THE VERY LONG BASELINE ARRAY GALACTIC PLANE SURVEY—VGaPS

VLBI can detect emission only from a compact component of a source, which should be bright enough to be detected above the noise background. Chances of finding a radio source sufficiently compact for VLBI detection are significantly enhanced if information about source spectra is available. For a majority of sources with flat spectra, i.e., with a spectral index of $\alpha > -0.5 \enskip (S\sim f^{\alpha })$, synchrotron emission from a compact core often dominates, so such sources are often compact. In regions more than 10° from the Galactic plane, comparison of surveys at different frequencies can be used to select the sources with flat spectra. However, in the crowded Galactic plane, source misidentification and confusion from surveys at different frequencies often result in ambiguous information about a source spectrum. Also, there are many extended sources with thermal emission, such as planetary nebulae and H ii regions, that have flat spectra but are too extended for VLBI detection.

To determine if a candidate source was sufficiently strong and compact to be a VLBI calibrator, we first used the VERA array at 22 GHz (Petrov et al. 2007a). In the survey ∼1400 objects within 6° of the Galactic plane and ∼1000 objects within 2° of known maser sources outside the Galactic plane were observed with the four-element VERA array at baselines of 1000–2000 km in two scans each, and approximately 20% were detected. Among the 533 detected sources, 305 objects are at Galactic latitudes of |b| < 10°, and 228 objects are at |b| > 10°. The list of detected sources included all objects with a probability of false detection <0.1. These objects formed the first set of candidate calibrators.

We also added 239 objects in the Galactic plane and 44 objects outside the Galactic plane, selected on the basis of their spectra using the Astrophysical CATalogs support System CATS (Verkhodanov et al. 2005). This database currently includes catalogs from 395 radio astronomy surveys. We selected all entries with sources within a 20'' radius and with measurements of flux density for at least three frequencies in the range 1.4–100 GHz. We fit a straight line to the logarithm of the spectrum and then estimated the spectral index and the flux density extrapolated to 24 GHz. We selected 451 objects with extrapolated correlated flux densities in the K-band >200 mJy, spectral indices flatter than −0.5, and |b| < 10°. Then, we scrutinized this list and removed sources within 30'' of known planetary nebulae or H ii regions and sources with anomalous spectra that indicated a possible misidentification. The remaining list was added to the pool of calibrators. This list also contained Sagittarius A* since it has never before been observed with VLBI in the absolute astrometry mode.

This set of sources formed the pool of 816 candidate objects to be followed up with the VLBA. Because of the large number of candidate objects, a priority from 1 to 4 was assigned to each source. The first priority was given to 180 new sources with correlated flux densities in the range 100–300 mJy detected with VERA that have never been observed with VLBI before. The second priority was given to sources in the Galactic plane detected with VERA, the third priority was given to other sources detected with VERA, and the least priority was given to sources outside the Galactic plane not detected with VERA. All new sources detected with VERA were scheduled. Sources outside of the Galactic plane were included for logistical reasons. Since the Galactic plane has an inclination with respect to the equator of 626, the density of candidate sources near the Galactic plane is a non-uniform function of right ascensions. We thus included sources outside the Galactic plane in the right ascension regions that had significantly fewer candidate sources than others in order to avoid gaps in the schedules.

In addition to the target sources, we also selected 56 objects from the K/Q survey (Lanyi et al. 2010) to serve as amplitude and atmosphere calibrators. These are the brightest sources in the K band with precisely known positions and publicly available images in FITS format.

A sufficiently good preliminary VLBA observing schedule for three 24 hr sessions was prepared automatically. Then it was manually adjusted in order to produce a more efficient final schedule.

The three sets of information needed in order to compile the schedule were: (1) the list of the 816 target sources with their J2000 positions and priority levels, (2) the list of the 56 calibration sources at 24 GHz, and (3) the two or three sidereal times ranges (at the array center at station PIETOWN) at which each source should be observed. These times were a function of the source declination. For example, a source with δ > 50° could be observed three times with very flexible time ranges, whereas a source with δ < −35°, must have tight ranges in order to be observed by at least eight VLBA antennas at an elevation higher than 10°.

The scheduling goal was to observe each source for three scans of 120 s, unless it was south of declination −25°, in which case only two scans were scheduled. With an average overhead of about 45 s between sources, about 1550 scans over the three days (72 hr) could be scheduled. Every 90 minutes, four scans were reserved for atmosphere calibrators. The algorithm then began filling in observing slots, taking the highest priority sources first, and those at the lowest declinations, since these have minimal scheduling flexibility. The algorithm scheduled sources that were relatively close to one another in the sky in order to minimize slew times, which can be as long as three minutes.

The slots containing the calibrators were chosen in order to maximize the range of elevations for the VLBA antennas. In practice, this meant maximizing elevations of observed sources at SC-VLBA, MK-VLBA and PIETOWN. It was important to have one low elevation observation for all telescopes, and this could be scheduled by observing a source either in the far north and/or in the far south. All three days were scheduled at the same time, since it did not matter on which day any of the three sidereal time slots occurred for a source.

The final schedule optimization was done "by hand" and consisted of several steps. First, some sources could be placed in only one or two slots and would have to be removed if another slot could not be found. Since about 10% of the slots could not be filled using the automatic algorithm because the source list was not uniformly distributed over the sky, additional slots were usually found to complete a source's schedule requirement. This often meant bending some of the rules or moving calibration sources or blocks by five to fifteen minutes. In regions where the target source density was large, scan integrations were decreased from 120 s to 110 s. All sources with priority 1 were included.

The next stage insured that the calibration scans were optimized in order to obtain good elevation coverage for the antennas. The purpose of these observations was (1) to serve as amplitude and bandpass calibrators, (2) to improve robustness of estimates of the path delay in the neutral atmosphere, and (3) to tie the source positions of new sources to existing catalogs such as the ICRF catalog (Ma et al. 1998). The final stage of optimization tweaked the observing schedule in obvious ways in order to save slewing time since the scheduling algorithm did not minimize slew times, and because of the above adjustments to insure proper observing coverage for each source. The schedule for each session was then carefully checked using the NRAO SCHED program to insure that each scan had sufficient integration time, and no more than one of the antennas was below 10° elevation (except for sources south of −35°).

The results of the scheduling and ultimate detection for each priority group is given in Table 1. The number of target sources selected in each group is given in Column 2, those scheduled in Column 3, those detected by the VLBA in Column 4, and the detection rate in the last column. The first two groups are sources detected in the VERA Fringe Search Survey; the last two groups contain other objects. The table does not include the 56 sources used as atmosphere calibrators.

The VGaPS observations were carried out in three 24 hr observing sessions at the VLBA on 2006 June 4, 2006 June 11, and 2006 October 20. Each target source was observed in several scans: three sources in one scan, 356 sources in two scans, one source in three scans, 124 in four scans and three sources in five scans. The scan durations were 100–120 s. In total, antennas spent 57% of time on target sources. In addition to these target objects, 56 strong sources previously observed in the K band were taken from the astrometric and geodetic catalog 2004f_astro.⁶

The data digitized at four levels were recorded with a rate of 256 Mbit s⁻¹ in eight 8 MHz wide intermediate frequencies (IF) bands spread over a bandwidth of 476 MHz (Table 2). The frequencies were selected to minimize the amplitude of sidelobes of the Fourier transform of the bandpass.

The data set then contains 128 data streams for each of the 45 baselines if all 10 VLBA antennas are operating: eight IFs, each with 16 equally space frequency channels. After the complex bandpass calibration, the phase difference among all of the frequency streams remains unchanged, and the relevant residual phase parameters associated with any scan are (1) the residual phase at the scan midpoint, (2) the average group delay (phase gradient with frequency), and (3) the average delay rate (delay gradient with time), for each antenna. These parameters are called the residual phase terms. These parameters are estimated with a fringe fitting procedure. Generally, one antenna is chosen as the reference—a well-behaved one near the array center—so that the set of residual phase terms is associated for each scan and all other antennas. These parameters are functions of many astrometric quantities (source position, site positions, antenna-based tropospheric path delays, Earth orientation parameters, etc.), which can be determined from analysis systems like Calc/Solve from data obtained from carefully prepared observing schedules. Source structure and dispersive phase effects (e.g., ionosphere) produce a non-linear phase/frequency relationship.

The algorithm that determines the residual phase terms is implemented in the AIPS task FRING, and in the past, all VLBA observations under the absolute astrometry and geodesy programs, such as VCS, RDV, and K/Q, were processed with the use of this software (Greisen 1988). When IFs are spread over a wide band with gaps, the AIPS algorithm determines the residual phase in two steps: first, for each of the eight IFs the residual phase, single-band group delay, and rates are independently obtained. Then, the residual phases of each IF are fit to a linear phase versus frequency term to produce the group delay, using the AIPS program MBDLY. This procedure is described in detail in Petrov et al. (2009).

