ON THE ESTIMATE OF MAGNETIC NON-POTENTIALITY OF SUNSPOTS DERIVED USING HINODE SOT/SP OBSERVATIONS: EFFECT OF POLARIMETRIC NOISE

The high-resolution spectro-polarimetric observations were made by the SP instrument working with SOT on board the Hinode space mission. We choose a regular isolated sunspot in NOAA 10933 during 2007 January 5, when it was located close to the disk center (μ = 0.99) during 07:20 UT. The sunspot was scanned with "fast-map" observing mode by the SP instrument. In this mode the spatial sampling along the slit is 032 per pixel and across the slit is 029 per pixel with an integration time of 1.6 s. The spectral lines used for polarimetric measurement are Fe i 6301.5 and 6302.5 Å line pair. The vector magnetic field for this sunspot was derived using the ME code named MERLIN, provided by the Community Spectro-polarimetry Analysis Center (CSAC; Lites et al. 2007). It is an inversion code based on the least-squares fitting of the observed Stokes profiles using the Levenberg–Marquardt algorithm and is quick due to parallel computing. It assumes a standard ME atmosphere to retrieve the magnetic field vector, line-of-sight velocity, source function, and Doppler broadening as well as the macro-turbulence and the stray light filling factor. For Monte Carlo estimation of error in magnetic parameters we use another ME inversion code named HeLIx (Lagg et al. 2004). We compared the magnetic field parameters retrieved by the two codes after inverting the SOT/SP data set used in the present study. We made difference maps of the retrieved magnetic field parameters, i.e., δM = M_MERLIN − M_HeLIx, where M = B (field strength), γ (inclination), or ψ (azimuth). Within the sunspot, the two codes retrieve identical magnetic field parameters giving zero mean of δM and (1σ) standard deviation of 50 G, 075 and 25 in field strength, inclination, and azimuth angle, respectively. These values are comparable to the standard errors of inversion by MERLIN, reported in Figure 3. Standard errors mean that if the same inversion code is repeatedly applied to a given observed (noisy) profile, it will lead to the same model atmosphere within the error bars given by the standard error estimate, provided the minimum of the merit function is reached (Bellot Rubio et al. 2000). So, the comparison suggests that the two codes are identical so that the differences in the output are of the same magnitude as the difference of a particular code with itself. The reason for choosing HeLIx over MERLIN in our Monte Carlo error estimation is the built-in routines for adding noise to the synthetic Stokes profiles in the HeLIx code which helps us save time as we avoid adding noise offline. The azimuthal ambiguity of the transverse component of the magnetic field vector was resolved using the acute angle method (Harvey 1969).

The uncertainties in the field parameters derived using ME inversion of observed Stokes profiles, purely as a result of polarimetric noise present in the observed data, are considered here. There are two ways of determining the uncertainties.

• 1.  
  Standard errors. Basically, in the Stokes profile inversions, a merit function is defined as

  $$\begin{equation*} \chi ^2 = \frac{1}{\nu } \sum _{k=1}^4\sum _{i=1}^{M} \big[I_k^{\rm obs}(\lambda _i)-I_k^{\rm syn}(a,\lambda _i)\big]^2\frac{w_{ki}^2}{\sigma _{ki}^2}, \end{equation*} \tag{ 3 }$$

  where k = 1 to 4 represent the four Stokes parameters, i = 1 to M represent the number of wavelength samples for observed and synthetic Stokes profiles, ν represents the number of degrees of freedom (i.e., the number of observables minus the number of free parameters), and the vector a consists of the parameters of the model, σ_ki represents the noise in the observations and w_ki represents the weight given to the data points. This merit function, χ², is then minimized by using the nonlinear least-squares method to obtain χ_min. The standard errors in the model parameters basically depend upon the curvature of the χ² function near the region of minimum in the parameter space (Press et al. 1986). More details about the estimation of the standard errors in the model parameters derived by Stokes inversion are given in Bellot Rubio et al. (2000).
• 2.  
  Monte Carlo errors. In the Monte Carlo method, we create several artificial realizations of the spectro-polarimetric data. This is done by generating the synthetic Stokes profiles and adding different realizations of noise to them. These artificial data sets are then inverted to obtain the magnetic parameters, leading to a collection of results. The errors are then determined by finding the spread or standard deviation in the derived parameters. It may be noted that the Monte Carlo errors have no obvious relationship with the standard errors.

Westendorp Plaza et al. (2001) have shown that in the case of SIR inversion of a single profile, the standard errors are slightly larger than the estimates from the Monte Carlo method. However, this depends on the distribution of the field vector, and so the comparison of errors by the two methods may vary from pixel to pixel. Further, the Monte Carlo error estimates depend upon the noise in the data as well as the noise in the process of inversion. Thus, these are useful in establishing the statistical significance of the results.

