Ricean Bias Correction in Linear Polarization Observation
 Author: Sohn Bong Won
 Organization: Sohn Bong Won
 Publish: Journal of Astronomy and Space Sciences Volume 28, Issue4, p267~271, 15 Dec 2011

ABSTRACT
I developed an enhanced correction method for Ricean bias which occurs in linear polarization measurement. Two known methods for Ricean bias correction are reviewed. In low signaltonoise area, the method based on the mode of the equation gives better representation of the fractional polarization. But a caution should be given that the accurate estimation of noise level, i.e. σ of the polarized flux, is important. The maximum likelihood method is better choice for high signaltonoise area. I suggest a hybrid method which uses the mode of the equation at the low signaltonoise area and takes the maximum likelihood method at the high signaltonoise area. A modified correction coefficient for the mode solution is proposed. The impact on the depolarization measure analysis is discussed.

KEYWORD
polarization , data analysis , Ricean bias , radio loud active galactic nuclei

1. INTRODUCTION
Synchrotron radiation from radio galaxies is known to be linearly polarized. Even unresolved sources show measurable linear polarization. Using this linear polarization property, I can measure the fractional polarization and the polarization angle of the radiation after traversing the Faraday medium (e.g. a magnetized thermal plasma between the polarized source and us). First investigations of the galactic Faraday medium were made by SimardNormandin et al. (1981) and others using the integrated polarization of extragalactic sources. With increased sensitivity and angular resolution of radio telescopes and telescope arrays, the investigation of the Faraday medium of radio galaxies has become one of the main interests of this field. Burn (1966) and Laing (1984) have shown that deciphering the geometry of the Faraday medium is very complex. In general, there are two basic geometric configurations.
 The Faraday medium is 'internal' to the source, and physically mixed with the emitting volume (e.g. a thermal cloud surrounded by a shell of relativistic electrons).
 A foreground Faraday medium (in front of the synchrotron source) has a fine structure that is unresolved by the telescope.
Decades of multifrequency observations have shown that the Faraday rotation is highly linear in λ^{2} when the spatial resolution is good enough. This linearity is the property of the resolved slab and the external (resolved)Faraday medium (Burn 1966). The difference between the internal slab geometry and the external medium becomes evident in the fractional polarization. In the internal case, I will see the secondary maxima of the fractional polarization in the lowfrequency regime. In the external case, I will see no depolarization at low frequencies.The observational results seem to be 'in between.' The achievement of full frequencysampled data is virtually impossible. While a depolarization trend is seen, it is not easy to distinguish between the internal slab and the unresolved external component.
The Faraday rotation of supernova remnants and spiral galaxies is an example of the first case. In this case, the Faraday rotation profile is nonlinear in λ^{2}. On the other hand, the observational results of radio galaxies over the last decades, especially multifrequency observations with high angular resolution, have proven that Faraday rotation in radio galaxies does follow a λ^{2} law. Therefore, it must be concluded that the Faradayrotating media do not permeate the lobes of radio galaxies. Recently obtained sensitive Xray images (e.g. with
ROSAT ,Chandra ) exhibit cavities in the hot gas towards the lobes of radio galaxies. This confirms that the thermal electrons of the Xray halos and the relativistic electrons of the radio lobes are not mixed. Physically, this could imply pressure balance between the thermal and the relativistic electrons. Alternatively, it could mean rapid acceleration of the thermal electrons in the radio lobes and rapid cooling of the relativistic electrons in the Xray halo. In any case, the λ^{2} behavior of the Faraday rotation of radio galaxies indicates that the differential depolarization (DP) and the rotation measure (RM) of radio galaxies are due to an 'external' Faraday medium. It also implies that the Faraday media are almost or even fully resolved. This should lead to a low differential depolarization.In the case of fully resolved Faraday cells and assuming that radio galaxies reside in the center of Xray halos with King profiles, I expect an RM asymmetry instead of a depolarization asymmetry. Although the scale length of the Faraday cells is not definitely known yet, the λ^{2} behavior suggests that I do resolve the Faraday screen. On the other hand, the asymmetry of the DP (i.e. the 'LaingGarrington effect' Laing [1988], Garrington et al. [1988]) is known to be the best proof of the relativistic beaming effect of jets in radio galaxies. The LaingGarrington effect means that a nonnegligible amount of (magnetic) energy is contained in smallscale (unresolved) magnetic field structures. In that case, Faraday rotation will deviate from the λ^{2} behavior and will be saturated at ？χ < π / 3 (Burn 1966).
The fractional polarization of a radio source is determined by the pitch angle distribution of the relativistic electrons, the geometry of the magnetic field, and by the energy distribution function of the relativistic electrons
The general powerlaw distribution is known from radio observations, cosmicray experiments, and from theory. The theoretical fractional polarization of the radio intensity is (almost) independent of frequency. On the other hand, the measured fractional polarization of the flux density is additionally dependent on the depolarization due to differential Faraday rotation, i.e. on the distribution of thermal electrons along the lineofsight, as well as the lineofsight component of the magnetic field. This way, the polarized component of the radiation undergoing Faraday rotation becomes frequency dependent.
For decades, onesided jets of active galactic nuclei have been observed. The most successful explanation of this phenomenon is Doppler boosting caused by the relativistic bulk motion of the jet material. Since the radio lobes are connected to, and fed by, the jets, large efforts have been made in order to look for independent projection effects which do or do not support the interpretation in terms of projection effects of the relativistic jets. As far as the radio lobes are concerned, the depolarization asymmetry, i.e. the LaingGarrington effect, is the strongest support of this view. The other asymmetries of the radio lobes, the lobe length and the spectral index asymmetry, do not show any clear connection to the projection effect of the relativistic jets. To conclude, the LaingGarrington effect is interpreted in terms of DP. In case of such an
internal DP, the foreground medium should also have magnetic field reversals along the lineofsight.2. RICEAN BIAS IN LINEAR POLARIZATION
In this Section, I discuss the Ricean bias in the polarization intensity and suggest a new correction method. The linearly polarized intensity is obtained from Stokes parameters U and Q according to
It follows a Ricean distribution (Vinokur 1965):
where
I_{pobs} is the observed polarized intensity,I_{p} is the intrinsic polarized intensity,B _{0} is the modified Bessel function of zero order and σ_{Ip} is the noise level. ForI_{p} ≫ σ_{ Ip} andI_{p} ~ σ_{ Ip} the Rice distribution is approximated by the well known statistical forms. In the case ofI_{p} ≫ σ, the Rice distribution ofI_{pobs} asymptotically approaches a Gaussian distribution (Fig. 1). Since the Gaussian is a symmetric distribution, the most probable value and the peak value,df / dI_{P} = 0 are the same. Therefore, the most probableI_{pobs} is the realI_{p} in this case (Fig. 1).In the other case,
