X-band

A quantity of interest is the rms variation of the observed visibilities. Since at X-band the field is nearly blank, there is really no need to even map the field to estimate the noise characteristics of the instrument (although it is always done anyway). The rms variation on a two-antenna, single-multiplier, correlation interferometer observing weak sources is given by (Crane and Napier 1994):

$$\Delta S = {{\sqrt{2} \, k_b \, T_{\rm sys}} \over {A \, \et... ... \eta_c \, \sqrt{\Delta t \, \Delta \nu}}} \quad , \eqno (1)$$

where $k_b$ is Boltzmann's constant, $T_{\rm sys}$ is the system temperature, $A$ is the physical aperture size, $\eta_a$ is the aperture efficiency, $\eta_c$ is the correlator efficiency, $\Delta t$ is the visibility integration time, and $\Delta \nu$ is the bandwidth of observation. Now, for a complex correlator, with real and imaginary outputs, each of the outputs will have the same amount of gaussian noise, characterized by the same standard deviation, $\Delta S$. Figure 3 shows a histogram plot of real, imaginary, and amplitude values of the visibilities for one of the Standard Field experiments. The real and imaginary distributions are clearly gaussian, with near 0 mean. Because of this, the amplitude distribution (which follows a Rice distribution in general) is Rayleigh distributed. However, there are generally some ``bad'' data points (from interference, e.g.), which need to be taken out of the visibility data set (flagged). In order to do this, a good estimate of the clipping level is needed. It is fairly simple to calculate the number of visibilities expected to have amplitudes greater than some value above the mean amplitude. For the Standard Field observations at X-band, the mean amplitude is given by (Thompson et al. 1991, no-signal case):

$$\langle z \rangle \ = \ \sqrt{\pi \over 2} \,\ \Delta S \quad , \eqno (2)$$

and the fraction with amplitudes greater than $\langle z \rangle \, + \, n \, \Delta S$, for $n$ = 2, 3, and 4 is: .005032, .000118, and .000001. So, given 50,000 visibilities (which is typical for these observations) in a data set, only 6 visibilities should have amplitudes greater than $\langle z \rangle \, + \, 3 \, \Delta S \, = \, (\sqrt{\pi / 2} \, + \, 3) \, \Delta S \, \sim \, 4.253 \, \Delta S$. For that reason, I use the criteria that a visibility is ``bad'' if its amplitude is $> \, 4.253 \, \Delta S$, where $\Delta S$ is measured from the data set itself, and live with the fact that I'm actually rejecting a few valid visibilities. After that clipping is performed, new values of $\Delta S$ can be estimated directly from the real and imaginary portions of the visibilities, and from that, the quantity:

$${{T_{\rm sys}} \over {\eta_a}} = {{\Delta S \, A \, \eta_c \,... ...{\Delta t \, \Delta \nu}} \over {\sqrt{2} \, k_b}} \eqno (3)$$

can be estimated. This quantity is a measure of the performance of the instrument. Table 1 shows the quantity $\Delta S$ measured in all of the Standard Field observations with the current setup, at X-band. Values of the parameters used were: $k_b = 1.3805 \times 10^{-23} \ {\rm J/K}, A = 491 \ {\rm m}^2, \eta_c = 0.79, \Delta t = 30 \ {\rm s}, \Delta\nu = 46 \ {\rm MHz}$. For each observation, a value of $\Delta S$ is calculated for each polarization (RR, LL, RL, LR) and IF (1 and 2) separately for the real and imaginary portion of the visibilities. An average of the resulting 16 values is then taken, and is what is shown in Table 1. The variation in values of $\Delta S$ across real/imaginary, polarization, and IF is very small. Also shown in Table 1 is the value of $T_{\rm sys}/ \eta_a$ for each observation.

Figure: Histogram of X-band visibilities for one Standard Field observation. Real, imaginary, and amplitude spectra are shown.

