They're measuring it by looking for phase differences in the received L-band (~2GHz) signals, rather than amplitude. That eliminates lots of noise. And they're looking for a particular pattern, which lets you get way below the noise floor. For example, the signal strength of the GNSS (GPS) signal itself might be -125 dBm, while the noise level is -110 dBm [2]. That means the signal is 10^-12 _milliwatts_, and the noise is about 30 times larger. But by looking for a pattern the receiver gets a 43 dB processing boost, putting the effective signal well above the noise.
[1] https://www.swpc.noaa.gov/phenomena/total-electron-content
>> They're measuring it by looking for phase differences in the received L-band (~2GHz) signals
The "L-Band signals" are GNSS signals, for example GPS L1 and L2, which use a carrier wavelength of 1575.42 MHz and 1227.6 MHz, respectively. Both L1 and L2 signals are emitted at the same time, but experience differing levels of delay in the ionosphere during their journey to the receiver. The delay is a function of total electron content (TEC) in the ionosphere and the frequency of the carrier wavelength. Since we already know precisely how carrier frequency affects the ionospheric delay, comparing the delay between L1 and L2 signals allows us to calculate the TEC along the signal path.
Another way to think of it is: we have an equation for signal path delay with two unknowns (TEC, freq). Except, it is only one unknown (TEC). Use two signals to solve simultaneously for this unknown. Use additional signals (like L5) to reduce your error and check your variance.
edit: The atmosphere between the surface and the ionosphere forms a natural low-pass filter as well. I imagine typical ocean waves as seen by us are way too high-frequency to make it up to the ionosphere.
There are other, natural disturbances in the ionosphere, such as traveling planetary waves[2], but they have a significantly longer wavelength. As such the paper[3] mentions filtering them out using a high-pass filter.
In the paper they show some preliminary results trying to invert the parameters in order to estimate the height of the tsunami based on the measured ionosphere disturbance based on synthetic data, and the baseline amplitude is 10cm (4 inches), which the model comes quite close to.
[1]: https://earthobservatory.nasa.gov/blogs/fromthefield/2014/04...
[2]: https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/201...
[3]: https://link.springer.com/article/10.1007/s10291-022-01365-6