3 Spatial sampling  

Two effects are specific to the observing scheme described above: the decrease in sensitivity due to reduced uv-sampling at each observing frequency, and uneven spatial samplings at different observing frequencies. In this section, we investigate the effect of these factors on VLBI data at different frequencies.

With a larger number of antennas, the overlapping parts of the uv-coverages at different observing frequencies constitute an increasingly larger fraction of the joint uv-population. This results in decreasing the flux density level at which the confusion effects dominate the results of spectral index calculations. The confusion level can be lowered further by applying specific weighting schemes to the data (uv-weighting), and by matching the ranges of the uv-coverages at both frequencies. In the spectral index maps produced using the above procedures, confusion occurs at the flux level of about $0.3-0.5\%$ of the mean peak flux density in the corresponding total intensity maps (Lobanov 1996).

3.1 Longest sampling intervals and largest detectable structures

For a given time sampling interval $\Delta t$, an estimate of the largest angular size, $\Omega_{\rm max}$, of structures that can be detected on a baseline B can be calculated from the time-average smearing (Bridle & Schwab 1989). The sensitivity reduction is greatest when the apparent motion of the source is perpendicular to the fringes associated with the selected baseline, and so we can assume a polar source and an East-West oriented baseline, in order to provide the most conservative estimates of $\Omega_{\rm max}$. This gives  

$$\Omega_{\rm max} = \frac{\nu_{\rm obs}}{\omega_{\rm e} c \Delta t (B_{X}^2 + B_{Y}^2)^{1/2}} \,,$$ (2)

with $\rm \omega_e$ denoting the Earth angular rotation speed. For a feature located at the sky coordinates l, m with respect to the phase-tracking center (Thompson et al. 1986), the corresponding average reduction in amplitude over a 12-hour period is  

$$<R_{\Delta t}\gt = <I/I_0\gt \approx 1 - \frac{\pi^2}{12 \Omega_{\rm max}^2} (l^2 + m^2 \sin^2 \delta) \,,$$ (3)

where $\delta$ is the source declination.

For every combination of wavelength and structure size, the left panel of Table 1 gives the corresponding maximum allowed sampling interval [in minutes] between individual scans. Dots indicate that a structure remains unresolved. Italics highlight the 50% decrease of sensitivity. The right panel of Table 1 gives the expected maximum size of detectable structure [in mas], for all combinations of wavelengths and sampling intervals. The calculations have been done for the longest available VLBA baseline (B_L=8600km). Here we postulated B_Z=0 and B_L² = B_X² + B_Y², to provide more restrictive estimates.

It follows from Table 1 that multi-frequency observations can provide satisfactory structure and flux sensitivities for bright sources with intermediate ($\approx 10$mas) extension. For such sources, full-scale spectral index mapping with 5 minutes-long scans can be done at frequencies lower than 43GHz (0.7cm).

3.2 Simulations of multi-frequency VLBA data

To study the effect of uneven uv-coverages on spectral imaging, we simulate visibility data at all frequencies available at the VLBA, using the routine "FAKE'' from CIT VLBI package (Pearson 1991). At all frequencies, the simulated data are produced from the same "CLEAN'' (Cornwell & Braun 1989) $\delta$-component model of the structure in the top panel of Fig. 1. To improve short-spacing coverage, we include one VLA antenna in the simulations. The simulated bandwidth, W=64MHz, corresponds to one of the standard VLBA observing modes (128Mbs^-1 data rate with 1 bit sampling; Romney 1992). We model the antenna efficiencies, $\eta_i$ (Crane & Napier 1989), using the antenna sensitivities, K_i (Crane & Napier 1989), and the mean VLBA values, $\eta_{\rm VLBA}$ and $K_{\rm VLBA}$, given in Napier (1995). Then, for each VLBA antenna, the resulting efficiency is:  

$$\eta_i = \eta_{\rm VLBA} (K_i/K_{\rm VLBA})\, .$$ (4)

The system equivalent flux densities, SEFD, are calculated from the antenna sensitivities and system temperatures, $T_{\rm sys}$: $SEFD_i = T_{{\rm sys},i}/K_i$ (Walker 1995; Crane & Napier 1989).

To make the simulated data as realistic as possible, we introduce four types of errors: the Gaussian additive noise, $\sigma_{\rm therm}$,Gaussian multiplicative noise $\sigma_{\rm m}$, gain scaling errors $\sigma_{\rm gain}$, and random station-dependent phase errors. Both $\sigma_{\rm m}$ and $\sigma_{\rm gain}$ are chosen to be at a 2% level, which is a good approximation of typical VLBA gain calibration errors.

