ITU Recommendation F.1487-0 defines methods for testing HF ionospheric paths for bandwidths up to 12 kHz. While ionospheric propagation can be complex, this document provides a starting point for the widely applicable basics.
It characterizes an HF channel with two parameters:
- multipath differential time delay, and
- Doppler spread.
The multipath differential time delay is the maximum difference in time of arrival between multipath components. Put another way, it's the length of the channel impulse response. When the length of a symbol is very long compared to this value, differential time delay has negligible effect on demodulation performance. The ITU document states that the differential time delay exceeds 5 ms 5% of the time. Given that most very weak signal communication modes will have symbols much longer than this, differential time delay is not likely a major detriment to performance in this case.
The other parameter, Doppler spread, quantifies how "spread out" the power spectra of the signal will become due to each path having a randomly changing Doppler shift. The worst environment described is "disturbed conditions at high latitudes", with a Doppler shift of 30 Hz.
If the objective is coherently detecting a very long symbol, Doppler spread may better be understood by its dual, coherence time. Coherence time $T_C$ can be defined as:
$$ T_c = {9 \over 16 \pi f_m} $$
where $f_m$ is the Doppler spread. This definition of coherence time gives the time where the correlation of the channel impulse response will be above 0.5. In other words, if one were to receive a signal at some time, and then an identical signal $T_c$ later, the correlation of those received signals will be on average 0.5.
For the worst case of 30 Hz, this works out to a coherence time of:
$$ {9 \over 16 \pi\ 30\:\mathrm{Hz}} = 5.97\:\mathrm{ms} $$
In other words, detecting a 6 ms symbol might work OK, but doubling the symbol length to 12 ms doesn't make the symbol twice as easy to detect since the second half of the symbol doesn't correlate perfectly with the first.
This is why polar paths are so challenging: Doppler spread can be extremely high.
WSPR-15 has a symbol rate of 0.1831 baud, whereas the ITU document gives a differential time delay of 0.5 Hz for "quiet conditions" at mid and low latitudes. From this we can already see the challenge: considered in the time domain, we can't count on an individual tone to maintain the same phase long enough that it won't start to cancel itself out. Or considered in the frequency domain, it's a challenge for WSPR-15 to resolve individual tones since the Doppler spread smears them together.
What can be done about it? I'm not entirely sure: I am after all answering my own question. But if the challenge is to establish communication even when slowing the symbol rate enough to approach the coherence time is insufficient, and transmitter power can't be increased, I'd guess the approach must be to take many shorter samples and add them noncoherently over a long time.
Consider the bad polar case where the coherence time is 6 ms: one could calculate an FFT every 6 ms and accumulate the magnitudes of each bin over a longer time. The Doppler spread means the received phase will be effectively random but a constant carrier, given enough time, will accumulate enough bias in the magnitude to become detectable above the noise. The short FFT duration will also mean the bins will be wider than necessary, which will introduce additional noise and require a wider tone spacing, but then if it was easy everyone would do it.
