Memo 113
LOFAR imaging capabilities and
system sensitivity
R.J. Nijboer
M. Pandey-Pommier
A.G. de Bruyn
07/09
www.skatelescope.org/pages/page_memos.htm
Table of contents:
1 Introduction
2 Station configurations
2.1
LBA station configuration
2.2
HBA station configuration
3 LOFAR imaging capabilities
4 Sensitivity of the LOFAR array
4.1
System Equivalent Flux Density
4.2
Sensitivities
5 Cautionary notes
6 Acknowledgements
7 References
8 Appendix: Plots of the LBA effective area lower bound
8.1
15 MHz
8.2
30 MHz
8.3
45 MHz
8.4
60 MHz
8.5
75 MHz
-1-
2
3
3
4
7
9
9
11
13
14
15
16
16
17
18
19
20
1 Introduction
This document describes the general astronomical capabilities of the LOw Frequency ARray (LOFAR). The
frequency range covered by LOFAR is split into two bands denoted as low band (LB, 10 - 80 MHz) and high
band (HB, 120 - 240 MHz). LOFAR stations are spread over a 100 km sized region in the northern part of the
Netherlands. In addition to the Dutch stations there will be European stations providing baselines between
200 km and 1000 km. Most of the results in this document, however, are limited to the Dutch array.
In section 2 the LBA and HBA station configurations are reviewed. Section 3 reviews the imaging capabilities
as station Full Width Half Maximum, station Field of View, and array resolution. In section 4 the system
sensitivity is considered and thermal noise levels in LOFAR images are given. Finally, in section 5 some
cautionary notes are collected. We advise you to read these notes carefully before using the numbers that
are presented in this report.
-2-
2 Station configurations
Each Dutch LOFAR station is capable of processing 48 dual polarization signal paths, whereas European
stations are capable of processing 96 dual polarization signal paths. For the Dutch stations in the low-band
96 single dipole antennas (LBA) are connected to 48 Receiver Unit boards, using the low-band-low and the
low-band-high connections. In this way out of 96 antennas 48 can be selected and used in an observation.
For the European stations all 96 antennas will be connected to either the low-band-low or the low-band-high
connection, so that for these stations all 96 antennas can be used simultaneously.
For the Dutch stations in the high band 48 4 by 4 dipole tiles are connected. For the core stations these are
split in a 2 times 24 configuration, in this way doubling the number of core stations. For the European
stations 96 tiles are connected.
2.1
LBA station configuration
In this section the different LBA antenna configurations are summarized. Out of 96 antennas 48 can be
selected. The array configuration and connection scheme described in LOFAR memo 241 by Wijnholds [1] is
used. The low band antennas are arranged in a randomized exponentially expanding configuration. This
ensures that the station can provide approximately the same sparseness factor and Field of View (FoV) at
different frequencies, which is convenient from the calibration perspective. Five antenna configurations will
be available: an all X configuration, an all Y configuration, an inner configuration, an outer configuration, and
a sparse configuration.
Figure 1 Array configuration of 94 LBA dipole antennas. Two outlier dipoles are not shown.
The inner configuration consists of the innermost 46 antennas in dual polarization + two outlier antennas for
station calibration purposes. The station beam can be constructed from 46 antennas or the total 48
antennas. Here we assume the beam is made out of the innermost 46 antennas. The outer configuration
consists of the outermost 48 antennas in dual polarization. The sparse configuration consists of a total of 48
antennas in dual polarization from the inner and outer configurations. Finally, the X and Y configurations
consist of all 96 antenna dipoles using the X or Y signal chain only.
-3-
In Table 1 the maximum and minimum baseline length for each of the array selections is given. The
maximum baseline length determines the FoV of the station beam. The minimum baseline length determines
the critical frequency for satisfying the spatial Nyquist criteria. Below this critical frequency there is still a λ / 2
spacing. Above the critical frequency the spacing of the dipoles is larger than λ / 2 implying that the station
beam sidelobe level will increase. For the outer array the critical frequency lies at 48.8 MHz. For the inner
and sparse arrays the critical frequency lies at 62.5 MHz. Note that due to the physical dimensions of the
dipole antennas it is not possible to have λ / 2 spacing above 62.5 MHz.