The two-step approach has a substantial shortcoming: it requires fringe detection for each individual IF independently within a narrow band. Using all N IFs simultaneously for coherent averaging, we can detect a source with an amplitude smaller than $\sqrt{N}$ using only one IF. This degradation of the detection limit does not pose a problem for most geodesy observations, or for absolute astrometry of bright sources, since the target SNR is usually very high, but it significantly impacts absolute astrometric experiments of weak sources. The traditional AIPS algorithm did not detect a sizeable fraction of the sources observed in the VGaPS experiment.

This limitation motivated us to develop an advanced algorithm for wide-band fringe fitting across all of the IFs within the band. For logistical reasons, instead of augmenting AIPS with the new task, we decided to develop a new software package called PIMA⁸ from scratch that is supposed to replace the AIPS for processing absolute astronomy and geodesy experiments. Here we outline the method of wide-band fringe search used for processing this experiment.

3.2.1. Spectrum Re-normalization

Digitization of the input signal and its processing with a digital correlator causes an amplitude distortion with respect to an ideal analogue system. As documented by Kogan (1995), many effects distort both cross-correlation and auto-correlation spectra exactly the same way. Therefore, if we divide the cross-correlation spectrum by the auto-correlation spectrum averaged over time and over frequencies within each IF, we will remove these distortions. However, there are two effects that affect cross- and auto-spectra differently because the amplitude of the cross-spectrum is very low and the amplitude of the auto-correlation spectrum is close to 1.

The non-linear amplitude distortion of the digitized signal was studied in depth by Kogan (1998) who derived a general expression for the correlation coefficient of the digitized signal as a function of the correlation coefficient of the hypothetical analogue signal. In the absence of fringe stopping, the output correlation coefficient ρ_out is expressed via the correlation coefficient for an analogue case ρ as

$$\begin{eqnarray} \rho _{{\rm out}} &=& 2 \kappa \int _0^\rho \frac{\strut 1}{\strut \sqrt{1 - \rho ^2}}\, d \rho\nonumber \\ & & + 2 \kappa (n-1) \int _0^\rho \frac{\strut 1}{\strut \sqrt{1 - \rho ^2}} \left(e^{-\frac{v^2}{2 (1-\rho ^2)}} + e^{-\frac{v^2_1}{2 (1-\rho ^2)}} \right) d \rho \nonumber\\ & & + \kappa (n-1)^2 \int _0^\rho \frac{\strut 1}{\strut \sqrt{1 - \rho ^2}}\nonumber\\ &&\times \left(e^{-\frac{v^2_1 - 2\rho v_1 v_2 + v^2_2}{2 (1-\rho ^2)}} + e^{-\frac{v^2_1 + 2\rho v_1 v_2 + v^2_2}{2 (1-\rho ^2)}} \right) d \rho, \end{eqnarray} \tag{ 1 }$$

where κ is the normalization coefficient, n is the ratio of the two levels of quantization, and v₁ and v₂ are the dimensionless digitizer levels in units of variance of the input signal. Their numerical values compiled from the paper of Kogan (1993) are presented in Table 3.

Table 3. Numerical Coefficients in Integral (1) for Three Cases of the Number of Bits per Sample: (1,1), (1,2), (2,2)

Comb. of Bits	n	v1	v2	κ
(1,1)	1.0	0.0	0.0	0.3803
(1,2)	3.3359	0.0	0.9816	0.05415
(2,2)	3.3359	0.9816	0.9816	0.07394

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When ρ ≪ 1, the dependence ρ_out(ρ) becomes linear: ρ_out = 2/π · ρ ≈ 0.6366ρ for the case of one-bit sampling, and ρ_out ≈ 0.8825ρ for the case of two-bit sampling. Therefore, distortion of the cross-spectrum is proportional to ρ_out/ρ, and dividing the cross-spectrum by 0.6366 or 0.8825 we eliminate the distortion of the fringe amplitude introduced by the digitization.

However, the amplitude of the auto-correlation spectrum is close to 1, and we cannot ignore non-linearity of ρ_out(ρ). To correct the auto-correlation spectrum for digitization distortion, we follow the procedure outlined by Kogan (1995). First, the auto-correlation is inverse Fourier transformed. It should be noted that the correlator provides the auto-correlation for N spectral channels for non-negative frequencies from 0 to N − 1. We restore the auto-correlation at the Nth channel by linear extrapolation using the N − 2 and N − 1 values of the spectrum and set the spectrum for N − 1 negative frequencies to zero. The dimension of the Fourier transform is 2N. Second, the result of the transformation, the auto-correlation coefficient versus time lag, is de-tapered, i.e., divided by the self-convolution of the weighting function. The VLBA correlator normally uses uniform weighting, i.e., uses weight 1 for all points. The self-convolution of the uniform weighting is a triangle function ∧(i):

$$\begin{eqnarray*} \wedge (i) = \left\lbrace \begin{array}{@{}l @{\qquad } l} \dsty \frac{1}{N}(N - |i|) & \mbox{if} \quad |i| < N \vspace{4.25pt}\\ \dsty 0 & \mbox{otherwise.} \end{array} \right. \nonumber \end{eqnarray*}$$

Third, the correlation function is divided by its maximum which is found at the zero lag. Fourth, each correlation coefficient is divided by ρ_out(ρ). Fifth, the correlation function is multiplied back by the stored value at the zero lag. Sixth, the correlation function is again tapered by multiplying it by ∧(i). Finally, we perform the Fourier transform of the corrected correlation function and get the re-normalized auto-spectrum, free from digitization distortion. The square root of the product of the auto-correlation spectra from two stations of a baseline, averaged over time and frequency within each IF, gives us an estimate of the fringe amplitude scale factor, but before dividing the cross-correlation spectrum by this scale factor, we have to take into account a specific effect of the hardware VLBA correlator.⁹ An insufficient number of bits in internal correlator registers resulted in a decrease of the amplitude when it was large, and when that happened, the auto-correlation spectra was corrupted. Kogan (1993) suggested the following model for accounting for this effect:

$$\begin{equation} F = 1 + \frac{\strut w}{\strut 4 S A V_s}, \end{equation} \tag{ 2 }$$

where w is the weight of the spectrum data equal to the ratio of processed samples to the total number of samples in an accumulation period, A is the accumulation period length, S is 2 when the correlator processed single polarization data and 1 if both polarizations were correlated, and V_s is the visibility scale factor provided by the correlator. We divide the auto-correlation spectrum by the factor F.

3.2.2. Coarse Fringe Search

The correlator output provides auto-correlation and cross-correlation spectra for each pair of baselines and each scan. The cross-correlation spectrum is computed at a uniform two-dimensional (2D) grid of accumulation periods and frequencies and is accompanied with weights that are the ratio of the number of processed samples in each accumulation period to the nominal number of samples.

The fringe fitting procedure searches for phase delay τ_p, phase delay rate $\dot{\tau }_p$, group delay τ_g, and its time derivative $\dot{\tau }_g$ which correct their a priori values used by the correlator model in such a way that the coherent sum of weighted complex cross-correlation samples c_kj,

$$\begin{eqnarray} &&\hspace{-1pc} C(\tau _p,\tau _g,\dot{\tau }_p,\dot{\tau _g}) = \sum _k \sum _j c_{kj} w_{kj} \nonumber \\ &&\qquad\times e^{i(\omega _0 \tau _p \,{+}\, \omega _0 \dot{\tau }_p (t_k\,{-}\,t_0) \,{+}\, (\omega _j \,{-}\, \omega _0) \tau _g \,{+}\, (\omega _j \,{-}\, \omega _0) \dot{\tau }_g (t_k \,{-}\, t_0))}, \end{eqnarray} \tag{ 3 }$$

reaches the maximum amplitude. Index k runs over time, and index  j runs over frequencies. ω₀ and t₀ denote the angular reference frequency within the band and the reference time within a scan and w_kj is weight. Function $C(\tau _p,\tau _g,\dot{\tau }_p,\dot{\tau _g})$ is essentially non-linear, and we need a really good starting value in order to find the global maximum by traditional optimization algorithms. We can notice that term 2πω₀τ_p in expression (3) does not depend on the summation indices, and $\dot{\tau }_g$ is usually small. Therefore, for the purpose of a coarse fringe search we simplify expression (3) to

$$\begin{eqnarray} \hspace{-10.0pt} C(\tau _p,\tau _g,\dot{\tau }_p) e^{-i2\pi \omega _0 \tau _p} \approx & \displaystyle \sum _k \sum _j c_{kj} \times e^{i(\omega _0 \dot{\tau }_p (t_k-t_0) \,{+}\, (\omega _j - \omega _0) \tau _g)}. \nonumber\\ \end{eqnarray} \tag{ 4 }$$

For the search of the maximum, the trial functions $C(\tau _p,\tau _g,\dot{\tau }_p)$ are computed on a dense grid of the search space $\tau _g,\dot{\tau }_p$. It follows immediately from expression (4) that $|C| = |{\cal {F}}(c_{kj})|$, where $\cal {F}$ denotes the 2D Fourier transform.

