Optimum estimate of delays and dispersive effects in lowfrequency interferometric observations
I. MartíVidal
MaxPlanckInstitut für Radiastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
Received 23 December 2009 / Accepted 3 May 2010
Abstract
Context. Modern radio interferometers sensitive to low
frequencies will make use of wideband detectors (with bandwidths of
the order of the observing frequency) and correlators with high data
processing rates. It will be possible to simultaneously correlate data
from many subbands spread through the whole bandwidth of the
detectors. For such wide bandwidths, dispersive effects from the
atmosphere introduce variations in the fringe delay which change
through the whole band of the receivers. These undesired dispersive
effects must be estimated and calibrated with the highest precision.
Aims. We studied the achievable precision in the estimate of the
ionospheric dispersion and the dynamic range of the correlated fringes
for different distributions of subbands in lowfrequency and wideband
interferometric observations. Our study is focused on the case of
subbands with a bandwidth much narrower than that of the total covered
spectrum (case of LOFAR).
Methods. We computed the formal statistical uncertainty of the
ionospheric delay, the delay ambiguity and the dynamic range of the
correlated fringes using four different kinds of distributions of the
subbands: constant spacing between subbands, random spacings,
spacings based on a powerlaw distribution, and spacings based on
Golomb rulers (sets of integers, n_{i}, whose sets of differences, n_{j}n_{i}, have nonrepeated elements).
Results. We compare the formal uncertainties in the estimate of
ionospheric effects in the data, the ambiguity of the delays, and the
dynamic range of the correlated fringes for the four different kinds of
subband distributions.
Conclusions. For a large number of subbands (>20, depending
on the delay window) spacings based on Golomb rulers give the most
precise estimates of dispersive effects and the highest dynamic ranges
of the fringes. Spacings based on the powerlaw distribution give
similar (but slightly worse) results, although the results are better
than those from the Golomb rulers for a smaller numbers of subbands.
Random distributions of subbands result in relatively large dynamic
ranges of the fringes, but the estimate of dispersive effects through
the band is worse. A constant spacing of subbands results in very bad
dynamic ranges of the fringes, but the estimates of dispersive effects
have a precision similar to that obtained with the powerlaw
distribution. Combining all the results, the powerlaw distribution
gives the best compromise between homogeneity in the sampling of the
bandwidth, precision in the estimate of the ionospheric dispersive
effects, dynamic range of the correlated fringes, and ambiguity of the
group delay.
Key words: atmospheric effects  instrumentation: interferometers  techniques: interferometric  telescopes
1 Introduction
Modern radio interferometers sensitive to low frequencies, like the LOw Frequency ARray (LOFAR, see, e.g., de Vos, et al. 2009 and references therein), will make use of wideband detectors (with a bandwidth of the order of the observing frequency) and correlators with a high data processing rate. For the case of LOFAR, it will be possible to correlate a bandwidth of 42 MHz, which can be divided into many subbands to be spread through a wide band (either 1090 MHz, 110190 MHz, 170230 MHz, or 210250 MHz; de Vos, et al. 2009). Therefore, the correlator will be able to simultaneously cover a large portion of the lowfrequency spectrum. However, atmospheric dispersion may introduce strong frequencydependent effects, which will be so different for each subband and must be estimated and calibrated with care. At low frequencies, the ionosphere is the main limiting factor in the quality of the interferometric observations. For the calibration of the ionospheric dispersion, the total electron content (TEC) must be estimated through the field of view (FoV) over each antenna and for each time. The TEC can be estimated, for instance, from globalpositioningsystem (GPS) data (e.g. Ros et al. 2000; Todorova, et al. 2006). With these techniques, however, it is not possible to obtain highresolution estimates of the ionospheric turbulent screen over the FoV. For that purpose, it is necessary to apply selfcalibration strategies, using therefore the interferometric data to derive the structure of the TEC screen over each station and its evolution (see Intema et al. 2009; Cohen & Rottgering 2009, for a discussion on the ionospheric screens).
