1. Berrilli F., Del Moro D., Giordano S., Consolini G., Kosovichev, A.
Topology of Supergranulation: Pair Correlation Function g2(r) and Information Entropy H’(l)
I show two stathistical method used in order to study the topology of supergranulation: the pair correlation function and the information entropy.
Berrilli F., Del Moro D., Giordano S., Consolini G., Kosovichev, A.

2. Introduction
The spatial configuration of enhanced magnetic fields on the surface of the Sun derives from the interaction of convective flows and magnetic field.
The global properties of the outer convective layers are important to fully understand heat transport and how it is influenced by magnetic field.
We have developed two methods able to derive topological information of SG field.
The spatial configuration of enhanced magnetic fields on the surface of the Sun derives from the interaction of convective flows and magnetic field.
The global properties of the outer convective layers are important to fully understand heat transport and how it is influenced by magnetic field.
We have developed two methods able to derive topological information of SG field

3. Normalized Information Entropy, H’(m)
Statistical method of analysis (not biased by subjective criteria).
It is able to discern characteristic dimensions of particles and particles clustering.
It allows perceiving of a lattice constant in quasi-ordered images. L
The first method is the normalized information entropy, introduced by Van Siclen in 1997.
It is a statistical method of analysis able to give a more objective characterization of supergranular structures
This method is able to give us information about clustering effects of particles and about spatial ordering.
Consider a two level image L X L and a square box of dimension m x m sliding along the image. For any m and for any position of the sliding box along the image we count the number of black pixel into the box. So we obtain an istogram. This istogram is then divided fpr the total number of shifts along the image. Finally we calculate the information entropy and compare this value to that obtained in the case of a random distribution of pixel.
If there’s order in the image the IE signal shows modulation.
L x L = image dimensions
m x m = sliding box dimensions
N = total number of black pixels
Order in the image → modulation of the H’(m) signal

4. Some Simulations Same as above + random displacement of max 7 pixels
15X15 Square Lattice of 3x3 Squares
Same as above + random displacement of max 7 pixels
As we are interested in the study of supergranular structures that are extracted through the cromospheric network I show some results obtained applying information entropy to structures( in black) and reticula(in red).
In the case of a perfect square lattice, both show modulation
In the case of a lattice with thermal displacement of strucutures around the rest position, the reticule results deformed, but the IE show a modulation and so the spatial ordering underlying the structures
As we want to apply this method on image with very large field of view, were the convexity of the field becomes important, in the last row i show the case of a reticule warped by projection over a sphere, and we found that IE still shows the clustering scales, but with a characteristic growth of the signal.

5. Pair Correlation Function, g2(r)
g2(r) shows density – density correlation.
g2(r) is the probability to find a particle in the volume element dr located at r if at r = 0 there is another particle.
This probability is given relative to what expected for a uniform random distribution of particles of the same density.
For isotropic system the g2(r) can be averaged over angles and computed from data by counting the mean number of particles at distance r – r+Δr from any given particle, <Nr>:
The last method is the pair correlation function that is able to measure the degree of association of a structure with others as a function of distance from its barycentre.
In particular it’s the probability to find a particle at distance r from another one. Also in this case this probability is normalized to a random distribution, so that the modulation shows in some way ‘order’.

6. Again Some Simulations
ω is an estimator of order at short range.
λ is an estimator of decorrelation.
Square Lattice.
A= 0.42 ± 0.02
λ= 340 ± 60
ω= 41.7 ± 0.1
K= ± 0.006
Same as above + thermal displacement
Also in this case I show you some simulations.
In the case of a perfect lattice the PCF shows some deltas at the locations of structures.
In the case of a thermal displacement of the structures around the rest position the pair correlation function shows a modulation with a damping of amplitude. If we fit with a damped cosine we obtain a decorrelation length much greater than the period of oscillation
In the last case we have a pseudo random distribution where the structures cannot stay closer than a fixed distance from each-other.The fit used shows a decorrelation length smaller than the period of oscillation and so shows a first neighborood interaction.
many difference between the seconda nd third image ca be discerned, but the PCF shows the first is a lattice, the second is a first neighbourhood interaction.
In fact, the damped cosine fit we used showed a decorrelation length much greater than the period of oscilaltion for the lattice,
and a decorrelation length smaller than the period for the pseudo-random distribution.
A= 1.0 ± 0.6
λ= 20 ± 5
ω= 25 ± 2
K= ± 0.004
Random distribution with imposed minimum distance.

7. MDI Heliotomography divergence maps
SOHO-MDI
FOV: 768”X768”
Pixel scale: 4”
ΔT: 8 h
Duration: imm  6 days
~2 Mm below solar surface
Identification of structures barycentres.
Identification of cell network.
Supergranulation fields are obtained from MDI heliotomography divergence maps.
19 images showing the convective flows at about 2 Mm below solar visible surface.
We then applied algorithms for structures barycentre location and identification of cells boundaries: i.e. the network.

8. MDI Heliotomography divergence maps
SOHO-MDI
FOV: 768”X768”
Pixel scale: 4”
ΔT: 8 h
Duration: imm  6 days
~2 Mm below surface
Identification of structures barycentres.
Identification of cell network.
Supergranulation fields are obtained from MDI heliotomography divergence maps.
19 images showing the convective flows at about 2 Mm below solar visible surface.
We then applied algorithms for structures barycentre location and identification of cells boundaries: i.e. the network.

9. MDI Heliotomography divergence maps
SOHO-MDI
FOV: 768”X768”
Pixel scale: 4”
ΔT: 8 h
Duration: imm  6 days
~2 Mm below surface
Identification of structures barycentres.
Identification of cell network.
Supergranulation fields are obtained from MDI heliotomography divergence maps.
19 images showing the convective flows at about 2 Mm below solar visible surface.
We then applied algorithms for structures barycentre location and identification of cells boundaries: i.e. the network.

10. Mean H’(m) of divergence maps
This is the mean information entropy signal of 19 divergence maps.
It shows a first strong peak indicating the supergranulation scale and then a rapid growth with a faint modulation.
Analyzing the first derivative of the IE signal we found a second maximum,at about 60 Mm showing a scale of clustering of SG
First peak at  30 Mm.
Strong increase of signal.
Analysis of the first derivative of the H’(m): second maximum at  60 Mm.

11. Mean g2(r) of divergence maps
λ= 23 ± 3
ω= 29 ± 1
K= 1.02 ± 0.01
Now I show the mean PCF of the 19 divergence maps.
The signal shows evident modulation and quite strong damping.
The period of ~29 Mm of the oscillation is related to mean supergranular size, and the short decorrelation length (23 Mm) suggest a first neighbourhood interaction.
the signal is very similar to that obtained in the case of a close pached hard disk cluster

12. H’(m) Conclusions
The capacity of Van Siclen method to work on reticular structures has been proved.
First peak shows the Supergranular structure mean scale at  30 Mm.
Analyzing the first derivative of the H’(m), a second faint maximum has been found, indicating the presence of an higher scale of clustering at  60 Mm.

13. g2(r) Conclusions Mean size of SG structures ~ 29 Mm
Decorrelation length ~ 24 Mm
The interpretation of the results leads to consider the topology of the divergence maps incompatible with a random field, being more similar to the topology of close packed, mutually impenetrable structures.