J. Astrophys. Astr. (1987) 8, 281–294
Ionospheric Refraction Effects on Radio Interferometer Phase
S. Sukumar
Radio Astronomy Centre, Tata Institute of Fundamental Research,
Post Box 8, Ootacamund 643001
Received 1986 November 22; revised 1987 May 24; accepted 1987 June 27
Abstract. The refraction of radio waves as they traverse through the
terrestrial ionosphere and troposphere introduces a differential phase path
which results for a radio interferometer in variations of the visibility phase.
Though refraction due to troposphere is significant for synthesis radio
telescopes operating at 1.0 GHz and above, ionospheric refraction is
dominant at lower frequencies. This problem is important in the case of
Ooty Synthesis Radio Telescope (OSRT) operating at 326.5 MHz, due to
its proximity to the magnetic equator. This paper deals with the nature of
phase variations suffered by OSRT due to refraction and explains the
methodology evolved to alleviate them.
Key words: radio interferometers—visibility—ionosphere—refraction
1. Introduction
“Atmosphere” refers to two sections viz., “troposphere” and “ionosphere”. The region
up to 50 km from the surface of earth is called troposphere and the ionosphere
typically extends from 70 km up to 3000 km above earth’s surface. For single antenna
radio telescopes, refraction suffered by radio waves in the atmosphere has generally
amounted to apparent positional shifts of astronomical sources, which could easily be
corrected in most cases. But interferometric observations suffer differential phase
paths between the radio waves received at different interferometer elements due to the
curved structure/shape of the refracting medium. Also irregularities within the refracting medium introduce further phase variations in the visibility phase observed
with a radio interferometer.
For the tropospheric refraction, it has been adequate to assume an exponential
variation for the refractive index as a function of height above ground and solve
analytically for the differential phase path. This procedure could further be simplified
by invoking a slab approximation for the troposphere (Brouw 1971; Hinder & Ryle
1971). For a radio telescope such as the Ooty Synthesis Radio Telescope (OSRT)
having antennas at different heights above ground, an additional phase path is
introduced due to tropospheric refraction between the signals received at various
antennas. These effects are relatively simpler to correct and hence we will not discuss
them. However, we will show that a simplified slab approximation does not adequately correct for ionospheric refraction effects and instead present a numerical
approach making use of electron density distributions in the ionosphere.
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One may assume, for the purpose of evaluating refraction corrections, that the
ionosphere consists of two components, (i) an undisturbed regular ionosphere without any movements of electron density irregularities and (ii) a disturbed ionosphere
having movements of electron density irregularities of varying spatial and time scales.
Though it is not possible to take care of the effects of disturbed ionosphere by simple
means, refraction effects due to regular ionosphere can be corrected by assuming
simple models for the ionosphere. Some analytical methods assuming simple gradients in the electron density of the ionosphere have been discussed by Komesaroff
(1960) and Hagfors (1976). The numerical method to correct ionospheric refraction
effects for the Westerbork Synthesis Radio Telescope has been described by Spoelstra
(1983). We discuss here a correction procedure developed for OSRT, located at 76° Ε
and 11° N. Owing to its proximity to the magnetic equator wherein abnormal
conditions of the ionosphere could often be encountered, a synthesis radio telescope
like OSRT operating at 327 MHz (Swarup 1984) especially requires an ionospheric
refraction correction procedure. Besides, the OSRT has phased arrays in the focal
lines of its antennas adding further complexities and without correcting for ionospheric refraction, it is difficult to evaluate the baselines to any reasonable accuracy. In
this paper, we discuss a methodology evolved to compute the differential phase path
corrections due to refraction through undisturbed ionosphere and present some
observational effects noticed with OSRT during disturbed ionospheric conditions.
2. Ionospheric refraction
The refractive index of ionosphere varies as a function of height depending upon the
electron density (number of electrons m– 3) and can be expressed as
(1)
Figure 1. Curved ionosphere geometry for differential phase path.
Ionospheric refraction effects
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where h is the height above earth’s surface (metres), n(h) the refractive index as a
function of height, N (h) the electron density as a function of height (m– 3), and f the
observing frequency (Hz).
From Equation 1, it can be realized that n(h) has a maximum value of unity and
generally it is less than unity. At f =327 MHz, the factor 81 N(h)/f 2 is very small
(<10– 3) and hence we can write,
n(h) = 1 – 40.5 N (h)/f2.
(2)
In Fig. 1, we show the rays reaching two stations A and Β on earth’s surface
traversing through the ‘curved ionosphere’ having uniform electron density. As seen,
the path lengths A2A1 and B2B1 are not equal and the difference between them is
termed as ‘differential phase path’ (DPP). It is important to realize that this effect
arises purely due to curvature of earth. In the case of plane-parallel ionosphere, the
incident and emergent rays at ionospheric boundaries will be parallel, making the
path lengths equal.
