predicted), respectively.
Values for D4 and D16 for Cassegrain Focus, Lateral Focus Off are from pointing measurements not reported here. Current data is insufficient for finding values for D4 and D16 under this configuration of optics.