The first step of the coarse fringe search is to compute the 2D fast Fourier transform (FFT) of the matrix of the cross-correlation spectrum. The first dimension of the matrix runs over time, and the second dimension runs over frequency. The sampling intervals are Δt/β and Δf/γ, where Δt and Δf are the duration of the accumulation period and the spectral resolution, respectively, and β and γ are integer oversampling factors. The elements of the matrix that do not have measurements or have discarded measurements are padded with zeroes. The dimensions of the matrix are chosen to have a power of 2 for gaining the maximum performance of the FFT numerical algorithm.

Oversampling factors greater than 1 are used to mitigate amplitude losses. The FFT produces estimates of |C| at a discrete grid of group delays and delay rates. If the maximum of the amplitude of the coherent sum of the cross-correlation function samples falls at group delays and delay rates between the nodes of the grid, its magnitude will be greater than the amplitude of |C| at the nearest grid point by a factor of L.

The amplitude loss factor L of the coarse search matrix is the integral average over time and frequency:

$$\begin{eqnarray} L &= & \frac{\strut 1}{\strut t_s} \displaystyle \int ^{t_s/2}_{-t_s/2} \cos 2\pi \left(\omega _0 \tau _p - \frac{\strut k}{\strut \beta t_s} \right) t dt \nonumber\\ && \times\frac{\strut 1}{\strut f_b} \displaystyle \int ^{f_b/2}_{-f_b/2} \cos 2\pi \left(\omega _0 \tau _g - \frac{\strut l}{\strut \gamma f_b} \right) f df, \end{eqnarray} \tag{ 5 }$$

where k and l are indices of the nearest grid nodes, t_s is the scan duration, and f_b is the total bandwidth. The integral (5) is easily evaluated analytically, and the maximum losses L = sinc(π/(2β)) · sinc(π/(2γ)) are achieved when $\dot{\tau _p}$ and τ_g happen to be just between grid nodes. In a case where the oversampling factor is 1, L = 4/π² ≈ 0.405. That loss factor effectively raises the detection limit by 1/L = 2.467 in the worst case. We used the grid 4096 × 4096 which corresponds to β = 4, γ = 4 for t_s = 120 s, f_b = 476 MHz. Therefore, the maximum loss factor for our experiment is 0.959, which results in raising the detection limit by no more than 4.1%.

The group delay and delay rate that correspond to the maximum of the discrete Fourier transform of the cross-correlation matrix provide the coarse estimates of group delay and delay rate. Their accuracy depends on the grid resolution. The next step is to refine their estimates. The first stage of the fine search is an iterative procedure that computes |C| at a progressively finer 3D grid in close vicinity of the maximum using the discrete 3D Fourier transform. The third dimension is group delay rate, omitted during the coarse search. Dimensions of the transform for the group delay, phase delay rate, and group delay rate are 3, 3, and 9, respectively. At the first step of iterations, the grid runs over ±1 element of the coarse grid for group delays and phase delay rates and in the range ±2 × 10⁻¹¹ for the group delay rate. After each step of iterations, the grid centered around the maximum element shrinks its step by 2. In total, eight iterations are run. The phase of the coherent sum of the cross-correlation function determined with the iterative procedure according to expression (4) is the fringe phase with the opposite sign.

3.2.3. Probability of False Detection

The significance of the fringe amplitude depends on the level of noise. In the absence of signal, the amplitude of the cross-correlation function a has a Rayleigh distribution of:

$$\begin{equation} p(a) = \frac{a}{\sigma ^2} e^{-\frac{a^2}{2\sigma ^2}}, \end{equation} \tag{ 6 }$$

where σ is the standard deviation of the real and imaginary part of the cross-correlation function and n is the total number of spectrum points. In a case where all points of the spectrum are statistically independent, the cumulative distribution function of the coherent sum over n points is (Thompson et al. 2001)

$$\begin{equation} P(a) = \left(1 - e^{- \frac{ a^2}{2\sigma ^2} } \right)^n. \end{equation} \tag{ 7 }$$

Differentiating this expression, we find the probability density function of the ratio of the amplitude of the coherent sum of spectrum samples of the noise to its standard deviation as

$$\begin{equation} p(a) = n \frac{\strut a}{\strut \sigma ^2} e^{-\frac{a^2}{2\sigma ^2}} \left(1 - e^{-\frac{a^2}{2 \sigma ^2}}\right)^{n-1}. \end{equation} \tag{ 8 }$$

Under the assumption that all samples are statistically independent, the variance of the noise of the coherent sum of N samples is scaled as $\sigma = \sigma _s/\sqrt{t_s S_r}$, where S_r is the sampling rate of the recorded signal (6.4 × 10⁷ in our experiment), t_s is the scan duration, and σ_s is the variance of an individual sample, 1 for a perfect system.

However, the assumption of statistical independence is an idealization. The presence of systematic phase errors, deviation of the bandpass from the rectangular shape, and other factors distort the distribution. The deviation from the statistical independence is difficult to assess analytically.

We evaluate the variance of the noise by estimating the variance over a sample of 32,768 random points in the region of the Fourier transform of the coherent sum of the cross-correlation that does not contain the signal. The indices of grid elements of the sample are produced by using the random number generator. The sample of amplitudes is ordered, and half of the points greater than the median are rejected. The variance over the remaining points is computed and an iterative procedure is launched that adds back previously rejected points in ascending order of their amplitudes and updates the mean value and variance. The iterations are run till the maximum amplitude of the next sample reaches 3.5σ_a. The initial rejection and consecutive restoration of points with amplitudes >3.5σ_a ensures that no points with signal from the source affect the computation of σ_a and 〈a〉. The rejection of the tail of the amplitude distribution causes a bias in estimate of the mean, but the magnitude of the bias is only −2 × 10⁻⁴, which is negligible. We define a signal-to-noise ratio as SNR = a/〈a〉. It follows immediately from expression (6) that $\langle a \rangle = \sqrt{\frac{\pi }{2}} \sigma$.

We assume that the a posteriori distribution of the SNR s = a/σ_a can be approximated as a function like expression (8) with effective parameters of σ_eff and n_eff:

$$\begin{equation} p(s) = \frac{\strut 2}{\strut \pi } \frac{\strut n_{\rm eff}}{\strut \sigma _{\rm eff}} s e^{-\frac{s^2}{\pi }} \left(1 - e^{-\frac{ s^2}{\pi }} \right)^{n_{\rm eff}-1}. \end{equation} \tag{ 9 }$$

These parameters σ_eff and n_eff can be found by fitting the left tail of the empirical distribution of the SNRs. Using their estimates, we can evaluate the probability of finding an amplitude less than a if no signal is present, i.e., the probability of false detection:

$$\begin{equation} P_f(s) = 1 - \int _0^s p(s)\, ds = 1 - \frac{1}{\sigma _{\rm eff}} \left(1 - e^{-\frac{s^2}{\pi }} \right)^{n_{\rm eff}}. \end{equation} \tag{ 10 }$$

The low end of the empirical SNR distribution and the table of the probability of false detection for VGaPS experiments are shown in Figure 1 and Table 4.

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Table 4. Probability of False Detection as a Function of SNR

SNR	Pf(s)
4.96	0.3
5.19	0.1
5.61	0.01
5.99	0.001
6.68	10−5

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3.2.4. Fine Fringe Search

Finally, the group delay, phase delay rate, group delay rate, and fringe phase at the reference frequency are estimated using LSQ in the vicinity of the maximum provided its amplitude exceeds the detection limit. The goal of the LSQ refinement is to obtain realistic estimates of statistical errors of fitted parameters and to account for possible systematic errors by analyzing residuals. All cross-correlation spectrum data points of a given observation are used in a single LSQ solution with weights reciprocal to their variance.

Determining the variance of the fringe phase of an individual point is trivial only in two extreme cases: when SNR ≪ 1, and when SNR ≫ 1. In the first case the fringe phase distribution becomes uniform in the range of [0, 2π] with a variance of $\pi /\sqrt{3} \approx 1.813$. In the second case expanding the expression for fringe phase in the presence of noise $\phi = \arctan \frac{S_i + n_i}{S_r + n_r}$, where S_i and S_r are the real and imaginary parts of the signal and n_i and n_r are the real and imaginary parts of the noise, into the Taylor series, neglecting terms O(n/S²), and evaluating the variance of the expansion, we get $\sigma _\phi = \sqrt{\vphantom{A^A}\smash{\hbox{${\frac{2}{\pi }}$}}}\frac{1}{\mbox{\sc {s/n}}}$. For the general case, the problem becomes more difficult, since the σ_ϕ depends on the variance of the noise and on the amplitude of the signal nonlinearly. An analytical solution requires evaluation of complicated integrals that are not expressed via elementary functions.

