The solar survey at Pic du Midi: calibrated data and improved images

At Pic du Midi observatory we carry a solar survey with images of the photosphere, prominences and corona. This survey, CLIMSO, is in the following spectral lines: Fe~XIII corona (1.075~$\mu$m), H-$\alpha$ (656.3~nm) and He~I (1.083~$\mu$m) prominences, H-$\alpha$ and Ca~II (393.4~nm) photosphere. All frames cover 1.3 times the diameter of the Sun with an angular resolution approaching one arc second. The frame rate is one per minute per channel (weather permitting) for the prominences and chromosphere, and one per hour for the Fe XIII corona. This survey started in 2007 for the disk and prominences, in 2015 for the corona. We have almost completed one solar cycle, and hope to cover several more, keeping the same wavelengths or adding others. We make the solar CLIMSO images easier to use and more profitable to the scientific community, by providing"Science ready"data. We have improved the contrast capabilities of our coronagraphs, which now provide images of the Fe~XIII corona, in addition to the previous spectral channels. We have also implemented an autoguiding system based on a diffractive Fresnel array for precise positioning of the Sun behind the coronagraphic masks. The images and films, are publicly available and downloadable through virtual observatories and dedicated sites: {\it e.g.} http://climso.irap.omp.eu. For the H-$\alpha$ and and Ca II channels we calibrate the data in physical units, independent of atmospheric or instrumental conditions: we provide solar maps of spectral radiances in $\rm {W m^{-2} sr^{-1} nm^{-1} }$. The instrumental improvements and the calibration process are presented in this paper.


Introduction
Pic du Midi de Bigorre in the French Pyrenées is the observatory where Bernard Lyot invented coronagraphy and obtained remarkable images in the 1930s and 1940s; see for example Lyot (1930Lyot ( , 1945Lyot ( , 1950. Since then, Pic du Midi has regularly provided high quality solar and coronagraphic data. In a Lyot coronagraph, the bright image of the solar photosphere is blocked by an occulting disk of same diameter in the focal plane. Furthermore, the light diffracted around the occulting disk, which would be too bright to allow for observations of the corona, is in turn blocked by a "Lyot stop" in a pupil plane. Such an optical setup associated with precise baffling allows for a high rejection factor of solar light in the surrounding field, defined as the "dynamic range" in this work, which reaches 10 6 . This ongoing survey aims to help the scientific community with solar studies by providing a large set of solar images and films with dense temporal sampling: 1 min between frames, and long span: several solar cycles. We have kept constant the observation conditions: frame rate, field, wavelengths, and bandpass, but that does not forbid improvements on the image quality, such as arcsecond angular resolution, and high contrast (high dynamic range) as shown in Figs. 1 and 2.
At present, five instruments contribute to the CLIMSO (CLIchés Multiples du SOleil or Christian Latouche IMageur SOlaire) solar survey: two coronagraphs, two solar telescopes (Dettwiller & Noens 2008), and an autoguider. The autoguider uses a diffractive Fresnel array as a focuser and provides images of the Sun to center the other instruments.
We have been carrying out this survey since 2007 thanks to the important contribution of the "Observateurs associés", the volunteer astronomers 1 who contribute in many ways, including instrumental and software development, financial support, and yearlong handling of image acquisition.
Among several upgrades, solar telescope l 1 now has high quality optics and a new Hα filter. We have improved the baffling compare with the simultaneous image in Ca II on the right. The original image covers the full disk in 2k × 2k pixels. Images are taken every minute, from one hour after sunrise to one hour before sunset, weather allowing. Right: solar chromosphere in the Ca II line at 393.4 nm. These two images were taken with l 1 and l 2 at Pic du Midi (2015-03-27). and especially the correction of the filter stack leakage in coronagraph c 2 . In 2015 this led to the 10 6 dynamic range allowing for images of the solar corona.
We define the dynamic range of a coronagraph as the ratio between the brightness in the image of the photosphere and the residual light in the surrounding field at the final focal plane. This could also be called "rejection factor".
In good weather conditions, a set of four images is taken every minute, and one image of the solar corona in the Fe XIII line at 1074.7 nm is taken per hour. All images are uploaded to the database each night, along with several movies made from these images. During daytime, a few images are selected for realtime uploading to the data base.