The distribution of subbands through the whole bandwidth of the detectors affects the scientific information that can be extracted from the observations, but is also important for a precise and accurate estimate of the atmospheric dispersion from the interferometric observables. The distribution of subbands has also an effect in delay space, since the interferometric fringes have a shape related to the Fourier transform of the bandpass (i.e., in our case, the distribution of subbands). Therefore, we should distribute the subbands in such a way that the spectral coverage and sampling are maximized, but the dynamic range of the fringes (i.e., the height of the fringe peak relative to that of the highest sidelobe) is also maximized to improve the sensitivity of the interferometer. Finding out the right distribution of subbands to achieve an optimum spectral sampling and fringe dynamic range is not that simple, and the answer may depend on each case, namely, the total bandwidth to be covered, the number of subbands, and the bandwidth of each subband. For instance, Petrachenko (2008) studied the performance of ``broadband delays'', which are computed from several bands (up to 5) spread from 23 GHz to 1114 GHz, as a function of the way these bands are distributed through the spectrum. Petrachenko (2008) concluded that the use of more than 2 bands, covering a total bandwidth as wide as possible, improves the performance of the interferometer.
In this paper, we report on a study of the spectral coverage, the precision in the estimate of atmospheric dispersive effects, the delay ambiguity, the dynamic range of the fringes, and the precision in the estimate of the source spectral index for different kinds of subband distributions and spectral configurations of an interferometer at low frequencies. The remainder of this paper is structured as follows: in Sect. 2 we describe the process of analysis. In Sect. 3 we describe the different subband distributions studied. In Sect. 4 we report on the results obtained and in Sect. 5 we summarize our conclusions.
2 Analysis
2.1 Spectral configuration of the interferometer
The spectral configuration of an interferometer can be characterized using
the following parameters: i) minimum observing frequency,
;
ii)
total bandwidth in units of the minimum observing frequency, i.e.,
where is the maximum observing frequency; iii) number of subbands, N (and their distribution); and iv) the bandwidth of each subband, . The ith subband is centered at frequency (with ), which can be written as
(2) 
being R_{i} a real number between and . The spectral configuration of an interferometric observation is then characterized by , , , and R_{i}. For simplicity, we assume the same for all subbands, and we also assume that (i.e., the bandwidth of the subbands is much narrower than the total covered bandwidth, so R_{i} is defined between 0 and 1). This latter assumption corresponds to the case of the LOFAR interferometer.
2.2 Dynamic range of the fringes
The response of a baseline of an interferometer to a set of observed sources is equal to the addition of several fringes, one fringe for each source. The amplitude of these fringes is equal to a shrinked version of the amplitude of the Fourier transform of the bandpass (e.g., Thomson, et al. 1991). Spectral effects in the sources are not considered here. In the case of several subbands spread through a wider band, different shapes can be obtained for the fringes, which may have relatively large sidelobes. These large sidelobes may lead to confusion in the estimate of the fringe peaks. In frequency space, this effect can be understood as the possibility of fitting different slopes of phase vs. frequency to the same dataset. Minimizing the height of the sidelobes and maximizing their distance to the fringe peak decreases the probability of confusion and, therefore, enhances the sensitivity of the interferometer.
For each of the studied band distributions (described in Sect. 3), the shape of the fringes, ,
was computed as a vector with elements given by
(3) 
i.e., the module of the direct Fourier transform (DFT) of the subband distribution (the sine term of the DFT is zero). The vector is thus computed assuming that the bandwidth of the subbands, is much narrower than the total bandwidth, . The dynamic range of the fringe is estimated as the ratio between the fringe peak (which corresponds to the element F_{1}) and the peak of the highest sidelobe. We call this ratio D. The ambiguity of the delay is computed as the distance between these two peaks, in units of the Nyquist time resolution (i.e., ). We call this quantity . For the results reported in the following sections, we used a vector of 512 elements.
If the width of the subbands is not much narrower than the total covered bandwidth, the dynamic range, D, is corrected by
(4) 
where .
2.3 Estimate of the atmospheric dispersion
The ionosphere introduces a change in the phase of the visibilities. For a given baseline
and source, this phase depends on frequency as (e.g., Thomson, et al. 1991)
The parameter K is related to the TEC of the ionosphere over the elements of the baseline in the lineofsight direction to the source. For a good calibration of the ionospheric dispersion, K must be precisely estimated for each baseline, time, and pointing direction over the FoV of the interferometer.