We used the Monte Carlo approach to compute these variances. Let us consider a random complex set s = A  +  n_r  +  in_i, where A is the amplitude of the simulated signal and n_r, n_i are independent random variables with Gaussian distribution. Their variance σ was selected in such a way that $A = \sqrt{\frac{\pi }{2}} \sigma {\rm SNR}$. Then for a given SNR, we can compute the variance of phase of s as a function of the normalized amplitude |s|/σ. This is done by generating a long series (1 billion points) of the simulated complex signal for a given SNR, computing the amplitude and phase of the time series, splitting the signal into a uniform grid of 128 bins over normalized amplitude that spans the interval [SNR − 4.5, SNR + 4.5], computing the variance of the phase of the simulated signal over all points that fall into each bin, and approximating the dependence of σ_ϕ(|s|/σ) with a smoothing B-spline of third order over six nodes. We computed σ_ϕ(|s|/σ) for SNRs in the range [0, 12.7] with steps of 0.1. Examples of this dependence for several SNRs are shown in Figure 2. The set of B-spline coefficients forms a 2D table with axes SNR and normalized amplitude which allows us to evaluate σ_ϕ for a given SNR and a given fringe amplitude.

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It should be noted that the SNR of the coarse search is related to the amplitude coherently averaged over all valid cross-correlation spectrum samples, ∑Ntk∑Nsjwkj, where Nt is the number of accumulation periods and Ns is the total number of spectral channels. The SNR of an individual accumulation period (the elementary SNR) is $\sqrt{ \sum _k^{N_t}\sum _j^{N_s} w_{kj} }$ times less. For our experiments, the typical reduction of the SNR is a factor of 340. This means that for almost all sources the elementary SNR will be less than 1. As we have seen previously, the distribution of fringe phases at very low SNRs is close to uniform with a variance of $\pi /\sqrt{3}$. The phase becomes uncertain due to the 2π ambiguity, and the LSQ estimation technique loses its diagnostic power. Therefore, the cross-correlation function with applied phase, phase delay rate, and group delays computed during the coarse fringe search have to be coherently averaged over time and frequency in segments large enough to have sufficiently high SNR over a segment to provide an ambiguous phase. As seen in Figure 3, the fringe phase distribution at SNR = 1 is already sharp enough for that. Therefore, the number of spectral channels and the number of accumulation periods within a segment is chosen in such a way that the SNR is at least 1. Marginally detected scans with SNR = 5 have 24 segments that average all spectral channels within an IF and over 1/3 of the scan interval.

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Using all segments, we determine four fitting parameters $\bf p$ using the LSQ:

$$\begin{equation} {\bf p} = ({{A}^{\!\bf \top }} {\cal W} A)^{-1} {{A}^{\bf \top }} {\cal W} \mbox{\boldmath $\phi _s$}, \end{equation} \tag{ 11 }$$

where A is the matrix of observations, ${\cal W}$ is the diagonal weight matrix and $\mbox{\boldmath $\phi _s$}$ is the vector phases of the cross-correlation function averaged over segments.

The mathematical expectation of the square of the weighted sum of residuals R in the presence of noise is

$$\begin{equation} E(R) = \rm Tr ({\cal W} {\rm Cov}\,\epsilon) - {\rm Tr} ({\rm Cov}\,{\epsilon } {\cal W} A ({{A}^{\bf \top }} {\cal W} A)^{-1} {{A}^{\bf \top }} {\cal W}),\quad \end{equation} \tag{ 12 }$$

which is reduced to n − m, where n is the number of equations and m is the number of estimated parameters if the weight matrix ${\cal W}$ is chosen to be (Cov)⁻¹.

The presence of additive errors, for example fluctuations in the atmosphere, will increase Cov and our estimate of the error variance based on the amplitude of the spectrum sample without knowledge of the scatter of the cross-spectrum phases is incomplete. A small additive noise with variance k times less than the amplitude of the signal affects the amplitude as O(k²), but it affects the phase as O(k). One of the measures of the incompleteness of the error model is the ratio of the square of the weighted sum of residuals to its mathematical expectation.

We can extend our error model assuming that the weight matrix is ${\cal W} = {\rm Cov\epsilon }^{-1} - q^2 I$, where I is the unit matrix of the same dimensions as ${\cal W}$ and q is the parameter. This model is equivalent to an assumption that the used squares of weight of each segment are less than the true one by some parameter q equal to all segments. The mathematical expectation of R for this additive weighting model is

$$\begin{equation} E(R) = n - m + q^2 [ {\rm Tr} ({\cal W}) - {\rm Tr} ({\cal W} (A {\cal W}^{2} A)^{-1} {{A}^{\bf \top }}) ]. \end{equation} \tag{ 13 }$$

Inverting Equation (13), we find q for a given E(R):

$$\begin{equation} q = \sqrt{ \frac{\strut E(R) - (n - m)}{\strut {\rm Tr} ({\cal W}) - {\rm Tr} ({\cal W} (A {\cal W}^2 A)^{-1} {{A}^{\bf \top }}) }. } \end{equation} \tag{ 14 }$$

Replacing the mathematical expectation of R with its value evaluated from the residuals, we can find the re-weighting parameter q for a given solution. Several iterations provide a quick convergence of $\frac{E(R)}{R}$ to 1. Applying an additive re-weighting constant results in an increase in estimates of parameter uncertainties. It may happen that q becomes imaginary. This means that ${\cal W}$ was overestimated. In our analysis we set q = 0 when that happens.

3.2.5. Complex Bandpass Calibration

Coherent averaging of the cross-correlation spectrum over frequencies assumes that the data acquisition system does not introduce a distortion of the recorded signal, but this is usually not the case. Each intermediate frequency has its own arbitrary phase offset and group delay that may vary with time. The imperfection of baseband filters results in a non-rectangular shape of the amplitude response.

To calibrate these effects, a rail of narrow-band phase calibrator signals with a spacing of 1 MHz was injected near the receivers. Two tones per IF were extracted by the data acquisition hardware, and their phase and amplitude are available for data analysis. When the phase of the phase calibration signal is subtracted from fringe phases, the result is referred to the point of injection of the phase calibration signal and this procedure is supposed to take into account any phase changes that occurred in the signal passing through the data acquisition terminal. However, we should be aware of several complications that emerge when we try to use the benefits of the phase calibration signal. First, the phases of two tones of the phase calibration signal separated by 6 MHz, as in our sessions, have ambiguities. Since the instrumental group delay may reach several phase turns over an 8 MHz IF, the second phase calibration tone is useless without resolving the ambiguity. Second, the phase calibration itself may have a phase offset or may become unstable if its amplitude is not carefully tuned. Therefore, we need to re-calibrate the phase calibration signal itself in order to successfully apply it to data.

We compute the complex bandpass function for each station, except the reference station, that describes the residual instrumental complex bandpass after applying the phase calibration signal. The cross-correlation spectrum needs to be divided by the complex bandpass in order to correct the instrumental frequency-dependent delay and fading of the amplitude. The procedure for evaluating the complex bandpass has several steps.

First, all data are processed applying the first tone of the phase calibration, i.e., the phase of the phase calibration signal is subtracted from each phase of the cross-correlation signal at a given IF.

Second, the bandpass reference station is chosen. Then, for each station, we find an observation at a baseline with the reference station that provided the maximum SNR during the first run. Then for each IF we average the residual spectrum over time and perform a linear fit to the residual phases to determine the instrumental group delay. Using this instrumental group delay, we extrapolate the phase of the phase calibration signal of the first phase calibration tone to the frequency of the second phase calibration tone and resolve its phase ambiguity. After that, we re-run the fringe search procedure for these scans by applying the phase calibration phase to each IF in the form of a linear function of phase versus frequency that is computed from two phase calibration tones with resolved phase ambiguity. The result of the new fringe search gives a new residual spectrum. Then we averaged the residual spectrum over time and over M segments within each IF (M = 2 in our experiment). The amplitude of the spectrum is normalized by dividing the average amplitude over all IFs. The phase and the amplitude of the residual averaged spectrum are approximated with a fifth degree Legendre polynomial. The result of this approximation as a complex function of frequency defines the so-called initial complex bandpass. Analysis of the residuals of rejected observations helps to diagnose malfunction of the equipment. For example, one or more video converters may fail, which may result in a loss of coherence. In that case, the part of the affected cross-correlation spectrum is masked out.

