Magnetogram from the Filtergraph (FG) observation K.Ichimoto, M.Kubo, Y.Katsukawa and SOT Team

1. Magnetogram from the Filtergraph (FG) observation K.Ichimoto, M.Kubo, Y.Katsukawa and SOT Team SOT#17 2006.4.17-20

2. Detection limit of FG for the weak magnetic fields, e = 0.001 I’: line profiles convoluted by TF transmission curve 2nd moments of s and p-components

3. Outline of this study In this report, we consider the algorithm to derive the magnetic field from the IQUV product of NFI with arbitrary number of observed wavelength points. • NFI observables -- I(li), Q(li), U(li), V(li), i = 1,,, N • Physical quantities derived from the observables • -- B field strength (G), • g inclination (deg.), • c azimuth (deg), • S Doppler shift (mA) • fill factor =1 • Other quantities responsible for line formation are assumed to • be those in typical quiet sun. • The algorithm to derive the magnetic field from the NFI observables utilizes the model Stokes profiles calculated beforehand.

4. Model Stokes profiles • are calculated as below. • Model atmosphere : Holweger & Muller (1974, Sol.Phys., 39, 19-30.) • Line : FeI6302.5, FeI5250A, others to be added Wavelength : -400 ~ 400mA (nwl=161, 5mA step) B : 0 ~ 3000G (nB=41) =[0, 0.5, 1, 2, 4, 10, 20, 40, 60, 80, 100, 150, 200, 300,,,,, 3000] g : 0 ~ 180deg. (ng = 19, 10deg step) c : 0deg. (41*19=779 profiles) Profiles are convolved by theoretical TF profile ‘Model NFI observables’ are generated when the wavelength points are specified I,q,u,v (S, B, g, li) S: Doppler shift -90 ~ 90mA (ns=37, 5mA step) Thus the LUT spans the parameter space of (S,B,g) ~ (37, 41, 19)

5. Model profiles (B,g ) I Q U V (41*19 =779 profiles) g (deg.)

6. The algorithm to derive the magnetic field vector from the NFI observables depends on the number of observed wavelength points. N = 1: 1-dimensional LUT for V/IBl, Q/I Bt individually N = 2: Rotate the frame to make U=0 (ignore MO effect) + search for the best fitting to model observable in (B, g, S) space N> 3: Initial guess with cos-fit algorithm + rotate the frame to make U~0 + search for the best fitting to model observable in (B, g, S) sub space

7. N = 1 v = V/I , q = Q/ I , u = U/ I c = tan-1(u, q)/2 q’ = q *cos(2c)+ u *sin(2c) vBl q’ Bt no Doppler information- LUT Dl = -80 mA saturation g = 0o g = 90o Polarization signals (V, Q) vs. magnetic field strength (Bl, Bt) with a Lyot filter (width =100mA) positioned at dl = -[10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120] mA from the line center. Thick curve is for dl = -80mA.

8. N > 3: flowchart Initial guess (B,g,c,S) from observables, rotate Q,,U by c, and then search best fit to model observables (I, Q’,U’,V) in (B,g, S) space around the initial guess point, finally calculate c from Q’,,U’ again. observable Io, Qo,Uo,Vo(i) (i = 1 ~ Nwp) Model profiles at c =0 Im, Qm, Um, Vm(l ,B,g) Expand in S (Dopp.shift) Sampling l position Initial guess Io, Qo,Uo,Vo B0, g0, c0, S0 Model observable table I’m, Q’m, U’m, V’m(i) (B, g, S) (i = 1 ~Nwp) Rotate Qo,Uoby -c0, normalize Qo,Uo by Io(0) Subspace around (B0,g0, S0 ) I’o, Q’o, U’o, V’o(i) (i = 1 ~Nwp) I’m, Q’m, U’m, V’m(B, g, S)(i =1 ~Nwp) d (S,B,g) =∑i{f*(I’o - I’m) 2 + (Q’o - Q’m) 2 + (U’o - U’m) 2 + (V’o - V’m) 2 } find (S, B, g ) that minimize d w/ interpolation by 2nd polynomial (Qo,Uo ) & (Qm,Um ) c