Table: X-band Standard Field noise measurements (visibility based)

date	config	$\Delta\, S$	$T_{\rm sys}/ \eta_a$	Rick's $K$	weather	flux
		(mJy)	(K)	(mJy)	comments	calibrator
1/1/93	A	11.82	87.24	10.35	50% stratus.	3C48
					fog.
3/29/93	B	9.59	70.78	8.40	50-100% cumulo.	3C48
					drizzle.
8/21/93	C	14.55	107.39	12.74	100% stratus.	3C48
11/24/93	D	8.36	61.70	7.32	10% stratus.	3C48
4/2/94	A$^*$	15.35	65.41	7.76	70% stratus.	3C286
4/23/94	A$^*$	16.45	70.10	8.32	50% cumulo.	3C286
8/18/94	B	8.72	64.36	7.64	20-35% cumulo.	3C286
					& strato.
11/12/94	C	18.25	134.70	15.98	100% strato.	3C48
					rain.
3/18/95	D$^*$	15.72	67.00	7.95	20% strato.	3C48 & 3C286
					fog.
8/9/95	A$^*$	15.02	64.00	7.59	50-80% cumulo.	3C48
10/27/95	B$^*$	15.88	66.10	7.84	clear skies	3C286

$^*$ these observations had $\Delta t$ = 10 s

The value of $\Delta S$ varies considerably, mostly due to weather, and hence increased effective $T_{\rm sys}$. However, even the best values observed in the Standard Field observations (under fairly good weather) are not as good as the values supplied in the VLA Observational Status Summary (OSS). The value of $T_{\rm sys}/ \eta_a$ can be derived from the supplied value $K$ given in the OSS (which is a value obtained from measurements of the noise characteristics at each band made by Rick Perley) and is given by: $T_{\rm sys}/ \eta_a = K / 0.1186$. For X-band, the OSS gives: $K = 5.6$ mJy (note that this was the value in the 1994 OSS, and was changed to 6.8 in the 1995 OSS), implying a value of $T_{\rm sys}/ \eta_a = 47.22$ K. Independent measurements of $T_{\rm sys}$ and $\eta_a$ yield values of $\sim 30$ K, and $\sim 0.62$, respectively (at zenith). These values agree well with the value of $T_{\rm sys}/ \eta_a$ of 47.22 K. However, these numbers are much lower than the values shown in Table 1, where the best (lowest) value is 60.35 K. The inferred value of Rick's $K$ parameter in each standard field experiment is shown in Table 1. Again, they are higher than the value of 5.6 (or 6.8) supplied in the OSS.
As a test, the rms variations in a map made from visibilities with a given $\Delta S$ should be:

$$\Delta I_m = {{\Delta S} \over {\sqrt{N_{\rm vis}}}} \quad , \eqno (4)$$

where $N_{\rm vis}$ is the number of visibilities which went into the map. As an example, in the D configuration in 1993, $\Delta S = 9.05$ mJy in the RR polarization of IF 1, and there were 42434 visibilities in that polarization/IF. This implies an image rms of: $\Delta I_m = 43.93\, \mu{\rm Jy/bm}$. The resultant dirty map (with natural weight) had a measured rms of: $\Delta I_m = 44.66 \, \mu{\rm Jy/bm}$, which is pretty close. By comparison, the OSS gives the following to calculate the rms noise in a map:

$$S_{\rm rms}= {{K} \over {\sqrt{N \, (N-1) \, (n \, \Delta t_{\rm hrs} \, \Delta \nu_{\rm MHz})}}} \quad , \eqno (5)$$

where $N$ is the number of antennas, $n$ is the number of IFs or spectral line channels, $\Delta t_{\rm hrs}$ is the total observing time in hours, and $\Delta \nu_{\rm MHz}$ is the observing bandwidth in MHz. The $K$ here is the same value as that described above. An equivalent form of the expression for $S_{\rm rms}$ is:

$$S_{\rm rms}= {{K} \over {\sqrt{2 \, N_{\rm vis} \, (n \, \Delta t'_{\rm hrs} \, \Delta \nu_{\rm MHz})}}} \quad , \eqno (6)$$