Guiding system
The fifth telescope is used for guiding. The main optics of this telescope consist of a Fresnel diffractive array, 2.5 m in focal length at λ = 633 nm, placed close to the c 1 coronagraph. The coronagraph and diffractive optics for the guider are fastened to each other, thus limiting differential flexion.
The Fresnel diffractive array shown in Fig. 4 is a copper foil 80 µm thick and 65 mm in diameter with several thousand holes, based on the diffraction principle Fresnel (1818) and Soret (1875). This foil, or diffractive array, acts as a Fresnel zone plate and forms a sharp image of the Sun on a camera, which sends a frame or frames to an image processor. The diffractive Fresnel array has a binary transmission: 0 or 1 (Koechlin et al. 2005). There is no optical material in this objective lens, only opaque material (in this case, a thin copper plate: transmission 0) with holes of precisely calculated shapes (where transmission is 1) that focus light by constructive interference. This setup is lightweight, steady, affordable, and yields high contrast images.
This optical concept provides diffraction-limited images. It works with broadband (non-monochromatic) light as it can be corrected from chromatic aberration. Fresnel arrays have supplied high contrast images of sky objects; see Koechlin et al. (2012) and Koechlin (2014). Although these authors refer to tests with 20 × 20 cm square Fresnel arrays, these two publications provide references of studies for larger diffractive arrays.
Diffraction focusing causes chromatism in the resulting images at prime focus, but we do not need to cover this in this work: we placed a narrowband filter (λ = 633 nm, ∆λ = 1 nm) near focus. This wavelength of He-Ne lasers does not correspond to a feature in the solar spectrum, so the images of the solar disk in the autoguider are prominence-free. This He-Ne filter is affordable; its spectral finesse (1 nm) is slightly too low for complete chromatism cancelation in this optical configuration, but the images are almost diffraction-limited.
To achieve a good contrast we had to block the light from unwanted diffraction orders; this mainly concerns order zero passing as a plane wave, unfocused through the Fresnel array. To block this order we used a central obscuration, casting its shadow on the focal plane. By geometric design, this also removes all positive and negative diffraction orders except the desired one. As the Sun is not a point source, the central obscuration (here 52 mm) needs to be larger than the required field for guiding. On the other hand, the outer diameter of the objective is limited by the size of the narrowest Fresnel rings in the diffractive array, which can be grooved and is itself limited by the waist of the UV laser beam used to manufacture it. The local machine tools that were used have a laser beam width of 20 µm, thus limiting the Fresnel array diameter to 62 mm for 2.5 m focal length at 633 nm. Recent machine tools with thinner laser beams can manufacture much wider Fresnel arrays with equal focal length.
An image of the Sun is taken every second by a SBIG-8300F camera at focus behind a small lens adapting the field scale to the camera sensor; this image is then sent to the autoguider software, which keeps the five instruments precisely pointed, even when The solar disk is processed to extract its edge (ring or arc), which in turn is correlated with a reference circle. The guiding software adapts the diameter of that circle to the apparent diameter of the Sun. The position of the reference circle can be either manually or automatically adjusted to any desired position. To reduce the processing time we use sparse matrix algorithms rather than fast Fourier transforms. The computed distance between the center of the reference circle and the center of the solar disk is used to determine a command sent to the equatorial mount. The source code in C++ and a detailed documentation of the autoguider are available on github 3 .
The guiding precision is better than 1 arcsec in good seeing conditions. Of course, because of the slow sampling rate of 1 s and the inertia of the equatorial mount, most of the tip-tilt agitation due to atmospheric seeing is not corrected. This limitation in performances is significant in bad seeing or high wind conditions. We plan to improve the algorithm that computes the commands to the equatorial mount based on the history of solar positions found; at present it is just a proportional response.