Max. baseline length (m) Min. baseline length (m)
Inner array
32.25
2.40
Outer array
81.34
3.08
Sparse array
81.34
2.40
Full array
81.34
2.40
Table 1 Maximum and minimum baseline length for the different array selections.
The choice for a particular LBA station array is determined by sensitivity (effective area), FoV, and beam
quality (uv sampling). For instance, survey speed is given by FoV * (Aeff / Tsys)2. At constant frequency this
scales like (Aeff / D)2, where D is the longest baseline within the selected array. In Table 2 below we give
Aeff / D for the three array selections. (For details on the effective area see section 4.)
Freq. (MHz) Inner array Outer array Sparse array
15
13.72
23.99
14.12
30
13.72
16.24
10.69
45
13.35
8.345
6.868
60
11.03
4.793
4.564
75
7.598
3.082
3.042
Table 2 Aeff / D, where D is the longest baseline in the array selection.
From Table 2 it follows that in terms of survey speed for a “narrow band” observation it is best to choose the
outer array below approximately 40 MHz, and the inner array above approximately 40 MHz. If, however, one
wants to do a broadband observation observing in some subbands above 50 MHz and in some other
subbands below 40 MHz, then the sparse array could be a good option.
2.2
HBA station configuration
The high band tiles are configured on a regular grid. Each tile measures 5 meter by 5 meter and contains 16
dipole antennas on a 4 by 4 grid. The tiles are separated by 15 cm. A core station consists of two times 24
tiles, both in a 2-4-6-6-4-2 configuration and has a size from edge to edge of 30.75 m (see Figure 2). A
remote station consists of 48 tiles in a 4-4-8-8-8-8-4-4 configuration and has a size from edge to edge of
41.05 m (see Figure 3). A European station consists of 96 tiles in a 5-7-9-11-11-10-11-11-9-7-5 layout and
has a size from edge to edge of 56.5 m (see Figure 4).
-4-
Figure 2 Core station layout.
Figure 3 Remote station layout.
-5-
Figure 4 European station layout.
-6-
3 LOFAR imaging capabilities
The Full Width Half Maximum (FWHM) of a LOFAR Station beam is determined by
FWHM = α 1
λ
D
,
α 1 = 1.3,
where λ denotes wavelength and D denotes the station diameter. The value 1.3 is representative for a WSRT
dish, where it depends on the dish illumination. For LOFAR it will depend on the final tapering of the station.
For the LBA probably no tapering will be applied. In that case it is expected that the value of α 1 will turn out
to be between 1.2 and 1.4. For reference, the value for the LOFAR Initial Test Station (ITS) was 1.42 (S.J.
Wijnholds, private communication).
The Field of View (FoV) of a LOFAR station is defined as
2
⎛ FWHM ⎞
FoV = π ⎜
⎟ .
2
⎠
⎝
The number of pointings to cover 2π steradians of sky in a Nyquist sampled way (on a square grid) is
approximately 64800 FoV . After weighted combination of the different pointings the effective sensitivity will
be better by a factor 1.5. Hence, one can choose to sample at slightly less than Nyquist to increase
surveying speed. Table 3 summarizes the FWHM, the FoV, and the number of pointings for the Dutch
LOFAR stations.
Freq
λ
Stat. diam.
FWHM
FoV
# pointings
D
all sky
(MHz) (m)
(m)
(deg)
(sq.deg.)