Third, we refine the complex bandpass. We select N more observations with the highest SNR from the first run at all baselines (N = 16 in our experiment) and re-run the fringe fitting procedure with applied phase (but not amplitude!) of the initial bandpass. We compute a residual spectrum averaged over time and M segments for each processed observation and normalize its amplitude. Then we fit a set of six coefficients of the Legendre polynomial for each station, except the reference one, for each IF, for both amplitude and phase to the phase and amplitude of the residual spectrum using a single LSQ solution. Then the residuals are computed and the observations with the maximum absolute value of residual phases and residual amplitudes are found separately. If the maximum absolute value of residual phase or residual amplitude exceeds the predefined limit, the affected observation is removed and the solution is repeated. Iterations are run until either the absolute values of all remaining residuals are less than the predefined limit or the number of observations at a given baseline drops below N/2. The fitted Legendre polynomial coefficients are added to the coefficients of the initial bandpass and the result defines the so-called fine complex bandpass.

Fourth, all observations are reprocessed with the fine bandpass applied: the phase of the fine complex bandpass of the remote station of a baseline is subtracted and the phase of the bandpass of the reference station of a baseline is added before fringe fitting, and the amplitude is divided by the square root of the products of bandpass amplitudes after fringe fitting. Examples of residual phases before and after calibration are shown in Figure 4.

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The results of the fringe search are residual phase and group delay as well as their time derivatives determined from analysis of an observation at a given baseline of a given scan with respect to the a priori delay model used by the correlator. We need to compute the total path delay related to a certain moment of time common to all observations of a scan, called a scan reference time (t_srt).

For logistical reasons, fringe searching at different baselines is performed independently and therefore, each observation has its own reference epoch, called fringe reference times (t_frt). This time epoch is computed as the weighted mean epoch of a given observation. Since stations usually start and end at slightly different times and the number of processed samples may be different, in general, t_frt is different at different baselines of the same scan. The scan duration may be significantly different at different stations either by a schedule design when antennas with different sensitivities participate in observations—this is often made in geodetic observations, or due to losses of some of the observing time at stations for various reasons. If we set t_srt as an average of t_frt over all baselines of a scan, it may happen that for some baselines the difference t_srt − t_frt may reach several hundred seconds. Setting a common t_srt for as many baselines as possible is desirable since it allows computation of delay triangle misclosures and some other important statistics. On the other hand, the uncertainty of the group delay estimate is minimal at t_frt. The uncertainty in group delay at t_srt becomes

$$\begin{equation} \sigma ^2_{\tau }(t_{\rm srt}) = \sigma ^2_{\tau }(t_{\rm frt}) \,{+}\, 2 {\rm Cov}(\tau, \dot{\tau }) (t_{\rm srt} \,{-}\, t_{\rm frt}) \,{+}\, \sigma ^2_{\dot{\tau }}(t_{\rm srt} \,{-}\, t_{\rm frt})^2,\quad \end{equation} \tag{ 15 }$$

which is undesirable. In our work, we set the tolerance limit for the growth of the uncertainty due to the differences between t_srt and t_frt to 0.1σ or 5 ps, whichever is less. Setting this limit, we find for each observation the maximum allowed |t_srt − t_frt| by solving quadratic Equation (15). In a case where all observations of a scan have overlapping intervals of acceptable scan reference times, we set it to the value that minimizes $2 \sum {\rm Cov}(\tau, \dot{\tau }) (t_{\rm srt} - t_{\rm frt}) + \sigma ^2_{\dot{\tau }}(t_{\rm srt} - t_{\rm frt})^2$ over all observations. In the case where there are observations that have intervals of acceptable t_srt that are not overlapping, the set of observations of a scan is split into several subsets with overlapping acceptable t_srt and the optimization procedure is performed under each subset. Finally, t_srt is rounded to the nearest integer second.

The VLBA correlator shifts the time tag of data streams from each station to the moment of time when the wavefront reaches the center of the coordinate system. This operation facilitates correlation and allows station-based processing. The a priori path delay is computed for this modified time tag. This shift of the time tag depends on the a priori parameters of the geometric models, and therefore the total path delay produced from such a modified quantity would depend on errors of the a priori model which would considerably complicate data analysis. Therefore, we have to undo this shift of the time tag for the reference station of a baseline for further processing.

The correlator delay model for the VLBA hardware and software correlators is computed as a sum of a fifth-degree polynomial fit to the geometric delays, the linear clock offset, and the coarse atmospheric model delay over intervals of 120 s. We find the a priori delay of the baseline reference station τ^rf_a related to the time tag at TAI of the wavefront arrival to its phase center from the implicit equation

$$\begin{equation} \tau ^{\rm rf}_a = \sum _{k=0}^{k=5} a^{\rm rf}_i \big(t_{\rm srt} - t_o - \big(\tau ^{\rm rf}_a - \tau ^{\rm rf}_{\rm cl} - \dot{\tau }^{\rm rf}_{\rm cl} \tau ^{\rm rf}_a - \tau ^{\rm rf}_{\rm atm} \big) \big) ^k,\quad \end{equation} \tag{ 16 }$$

which is solved by iterations. Here, t_o is the TAI time tag of the polynomial start time and τ^rf_cl and τ^rf_atm are the clock model and the atmosphere contribution of the a priori path model. We set τ^rf_a on the right-hand side of expression (16) to zero for the first iteration. Three iterations are sufficient to reach the accuracy of 0.1 ps. Using the value τ^rf_a found at the last iteration, we compute the a priori path delay for the remote station of the baseline:

$$\begin{equation} \tau ^{\rm rm}_a = \sum _{k=0}^{k=5} a^{\rm rm}_i \big(t_{\rm srt} - t_o - \big(\tau ^{\rm rf}_a - \tau ^{\rm rm}_{\rm cl} - \dot{\tau }^{\rm rm}_{\rm cl} \tau ^{\rm rf}_a - \tau ^{\rm rm}_{\rm atm} \big) \big) ^k. \end{equation} \tag{ 17 }$$

The a priori delay rate is computed using an expression similar to Equation (17). Finally, we compute the total path delay by extrapolating the residual delay to the scan reference time:

$$\begin{equation} \tau _{{\rm tot}} = \tau ^{\rm rm}_{\rm apr} - \tau ^{\rm rf}_{\rm apr} + \tau _{\rm res} + \dot{\tau }_{\rm res} (t_{\rm srt} - t_{\rm frt}). \end{equation} \tag{ 18 }$$

The delay produced this way is the difference between the interval of proper time measured by the clock of the remote station between events of arrival of the wave front to the reference point of the remote antenna and clock synchronization in TAI and the interval of proper time measured by the clock of the reference station between events of arrival of the wave front to the reference point of the reference antenna and clock synchronization in TAI.

Our computation of theoretical time delays in general follows the approach presented in detail by Sovers et al. (1998) with some refinements. The most significant ones are the following. The advanced expression for time delay derived by Kopeikin & Schäfer (1999) in the framework of general relativity was used. The displacements caused by the Earth's tides were computed using the numerical values of the generalized Love numbers presented by Mathews (2001) following a rigorous algorithm described at Petrov & Ma (2003) with truncation at a level of 0.05 mm. The displacements caused by ocean loading were computed by convolving the Green's functions with ocean tide models using the NLOADF algorithm of Agnew (1997). The GOT00 model (Ray 1999) of diurnal and semi-diurnal ocean tides, the NAO99 model of ocean zonal tides (Matsumoto et al. 2000), the equilibrium model (Petrov & Ma 2003) of the pole tide, and the tide with a period of 18.6 years were used. Atmospheric pressure loading was computed by convolving the Green's functions with the output of the atmosphere NCEP Reanalysis numerical model (Kalnay et al. 1996). The algorithm of computations is described in full detail in Petrov & Boy (2004). The empirical model of harmonic variations in the Earth orientation parameters heo_20091201¹⁰ derived from VLBI observations according to the method proposed by Petrov (2007) was used. The time series of UT1 and polar motion from the Goddard operational VLBI solutions were used as a priori.

The a priori path delays in the neutral atmosphere in directions toward observed sources were computed by numerical integration of differential equations of wave propagation through the heterogeneous media. The four-dimensional field of the refractivity index distribution was computed using the atmospheric pressure, air temperature, and specific humidity taken from the output of the Modern Era Retrospective-Analysis for Research and Applications (MERRA; Schubert et al. 2008). That model presents the atmospheric parameters at a grid 1/2° × 2/3° × 6^h at 72 pressure levels.

