Seeing estimation

Helia can estimate the seeing from the capture itself, with no reference star (#49). It gives two numbers that answer different questions:

• r0, the Fried parameter, from the spectral ratio. An absolute figure: the diameter of the patch of atmosphere over which the wavefront is flat to about a radian, at the capture's wavelength and scaled to 500 nm. D / r0 (the aperture over r0) says how many seeing cells the telescope looks through, and with it how rare a lucky frame is. It needs the telescope aperture.
• A motion index from the alignment points. A relative figure: how much neighbouring alignment points move against each other from frame to frame. It needs nothing but the capture, and is not an absolute r0.

Switch it on with --seeing (add --aperture-mm 150 for r0), seeing and aperture_mm in an API job, or set the telescope aperture under Advanced settings in the app (the Stack step then shows a seeing line). It reads about 120 more frames, spread evenly over the capture, after the stack: about two seconds on the project's captures. The result is printed after the stack, written into the FITS header (SEEMOTPX, SEEMOTAS, SEEDR0, SEER0500, SEEGRADE) and returned by the API as summary.seeing.

$ helia stack Sun_Test.ser -o sun.fits --preset na --seeing --aperture-mm 80
...
seeing:  followed 16 tiles of 64 px through 120 frames spread over the capture
seeing:  motion index:   0.274 px (0.59 arcsec) RMS differential motion per axis between points 72 px (154 arcsec) apart (14 pairs); relative, not r0
seeing:  spectral ratio: D/r0 8.2, r0 1.0 cm at the capture's wavelength, 0.8 cm at 500 nm (18 rings, 16 tiles, tiles differ by 25%)
seeing:  note:           a 64 px tile spans 138 arcsec, more than one isoplanatic patch (a few arcseconds); the method is validated for one transfer function per tile, so r0 is indicative and no grade is given
seeing:  note:           the cut-off (1.42 cycles/px) is above the sampling limit, so fine detail is aliased and the r0 is indicative; a scale of 0.76 arcsec per pixel or finer avoids it

(The aperture here is a guess: the project's captures do not say what telescope they were taken with.)

The spectral ratio

For an object that does not change during the burst, the Fourier transform of a tile of frame k is F_k = O S_k: the object's transform times the transfer function of that frame's atmosphere and telescope. Average over the frames, once aligned, and the ratio |<F>|^2 / <|F|^2> equals |<S>|^2 / <|S|^2>. The object cancels (von der Luehe 1984), leaving a curve that depends only on D / r0 and on the frequency in units of the telescope's cut-off D / lambda. Helia matches the measured curve to the one a Kolmogorov atmosphere gives, which it holds as a table made by Monte Carlo over phase screens (cargo run --release -p helia-core --example seeing_table; the structure of the screens is checked against Noll's tilt-removed variance, 0.134 (D / r0)^(5/3), to within 1%).

In practice: up to 16 tiles of 64 x 64 pixels, the ones with the most detail and wholly inside the disc, are followed through the sample by correlation. Each tile is cut at its integer position, the sub-pixel remainder removed in Fourier space, and the mean transform and the mean power accumulated. The noise is measured on the tiles (Immerkaer's estimator) and its power taken out of the denominator; the finite-sample bias of |<F>|^2 is taken out of the numerator. Rings of the spectrum with a signal to noise above three, between 5% and 90% of the cut-off (and below the sampling limit), are fitted by least squares.

The frames are a sample of all the frames, not the best ones: this is the seeing of the session, not of its lucky moments.

Validation on synthetic data

Kolmogorov phase screens, a granulation-like object, known D / r0 and noise (cargo test -p helia-core seeing):

It loses sensitivity as D / r0 grows (above about 12 only the lowest frequencies carry any signal, and the estimate reads low); the report says so.

