Looking for a math wiz!

[Chad Speer]

Okay, I know I've seen a lot of engineers and technical folks on here.
I have a complex math problem relating to the classic wind triangle
that I posted on sci.math and received little response. I don't know
if they're stumped or just not interested. :-)

Here is a copy of my original post and the only useful response I
received. Anyone have a solution?

(For the controllers here, this is an enhancement we are trying to add
to the Falcon program that centers will see next year, which was
developed by a controller here at ZKC.)


Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC



************************************************** *
************************************************** *

My original post:
*****
I am helping someone with a program that estimates wind speed and
direction using radar data from aircraft. I need help finding a
formula that can determine the wind speed and direction when given the
following information for multiple aircraft:

direction of travel
speed across the ground
speed through the air

The direction of travel and the speed across the ground are taken from
the radar data. The speed through the air is taken from the pilot's
flight plan. We're air traffic controllers trying to improve our
training tools, so we get access to all the goodies.

I know that with information from just one aircraft, the possibilities
are endless for the wind speed and direction. I think it is possible
to use the same data from two or more aircraft to determine the wind
speed and direction.

I thought I could come up with a formula to solve this, but the need to
reference everything to north in order to achieve actual directions
instead of just angles took it way above my head.

Basically, you are given the lengths of two adjacent sides of many
different triangles. You also know the angle of one of those sides
(aircraft direction) with respect to a known reference (north). The
properties common to all of the triangles, which are unknown, are the
length of the third side (wind speed) and the angle of that third side
(wind direction) with respect to a known reference (north).

If I have not described this well enough, I can upload some diagrams to
a web page to simplify the problem. I would really be grateful if
someone is able to solve this!
*****



A useful response:
*****
Interesting problem.

For the single aircraft, case, let:
Wd, Ws be wind direction and speed.
Ad, As be aircraft direction and speed (relative to the air, not the
ground).
Gd, Gs be aircraft direction and speed (relative to the ground).

Representing each vector as a 2-tuple of (direction, magnitude) gives:
(Gd, Gs) = (Ad, As) + (Wd, Ws)

The unknowns are Ad, Ws, and Wd. (Note that As is known due to filed
information, i.e. a pilot will know how fast his aircraft cruises.)

Splitting into x- and y-components will result in 2 equations with 3
unknowns. I agree that there are infinitely many solutions in the
single-aircraft case.

For the two-aircraft case (and I'll just suffix with 1 and 2), we have:
(Gd1, Gs1) = (Ad1, As1) + (Wd, Ws)
(Gd2, Gs2) = (Ad2, As2) + (Wd, Ws)

where of course we assume that the wind affecting each aircraft (since
they are presumably not too many tens of miles apart) is the same.
Breaking into x- and y-components leads to 4 equations and 4 unknowns
(Ad1, Ad2, Wd, Ws). The 2-aircraft case probably has a unique solution.
We have:

Gs1 * cos(Gd1) = As1 * cos(ad1) + Ws * cos(Wd)
Gs1 * sin(Gd1) = As1 * sin(ad1) + Ws * sin(Wd)
Gs2 * cos(Gd2) = As2 * cos(ad2) + Ws * cos(Wd)
Gs2 * sin(Gd2) = As2 * sin(ad2) + Ws * sin(Wd)

One can probably square equations, add, and make use of the
relationship that sin^2(x) + cos^2(x) = 1. I'm not sure what the form
of the solution would be. I'm too lazy to work it out. It will be
messy.

I believe at first glance that the 2-aircraft case has a single unique
solution (4 equations, 4 unknowns).

However, moving on to more than 2 aircraft ...

If there are more than 2 aircraft, the system is "overspecified" (there
is a mathematical term for this, but it has been so long ...). You
probably want a way to pick a single best solution among the infinitely
many, assuming that you have some "noise" in the data.

The style of solution you want is probably about the same as a
"least-squares" solution to a system of linear equations, i.e.

http://www.mathresource.iitb.ac.in/linear%20algebra/mainchapter8.5.html

I would need to do some thinking about how to phrase this problem as a
least-squares problem (the sines and cosines above put doubts in my
head), but there is probably a way to do it. So, out of a group of
data for at least 2 aircraft, you should be able to grind out a
solution that is unique according to some constraints and assumptions.

To summarize my thoughts:

a)1 aircraft --> system not solvable.
b)2 aircraft --> system has one solution, but I'm too lazy to do the
algebra.
c)3 or more aircraft --> system is overspecified, and some least
squares approach should give a solution.

One more thought: I've spent a fair amount of time in little Cessnas.
It has been my experience that wind direction and speed won't vary too
much over distance, but may vary EXTREMELY with altitude. I've flown
on days when the winds at 3,000 feet were 15 knots and the winds at
6,000 feet were 50 knots (or at least this is my memory). I accept the
assumption that winds affecting aircraft at the same altitude that are
within maybe 20NM of each other are about the same. But I do not
accept the assumption that winds at different altitudes are similar --
my experience says otherwise. If you agree, this adds yet another
dimension to the problem.

