FPA correction during Cold Weather approaches
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However the drawings aren't representative of what's actually happening
I've tested my way of calculating with a specific example and compared it to the result when using the formula BuzzBox gave and although the difference is small indeed, it is there. LOWW RWY 34, 2400ft height on a 3° GP yields 3,27° vs 3,22°.
The reason I'm interested in getting the most accurate way of calculating this whole bit is because I'd like to write a small app that does those corrections for me. So all inputs are highly welcome ...
Only half a speed-brake
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What we have is an approximation assuming a linear geometric flight path from (linearly) corrected altitudes.
But that's not what's actually happening. With corrected altitude the plane is physically flying at H, not (H+dh) as in the graphs. What has changed is the air density, and the density gradient is nonlinear.
Plus the gradient itself changes with non-ISA conditions, while an uncorrected FMS will always assume ISA pressure gradients.
So basically the plane is flying a non-linear "barometric" flight path using the "wrong" (ISA) pressure gradient. Hence a geometric interpretation based on straight ratios can only yield approximations.
Here's a table from Transport Canada showing true descent angles for some non-ISA temperatures (sea level):
What we want to find is the "inverse" of the above, but I think you will find our simple linear geometric models don't quite "match up" to the actual angles.
But that's not what's actually happening. With corrected altitude the plane is physically flying at H, not (H+dh) as in the graphs. What has changed is the air density, and the density gradient is nonlinear.
Plus the gradient itself changes with non-ISA conditions, while an uncorrected FMS will always assume ISA pressure gradients.
So basically the plane is flying a non-linear "barometric" flight path using the "wrong" (ISA) pressure gradient. Hence a geometric interpretation based on straight ratios can only yield approximations.
Here's a table from Transport Canada showing true descent angles for some non-ISA temperatures (sea level):
What we want to find is the "inverse" of the above, but I think you will find our simple linear geometric models don't quite "match up" to the actual angles.
Only half a speed-brake
Kaz: Easy on yourself, use distance to THR and TCH of 50'. It follows the way the approaches are designed and actually flown - so not an oversimplification at all.
FPA.corrected = arctan [ (x+y) / dist.to.threshold ] where
y = FAF.alttitude - (THR.elevation+50ft)
x = ALT.correction.over.FAF /properly calculated from FAFalt-ADelev/
FPA.corrected = arctan [ (x+y) / dist.to.threshold ] where
y = FAF.alttitude - (THR.elevation+50ft)
x = ALT.correction.over.FAF /properly calculated from FAFalt-ADelev/
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Thx for the inputs!
@peekay4
As far as the density gradient not being linear is concerned, there's probably no way around that approximation. After all, the ICAO doc 8168 seems to create its height correction tables on an assumed linearity as well, so I guess this is will be close enough for a corrected FPA too.
I still would think that even when some of these calculations are based on approximations they still should be more accurate than a simple rule of thumb.
Your "What we want to find is the "inverse" of the above" got me thinking though (tell me if that's closer to what you suggested): if I want to correct the (wrong) height to reach the charted height, I've got to tackle the problem "from below" (as in, how much do I need to increase the altitude on my altimeter to actually fly the charted height), so instead of putting the dh correction on top of H, I've changed my drawings to add dh to the pressure altitude so as to reach H after correction. And after adding the threshold-to-TD correction I get similar results to those from the more simple formula suggested earlier.
@FlightDetent
Isn't going all the way to TD the more clean way to calculate this? Going all the way down to height 0 eliminates the need for another correction at threshold height (again, based on approximations, however small).
Funny thing about these FPA corrections is that our company has removed the table from the QRH, apparently after someone had messed up after using it. We're now to make the corrections to the FPA "as appropriate"
@peekay4
As far as the density gradient not being linear is concerned, there's probably no way around that approximation. After all, the ICAO doc 8168 seems to create its height correction tables on an assumed linearity as well, so I guess this is will be close enough for a corrected FPA too.
I still would think that even when some of these calculations are based on approximations they still should be more accurate than a simple rule of thumb.
Your "What we want to find is the "inverse" of the above" got me thinking though (tell me if that's closer to what you suggested): if I want to correct the (wrong) height to reach the charted height, I've got to tackle the problem "from below" (as in, how much do I need to increase the altitude on my altimeter to actually fly the charted height), so instead of putting the dh correction on top of H, I've changed my drawings to add dh to the pressure altitude so as to reach H after correction. And after adding the threshold-to-TD correction I get similar results to those from the more simple formula suggested earlier.
@FlightDetent
Isn't going all the way to TD the more clean way to calculate this? Going all the way down to height 0 eliminates the need for another correction at threshold height (again, based on approximations, however small).
Funny thing about these FPA corrections is that our company has removed the table from the QRH, apparently after someone had messed up after using it. We're now to make the corrections to the FPA "as appropriate"
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Kazume in the 8168 correction formulas the temperature is assumed to be linear with height, but not the pressure / density factor (which it is essentially calculating).
Hence the formulas aren't in the form y = mx + b, and if you look at the correction tables, the resulting values are non-linear.
I think the idea is to make a conservative estimate and leave it at that.
Hence the formulas aren't in the form y = mx + b, and if you look at the correction tables, the resulting values are non-linear.
I think the idea is to make a conservative estimate and leave it at that.
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As far as the FPA is concerned, does that mean - in your opinion - that it is better to guess the corrected value rather than calculate it using a formula (with linearity assumption)?
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Ideally neither. The FMS should handle cold weather compensation, including flight path corrections as necessary. In fact here in Canada it's required that all new or updated FMSs have cold weather compensation built-in.
As such there is an accepted ICAO-blessed algorithm used by FMSs to compute these corrections, published as part of the PBN RNP standards. Unfortunately I don't have access to these standards to read what they say, but may be you can get them from your company. The algorithm might be a simple linear approximation, but maybe not.
As such there is an accepted ICAO-blessed algorithm used by FMSs to compute these corrections, published as part of the PBN RNP standards. Unfortunately I don't have access to these standards to read what they say, but may be you can get them from your company. The algorithm might be a simple linear approximation, but maybe not.
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I'll see what I can get from my company (the best I got so far is "it's really complicated" and getting redirected to someone else). I'm finding this a very intriguing topic. The deeper you dig, the more you realise it seems to be a really deep pit :P
