Quiet Sun intensity distributions

C. A. Young1, D. Bewsher2 , J. Ireland1

1L3Com/GSI, NASA’s GSFC, Code 612.5, Greenbelt MD 207712Department of Physics, Astronomy and Mathematics, UCLAN,

Preston, PR1 2HE, United Kingdom.

This work was supported by NASA contract # NAS5-00220.

IntroductionThe statistical properties of the solar atmosphere are a result of the physical processes present. There have been several attempts to describe the intensity distribution intensity in the quiet Sun. Knowledge of the intensity distribution may yield information on its formation, or the number of identifiable components to the quiet Sun.

Two recent studies have looked at the quiet Sun intensity distribution with two very different purposes in mind. Gallagher et al. (1998) sought to define a threshold distinguishing network from internetwork, whilst Pauluhn et al. (2000) sought the best model description for the entire distribution. We examine these studies in more detail.

Gallagher et al. (1998) used

• SOHO1-Coronal Diagnostic Spectrometer (CDS) quiet Sun data• Every pixel in each raster.• Mixture modeling to find the number of Gaussian distributions present in the histogram of observed intensities.• Fitting of the sum of Gaussians to the histogram of observed densities to find the Gaussian parameters.However

• Neighbouring pixels are not statistically independent due to the CDS point spread function.• The binning of the histogram is arbitrary, which necessarily affects the fitting.• The choice of using Gaussians to model the histogram is arbitrary.• The purpose of the study is to find a network/internetwork threshold, not to model the distribution.1Solar and Heliospheric Observatory

Pauluhn et al. (2000) used

• Coronal Diagnostic Spectrometer (CDS) and Solar and Ultraviolet Measurements of Emitted Radiation (SUMER) quiet Sun data.• Every pixel in each raster.• Fitting of a variety of arbitrary test distributions to the histogram of observed intensities.

However

• Neighbouring pixels are not statistically independent due to the CDS and SUMER point spread function.• The binning of the histogram is arbitrary, which necessarily affects the fitting.• The choice of test distributions to model the histogram is arbitrary.• The purpose of the study is to model the distribution.

Both studies use data which are not statistically independent and make quality judgments on the best model based on a fitting technique which

includes an arbitrary parameter.

The point spread function inherent in many instruments means that neighbouring pixels are unlikely to be statistically independent. This can be circumvented by undersampling the image.

Fitting a distribution, or mixture of distributions, to a histogram clearly depends on the histogram binning, introducing subjectivity. This approach can be sidestepped completely by using an Expectation-Maximization (EM) algorithm.

The choice and mixture of distributions can be motivated by physical concerns, through considering fragmentation mechanisms and mixture modeling.

Mixture Modeling

αmm=1

k

∑ = 1

p(y(i ) |θm) =(2π )−

12

σm

exp(−12(y(i ) −μm)

σm2 )

p(y(i ) |θm) =(2π )−

12

yσm

exp(−12(logy(i ) −μm)

σm2 )

p(y |θ) = αmp(y|θm)m=1

k∑The distribution of the n data, y, is described by a k-component finite mixture distribution, . The αms are the mixing probabilities, and θ is the complete set of parameters needed to specify the distributions.

For the component distributions we use either a normal or a lognormal:

An EM algorithmIn order to determined the parameters in the mixture distribution we use the maximum likelihood estimate,

This cannot be found analytically so we use a modified form of the EM or Expectation-Maximization algorithm.

• E-step: compute the conditional expectation the complete log-likelihood, the so-called Q function. • M-step: Iteratively estimate the parameters that maximize the Q function.

Q(θ,)θ (t)) ≡E[log p(y,z|θ) |y,

)θ (t))]

)θ (t+1)=arg max

θ Q(θ ,

)θ (t))

ML

)θ =arg max

θ{ log p(y|θ)}

We use a modified method that is unsupervised in that it determines the number of components.

Lognormal distributions and fragmentation

Pauluhn et al. (2000) show that distributions of EUV intensities in quiet Sun SOHO-CDS1 and SOHO-SUMER2 data can be well represented by a lognormal distribution

€

1

xexp −

1

2

ln x −μ

σ

⎛

⎝ ⎜

⎞

⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥

where x is the EUV intensity, μ is the location parameter and σ is the shape parameter. This single distribution is found to fit the observed distribution better than a two Gaussian distribution, as found by Gallagher et al. (1998). The presence of one distribution, rather than two, implies that a single process may be occurring, as opposed to two processes. Lognormal distributions also arise in the distribution of sunspot areas (Bogdan et al., 1988) and decay rates (Martínez-Pillet et al., 1993).

A lognormal distribution arises in the fragmentation of a quantity A into two fractional pieces A(1-xi) and Axi. After n fragmentations

€

An = A 1− x i( )i=1

n

∏

If the set xi are independent random variables drawn from the same distribution p(x) then, under certain conditions on p, the logarithm of the distribution of fragmented areas is normally distributed.

Given that lognormal distributions are observed in the quiet Sun, and that a reasonable physical mechanism may be responsible for their presence, we model the observed intensity distributions using a mixture of lognormals. We compare these results to using a mixture of Gaussian distributions. The data used are the same as in Gallagher et al. (1998).

The Dataset

• Instrument: CDS/NIS• Observed lines: He I, He II, O III, O

IV, O V, Ne VI, Mg IX, Mg X• Area imaged: 240 x 240 arcsec2

• Number of images: 10• Subsampling: use every third pixel

in x and y

Example images

true CDS raster image undersampled image

Example mixture for fully sampled image

Gaussian mixture lognormal mixture

Example mixture for subsampled image

Gaussian mixture lognormal mixture

Results:

Gaussian vs.

lognormal for fully

sampled data

Results:

Gaussian vs.

lognormal for

under-sampled

data

Discussion

The presence of lognormal distributions is taken as a signature of the presence of a fragmentation mechanism. However, it is not clear what may be fragmenting to cause the observed intensity distribution.