The formal statistical uncertainty in the estimate of K is that of the slope of the linear fit of
vs.
.
It is straightforward to show that this uncertainty is
i.e., the precision in K is proportional to the standard deviation of the distribution of the inverse of the central frequencies of the subbands, . This standard deviation maximizes when half of the subbands gather close to and the other half gather close to . In this case
However, this distribution of subbands is not, by far, optimum, since the sampling of the total band is very poor and, moreover, spectral effects in the sources, which would not be well sampled through the bandwidth, could introduce important systematics in the estimate of the ionospheric dispersion using Eq. (5). Additionally, there could be several undetermined cycles of the phase drift caused by the ionosphere between and , so, for this distribution, it would not be possible to connect the phases between all subbands to obtain a correct estimate of K.
The uncertainties in the estimate of the ionospheric dispersion, which we analyze in Sect. 4, were estimated as (computed from Eq. (6)) in units of its minimum possible value, (i.e., that one corresponding to , which is given in Eq. (7)).
3 Distributions of subbands
Figure 1: Examples of the four kinds of distributions studied in this paper (using 16 subbands and setting , see Eq. (1)). a) corresponds to the uniform (i.e. constant) distribution; b) to the random distribution; c) to the powerlaw distribution; and d) to the distribution based on the Golomb ruler. 

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3.1 Uniform (i.e., constant) distribution
This is the most simple spectral configuration of the interferometer.
The central frequencies of the subbands are distributed as
i.e., the frequency spacing between subbands is constant. We show an example of this distribution in Fig. 1a.
3.2 Random distribution
In this case, the subbands are randomly distributed over the bandwidth, i.e.,
where U_{i} is a random real number between 0 and 1, following a uniform statistical distribution. We show an example of this distribution of subbands in Fig. 1b. Different sets of U_{i} may translate into different fringe dynamic ranges and precisions in the estimate of ionospheric dispersion (usually, a higher precision in the estimate of the atmospheric dispersion translates into a lower dynamic range of the fringes). We estimated the quantities , D, and as the averages of those computed from 100 different sets of U_{i}. Therefore, the results reported in Sect. 4 for the random distribution correspond to an averaged behavior of this distribution.
3.3 Powerlaw distribution
The ionosphere introduces larger phase drifts at lower frequencies.
Therefore, it is plausible
that a distribution that samples better the region of lower frequencies
will give more precise estimates of the ionospheric dispersion, since
the phase drifts will be better sampled in the region of the spectrum
where the ionospheric effects are larger. A natural distribution to
obtain this kind of sampling is setting the density of subbands
proportional to a power law of frequency,
,
where
is a given (negative) constant, i.e.,
Therefore,
For the special case of we have instead
Equation (10) becomes Eq. (8) for . It can be shown that the standard deviation of (i.e., the precision in the estimate of the ionospheric dispersion) is maximum for in the case of a large number of subbands. We show in Fig. 2 the optimum as a function of the number of subbands, N. These values of were computed by finding numerically the minimum of (using Eq. 6) as a function of . The values of shown in Fig. 2 can be estimated using the model (which is also shown in the figure)
which tends to 5/3 for large N. These are the values of that were used to obtain the results reported in the next section. We show an example of this powerlaw distribution in Fig. 1c.
3.4 Golomb rulers
A Golomb ruler is a set of n_{i} integers such that the set of differences,
d_{ij} = n_{i}  n_{j}, has
no repeated elements (see, e.g., Atkinson, et al. 1986).