To consider the contribution of the ionosphere to the phase of the cross-correlation spectrum, note that the electromagnetic wave propagates in a plasma with phase velocity

$$\begin{equation} v_p = \frac{\strut c}{\strut \sqrt{ 1 - \frac{\strut N_v e^2}{\strut m_e \epsilon _o \omega ^2} }}, \end{equation} \tag{ 19 }$$

where N_v is the electron density, e is the charge of an electron, m_e is the mass of an electron, _o is the permittivity of free space, ω is the angular frequency of the wave, and c is the velocity of light in a vacuum. Phase velocity in the ionosphere is faster than the velocity of light in a vacuum.

After integration along the ray path and expanding expression (19), withholding only the term of the first order, we get the following expression for additional phase rotation caused by the ionosphere:

$$\begin{equation} \Delta \phi = {-} \frac{\strut a}{\strut \omega }, \end{equation} \tag{ 20 }$$

where ω is the angular frequency and a is

$$\begin{equation} a = \frac{\strut e^2}{\strut 2 c m_e \epsilon _o } \left(\int N_v d s_1 - \int N_v d s_2 \right). \end{equation} \tag{ 21 }$$

Here, s₁ and s₂ are the paths of wave propagation from the source to the first and second stations of the baseline. If ∫N_vds is expressed in units of 1 × 10¹⁶ electrons m⁻² (so-called total electron contents (TEC) units), then after having substituted values of constants, we get a = 5.308018 × 10¹⁰ s⁻¹ times the difference in the TEC values at the two stations.

Since the ionosphere contribution is frequency-dependent, it distorts the fringe-fitting result. Taking into account that the bandwidth of the recorded signal is small with respect to the observed frequency, we can linearize Equation (20) near the reference frequency ω_o: ϕ = −a/ω_o + a(ω_i − ω)/ω²_o. Comparing it with expression (4), we see that the first frequency-independent term contributes to the phase delay and the second term, linear with frequency, contributes to the group delay. The fine fringe search is equivalent to solving the LSQ for τ_p and τ_g using the following system of equations:

$$\begin{equation} \tau _p \omega _o + \tau _g (\omega _k - \omega _o) = \phi _i + \frac{\strut a}{\strut \omega _i}, \end{equation} \tag{ 22 }$$

where index i runs over frequencies and index k runs over accumulation periods.

A solution of the 2 × 2 system of normal equations that originates from Equation (22) can be easily obtained analytically. Gathering terms proportional to a, we express the contribution of the ionosphere to phase and group delay as τ^iono_p = −a/ω_p² and τ^iono_g = a/ω_g², where ω_p and ω_g are effective ionospheric frequencies:

$$\begin{eqnarray} &&\hbox{\fontsize{11}{13}\selectfont$\omega _p = \sqrt{ \omega _o \frac{\strut \sum _i^n w_i \,{\cdot}\, \sum _i^n w_i (\omega _i - \omega _o)^2 \,{-}\, \big(\sum _i^n w_i(\omega _i \,{-}\, \omega _o) \big) ^2 }{\strut \sum _i^n w_i (\omega _i \,{-}\, \omega _o) \sum _i^n w_i \frac{(\omega _i \,{-}\, \omega _o)}{\omega _i} \,{-}\, \sum _i^n w_i (\omega _i \,{-}\, \omega _o)^2 \,{\cdot}\, \sum _i^n \frac{w_i}{\omega _i} } },$} \nonumber \\ &&\hbox{\fontsize{11}{13}\selectfont$\omega _g = \sqrt{\frac{{\strut \sum _i^n w_i \,{\cdot}\, \sum _i^n w_i(\omega _i \,{-}\, \omega _o)^2 \,{-}\, \left(\sum _i^n w_i(\omega _i \,{-}\, \omega _o) \right) ^2 }}{{\strut \sum _i^n w_i(\omega _i \,{-}\, \omega _o)^2 \sum _i^n \frac{w_i}{\omega _i} \,{-}\, \sum _i^n w_i \,{\cdot}\, \sum _i^n w_i \frac{(\omega _i \,{-}\, \omega _o)}{\omega _i} }}}, $}\nonumber\\ \end{eqnarray} \tag{ 23 }$$

here w_i is the weight assigned to the fringe phase at the ith frequency channel.

They have a clear physical meaning: if the wide-band signal was replaced by a quasi-monochromatic signal with a group or phase effective ionosphere frequency, then the contribution to group or phase delay of the wide-band signal would be the same as the contribution of the quasi-monochromatic signal at those effective frequencies.

To compute the contribution of the ionosphere, we used the TEC maps from analysis of linear combinations of GPS observables made at two frequencies, 1.2276 and 1.57542 GHz. Using GPS-derived TEC maps for data reduction of astronomic observations, first suggested by Ros et al. (2000), has became a traditional approach for data processing. Analysis of continuous GPS observations from a global network comprising 100–300 stations makes it feasible to derive an empirical model of the TEC over the span of observations using the data assimilation technique. Such a model has been routinely delivered by GPS data analysis centers since 1998. For our analysis we used the TEC model provided by the GPS analysis center CODE (Schaer 1998). The model gives values of TEC in the zenith direction on a regular 3D grid with resolutions 5° × 5° × 2^h.

For the purpose of modeling, the ionosphere is considered a thin spherical layer with constant height H above the Earth's surface. The typical value of H is 450 km. In order to compute the TEC from GPS maps, we need to know the coordinates of the point at which the ray pierces the ionosphere—point J in Figure 5. First, we find the distance from the station to the ionosphere piercing point D = |AJ| by solving triangle OAJ. Noticing that |OA| = R_⊕ and |OJ| = R_⊕ + H, we immediately get

$$\begin{eqnarray} \beta & = & \arcsin \dsty\frac{\strut \cos E }{\strut 1 + \frac{H}{R_\oplus }},\nonumber\\ D & = & R_\oplus \sqrt{ 2 \frac{\strut H}{\strut R_\oplus } \left(1 - \sin \left(E + \beta \right) \right) + \left(\frac{\strut H}{\strut R_\oplus } \right) ^2 }, \end{eqnarray} \tag{ 24 }$$

where E is the elevation of the source above the horizon.

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Then the Cartesian coordinates of point J are $\vec{r} + D\vec{s}$. Transforming them into polar coordinates, geocentric latitude and longitude, we get arguments for interpolation in the 3D grid. We used the 3D B-spline interpolation by expanding the TEC field into the tensor products of basic splines of the third degree. Interpolating the TEC model output, we get the TEC through the vertical path |JI_o|. The slanted path |JI₁| is |JI_o|/cos β. Therefore, we need to multiply the vertical TEC by 1/cos β(E), which maps the vertical path delay through the ionosphere into the slanted path delay. Here we neglect the ray path bending in the ionosphere. We also neglect Earth's ellipticity, since the Earth was considered spherical in the data assimilation procedure of the TEC model.

Combining equations, we get the final expression for the contribution of the ionosphere to path delay:

$$\begin{equation} \tau _{{\rm iono}} = \pm \frac{\strut a}{\strut 4 \pi ^2 f^2_{\rm eff} } \mbox{TEC} \frac{\strut 1}{\strut \cos {\beta (E)}} {,} \end{equation} \tag{ 25 }$$

where f_eff is the effective cyclic frequency, the plus sign is for the group delay, and the minus sign is for the phase delay.

Computation of the theoretical path delay and its partial derivatives over model parameters is made using VTD software.¹¹

Astrometric analysis is made in several steps. First, each individual 24 hr session is processed independently. The parameter estimation model includes estimation of (1) clock functions presented as a sum of a second-degree polynomial and a linear spline over 60 minutes, (2) residual zenith atmosphere path delay for each station presented as a linear spline, (3) coordinates of all stations, except a reference station, and (4) coordinates of the target sources. The goal of the coarse solution is to identify and suppress outliers. The main reasons for outliers are (1) errors in determining the global maximum of the fringe amplitude during fringe search and (2) false detections. Both errors decrease with increasing the SNR. Because of this, we initially run our solution by restricting to SNR ⩾ 6, and then restore good detections with SNR in the range of [5, 6].

After identifying outliers and removing them from our solution, we apply estimates of the parameters of the a posteriori model to outliers, which allows us to predict the path delay with accuracy better than 500 ps, except for sources that had fewer than two detections. Then we re-run the fringe search for outliers and restrict the search window to ±1000 ps with respect to predicted delay. We also lower the SNR detection limit to 4.8, since the number of independent samples in the restricted search window, and therefore, the probability of false detection at a given SNR is less. This procedure allows to restore from 50% to 80% of observations marked as outliers in the previous step. The weights of observables were computed as $w = 1/\sqrt{\sigma _o^2 + r^2(b)}$, where σ_o is the formal uncertainty of group delay estimates and r(b) is the baseline-dependent re-weighting parameter that was evaluated in a trial solution to make the ratio of the weighted sum of the squares of residuals to its mathematical expectation to be close to unity using the technique similar to that used for fine fringe search.