where $\Delta t'_{\rm hrs}$ is now the individual visibility integration time (still in hours). So, given $K = 5.6 \ {\rm mJy}, n = 1, N_{\rm vis} = 42434, \Delta t = 30 \ {\rm s} = .5/60 \ {\rm hr}, \Delta \nu_{\rm MHz} = 46$, then the calculated $S_{\rm rms}= 31.05 \, \mu{\rm Jy/bm}$. This is quite a bit lower than that observed (by a factor of $\sim 30\%$) [using $K = 6.8 \rightarrow S_{\rm rms} = 37.70 \, \mu{\rm Jy/bm}$, still lower than observed by $\sim 15\%$].
I have since been supplied with 2 more independent verifications of the high values of $T_{\rm sys}/ \eta_a$. The first was an observation done by Rick Perley to test this on 2/8/95, when the array was in the DnC configuration. In this observation, Rick simply looked at 3C286 and then a nearby presumed blank field. The approximate elevation of the field was $37^\circ$ at the time of the X-band observations. The measured value of $\Delta S$ was 13.89 mJy. The derived value of $T_{\rm sys}/ \eta_a$ is thus 59.19 K (with $\Delta t = 10$ s and other values as above). This is slightly better than any of the standard field observations, but still significantly higher than 47.2 K (from the OSS values, and independent $T_{\rm sys}$ and $\eta_a$ measurements). The second verification was in sensitivity numbers from one of a number of experiments done by Ed Fomalont. This particular observation was done on 11/6/94, where a blank field near $\delta = +42^\circ$ was tracked for $\sim 10$ hours. Absolute flux calibration was done with an observation of 3C286, and the phase calibrator 1244+408 was used to calibrate the complex gains. Table 2 shows the resultant measured values of $\Delta S$, as a function of elevation throughout the observation. Note that $\Delta S$ vs. elevation is not symmetric about zenith, as sunrise occurred near the middle of the experiment. At any rate, the value of $\Delta S$ near zenith is $\sim 16.4$ mJy, implying a value of $T_{\rm sys}/ \eta_a$ of $\sim 69.89$ K. According to Ed, this was typical of values he got on other ``good weather'' nights. This number is very similar to the best numbers in Table 1, and again, much higher than 47.2 K. Note also that a gross estimate of how the elevation of the standard field observations is affecting the values derived from them can be obtained from Table 2 (at least for relatively good weather).

Table: Ed's X-band noise measurement

elevation $(^\circ)$	44.5	50.5	55.5	60.5	65.5	70.5	76.0	79.5	81.0	79.5	75.0
$\Delta\, S$ (mJy)	16.8	16.5	16.4	16.4	16.5	16.4	16.4	16.4	16.4	16.4	16.4
elevation $(^\circ)$	70.5	65.5	60.0	55.5	49.5	44.5	40.0	34.5	30.0	26.0
$\Delta\, S$ (mJy)	16.4	16.5	16.6	16.8	17.2	17.9	18.2	18.6	19.1	19.6

In order to investigate what is causing the value of $T_{\rm sys}/ \eta_a$ to be relatively high in our measurements, I've gone back and tried to recover the values of $T_{\rm sys}$ for 2 of the observations (B and C configurations of 1994). If the data is FILLMed with the proper parameters (CPARM(2)=2), values are written into the TY table which can be used to recover the value of $T_{\rm sys}$ at the time. At every source change, the on-line system calculates the quantity:

$$I_{\rm sens} = {{21.59 \ \eta_a} \over {T_{\rm cal}\ g}} \eqno (7)$$

for each antenna and IF, where $\eta_a$ is the dish efficiency at the observed band, $T_{\rm cal}$ is the assumed noise tube temperature (in K) for that antenna/IF, and $g$ (the so-called ``peculiar gain'') is a fudge factor (see below). The 21.59 is a constant that subsumes the area of the dish, Boltzmann's constant, the front end gain, and other radiometric constants (note that for observations done prior to 1989, this value was 24.32). Now, every 10 seconds, the on-line system calculates the following quantity (the so-called ``nominal sensitivity''):

$$I_{\rm corf} = {{3} \over {V_{\rm sd} \ I_{\rm sens}}} = {3 ... ...{T_{\rm cal}\ g} \over {\eta_a}} \right) \quad , \eqno (8)$$

where $V_{\rm sd}$ is the front end synchronous detector voltage for each antenna/IF. For each correlated visibility, the geometric mean of $I_{\rm corf}$ for the two antennas/IFs is used as a multiplicative factor to convert correlation coefficient to 10's of Janskys. This value is what is written to the archive tape, and subsequently to the TY table by FILLM. The values of $T_{\rm cal}, \ \eta_a, \ {\rm and} \ g$ are retrieved from files on the on-line system. Now, the values of $g$ are adjusted regularly, so that the observed correlation coefficient converts to the proper number of Janskys for 3C286 or 3C48. Apparently the values of $g$ at X-band are quite stable, and near 1. As an example, during the first week of January 1995, the values of $g$ from the file had maximum and minimum values of 1.46 and 0.89 (out of 112 values, from 28 antennas and 4 IFs). The mean value was 1.022, with a standard deviation of 0.011. By contrast, at this same time, the values of $g$ from the L-band file had maximum and minimum values of 2.54 and 0.79, with mean and standard deviation of 1.526 and 0.211.
Now, the system temperature is given by:

$$T_{\rm sys}= {{15 \ T'_{\rm cal} \ V_{\rm TP}} \over {V_{\rm sd}}} \quad , \eqno (9)$$