3. Computation of differential phase path
3.1 Slab Model for the Ionosphere
In this model, the ionosphere is simply replaced by a slab of uniform electron density
whose thickness is given by
S = N t/N max
(3)
where S is the slab thickness in metres, Nt the total electron content (number of
electrons contained in a column of one square metre cross-sectional area extending all
along the ionosphere), and N max the maximum electron density in the ionosphere.
The approximate formula for the differential phase path in the slab model is given
by
(4)
where L is the baseline length in metres, λ the observing wavelength in metres, n the
refractive index of ionosphere, Ζ the zenith angle of the source, A the angle between
source azimuth and baseline azimuth, and φ1, φ2 the angles subtended at the centre of
earth by normals drawn at the earth’s surface, lower and upper boundaries of the
ionosphere where the ray intersects (refer Fig. 1). Knowing the lower and upper limits
of the slab ionosphere these angles can be calculated.
3.2 Ray-Tracing Through Electron Density Profiles
The lengths A2A1 and B2B1 may also be evaluated by ‘ray-tracing’ along a straight
line through the ionosphere. Actually, this is only an approximation as the real
electron density profile cannot be measured in the direction to the radio source and
during the observations. The complete ray-tracing implies computation of refractive
index at each point in the ionosphere, evaluating the refractive bending suffered by the
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electromagnetic waves and iterating the computation all along the curved path
traversed by the ray. Since this procedure involves enormous amount of computing
time, we have approximated the curved path by a simple straight line. This approximation will not introduce errors greater than a few percent and will be adequate for
our purpose. The excess length travelled by the ray is given by (at station A)
(5)
where ds is the line element along the ray, given by
(6)
where dh is the vertical component of ds and ZA, the zenith angle of the source,
measured at the subionospheric point over station A. Similarly for station Β we write
(7)
The differential phase path is given by
(8)
To evaluate the integrals given by Equations 5 and 7 we should know the functional
form of n(h) which in turn is derived from the electron density distribution of the
ionosphere.
4. Ionosphere models
Many models have been proposed to predict the electron density distribution of
ionosphere at low geo-magnetie latitudes (equatorial regions). Some of them are
theoretical (Hunt 1973), some are partly theoretical, partly observational (Somayajulu
& Ghosh 1979), the rest are purely empirical (Bent et al. 1972; Bent et al. 1975;
Rajaram & Rastogi 1977). Apart from these, International Reference Ionosphere (IRI)
models are also available for low and mid geomagnetic latitudes.
All these models essentially use three parameters, namely hm F2 (height above
ground at which maximum electron density occurs), N m F2 (maximum electron
density in the F2 region), and Y m (semi-thickness of bottomside ionosphere) to
generate the electron density profiles. The parameters vary as functions of (i) geomagnetic latitude of the place, (ii) hour of the day, (iii) season (equinox, winter and
summer), and (iv) solar activity.
Most of the models also describe the ways of deriving h m F 2 , N m F 2 and Y m .
Measured parameters like M (3000)F2 (maximum usable frequency over a distance of
3000 km on ground using reflection of the electromagnetic waves from the F2 layer of
the ionosphere), f0F2 (critical frequency of the F2 region) are generally used to derive
these parameters. In our case, the choice of a particular model is based on how good it
accounts for the observed total electron content and considerations of computational
ease, availability and reliability of the basic parameters used to generate the model etc.
We have chosen Bent ionospheric model. This empirical model is based on a vast
amount of satellite measurements, F2 peak layer measurements and profiles from
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ground stations. The prime objective of the Bent model is to keep the total electron
content as accurate as possible in order to obtain reliable values of the path length and
directional changes of a wave due to refraction. Tests with Faraday rotation data
show that 75–90 per cent of the ionospheric effects can be accounted for by the model
(Llewellyn & Bent 1973). The Bent model describes the mean ionosphere in terms of
simple biparabolic, exponential layers and this simplicity results in less computing
time to generate the model, thus making it more attractive. The complete description
of this model can be found in Bent, Llewellyn & Schmidt (1972), Bent et al. (1975) and
a brief description of it as applicable for OSRT can be found in Sukumar (1986).
