Kalibrace magnetografických měření

Primary Processing of Data Obtained by the Ondrejov Magnetograph

Algorithms for processing of brightness fields

a) Correction for atmospheric transparency changes
We use for this procedure the intensity of light in the center of the solar disk, registered during the measurements of the chosen area.

I_k=I(x, y)*I_s(0)/I_s(t)
I_k(x,y) corrected intensity value in the point (x,y)
I(x,y) measured intensity value in the point (x,y)
I_s(t) light intensity in the center of solar disk in the time of actual measurement in the point (x,y)
I_s(0) light intensity in the center of solar disk at the beginning of measurement

b) Correction for the limb darkening
Using the method of a rolling straight line we lead an envelope through all data of each individual scanned line in such a way that none of the measured values occurs above this straight line. These envelopes belonging to individual lines form a reference intensity surface Ir(x,y), against which the contrast C(x,y) is estimated:

C(x,y) = I_r(x,y) - I_k(x,y)

For an easy comparison of different photograms we introduce a normalized contrast CN(x,y):

C_N(x,y) = I_rN{1 - [I_k(x,y)]/[I_r(x,y)]},

where I_rN represents the standard level, independent on the coordinates x,y; in our case we use I_rN = 10000.

c) Correction for the scattered light
The use of contrast enables a simple correction for the scattered light:

C_sc(x,y) = C_N(x,y)*C_Nmax/C_N(x,y)_max

C_sc(x,y) - contrast corrected for the scattered light
C_Nmax - requested value of the maximal contrast in the point of maximal measured contrast CN(x,y)max

d) Photospheric isodensity curves
It is advantageous to use for the drawing of photospheric isodensity curves the values of the normalized contrast, because the quantization level neariest to zero - describing a considerable part of the photosphere - is well definable. On the contrary, the change of a scale by the quantization of intensity values may caused large, although topologically insignificant effects.

Algorithm for processing of Doppler velocities

Signal from the channel of Doppler velocities is proportional to the angle of turning of the planparallel plate of the compensator. Thanks to the small deviation from linearity we may express the Doppler velocity D_r(x,y) for the case of our compensator using the relation

D_r(x,y) = [126*N - 14*D_s(x,y)/N],
where N indicates the degree of numerical integration and D_s(x,y) the measured value from the Doppler channel. These measurements do not give the absolute values of Doppler velocities, the zero position depends on the setting of the spectral line. To enhance the sensitivity and the space resolution we make a numerical correction of Doppler velocity values. This velocity correction d(x,y) is based on the computation of the geometrical deviation of the spectral line from its compensated position. By the computation of this correction we get out of the difference between the signals of intensity I₁(x,y) and I₂(x,y) from both wings of the spectral line:

d(x,y) = D₀[I₁(x,y) - K*I₂(x,y)]/[I₁(x,y) + K*I₂(x,y)]

D₀ - constant characterizing the sensitivity of the compensation system In our case D₀ = 20000. The coefficient K takes away the data asymmetry in the compensated intensity channels:

K =	S	I₁(x,y) /	S	I₂(x,y)
	(x,y)		(x,y)

Then for the resulting value of Doppler velocity D(x,y) the following term is valid:

D(x,y) = D_r(x,y) - d(x,y).

From the analysis of the second relation it follows that the quoted compensation does not depend on the changes of the light intensity, but it is related to the geometrical shift of the spectral line from its central position only. This is the reason why this compensation works with reliability also in sunspots, where the compensator of Doppler shifts looses its sensitivity. This correction is that effective that it is able to reconstruct the velocity field even in the case of a switched off compensator.

Methodology of the magnetic field computation

a) Calibration of the magnetograph
The calibration is made always at the beginning of each measurement. It means that we register signals from all channels measuring the circular polarization and the channels dark currents. Than we calculate the calibration coefficient KC:

K_C = (I_C - I_OC)/(V_C - V_OC) * Z_C/Z_m

V_C, I_C - calibration signals during the measurement of circular polarisation
I_OC, V_OC - dark currents in channels V and I during the calibration mode
Z_C, Z_m - amplifications in channel V during the modes of calibration and measurement

b) Computation of magnetic induction
On the base of a characteristic of the solar atmosphere Staude (Bachmann et al., 1975) derived the interdependence between the Stokes parameters V and I and the magnetic induction. We calculate first for each point in the measured area the parameter V/I:

V/I = K_C*[V(x,y) - V_Om] / [I(x,i) - I_Om]

V(x,y), I(x,y) - signals of channels V and I in the measured point
V_Om, I_Om - dark currents in the channels V and I during the mode of measurement
Using the numerical calculation, following the derived dependences, we establish the longitudinal component of magnetic field. We delineate the area of a sunspot - which differs from the surrounding photosphere - on the base of evaluation of the field of the normalized contrast C_N, we obtained from the values of the Stokes parameter I in the proper wing of the spectral line.

