2 Data and analysis procedure

2.1 Data sets

The Solar Oscillations Investigation/Michelson Doppler Imager (SOI/MDI) instrument is a state-of-the-art helioseismology experiment and magnetograph on board the SOHO spacecraft, devoted to study the interior structure and dynamics as well as the surface magnetic field of the Sun. This instrument gives an image of the Sun on a $1024\times1024$ CCD camera, and can observe in two spatial resolution modes, full disk and high-resolution of the central part of the disk (HR). We are interested in the full disk measurements, which have a field of view of $34\times34\mbox{\arcmin}$ and a pixel size of $2\times2\mbox{\arcsec}$. Two tunable Michelson interferometers allow MDI to record filtergrams centered at five wavelengths across the Ni I 6768 Å absorption line. From the filtergrams, MDI computes the following six observables: Doppler velocity, continuum intensity, line depth, longitudinal magnetic field, horizontal velocity and limb position. The SOI/MDI instrument is described in detail by Scherrer et al. (1995).

The products of interest for our work are the full disk magnetograms and continuum intensity images. Magnetograms only measure net magnetic flux per resolution element, therefore the signal is not the true magnetic field strength B, inside a flux tube, but its longitudinal component, $\langle \vert\vec B\vert \cos\gamma\rangle$, averaged over the pixel, where $\gamma$ is the angle between the magnetic vector and the line of sight. For simplicity, we hereafter refer to $\langle \vert\vec B\vert \cos\gamma\rangle$ as B. In a 2-component model of the magnetic field, with magnetic flux tubes of field strength B covering a fraction $\alpha$ of the solar surface separated by a field-free component covering $(1-\alpha)$ of the surface, we can write $\langle \vert\vec B\vert \cos\gamma\rangle$ as $\alpha \vert\vec B\vert \cos\gamma$. Since the true field strength $\vert\vec B\vert$ lies in a relatively narrow range of 1000-1500 G for all magnetic features except intranetwork elements (Solanki et al. 1999), and $\cos\gamma\approx\cos\theta\approx\mu$ is a reasonable approximation ($\theta$ is the heliocentric angle), the strength of the magnetogram signal mainly provides information on the magnetic filling factor $\alpha$.

MDI magnetograms are usually obtained every 96 min, with the exception of periodic campaigns in which 1-min cadence measurements are available. The 1-$\sigma$ noise level for a one-min longitudinal magnetogram is 20 G. Full disk continuum intensity images are taken each minute with a noise level of 0.3%.

The analyzed data set consists of nearly simultaneous magnetograms and continuum intensity images recorded on 10 days in the period February to October, 1999, as shown in Table 1. The time of the observations is given for the averaged magnetograms (see Sect. 2.2). It corresponds to the middle of the 20 min integration time. These days were chosen because they belong to a high activity period so that everything from quiet network to intense active-region plage was present on the solar surface. The sample contains active regions spread over almost all $\mu=\cos\theta$ values. They are also generally well separated in time in order to avoid duplication.

 

 

Table 1: Selected days and times (hours/min/sec) during 1999 at which the averaged magnetograms analyzed here were recorded (see text for details).

1999 observation dates	Time (UT)
February 13	00:10:02
February 20	04:10:02
May 14	00:10:03
May 28	06:10:03
June 25	01:10:02
July 2	03:10:02
July 10	01:10:02
August 7	00:10:02
October 12	09:10:03
October 15	06:10:03

   
2.2 Reduction method and analysis

Figure 1: Example of a 20 min averaged MDI magnetogram (top panel), the corresponding intensity image after removal of limb-darkening (middle panel) and the resulting contrast mask (lower panel) for October 12, 1999.

We employ averages over 20 single magnetograms, taken at a cadence of 1 per minute, in order to reduce the noise level sufficiently to reliably identify the quiet network. The individual magnetograms were rotated to compensate for the time difference before averaging. Intensities are standard 1-min images. Care has been taken to use intensity images obtained as close in time to the magnetograms as possible. In all cases but one, the two types of images were recorded within 30 min of each other, with 37 min being the highest difference. The intensity images have been rotated to co-align them with the corresponding average magnetogram. Intensity images have also been corrected for limb-darkening effects using a fifth order polynomial in $\mu$ following Neckel & Labs (1994). Our final data sets are pairs of co-aligned averaged magnetograms and photospheric continuum intensity images for each of the 10 selected days. Both types of images can be compared pixel by pixel. An example magnetogram and the corresponding intensity image recorded on October 12, 1999 are shown in Fig. 1 (top and middle panels).

