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TRUTH or CONSEQUENCES: episode I

"ALFA spectral parameters & calibration"


--desh 18-Nov-2004



Abstract:

A set of ALFA spectral scans of apparent SEFD, T_sys
& telescope Gain are presented mainly to highlight
the ``relative'' variation of relevant quantities as
a function of frequency, beam and polarization channel.
The absolute scales associated with these estimates are likely
to have several possible uncertainties (as discussed below)
based on the ``calibration" used, and therefore should
not be taken at their face value. On the other hand, the
relative variations are expected to be less prone to
systematic uncertainties. Considering the various
sources of systematic uncertainties and random errors,
the overall accuracy of the estimation is expected to
be about (and no better than) 5% over most of the spectral
span, while the formal uncertainties are believed to
be about 3%.

OBSERVATION DETAILS:

The data analyzed here were acquired with the WAPP back-ends
measuring auto & cross-correlations between the two orthogonal
linear polarization signals of 100-MHz bandwidth after
3-level digitization. However, in this analysis only the
XX & YY products are used. The correlations (at up to
typically 1024 lags) were recorded at intervals of
~1 millisecond using the pulsar (SEARCH) mode of the WAPPs.
In most cases, a high-CAL (nominally ~ 11 K) noise diode
setting, winking at 25 Hz rate was used to provide a
correlated calibration signal in the two polarization channels.
Since only a 100-MHz bandwidth could be sampled at a time,
measurements were made at five different center frequencies
differing in steps of 75 MHz (e.g., 1225, 1300, 1375, 1450
& 1525 MHz), allowing adequate overlap between the successive
bands.

Each measurement for a chosen ALFA beam consists of a (slew)
scan from an OFF-source position (1 degree away in azimuth)
to an ON-source position of a selected continuum
flux-calibration source, and subsequent tracking of the
ON-source position. The slew-scan is thus in azimuth,
rather than zenith angle, avoiding additional possible
effects due to the usually stronger variations in the
relevant telescope response as a function of zenith angle.
Such measurement sets were repeated for each of the 7 beams.

ANALYSIS:

The correlation functions corresponding to CAL-ON and CAL-OFF
states over each intervals of 1 second are averaged separately,
and the corresponding power (or cross-)spectra computed after
van-Vleck correction appropriate for the 3-level digitization.
The data in time intervals associated with the transitions in
the CAL state are excluded (within a 2-millisecond window to
be on the safer side). The sets corresponding to the two CAL
states are stored as successive (time) samples for further analysis.
No additional smoothing is applied to the spectra. In cases
where the winking CAL was not used, simple 1-second averages were
estimated as corresponding to the CAL-OFF state, and naturally
had no transitions to be excluded from the integration.

The 4 kinds of plots made available presently are listed under
their respective tags, e.g., LOADD, SEFD, Tsys & Gain.

In the text below, we describe the processing leading to the
respective results, and discuss possible sources and magnitudes
of uncertainties affecting these estimates.

(In cases where a particular panel shows two color-line plots,
the black-line corresponds to data on the product XX, and the
other to the product YY).

In general, four different average levels are estimated from
a typical source on-off measurement, with CAL-ON and CAL-OFF
states. In the discussion below These will be represented by
S_OFF_CAL_OFF, S_OFF_CAL_ON, S_ON_CAL_OFF & S_ON_CAL_ON.

The CAL step estimates (in the off-source and on-source cases)
are then

CAL_STEP_S_OFF = S_OFF_CAL_ON - S_OFF_CAL_OFF
CAL_STEP_S_ON = S_ON_CAL_ON - S_ON_CAL_OFF

And the SOURCE step (or deflection) is estimated as

SOURCE_STEP = CAL_STEP_S_OFF*(R_S_ON - R_S_OFF)
= S_ON_CAL_OFF*R_CAL_STEPS - S_OFF_CAL_OFF

where R_S_ON = S_ON_CAL_OFF/CAL_STEP_S_ON,
R_S_OFF = S_OFF_CAL_OFF/CAL_STEP_S_OFF
and R_CAL_STEPS = CAL_STEP_S_OFF/CAL_STEP_S_ON

The source step, defined as above, accounts for any
first-order differences in the receiver gain in the on-
and the off-source states, and retains the desired
correspondence with the relevant off-source levels.

