OK, I sort-of answered my own question originally posted below. I just ran a numerical simulation which calculates the rate of change of path length from station A to the plane and the plane to station B using simple three-dimensional geometry. It then uses the bistatic doppler equation to find the instantaneous doppler shift at +-30s from the crossing point. As the plane height is potentially significant, at least on shorter paths, I took the height of the plane above the direct line from station A to station B – which obviously goes through the earth to the depth of the earth curvature on the path – along with the angle of the plane to the direct path and the speed of the plane, and found that the rate of change of the rate of change of the path length is indeed just about constant over a 200-600km path with an offset of 10-15km from the direct path for constant plane speed when the angle to the path is around 90 degrees. At sharper angles, it all gets a bit asymmetric and leads to curved tracks on the waterfall, particularly where the angle is less than 30 degrees or more than 160, and when the crossing point is less than 15% from either end of the direct path.

Magically, it works out to be within a few percent of the measured value of the rate of change of frequency on a real A/S reflection from a plane using the velocity vector reported on Flightradar24. For the example below, it gives 4.65Hz/sec, against a measured 4.9+-0.2 which is probably within the error margins for speed and heading and location of the plane.

Now all I have to do it turn the numerical model into an analytical one, so I can enter the frequency, path length, distance from A to B, distance of A to the plane, angle of the plane to the direct path, plane height and speed, and it will tell me the expected rate of change of frequency.

Next level is to use the rate of change of doppler and the zero-shift time to work out unambiguously which plane caused which reflection when there are 3-5 planes in the zone as reported by Airscout, and try to extract from that historic data some way to prioritise which path to try next, when there are five potential DX stations to work and ten possible planes in the next fifteen minutes. It would be good to know that the 21:35 crossing of the Reykjavik flight to a station in GM is likely to be better than the 21:34 Bratislava flight for a path into DL.
On 29/01/2017 03:00, Neil wrote:
>
> Tonight, I watched as a plane crossed the patch between me and GB3ZME on 13cm. Plane’s path was at about 70 degrees to the direct line between me and ZME, and was within 2km of the midpoint of the path. Plane was at a constant height of 8530m and travelling at 169m/s. I saw the reflection as usual, with the doppler changing by 4.9Hz/sec over the 21.5 seconds of the reflection. That inspired me to do the trigonometry and see if I could understand why the rate of change of frequency was constant and had that particular value.
>
> Now back in the day I remember doing something at college for the doppler effect from a moving mirror. So long as you aren’t at relativistic speeds it looks simple enough if you treat the plane as a mirror travelling perpendicular to the line of sight path between you and the source. The equation looks like:
>
> where v is the plane’s ground speed, c is the speed of light and α is 90 degrees minus the angle between the plane and the direct line of sight. Now v^2/c^2 is very very close to zero for a plane at 328 kts, so the doppler shift is very nearly just (2fv/c) cos(α) where f is frequency (in the paper that equation comes from, the author used omega instead of nu or f, but I *think* he meant frequency not angular frequency as it comes from the photon energy with Planck’s constant)
>
> So, working out using the full equation for 10 seconds either side of the line of sight, ignoring the height of the plane and the fact it isn’t at 90 degrees, the rate of change of doppler shift should be a constant 2.7Hz/sec. That is a lot different from the measured 4.9Hz/sec. I took the plane’s speed from Flightradar24, so it is possible that it was wrong, for example if it was really doing about 440kts, the rate of change would be just about right.
>
> However, it would be odd for the site to be that far wrong. I am pretty sure the component of velocity of the “mirror” along the line between me and the source must cancel out for the two halves of the path, but perhaps the angle between the source-plane and plane-me paths matters.
> Can anyone point me at a paper or reference which will fill in the missing elements of the height of the plane and its angle to the direct path and put me out of my misery please? All the papers on passive bistatic/multistatic radar seem to be behind paywalls or only talking about the path-loss implications of the bistatic radar equation and forward-scatter, and don’t have anything quantitative about the doppler shift.
>
> Neil G4DBN
>
> https://arxiv.org/pdf/1207.0998.pdf
>
> http://www.vk3hz.net/aep/vk7mo_2000.pdf
>
> http://www.nitehawk.com/w3sz/W3SZ-NEW-AirCraftScatterNEWS2014.pdf
>
>