where $T'_{\rm cal}$ is the actual (as opposed to assumed) noise tube temperature (in K) for a given antenna/IF, and $V_{\rm TP}$ is the total power voltage input to the correlator. The ALC's constrain $V_{\rm TP}$ to be near 3 V, so this is nearly a constant value. The factor of 15 is strictly an electronics gain factor. So,

$$T_{\rm sys}\sim {{45 \ T'_{\rm cal}} \over {V_{\rm sd}}} \quad , \eqno (10)$$

or,

$$V_{\rm sd} \sim {{45 \ T'_{\rm cal}} \over {T_{\rm sys}}} \quad . \eqno (11)$$

Substituting this into the equation for $I_{\rm corf}$ yields:

$$I_{\rm corf} \sim {{3 \ T_{\rm sys}} \over {45 \ T'_{\rm cal}... ... {{T_{\rm cal}\ g} \over {\eta_a}} \right) \quad , \eqno (12)$$

or,

$$T_{\rm sys}\sim 323.85 \ {{\eta_a \ T'_{\rm cal}} \over {g \ T_{\rm cal}}} \ I_{\rm corf} \quad . \eqno (13)$$

Now, the adjustments to $g$ mentioned above might imply that $g \ T_{\rm cal}\sim T'_{\rm cal}$, in which case,

$$T_{\rm sys}\sim 323.85 \ \eta_a \ I_{\rm corf} \quad . \eqno (14)$$

The value of $\eta_a$ is again taken from the same file which contains the values of $g$ (and $T_{\rm cal}$). These values are the ``standard'' numbers, i.e., $\eta_a = 0.62$ for X-band, and 0.51 for L-band. Given this value, the values of $T_{\rm sys}$ can be derived directly from the values written to the TY table ($I_{\rm corf}$). Note that uncertainties in the value of $\eta_a$ are unimportant, as long as the $\eta_a$ which was used by the on-line system is used. Errors are due to fluctuations in $T'_{\rm cal}$, and in $V_{\rm TP}$. Of these, fluctuations in $T'_{\rm cal}$ should dominate. There is no good knowledge of how these values fluctuate over short or long time scales, however, current wisdom is that the values are relatively stable (to $\sim 10\%$, see Bagri and Lilie 1993, and Lilie 1992). Therefore, estimating the value of $T_{\rm sys}$ from the values in the TY table should be accurate to $\sim 10\%$ for a given antenna, and might be as accurate as a few percent for an average over all antennas. I've made an AIPS task which does the conversion from $I_{\rm corf}$ to $T_{\rm sys}$ (in K) in the TY table, called TYCNV. Figure 4 shows the results of performing this conversion on the TY table for the B configuration experiment in 1994. Table 3 shows the value of $\overline{T_{\rm sys}}$ for each of the IFs, which is the value of $T_{\rm sys}$ averaged over all antennas and elevations for that IF. The rms is strictly the data scatter, and doesn't take into account the possible fluctuations in $T'_{\rm cal}$. The fact that the values of $T_{\rm sys}$ make sense is a very loose verification of the conversion algorithm (and TYCNV). However, this then implies that the aperture efficiency, $\eta_a$, is not the 0.62 that is advertised at X-band. If $T_{\rm sys}$ is indeed $\sim 30$ K, and the value of $T_{\rm sys}/ \eta_a \sim 62$ K (best value from Table 1), then the inferred aperture efficiency is: $\eta_a \sim 0.48$, at X-band.

Table: Derived values of $T_{\rm sys}$ for an X-band Standard Field observation

IF	$\overline{T_{\rm sys}}$ (K)	$\sigma_{T_{\rm sys}}$ (K)
A	28.90	2.98
B	28.55	2.51
C	28.63	2.62
D	28.69	2.44

Figure: Plot of X-band system temperature ($T_{\rm sys}$) vs. Elevation for one Standard Field observation. Each antenna is plotted separately in each IF.