(BTW, near the end of my student training, I used to like to fly in
heavy crosswinds to practice technique. We have an airport about 20NM
North of our local airport with a roughly perpendicular runway, and I
discovered that if the wind was blowing straight down the runway here I
could get a perfect crosswind to practice with just by going N and
using the other airport. The surface winds were always about the same
at both airports. That is why I'm comfortable with the assumption that
winds don't vary much over relative short distances.)

Good problem.

I am not a mathematician. I hope others can add more insight.
*****

[d&tm]

"Chad Speer" > wrote in message
ups.com...
> Okay, I know I've seen a lot of engineers and technical folks on here.
> I have a complex math problem relating to the classic wind triangle
> that I posted on sci.math and received little response. I don't know
> if they're stumped or just not interested. :-)
>
> Here is a copy of my original post and the only useful response I
> received. Anyone have a solution?
>
> (For the controllers here, this is an enhancement we are trying to add
> to the Falcon program that centers will see next year, which was
> developed by a controller here at ZKC.)
>
>
> Chad Speer
> PP-ASEL, IA
> ATCS, Kansas City ARTCC
>
>
>
> ************************************************** *
> ************************************************** *
>
> My original post:
> *****
> I am helping someone with a program that estimates wind speed and
> direction using radar data from aircraft. I need help finding a
> formula that can determine the wind speed and direction when given the
> following information for multiple aircraft:
>
> direction of travel
> speed across the ground
> speed through the air
>
> The direction of travel and the speed across the ground are taken from
> the radar data. The speed through the air is taken from the pilot's
> flight plan. We're air traffic controllers trying to improve our
> training tools, so we get access to all the goodies.
>
> I know that with information from just one aircraft, the possibilities
> are endless for the wind speed and direction. I think it is possible
> to use the same data from two or more aircraft to determine the wind
> speed and direction.
>
> I thought I could come up with a formula to solve this, but the need to
> reference everything to north in order to achieve actual directions
> instead of just angles took it way above my head.
>
>snip

Chad
if you know HDG ( ie where you are pointing), GS and TAS then there is only
1 possibility for the wind speed and direction. these can be calculated
from the cosine rule. If you know the cosine rule and the sine rule for
triangles you can calculate a lot of things.
to apply both of these rules draw yourself a little triangle and mark the
sides small a,b and c.
then mark the angles capital A,B and C where angle A is opposite side a
and angle B is opposite side b.
cosine rule a^2 = b^2 +c^2 - 2bc cos( A)
sine rule a/sin A = b/sin B = c /sin C

in the case of the NAV triangle

WS= SQRT( GS^2+TAS^2 =2*GS*TAS*cos(HDG-TR))
where WS= windspeed
GS = ground speed
TAS = airspeed
HDG = heading(where you are pointing)
TR = track ( where you are going)

If you want I can email you an excel spreadsheet that has this already
coded. you just enter your TAS, GS and HDG and it will give you the WS and
Wind direction.

Terry
PPL

[Stefan]

d&tm schrieb:

> if you know HDG ( ie where you are pointing), GS and TAS then there is only
> 1 possibility for the wind speed and direction.

Actually, there are two.

But the question was a different one. It has already been answered
pretty well, all what remains is to do the dirty work and shuffling some
formulas.

Stefan

[Chad Speer]

Terry - thanks for the reply, but heading is not known.

Stefan - we need to be able to plug these known values into a formula
and kick out a result. Assuming the original responder was on the
right track, I still don't know what to do with his suggestion. Any
ideas on that? I'm not lazy, this just went over my head a long time
ago. :-)

Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC

[Kev]

Stefan wrote:
> d&tm schrieb:
>
> > if you know HDG ( ie where you are pointing), GS and TAS then there is only
> > 1 possibility for the wind speed and direction.
>
> Actually, there are two.

Eh? Not to sidetrack the thread too much, but how could there be two
wind answers?

For example on the E-6B, to solve this, you'd set TRACK up, grommet
over GS, and then look for where your TAS arc meets your drift
correction angle (HDG-TRACK). The vector back to the grommet is the
single direction and speed for the wind.

Thanks, Kev

[d&tm]

"Stefan" > wrote in message
...
> d&tm schrieb:
>
> > if you know HDG ( ie where you are pointing), GS and TAS then there is
only
> > 1 possibility for the wind speed and direction.
>
> Actually, there are two.

I give up, can you please explain how there can be 2 ?

[d&tm]

"Chad Speer" > wrote in message
oups.com...
> Terry - thanks for the reply, but heading is not known.
>
> Stefan - we need to be able to plug these known values into a formula
> and kick out a result. Assuming the original responder was on the
> right track, I still don't know what to do with his suggestion. Any
> ideas on that? I'm not lazy, this just went over my head a long time
> ago. :-)
>
Chad, sorry misread the question. But I like a challenge so I had another
go.
Firstly I dont believe the equations given by the original poster are
correct. if you apply the cosine rule to the wind triangle for 2 aircraft
you get the following 2 equations.