5. Correction procedure
5.1 Parameters of Equatorial Ionosphere
As noted by Köhnlein (1978), the capability of Bent model can be improved if
measured values of the total electron content or critical frequency f0F2 is incorporated along with the observation station and time information. The OSRT is situated
close to magnetic equator at a geomagnetic latitude of 4° Ν and is well within the
region exhibiting equatorial anomaly. Depending upon the presence of equatorial
electrojet or counter electrojet the peak electron density region shifts towards the
magnetic equator. This effect drastically changes the total electron content as well as
other characteristics of the ionosphere over OSRT location. Hence it is preferable to
generate Bent ionospheric profiles based on the parameters derived from ionograms
obtained on the OSRT observational days from a closely situated ionospheric
research station (B. V. Krishnamurthy, personal communication) such as Vikram
Sarabhai Space Centre at Thumba or the Solar Observatory, Kodaikanal.
However, we also note that it is just not sufficient to generate ionospheric profiles
for OSRT location alone since over the hour-angle and declination coverage of OSRT
antennas, the rays reaching them meet the ionosphere at totally different subionospheric latitudes belonging to different geographic locations. The ionospheric
parameters at these locations could be quite different from that over OSRT. In
principle, the different ionospheric profiles could be extrapolated from the ionograms
obtained from Thumba provided the magnetic field variation over a large region is
also known. This involves reduction of several ionograms and magnetograms obtained on the observational days, which is rather cumbersome. Besides, these differences show only second order effects resulting in deviations of about 10–20 per cent
from the observed values of the ionospheric parameters. Hence we have resorted to a
simplified procedure of deriving the ionospheric parameters from the monthly ionospheric predictions released by National Physical Laboratory (NPL), New Delhi
(Laxmi, Shastri & Reddy 1981). The uncertainties that could arise due to this
simplified method are briefly discussed later.
5.2 Equivalence Method
After generating the required profile, Equation 8 is numerically evaluated by raytracing along a straight line through the ionosphere. The cosine of the difference
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between the source azimuth and baseline azimuth is multiplied with this to get the
differential phase path. When we extend the phase correction program to different
antennas (baselines) the computations become quite tedious. This is mainly because
Equation 8 involves calculating zenith angles for different interferometer elements
using the latitude, longitude of each antenna which proved impracticable. This
problem has been circumvented by invoking the equivalence of differential phase path
with the apparent change in zenith angle of the source due to refraction which is a
baseline independent parameter (A. Pramesh Rao, personal communication). We
briefly outline the method below.
The effect of refraction by a curved atmosphere on the observations of stars has
been studied by Smart & Green (1977). The formula for the angle of refraction Δ is
(9)
where R0 is the radius of earth, μ0 the refractive index at the surface of the earth, Ζ the
zenith angle of the star, μ the refractive index at a distance r from the centre of the
earth at height h, and r = R0 + h.
The integration limits are from μ = 1; (h = ∞) to μ = μ0; (h = 0).
For the ionosphere
where N (h) is the electron density as a function of height and we have assumed that
μ(h) –1 1. Thus,
Inserting this in Equation 9 and setting μ0 ≃ μ ≃1, we get
(10)
Integrating Equation 10 by parts we get,
Since for the ionosphere N (h) = 0 at h = 0 and h= ∞ the first term is identically equal
to zero leaving
(11)
where α = R0/(R0 + h).
If we consider a plane formed by observer, the centre of the earth and the source, the
effect of refraction is to displace the source from the zenith by an amount Δ. If we
consider an interferometer with baseline L to be located on the surface of the earth in
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287
this plane, the effect of refraction is to introduce an error in the position of the source.
From Figs 2a and 2b, the path difference between the rays reaching the two antennas
is given by
Inserting for Δ from Equation (11)
Including the difference between source azimuth As and baseline azimuth Ab we get,
(12)
The integral in Equation (12) is solved numerically assuming the factor L cos (As
– Ab) to be unity and evaluating the zenith angle at the hmF2. The actual baseline
length and the azimuth factor are later evaluated from the baseline parameters and
appropriately multiplied with. The description of the computer programs written for
this purpose can be found in Sukumar (1986).
5.3 Results and Limitations
In Fig. 3, we show the phase variation observed with OSRT for short and long
baselines before and after correcting for ionospheric refraction.
Our calculations also show
(a) slab model overestimates the corrections by about 40 per cent. Table 1 shows
the difference between differential phase path calculated by slab model and Bent
ionospheric model;
Figure 2. (a) source–observer geometry, (b) source–baseline geometry, and (c) ionosphere–
baseline geometry.
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Figure 3. Phase variations observed during undisturbed conditions of the ionosphere.
Ionospheric refraction effects
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Figure 3. Continued.
Table 1. Comparison of differential phase path for slab and Bent models of the
ionosphere.
(b) the numerical calculations are very sensitive to any variation in hmF2 and
NmF2. Any change in these parameters changes the differential phase path
approximately in the same proportion whereas in the case of Ym, it is less by a
factor of 2.