A good question to ask is: ``why didn't Dwaraka see this?'' Well, to check on this, I've gone back through his notes. Table 4 shows values which he used for the parameter $K$ in each observation, the implied theoretical values of the noise $S_{\rm rms}$, and the measured value of $S_{\rm rms}$ for that observation. Apparently, the value of $S_{\rm rms}$ was estimated from a map made in Stokes I, with both IF's. So, it appears that he was regularly measuring higher noise levels than predicted. There are two reasons why he didn't see an even larger discrepancy. First, you can clearly see that the values of $K$ which he used are higher than what I've used (I'm using the value of 5.6 from the 1994 OSS). The second reason is that Dwaraka used a significantly lower value for the amplitude cutoff when flagging ``bad'' visibilities. As an example, the clipping level in the A configuration measurements of 1/1/93 was set at 20 mJy, which is only about 5 mJy above the mean value of the amplitudes. Therefore, a significant portion of the tail of the noise distribution was being chopped off, and the measured noise in the final map was necessarily biased down. For comparison, I made a map of that A configuration data, when a clipping level of 50 mJy was used on the visibility amplitudes. The measured rms in the map was: 26.3 $\mu$Jy/bm. Using Stokes V yielded 26.5 $\mu$Jy/bm. So, all of the values in columns 5-7 in Table 4 are probably about 10% too low. At any rate, it is clear that the inferred values of $T_{\rm sys}/ \eta_a$ agree well with the values in Table 1, and are again higher than presented in the OSS.
So, all indications are that the value of $T_{\rm sys}/ \eta_a$ at zenith is higher than currently advertised for the VLA at X-band. Taking into account the variation with elevation indicated from Ed's data, the best value of $T_{\rm sys}/ \eta_a$ at zenith for the standard field observations in the last two years was $\sim 56$ K. This is about 15% different than the number obtained by taking the nominal values of $T_{\rm sys}= 30$ K, and $\eta_a = 0.62$. A better value to use is more like $T_{\rm sys}/ \eta_a \sim 66$ K, which is an average of all of the measurements presented here excepting the 1993 A configuration data and both epochs of the C configuration data. This implies a value of 7.8 for Rick's $K$ parameter.

Table: X-band Standard Field noise measurements (map based)

			theoretical	measured	inferred	inferred
date	configuration	$K$ (mJy)	$S_{\rm rms}$ ($\mu$Jy/bm)	$S_{\rm rms}$ ($\mu$Jy/bm)	$K$ (mJy)	$T_{\rm sys}/ \eta_a$
1/1/93	A	7.4	17.2	23.9	10.3	86.7
3/29/93	B	7.4	17.0	20.0	8.7	73.4
8/21/93	C	6.6	16.0	31.1	12.8	108.2
11/24/93	D	6.6	17.7	20.6	7.7	64.8
4/2/94	A	6.3,7.6	32.5,39.6	46.7	9.0	75.6
4/23/94	A	6.3,7.6	22.6,27.6	28.6	7.9	66.4

truein Note: Durga Bagri has made some measurements which indicate that the ``system efficiency'' in interferometric observations seems to be lower than would be expected from the straightforward product of the aperture efficiency and the correlator efficiency (presented in the VLA test meeting of March or April 1995). i.e., the value of $\eta_a \ \eta_c$ in equation (1) should be replaced by some system value, $\eta_s$, where $\eta_s = \eta_a \ \eta_c \ \eta_o$, with $\eta_o$ being ``other'' system losses, e.g. LO coherence. This would explain the discrepancy between the rms variations being measured and what we expect theoretically from measurements of $\eta_a$, and $T_{\rm sys}$, and expected values of $\eta_c$, if $\eta_o \sim .85$. Durga indicated that the difference in the two efficiencies (single-dish vs. interferometric) was about 12-13%, which agrees well with what the numbers presented here indicate.
truein A small note on interference at X-band. It was brought to my attention by Ed Fomalont that he has seen some amount of interference during X-band observations in the C and D configurations when any relatively short N-S baseline involves antenna 6. The interference occurs in only 1 IF-pair (AC). Ed also brought this to the attention of Clint Janes, who is investigating the cause, I believe. At any rate, this effect shows up clearly in the standard field data from C and D configurations in 1993. The effect is much worse in the D configuration. To give a feel for the numbers, remember that the rms variation in the visibilities from that experiment (1993 D) was about 8.36 mJy (Table 1). However, in the corrupted IF, on baseline 6-1, the RR visibilities were apparently edited out by the on-line system, the LL and RL visibilities had an rms variation of $\sim 10$ mJy, while the LR visibilities had an rms variation of $\sim 70$ mJy. The effect does not show up in the C configuration data from 1994, the reason being that antenna 6 was at the end of the southeast arm (pad E18), and hence had no short N-S baselines. The effect shows up clearly in the data taken by Rick on 2/8/95, however, even though antenna 6 was still at the end of the SE arm (pad E9). Presumably the N-S baseline between antennas 6 and 17 (on pad E8) was short enough for the interference to occur. I don't know if it's really proper to use the term ``interference'' to describe this effect, but am merely using the term passed on to me by Ed.