The selfcohering tiedarray
P. Fridman
ASTRON, Oudehoogeveensedijk 4, Dwingeloo, 7991PD, The Netherlands
Received 6 October 2009 / Accepted 26 November 2009
Abstract
Context. Large radio astronomy multielement interferometers
are frequently used as single dishes in a tiedarray mode when signals
from separate antennas are added. Phase shifts arising during wave
propagation through a turbulent atmosphere can significantly reduce the
effective area of an equivalent single dish.
Aims. I aim to give estimates of the impact of the ionosphere
and troposphere on the effectiveness of a radio interferometer working
in tiedarray mode.
Methods. Statistical estimates of the effective area are
calculated and the powerlaw of turbulent atmosphere irregularities has
been used. A simple method of tiedarray calibration using optimization
techniques is proposed.
Results. The impact of phase errors on the effectiveness of
tiedarrays are given for low and high frequencies. Computer
simulations demonstrate the efficacy of the proposed calibration
algorithm.
Key words: instrumentation: interferometers  methods: data analysis  methods: statistical
1 Introduction
Large radio astronomy multielement interferometers (VLA, WSRT) are frequently used in the tiedarray mode where signals from separate antennas are added (Thompson et al. 2001, Chap. 9.9). The output sum signal can be used in VLBI, pulsar and transients observations (Cordes et al. 2004a,b), SETI signals detection and the directtoEarth (DtE) reception of signals from cosmic apparatus (Jones 2004). In all these cases a radio interferometer works as a singledish antenna with one output. Partial signals from antennas are properly phased to collect emission from a pointlike radio source in the sky and track it during its siderial movement. Standard calibration procedure using a correlator is employed to provide the necessary phase corrections for each individual antenna. Random phase perturbations such as phase shifts arising during wave propagation through the turbulent atmosphere can occur in the course of such observations. These phase errors reduce the total effective area of the tiedarray and must be compensated for in real time. Although it is possible to store baseband data for processing after observations have taken place, the amount of data to be stored places a firm limit to the number of antennas that can be used in this manner.
New large scale projects such as SKA and LOFAR will also be operating in tiedarray mode. The impact of ionospheric and tropospheric phase errors on the tiedarray is calculated in this paper. A simple method of correcting these errors using the output signal of the tiedarray is also proposed here.
2 Tiedarray with random phase errors
Voltage produced by the source at the output of the planar tiedarray is
(1) 
where is the source vector, , is the wavelength, f is the frequency, a_{s,n}  is the signal amplitude at nth array element, r_{s,n}  is the distance between the source and nth array element, is the number of antennas in the array, is the instrumental phase shift. The distance r_{s, n} can be represented as the module of the vector difference , where is the position vector of nth array element:
r_{s,n}  =  
=  (r_{s}^{2}2p_{n,x}s_{x}2p_{n,y}s_{y}+p_{n,x}^{2}+p_{n,y}^{2})^{1/2}  
=  (2) 
s_{x}, s_{y}, s_{z} are the components of vector and p_{n,x}, p_{n,y} are the components of vector , .
With these new notations (1) can be rewritten as:
(3) 
An array is directed at when these conditions are satisfied:
(4) 
The manifold of signals received in other directions ( ) forms the field pattern:
(5) 
The power pattern is the expected value of the product
=  
=  (6) 
where denotes a complex conjugate and denotes a time average.
In the absence of phase errors and in the direction of the radio source, i.e., for . The dc component proportional to the sum of the system noise power at each antenna is omitted here and considered to be a constant value and therefore not relevant.
In the presence of phase errors produced by, for example, the atmosphere, there are additional phase terms
in (6) and the power received in the direction
is:
=  
=  (7) 
where is the variance of the random phase difference for the baseline :
(8) 
It is assumed that the random phase difference has normal probability distribution with zero mean and variance . In this case (7) was obtained using this relation:
=  
=  (9) 
Therefore, the loss produced by the phase errors is equal to:
(10) 
The variance is the structure phase function of the turbulent atmosphere. The powerlaw (Kolmogorov) model will be used in the following sections to describe both for the ionosphere and troposphere phase fluctuations (Tatarskii 1978).
2.1 Ionosphere
Phase fluctuations due to the irregular spatial distribution of the
refraction index during wave propagation through the ionosphere are
described with the powerlaw model of the turbulence spectrum. The
electron density N
in the ionosphere, considered as a function of spatial coordinates, has
variations which are characterized by a structure function of electron
density D_{N}(b) (Thompson et al. 2001, Chap. 13):
(11) 
where , C^{2}_{N} is the constant, b is the baseline. This formula can be rewritten for the structure function of the refraction index:
(12) 
where m (electron radius), is the wavelength. Finally, the ionosphere phase structure function is:
(13) 
where h is the total propagation length through the irregularities of the ionosphere, and b must be substituted in meters. The value C^{2}_{N} can be estimated from (11) assuming that the ionosphere irregularities of electron density have a maximum dimension equal to L_{0}:
(14) 
For example, for and (day time) we have for , for and . Figure 1a shows the square root of the ionosphere structure function for , and calculated for three frequencies: 50 MHz, 100 MHz and 200 MHz.
Figure 1: a) Upper panel: square root of the ionosphere structure function of electrical length, in cm; , and L_{0}=10 km calculated for three frequencies: 50 MHz, 100 MHz and 200 MHz; b) lower panel: Fried length for different values of N, L_{0},h. 