All these quantities are implicit functions of the RF
frequency, f, unless mentioned otherwise. In the rest of
this document, wherever relevant, the values for the
flux-densities of the calibration sources used here come
from careful spectral modeling that Chris Salter has
performed on the multi-frequency measurements available
in the literature.



DETAILS OF THE VARIOUS PLOTS POSTED AND THE ESTIMATIONS :


LOADD: The dynamic spectra for the two products XX & YY,
... indicating the quality of the data used.
X-axis: time from start of the observation (seconds)
Y-axis: product number (combined spectral channel
number in units of the number of channels
per product)
Z-axis: apparent deflection (proportional to signal
power)

The central panel shows the dynamic spectra, and
bottom & right panels show the band- & time-averages
respectively. The spectra corresponding to the
upper and lower bounds are also shown in the right panel.
The alternate ups & downs correspond to the CAL-ON and
the CAL-OFF states, respectively. The overall step
changes in the levels are, of course, due to the
transition from OFF-SOURCE to ON-SOURCE, or the other
way round, during the azimuth slew.


SEFD: The spectral variation of apparent System(-temperature)
Equivalent Flux Density (SEFD; or S_sys).

X-axis: Frequency (MHz)
Y_axis: Apparent S_sys (Jy)

The spectral variation of apparent SEFD are estimated
in the usual way using On-Off measurements on a flux-
density calibration source, i.e.,

S_sys(Jy)[f] = S_source(Jy)[f]
*(S_OFF_CAL_OFF[f]/SOURCE_STEP[f])

Here, the possible uncertainties in the assumed flux densities
of the calibrators (i.e. S_source(Jy)[f]) are the main source
of error (typically less than 5%). The off_source_level (and
the source_deflection to a lesser fractional level) may, in
principle, have interference from local reflections, resulting
in the so-called ``standing-wave ripple" across the band
(typically a 3% effect; a memo by Tapasi Ghosh & Chris Salter
http://www.naic.edu/~astro/aotms/2001-02.ps has more details).
Although the SEFD estimates do not use/need CAL data and are
therefore free of uncertainties in the CAL temperature, they
are affected by the low gain of the IF band-pass at the
band-edges, as well by the presence of RFI. Also, the value
could be over-estimated if there is a pointing error. An
additional apparent over- and under-estimation of this
quantity for different polarization channels would occur if
the astronomical source (or the reflection mentioned above)
happens to be polarized (even partially). The related S_sys
deviations apparent in one polarization channel, due to using
a common default value of S_source, would be anti-correlated
with those in the orthogonal polarization channel, such that
estimates corresponding to a well-calibrated Stokes I would
be free of this effect. Use of appropriately differing
S_source[f] values for the two polarization channels, taking
in account the degree of linear polarization, as well as
the frequency dependent difference between the apparent
polarization position angle of the source and the feed
orientation, would also remove the polarization imprint.
The magnitude of these differences would less than or equal
to twice the %age linear polarization. The profiles on
B1040+123 show clear signs of the effect due the source
being at least partially polarized (and in this particular
case, the degree of linear polarization is 5.6% [ref. NVSS;
thanks to Chris who checked this detail for this and other
sources in our list]). B1140+223 has 1% linear polarization,
while for B2338+132 the value is nominally 0%, making the
latter a preferred source for modeling the spectral
variation in SEFD and Gain.

Tsys: Spectral Profile of apparent System-Temperature (T_sys)

X-axis: Frequency (MHz)
Y_axis: Apparent T_sys (K)

Variation of apparent system-temperature across the spectrum
is computed, in the usual way, using an off-source measurement
and our present knowledge of the CAL temperature (T_cal), i.e.,

T_sys(K)[f] = T_cal(K)[f]
*S_OFF_CAL_OFF[f]/CAL_STEP_S_OFF[f]

As is well known, this estimation does not use/need the
on-source data, but needs the data in CAL-ON and CAL-OFF
states and thus is affected by uncertainties in our knowledge
of the T_cal values. Also, the increased quantization noise
contribution at the band edges and the presence of RFI adversely
affect the estimation. Some of the smooth modulations are a
result of systematic uncertainties in the modeling of the
apparent CAL-temperature, but are expected not to exceed
5% of the average value. Here again, the off_source_level
may, in principle, have interference from local reflections,
resulting in the so-called ``standing-wave'' ripple across
the band. Additional differential deviations would be caused
if the reflections happen to be some what polarized, as they
in principle would be.