Nyquist sampled
15
20.0 32.3 81.3 46.2 18.3 1676 263
38.7
246
30
10.0 32.3 81.3 23.1 9.16 419 65.8
155
984
45
6.67 32.3 81.3 15.4 6.10 186 29.3
348
2214
60
5.00 32.3 81.3 11.6 4.58 105 16.5
619
3936
75
4.00 32.3 81.3 9.24 3.66 67.0 10.5
967
6149
120
2.50 30.8 41.1 6.06 4.54 28.8 16.2 2250
4009
150
2.00 30.8 41.1 4.84 3.63 18.4 10.3 3515
6265
180
1.67 30.8 41.1 4.04 3.02 12.8 7.18 5062
9021
210
1.43 30.8 41.1 3.46 2.59 9.40 5.28 6890
12279
240
1.25 30.8 41.1 3.03 2.27 7.20 4.04 8999
16038
Table 3 Primary beams of LOFAR. The five LBA configurations yield two different station sizes. The HBA
numbers are given for core and remote station sizes. The two columns for FWHM, FoV, and # pointings
correspond to the two different station diameters for the five LBA configurations and the two HBA
configurations.
The resolution of the LOFAR array is given by
α2
λ
L
,
α 2 = 0.8,
where L denotes the longest baseline. The value 0.8 depends on the array configuration and the weighting
scheme that is used during imaging (natural, uniform, robust, …). The current 0.8 is approximately the value
for the WSRT with standard tapering and it is likely that for LOFAR a different value will be used. Values of
the resolution of LOFAR based on a value of 0.8 can be found in Table 4.
-7-
Freq.
(MHz)
15
30
45
60
75
120
150
180
210
240
Table 4 Resolution of LOFAR
λ
(m)
20.0
10.0
6.67
5.00
4.00
2.50
2.00
1.67
1.43
1.25
Resolution
L = 2 km
(arcsec)
1650
825
550
413
330
206
165
138
118
103
Resolution
L = 10 km
(arcsec)
330
165
110
82.5
66.0
41.3
33.0
27.5
23.6
20.6
Resolution
L = 80 km
(arcsec)
41.3
20.6
13.8
10.3
8.25
5.16
4.13
3.44
2.95
2.58
Amitesh Omar investigated the effect of weighting schemes on the resolution and sensitivity of the LOFAR
array. Table 5 and Table 6 are reproduced from his results. The 20 station configuration uses a maximum
baseline of 15 km, whereas the 50 station configuration is based on a maximum baseline of 80 km. From
Table 5 it can be seen that for uniform weighting one achieves better resolution at the expense of higher
PSF sidelobes and lower sensitivity, as expected.
Table 5 Resolution-sensitivity for declination = +40 deg., fractional BW = 0.133 and 4 hrs observation. Note:
Robust is as parameterized in AIPS (+ integer means towards natural weighting, - integer means towards
uniform weighting). Rel. rms is relative to natural weighting, f is freq in MHz, maximum baselines for
LOFAR20 equals 15 km, an elliptical Gaussian beam is used, typical accuracy in the parameters = 10%. (A.
Omar [2]).
In Table 6 the resolution of LOFAR using robust weighting with a value of -2 (resolution equals 600 / f; see
Table 5) is given. The resolution given in Table 6 and is in good agreement with the values given in Table 4.
Table 6 Resolution for LOFAR50 using robust equals -2 weighting (A. Omar).
-8-
4 Sensitivity of the LOFAR array
4.1
System Equivalent Flux Density
The System Equivalent Flux Density (or system sensitivity) is defined as
S sys =
2ηk
Tsys
Aeff
where k denotes Boltzmann’s constant, η denotes the system efficiency factor (~ 1.0) , Tsys denotes the
system noise temperature, Aeff denotes the total collecting area, see [3], [4].
The system noise temperature consists of a sky brightness component and an instrumental component
Tsys = Tsky + Tinstr .
For all LOFAR frequencies the sky brightness temperature is dominated by the Galactic radiation which
depends strongly on the wavelength
Tsky = Ts 0 λ2.55
and Ts 0 = 60 ± 20 K, for Galactic latitudes between 10 and 90 degrees, [4]. The instrumental noise
temperature follows from measurements or simulations. For the LOFAR LBA dipoles and HBA tiles it is
shown in Figure 5. The lines in Figure 5 show results that are averaged over azimuth and elevation. The LBA
line is based on measurements in the field and is in addition averaged over 48 dipoles.