For a bandwidth of W[MHz] and integration time of $\tau_{\rm int}$[s], the additive Gaussian noise can be calculated for each antenna, using the antenna zenith system temperatures, $T_{\rm sys}$, and antenna efficiencies $\eta$.We have  

$$\sigma_{\rm therm} = 5 T_{\rm sys} / (\eta\epsilon_{\rm pt} D_{\rm ant}^2 \sqrt{\tau_{\rm int} W})\, ,$$ (5)

where $D_{\rm ant} = 25$m is the antenna diameter, and $\epsilon_{\rm pt}$is the pointing efficiency. The pointing efficiency can be estimated from the ratio of pointing errors, $\sigma_{\rm pt}$, to the half-power beamwidth of an antenna at a given frequency. The typical non-systematic pointing errors of VLBA antennas are within $8-14\hbox{$^{\prime\prime}$}$ (Romney 1992), which results in $\epsilon_{\rm pt} \sim 85-99\%$ at most of the VLBA observing frequencies. The overall parameters used in the data simulations are summarized in Table 2 for a typical VLBA (Wrobel 1997) and VLA antennas.

 

Table 2: Parameters of a typical VLBA and VLA antennas 

 

Figure 2: Baseline visibility amplitudes on the baseline between VLBA antennas at Los Alamos and North Liberty. Top panel shows data from a real observation; simulated data are shown in the bottom panel

Using the parameters from Table 2, and adding the errors described above, we simulate VLBA visibility datasets that would be obtained in an observation with a 5/15 duty cycle corresponding to observing at 3 frequencies, with 5 minutes long scans at each frequency. We simulate a 17 hours-long observation of 3C345, with 30 seconds averaging time for individual data points--similar to the typical duration and averaging time of real VLBI observations. The simulated data at 5GHz are compared in Fig. 2 with the data from a real VLBI observation, for a VLBA baseline Los Alamos - North Liberty. One can see that the noise levels are comparable in the real and simulated data.

3.3 Spatial sampling at different frequencies

To study the effects of uneven uv-coverages in VLBI data, we image the simulated datasets, following the procedure described in Sect. 2. An image obtained from the simulated data at 5GHz is shown in the lower panel of Fig. 1. Flux density and spectral index errors due to differences in spatial samplings can be estimated from comparison of the images made from the simulated data at different frequencies. In the ideal case, the flux ratio measured between any two images should remain unity in every pixel, and the corresponding spectral index should be zero across the entire image. Since all images are produced from the same source model, we ascribe all deviations from zero spectral index to the errors due to different spatial samplings and random errors, and choose to present these errors as a function of pixel SNR measured with respect to the self-calibration noise. The latter is taken to be equal to the largest negative pixel in the map, and is approximately 5-10 times bigger than the formal RMS noise of the image. In the simulated data, the average self-calibration noise is $\approx 5$mJy.

 

Figure 3: Fractional errors due to different uv-coverages. The errors are plotted against the pixel flux scaled to the average self-calibration noise, $\sigma=5$mJy

Figure 3 shows the fractional errors as a function of pixel SNR. We plot the results from all image pairs together (5, 8, 15, 22, and 43GHz data are used). The curved lines represent power law fits to the individual image pairs; the frequency ratio of each pair is given in the legend. The errors increase significantly at SNR $\le 7$. We find that pixel SNR is the main factor determining the derived errors, although the errors in pixels with comparable SNR tend to increase slightly at larger distances from the phase center. This increase however is considerably weaker compared to the increase of errors due to lower pixel SNR.

 

Figure 4: Errors in spectral index due to different uv-coverages. The errors are plotted against the pixel flux scaled to the average self-calibration noise, $\sigma=5$mJy

The spectral index errors are shown in Fig. 4 for the same image pairs. Similarly to the fractional errors, the magnitude of spectral index errors increases rapidly at low SNR. The main difference is that the errors become progressively smaller at larger frequency separations, which follows obviously from the definition of the spectral index.

From the error distributions shown in Figs. 3-4, we conclude that multi-frequency VLBA observations with the time sampling interval of 10 minutes at each frequency can be compared with each other, for pixels which are located at moderate ($\sim10-15$mas) distances from the phase-tracking center, and have a sufficiently high SNR ($\mathrel{\raise .4ex\hbox{\rlap{$\gt$}\lower 1.2ex\hbox{$\sim$}}}5$). Within these limits (and with the applied observing and data reduction strategy), the fractional errors should not exceed $\sim$10%, a precision that can be sufficient for several purposes including spectral index and turnover frequency mapping in the nuclear regions of parsec-scale jets. Most of the large amplitude errors occur at jet edges where the effects of uneven spatial samplings are most pronounced. This effect is readily confirmed by Fig. 5 in which the distribution of fractional errors is plotted for the 5-15GHz image pair. The contours outline areas in which the errors are larger than 5%. These areas are concentrated at the jet edges, and cover a fairly small fraction of the entire source structure.

 

Figure 5: Distribution of fractional errors in a 5-15GHz image pair. Contours outline the regions with errors larger than 5%