We have determined the noise level of the MDI magnetograms and continuum images as a function of position over the CCD array. The standard deviation for the magnetic signal has been calculated using a running $100\times100$ pixel box over the solar disk, with the exception of the limbs, which were avoided by masking out an outer ring of 75 pixels width. This process was applied to several 1996 low activity magnetograms, in order to avoid artifacts introduced by the presence of active regions. After that, their median was determined to eliminate the possible remaining activity. A second order surface was then fitted to the result and extrapolated to cover the whole solar disk. The resulting noise level, $\sigma_{{\rm mag}}$, shows an increase towards the SW limb that probably includes some velocity signal leakage. In Fig. 2 we show the calculated standard deviation for the 20-min averaged magnetograms. Note that when applying this noise surface to our data we have assumed that the MDI noise level has remained unchanged between 1996 and 1999.

Figure 2: Standard deviation (in G) of the 20-min averaged MDI magnetograms. An increase of the noise in the direction of the SW limb is evident. The shadowed contours indicate some of the values. Both the surface and the contours represent the standard deviation.

A similar procedure has been used to determine the mean and standard deviation of the quiet Sun continuum intensity for each selected day, $\langle I_{{\rm qs}}\rangle$ and $\sigma_{I{\rm qs}}$ respectively, where the subscript ${\rm qs}$ denotes "quiet Sun". Every pixel in the running mean box with an absolute magnetic signal value below 0.5 times $\sigma_{{\rm mag}}$ (i.e. pixels with corresponding magnetogram signal between approximately -2.5 and 2.5 G) has been considered as a quiet Sun pixel.

The surface distribution of solar magnetic features that produce a bright contribution to irradiance variations, is identified by setting two thresholds to every magnetogram-intensity image pair. The first threshold looks for magnetic activity of any kind, and is set to $\pm3\sigma_{{\rm mag}}$, which corresponds, on average, to 15 G. As we are only interested in bright magnetic features, the second threshold masks out sunspots and pores by setting all pixels with a continuum intensity $3\sigma_{I{\rm qs}}$ below the average to a null value. To reduce false detections, even at the risk of missing active pixels, we reject all isolated pixels above the given thresholds assuming that they are noise. $3 \times 10^4$ out of 10⁷ analyzed data points are rejected in this way. After this step, we find that 6% of the pixels satisfy both criteria. Using both thresholds we construct a mask of the contrast of bright features for each day. The result of applying the mask derived from the magnetogram (top panel) and intensity image (middle panel) shown in Fig. 1, is displayed in the bottom panel of that figure. Note that only features that lie above the given thresholds in the magnetogram and the intensity image are indicated by white pixels. Sunspots near the NE limb, for example, do not appear in the mask, but faculae surrounding those sunspots are well identified. Smaller features belonging to the magnetic network are also pinpointed outside of the active regions, although weaker elements of the network may well be missed. For each pixel with coordinates (x,y), the contrast $C_{{\rm fac}}$ is defined as:

$$C_{{\rm fac}}(x,y) = \frac{I(x,y) - \langle I_{{\rm qs}}\rangle(x,y)}{\langle I_{{\rm qs}}\rangle(x,y)}\cdot$$ (1)

Contrast, magnetic field strength averaged over the pixel and position, represented by the heliocentric angle $\mu=\cos(\theta)$, are calculated for each selected pixel. Finally, for each of these parameters the pixels above the thresholds for each of the 10 selected days are put together into vectors of about $6 \times 10^5$ elements, which should provide adequate statistics for a detailed study of the facular and network contrast.

The method used in this work resembles that employed by Topka et al. (1992, 1997), although our magnetic threshold is much lower due to the less noisy magnetograms used. The angular resolution, however, is also considerably lower, but it is constant.