Principle
Our photometric calibration is based on the fact that in an image of the Sun made from the ground, the sum of all pixels in the solar disk represents the total light received from the Sun through the spectral bandpass of the instrument. If we have an independent measure of the corresponding solar power above the atmosphere, such as a calibrated solar spectrum obtained by a space-borne system, and if we make the approximation that the solar absolute magnitude can be considered constant, we can deduce a calibration factor for any image made under 3 https://github.com/maelvalais/climso-auto the atmosphere. This calibration globally takes into account the atmospheric and instrumental transmission for that particular image.

Preprocessing
This calibration process concerns the past, present and future images of the ongoing survey. This process is applied to the images each day, just before the data are uploaded to the database.
In the first steps of this process, we correct the images from dark and flat-field effects induced by the cameras: chargecoupled device (CCD) on solar telescopes l1 and l2, and complementary metal oxide semiconductor (CMOS) on coronagraphs c1 and c2. As the exposure times are all very short, we considered that the bias and dark images are equivalent. The pixel scale is calibrated from the angular diameter of the Sun each time a modification is made to the optics. The rotation angles of the cameras and their variations during the day are also calibrated regularly, and the images are de-rotated so that the north solar pole is up.
We do not correct the geometrical distortion, but this error is very low: the optics in the coronagraphs and solar telescopes all have a long f-ratio and they work at small angles. For l1 and l2 we perform a photometric calibration explained in the following. The result is a spectral radiance map of the Sun.
At this point the cameras provide raw images of the solar disk that represent its local radiance, but these images are affected by (1) the Sun-Earth distance, (2) the absorption due to airmass and atmospheric conditions; and (3) the instrument, i.e., the optics transmission and camera response.
We calibrated those three parameters globally to obtain calibrated images that represent physical conditions at the surface of the Sun or its vicinity, so the pixel "brightness" values are in physical units. The choice of a relevant unit is not simple since the Hα, He i, Ca ii K, and Fe xiii light forming the images is emitted from volumes of solar plasma (chromosphere, prominences, and corona) with various optical thicknesses and altitudes.

Photometric units
To make the following description of our calibration procedure as clear as possible, we introduce four photometric definitions that we use. They are described as follows.
In emission, (1) the radiance L(x, y, α, δ), expressed in W m −2 sr −1 , is the flux emitted. This emission is in a given direction (α, δ), per unit of solid angle (sr), corresponding to the line of sight. This emission is from a point (x, y) of a surface, per unit area, (m 2 ) projected on a plane (the sky plane) normal to the considered direction (the line of sight). Also in emission, (2) the spectral radiance L(x, y, α, δ, λ), expressed in W m −2 sr −1 nm −1 . We considered the emission toward the Earth, i.e., the emission quasi-perpendicular to an emissive surface (in the plane of the sky), but not necessarily parallel to the local solar surface. Thus we can express the spectral radiance in our direction (α, δ) as L(x, y, λ).
In reception, (3) the illuminance I(x, y) is the flux received from all directions, at a given point (x, y) of a collecting surface, per unit area. Also in reception, (4) the irradiance corresponds to the illuminance when the collecting surface is perpendicular to mean propagation, which is our case. The irradiance I(x, y) is expressed in W m −2 , and the spectral irradiance I(x, y, λ) is expressed in W m −2 nm −1 .
To link the radiance L emitted from a point S above the Sun to the irradiance L received on Earth at P, we used Clausius' theorem found in Born & Wolf (1980), Klein (1970), written as where T is the transmission factor associated with the beam, when we neglect the effect of diffusion on the optical path, and assume the refraction index n 1 at S and P. For simplicity, when P is the image of S by an instrument, we used the same notation (x, y) for coordinates of S and P. As briefly explained in the first paragraph of this Sect. 3, we found a two-step approach to obtaining a radiance map of the Sun through the transmission of a telescope: first, the value of a pixel in a Lambertian situation is proportional to the radiance of the corresponding element of the Sun; second, the proportionality factor is uniform over the whole solar disk. Hence, the relative value of a pixel compared to the average all over the Sun is also the relative radiance, compared to the mean radiance of the whole Sun, which is known from airborne measurements.