It may be noted that about 60 per cent of the ionospheric refraction occurs in a
400 km thick region placed symmetrically about hmF2. About 25 per cent occurs in the
bottomside and the rest occurs in the topside between 1000–3000 km of the ionosphere.
Recently, Vikram Kumar & Rao (1985) have shown that the Bent model for
equatorial daytime ionosphere is generally consistent with the observations in the
topside, but does not agree in the bottomside. It uses the same semithickness for the
bottomside parabolic segments while observations show that the semithickness for
the bottomside is greater by a factor of 1.5 to 2.5. Hence we may introduce substantial
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errors if the assumed Ym deviates much from the true value. The optional approach to
compute actual ionograms and use the values of Ym derived from them will avoid this
problem.
The NPL ionospheric predictions suffer from uncertainties greater than 25 per cent
during sunrise and this effect is more pronounced for regions close to magnetic
equator (refer Fig. 4). Accordingly, the correction coefficients computed from these
predictions also inherit the errors and leave residual phase errors in the visibility data.
This could perhaps be alleviated only by special calibration procedures like selfcalibration, infield and frequent calibrations currently in wide use for reducing radioastronomical data.
In our approach we are ray-tracing along a straight line which is not an ‘actual raytracing’ in the true sense. We need to compute the complicated path the ray will travel
inside the ionosphere, which we have not attempted to do since the residual phase
errors may be taken care of by suitable radio astronomical calibrations like infield and
self-calibration.
6. Short period phase variations
Apart from the refraction of radio waves by regular undisturbed ionosphere, which
introduces slow phase variations of about 2π radians (equivalent to path variation of
one wavelength) over 6 hours period, we also encounter, short-period variations in the
interferometer phase during disturbed conditions of the ionosphere. These phase
variations result from movements of electron density irregularities having various
scale sizes. The generation of ionospheric irregularities are attributed to the presence
Figure 4. Variation in f0F2 during sunrise for low latitude and mid latitude stations (reproduced from Laxmi et al. 1979).
Ionospheric refraction effects
291
of acoustic gravity waves induced by various physical mechanisms (Yeh & Liu 1974).
There are two kinds of irregularities which affect the interferometer phase:
(i) The medium scale travelling ionospheric disturbances (TIDs) have scale sizes
ranging from 100 to 200 km and move with phase velocities of 100–200 m s– 1. We
have generally observed phase variations of about 60° peak-to-peak in the visibility
phase obtained over an interferometer baseline of 4 km, typically having periods of
12–20 min (see Fig. 5). Sometimes TID movements have been observed almost
Figure 5. Phase variations observed during movements of medium scale traveling
ionospheric disturbances in the ionosphere.
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Figure 6. Phase variations observed during ionospheric scintillations.
throughout the day. The effects of such TIDs on the visibility phase can be minimized
only by special calibration procedures, mentioned earlier.
(ii) The small-scale irregularities have average scale sizes of 600 m, travel at an
average phase velocity of 500–600 m s–1 and introduce rapid fluctuations in the
Ionospheric refraction effects
293
signals received by an antenna. These fluctuations, also known as ‘ionospheric
scintillations’, have been observed to occur over timescales in the range of 40 s to a few
minutes. Generally, the scintillations are present during night times of the equinox
months and are much dependent upon the solar activity. When the scintillations are
strong, S2 (ratio of rms fluctuations to the mean source strength) > 0.25, they result in
severe decorrelation (depending upon the time constant of the radio receiver) among
the radio signals received at different antennas. Only over short spacings of the
antennas, the effect of scintillations are correlated. This could be seen from Fig. 6,
wherein effects of both small-scale irregularities and movements of TIDs are shown
together for different baselines. Though frequent calibration can remove some of the
phase fluctuations observed with short spacings, it may not be straightforward to
estimate the amplitude decorrelations during severe scintillations and hence the
calibrations may be improper. Only a detailed ray-tracing through the disturbed
ionosphere could solve this problem, which in any case is not simple to perform.
Acknowledgements
The author is very grateful to Professor Ν. C. Mathur, Indian Institute of Technology,
Kanpur, for introducing him to the subject of ionosphere and also for his valuable
guidance. This work could not have been completed but for the numerous suggestions
and guidance the author received from many ionospheric researchers working in the
National Physical Laboratory, New Delhi, and Vikram Sarabhai Space Centre,
Thumba, Trivandrum. He also thanks Drs T. Velusamy, A. Pramesh Rao and
T. Spoelstra for many interesting discussions and valuable suggestions. He is thankful
to an anonymous referee for his various suggestions to improve the contents of this
paper.
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