TAS1^2=WS^2+GS1^2-2WSGS1COS(180-ABS(WD-TR1)
TAS2^2=WS^2+GS2^2-2WSGS2COS(180-ABS(WD-TR2)

WHERE TAS1 AND TAS2 ARE TRUE AIRSPEEDS FOR AIRCRAFT 1 AND 2
WS = WIND SPEED
WD =WIND DIRECTION IN DEGREES MAG
TR1 AND TR2 ARE TRACKS MAGNETIC FOR AIRCRAFT 1 AND 2
GS1 AND GS2 ARE GROUND SPEEDS FOR AIRCRAFT 1 AND 2.
(180-ABS(WD-TR) WILL GIVE YOU THE ACTUAL ANGLE BETWEEN WIND DIRECTION AND
TRACK.

Now we have 2 equations with 2 unknowns ( WS and WD) which should be
solvable but I am stuggling to get it out. However I did a little exercise
which suggests the end result may not be very useful. because small errors
in the ground speeds will cause very large errors in the calculated wind
speed.

By subtracting the 2 equations above you get
WS=(GS2^2-GS1^2-TAS2^2+TAS1^2)/(2(GS2COS(180-ABS(WD-TR2)-GS1COS(180-ABS(WD-T
R1))
What I then did was calculate the ground speeds for 2 aircraft for a known
wind of 20 kts from 220 M

aircraft 1 aircraft 2
TAS 120 100
GS 111.7 91.4
TR 290 150

If I then use my above equation for WS after setting wind direction to 220,
I get 20 kts as expected. However if I round off the GS to 112 and 91 kts
the Wind speed changes from 20 to 9.5kts which suggests the ground speeds
have to be super accurate to get anywhere near the right wind speed.. Given
that also you are taking the TAS from a flight plan, which will vary with
density altitude and RPM setting etc. suggests your objective is going to be
rather difficult to achieve in practice.

Hope this helps and good luck. If you come up with the solution I would love
to see it.
terry

[d&tm]

"d&tm" > wrote in message
...
>
> "Chad Speer" > wrote in message
> oups.com...
> > Terry - thanks for the reply, but heading is not known.
> >
> > Stefan - we need to be able to plug these known values into a formula
> > and kick out a result. Assuming the original responder was on the
> > right track, I still don't know what to do with his suggestion. Any
> > ideas on that? I'm not lazy, this just went over my head a long time
> > ago. :-)
> >
> Chad, sorry misread the question. But I like a challenge so I had another
> go.
> Firstly I dont believe the equations given by the original poster are
> correct. if you apply the cosine rule to the wind triangle for 2 aircraft
> you get the following 2 equations.
>
> TAS1^2=WS^2+GS1^2-2WSGS1COS(180-ABS(WD-TR1)
> TAS2^2=WS^2+GS2^2-2WSGS2COS(180-ABS(WD-TR2)
>
> WHERE TAS1 AND TAS2 ARE TRUE AIRSPEEDS FOR AIRCRAFT 1 AND 2
> WS = WIND SPEED
> WD =WIND DIRECTION IN DEGREES MAG
> TR1 AND TR2 ARE TRACKS MAGNETIC FOR AIRCRAFT 1 AND 2
> GS1 AND GS2 ARE GROUND SPEEDS FOR AIRCRAFT 1 AND 2.
> (180-ABS(WD-TR) WILL GIVE YOU THE ACTUAL ANGLE BETWEEN WIND DIRECTION AND
> TRACK.
>
> Now we have 2 equations with 2 unknowns ( WS and WD) which should be
> solvable but I am stuggling to get it out. However I did a little
exercise
> which suggests the end result may not be very useful. because small errors
> in the ground speeds will cause very large errors in the calculated wind
> speed.
>
> By subtracting the 2 equations above you get
>
WS=(GS2^2-GS1^2-TAS2^2+TAS1^2)/(2(GS2COS(180-ABS(WD-TR2)-GS1COS(180-ABS(WD-T
> R1))
> What I then did was calculate the ground speeds for 2 aircraft for a
known
> wind of 20 kts from 220 M
>
> aircraft 1 aircraft 2
> TAS 120 100
> GS 111.7 91.4
> TR 290 150
>
> If I then use my above equation for WS after setting wind direction to
220,
> I get 20 kts as expected. However if I round off the GS to 112 and 91 kts
> the Wind speed changes from 20 to 9.5kts which suggests the ground speeds
> have to be super accurate to get anywhere near the right wind speed..
Given
> that also you are taking the TAS from a flight plan, which will vary with
> density altitude and RPM setting etc. suggests your objective is going to
be
> rather difficult to achieve in practice.
>
> Hope this helps and good luck. If you come up with the solution I would
love
> to see it.
> terry

should have mentioned
in 180- ABS( WD-TR)
ABS is absolute , ignore negatives.
terry

[Denny]

Lots of fun to play with the formulas... But, the fly in the ointment
here is that the airspeed filed by the pilot is a canned number and has
nothing to do with reality... I fly IFR, I file the same airspeed - 130
kt - for all flights regardless of load, altitude, and what the power
setting winds up to be... If the actual airspeed in your problem is
within 15% of the filed number you will be lucky...

denny

[Stefan]

Kev schrieb:

> Eh? Not to sidetrack the thread too much, but how could there be two
> wind answers?

Mathematically: There are always two square roots which solve the
equation: A positive and a negative.

Physically: If you only know GS, TAS and HDG, then you don't know
whether the wind blows from the let or from the right. (If you also know
the track, then of course there's only one solution.)