Gain: Spectral Profile of apparent Gain of the Telescope

Gain(K/Jy)[f] = T_cal(K)[f]*SOURCE_STEP[f]
/(CAL_STEP_S_OFF[f]*S_source(Jy)[f])

X-axis: Frequency (MHz)
Y_axis: Apparent Gain (K/Jy)

As can be appreciated from the above equation and the
quantities involved, the estimation of the telescope gain
is prone to uncertainties of the entire variety we have
discussed in the context of both the SEFD and T_sys.
After all, the Gain estimate is merely the ratio of
T_sys to S_sys, with S_OFF_CAL_OFF being the only
common factor taken out. The primary sources of
uncertainties are thus those in S_source & T_cal, and
the effects related to polarization (intrinsic or otherwise),
local reflections, pointing, RFI and the higher
quantization noise at the band edges continue to be relevant.
Many of these effects result in systematic smooth variations
across the band, and may show finite correlation between the
estimates for the two polarization channels.
The profiles on B1040+123 provide a useful illustration of
the polarization imprint.

============================================================================
Estimation of apparent CAL-temperature values:
============================================================================

Two sets of HOT-load (~ 300 K) measurements were conducted
with the help of Ganesan and other members of the electronics
group, where a large size absorber was held in front of each
of the horns and 5 measurements, each spanning 100-MHz bandwidth
(with 75 MHz steps), were recorded using WAPP. Winking high
CAL (switching at 25-Hz rate) was injected during the data
taking. The general data reduction leading to average spectra
corresponding to CAL_OFF & CAL_ON states is similar to that
described above in a more elaborate situation. The CAL
contribution is estimated simply as CAL_STEP=CAL_ON-CAL_OFF.
The T_cal as a function of frequency is estimated as

T_cal[f] = (T_hot_load[t] + T_receiver[f])
*CAL_STEP[f]/CAL_OFF

The T_receiver values used here are based on the measurements
made by the ATNF/CSIRO group prior to ALFA arrival at Arecibo.
Their estimates (based on HOT-Load & Sky measurements) were
used to model the spectral dependence across the entire band
and the best-fit results are made available in the set mentioned
below. The contribution from the rest of the electronics
following the front-end is estimated to be ~2 K and is added to
the T_receiver value to be used above. The absorber temperature
was monitored at regular intervals during the measurement, and
appropriate T_hot_load values are used for the different
measurements taken at different time.

Given the generally small value of the ratio of the CAL_STEP to
the CAL_OFF level, a 1-K error in the assumed T_hot_load or
T_receiver values would cause only a 0.03 K error in the T_cal
estimation, i.e. 0.3% or less. In any case, adequate care was
taken to avoid such systematic errors as far as possible.

The estimates from the various sub-bands of ALFA are combined
together for each of the 14 pipelines of ALFA (7 pixels times
two polarization channels), and are modeled using high-order
polynomial/harmonic functions. These models are made available
in the tar set mentioned below. The results from the two sets
of HOT-Load measurements are included there. It is worth
pointing out that even in such apparently simple measurements,
there can be significant errors depending on the placement of
the absorber close to the horn-face, and the resultant
standing-waves due to local reflections plus other sources of
mis-match. In view of this, it is reassuring to note that the
T_cal estimates from the two sets are mutually consistent
within about 5%, and the average of the two sets would
be expected to be closer to its true value, within 3% or so.

To assess the above mentioned accuracy and consistency
with an alternative method of T_cal estimation, the HOT-Load
measurement set was combined with an equivalent Sky-Load
data extracted from other available measurements. For this
purpose, a coherent set of several Off-source measurements
spanning the ALFA band was combined suitably after appropriate
accounting of zenith-angle (ZA) dependence and a set of
wide-spectral profiles corresponding to the T_sys/T_cal ratio
(say, R_sky = S_OFF_CAL_OFF/CAL_STEP_S_OFF; consistent with the
terminology used above in the T_sys expression) were obtained.
Model fits to this set were obtained for use here and are also
made available for any future use. Using the R_sky ratio along
with the corresponding R_hotload ratio (= CAL_OFF/CAL_STEP[f];
for the HOT-Load measurements), an alternative estimate of
T_cal was obtained following the well-known procedure, i.e.,

T_cal[f] = (T_hot_load[t] - T_sky)
/(R_hotload[f] - R_sky[f])

A frequency independent value was assumed for the T_sky
(= 20 k), and the estimates were obtained using both the sets of
HOT-Load measurements separately. Given the average magnitudes of
of the relevant quantities used above, a 1-K error in the assumed
T_sky value would translate to only a 0.4% or so error in the
T_cal estimate.