We use the Monte Carlo approach to determine the statistical spread or uncertainty in the magnetic field parameters as well as the non-potentiality parameters of the active region arising due to polarimetric noise. The approach is as follows.

• 1.  
  First, we take real observations of a sunspot by Hinode SOT/SP and invert it with an ME code to retrieve the magnetic field vector.
• 2.  
  Resolve the azimuthal ambiguity in the transverse field using the acute angle method.
• 3.  
  Consider this vector field as the "true field" [B₀, γ₀, ψ₀], and the derived quantities α(B₀, γ₀, ψ₀) and $\widehat{\Psi }(B_0,\gamma _0, \psi _0)$ as "true non-potentiality parameters" [α₀,$\widehat{\Psi }_0$], for reference.
• 4.  
  Using the "true field" value at each pixel we generate synthetic Stokes profiles and add a polarimetric noise of level 0.5% of continuum intensity. These profiles are then inverted again under ME approximation to derive field vectors [B, γ, ψ] and [α_g, $\widehat{\Psi }$]. This process is repeated 100 times, each time with a different realization of polarimetric noise. The uncertainty in the field strength, inclination, and azimuth for each pixel and the uncertainty in the parameters α_g and $\widehat{\Psi }$ for the vector map are then estimated from the 1σ standard deviation in the 100 values of these parameters. This standard deviation is called the Monte Carlo error.

This simulation gives us an idea about the spread (standard deviation) in the derived field vector and the non-potentiality parameters of a real sunspot, arising due to polarimetric noise. This knowledge of standard deviation in the magnetic non-potentiality parameters α_g and $\widehat{\Psi }$ is important in determining whether the observed changes in these parameters, for example in relation to flares, are significant or not. Such studies are yet to be done and the present work will establish the level of uncertainties in the parameter $\widehat{\Psi }$ or α_g due to polarimetric noise in modern observations, such as from Hinode.

We first generate synthetic Stokes profiles corresponding to the "true field" using ME-based Stokes profile synthesis and the inversion code named HeLIx (Lagg et al. 2004). To these synthetic profiles we add normally distributed random noise with the 3σ level of 0.5% of continuum intensity (I_c). The noise, N(λ), is added to the synthetic Stokes profiles, S_syn(λ), as follows.

• 1.  
  First, a pseudo-random-number sequence is generated which is normally distributed with a zero mean and a 3σ standard deviation of given level, say L. In our case, L = 0.5% of I_c

  $$\begin{equation*} N(\lambda) = {\tt Random\_Number}({\rm Seed},L). \end{equation*} \tag{ 4 }$$
• 2.  
  This sequence is then added to the synthetic Stokes profiles to yield the noisy Stokes profile, S_noi(λ),

  $$\begin{equation*} S_{\rm noi}(\lambda)=S_{\rm syn}(\lambda) + N(\lambda). \end{equation*} \tag{ 5 }$$

The noise level is estimated from the observed signals in the continuum of the Stokes spectra. A continuum window between 6302.83 Å and 6303.28 Å is selected for monitoring the noise in the Stokes signal. The left panel of Figure 1 shows the histogram of noise in the observed Stokes profiles for a large number of pixels (1024 pixels) for a typical SOT/SP scan in "fast-map" observing mode. A Gaussian fit to this distribution yields a 3σ value of 0.5% of I_c. This is the noise level that we used in our simulations. The histogram in the right panel of Figure 1 shows the distribution of artificial noise that we add to the synthetic profiles. These profiles are then inverted with the HeLIx code. This process is repeated 100 times and so for each pixel we have 100 values distributed around a mean value.

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While modern spectro-polarimetric observations typically have noise levels of the order of 0.5% of I_c, archived observations from the ground-based instruments, which might be used for synoptic studies, may have higher levels of noise. Therefore, we also carried out an exercise to check the variation in the magnetic as well as non-potentiality parameters, $\widehat{\Psi }$ and α_g, with increasing polarimetric noise levels. We added four different levels of noise, i.e., 0.01%, 0.1%, 0.5%, and 1% of continuum intensity.