Figure 5 Left: measured Tsky / Tsys for a LOFAR LBA antenna. Right: simulated Tsys for a LOFAR HBA
antenna tile consisting of 16 dual dipoles; sky noise (red), noise from the HBA unit (blue), and total receiver
noise (HBA unit + coax + receiver unit; green).
The generic formula for the effective area of a single dipole is
Aeff ,dipole =
-9-
λ2
Ωe
where λ denotes the observed wavelength and Ω e denotes the effective solid angle covered by the antenna
beam. This factor is the integral over the sky weighted by the power beam pattern and normalized to one [1].
Thus, for an isotropic beam pattern, Ω e = 2π. For an isolated dipole, Ω e is approximately 3 and for a dipole
in a half wavelength spaced dense array, Ω e = 4. Since the effective area per dipole is constrained by the
available space, i.e. by the presence of its neighbours, in first order calculations, good results can be
obtained by using
⎫
⎧ λ2
Aeff ,dipole = min ⎨ , < available _ physical _ area > ⎬ .
⎭
⎩3
The effective area of a station is then the summation of the effective areas of the composing dipoles
Aeff , station = ∑ Aeff ,dipole ,i .
i
In Table 7 a lower limit for the effective area of the different LBA array selections is given. The upper limit is
given by the number of dipoles times λ2 / 3 and is given in the last column (for 48 dipoles). Note that the
inner array consists of only 46 dipoles. The effective area of each dipole in the array is determined by its
distance to the nearest dipole (d) within the full array:
⎧ λ2 πd 2 ⎫
Aeff ,dipole = min ⎨ ,
⎬.
⎩3 4 ⎭
This value is a lower limit of the actual effective area. Plots for the lower bound per dipole can be found in the
Appendix.
Freq (MHz) λ (m) Aeff inner (46) Aeff outer (48) Aeff sparse (48) 48 * Aeff_dipole
15
20.0
419.77
1973.4
1148.6
6400.0 *
30
10.0
419.77
1343.5
869.14
1600.0
45
6.67
415.83
693.61
558.64
711.11
60
5.00
347.37
398.18
371.19
400.00
75
4.00
239.67
256.00
247.43
256.00
Table 7 Effective area (m2) for the different LBA array selections as function of frequency (MHz) or
wavelength (m). (*) an 81.34 m station has an area of 5196.3 m2.
For an HBA dipole the effective area is limited by the available area in a tile. There are 16 dipole antennas
within one 5 m by 5 m tile, hence, the dipole effective area is given by
Aeff ,dipole
⎧ λ2
⎫
= min ⎨ , 1.5625⎬ .
⎩3
⎭
The resulting HBA station effective area is given in Table 8.
Freq. (MHz) λ (m) Core Remote European
120
2.50 600
1200
2400
150
2.00 512
1024
2048
180
1.67 356
711
1422
200
1.50 288
576
1152
210
1.43 261
522
1045
240
1.25 200
400
800
Table 8 Effective area (m2) for the HBA array as function of frequency (MHz) or wavelength (m). At 120 MHz
the effective area is limited by the tile size.
In Table 9 the resulting SEFD’s (in single polarization) for the different LOFAR stations are given, where the
above results for the sky temperature, the system temperature, and the effective area have been used. At 15
-10-
MHz and 30 MHz the outer array is considered, whereas at 45 MHz, 60 MHz, and 75 MHz the inner array is
used. For the LBA the system sensitivities are based on an average between the minimum and maximum
effective area as given in Table 7. The values for the European LBA station are based on a 65 m station.
From the Table 9 it can be seen that in the LBA LOFAR is most sensitive at 60 MHz and in the HBA LOFAR
is most sensitive at 150 MHz.
Freq NL-Core NL-Remote EU-Remote
(MHz)
(kJy)
(kJy)
(kJy)
15
483
483
519
30
89
89
41
45
48
48
19
60
32
32
15
75
51
51
25
120
3.6
1.8
0.89
150
2.8
1.4
0.71
180
3.2
1.6
0.81
210
3.7
1.8
0.92
240
4.1
2.0
1.0
Table 9 System Equivalent Flux Densities for the different LOFAR stations in single polarization.