It is intuitive that Golomb rulers are a good choice to maximize the
dynamic range of the fringes, since all pairs of subbands are
separated incoherently one respect to the other. Therefore, the
sidelobes of the Fourier transform of the bandpass are minimum. The
improvement
in the fringe dynamic range when the subbands are distributed
according to Golomb rulers has been previously reported for the case of
8 subbands (Mioduszewski & Kogan 2004). Here we generalize
the study to different
number of subbands and also analyze the impact of this kind of distribution in the precision of the
estimate of the ionospheric dispersion. The central frequencies, ,
of the subbands are computed
using the equation
where n_{i} is the ith element of a Golomb ruler of N elements (by convention, n_{1} = 0). The Golomb rulers for N < 24 were taken from the OGR project at http://distributed.net/ogr, and the others from Atkinson, Santoro & Urrutia (1986). We show an example of this distribution in Fig. 1d.
4 Results
Figure 1 shows that the random and Golombrulers distributions tend to poorly
sample some regions of the spectrum and oversample others. On the contrary, the constant and the
powerlaw distributions sample the bandwidth in a more homogeneous way. A more homogeneous sampling of
the spectrum is preferable to obtain information from as many regions of the bandwidth as possible.
Moreover, a more homogeneous sampling makes easier the the connection of the phases between
the subbands, since an unsampled wide lag in the spectrum could contain a number of
phase cycles that could introduce biases in the data analysis. To
better understand this
statement, let us consider, for example, a nondispersive delay, ,
added to the fringe.
The differential phase between subbands i and j, due to ,
would be
For larger values of (i.e., for wider lags between subbands), the differential phase between subbands is larger. Therefore, the probability of the differential phase to be larger than 2 is higher for wider lags between subbands.
Figure 2: Boxes, optimum values of (i.e., those that minimize the formal uncertainty in the estimate of the ionospheric dispersion) as a function of the number of subbands. Line, model corresponding to Eq. (12). 

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From this point of view, the uniform and/or the powerlaw distributions would be the best frequency configurations for the interferometer. However, we must also take into account the dynamic range of the fringes and the quality in the estimate of the atmospheric dispersion, which are analyzed in the following subsections.
4.1 Ionospheric dispersion
In Fig. 3a, we show the uncertainty in the estimate of the ionospheric dispersion, , in units of the minimum possible uncertainty (i.e., , computed from Eqs. (6) and (7)), as a function of the number of subbands, for a total bandwidth of (see Eq. (1)). The four different distributions are shown. For all distributions, the uncertainty (relative to the minimum possible one) increases with the number of subbands. It can be seen that the distribution based on Golomb rulers give the most precise estimates of the ionospheric dispersion, followed by the powerlaw distribution (with an uncertainty 3% larger, depending on the number of subbands), and the random and uniform distributions (with uncertainties 6% larger, also depending on the number of subbands).
In Fig. 4, we show the uncertainty in the estimate of the ionospheric dispersion, in units of the minimum possible uncertainty, as a function of the bandwidth (i.e., , see Eq. (1)) using a total of 32 subbands to cover the bandwidth. Golomb rulers yield again the most precise estimates of the ionospheric dispersion, although the uncertainty increases as the bandwidth increases. On the contrary, the powerlaw distribution keeps the uncertainty roughly constant as a function of the bandwidth. Both the constant and random distributions also increase the uncertainty in the atmospheric dispersion as the bandwidth increases, being this uncertainty 5% larger than that obtained with the Golomb rulers.
We conclude that the powerlaw distribution and that based on Golomb rulers give higher precisions in the estimate of the ionospheric dispersion. Although the difference between uncertainties from all the distributions is not so large (lower than 10% in all cases), its optimization may be important to obtain highcontrast images.
Figure 3: a) Formal uncertainty in the estimate of the ionospheric dispersion, in units of the minimum possible uncertainty; b) dynamic range of the fringes; and c) delay ambiguity (i.e., distance between the fringe peak and the closest sidelobe) in units of the Nyquist time resolution. All these quantities are shown as a function of the number of subbands for (see Eq. (1)). 

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Figure 4: Formal uncertainty in the estimate of the ionospheric dispersion, in units of the minimum possible uncertainty, as a function of (see Eq. (1)) for the case of 32 subbands. 

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4.2 Fringe dynamic range and delay ambiguity
In Fig. 3b, we show the dynamic range of the fringes as a function of the number of subbands for a total bandwidth of (see Eq. (1)). We notice, however, that the dynamic range of the fringes is independent of , since a change in is equivalent to a change in the delay scaling of the fringes (regardless of a phase factor that depends on ). Two regions in the subband space can be readily seen.