Photometric calibration
To obtain a calibrated image of the solar disk, the physical data we want for a pixel is the radiance toward the Earth: L(x, y) of the corresponding point on the solar disk, through the normalized transmission function of our filter T (λ) (whose maximum is T max ), plotted in Fig. 5, i.e., where L(x, y, λ) is the local spectral radiance on the Sun; in a plane close to the surface of the Sun and perpendicular to the line of sight, as discussed in the last paragraph of Sect. 3.3 and in Sect. 3.2. We made the following approximations: (1) On the image plane we are in a Lambertian situation, close to the optical axis. (2) T (λ) does not depend on (x, y), i.e., the filter is uniform in the field. (3) The instrument and camera defects are flat-fielded out, but in addition the atmospheric transmission is uniform over the image field. That is valid in clear sky conditions and can be an approximation for a uniform haze or cirrus. (4) Our cameras have a linear and uniform response (once corrected from dark and flat-field); we assume that a raw pixel value v(x, y), measured in arbitrary units adu for a point P(x, y) in the image, is proportional to the local irradiance L (x, y) in that image.
The raw pixel value v(x, y) is given by -C is a transmission factor (constant over the image field, but variable in time) driven by the instrument, the Sun-Earth geometry, and the atmosphere, which is canceled out during the calibration process. -The spectral transmissions of the atmosphere and optics, and the camera response contribute to C but they are slowly wavelength-dependent compared to the narrow filter bandpass T (λ), so C can be considered wavelength independent in the relevant spectral band. We take the variations of C with time, caused by changes of air mass or haziness, into account by the calibration process, as shown below. Changes in the instrument, such as aging of filters or cameras, which affect the global response, are also calibrated out by the same process. Furthermore, this will also be applied retrospectively to data that had not been calibrated when it was acquired. Care should be taken, however, that filter wavelengths are controlled and remain correct.
We know the equivalent bandpass ∆λ from the transmission function T , for l 1 , is written as The filter bandwidth and its precision are given by the manufacturer (Daystar). We take, from the literature and documentation, the following data: a standard solar spectrum I λ,m (averaged over a solar cycle), itself implicitly based on the solar constant, i.e., the standard solar irradiance at 1 AU; and spectral transmission of our filter T (λ).
As we use narrowband filters, we need a calibrated standard solar spectrum at high spectral resolution. We start from an uncalibrated standard spectrum at high resolution: the Meudon Solar Atlas Delbouille et al. 4 , then calibrate it with a calibrated standard solar spectrum at lower resolution 5 : From that we compute I m the standard solar irradiance above the Earth atmosphere integrated through the spectral profile of our filter, and we find its numerical value for l 1 (also in Table 1), i.e., The ±0.001 error bar comes from the solar spectrum calibrations and the filter spectral curve: we have numerically integrated Eq. (5) to obtain the numerical value in W m −2 . From this external data, and neglecting the cosine in the relation between the radiance and the elementary flux, we deduce for the current image L m , the standard radiance averaged over the solar disk (D m ), and integrated through the transmission of the instrument, where Ω m 0.680 × 10 −6 sr is the solid angle of the solar disk at 1 AU. The ±5% uncertainty resulting from our estimation of I m in Eq. (5) is much greater than the errors due to the effects of the cosine factor neglected in Eq. (6) and the distorsion. We now compare L m with the average of the raw pixel values v(x, y), over the N pixels covered by the solar disk image, This average solar pixel valueν corresponds to the apparent brightness of the solar disk through all the instrumental chain.
From an independent source we have its calibrated brightness: L m , calculated with Eq. (6), and we deduce the calibration factor C with This allows us to eliminate C, and we finally find where v r (x, y) := v(x, y)/ν is the relative value of a pixel compared to the average, and where K := L m /∆λ. At present we calibrate the images similarly for our other solar telescope l2 in the Ca ii line. We have a relatively inaccurate knowledge of the spectral transmission for the 10-year-old filter in telescope l2, in addition to the variability of the line, so we provide "calibrated" images with low precision. We plan to buy better filters in a few months.
The numerical values are listed in Table 1. In the third and fourth line of this table we have, corresponding to each filter, the mean spectral irradiance from the Sun above the Earth atmosphere, and then this spectral irradiance is integrated through the spectral curve specific to the filter. The calibration in Ca II is less precise because of the solar variability in that line, in addition to the aging of our Ca II filter. These are used to generate the calibrated images provided by the public database 6 . To keep a sufficient dynamic range in the 16 bit integer data, the pixel values in the database may be adjusted by a factor of 10. This will be documented in the Flexible Image Transport System (FITS) header. At present only the recent images have a calibrated counterpart, but all images since 2007 are going to be reprocessed and calibrated.