What limits it on real data

• The aperture must be right. It sets the cut-off, and r0 is D over the fitted D / r0. A wrong aperture gives a wrong r0 in proportion.
• Sampling. When the telescope's cut-off D / lambda is above 0.5 cycles per pixel (for 80 mm at 589 nm, a scale coarser than 0.76 arcsec per pixel) detail aliases. The synthetic test above shows aliasing does not by itself spoil the estimate, but the report says so as a caution.
• Tile size. The method assumes one transfer function per tile, that is a tile within the isoplanatic patch, a few arcseconds in daytime. A 64 pixel tile is that only at about 0.1 arcsec per pixel or finer. When a tile spans more than 15 arcseconds (a scale coarser than 0.23 arcsec per pixel) the report gives r0 as indicative and no grade, because it has not been validated there: the simulation above covers only the limiting case of independent patches, and real neighbouring patches are partly correlated.
• Few frames. Under 40 frames the report makes no r0.
• The object must not change during the burst. Granulation evolves in minutes; a capture of a minute or two is fine.

The grade

From r0 at 500 nm: 10 cm or more is excellent, 7 to 10 good, 4 to 7 fair, 2.5 to 4 poor, under 2.5 bad. These are day-time solar thresholds, not a standard.

The alignment point motion index

Tip and tilt of the whole frame (and tracking error and wind shake) move every patch of the Sun together, so they cancel when the shifts of two neighbouring alignment points are subtracted. What remains is the differential image motion, which turbulence causes (Scharmer and van Werkhoven 2010). Helia takes the tiles above, measures their shift in each of the sampled frames, subtracts neighbours (points up to one and a half grid spacings apart), removes each pair's mean difference (the static offset between the reference and the scene) and takes the RMS of what is left, per axis, in pixels and in arcseconds.

It is a relative index, for comparing captures taken through the same optics at the same scale. The patches look through the atmosphere at different field angles, so it depends on how high the turbulence is: for the same r0, turbulence at 5 km gives more differential motion than turbulence at the ground. On synthetic phase screens at D / r0 6, with tiles 16 pupil pixels apart at height:

and for turbulence at height it grows monotonically with D / r0 (3 to 12) with a log-log slope of 0.83, the 5/6 the theory gives for the RMS of a differential tilt.

On the real captures

The motion index of the whole of each capture (a sample of 120 frames; the fourth capture is too blocky to lock tiles):

The ranking agrees with the warp --ap-auto measures on a different principle (the spread of a point's shift about its frame's median).

The independent check on the real data: the quarter of the frames the stacker ranks sharpest (a half for the 100-frame capture) should read as better seeing than the quarter it ranks worst, in the same capture. cargo test -p helia-core --test seeing_real with HELIA_REAL_DATA set (Na D, green channel, 240 frames, the aperture taken as 80 mm, so only the comparison means anything):

Both rank the best frames as the better seeing on all three captures. On 15_32_09.avi the difference is small (a few per cent): the capture is short and steady.

The absolute values are not to be trusted on these captures: at 1.4 to 4.3 arcsec per pixel a 64 pixel tile is 90 to 280 arcsec wide, far beyond an isoplanatic patch, and Helia says so and gives no grade. The method is meant for scales near 0.1 arcsec per pixel; on a capture like that --seeing --aperture-mm is the one to use. There is no such capture in the test set yet, so the absolute r0 is validated on synthetic data only.

Using it

• To compare nights or settings with the motion index: same telescope, same scale, same filter.
• For r0 and D / r0: give the aperture, use a scale of 0.1 to 0.3 arcsec per pixel, and expect a grade.
• To choose how many frames to keep: the lower D / r0 is, the larger the share of good frames (Fried 1978). D / r0 under 4 makes a high keep share (25% or more) reasonable; above 10, keep fewer. Automatic keep share from this number is future work (#21, #25).

Sources

• von der Luehe, O. (1984), JOSA A 1(5), 510-519: estimating Fried's parameter from a time series of an arbitrary resolved object imaged through atmospheric turbulence.
• Scharmer, G. B. and van Werkhoven, T. I. M. (2010), A&A 513, A25: S-DIMM+ height characterisation of day-time seeing using solar granulation.
• Noll, R. J. (1976), JOSA 66, 207: Zernike polynomials and atmospheric turbulence.
• Immerkaer, J. (1996), Computer Vision and Image Understanding 64, 300: fast noise variance estimation.
• Lane, R. G., Glindemann, A. and Dainty, J. C. (1992), Waves in Random Media 2, 209: simulation of a Kolmogorov phase screen.