For N < 20, the random and the powerlaw distributions give higher dynamic ranges. Suprisingly, for these values of N, the distribution based on Golomb rulers give dynamic ranges 1. Why? Golomb rulers are sets of integer numbers. Therefore, the Fourier transforms of these subband distributions are periodic. If the delay window is larger than the period of the Fourier transform, there will be more than one peak in the fringe. The fringe period depends on each ruler and increases with the number of channels. For N<20, the fringe period is shorter than our delay window (1024 times the Nyquist time resolution, i.e., 512 channels in each direction of the delay) so there is more than one peak in the fringe. For N>20, the fringe period is larger and only one peak remains in the delay window. For N > 20, Golomb rulers give the highest dynamic ranges (around 2030% higher than those based on the random and powerlaw distributions). In all cases, the uniform distribution gives very poor dynamic ranges, 1, as it is indeed expected, since the Fourier transform of the bandpass is a periodic function with a very short period. For the case of the delay ambiguity, strong changes are seen as a function of the number of subbands for the Golomb rulers (the ambiguity ranges between 20 and 512 channels) and the powerlaw distribution (the ambiguity ranges between 150 and 512 channels, although the lower limit increases with N). These changes in the delay ambiguity are due to several sidelobes with similar peak values. Changing N also changes the relative height of the sidelobe peaks. As a consequence, for different values of N, different sidelobes are selected as the highest and, therefore, very different delay ambiguities are obtained. On the contrary, the random distribution has a delay ambiguity of 250 channels for all values of N (we notice, however, that, for this distribution, the figure shows the average for 100 different fringes). The uniform distribution, as expected, has a very small delay ambiguity (lower than 100 channels). This ambiguity increases with N, also as expected, since the spacing of subbands (which is shorter for larger N) is inversely proportional to the period of the fringe.
A first conclusion is that the uniform distribution is not a good choice from the point of view of the quality in the estimate of the group delay. The Golomb rulers are a good choice when the number of subbands is large (N > 20, although this number decreases if the width of the delay window decreases). The powerlaw distribution is, in general, a good choice for all N. It gives the best compromise between homogeneity in the sampling of the bandwidth, precision in the estimate of the ionsopheric dispersive effects, dynamic range of the correlated fringes, and ambiguity of the group delay. Therefore, this would be the preferable subband distribution to use in lowfrequency (wideband) interferometric observations.
Nevertheless, these conclusions are based on a number of subbands up to 64. If the number of subbands is large (say, N=512) there are no available Golomb rulers to work with, but we can still compare the results obtained from the uniform, random, and powerlaw distributions.
Setting N=512, the dynamic range of the fringes is similar for the three distributions, if we use a delay window of 1024 Nyquist channels (using 512 subbands, the period of the fringe corresponding to the uniform distribution is longer than the delay window). However, the powerlaw distribution still gives lower formal uncertainties in the estimate of the ionospheric dispersion (around 10% lower than the other distributions for and 4% for ). Therefore, the powerlaw distribution is still the best choice with a number of subbands as large as 512.
4.3 Source spectral index
Wideband observations allow to precisely determine the spectral
indices and spectral curvatures of radio sources. The different distributions of
subbands may also affect the achievable precision in the
estimate of the spectral properties of the radio sources. For the case of the
spectral index,
(being the flux density,
), the
formal uncertainty,
depends on the distribution
in the form
We show in Fig. 5 the formal uncertainty in the estimate of , in units of that of the uniform distribution, , as a function of the bandwidth, , for the case of 64 subbands. It can be seen in the figure that the uniform, random, and powerlaw distributions give similar precisions in the estimate of the spectral index (although the precision for the powerlaw distribution slightly increases with the bandwidth). Surprisingly, the Golombruler distribution gives a precision 5% higher than the other distributions for all bandwidths.