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Figure 2: a) Upper panel: random array configuration; b) lower panel: histogram of the baselines. 

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Figure 3: Loss produced by phase fluctuations in the ionosphere calculated for three frequencies: 50, 100 and 200 MHz. The structure function from Fig. 1a is used. 

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Phase fluctuations can be also characterized by the Fried length:
(15) 
where d_{0} is the baseline at which . For the parameters used in Fig. 1a km for 50 MHz, km for 100 MHz and km for 200 MHz. Figure 1b shows how Fried length depends on the frequency for the different values of N, L_{0},h.
The minimal time interval at which it is necessary to repeat calibrations can be calculated from Fried length: where is the wind velocity. Thus, for example, for km and , s.
Now the loss produced by ionosphere random phase errors can be calculated in the example of the array whose configuration is shown in Fig. 2a. It is the random planar 100element array with the coordinates x_{i} and y_{i} represented by random normal values with zero mean and standard deviation SC. For the array shown in Fig. 2a, SC=1000 m, therefore the maximum baseline is 5000 m. The distribution of baselines (histogram) is shown in Fig. 2b. Phase errors are maximal for the largest baselines but their relative number is less than the medium size baselines, therefore the signal loss for the array should take this particular distribution of baselines into account. Figure 3 demonstrates the dependance of loss versus array size . The curves are calculated for three frequencies: 50, 100 and 200 MHz.
2.2 Troposphere
Phase fluctuations due to the irregular spatial distribution of the
refraction index during wave propagation through the troposphere are
also described in the frame of the powerlaw spectrum turbulence model.
The troposphere phase structure function is (Stotskii 1973; Carilli et al. 1999):
=  2.91k^{2}C_{l}^{2}b^{5/3},L_{0}<b<L_{1}  
=  2.91k^{2}C_{L}^{2}b^{2/3}, L_{1}<b<L_{2}  
=  2.91k^{2}C_{L}, L_{2}<b,  (16) 
where L_{0} and L_{1} are the internal and external scales, respectively, of the isotropic threedimensional turbulence model, L_{0}=0.11 cm, L_{1}=5.6 km and L_{2}=20003000 km, the latter is determined by global meteorological variations.
Factors C_{l}^{2} and C_{L}^{2} depend on the local content of water vapor and oxygen in the troposphere (weather conditions) and the values chosen for the purpose of calculation are and .
Figure 4a represents the structure function (16) and Fig. 4b shows the Fried length as a function of the baseline. Figure 5 demonstrates the dependance of loss versus array size . The curves are calculated for three frequencies: 1400, 5000 and 8400 MHz.
Figure 4: a) Upper panel: square root of the troposphere structure function of electrical length, in cm; b) lower panel: Fried length as a function of frequency. 

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Figure 5: Loss produced by phase fluctuations in the troposphere calculated for three frequencies: 1400, 5000 and 8400 MHz. The structure function from Fig. 4a is used. 