Stefan

[Michael Ware]

"d&tm" > wrote in message
...
>
> "Stefan" > wrote in message
> ...
>> d&tm schrieb:
>>
>> > if you know HDG ( ie where you are pointing), GS and TAS then there is
> only
>> > 1 possibility for the wind speed and direction.
>>
>> Actually, there are two.
>
> I give up, can you please explain how there can be 2 ?
>

There are two possible situations for the wind correction. You do not know
the direction of the correction for wind ( i.e. is the plane crabbing left
or right to compensate for x-wind) you only know the magnitude (wind speed).
Think of the triangle that is formed by vectors on the e6b. Without the
direction, you have an ambiguous answer, looks like two similar triangles, a
lefty and a righty.
Someone else could probably explain this better, that's the basic idea.

Mike

[Aluckyguess]

Can you email that to me also.




"d&tm" > wrote in message
...
>
> "Chad Speer" > wrote in message
> ups.com...
>> Okay, I know I've seen a lot of engineers and technical folks on here.
>> I have a complex math problem relating to the classic wind triangle
>> that I posted on sci.math and received little response. I don't know
>> if they're stumped or just not interested. :-)
>>
>> Here is a copy of my original post and the only useful response I
>> received. Anyone have a solution?
>>
>> (For the controllers here, this is an enhancement we are trying to add
>> to the Falcon program that centers will see next year, which was
>> developed by a controller here at ZKC.)
>>
>>
>> Chad Speer
>> PP-ASEL, IA
>> ATCS, Kansas City ARTCC
>>
>>
>>
>> ************************************************** *
>> ************************************************** *
>>
>> My original post:
>> *****
>> I am helping someone with a program that estimates wind speed and
>> direction using radar data from aircraft. I need help finding a
>> formula that can determine the wind speed and direction when given the
>> following information for multiple aircraft:
>>
>> direction of travel
>> speed across the ground
>> speed through the air
>>
>> The direction of travel and the speed across the ground are taken from
>> the radar data. The speed through the air is taken from the pilot's
>> flight plan. We're air traffic controllers trying to improve our
>> training tools, so we get access to all the goodies.
>>
>> I know that with information from just one aircraft, the possibilities
>> are endless for the wind speed and direction. I think it is possible
>> to use the same data from two or more aircraft to determine the wind
>> speed and direction.
>>
>> I thought I could come up with a formula to solve this, but the need to
>> reference everything to north in order to achieve actual directions
>> instead of just angles took it way above my head.
>>
>>snip
>
> Chad
> if you know HDG ( ie where you are pointing), GS and TAS then there is
> only
> 1 possibility for the wind speed and direction. these can be calculated
> from the cosine rule. If you know the cosine rule and the sine rule for
> triangles you can calculate a lot of things.
> to apply both of these rules draw yourself a little triangle and mark the
> sides small a,b and c.
> then mark the angles capital A,B and C where angle A is opposite side a
> and angle B is opposite side b.
> cosine rule a^2 = b^2 +c^2 - 2bc cos( A)
> sine rule a/sin A = b/sin B = c /sin C
>
> in the case of the NAV triangle
>
> WS= SQRT( GS^2+TAS^2 =2*GS*TAS*cos(HDG-TR))
> where WS= windspeed
> GS = ground speed
> TAS = airspeed
> HDG = heading(where you are pointing)
> TR = track ( where you are going)
>
> If you want I can email you an excel spreadsheet that has this already
> coded. you just enter your TAS, GS and HDG and it will give you the WS
> and
> Wind direction.
>
> Terry
> PPL
>
>
>
>

[Orval Fairbairn]

In article . com>,
"Denny" > wrote:

> Lots of fun to play with the formulas... But, the fly in the ointment
> here is that the airspeed filed by the pilot is a canned number and has
> nothing to do with reality... I fly IFR, I file the same airspeed - 130
> kt - for all flights regardless of load, altitude, and what the power
> setting winds up to be... If the actual airspeed in your problem is
> within 15% of the filed number you will be lucky...
>
> denny

Another problem is that the radar data provide only the component of the
wind affecting the direction of flight -- there is no means of getting
the sideslip component without getting the aircraft's heading info --
presumably inaccessible.

[d&tm]

"Stefan" > wrote in message
...
> Kev schrieb:
>
> > Eh? Not to sidetrack the thread too much, but how could there be two
> > wind answers?
>
> Mathematically: There are always two square roots which solve the
> equation: A positive and a negative.
>
> Physically: If you only know GS, TAS and HDG, then you don't know
> whether the wind blows from the let or from the right. (If you also know
> the track, then of course there's only one solution.)
>
Of course you are correct, I meant to include Track in the knowns, (
afterall if you dont know the track whether you are the pilot or the ATC you
are really in trouble.)
terry

[[email protected]]

[followups set to r.a.piloting]

In rec.aviation.piloting Chad Speer > wrote:
> Anyone have a solution?

I don't know what the solution is but I can certainly admire the
problem. :)

Actually, I think I pretty much agree with the one response you posted,
in part:

>> a)1 aircraft --> system not solvable.
>> b)2 aircraft --> system has one solution, but I'm too lazy to do the
>> algebra.