The formal uncertainties of the uniform, random, and powerlaw distributions for a large number of subbands (N=512) are similar to those shown in Fig. 5. Therefore, for wideband observations using many subbands (i.e. where no Golomb rulers are available), the use of the powerlaw distribution is the best choice, allowing for a 12% higher precision in the estimate of the spectral index of the sources. With a smaller number of subbands, the use of Golombruler distributions would improve the precision in the estimate of by 5%.
Figure 5: Formal uncertainty in the estimate of the source spectral index, , in units of that of the uniform distribution ( ), as a function of the bandwidth, (see Eq. (1)), for the case of 64 subbands. 

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4.4 Other contributions to the optimum powerlaw distribution
Other contributions to the visibility phases (either due to the electron plasma of the ionosphere or to chromaticity in the structure of the observed sources), as well as the contribution of the galactic radiation to the visibility noise, have not been considered in the previous sections. In this section, we study how these contributions may affect the optimum sampling of the ionospheric dispersion using the powerlaw distribution of subbands.
4.4.1 Plasma frequency of ionospheric electrons
Equation (5) holds in the region of frequencies much higher than the plasma frequency,
,
of the ionospheric electrons. The electron density in the ionosphere takes values around
10^{4}10^{6} e^{} cm^{3}. This translates into a plasma frequency in the range 110 MHz (e.g., Pacholczyk 1970, Eq. (2.72)). From the refraction index
of a plasma (e.g., Pacholczyk 1970, Eqs. (2.77) and (2.78)), the phase drift in the case of
is
where is the Larmor electron frequency (for the Earth magnetic field it takes the value 1 MHz) and the signs correspond to the two possible circular polarizations of the radiation. We notice that the effect of in the computations reported in this section is negligible. K' is related to the TEC above each element of the baseline in the lineofsight to the source.
We computed the optimum values of (i.e., the exponent of the powerlaw distribution of subbands) that optimize the sampling of the ionospheric phase drifts (i.e., minimize the formal uncertainty in the estimate of K') for observing frequencies close to . We call these values , to distinguish them from the values without the effect of the plasma frequency (i.e., , which are shown in Fig. 2 and given in Eq. (12)). In Fig. 6 we show the ratios as a function of the minimum observing frequency, , in units of the plasma frequency, . For instance, using a bandwidth of (i.e., ) a plasma frequency MHz, and a minimum frequency MHz (i.e, 10 times the plasma frequency) results in values of equal to 0.86 times those of . For a large number of subbands (i.e., for ) this results in .
We notice that the exponent approaches zero as the minimum frequency approaches to the plasma frequency. Therefore, the optimum powerlaw distribution approaches the constant distribution at very low frequencies. Increasing the bandwidth tends to compensate a bit the decrease in the absolute value of (i.e., the ratios increase when increases) although this effect is small, as it can be appreciated in Fig. 6.
Figure 6: Optimum values of , in units of , as a function of the minimum observing frequency, , in units of the plasma frequency, . In each figure, the values are computed for five different values of (2, 4, 6, 8, and 10 MHz). Each figure corresponds to a different bandwidth, (0.1, 0.5, 1.0, and 1.5). 

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4.4.2 Frequencydependent source structure
A source structure which is independent of the observing frequency is a strong assumption over the broad frequency ranges considered in the previous sections. The contribution of a possible source chromaticity in the visibility phases can be divided in two parts. One is related to the source being intrinsically different at different frequencies (this contribution might be especially important for extended sources) and the other one is related to the position of the source (or that of its brightest feature) being different at different frequencies (as it is the case, for instance, of a selfabsorbed corejet structure).
On the one hand, the contribution of the source structure to the visibility phases can be determined from the image of the source at each frequency. This image depends on the visibility calibration, but can also be used to refine such calibration. Therefore, it should be possible, in principle, to decouple the source structure (which introduces baselinedependent phases) from the ionospheric dispersion (which introduces antennadependent phases), with the help of iterative and elaborated imagingcalibration algorithms. The details of these algorithms and their impact in the precision of the estimated ionospheric contribution, as it is decoupled from the sourcestructure contribution after the imaging, is beyond the scope of this paper.