Fragmentation has been well studied in the kinetics of polymer degradation. Cheng and Redner (1990) describe a model equation for the evolution of a distribution of particles c(x,t) that fragment independently under the influence of external forces,

The unsupervised expectation maximization analysis shows (for the CDS lines chosen in the quiet Sun) that the intensity distribution is more economically modeled with lognormal distributions in preference to Gaussian distributions, suggesting that fragmentation mechanisms may be operating in the quiet Sun. Undersampling also reduces the number of components in the mixture.

The first term on the RHS is number of particles of mass x lost due to their breakup: the 2nd term is represents the number of particles created from particles of larger mass. Conserving mass and assuming

€

∂c(x, t)∂t

= −a(x)c(x, t) + c(y, t)a(y) f (x | y)dyx

∞

∫ (1)

Here a(x)dt is proportional to the probability that a particle of mass x breaks in a time interval dt, while f(x|y) is the conditional probability at which x is produced from the breakup of y.

€

a(x)∝ x λ

€

c(x, t)∝ s(t)−2φ x /s(t)( )

that is, the overall breakup rate depends on the fragment mass only. A scaling ansatz is introduced,

(2)

(3)

Cheng and Redner (1990) show the existence of an asymptotic lognormal solution for small fragment size x,

At large fragment size x, the distribution behaves as

€

φ(x, t0)∝ exp −c0 ln x( )2

[ ]

for some constant c0, fixed time t0, and a conditional probability that has cut off (zero probability of fragmentation) at small fragment sizes - a minimum fragment size.

€

φ(x, t0)∝ x c1 exp −c2xλ

[ ]

for some constants c1, c2. It can be seen from this analysis that fragmentation distributions can be more complex than just lognormal.

(4)

(5)

Equation (1) describes the evolution of a population undergoing fragmentation only. There are many other processes going on in the quiet Sun; for example, Schrijver at al. (1997) describe the equations of magnetochemistry which govern the distribution of positive and negative magnetic flux (magnitude ) in the quiet Sun:

€

∂N±

∂t= +S±

€

−N± N±(x)l(φ,x)dx0

φ

∫

€

+1

2N±(x)N±(φ − x)l(x,φ − x)dx

0

φ

∫

€

+2 N±(x)k(φ,x)dxφ

∞

∫

€

−N± k(x,φ − x)dx0

φ

∫

€

+ N±(φ + x)Nm(x)m(φ + x,x)dx0

∞

∫

€

−N±(φ) Nm(x)m(φ,x)dx0

∞

∫

N± is the number of positive (negative) flux fragments. A source S± is present. The red terms describe the gain and loss by like fragments merging respectively, controlled by the rate l.

(6)

The blue terms describe gain and loss by binary fragmentation (2 daughter fragments produced in each fragmentation; equation 1 permits more than 2 fragments to be created per fragmentation) respectively, mediated by the rate k. The green terms describe the gain and loss by flux cancellation respectively, governed by the rate m.Equation 6 has solutions in special cases. Parnell (2002) measures the distribution of flux concentrations in SOHO Michelson Doppler Imager (MDI) data and finds that a Weibull distribution

fits the data much better than a simple power law (where x is the flux in a flux concentration, is the shape parameter and is the scale parameter). This distribution arises in the study of fractures in materials. Parnell (2002) derive

€

xγ −1 exp −x

β

⎛

⎝ ⎜

⎞

⎠ ⎟

γ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

(7)

functional forms for k,l and m from the measured Weibull distribution plus assumptions about the form of solution to (6).

Brown and Wohletz (1995) point out that the lognormal and Weibull distributions are very similar for certain parameter ranges which may lead to the misidentification of distributions. Although the Weibull distribution can be motivated from fragmentation concerns, Parnell (2002) did not compare the Weibull and power law fits with a lognormal distribution.It is hoped that by (1) applying more advanced data analysis techniques and (2) elements of fragmentation theory - both of which we have introduced here - we can arrive at a more complete understanding of the statistical properties of the quiet Sun and the mechanisms that create them.

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Brown, W. K., Wohletz, K. H., J. Appl. Phys., 48(4), 2758, 1995.

Cheng, Z., Redner, S., J. Phys. A: Math. Gen., 23, 1233, 1990.

Figueiredo, M., Jain, A., IEEE Trans. Patt. and Mach. Int., 24, 381, 2002.

Gallagher, P. T., Phillips, K. J. H., Harra-Murnion, L. K., Keenan, F. P., A. & A., 335, 733.

Martínez Pillet, V., Moreno-Insertis, F., Vázquez, M., A. & A., 274, 521, 1993.

Parnell, C. E., MNRAS, 335, 389, 2002.

Pauluhn, A., Solanki, S. K., Rüedi, I., Landi, E., Schüle, U., A. & A., 362, 737, 2000.

Schrijver, C. J., Title, A. M., van Ballegooijen, A. A., Hagenaar, H. J., Shine, R. A., ApJ, 487, 424, 1997.

Modified EMThe EM algorithm is very successful, however there are several drawbacks to the standard method.

• EM does not determine the number of components, only the parameters.• EM may converge to the edge of parameter space, i.e. one of the αs may approach zero causing some of the parameters to become singular.

We use a robust new modified method that is unsupervised (determines the number of components and is insensitive to initial conditions) and avoids the convergence problems.

(Figueiredo and Jain 2002)

Modified EM

(Figueiredo and Jain 2002)