4.2
Sensitivities
The sensitivity ΔS (in Jy) of a single dipole (or half an “antenna”) is defined as follows [3]
S sys ,dipole
ΔS dipole =
2δυδt
where δν denotes the bandwidth (in Hz) and δt (in s) denotes the total integration time. An antenna that
consists of two (equal) dipoles placed perpendicular to each other has a sensitivity of
ΔS antenna =
ΔS dipole
.
2
For a station the overlap in effective area from different dipoles has to be taken into account. Using the
SEFD of a station, S sys , station , given in section 4.1 for a single polarization, the sensitivity of a station (in
single polarization) is given by
ΔS station =
S sys , station
2δυδt
.
These sensitivities are somewhat artificial, since they are usually not measured. However, the noise level on
a baseline (i,j) correlation can be derived from it
ΔS ij = ΔS i ΔS j .
In an image signals from different baselines and two polarizations are combined. For LOFAR this includes
signals from stations with different effective areas. The noise level in a LOFAR image is given by
ΔS =
W
⎧ N ( N − 1) / 2
Nc Nr
N ( N − 1) / 2 ⎫
+
+ r 2r
2(2δυδt )⎨ c c2
⎬
S core S remote
S core
S remote
⎩
⎭
where W denotes a factor for increase of noise due to the weighting scheme (which depends on the type of
weighting: natural, uniform, robust, …), N c and N r denote the number of core and remote stations
-11-
respectively,
S core and S remote denote the SEFDs for the core and the remote stations respectively. This
expression can easily be extended to include European stations.
In case of an array having N equal stations (or dishes) the following familiar result is retrieved [3]
ΔS =
S sys
N ( N − 1)2δυδt
In Table 10 the expected noise levels in LOFAR images are given. The images are based on 1 hour
integration time, 2 polarizations, and 4 MHz bandwidth (based on results from LOFAR CS1 this is effectively
3.57 MHz i.e. 89.25% of 4 MHz). A weighting factor W of 1.3 is applied to incorporate the effect of weighting
(A.G. de Bruyn private communication; results from A. Omar indicate that this value might be as large as 2,
see section 3). SEFD’s of Table 9 have been used.
Freq.
λ
ΔS13+7
ΔS13+7
Tapered
(mJy)
ΔS18+18
ΔS25+25
(mJy)
(mJy)
(mJy)
(MHz) (m)
15
20.0
201
110
79
30
10.0
37
20
15
45
6.67
20
11
7.8
60
5.00
13
7.2
5.2
75
4.00
21
12
8.4
120
2.50
0.74
0.89
0.41
0.29
150
2.00
0.58
0.71
0.32
0.23
180
1.67
0.67
0.81
0.37
0.26
210
1.43
0.76
0.91
0.42
0.30
240
1.25
0.84
1.0
0.46
0.33
Table 10 LOFAR sensitivity for 1 hour integration time, an effective BW of 3.57 MHz, and dual polarization. A
weighting factor of 1.3 is applied. 13+7 denotes 13 core + 7 remote stations, while 18+18 denotes 18 core
stations + 18 remote stations and 25+25 denotes 25 core stations + 25 remote stations. The “Tapered”
column has the remote stations tapered to core size for use during the Million Source Shallow Survey
(MSSS).
-12-
5 Cautionary notes
Although great care has been given to deriving the numbers in this document, still they have to be treated
with some caution. Based on discussions with H. Röttgering the following list of arguments was collected,
which can also be found in [5]:
• The sensitivity numbers have been calculated taking a representative out of the plane value for the
sky brightness (section 4.1). However, the sky brightness varies with factors of a few (e.g. see the
150 MHz image from Landecker and Wielebinski [6]) and this needs to be taken into account.
Furthermore, the very bright galactic plane will contribute quite a lot of power through side and,
especially at frequencies above 170 MHz, grating lobes increasing the effective visibility noise.
• The raw sensitivity of a phased array like LOFAR is less at low elevation due a number of reasons.