I_{p} ~ σ_{Ip} , the Rice distribution ofI_{pobs} is a Rayleigh distribution (Fig. 2). In this case, the most probable value is larger than the peak value,df / dI_{P} = 0,The reason is that the Rayleigh distribution is an asymmetric distribution. This has become known as the Ricean bias in the literature, and is caused by the positivity of the noise in
Different from the polarized intensity, the distribution function of the polarization angles remains symmetric, and no Ricean bias correction is needed (Wardle & Kronberg 1974), since there is no positivity problem in
χ = tan^{1} (U /Q ).2.1 The Two Known Solutions
In practice, two solutions of the Ricean bias correction are widely used.
2.1.1 The solution of Wardle & Kronberg
Wardle & Kronberg (1974) have suggested a solution based on the mode of the equation, namely,
A good approximation of this solution is
Conventionally,
with
c_{1} = 1.2. As seen above,is the Rayleigh distribution.
2.1.2 The solution of Killeen et al. (1986)
The other solution which is based on a more solid statistical reasoning, viz. the maximum likelihood method,
was suggested by Killeen et al. (1986). With increasing signaltonoise, the Maximum Likelihood estimate approaches the value of
I_{P} more rapidly:A good approximation of this solution is
which is implemented in the AIPS task POLCO as such.
2.1.3 Pros and cons of the solutions
The performance of the maximum likelihood estimation is better at high signaltonoise,
I_{P} / σ_{IP} > 5, but does not yield any significant effect, since the difference to the solution of Wardle & Kronberg (1974) is much less than 1% (Figs. 3 and 4). This is because the probability distribution ofI_{pobs} becomes a (symmetric) Gaussian at highsignaltonoise. On the contrary, the difference at low signaltonoise, i.e. the overestimate of the maximum likelihood solution, is significant. As mentioned in the help tool to AIPS task POLCO, one should assume an underestimate of up to about a factor of 2 of the noise level, due to the socalled 'magic blanking'. Therefore, if one wants to go down to a 3 σ
_{Ip} level, the choices of the solution and ofc_{1} become critical.In the Wardle & Kronberg (1974) solution,
,
c_{1} should be carefully selected. As I am not interested in values ofI_{P} as low asI_{pobs} ~ σ_{Ip} ,c_{1} should be lower thanfor a better performance in the range 3 <
I_{pobs} / σ_{Ip} < 6 ~ 10. Above 6σ_{Ip} ··· 10σ_{Ip} , I can expect the best performance using the ML solution. From our experiments, I conclude thatc _{1} = 0.9 is a reasonable choice (Figs. 3 and 4)3. THE 'HYBRID' SOLUTION
Our goal is to improve the performance of the Ricean correction such that
should be reliable down to 3σ
_{Ip} and 3σ_{Ip} . In the next subsection, I will show the importance of the areal meanm' and the position information of DP. Our scheme is simple. I make a hybrid solution of the two solutions explained in the former subsection. At high signaltonoise, I will still use the scheme of Killeen et al. (1986). At low signaltonoise, the solution of Wardle & Kronberg (1974) is better, but withc _{1} = 0.9.The selection of
c _{1} has been done numerically, considering the fact that one should take account of the underestimation of the noise by up to factor of 2 in the intensity (National Radio Astronomical Observatory 2011), due to the blanked values. Further minor estimation errors could be caused, for example, through a wrong selection of the noise estimation area, which could further underestimate σ_{Ip} . I combine the two solutions (Wardle & Kronberg 1974, Killeen et al. 1986). Our solution accepts the Maximum Likelihood overI_{pobs} ~ 6σ_{Ip} ; below this, it useswith
c _{1} = 0.9, which is a good choice for σ_{Ip} /I_{pobs} between 1.5 to 6. Using this new fractional polarization, I can estimate DP with the positional information and can study any correlation between RM and DP.4. DISCUSSION
4.1 Integrated Fractional Polarization
Before working out a new method to estimate the fraction of polarization, I discuss the integrated fractional polarization as published in the literature, motivating this new correction method. There are two ways to estimate the fractional polarization of a region of interest. One can first integrate the polarized and the total intensity and divide them:
There are pros and cons to this integrated estimate. The pro is that the Ricean bias problems are largely removed by the integration. The obtained
m' reflects the fractional polarization of bright source structures, such as cores, hot spots and jets. The con is the loss of the positional information. Conventionally, DP maps are presented after one of the Ricean bias corrections, i.e. (Wardle & Kronberg 1974, Killeen et al. 1986), down to 3σ_{Ip} of the total intensity. This will lead to significant over and underestimates of the low  σ_{I,P } regions, such as the lobes. Therefore, in order to estimate the fractional polarization including the positional information, an estimateis desirable.
In polarization studies, other important informations are the polarization angle and the RM. Since they are independent of the brightness, the use of DP as obtained from
could lead to a wrong interpretation.
4.2 Underestimates of σ
_{Ip} in the LiteratureIn this subsection, I discuss the propagation of the systematic error as due to the insufficient Ricean bias correction in published DP maps. The discussion is based on the solution of Killeen et al. (1986). For the solution of Wardle & Kronberg (1974), this should be interpreted in an opposite sense. In that case, the systematic error in regions with low polarization intensity is less important, unless the signaltonoise is too low in the absence of any σ
_{Ip} cutoff.I consider the problem of underestimating σ
_{Ip} for two cases, namely for well resolved and for unresolved sources. I assume a powerlaw distribution forIν . When the source and the Faraday medium are resolved, the overestimate caused by the 3σ_{Ip} cut is serious at the highest frequencies. In this case, the fractional polarization of regions with lowI_{p} could be overestimated. The other case is that of low angular resolution. Because of the Faraday rotation in the foreground medium, depolarization will be significant. In the most depolarized regions, a cut at 3σ_{Ip} is serious at the lowest frequencies. Then the fractional polarization of such regions is overestimated at the lowest compared to the highest frequencies. In effect, the depolarization trend will thus be reduced.In depolarization studies, the integrated DP is rather independent of the polarized and total intensity, but rather depends on
where S is the projected surface. The first case (resolved source) overestimates DP, while the second (unresolved source) underestimates it. Let us discuss this in the light of the LaingGarrington effect. If one can assume that the jet side is brighter and more polarized, the two arguments will be more important for the counterjet lobe. In the first case, the LaingGarrington effect will be emphasized through the overestimate of DP, whereas it will be reduced in the second case because of the underestimation.
One more word of caution seems to be necessary regarding DP structures. Patchy distributions of DP in regions with low
I_{p} which are frequently seen should be interpreted with utter care. One can misinterpret such a patchy structure as real turbulence or fluctuations. Unless such patchy structures are confirmed independently, e.g. also in RM maps, they could be solely statistical.

