Este documento presenta un proyecto de tesis doctoral sobre el estudio de regiones activas solares a alta resolución utilizando técnicas de diversidad de fase. El objetivo es aprender métodos de restauración de imágenes y observar la dinámica fina en regiones activas aplicando técnicas como corrección de distorsión y filtrado. Se realizaron observaciones con el telescopio sueco en La Palma y el plan de trabajo incluye análisis de velocidades de flujo, mapas magnéticos y publicación de

1. Estudio en Alta Resolución de Regiones Activas Solares a través de Técnicas de Diversidad de Fase La Laguna - Tenerife, Septiembre de 2006 Santiago Vargas Domínguez PROYECTO DE TESIS DOCTORAL Directores: Jose A. Bonet & Valentín Martínez Pillet

4. 1.1 Restauración de imágenes solares Blurring: Pérdida de detalle en las estructuras. Movimientos de imagen: Desplazamientos globales de la imagen. Stretching: Distorsión de las estructuras causadas por movimientos diferenciales en diferentes partes de la imagen.

5. A feeling on the influence of the telescope aperture, the AO and the restoration codes SVST (ORM) Ø50 cm AO non active Jul 7, 1999 G-band 4305 Å Rebined x2 to have 0.041”/pix SST (ORM) Ø100 cm AO active Jul 15, 2002 G-band 4305 Å pix=0.041” Both images restored with Phase diversity before after restoration SST (ORM) Ø100 cm AO active + PD correction Sep 30 , 200 3 G-band 4305 Å pix=0.041”

6. Formación de imagen con luz incoherente Point Spread Function (PSF)

7. El problema inverso La restauración de imágenes son un caso particular del llamado problema inverso en Física que en general se refiere a la solución de la ecuación integral inhomogénea de Fredholm de primer orden. where es el kernel en la integral En nuestro caso particular ésta es una ecuación de convolución que se conoce frecuentemente como una deconvolución.

8. Image formation (formal problem) Imaging through turbulence with incoherent illumination The earth atmosphere is a turbulent medium varying randomly at at a time rate of a few milliseconds. Short exposure (a few milliseconds) : Solar imaging can be performed with short exposure but the instantaneous cannot be predicted by theoretical models Long exposure (seconds to hours) : can be predicted by theoretical models or measured by using a calibration point source Optical Transfer Function (OTF)

9. The formal solution of the problem is quite simple if is known: Restoration of the observed scene (formal problem) The retrieval of the ´´true´´ object implies the restoration of both amplitude and phase of the observed scene (i.e. has to be entirely characterized: modulus and phase). Note that we can only get an estimate of the true object: ideal telescope empirical short exposure (8 ms) long exp. (atm.model + ideal tel.) image deconvolution

10. Image formation (real problem) Any observed image will contain noise: photon noise, reading noise, etc... Noise uncorrelated with the level of the signal is often a good approximation. Thus, for a fixed time (short exposure): And the restoration gives: The term represents an amplification of the noise Disastrous effect particularly at high frequencies where the SNR is small. Therefore, noise filtering is demanding prior to any restoration process:

11. The “optimum” noise filter, is a real function which weights according to its SNR at each frequency Condition defining the filter: (where ε denotes the residual error), and by substitution of results: Considering that i ( q ) and n ( q ) are not correlated, and developing the squared modulus: A standard calculus of variations leads to: or equivalently: The non correlation between i ( q ) and n ( q ) has allowed to use:

12. Thus, the filter is a weighting real function determined by the ratio of power between the observed signal and the noise Dibujo a mano alzada del filtro Y 3 panels de la fig. De la tesis e Monica Decir aqui lo de os modelos suaves analiticos

13. The “optimum” restoration filter, Combining noise filtering and deconvolution we get the optimum restoration filter also known as the Wiener-Helstrom filter: where is the signal-to-noise ratio But neither nor even an estimate of it, are known a priori. Just the goal of our global problem is precisely to determine !!!! To circumvent this problem some models for SNR( u ) are often used, e.g.: Model 1: SNR( u ) = C (constant fixed by trial & error) Model 2: Note that the azimuthal symmetry assumed in these models is not necessarily true.