c) Correction of the zero position
Evaluation of magnetic fields from both wings of the spectral line leads to slightly different results, caused by the line asymmetry and by the influence of the light linear polarization in the telescope. For the clearing of this effect three methods have been used:
c-1) Polarity mode (P)
Measuring the magnetic field intensity from both wings of the spectral line we shift the zero position in such a way that the number of points in both polarities is the same.
c-2) Magnetic field averaging mode (M)
We calculate the magnetic field strength for both wings of the spectral line independently. than the resulting field we obtain by averaging the intesity values of both wings in each poit of the measured area.
c-3) Signal averaging mode (S)
In each point of the scanned area we average the signals of channels V and I from both wings of the line and the mean values of the calibration constant. Our experience demonstrates that the resulting data sets, obtained by different modes of zero correction, do not differ almost at all. From practical reason we use mostly the most rapid correction, the mode S.

Calibration curves

Dependency of Stokes parameters V/I on the longitudinal component of magnetic induction B_L for two spectral lines. We use Stelbmacher-Wiehr model for a sunspot and HSRA-Gingerich model for the quiet sun.

a)	Correction for atmospheric transparency changes We use for this procedure the intensity of light in the center of the solar disk, registered during the measurements of the chosen area. I_k=I(x, y)*I_s(0)/I_s(t) I_k(x,y) corrected intensity value in the point (x,y) I(x,y) measured intensity value in the point (x,y) I_s(t) light intensity in the center of solar disk in the time of actual measurement in the point (x,y) I_s(0) light intensity in the center of solar disk at the beginning of measurement

b)	Correction for the limb darkening Using the method of a rolling straight line we lead an envelope through all data of each individual scanned line in such a way that none of the measured values occurs above this straight line. These envelopes belonging to individual lines form a reference intensity surface Ir(x,y), against which the contrast C(x,y) is estimated: C(x,y) = I_r(x,y) - I_k(x,y) For an easy comparison of different photograms we introduce a normalized contrast CN(x,y): C_N(x,y) = I_rN{1 - [I_k(x,y)]/[I_r(x,y)]}, where I_rN represents the standard level, independent on the coordinates x,y; in our case we use I_rN = 10000.

c)	Correction for the scattered light The use of contrast enables a simple correction for the scattered light: C_sc(x,y) = C_N(x,y)*C_Nmax/C_N(x,y)_max C_sc(x,y) - contrast corrected for the scattered light C_Nmax - requested value of the maximal contrast in the point of maximal measured contrast CN(x,y)max

d)	Photospheric isodensity curves It is advantageous to use for the drawing of photospheric isodensity curves the values of the normalized contrast, because the quantization level neariest to zero - describing a considerable part of the photosphere - is well definable. On the contrary, the change of a scale by the quantization of intensity values may caused large, although topologically insignificant effects.

a)	Calibration of the magnetograph The calibration is made always at the beginning of each measurement. It means that we register signals from all channels measuring the circular polarization and the channels dark currents. Than we calculate the calibration coefficient KC: K_C = (I_C - I_OC)/(V_C - V_OC) * Z_C/Z_m V_C, I_C - calibration signals during the measurement of circular polarisation I_OC, V_OC - dark currents in channels V and I during the calibration mode Z_C, Z_m - amplifications in channel V during the modes of calibration and measurement

b)	Computation of magnetic induction On the base of a characteristic of the solar atmosphere Staude (Bachmann et al., 1975) derived the interdependence between the Stokes parameters V and I and the magnetic induction. We calculate first for each point in the measured area the parameter V/I: V/I = K_C*[V(x,y) - V_Om] / [I(x,i) - I_Om] V(x,y), I(x,y) - signals of channels V and I in the measured point V_Om, I_Om - dark currents in the channels V and I during the mode of measurement Using the numerical calculation, following the derived dependences, we establish the longitudinal component of magnetic field. We delineate the area of a sunspot - which differs from the surrounding photosphere - on the base of evaluation of the field of the normalized contrast C_N, we obtained from the values of the Stokes parameter I in the proper wing of the spectral line.

c)	Correction of the zero position Evaluation of magnetic fields from both wings of the spectral line leads to slightly different results, caused by the line asymmetry and by the influence of the light linear polarization in the telescope. For the clearing of this effect three methods have been used: c-1) Polarity mode (P) Measuring the magnetic field intensity from both wings of the spectral line we shift the zero position in such a way that the number of points in both polarities is the same. c-2) Magnetic field averaging mode (M) We calculate the magnetic field strength for both wings of the spectral line independently. than the resulting field we obtain by averaging the intesity values of both wings in each poit of the measured area. c-3) Signal averaging mode (S) In each point of the scanned area we average the signals of channels V and I from both wings of the line and the mean values of the calibration constant. Our experience demonstrates that the resulting data sets, obtained by different modes of zero correction, do not differ almost at all. From practical reason we use mostly the most rapid correction, the mode S.