I worked out a little of the algebra for b) and the equations just
seem to be getting longer and longer instead of heading for a solution,
so I stopped.

>> c)3 or more aircraft --> system is overspecified, and some least
>> squares approach should give a solution.

I may be saying the same thing he is, but here is possibly another way
to look at case for 3 or more aircraft. If you know how to solve the
problem for 2 aircraft, and you have more aicraft than that, you can
pick any two and solve the problem for those two aircraft, yielding
a wind speed and direction. Then you can pick a different pair of
aircraft and solve the problem again -- you should get something close
to the same answer you got the first time. If you do this for all the
possible pairs of aircraft, you will _probably_ end up with a range of
answers that are somewhat grouped around a middle point. This does
result in a lot of calculations - 380 pairs for 20 aircraft or 9900
pairs for 100 aircraft - but this is the kind of thing computers are
good at.

>> It has been my experience that wind direction and speed won't vary
>> too much over distance, but may vary EXTREMELY with altitude.

I definitely agree with this. When you are picking pairs of airplanes,
it may be helpful (in terms of coming up with meaningful numbers) to
pick ones that are sort of close to the same altitude.

[from earlier in your post:]
> I have a complex math problem relating to the classic wind triangle
> that I posted on sci.math and received little response.

Here is some complete speculation on why it didn't get much response:

1) The folks there saw the magic words "air traffic control" in your
post and figured that if they helped you with it, they'd probably
get sued any time a plane crashes for the next 50 years.

2) The folks there saw the magic words "air traffic control" in your
post and figured out that you really do work for the FAA and
therefore have unlimited amounts of money and should give them a
grant to study this problem, rather than them answering for free on
Usenet.

Understand that I'm not saying that you shouldn't have said the magic
words - it's often quite helpful to understand the basic problem
somebody is trying to solve. And maybe neither of my speculations are
accurate.

Some other ideas on places to ask for help:

The halls of academentia. Go down to UMKC, find the math department,
and see if one of the professors can help you. They might also refer
you to a grad student who is good at turning food into solved math
problems. :) One minor problem with this is timing - they may have
all bugged out for the holidays.

NWS/NOAA. They might have solved this problem themselves at some point
and might be able to give you some code. My first two guesses at where
to try would either be the regular office in Pleasant Hill, MO, or the
Severe Storms Lab in Norman, OK.

You probably know about this, but you can cheat by pointing a radar
straight up and letting it figure out what the winds are doing:
http://www.profiler.noaa.gov/npn/profiler.jsp?options=full
But it sounds like you might be working on a (partially?) "canned"
training scenario and current real-world data is not exactly what
you need.

I hope this helps!

Matt Roberds

[Everett M. Greene[_2_]]

"Michael Ware" > writes:
> "d&tm" > wrote
> > "Stefan" > wrote
> >> d&tm schrieb:
> >>
> >> > if you know HDG ( ie where you are pointing), GS and TAS then there is
> >> > only 1 possibility for the wind speed and direction.
> >>
> >> Actually, there are two.
> >
> > I give up, can you please explain how there can be 2 ?
>
> There are two possible situations for the wind correction. You do not know
> the direction of the correction for wind ( i.e. is the plane crabbing left
> or right to compensate for x-wind) you only know the magnitude (wind speed).
> Think of the triangle that is formed by vectors on the e6b. Without the
> direction, you have an ambiguous answer, looks like two similar triangles, a
> lefty and a righty.
> Someone else could probably explain this better, that's the basic idea.

And a simple explanation of the whole process is that
the wind triangle has three (vector) components:
heading, course, and wind. The vector sum of heading
and wind gives course which is the problem that pilots
are accustomed to solving. Rearranging the equation
so as to compute wind given heading and course is not
at all difficult. The law of cosines allows determination
of the third side of a triangle given two sides and
the included angle. The law of sines allows determining
the other two angles given the three sides. There is no
left/right ambiguity given the course and heading.

[Chad Speer]

Denny wrote:
*****
Lots of fun to play with the formulas... But, the fly in the ointment
here is that the airspeed filed by the pilot is a canned number and has
nothing to do with reality... I fly IFR, I file the same airspeed - 130
kt - for all flights regardless of load, altitude, and what the power
setting winds up to be... If the actual airspeed in your problem is
within 15% of the filed number you will be lucky...
*****


You're right, Denny. When you get down to the piston single/twin
level, there is great variation. This will be used mostly with high
altitude traffic where, with the exception of an occasional turboprop,
the true speeds are 400 knots plus. Most commercial operators file to
the knot and, in my experience, fly within ten knots. That is probably
going to be an acceptable variation, but time will tell. We will also
be polling some pilots initially to determine their heading, true
airspeed, and observed winds to validate our results.

Chad Speer
PP-ASEL, IA
ATCS, Kansas City Center

[Chad Speer]

Terry wrote:
*****
By subtracting the 2 equations above you get

WS=(GS2^2-GS1^2-TAS2^2+TAS1^2)/(2(GS2COS(180-ABS(WD-TR2)-GS1COS(180-ABS(WD-TR1))

What I then did was calculate the ground speeds for 2 aircraft for a
known wind of 20 kts from 220 M

aircraft 1 aircraft 2
TAS 120 100
GS 111.7 91.4
TR 290 150

If I then use my above equation for WS after setting wind direction to
220, I get 20 kts as expected. However if I round off the GS to 112
and 91 kts the Wind speed changes from 20 to 9.5kts which suggests the
ground speeds have to be super accurate to get anywhere near the right
wind speed.. Given that also you are taking the TAS from a flight
plan, which will vary with density altitude and RPM setting etc.
suggests your objective is going to be rather difficult to achieve in
practice.