On the other hand, the contribution of a frequencydependent source position on the visibility phases can be
studied if some assumptions are considered. Porcas (2009) reported on the effects of source chromaticity
in the source position estimates through VLBI astrometry, performed using either phase delays or group delays, for
the case of a corejet structure following the model of Blandford & Königl (1979). The
contribution of the chromatic coreshift to the interferometric phase is
where K_{s} depends on the physical conditions in the jet and the angle of the projected baseline with respect to that of the jet. The parameter also depends on the physical conditions in the jet and may take values between 0 and 2. If we combine Eqs. (5) and (16), we obtain the total drift of the visibility phases through the subbands when both effects, ionospheric dispersion and source chromaticity, are taken into account:
Therefore, the effect of a frequencydependent source position is equivalent to the addition of an extra term coupled to the parameter K of the ionospheric dispersion. For (i.e., no coreshift), K_{s} translates into a contribution to equal to a constant (i.e., nondispersive) group delay, with the phase proportional to the frequency. This group delay is equivalent to a shift of the source, which is the same at all frequencies. This shift can be easily fixed in the calibration if the source position is known. For , the effect of the source chromaticity on is just adding a constant, so it does not affect (see Eq. (6)). Therefore, the subband distribution for the optimum estimate of ionospheric dispersion will be the same as with no chromatic effect. However, for the (physically unrealistic) case , there is a complete coupling between the ionospheric dispersion, K, and the source chromaticity, K_{s}, so it is especially difficult to calibrate the ionospheric delay using these sources, regardless of the distribution of subbands used. Real sources may have values of falling between the values here analyzed (0, 1, and 2). It is thus expected to find intermediate cases in which the effect of the frequencydependent source position will be a combination of either a nondispersive group delay, no effect in the estimate of ionospheric dispersion, and a complete coupling between that estimate and the source position. Additionally, all the corejet sources detected in a given FoV may have different values of , so different calibration issues will appear in the same image depending on the coordinates of each source and the values of .
In any case, we notice that K_{s} depends on the direction of the projected baseline relative to that of the jet, so it is a baselinedependent quantity. However, the ionospheric contribution, K, is antennadependent. This different behavior of K_{s} and K, depending on the pair of stations selected, should allow for a robust decoupling of K_{s} from K, provided the number of observing stations is large enough. Therefore, any chromaticity in the source structure and/or position should not affect the results reported in this paper, as long as the baselinedependent chromatic effects are decoupled from the antennadependent ionospheric contribution using the appropriate calibration algorithms.
4.4.3 Radiation from the Galaxy
For frequencies below 400 MHz, the sky brightness temperature is dominated by the Galactic radiation, which depends strongly on the observing frequency ( with ). It means that in LOFAR wideband observations, the noise in the lowfrequency subbands will be higher than that in the highfrequency subbands. In the cases of observations dominated by the radiation from the Galaxy, Eq. 6 must be adapted to take into account the different uncertainties in each subband.
Thermal noise from the sky brightness temperature translates into a Gaussianlike noise in the real and
imaginary parts of the visibilities, with a value of
proportional to the equivalent flux density of the
system, which is in turn proportional to the total (i.e., receivers plus source) temperature (e.g., Thomson,
et al. 1991).
If the observed sources are strong, the noise in the amplitudes and
phases can also be approximated as being Gaussianlike, with a
proportional to that of the real and imaginary parts of
the visibilities. If we take this approximation into account and assume that the galactic radiation dominates
the system temperature (i.e.,
), then the uncertainty in the visibility
phase of the ith subband is
and Eq. (6) becomes
The values of in Eq. (10) that minimize the given by Eq. (18) (we call these values ) are shown in Fig. 7 as a function of the normalized bandwidth, . For small values of , we find that is positive. It means that the noise in the phases at the lower frequencies is such high, that a better sampling of the highfrequency region of the band yields more precise estimates of the ionospheric dispersion. However, this effect is less important for wider bands and/or higher minimum frequencies, as it can be seen in Fig. 7. If the band is wide enough (for instance, for a minimum frequency of 200 MHz) the optimum value is negative. The 2090 MHz band of LOFAR corresponds to a value of as large as 3.5, for which we find (setting MHz).