Finally, we run a global VLBI solution that uses all available observations to date, 7.56 million, from 1980 April through 2010 August in a single LSQ run. The estimated parameters are as follows.

• 1.  
  global (over the entire data set): coordinates of 4924 sources, including target objects in the VGaPS campaign, positions and velocities of all stations, coefficients of B-spline expansion that model non-linear motion of 17 stations, coefficients of harmonic site position variations of 48 stations at four frequencies: annual, semi-annual, diurnal, semi-diurnal, and axis offsets for 67 stations.
• 2.  
  local (over each session): tilts of the local symmetric axis of the atmosphere (also known as "atmospheric azimuthal gradients") for all stations and their rates, station-dependent clock functions modeled by second-order polynomials, baseline-dependent clock offsets, and the Earth orientation parameters.
• 3.  
  segmented (over 20–60 minutes): coefficients of linear splines that model atmospheric path delays (20 minutes segment) and clock functions (60 minutes segment) for each station. The estimates of clock functions absorb uncalibrated instrumental delays in the data acquisition system.

The rate of change for the atmospheric path delays and clock functions between adjacent segments was constrained to zero with weights reciprocal to 1.1 × 10⁻¹⁴ and 2 × 10⁻¹⁴, respectively, in order to stabilize solutions. We apply no-net rotation constraints on the positions of 212 sources marked as "defining" in the ICRF catalog (Ma et al. 1998) that requires the positions of these sources in the new catalog to have no rotation with respect to the position in the ICRF catalog to preserve continuity with previous solutions.

The global solution sets the orientation of the array with respect to an ensemble of ∼5000 extragalactic remote radio sources. The orientation is defined by the series of Earth orientation parameters and parameters of the empirical model of site position variations over 30 years evaluated together with source coordinates. Common sources observed in VGaPS as atmosphere and amplitude calibrators provide a connection between the new catalog and the old catalog of compact sources.

As follows from the Gauss–Markov theorem, the estimate of parameters has minimum dispersion when observation weights are chosen reciprocal to the variance of errors. The group delays used in the analysis have errors due to the thermal noise in fringe phases and due to mismodeling theoretical path delay in the atmosphere

$$\begin{equation} \sigma ^2 = \sigma ^2_{{\rm th}} + \sigma ^2_{{\rm io}} + \sigma ^2_{{\rm na}}, \end{equation} \tag{ 26 }$$

where σ²_th is the thermal noise, and σ²_io and σ²_na are the contribution of the ionosphere and the neutral atmosphere to the error budget, respectively.

3.6.1. A Priori Errors of the GPS Ionosphere Model

The first term, σ²_th, was estimated during the fringe fitting. The second term can only be evaluated indirectly. Sekido et al. (2003) used six dual-band intercontinental VLBI sessions at 10 stations in 2000 July and compared TEC values from GPS with TEC estimated from VLBI observables. They drew a conclusion that the errors in path delay derived from the GPS TEC model were at the range of 70 ps in the zenith direction at 8.6 GHz. However, the ionosphere path delay is a non-stationary process. Therefore, great caution should be taken in an attempt to generalize conclusions made from analysis of a small network over a short time period. The non-stationarity of ionospheric fluctuations implies that an exact expression for the variance of the ionosphere fluctuations during any given period does not exist, and any expression for the variance is an approximation.

Since 1998 June when the GPS TEC maps became available through 2010 August, more than 2000 dual-band S/X VLBI sessions under geodesy and absolute astrometry programs were carried out, including 92 sessions at the VLBA. It can be easily shown that the contribution of the ionosphere in the X band, τ_xi, can be found from the linear combination of group delay observables with coefficients that are expressed through effective ionosphere frequencies at these bands, ω_x and ω_s, defined in expression (23):

$$\begin{equation} \tau _{xi} = (\tau _x - \tau _s) \frac{\strut \omega ^2_s}{\strut \omega ^2_x - \omega ^2_s}. \end{equation} \tag{ 27 }$$

We used this data set to evaluate the errors of the contribution of the ionosphere to group delays derived from GPS TEC maps, considering the ionosphere contribution from dual-band VLBI observations as true for the purpose of this comparison. We computed the ionosphere contribution from the GPS model and from VLBI observations for each session. The root mean squares (rms) of the differences of the contribution VLBI–GPS was computed for all sessions and all baselines. We sought regressors that can predict rms(VLBI–GPS). We expect the short-term variability of the ionosphere at scales less than several hours to dominate the errors of the GPS model. The sparseness of the GPS network and limited sky coverage result in missing high frequency spatial and temporal variations of the ionosphere. The turbulent nature of the ionosphere path delay variations suggests that the rms of the ionosphere model errors due to missed high frequency variations will be related to the rms of the low frequency variations either as a linear function or as a power law. After many tries we found that the following parameter can serve as a regressor: RG = $\sqrt{\vphantom{A^A}\smash{\hbox{${\mbox{rms}_{g1}^2 + \mbox{rms}_{g2}^2}$}}}$, where rms_gi is the rms of the GPS path delay at the ith station of a baseline during a session. The rms_g1 and rms_g2 are highly correlated at short baselines and $\sqrt{\vphantom{A^A}\smash{\hbox{${\mbox{rms}_{g1}^2 + \mbox{rms}_{g2}^2}$}}} > \mbox{rms(g2 -- g1)}$. At long baselines the ionosphere contribution de-correlates. Therefore, the dependence of rms(VLBI–GPS) versus RG will depend on the baseline length and possibly on other parameters. Figure 6 shows examples of this dependence for a short baseline and for a long baseline.

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It is remarkable that rms(VLBI–GPS) versus RG fits reasonably well with a linear function. We computed rms(VLBI–GPS) for each baseline and fitted it to the linear model F + S · RG. The floor of the linear fit has a mean value around 20 ps (Figure 7). As expected, the slope of the fit increases with baseline length as shown in Figure 8. The growth is linear up to baseline lengths of 2000 km, which apparently correspond to the decorrelation of the paths through the ionosphere. The growth of the slope beyond baseline lengths of 2000 km is slower and it shows more scatter.

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We use this dependence to predict the rms of the GPS ionosphere model errors. For each station that participated in the experiment we computed the rms of the ionosphere variations in zenith direction. Then we express the predicted variance of the GPS ionosphere model errors as

$$\begin{eqnarray} \sigma _{i}^2 & = & \left(F_b \frac{\strut f^2_t}{\strut f^2_{\rm eff}} \right) ^2 + \left(\tau _{\rm iono,1} - \frac{\bar{\tau }^z_1}{\cos \beta (E_1)} \right) ^2 S_b^2\nonumber \\ & & + \left(\tau _{\rm iono,2} - \frac{\bar{\tau }^z_2}{\cos \beta (E_2)} \right) ^2 S_b^2, \end{eqnarray} \tag{ 28 }$$

where τ_{iono, i} is the ionosphere path delay at the ith station computed using the GPS TEC maps, $\bar{\tau }^z_1$ is the zenith ionosphere path delay from GPS TEC maps averaged over a 24^h period with respect to the central date of the session, F_b and S_b are the parameters of the linear model rms(VLBI–GPS) versus RG for a given baseline, f_t is the frequency for which the model was computed (8.6 GHz), and f_eff is the frequency of the experiment for which the model is applied (24.5 GHz). The term $\tau _{\rm iono,i} - \frac{\bar{\tau }^z_i}{\cos \beta (E_i)}$ is the difference between the ionosphere path delay from GPS at a given direction and the average ionosphere path delay scaled to take into account the elevation dependence. This term is an approximation of rms_g1 used for computation of RG.

3.6.2. A Priori Errors of the Path Delay in the Neutral Atmosphere

Rigorous analysis of the errors of modeling the path delay in the neutral atmosphere is beyond the scope of this paper. Assuming the dominant errors of the a priori model are due to high frequency fluctuations of water vapor at scales less than 3–5 hr, we seek a regression model in the form of dependence of the rms of errors on the total path delay in the non-hydrostatic component of the path delay. We made several trial runs using all 123 observing sessions at the VLBA under geodesy and absolute astrometry programs with reciprocal weights modified according to

$$\begin{equation} \sigma ^2_{{\rm used}} = \sigma ^2 + \left(a \cdot \frac{\strut \tau _{s}}{\strut \tau _z} \right) ^2. \end{equation} \tag{ 29 }$$

Here, τ_s is the contribution of the non-hydrostatic constituent of the slanted path delay and τ_z is the non-hydrostatic path delay in the zenith direction computed by direct integration of the equations of wave propagation through the atmosphere using the refractivity computed from the MERRA model, and a is the coefficient. We found that when coefficient a = 0.02 is used, the baseline length repeatability, defined as the rms of the deviation of baseline length with respect to the linear time evolution, reaches the minimum. We adopted the value 0.02 in our analysis of VGaPS experiments. For typical values of τ_z, the added noise is 8 ps in the zenith direction and 80 ps at 10° elevation.