Hope this helps and good luck. If you come up with the solution I would
love to see it.
*****

Thanks for taking a crack at that, Terry. I hadn't considered that
such a small inaccuracy in aircraft speed could cause such an error.

Wouldn't the situation you describe above be a near worst case scenario
for error, i.e. only two aircraft and a nearly direct crosswind for
each? How would rounding those speeds affect the calculated wind speed
if the aircraft headings were 260 and 020? Isn't that a bit like
plotting position based upon two nearly same or reciprocal bearings,
rather than two that are near 90 degrees offset?

Also, since data will be available for many aircraft, we will probably
be able to calculate winds using at least five or six aircraft within
2000 feet of each other, sometimes many more.

This tool is basically meant to be a replay of a high altitude training
session, where the trainer can say "Okay, let's see what would have
happened if you had turned AAL460 fifteen left rather than descending
him." As it exists now, it is really cool, but a bit unrealistic. If
the winds are strong enough, a fifteen degree turn can easily add
twenty knots to a groundspeed. It is common for a new (or weak)
controller to turn an aircraft behind another, only to see that same
aircraft pick up a bunch of speed. The turn has nearly no effect on
separation, so they turn them some more. Within minutes, the turn is
fifty degrees and the controller is sucking up a seat cushion. Fun to
watch, but it would be nice to train that kind of stuff out of the
workforce. :-)

With even a close approximation of the winds, the realism is greatly
enhanced. Otherwise, the what-if's are reduced to a bland simulation
and utility is reduced.

Thanks again for your thoughtful reply.


Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC

[Chad Speer]

Matt wrote:
*****
If you know how to solve the problem for 2 aircraft, and you have more
aicraft than that, you can pick any two and solve the problem for those
two aircraft, yielding a wind speed and direction. Then you can pick a
different pair of aircraft and solve the problem again -- you should
get something close to the same answer you got the first time. If you
do this for all the possible pairs of aircraft, you will _probably_ end
up with a range of answers that are somewhat grouped around a middle
point. This does result in a lot of calculations - 380 pairs for 20
aircraft or 9900 pairs for 100 aircraft - but this is the kind of thing
computers are good at.
*****

Exactly how I hope this plays out.

*****
When you are picking pairs of airplanes, it may be helpful (in terms of
coming up with meaningful numbers) to pick ones that are sort of close
to the same altitude.
*****

I didn't specify in my original post because I didn't expect the
question to be raised, but we will be using aircraft within a 2000 foot
window. At the higher altitudes, that rarely involves a difference of
more than a few degrees and maybe six knots of wind.

*****
1) The folks there saw the magic words "air traffic control" in your
post and figured that if they helped you with it, they'd probably get
sued any time a plane crashes for the next 50 years.
*****

I hadn't considered that. Hell, even *I* don't trust the FAA. :-)

*****
2) The folks there saw the magic words "air traffic control" in your
post and figured out that you really do work for the FAA and therefore
have unlimited amounts of money and should give them a grant to study
this problem, rather than them answering for free on Usenet.
*****

I wish I could offer someone money. This whole system was designed by
a controller who realized the data was just sitting there and decided
to make something useful with it. Now, it's being deployed nationwide.
If the FAA really gets involved, this will be a useless program.
Never fails.

I really have no involvement in this. He briefed me on his work and I
told him I thought I could produce a formula for the wind. We'll see.
I may have bitten off too much. :-)

*****
The halls of academentia. Go down to UMKC, find the math department,
and see if one of the professors can help you. They might also refer
you to a grad student who is good at turning food into solved math
problems. :) One minor problem with this is timing - they may have
all bugged out for the holidays.
*****

This was my original thought. We even have an aerospace engineering
program nearby (University of Kansas) where I could probably shame
someone into a solution. "The guys in the math department said you
couldn't handle the trigonometry."

I really like the open discussion of Usenet and would love to make this
solution an eternal part of rec.aviation. If that doesn't happen, I'll
bribe some grad students...

*****
NWS/NOAA. They might have solved this problem themselves at some point
and might be able to give you some code. My first two guesses at where
to try would either be the regular office in Pleasant Hill, MO, or the
Severe Storms Lab in Norman, OK.
*****

Now there's an idea I will consider. We even have meteorologists on
staff who could probably grease some wheels there.

Thanks for the discussion!


Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC

[Kev]

Chad Speer wrote:
> Thanks for taking a crack at that, Terry. I hadn't considered that
> such a small inaccuracy in aircraft speed could cause such an error.
>
> Wouldn't the situation you describe above be a near worst case scenario
> for error, i.e. only two aircraft and a nearly direct crosswind for
> each? [...]

