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

2 Wavelength (nm)g eff Ｇ Pol. Sensitivity (diagonal element of X) Detection limit for B (Gauss) VQUBlBl BtBt MgI FeI FeI NaI FeI HI >5000 Detection limit of FG for the weak magnetic fields,  = nd moments of  and  -components I’: line profiles convoluted by TF transmission curve

3 NFI observables -- I( i ), Q( i ), U( i ), V( i ), i = 1,,, N Physical quantities derived from the observables --B field strength (G),  inclination (deg.),  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. 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. Outline of this study

4 are calculated as below. Model atmosphere :Holweger & Muller (1974, Sol.Phys., 39, ) Line :FeI6302.5, FeI5250A, others to be added Wavelength :-400 ~ 400mA (n wl =161, 5mA step) B :0 ~ 3000G (n B =41) =[0, 0.5, 1, 2, 4, 10, 20, 40, 60, 80, 100, 150, 200, 300,,,,, 3000]  :0 ~ 180deg. (n  = 19, 10deg step)  : 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, , i ) S:Doppler shift -90 ~ 90mA (n s =37, 5mA step) Model Stokes profiles Thus the LUT spans the parameter space of (S,B,  ) ~ (37, 41, 19)

5 Model profiles (B,  )  (deg.) QVIU (41*19 = 779 profiles)

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  B l, Q/I  B t individually N = 2: Rotate the frame to make U=0 (ignore MO effect) + search for the best fitting to model observable in (B, , 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, , S) sub space

7 vB l q’ B t no Doppler information- Polarization signals (V, Q) vs. magnetic field strength (B l, B t ) with a Lyot filter (width =100mA) positioned at d = -[10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120] mA from the line center. Thick curve is for d = - 80mA.  = 0 o  = 90 o saturation LUT v = V/I, q = Q/ I, u = U/ I  = tan -1 (u, q)/2 q’ = q *cos(2  )+ u *sin(2  )  =  80 mA N = 1

8 N > 3: flowchart observable I o, Q o,U o,V o (i) (i = 1 ~ N wp ) Initial guess (B, ,S) from observables, rotate Q,,U by , and then search best fit to model observables (I, Q’,U’,V) in (B,  S) space around the initial guess point, finally calculate  from Q’,,U’ again. I’ m, Q’ m, U’ m, V’ m (B,  S  (i =1 ~ N wp ) I’ o, Q’ o, U’ o, V’ o (i) (i = 1 ~ N wp ) Rotate Q o,U o by -  0, normalize Q o,U o by I o (0) Initial guess I o, Q o,U o,V o  B 0,  0  0, S 0 Subspace around (B 0,  0  S 0 ) Expand in S (Dopp.shift) Sampling position Model profiles at  =0 I m, Q m, U m, V m (,B,  ) find ( S, B,  ) that minimize d w/ interpolation by 2nd polynomial (Q o,U o ) & (Q m,U m )   d (S,B,  =∑ i {f*(I’ o - I’ m ) 2 + (Q’ o - Q’ m ) 2 + (U’ o - U’ m ) 2 + (V’ o - V’ m ) 2 } Model observable table I’ m, Q’ m, U’ m, V’ m (i) (B, , S) (i = 1 ~ N wp )

9 Least square fitting is applied to the absorption line (=observable) using the following sinusoidal function then Doppler shift ( 0 ) is obtained by 0 =tan — 1 (b 1, a 1 ) 1) S 0 (Doppler shift) cos fitting for (I + V) i  S + “ (I  V) i  S - Initial guess method (B 0,  0  0, S 0 )  S 0 = (S + + S - )/2 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 cos fitting method  0 (inclination) Weak field approximation

10 3) B 0 (field strength) 1) Case |  - 90 o | > 20 o (S + and S - have significant separation) 2) Case |  - 90 o | < 20 o (nearly horizontal field) 3) Case |  - 90 o | 1000G (nearly horizontal and saturation ) = 6303A, w=0.075A, d=0.5,g eff =2.5   1 = 3230 with some correction from the numerical experiments  2 is determined experimentally from model profiles For FeI 6303A,  2 = weak field approximation 4)  0 (azimuth) q’ i, u’ i are Q i ’/I i ’, U i ’/I i ’ from the model at  = 0 o, (B 0,  0  S 0 )

11 Search space of (B,  S ) for fitting (I’, Q’, U’, V’) i B G  S 180deg. +90mA -90mA (B 0,  0  S 0 ) S : S ~ S 0 *3 +20 mA B: B 0 * ~ B 0 * G  :  0 * ~  0 * deg.