[Fig.1.] A Gaussian distribution.

[Fig. 2.] A Rayleigh distribution.

[Fig. 3.] The simulation shows the case σ=σIpobs an ideal case. The abscissa is Ipobs /σIpobs and the ordinate is Ipobs/ Ip. 'WK' denotes the solution of Ipobs following Wardle & Kronberg (1974) 'KBE' that of Killeen et al. (1986) and 'this work' that of this work. In practice one should bear in mind that Ipobs will NOT be the mathematical solution of the former section but rather the distribution around this mean value. Therefore fluctuations of differential depolarization and FPOL if obtained from low polarized intensities are largely purely statistical but not physically meaningful. The new solution shows an enhanced performance at low signaltonoise. Between 4 < Ip/σIp the old solutions are better but the difference is less than 0.1 percent.

[Fig. 4.] The simulation shows the case σIp = 2σIpobs. A underestimate of σ by a factor of 2 of σIp could be possible due to the blanked values (AIPS cookbook [National Radio Astronomical Observatory 2011] see also AIPS HELP). In view of the underestimate of σIp our new solution is the best one. The solution implemented into AIPS task POLCO Killeen et al. (1986) strongly overestimates the polarization intensity while the Wardle & Kronberg (1974) solution underestimates it. The difference is much larger than that to our solution visible in Fig. 3.