The computation of the non-potentiality parameter α_g and $\widehat{\Psi }$ is done as proposed by Tiwari et al. (2009a, 2009b). The expression for computation of these parameters is given in Equations (1) and (2) in Section 1. Only those pixels which have field values above a certain noise level are analyzed. This filtering is done in the following way: we select a quiet region on the Sun and evaluate 1σ deviation in the three vector field components B_x, B_y, and B_z separately. The box selected for this estimation is shown in the top-left panel of Figure 2. For transverse vector fields, we take the 1σ noise level as the resultant deviations obtained in B_x and B_y. Only those pixels where the transverse and line-of-sight (LOS) fields both are together greater than twice the above mentioned noise level of 1σ are analyzed. The 1σ value for longitudinal and transverse field inside the box is 380 and 258 G, respectively.

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Figure 2 shows the effect of noise on magnetic field parameters for the NOAA 10933 sunspot. The top row shows the initial magnetic field parameters [B₀, γ₀, ψ₀] derived by ME inversion of Stokes profiles obtained by Hinode SOT/SP scan during 07:20 UT on 2009 January 5. The middle row shows the map of Monte Carlo error, i.e., the 1σ standard deviation in the magnetic field parameters, derived using the Monte Carlo method. The bottom row shows the map of standard error in the field parameters from the least-squares fit of the Stokes profiles. The maximum error (taking into account both the standard error and the Monte Carlo error estimate) inside the sunspot is less than 50 G for the field strength, while for inclination and azimuth it is less than a few degrees. The azimuth errors are typically largest in the umbral and plage region where the field is almost vertical and the azimuth is not well defined.

In Figure 2, we isolate the umbral and penumbral regions by using continuum intensity thresholds. We compare the standard errors and the Monte Carlo errors in these two regions in Figure 3. The number of pixels in the umbral and penumbral regions plotted in Figure 3 is 1953 and 13615, respectively. It may be seen that

• 1.  
  In the umbral and penumbral regions the Monte Carlo errors are typically less than 5 G and 8 G for field strength, less than 1 and 05 for field inclination, and less than 5° and 2° for field azimuth, respectively.
• 2.  
  The standard error is larger than Monte Carlo error estimates in both the umbral and penumbral regions for field strength as was found by Westendorp Plaza et al. (2001) in the case of the inversion of a single profile. For field azimuth and inclination it has an opposite relation, except for the middle panel of the bottom row.

The standard error depends upon the sensitivity of the χ² to the changes in a particular parameter. It can happen that the Stokes profiles might not have significant sensitivity to the changes in that parameter. This results in a large standard error. On the other hand, the Monte Carlo method examines the changes in the derived parameters resulting from the changes in the Stokes profiles arising due to random fluctuations in the profile as a result of different realizations of noise being added. Since both errors depend upon different origins of the profile fluctuations, they need not produce the same results.

In general, considering all the panels in Figure 3, it may be concluded that the standard errors are larger than the Monte Carlo errors. Here, one should bear in mind that standard errors shown in Figure 3 are for inversion of real Stokes profiles which possess asymmetry, apart from the polarimetric noise, while Monte Carlo errors correspond to repeated inversion of purely synthetic, and therefore symmetric (or antisymmetric for Stokes-V) profiles with polarimetric noise. Therefore, the main source of difference between the two types of errors could be attributed to the presence of Stokes asymmetry in the real Stokes profiles leading to a large value of the standard error. Ideally, one would like to perform a Monte Carlo error estimate by obtaining several simultaneous observational data sets of a real sunspot where the Stokes profiles would have inherent Stokes asymmetry as well as different realization of polarimetric noise. However, in the absence of such a possibility we used the present method to study the effect of polarimetric noise alone, neglecting Stokes asymmetry. The results of the present study are therefore only the lower limits of the errors that can occur practically in real observations as well as in the non-potentiality parameters $\widehat{\Psi }$ and α_g.

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It is well known that the Zeeman effect diagnostics cannot detect the direction of the transverse field component and so a 180° ambiguity remains in the determination of the field azimuth. Various methods, however, have been developed by researchers to resolve this ambiguity using different arguments (Metcalf et al. 2006). One of the most common and widely applicable methods is the so-called acute angle method. In this method, the angle between the observed and potential transverse fields, i.e., θ = acos(B^obs_o.B^pot_t)/(∣B^obs_o ∣ ∣B^pot_t∣), is computed and the solution for which the value of θ is an acute angle is considered as the correct solution. However, in the presence of polarimetric noise in the observations, this method can also fail, especially in situations where the observed field is highly sheared. Such highly sheared regions are found near the polarity inversion line (PIL) of the active regions.

