I gave relatively brief answers the last time, and didn't word things too clearly, let me try again.
So here is how I view the topic, tell me where I am right or wrong:
and let's ignore the ISO-invariant aspect for the time being just to simplify the discussion.
1. The optimum strategy is to collect as much light as possible, since that will maximize SNR based on photon noise.
Yes. It will also have no direct effect on the noise added by the camera. However, as the exposure goes up, the amount of shot noise also goes up, making the extra noise added by the camera less and less significant as a part of the total noise. The signal is the amount of light captured. SInce the shot noise goes up with the square root of the signal, the signal to noise ratio declines and the image looks less noisy, even though it has more total noise.
Ideally, we'd expose as far to the right as possible without clipping highlights,
OK.
at the lowest ISO possible (base ISO),
We use the lowest possible ISO if we have already got the maximum exposure that will not blow highlights.
collecting the maximum amount of light possible.
OK
This produces the best SNR overall.
Not quite. It produces the best ratio of signal to
shot noise. It may not produce the best ratio of signal to
total noise. In cases where the highest exposure we can get is below the theoretical exposure that would have been the hightest not to blow highlights, we can often reduce the ratio of signal to camera-added noise, without affecting the ratio of signal to shot noise, by increasing ISO.
2. if we are limited so that we can't achieve the goals of #1, for example, we can't slow the shutter speed low enough for whatever reason. Then as we increase ISO our signal is amplified to levels similar to #1, just below clipping, but at some ISO other than the base ISO.
Yes, as long as the ISO increase is implemented through analog gain. The signal is amplified. The shot noise is amplified by the same amount. Any noise added by the camera before the gain stage is also amplified by the same amount. So the signal to noise ratio for all that noise remains the same. However, any noise added by the camera after the gain stage is not amplified. So the SNR for that particular noise is reduced: the signal was amplified, but that portion of noise was not. When you reduce the SNR for one component of noise, while leaving the SNR for all other noise the same, the SNR for total noise is also reduced.
Let's assume ISO 100 for case #1 and ISO 400 for case #2.
OK.
Just to clarify, in case 2, the signal level before any amplification due to an ISO increase is 1/4 what it was in case 1, right?
Case #2 has higher absolute output read noise than case #1 because the pre-amplification read noise was multiplied by 4x, and in case #1 that same noise was multiplied by 1x.
OK
The post-amplification read noise is the same in both cases.
OK
The signal levels of both cases are essentially the same,
Post amplification? Yes. Pre-amplification? No.
just below highlight clipping. So in my mind, case 2 has lower output SNR because of the amplified read noise component, and I don't think you disagree with that statement.
It has a lower SNR because some of the noise wasn't increased by a factor of four but all of the signal was.
Now, I get that as you amplify the signal more with higher ISO settings, the post-amplification read noise stays constant.
Yes
And at higher ISO/gain settings fewer electrons are required for a given ADC value,
I'm not sure about this switch to electrons. Is it equivalent to voltage? If so, fewer electrons coming off the sensor, yes. Not fewer going to the ADC.
so the post-amp read noise (in ADC units) when expressed in terms of electrons becomes lower.
How does it diminish? It is the same as it always was. It wasn't amplified but it wasn't reduced either. It is smaller relative to the amplified signal.
The math part doesn't confuse me, what I'm struggling with is how that has any practical application. Ideally, I'd like to change the exposure to get closer to case 1, since that will increase my SNR. But if I can't, then I'm going to select the ISO that provides the highest signal levels without clipping, it wouldn't make sense to do anything else. The only scenario I can think of is if a "properly" exposed image in case 2 would actually be at ISO 200 according to the camera's metering, but I have enough headroom to push it to ISO 400 and ETTR even further. In that case the SNR would improve between ISO 200 and 400, is that what we're talking about?
I don't think so.
Let's try an example with numbers. Let's say that highlights will be blown at 420. Noise added before the gain stage is 4 and after the gains stage is 1.
In case A, where we get maximum exposure at ISO 100, we get a signal of 400. Shot noise is therefore 20 (Sqrt(400)).
Noise components are added in quadrature. So going into the ADC we have signal of 400 and noise of sqrt(20^2 + 4^2 + 1^2) = sqrt (400 + 16 +1) = 20.42. This gives us an SNR of 19.588.
In the second case, exposure is only 1/4 what it was in case 1, so signal is 100, and shot noise is 10. If we do not raise ISO to 400, what goes to the ADC is a signal of 100 and total noise of sqrt(10^2 + 4^2 + 1^2) = sqrt(100 + 16 + 1) = 10.82. This gives an SNR of 9.245. As expected, this is a bit under 1/2 of case 1.
Now, if we increase ISO to 400 through analog gain, then the signal is amplified to 400, the shot noise is amplified to 40, pre-gain camera-added noise is amplified to 16 and the post-gain noise is still 1. Total noise is sqrt(40^2 + 16^2 + 1^2) = sqrt(1600 + 256 + 1) = 9.282.
The SNR has increased because of the ISO increase, but not by very much. For an increase in ISO to have a significant effect on SNR, the camera-added noise needs to be close to as large as the shot noise or larger (only happens in very low exposures), and the noise added after the gain stage has to be a large component of all camera added noise (as it was for instance in older Canon systems).