Calculating winds this way is a slight variation of the old Wind Star
method developed about 90 years ago for aviation. In the old texts,
it was emphasized that a small angle difference in the aircraft vectors
would cause large errors. So you'd like to use aircraft that are
flying at as close to 90/270 degrees apart as possible. See pages
430-431 here (the pdf starts at page 380):

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930091494_1993091494.pdf

> This tool is basically meant to be a replay of a high altitude training
> session, where the trainer can say "Okay, let's see what would have
> happened if you had turned AAL460 fifteen left rather than descending
> him." As it exists now, it is really cool, but a bit unrealistic. [...]

Question: isn't ACARS data available to you? With all the airliners
reporting wind and temps aloft every few minutes, I'd have thought an
FAA system could use that.

Very interesting thread, btw.
Kev

[Kev]

> "Stefan" > wrote in message
> ...
> > Mathematically: There are always two square roots which solve the
> > equation: A positive and a negative.
> >
> > Physically: If you only know GS, TAS and HDG, then you don't know
> > whether the wind blows from the let or from the right.

While that statement is true, it doesn't cover all the possibilities.
We're talking about the case of a single aircraft? We know its HDG,
GS, TAS, but nothing else? That's only three of the six wind triangle
variables. We must know at least four to get an exact answer. With
only three knowns, there are a whole range of possible answers, since
we don't know what the wind correction angle is without knowing a CRS.

For example, for the following HDG, TAS and GS:

HDG 0
TAS 120
GS 100

All the following CRS, WD an WS are valid solutions, plus many more
in-between:

CRS WD WS
=== === ===
310 054 095
330 056 060
350 039 028
000 000 020 CRS=HDG
010 321 028
030 304 060
050 306 095

I think you were visualizing a triangle, and thought of the two obvious
solutions. But there are a lot more. Again, use the E6B method and
you'll see that any drift angle along the TAS arc contains a valid WS
and WD answer.

Regards, Kev

[Danny Deger]

"Chad Speer" > wrote in message
ups.com...

> Chad Speer
> PP-ASEL, IA
> ATCS, Kansas City ARTCC
>
>
>
> ************************************************** *
> ************************************************** *
>
snip

> direction of travel
> speed across the ground
> speed through the air
>

If you can get heading also, it is a fairly simple equation as others have
posted.

Danny Deger

[Danny Deger]

"Denny" > wrote in message
oups.com...
> Lots of fun to play with the formulas... But, the fly in the ointment
> here is that the airspeed filed by the pilot is a canned number and has
> nothing to do with reality... I fly IFR, I file the same airspeed - 130
> kt - for all flights regardless of load, altitude, and what the power
> setting winds up to be... If the actual airspeed in your problem is
> within 15% of the filed number you will be lucky...
>
> denny
>

Now I see why heading is not known. I agree, calculating wind on filed
airspeed would not be a good idea. Maybe taking the average of many
airplanes might be OK, but the difference between filed and actual can be
HUGE.

Danny Deger

[Danny Deger]

"Chad Speer" > wrote in message
ps.com...
> Denny wrote:
> *****
> Lots of fun to play with the formulas... But, the fly in the ointment
> here is that the airspeed filed by the pilot is a canned number and has
> nothing to do with reality... I fly IFR, I file the same airspeed - 130
> kt - for all flights regardless of load, altitude, and what the power
> setting winds up to be... If the actual airspeed in your problem is
> within 15% of the filed number you will be lucky...
> *****
>
>
> You're right, Denny. When you get down to the piston single/twin
> level, there is great variation. This will be used mostly with high
> altitude traffic where, with the exception of an occasional turboprop,
> the true speeds are 400 knots plus. Most commercial operators file to
> the knot and, in my experience, fly within ten knots. That is probably
> going to be an acceptable variation, but time will tell. We will also
> be polling some pilots initially to determine their heading, true
> airspeed, and observed winds to validate our results.
>
> Chad Speer
> PP-ASEL, IA
> ATCS, Kansas City Center

Each plane gives you an estimate of the head/tail wind, but no information
on the cross wind. You need a way to combine all the different airplanes
wind data into a single estimate of the wind. The error in filed airspeed
vs. actual airspeed could be averaged out over many airplanes.

I am sure a Kalman filter would work (I think :-), but it has been a while
since I have worked on a Kalman Filter. The math is not trivial, and this
particular filter would definately not be trivial. You could start the
filter with the estimate of wind from the weather guy -- or would this be
considered cheating.

Danny Deger

>

[Danny Deger]

"d&tm" > wrote in message
...
>
> "Chad Speer" > wrote in message
> oups.com...
>> Terry - thanks for the reply, but heading is not known.
>>
snip

> TAS1^2=WS^2+GS1^2-2WSGS1COS(180-ABS(WD-TR1)
> TAS2^2=WS^2+GS2^2-2WSGS2COS(180-ABS(WD-TR2)
>
> WHERE TAS1 AND TAS2 ARE TRUE AIRSPEEDS FOR AIRCRAFT 1 AND 2
> WS = WIND SPEED
> WD =WIND DIRECTION IN DEGREES MAG
> TR1 AND TR2 ARE TRACKS MAGNETIC FOR AIRCRAFT 1 AND 2
> GS1 AND GS2 ARE GROUND SPEEDS FOR AIRCRAFT 1 AND 2.
> (180-ABS(WD-TR) WILL GIVE YOU THE ACTUAL ANGLE BETWEEN WIND DIRECTION AND
> TRACK.
>
> Now we have 2 equations with 2 unknowns ( WS and WD) which should be
> solvable but I am stuggling to get it out. However I did a little
> exercise
> which suggests the end result may not be very useful. because small errors
> in the ground speeds will cause very large errors in the calculated wind
> speed.