Figure 7: Optimum value of that minimizes the given in Eq. (18) for different values of the minimum frequency, . These values of have been computed assuming that the Galaxy radiation dominates the noise of the visibilities. A line at is also shown for clarity. 

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5 Summary
We studied the achievable precision in the estimate of the ionospheric dispersion, the ambiguity of the group delay, the dynamic range of the correlated fringes, and the precision in the estimate of the source spectral index in lowfrequency and wideband interferometric observations for four different distributions of the subbands through the total bandwidth of the detectors: constant spacing between subbands, random spacings, spacings based on a powerlaw distribution, and spacings based on Golomb rulers.
For a large number of subbands, spacings based on Golomb rulers give the most precise estimates of dispersive effects and the highest dynamic ranges of the fringes. Spacings based on the powerlaw distribution give similar (but slightly worse) results, although the results are better than those from the Golomb rulers for a smaller numbers of subbands. Random distributions of subbands result in relatively large dynamic ranges of the fringes, but the estimate of dispersive effects through the band is worse. A constant spacing of the subbands results in very bad dynamic ranges of the fringes, but the estimates of dispersive effects have a precision similar to that obtained with the powerlaw distribution.
From all combinations of the number of subbands and the total covered bandwidth, the powerlaw distribution (with given by Eq. (12)) gives the best compromise between homogeneity in the sampling of the bandwidth, precision in the estimate of the ionsopheric dispersive effects, dynamic range of the correlated fringes, and ambiguity of the group delay. Therefore, this would be the preferable subband distribution to use in lowfrequency (wideband) interferometric observations.
Finally, we study how the powerlaw distribution that optimally samples the ionospheric dispersion is affected in the cases of 1) observing frequencies close to the plasma frequency of the ionospheric electrons; 2) chromatic effects in the structure of the sources; and 3) nonnegligible noise coming from the Galaxy radiation.
The author is a fellow of the Alexander von Humboldt Foundation in Germany. The author is very thankful to Eduardo Ros and the anonymous referee for their useful comments and suggestions to improve this paper. The author also acknowledges the collaboration of Nicolás MartíDunca during the preparation of this paper.
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All Figures
Figure 1: Examples of the four kinds of distributions studied in this paper (using 16 subbands and setting , see Eq. (1)). a) corresponds to the uniform (i.e. constant) distribution; b) to the random distribution; c) to the powerlaw distribution; and d) to the distribution based on the Golomb ruler. 

Open with DEXTER  
In the text 
Figure 2: Boxes, optimum values of (i.e., those that minimize the formal uncertainty in the estimate of the ionospheric dispersion) as a function of the number of subbands. Line, model corresponding to Eq. (12). 

Open with DEXTER  
In the text 
Figure 3: a) Formal uncertainty in the estimate of the ionospheric dispersion, in units of the minimum possible uncertainty; b) dynamic range of the fringes; and c) delay ambiguity (i.e., distance between the fringe peak and the closest sidelobe) in units of the Nyquist time resolution. All these quantities are shown as a function of the number of subbands for (see Eq. (1)). 

Open with DEXTER  
In the text 
Figure 4: Formal uncertainty in the estimate of the ionospheric dispersion, in units of the minimum possible uncertainty, as a function of (see Eq. (1)) for the case of 32 subbands. 

Open with DEXTER  
In the text 
Figure 5: Formal uncertainty in the estimate of the source spectral index, , in units of that of the uniform distribution ( ), as a function of the bandwidth, (see Eq. (1)), for the case of 64 subbands. 

Open with DEXTER  
In the text 
Figure 6: Optimum values of , in units of , as a function of the minimum observing frequency, , in units of the plasma frequency, . In each figure, the values are computed for five different values of (2, 4, 6, 8, and 10 MHz). Each figure corresponds to a different bandwidth, (0.1, 0.5, 1.0, and 1.5). 

Open with DEXTER  
In the text 
Figure 7: Optimum value of that minimizes the given in Eq. (18) for different values of the minimum frequency, . These values of have been computed assuming that the Galaxy radiation dominates the noise of the visibilities. A line at is also shown for clarity. 

Open with DEXTER  
In the text 
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