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3 Selfcohering
Observations in the tiedarray mode (VLBI, transients, DtE) are preplanned at any time and it is impossible to postpone them in order to choose better atmospheric conditions (for example, night time in the case of the ionosphere). The effectiveness of the synthesized aperture must be maximal during observations which means that periodical calibrations are necessary. Traditional methods of radio interferometer calibration can be applied using the grid of calibration point sources. This calibration must be made in real time with the help of available correlators which must work in parallel with the tiedarray adder. Here, another method is proposed which uses the points in the direct images of the fieldofview with calibration sources.
It is presumed that a full calibration with the correlator has already been performed before each tiedarray observation. During subsequent observations the total power output of the tiedarray is used as a tracking tool and the proposed algorithm will introduce small phase corrections at short time intervals therefore keeping the amplitude of the calibration source at a prescribed level. This has similarities to the approach of Muller & Buffington (1974) which is also described in Tyson (1991).
The choice of calibration sources is the same as in traditional methods.
Equation (7) corresponds to the synthesized beam when there are phase errors
produced during propagation through the turbulent atmosphere.
To eliminate
,
compensation phase shifts
are introduced at each nth array element. The phase of the signal corresponding to the direction
at the nth array element is
(17) 
where is the compensation phase shift introduced for the correction of  atmospheric phase error. The value of the signal power in the prescribed direction in the image with one or several calibration sources available in the fieldofview (FoV) can be obtained by convolution of the sky intensity with the synthesized beam
(18) 
Figure 6: 60element tied array configuration. 

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Figure 7: Spatial phase error distribution, projected on the array plane. 

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Figure 8: Phase errors as function of baseline length. 

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(19) 
Figure 9: Left panel: synthesized image without phase errors; middle panel: synthesized image with phase errors; right panel: synthesized image after correction. 

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Figure 10: Contour presentations, left panel: synthesized image without phase errors; middle panel: synthesized image with phase errors; right panel: synthesized image after correction. 