14. Straightforward calculation of from the equation: A nice property of the convolution of h(x) with a given function f(x) : Applying this property to i(x, 0 ) : Once has been determined the calculation of its Fourier transform gives: Which is the section of the desired OTF along the x-frequency axis. Assuming azimuthal symmetry, a estimate of the OTF is obtained by rotation of .

15. Diversidad de Fase The method is based on at least two simultaneous images of the same object: Conventional focal-plane image Simultaneous image with a known extra aberration called: phase diverse unknown unknown known A defocus is the simplest way to induce a known aberration Outputs of the method: Phase retrieval (wavefront sensing). Image restoration.

16. Mathematical formulation of the problem Image formation of extended incoherent objects: 2 unknowns : Noise terms imply a statistical solution to the problem. A maximum-likelihood estimate of the object in case of Gaussian noise leads to the solution of the following least-squares error fit :

17. The error metric in the Fourier domain: OTF Generalized pupil function Joint phase aberration caused by telescope and turbulence

18. In principle, numerical non-linear optimization techniques are suitable to minimize but there are too many unknowns !!! N x M pix in and J aberr.coeffs in Fortunately, part of the minimization of can be done analytically !!! substitution in gives a modified error metric: Object estimate Modified error metric where the only explicit unknowns are the J aberr.coeffs. α Minimization of wavefront sensing

19. Minimization of : posing the problem The conditions of minimum with respect to the free parameters defining the model of S : (the aberration coeffs.), give rise to a non-linear system of equations : ( J equations)

20. Linearization of the problem A more compact notation: A Taylor´s series expansion of E around α gives:

21. Defining the inner-product of two functions as: the conditions of minimum can be expressed in matricial form: In compact form: The cost of linearizing the system of equations - swapping of unknowns: α by δ α - need to proceed iteratively: ( n -th iter.)

22. Image formation (real problem) Any observed image will contain noise: photon noise, reading noise, etc... Noise uncorrelated with the level of the signal is often a good approximation. Thus, for a fixed time (short exposure): And the restoration gives: The term represents an amplification of the noise Disastrous effect particularly at high frequencies where the SNR is small. Therefore, noise filtering is demanding prior to any restoration process:

24. Estructura fina en regiones activas solares Campo magnético que se genera bajo la zona de convección emerge a la superficie en las regiones activas . Manchas solares Poros SST - La Palma

25. Thomas et al. (2002) Tubos de flujo (flux tubes)

26. Flujo Evershed Normal, a nivel fotosférico : Flujos horizontales predominantemente radiales de la mancha hacia sus alrededores. Inverso, a nivel cromosférico : El material fluye dirigido hacia la umbra.

32. 3.1 Observaciones Se hicieron con la Torre Solar Sueca (SST), La Palma. Apertura: 1 m Distancia focal primaria: 20.35 m Distancia focal secundaria: 45.8 m Escala de imagen: 4.5 /mm AO de bajo orden

33. Diversidad de Fase Setup SST

36. Restauraciones G-band

37. Restauraciones Imágenes que se usaron:

38. Corrección de distorsión y Filtrado

39. Corrección de distorsión y Filtrado

41. 3.3 Análisis y Resultados Movimientos de flujo en la penumbra y sus alrededores Técnicas de correlación local (LCT) usando algoritmo de November & Simon (1988) para inferir velocidades horizontales.

42. Mapa de velocidades horizontales

44. Mapa de velocidades verticales Gránulos explosivos > 24 m/s Sumideros < -8 m/s

45. Moats Flujo estacionario

46. Evolución de los centros de divergencia

48. Futuros estudios sobre el tema Analizar más dias de la campaña de 2005 Usar magnetogramas y dopplergramas para inferir las propiedades y comportamiento del campo magnético solar en el campo con la línea neutra.

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