12 N = 2: ( ignore MO effect, search entire (S, B,  ) space) observable (i), I o, Q o,U o,V o (i) (i = 1 ~ N wp ) d (S,B,  =∑ i {f*(I’ o - I’ m ) 2 + (Q’ o - Q’ m ) 2 + (V’ o - V’ m ) 2 } find ( S, B,  ) that minimize d w/ interpolation by 2nd polynomial Calculate azimuth from Q,U, rotate frame to get Q’ with U’=0 Search best fit for ( I,Q’,V ) in (S,B,  ) space of LUT fgmag V0.1 fgmags Model profiles I m, Q m,V m (,B,  ) Model observable I’ m, Q’ m,V’ m (B, , S) (i = 0 ~ N wp -1)  = tan -1  U o *, Q o *  Q’ o, = Q o cos2  + U o sin2  V’ o, = V o  I o, I’ o = I o  I o (0) I’ o, Q’ o, V’ o ( i ) (i = 1 ~ N wp ) f = 0.2; weight for I Expand in S (Dopp.shift) Sampling position

13 Q Q/I (Q,U)   Ignore MO effect.. > 0 Among 2 wavelength points, which (Q i, U i ) should be used to get  P i, = √(Q i 2 + U i 2 ) Index of max(P i )  imax, Index of 2 nd max(P i )  imax2 if | Q imax2 | > | U imax2 |  use Q else use U in the following if Q imax * Q imax2 max(I i ) *  (noise) then use (Q i, U i ) with larger I i （ avoid line core ） else use (Q imax, U max )  Q i *, U i *

14 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 <  < 180 deg. -90 <  < +90 deg. -90 < S < +90 mA Numerical experiment

15 No Doppler info. N = 1 at dl = -80mA, Simulation result Sample observable, 1000points B < 2000GB >2000G |S| < 60mAblackblue |S| > 60mAgreenred

16 Initial guessAfter fitting B < 2000GB >2000G |S| < 60mAblackblue |S| > 60mAgreenred N = 4 at d = [-105, -35, 35,105] mA, simulation result

17 Initial guess After fitting B < 2000GB >2000G |S| < 60mAblackblue |S| > 60mAgreenred N = 4 at d = [-110, -70, 70,110] mA, simulation result

18 Initial guess After fitting B < 2000GB >2000G |S| < 60mAblackblue |S| > 60mAgreenred N = 3 at d = [-80, 0, 80] mA, simulation result

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

20 alternative method: - ignoring MO effect - search entire (S, B,  ) space B < 2000GB >2000G |S| < 60mAblackblue |S| > 60mAgreenred N = 2 at d = [-80, 80] mA, simulation result

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 Without ripple With ripple More simulation with the filter ripple

24 More simulation with other atmospheric models ME parametersQuiet regionPenumbraUmbra η 0 : Line strength B0 : Source function (τ= 0) B1 : Source function gradient  v D : Doppler width [mA] a : Damping parameter  v m : Macroturbulence [km/s] 0.8 B: field strength [G] 100 ～ 3000 (Δ = 200)  : inclination [deg] 0 ～ 180 (Δ = 15)  : azimuth [deg] -90 ～ +90 (Δ = 15) S : Doppler shift [km/s] -1.5 ～ +1.5 (Δ = 0.25) Using the ASP data, typical ME parameters for FeI 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 input result B [kG]  [deg.]  [deg.] S [km/s] penumbra umbrae Quiet region More simulation with other atmospheric models Å [-105, -35, +35, +105] mÅ

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- pos., w/ Pentium-D 2.8MHz PC)  ~30min/2kx2k magnetogram Only available for FeI6302.5A and A at this point.

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

28 Summary Initial version of the program for reducing the magnetic fields from the FG observables is ready for FeI and A. 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 In case of x k = [-3, -1, 1, 3]*  /4 i.e. cos(x k ) = [-1, 1, 1, -1]/sqrt(2) sin(x k ) = [-1, -1, 1, 1]/sqrt(2) Suppose that I k (k=1,2,,,N) is observed intensity at wavelength position λ k. Fit them with a sinusoidal function: I k = a 0 + a 1 cos(x k ) + b 1 sin(x k ), where x i is the phase of λ k in a certain wavelength range. Least square solution is derived as follows: The phase of the sinusoidal curve is calculated by tan — 1 (b 1, a 1 ) 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 2 nd row of T is and, in the same way, the sum of 3 rd row is ∑T 3k = 0. Suppose that I k contains a dark noise, namely I k = I ’ k + d, then Thus the dark bias in I k is automatically corrected in obtaining a 1 and b 1.

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