3.6.3. Ad Hoc Added Variance of the Noise

Positions of two sources, J0432+4138 (also known as 3C119) and J2020+2942 (3C410) are found to be significantly off, namely 38.3 and 38.0 mas, respectively. The significance of this offset, 55 and 72 times the inflated uncertainties, is too high to be explained by known error sources. RATAN-600 observations have shown that both of them continuously show steep radio spectra and are slowly variable. On VLBI scales, they were found to have significantly extended structures (e.g., Figures 17 and 18).

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One of these compact steep spectrum radio sources, the object J0432+4138, is well studied at parsec scales. Recent high-dynamic range images at 5 and 8.4 GHz and a full bibliography on historic observations can be found in Mantovani et al. (2010). This source shows several bright components at 24 GHz. Two of them, namely components "A" and "C" (Figure 17), are located 40.6 mas apart. The feature C is stronger, with a total flux density of 0.54 Jy, but more extended. The FWHM of a circular Gaussian component fitted to the feature is found to be 1.4 mas. The feature A is dimmer, 0.13 Jy, but more compact—0.3 mas. We note that here and below in this section the total flux density reported for different source features is calculated as a sum of all CLEAN components representing the corresponding structure. The Gaussian components fit is being performed in the visibility plane.

It is evident that the VGaPS 24 GHz observations referred to the position of the weaker A component, while the S/X observations referred to the position of the C component. This is counter-intuitive. Since component C is resolved, its contribution to the fringe amplitude is small at long baseline projections. At short baseline projections component C dominates; at long baseline projections component A dominates, but since the partial derivative of group delay with respect to source position is proportional to the baseline length projection on the source tangential plane, the contribution of long baseline dominates in estimates of source position. According to Mantovani et al. (2010), the total flux density of components A and C at 8.4 GHz in 2001 was 70 mJy and 1121 mJy, respectively. This large difference between components' flux densities is also supported by the previous VCS1 observations of this object in the S/X bands. Component C is less resolved at 8.4 GHz and it dominates over A even at long baselines. Therefore, coordinates of a source in the X band are closer to the C component. We checked this directly by suppressing observations at baselines longer than 800 km in a K-band trial solution. The position estimate became close to the S/X position. It is worth noting that the difference in position in the K and X bands are 29.2 ± 0.4 mas in right ascension and 24.1 ± 0.4 mas in declination, while the offset of component A with respect to component C in the K band is slightly larger: 30.25 ± 0.10 mas and 27.05 ± 0.15 mas. The K-band and X-band positions are not exactly the position of components A and C, since in both solutions the contribution of the second component is small but not entirely negligible.

The nature of the difference in positions of J2020+2942 is similar. It was observed in the VCS2 experiment on 2002 May 15 (Fomalont et al. 2003). It had 63 detections in the S band and 72 detections in the X band. The position of this source in the S band from analysis of only S-band data applying the ionosphere contribution from the GPS TEC model shows a very large offset of 05 with respect to the X-band position (refer to Table 11). The errors of the ionosphere contribution in the S-band during the solar maximum affected position estimates considerably, however, not to that extent. A comparison of the positions of 130 sources with δ > 0 from the solution that used only S-band group delay observables with respect to the X/S solution in that experiment showed differences in the range 2–7 mas. It is remarkable that the X/S position is away from both X- and S-band positions, although intuitively we can expect them to be between X and S positions. This can be explained if we surmise that the source J2020+2942 has two components, 048 apart, one of which is visible in the X band, but not visible in the original S-band image, and another which is visible in the S band, but not in the X band. An ionosphere-free linear combination of X- and S-band observables is used in the X/S solution: (1 + β)τ_gx − βτ_gs, where β = 1/(ω²_x/ω²_s − 1) as it follows from Equation (27). In the case where the position in the S band, $\vec{k}_s$, is shifted with respect to the X-band position vector, $\vec{k}_x$, the ionosphere free linear combination can be written as

$$\begin{equation} \tau _{if} = (1+\beta) \tau _{gx}(\vec{k}_x) - \beta \tau _{gs}(\vec{k}_x) - \beta \frac{\strut \partial \tau }{\strut \partial \vec{k} } (\vec{k}_s - \vec{k}_x). \end{equation} \tag{ 32 }$$

The first two terms correspond to a case with no offset between X- and S-band positions. Therefore, estimates of the source position from the ionosphere-free linear combination of observables will be shifted at −β with respect to the offset $(\vec{k}_s - \vec{k}_x)$, i.e., in the opposite direction. Parameter β was 0.076153 in the VCS2 experiment. Therefore, if our hypothesis that the X-band and S-band observations detected emission from two components is true, then the shift of the X/S position with respect to the X-band position should be −1.96 mas in right ascension and −36.46 in declination, just within 0.3 mas from the reported S/X positions! The K-band position is within 1.5σ of the X-band position.

In order to check our hypothesis, we have re-imaged VCS2 observations of J2020+2942 in a wide field and have detected several previously unknown features "B," "C," and "D," on a distance up to about 500 mas from the dominating extended structure "A" in the S band (Figure 18). The total flux density of the features A and D is 1.40 Jy and 0.09 Jy, respectively. Feature A is significantly more extended than D. We have fitted two circular Gaussian components to the uv-data in order to determine the positions of features A and D. We note that the accuracy of position determination for component A from the image is very poor since its structure is extended over at least 40 mas and is not well represented by a single Gaussian component. The distance between Gaussian components A and D is 479 mas, while the positional difference in right ascension and declination is 25 and 479 mas, respectively—in very good agreement with independent astrometric measurements (Table 11).

X-band and K-band wide field imaging (Figure 19) did not reveal components with wide separation. We conclude from the astrometric analysis presented above that X- and K-band images represent the more compact component D. In this case we could also analyze its spectrum on the basis of simultaneous S/X-band observations. Its total flux density in the X-band is found to be 0.25 Jy which provides the 2.3–8.6 GHz spectral index estimate α = +0.8 (flux density ∝ν^α)—an indication of synchrotron emission with significant self-absorption. The features A, B, and C become too weak and/or too resolved for us to detect in the snapshot VLBA images with a limited dynamic range and uv-coverage.

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Position differences for other objects do not show peculiarities. For instance, no declination-dependent systematic differences similar to those reported by Lanyi et al. (2010) were found. The wrms of the differences is 0.46 mas in right ascension scaled by cos δ and 0.61 mas in declination. We have computed the normalized distances by dividing them by $\sqrt{\vphantom{A^A}\smash{\hbox{${e_{k}^2 + e_{xs}^2}$}}}$, where e_k is the projection of the error ellipse of the K-band position to the direction of position difference and e_xs is the similar projection of the error ellipse of the S/X position. In the case if position errors from K-band and S/X catalogs are independent and Gaussian with the variance equal to reported uncertainties, the distribution of normalized distances will be Rayleigh with σ = 1. The average of the normalized distances over all sources, except J0432+4138 and J2020+2942, is 1.276, only 2% greater than the mean of the Rayleigh distribution, $\sqrt{\pi /2}$. However, a close examination of the distribution (see Figure 20) reveals a slight deviation of its shape from the shape of the Rayleigh distribution. The Rayleigh distribution that best fits the distribution of normalized distances has σ = 0.90. This is an indication of a deviation of parent distributions from Gaussian.

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We can make several conclusions from this test. First, on average, reported formal uncertainties are correct within several percent. Second, the effect of the core-shift is too small to contribute significantly to results of single-epoch surveys. According to Kovalev et al. (2008), the typical apparent core-shift is expected to be 0.4 mas between the S and X bands. Porcas (2009) stressed that when the core-shift is proportional to f⁻¹, a source position derived from ionosphere-free linear combinations of X- and S-band group delays is not sensitive to the core-shift and corresponds to a true position of the jet base. The f⁻¹ core-shift dependence is expected for a conical jet with synchrotron self-absorption in the regime of equipartition between the jet particle and magnetic field energy densities (Lobanov 1998). If we assume that this is indeed the case for the majority of the sources (see also Sokolovsky et al. 2011), the average core-shift between K-band and effective S/X positions is reduced to $0.4 \times \frac{f_\mathrm{x} f_\mathrm{s}}{f_\mathrm{k} (f_\mathrm{x} - f_\mathrm{s})} = 0.06$ mas. Our observations would allow detection of the core-shift between positions from S/X and K-band observables the 95% confidence level of a sample of 190 objects if the variance of the core-shift is greater than 1 mas. VGaPS observations set the upper limit of this variance to 1 mas, which does not contradict the result of core-shift measurements and predictions.