9. Initial guess method (B0, g0, c0, S0 ) 1) S0(Doppler shift) cos fitting for (I + V)i S+ “ (I-V)i S-  S0 = (S+ + S-)/2 cos fitting method Least square fitting is applied to the absorption line (=observable) using the following sinusoidal function then Doppler shift (l0) is obtained by l0 =tan—1(b1, a1) Fitting for I is less accurate than fitting for I+V and I-V, since I becomes broader than I+V and I-V for large B 2) g0 (inclination) Weak field approximation

10. 3) B0 (field strength) 1) Case |g - 90o| > 20o(S+ and S- have significant separation) 2) Case |g - 90o| < 20o (nearly horizontal field) weak field approximation • = 6303A, w=0.075A, d=0.5,geff=2.5  b1 = 3230 with some correction from the numerical experiments 3) Case |g - 90o| < 20o and if B > 1000G (nearly horizontal and saturation) b2 is determined experimentally from model profiles For FeI 6303A, b2 = 15000 4) c0 (azimuth) q’i, u’i are Qi’/Ii’, Ui’/Ii’ from the model at c = 0o , (B0, g0, S0 )

11. Search space of (B, g, S) for fitting (I’, Q’, U’, V’)i S : S0-20 ~ S0*3 +20 mA B: B0*0.7 - 100 ~ B0*1.5 + 100 G g: g0*0.8 - 20 ~ g0*1.2 + 20 deg. 3000G B (B0, g0, S0) 180deg. g -90mA 0 S +90mA

12. N = 2: ( ignore MO effect, search entire (S, B, g ) space) Calculate azimuth from Q,U , rotate frame to get Q’ with U’=0 Search best fit for ( I,Q’,V ) in (S,B,g) space of LUT fgmag V0.1 fgmags Model profiles Im, Qm,Vm(l ,B,g) observable l(i), Io, Qo,Uo,Vo(i) (i = 1 ~ Nwp) • = tan-1(Uo*,Qo*)/2 Q’o, = Qocos2c + Uosin2c V’o, = Vo/ Io , I’o = Io/ Io (0) Expand in S (Dopp.shift) Sampling l position Model observable I’m, Q’m,V’m(B,g, S) (i = 0 ~Nwp-1) I’o, Q’o, V’o(li) (i = 1 ~Nwp) d (S,B,g) =∑i{f*(I’o - I’m) 2 + (Q’o - Q’m) 2 + (V’o - V’m) 2 } f = 0.2; weight for I find (S, B, g ) that minimize d w/ interpolation by 2nd polynomial

13. Q/I Q (Q,U)  c Ignore MO effect.. > 0 Among 2 wavelength points, which (Qi, Ui) should be used to get c Pi, = √(Qi2+ Ui2) Index of max(Pi )imax, Index of 2nd max(Pi )imax2 if | Qimax2 | > | Uimax2 |  use Q else use U in the following if Qimax * Qimax2 < 0 （opposite sign） andQimax2 > max(Ii ) *e (noise) then use (Qi, Ui ) with larger Ii （avoid line core） else use (Qimax, Umax )  Qi*, Ui*

14. Numerical experiment • To test the performance of the algorithm, numerical simulations are made using ‘sample observables’ (1000 sets) calculated with the same atmospheric model and random physical parameters in a range of • 0 < B < 3000 G • 0 <g< 180 deg. • -90 <c< +90 deg. • -90 < S < +90 mA

15. N = 1 at dl = -80mA, Simulation result No Doppler info. Sample observable, 1000points

16. N = 4 at dl = [-105, -35, 35,105] mA, simulation result Initial guess After fitting

17. N = 4 at dl = [-110, -70, 70,110] mA, simulation result Initial guess After fitting

18. N = 3 at dl = [-80, 0, 80] mA, simulation result Initial guess After fitting

19. N = 2 at dl = [-80, 80] mA, simulation result Same as N>3 case, but S+/- are obtained from the formulae () applied to I+V