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The image containing three point sources is represented in Fig. 9 (left panel) and the synthesized image in the presence of the phase errors (Fig. 8) is shown in Fig. 9 (middle panel) (isoplanicity being presumed).
The value of the synthesized image in the direction of the largest source (lower left in the image) was used as the cost function. The genetic algorithm (GA) was applied because of the strong multimodality of the cost function (19) and this algorithm finds the global maximum successfully. Genetic algorithms search for a solution to a set of variables by the use of simulated evolution, i.e., the survival of the fittest strategy. In contrast to calculusbased algorithms (conjugate gradients and quasiNewtonian methods), GA, first introduced in (Holland 1975), exploit a guided random technique during optimization procedure (Goldberg 1989; Michalewicz 1994; Charbonneau 1995).
GA optimizers are particularly effective when the goal is to find an approximate global maximum in a highdimension, multimodal function domain in a nearoptimal manner. They are also largely independent of the starting point or initial guess. There is parallelism which allows for the exploitation of several areas of the solution space at the same time. This parallelism can be very useful in the implementation of GA on the multicore platform and FPGA. In this article, computer simulation has been done on a PC (Intel Pentium, 2.5 GHz, 1 GB RAM) using Matlab 7.6.0. Specific GA operations (selection, crossover and mutation) have taken approximately of the total computing time: 150 s for 100 iterations (each iteration is the full cycle of these operations). The rest of the computing time was spent on the calculation of the cost function (formation of the beam with corrected phases > convolution with the image > total power output). But these calculations are necessary only in computer simulations: in reality the values of the cost function are supplied by the tiedarray itself (``Nature'' does the job).
After applying the optimization procedure and introducing the resulting compensation phases the corrected image is shown in Fig. 9 (right panel).
The contour presentations in Fig. 10 correspond to the undistorted image (left panel), the image with phase errors (middle panel) and the image after correction (right panel), respectively.
The corresponding synthesized beam is restored up to 0.94 of its undistorted value.
There are some peculiarities in image processing with nonplanar arrays (Perley 1999) but the tiedarray mode concerns only pointlike sources. Therefore, there is no difference in planar and nonplanar arrays in the context of this article (phase irregularities due to atmospheric turbulence), especially for the adaptive calibration procedure described here.
4 Conclusions
 1.
 The effective area of tied arrays may be significantly reduced by ionospheric and tropospheric phase irregularities at low and high frequencies, respectively.
 2.
 Observations are made at times (VLBI, transients monitoring, DtE) when it is impossible to choose quiet atmospheric conditions and realtime calibration is necessary and has to be fulfilled in parallel with observations.
 3.
 The total power at the auxiliary outputs of the tiedarray, phased in the direction of calibration sources, can be used on a level with traditional calibration methods using correlators. Multibeam facilities are necessary for creating these auxiliary outputs. Optimization algorithms (genetic algorithms, simulated annealing) can be used to compensate for propagation phase errors by maximizing the amplitude of a chosen calibration source. The tied array can preserve its correctly phased state during lengthy observations using one or several auxiliary outputs, therefore working in the selfcohering regime. The proposed scheme does not exclude traditional methods of calibration  it is complementary to them.
I am grateful to Roy Smits whose comments were very helpful.
References
 Carilli, C. L., Carlstrom, J. E., & Holdaway, M. A. 1999, in Synthesis Imaging in Radio Astronomy II, ed. G. B. Taylor, C. L. Carilli, & R. A. Perley (ASP) (In the text)
 Charbonneau, P. 1995, APhS, 101, 309 (In the text)
 Cordes, J. M., Kramer, M., Lazio, T. J. W., et al. 2004a, New Astron. Rev., 48, 1413 [NASA ADS] [CrossRef] (In the text)
 Cordes, J. M., Lazio, T. J. W., & McLaughlin, M. A. 2004b, New Astron. Rev., 48, 1459 [NASA ADS] [CrossRef] (In the text)
 Jones, D. L. 2004, New Astron. Rev., 48, 1543 [NASA ADS] [CrossRef] (In the text)
 Goldberg, D. E. 1989, Genetic Algorithms in Search, Optimization and Machine Learning (AddisonWesley) (In the text)
 Holland, J. H. 1975, Adaptation in Natural and Artificial Systems (Ann Arbor: University of Michigan Press) (In the text)
 Michalewicz, Z. 1992, Genetic Algorithms + Data structures = Evolution Programs (Springer) (In the text)
 Muller, R. A., & Buffington, A. 1974, JOSA, 64, 1200 [NASA ADS] [CrossRef] (In the text)
 Perley, R. A. 1999, in Synthesis Imaging in Radio Astronomy II, ed. G. B. Taylor, C. L. Carilli, & R. A. Perley (ASP) (In the text)
 Stotski, A. A. 1973, Radiophys. Quant. Electron, 16, 620 [NASA ADS] [CrossRef] (In the text)
 Tatarskii, V. I. 1978, Wave Propagation in Turbulent Media (John Wiley & Sons, Inc.) (In the text)
 Thompson, A. R., Moran, J. M., & Swenson, G. W. 2001, Interferometry and Synthesis in Radio Astronomy (John Wiley & Sons, Inc.), Chaps. 9.9, 13 (In the text)
 Tyson, R. K. 1991, Principles of Adaptive Optics (SanDiego: Academic Press) (In the text)
 Wright, M. 2004, SKA Memo, 46 (In the text)
All Figures
Figure 1: a) Upper panel: square root of the ionosphere structure function of electrical length, in cm; , and L_{0}=10 km calculated for three frequencies: 50 MHz, 100 MHz and 200 MHz; b) lower panel: Fried length for different values of N, L_{0},h. 

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In the text 
Figure 2: a) Upper panel: random array configuration; b) lower panel: histogram of the baselines. 

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In the text 
Figure 3: Loss produced by phase fluctuations in the ionosphere calculated for three frequencies: 50, 100 and 200 MHz. The structure function from Fig. 1a is used. 

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In the text 
Figure 4: a) Upper panel: square root of the troposphere structure function of electrical length, in cm; b) lower panel: Fried length as a function of frequency. 

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In the text 
Figure 5: Loss produced by phase fluctuations in the troposphere calculated for three frequencies: 1400, 5000 and 8400 MHz. The structure function from Fig. 4a is used. 

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In the text 
Figure 6: 60element tied array configuration. 

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In the text 
Figure 7: Spatial phase error distribution, projected on the array plane. 

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In the text 
Figure 8: Phase errors as function of baseline length. 

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In the text 
Figure 9: Left panel: synthesized image without phase errors; middle panel: synthesized image with phase errors; right panel: synthesized image after correction. 

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In the text 
Figure 10: Contour presentations, left panel: synthesized image without phase errors; middle panel: synthesized image with phase errors; right panel: synthesized image after correction. 

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In the text 
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