These include (i) smaller gains of the dipoles, (ii) reduced projected collecting area of the phased
array, (iii) longer paths length through the ionosphere, (iv) larger separation of the ionospheric pierce
points of the calibrator sources. Considering the first two points currently sensitivity values that are
averaged over azimuth and elevation have been used. The latter two points imply that remaining
calibration errors will be larger at low elevations. Hence the effectively reached noise levels are
bound to go up at lower declinations.
• The inner sidelobes of a (HBA) station need to be suppressed to reduce scattered sidelobe noise.
This can, and probably will, be done by tapering the station beam which has the drawback that the
sensitivity will be reduced by 30-50%. The tapering will result in a larger station beam, leading to an
increase in survey speed. Note however, that tapering will always result in a loss of survey speed.
• One of the most challenging aspects of deep LOFAR observations is to obtain the needed enormous
dynamic range [7], [5]. In a 4 hour observation using 4 MHz of bandwidth the dynamic range needs
to be on average 100,000 to 1. For some deep fields (using more than 4 hours of data and / or more
than 4 MHz of bandwidth) it may need to be 1,000,000 to 1. Residual phase errors in solving for the
ionosphere and the beams are likely to limit the dynamic range. To what extent needs to be
determined during commissioning.
• Definite values for α 1 and α 2 (see section 3) have to be determined during the commissioning of
LOFAR.
• Final values for Aeff / Tsys have to be determined during the commissioning of LOFAR. From the
commissioning also the effect of mutual coupling must be determined. The LBA numbers in this
report are based on measurements using a 48 dipole set-up. The effect of mutual coupling will be
present in the data, although it probably will be averaged out due to the averaging over azimuth,
elevation, and all the dipoles. The final LBA station is based on a 96 dipole configuration. This
means that mutual coupling in the final station is likely to be different from the present data and its
variation with azimuth and elevation needs to be determined. Current data in the HBA is based on
single tile simulations, so any tile-tile couplings are not taken into account. Also for the HBA this is
something to be determined in the commissioning of the final stations.
-13-
6 Acknowledgements
The authors would like to acknowledge H. Röttgering, S.J. Wijnholds, A. Omar, and I. van Bemmel for
providing important feed back in the preparation of this document.
-14-
7 References
[1] S.J. Wijnholds, LBA station configuration, LOFAR-ASTRON-MEM-214, 4 Dec. 2007.
[2] A. Omar, Effect of weights on the LOFAR sensitivity, Leiden University, 25 Jan. 2008.
[3] Taylor, Carilli, and Perley (eds.), Synthesis Imaging in Radio Astronomy II, ASP, San Fransisco, USA,
1999.
[4] J. Bregman, Design concepts for a sky noise limited Low Frequency Array, in Perspectives on Radio
Astronomy – Technologies for Large Antenna Arrays (ed. A.B. Smolders and M. P. van Haarlem),
ASTRON, The Netherlands, 1999.
[5] H.J.A. Röttgering et al., Proposed LOFAR Survey specifications, version 0.9.3, 2 Oct. 2008.
[6] T.L. Landecker and R. Wielebinski, The Galactic Metre Wave Radiation: A two-frequency survey
between declinations +25o and -25o and the preparation of a map of the whole sky, Austrialian Journal
of Physics Supplement, Vol. 16, p. 1, 1970.
[7] A.G. de Bruyn and J.E. Noordam, Dynamic range requirements in LOFAR imaging, LOFAR-ASTRONMEM-213, 8 May 2006.
-15-
8 Appendix: Plots of the LBA effective area lower bound
8.1
15 MHz
Figure 6 Left to right, top to bottom, Full array, inner array, outer array, sparse array.
-16-
8.2
30 MHz
Figure 7 Left to right, top to bottom, Full array, inner array, outer array, sparse array.
-17-
8.3
45 MHz
Figure 8 Left to right, top to bottom, Full array, inner array, outer array, sparse array.
-18-
8.4
60 MHz
Figure 9 Left to right, top to bottom, Full array, inner array, outer array, sparse array.
-19-
8.5
75 MHz
Figure 10 Left to right, top to bottom, Full array, inner array, outer array, sparse array.
-20-