These alternative estimates compare very well with those based
on the HOT-Load data alone, and the agreement over most of the
spectral span is within about 3%, with the lower frequency end
being less consistent (but within about 6%).

Given these consistency checks and the results, the T_cal
estimates may be treated as accurate within 3% of their true
values.



============================================================================
Easy access to the entire set of spectral scans:
composing spectral-scans over the wide ALFA-band
============================================================================



Using the data and estimations described above, a full
ALFA-bandwidth spectral profile is synthesized from
5 measurements, each sampling a 100-MHz band in
75-MHz steps.

Such profiles for the relevant quantities, i.e.,
SEFD, T_sys & Gain are modeled using a relatively
low-order polynomial/harmonic function, and the results
of these fits are now available in ALFA_POLY_FITS.tar
which includes a Fortran code to access the information.
While combining the measurements/estimates from different
sections of the wide-band, the ZA-dependence is corrected
for as far as possible such that the combined profile
corresponds to estimates at ZA=5 deg. The applied corrections
are based on the beam-0 measurements of band-averaged SEFD,
t_sys and Gain (carried out by Chris Salter and Tapasi Ghosh
in May 2004) over a range of zenith angles. The zenith-angle
dependence of the relevant quantities is also modeled and
the results are also made available in the tar file mentioned
above.

While applying the zenith angle correction as described above,
we make an implicit, but justifiable, assumption that the
spectral and zenith angle dependences of the relevant
parameters are mutually independent (i.e. are uncoupled).
This assumption is supported by the apparent consistency
(and absence of any significant discontinuities) across
the wider spectral scan that are synthesized from sections
corresponding to observations at differing zenith angles.

We expect to update this ZA-dependence model later when
more detailed data on other beams becomes available.
Thus by accessing a relevant spectral profile model and
applying suitable scaling from the ZA-dependence model,
it is possible to obtain an estimate of the mentioned ALFA
parameters as function of frequency and zenith angle for
any of the 14 pipelines of ALFA. For obtaining CAL temperature
values, two sets of best-fit models based on the two separate
methods (using only Hot-LOAD, and using both Hot-LOAD and
Sky-LOAD measurements) are available.

A tar file of the entire set of models mentioned so far,
and of other relevant files, is available at the ALFA
web-site. These sets may be updated appropriately, based on
possible future measurements. For example, the ripple
apparent in the the gain scans of beam 5. Although small,
it's origin needs to be understood adequately if its
reality is confirmed. Naturally, such issues/details may
prompt additional measurements.

===============================================================================
Calibrating data in temperature or flux-density units:
===============================================================================

The ratio of any observed deflection to the corresponding CAL_STEP
is to be multiplied by a factor equal to

T_CAL[f] for calibration to temperature units,

and

(T_CAL[f]/Gain[f]).(Gain_ZA[5 deg]/Gain_ZA[ZA])

for calibration to flux-density units,

where T_CAL[f] is from the models in

ALFA_TCAL_POLY_FIT.PARAMETERS
or ALFA_TCAL_POLY_FIT_ALTERNATIVE.PARAMETERS

Gain[f] is from the models in

ALFA_GAIN_POLY_FIT.PARAMETERS
or ALFA_GAIN_POLY_FIT_ALTERNATIVE.PARAMETERS


Gain_ZA[ZA] (& Gain_ZA[5 deg]) is (are) from models in

Gain_Vs_ZA_beam0_olddata_fit.parameters
or a similar future set with data on all beams.


These and other files containing various models can be accessed
using a supplied subroutine fetch_from_best_fit_model.f, or
other appropriate codes/routines.


Note that the frequency argument [f] relevant to these models
refers to the RF frequency in the observer's frame.


===============================================================================

It is a pleasure to acknowledge contributions from Chris Salter,
Tapasi Ghosh, Jeff Hegan, Ganesan and others in the electronics
group to this characterization effort. Thanks to Chris & Tapasi
also for their many useful comments on this write-up.

===============================================================================


Any questions or comments concerning this write-up, or
the analysis of the spectral-scans, may be directed to
desh@naic.edu

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