20. N = 2 at dl = [-80, 80] mA, simulation result alternative method: - ignoring MO effect - search entire (S, B, g ) space

21. More simulation with random noise Without noise With 0.5% (rms) noise

22. More simulation with the filter ripple TF ripple model: Transmission = [1.00, 0.99, 0.85, 0.83 ]

23. More simulation with the filter ripple Without ripple With ripple

24. More simulation with other atmospheric models Using the ASP data, typical ME parameters for FeI 6302.5 are determined for quiet, prnumbra and umbral regions individually, and sample ME Stokes profiles are calculated for each region to test the algorithm (by M.Kubo).

25. More simulation with other atmospheric models 6302.5Å [-105, -35, +35, +105] mÅ Quiet region penumbra result umbrae g [deg.] c [deg.] B [kG] S [km/s] input

26. IDL program IDL> fgmag, wlp, I, Q, U, V, B, gam, xai, S, $ modelfile=modelfile ; INPUTS : ; wlp(n) -- sampled wavelength from line center, [mA] ; I(*,*,n), Q(*,*,n), U(*,*,n), V(*,*,n) ; -- observed IQUV in any dimension ; OUTPUT : ; B(*,*) -- magnetic field stregth, [G} ; gam(*,*) -- magnetic field inclination, [deg.] ; xai(*,*) -- magnetic field azimth, [deg.] ; S(*,*) -- Doppler shift, [mA] ; KEYWORD : ; modelfile -- IDL save file containing the model Stokes profiles Evaluated speed ~ 0.46ms/pix (4-lpos., w/ Pentium-D 2.8MHz PC)  ~30min/2kx2k magnetogram Only available for FeI6302.5A and 5250.2A at this point.

27. Atlas profile Model profile with semi-LTE code Plan for MgI 5172A Non-LTE source function P.J.Mauas, E.H.Avrett and R.Loeser, 1988, ApJ, 330, 1008. Damping constant as a function of depth It is straightforward to calculate the model Stoke profiles for MgI 5172A..

28. Summary • Initial version of the program for reducing the magnetic fields from the FG observables is ready for FeI6302.5 and 5250.2A. • More tests are needed with actual sun data; we plan to make comparison with the ME inversion using ASP data. • Tuning of parameters will be necessary with the real SOT data after launch of Solar-B. • We plan to do with MgI 5172A and NaI 5896A a slightly modified LTE calculation to fit the profile, but more realistic model may be desirable (we have no idea at this point). • Reduction program for IV-mode of FG data is to be created.

29. Appendix-1. Cosine fitting algorithm Suppose that Ik(k=1,2,,,N) is observed intensity at wavelength position λk. Fit them with a sinusoidal function: Ik= a0 + a1 cos(xk) + b1 sin(xk), where xi is the phase ofλk in a certain wavelength range. Least square solution is derived as follows: In case of xk = [-3, -1, 1, 3]*p/4 i.e. cos(xk) = [-1, 1, 1, -1]/sqrt(2) sin(xk) = [-1, -1, 1, 1]/sqrt(2) The phase of the sinusoidal curve is calculated by tan—1(b1, a1) and we obtain the familiar formulae for Dopplergram.

30. Thus T is a N x 3 matrix. The inverse matrix is written as where D is the determinant of the source matrix. The sum of 2nd row of T is and, in the same way, the sum of 3rd row is ∑T3k = 0. Suppose that Ikcontains a dark noise, namely Ik = I’k + d , then Thus the dark bias in Ik is automatically corrected in obtaining a1 and b1.

31. Example of cos fitting Sinusoidal curve fitting for the FeI5576A line at lk = [-90, -30, 30, 90] mA. Thick curve is atlas spectrum convolved with the transmission profile of tunable filter. Same as upper panel but with lk = [-120, -40, 40, 120] mA.