Rates of Change
on February 13, 2017

One of our analysts asked if I had written a blog about how we calculate vertical speed, and I said “of course I did, ages ago”. There followed a 15-minute hunt for the subject and eventually I had to admit that it was missing, so here goes with a quick blog to plug a gap in the topics covered.

Rate of Change – The Mathematics

Don’t panic, we will keep it simple. For the people who remember differential calculus from their schooldays, we are going to compute the first differential with time:

image 1

OK – back in the real world, we are going to measure the change in height (or any other parameter) with time, and the easy way to do this is to subtract two consecutive measurements of height and divide by the time between them.

Rate of Change – Vertical Speed

Taking a sample of real data from a descent, here we have the differences with one sample subtracted from the previous sample one second previously, scaled to fpm.


image 2












The quantization steps of 240 fpm come about because this aircraft has a recorded altitude signal with 4ft resolution and so the difference signals can be -44ft = -2640fpm or -48ft = -2880 fpm. The problems of this approach are that the quantization is significant, the computation introduces a delay (effectively every value is an average over the preceding second, making it look half a second late). We first improve on this by taking the difference over a period of four seconds. For the moment in time when we want to know the height rate we look two seconds back in time and two seconds forward in time to create our four-second spread. This results in a smoother and not-delayed signal:


image 3













This period gives a good trade-off between noise and response to short term effects, and happens to be the same period as recorded on Airbus aircraft. The noise problem is worse on some older aircraft, and if you are operating Hercules, BAE 146, Boeing 747-200 or a Boeing 737 Classic on the older -6 frame, special noise reduction techniques are used. See our github repository for details.

Now in fact pilots are unable to control vertical speed of the aircraft rapidly, so we start by smoothing the initial altitude signal to remove transient spikes and this gives a signal which is reasonable to expect the crew to control. Here is the final version, with smoothing applied.


Image 4













So you can see that the short-term minimum of -3120 fpm was recomputed using a longer period at -2880 fpm and smoothed to a minimum of -2742 fpm.

(We apply the smoothing process to the altitude signal first, so that safety events relating to altitude are also less susceptible to transients. The effect on the differentiated signal is the same as though we had added this processing afterwards).

Of Roll and Pitch

Roll Rate and Pitch Rate are computed in similar ways, using periods of 2 seconds and without an additional stage of smoothing, because we are more concerned about short term issues than when looking at vertical speed.

Recorded Rates

Some aircraft record vertical speed or the roll rate or pitch rate, and often these are worse than the signals we compute. For example the quantization may be poor or the sample rate may be lower. For these reasons we sometimes force the system to compute a better value. This is done by renaming the recorded signal, for example as “Roll Rate Recorded”, so that it is available to view, but the computed Rate parameter is then used in computation and exceedance detection.





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