Fortunately, in a normal round sunspot, like the one used in our simulations, there are no high-shear regions and therefore the azimuthal ambiguity is easily solved with the acute angle method. Nevertheless, we need to check the effect of polarimetric noise on the azimuth ambiguity resolution in our simulations before we examine the uncertainty in the $\widehat{\Psi }$ and α_g parameters of the sunspot in 100 realizations. After resolving the azimuthal ambiguity using the acute angle method for the 100 realizations of the sunspot vector maps, we made a map of 1σ standard deviation for the azimuth angle, as shown in the middle-right panel of Figure 2. If the ambiguity is not resolved properly in these 100 realizations, there will be fluctuations of the order of 180°, leading to a large value of standard deviation in azimuth value for a given pixel. However, in the right panel of the middle row in Figure 2 we see the following.

• 1.  
  The resolution of the azimuth angle is quite stable for the most part within the sunspot, especially in the penumbral region. Outside the sunspot, in the quiet and facular areas, the errors are large. This is mainly due to poor S/N in these areas as a result of weaker and/or vertical fields.
• 2.  
  As we go toward the umbra the values of standard deviation are large. We examined the values of the field azimuth in these regions and found that the values do not vary to the extent of 180° and therefore the large errors in the umbral region is not due to the ambiguity solver but due to poor S/N in Stokes-Q and U observations. Only in the very central part of the umbra, where the field inclination is close to 90°, the azimuth loses its meaning and so we see a large spread.

Further, we also checked the effect of the ambiguity solver with increasing noise, i.e., four different levels of noise of 0.01%, 0.1%, 0.5%, and 1% of continuum intensity. Here also the azimuth ambiguity resolution is stable for most part of the sunspot.

Figure 4 shows the histogram of $\widehat{\Psi }$ and α_g corresponding to 100 Monte Carlo realizations of the vector field. The following can be noticed.

• 1.  
  The $\widehat{\Psi }$ is not affected much by noise. The distribution of $\widehat{\Psi }$ corresponding to 100 realizations show less scatter (∼1%) with $\widehat{\Psi }$ = −1.68 ± 0014.
• 2.  
  The values of α_g are more affected. The distribution shows large scatter (∼10%) in values with α_g = −3.5 ± 0.37(×10⁻⁹ m⁻¹).

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These results show that the $\widehat{\Psi }$ may be more reliable as compared to α_g for typical polarimetric noise present in modern spectro-polarimetric observations as it is a lower dispersion parameter.

We also check the effect of increasing noise in the polarimetric measurements on the magnetic as well as non-potentiality parameters of the same sunspot. In order to compare the effect of increasing noise on the field parameters in the sunspot we isolate the umbral and penumbral regions, as was done in Figure 2, and do a scatter plot between the input and output field parameters, for different levels of noise. The scatter plot for umbral and penumbral regions is shown in Figures 5 and 6, respectively. It can be seen that, in the umbra, where the field is mostly vertical, the azimuth determination is more affected with increasing noise than the field strength and inclination. Especially for 0.5% and 1% noise levels the spread is large in the azimuth and inclination values. In comparison, the scatter plot for the penumbral region, where the field is not so vertical, shows that the azimuth is determined with less spread even for 0.5% and 1% noise levels, respectively.

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Table 1 lists the non-potentiality parameters, derived from the ME inverted magnetic field vector, after adding noise of different levels in the synthetic Stokes profiles. In Table 1, the spatial fluctuations of the shear angle Ψ as well as local α in the map, are represented as σ_Ψ and σ_α. Starting with no noise and then adding random noise of 0.01%, 0.1%, 0.5%, and 1% of continuum intensity in the Stokes profiles we get the following results: (1) the sign of the twist in the sunspot magnetic field, as inferred by both α_g and $\widehat{\Psi }$, is negative and is not affected even when the S/N of Stokes profiles is poor; (2) the absolute value of $\widehat{\Psi }$ tends to decrease systematically with increasing noise in the Stokes profiles; and (3) the fluctuation of 2.5% in the $\widehat{\Psi }=-1.65\pm 0\mbox{.\!\!^\circ $}04$ is less than the fluctuation of about 5% in α_g = 3.15 ± 0.17(×10⁻⁹ m⁻¹).

Table 1. Effect of Increasing Polarimetric Noise in the Estimation of $\widehat{\Psi }$ and α_g

Noise (% of Ic)	$\widehat{\Psi }$ (°)	σΨ (°)	αg(m−1)	σα(m−1)
0	−1.692	15.724	−3.334 × 10−9	3.572 × 10−7
0.01	−1.688	15.726	−3.360 × 10−9	3.573 × 10−7
0.1	−1.685	15.702	−3.221 × 10−9	3.574 × 10−7
0.5	−1.648	15.636	−2.976 × 10−9	3.562 × 10−7
1	−1.604	15.364	−3.026 × 10−9	3.539 × 10−7

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