Future calibration of the Hα prominences
The spectral bandpass of the Hα filter in coronagraph c 1 is ∆λ c1 = 0.25 nm. This can be considered to be wider than the emission lines of prominences, but narrow enough so that no significant emission from another phenomenon contributes to the pixel value c 1 (x, y) and where L Hα (x, y) is the radiance at (x, y) over the Hα line of the prominences. The images from coronagraph c 1 in the left panel of Fig. 2 cannot be calibrated similarly to l 2 and l 1 because it would require sampling the solar disk, which is too bright for c 1 and not adapted to its ring-shaped field. However, as c 1 and l 1 are centered on the same wavelength and both see the prominences (although c 1 is much more sensitive), a cross-calibration is possible. It is not simple to implement, but we propose the following approximations. First, the images on c 1 and l 1 can be considered Fig. 6. Solar irradiance above the atmosphere in W m −2 nm −1 as a function of wavelength in nm. The blue curve is the FTS high resolution solar spectrum near Hα, Kurucz et al. (1984), which we use to calibrate the images taken with l 1 (λ = 656.28 nm; ∆λ = 50 pm). The red curve is shown for comparison. It is a calibrated solar spectrum taken at the ISS by Meftah et al. (2018). The gray zone around the red curve represents the area covered by its error bars. The wavelength axes of these two curves have been adjusted to each other. The superposition of the red and blue curves indicates a good agreement between data from the International Space Station and our calibration of the high resolution spectrum. simultaneous as they are both taken within a few seconds interval. Second, the instrument characteristics and camera sensitivities evolve slowly and the average ratio between the responses of c 1 and l 1 can be considered constant for a few hours.
This image calibration for c 1 is not done yet, but is planned for the near future: we will use sets of prominences that have brightnesses within the linear response of both instruments l 1 and c 1 : we find by least-squares fit the ratio Pro(l 1 )/Pro(c 1 ) between images made by l 1 and images by c 1 of the same prominences.
As the radial velocities may drive some of the emission beyond the narrow bandpass of l 1 while still within the wider bandpass of c 1 , special care will be taken in the algorithm to extract a precise brightness ratio from the histogram of ratios found.
3.6. Calibration for coronagraph c 2 is forthcoming C 2 looks into the He I line at 1.0830 µm (Fig. 2, right; Fe XIII at 1.0747 µm, Fig. 3.) It cannot be calibrated yet as we have no solar photometer at that wavelength. We could compare the optics transmission and camera response at different wavelengths and then estimate a ratio for the air-mass effect, but this would be prone to many cumulative errors. For the moment and pending the implementation of a photometer in the corresponding waveband, for c 1 and c 2 we upload merely normalized images of prominences and corona.

Conclusions
We hope that our calibrated images contribute to the development of solar physics. We do not claim a high photometric precision immediately, but our calibrated images should be useful.
This calibration uses the solar spectrum, itself based on the solar constant; hence it can be done for any image, even if the image was obtained long ago, as long as it contains the solar disk (or a known fraction of this disk) with known wavelength and bandpass. To improve precision, we plan to reject spots, filaments, and eruptive regions when computing the solar disk average of each image. This CLIMSO database was started in 2007 and we hope to carry on for several solar cycles, and to complement other ground-based or space surveys. You are welcome to use it!