Sony A7S Concentric Banding and Histogram Gaps

I have identified a possible mechanism that causes a "bullseye" concentric banding on the Sony A7S and which may also apply to other Sony A7 series cameras.

I need to point out right at the outset that the in-camera vignetting correction ("shading compensation") is switched off.

In astrophotography the desire to tease out very faint data means very extreme processing is applied to the data. Master flat frames are used for vignetting correction and I first noticed a coloured rings problem when dividing a single flat frame by a master flat:

Using RawDigger I also noticed that the histogram of the single flat file had regular histogram dips (highlighted in yellow here):

Note that I am not talking about the infamous lossy Sony data compression that leads to all the odd values being omitted at low data values.

I took the green channel from the coloured rings image:

I then highlighted the data values corresponding to the histogram dips:

The correlation between the two was remarkable. Whatever was causing the dips in the histogram probably also caused the concentric rings. A possible explanatory hypothesis is that the 14bit integer data is read from the Sony sensor and then (for some obscure reason) it is then multiplied by a factor of approximately 1.026 and stored back as integers. This would lead to missing values in the histogram in approximately the observed places i.e. with a period of approx 38. When the lossless compression is applied it leads to a dip in the histogram instead of an actual missing value.

The scaling would also lead to a "sawtooth" being imposed on a graph of data values, again with a period of 38. This sawtooth explains the concentric ring pattern. For low data values (i.e. underexposed parts of the image) and with low ISO, the shot noise (plus read noise) is unable to adequately dither this sawtooth and so we end up with coloured rings.

Here is an image I took at ISO 100 through a white diffuser with 4-5 stops of underexposure. The exposure was pushed in post-processing and colour saturation applied to bring out the coloured rings:

Of course this is a contrived example and similar effects have been noticed in contrived examples in the past:
https://www.dpreview.com/forums/thread/3588565

However, here is a real image where the same banding is observed:
https://diglloyd.com/blog/2014/20141109_2120-SonyA7R-ZeissLoxia35f2-aseries-SoloBristleconeDuskEarthShadow.html

To conclusively demonstrate that a 1.026 scaling can actually cause this problem and that it not dithered away by the shot noise I wrote a simulation that does the following:
1) Creates a simple vignetting function
2) Creates integer data according to this vignetting function and with the red and green channels have lower intensity - consistent with the white balance.
3) Shot noise and read noise (assuming a gain of 10e/data_value which is about right for the Sony A7S at ISO 100) are added to the data before conversion to integer.
4) The resulting integer values are multiplied by 1.026 and re-stored as integers (nearest integer).

During "post processing" of the data it is scaled up and then white balance is applied (2.45 : 1.0 : 1.65 for the A7S). Saturation is then increased. Here is a typical result:

If the above step 4 (scaling by 1.026) is omitted then no coloured banding takes place. This again shows the scaling to be a potential causal mechanism.

Assuming my hypothesis is correct then I still have some unanswered questions:

• The A7Sii (and some other newer A7 series cameras) have the ability to produce uncompressed raw 14bit files. Do these files show regular missing histogram values?
• The A7S shows histogram dips with a period of 38. Do other A7 cameras have regular histogram dips and/or missing values and what is their period?
• Does anyone else have example images that may be showing this problem?  Preferably with shading compensation switched off.

Mark