The experienced and mysterious audio engineer "NwAvGuy" [0] praised the virtue of using two gain stages and moving the volume control away from the first input to reduce Johnson noise in audio amplifier designs [1]. It's a good example of how the basic principle applies both to mundane audio and cutting-edge science: the system noise is dominated by the first amplifier stage. Adding some noise before the first stage significantly degrades signal-to-noise ratio, but adding the same noise after the first stage is often acceptable since the signal is much stronger now. To reduce noise, you move the noise-generating resistor away in an audio amp, or cryogenically cool the resistor in a radio telescope front-end.

> One of the big claims for many audiophile op amps is lower noise. The chip manufactures make a big deal about it and audiophiles, not surprisingly, have jumped on the bandwagon. But, in reality, it’s often the Johnson Noise that limits the noise performance of a headphone amp, not the op amps. Johnson Noise is, literally, self generated noise that’s present in any resistor. The larger the resistor value, the more noise you get. Many DIY headphone amp designs have the volume control at the input to the gain stage. And it’s, at the lowest, usually 10,000 ohms. By comparison the O2 has 274 ohms in series with the input. That’s a huge difference in Johnson Noise. The way volume controls work, the noise is typically worst at half volume where you have 5000 ohms in series with the source and 5000 ohms to ground. So, at typical volume settings, you get a fair amount of Johnson Noise from the volume control that’s amplified by whatever gain your amp has. That noise typically exceeds the op amp’s internal noise. If you put the volume control after the gain stage its Johnson Noise is no longer amplified. And, as a bonus, the volume control at lower settings now attenuates noise from the gain stage. For more, see O2 Circuit Description and Circuit Design.

> To put these numbers in perspective, referenced to the old 400 mV they’re –105.3 dBr and –108.2 dBr. On the exact same test, at half volume, the Mini3 had nearly 11 dB more noise and measured –94.5 and –97.5 dB. Noise of –113 dB below 1 volt is under 3 microvolts.

[0] https://spectrum.ieee.org/tech-history/silicon-revolution/nw...

[1] https://nwavguy.blogspot.com/2011/07/o2-headphone-amp.html