I think the head/tail wind portion of the estimate should be pretty good,
but the crosswind component would have so much error as to be useless. We
need a way to use the head/tail wind components from different airplanes,
flying different headings and NOT use the less acurate crosswind estimates.
I think a Kalman filter would work.

Danny Deger

[Travis Marlatte]

Chad,

I think that some clarification of the use might help. I can imagine two
scenarios:

1) enroute - in this case, you know the track and ground speed from radar.
You have the filed TAS. You have no idea of the crab angle or heading. This
sounds like the information you presented in your question. However,
enroute, who cares what the wind correction is. I am having trouble
imagining how this would help a controller - in training or not.

2) terminal - in this case, you still know the track and ground speed from
radar. However, now the filed TAS is probably no longer valid since it is
probably at a lower altitude than filed and used to compute the TAS. The
plane may also be under speed constraints. On the other hand, the controller
may be vectoring so now you have an assigned heading and maybe even an
assigned speed. You also have the pressure for the area and the TAS can be
computed from that and radar data. If this were an automated computation, it
could be done on every plane that was given an assigned speed (or even IAS
queried from the pilot) and an assigned heading (or even queried from the
pilot). Now maybe not every plane can provide all of the data but certainly
some could. Enough to update the calculated wind periodically.

Scenario 2 seems to me the one that would make knowledge of the wind most
desirable. It is also the scenario that could provide additional data and
certainly provides different data than what was posed in your question.

What say you?

--
-------------------------------
Travis
Lake N3094P
PWK

[Jose[_1_]]

> It is common for a new (or weak)
> controller to turn an aircraft behind another, only to see that same
> aircraft pick up a bunch of speed. The turn has nearly no effect on
> separation, so they turn them some more. Within minutes, the turn is
> fifty degrees and the controller is sucking up a seat cushion. Fun to
> watch, but it would be nice to train that kind of stuff out of the
> workforce. :-)
>
> With even a close approximation of the winds, the realism is greatly
> enhanced. Otherwise, the what-if's are reduced to a bland simulation
> and utility is reduced.

Do the controllers know (and have easily available, say, on the
terminal) the wind speed forecasts? Would they be sufficient to greatly
reduce the above scenario?

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.

[Jose[_1_]]

> I really like the open discussion of Usenet

Try some of the math and CS newsgroups. You might get more traction.

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.

[Jose[_1_]]

> Now I see why heading is not known.

Yanno, each airplane that has GPS could transmit its data to Center,
which could then sort it out. This is possibly much more useful than
the Mode S ID or the N-number.

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.

[Kev]

Chad Speer wrote:
> This tool is basically meant to be a replay of a high altitude training
> session, where the trainer can say "Okay, let's see what would have
> happened if you had turned AAL460 fifteen left rather than descending
> him." As it exists now, it is really cool, but a bit unrealistic. If
> the winds are strong enough, a fifteen degree turn can easily add
> twenty knots to a groundspeed. [..]

Chad, I think we need to stop and get the requirements, so we can help
better. I'm guessing how the program will work:

You have data from a training session. It has a recording of many
aircraft movements that you can play back. Perhaps you add one new
and controllable craft to the mix (AAL460 above?) and that's the one
that gets vectored by the trainee, to see what kind of trouble he can
get in/out of. Since all the other aircraft had been affected by wind,
the controllable plane needs the same flight input. Is this a good
description?

If so, I'm thinking that the wind within the 2000' block you talked
about, is a single fixed value for the entire training session. Would
that be fair to say? If that is true, then you don't need to
calculate the wind except once at the beginning. Can this be stored
for later use each time?

Just trying to get a handle on things,
Kev

[Chad Speer]

Jose wrote:
*****
Do the controllers know (and have easily available, say, on the
terminal) the wind speed forecasts? Would they be sufficient to
greatly reduce the above scenario?
*****

Most center controllers have a graphical display of forecast winds
aloft that is interpolated to provide estimates for every 1000 foot
level. That is very useful. We also have the ability to estimate
winds based upon our observation of the radar tracks at the sector.
This is something that takes a while to not only learn, but to remember
to look for it in the first place.

Each source of information is useful to prevent poor controlling in the
same way the gear handle is useful to prevent gear up landings.
There's no guarantee we'll use either. :-)


Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC

[Chad Speer]

Kev wrote:
*****
You have data from a training session. It has a recording of many
aircraft movements that you can play back. Perhaps you add one new
and controllable craft to the mix (AAL460 above?) and that's the one
that gets vectored by the trainee, to see what kind of trouble he can
get in/out of. Since all the other aircraft had been affected by wind,
the controllable plane needs the same flight input. Is this a good
description?

If so, I'm thinking that the wind within the 2000' block you talked
about, is a single fixed value for the entire training session. Would
that be fair to say? If that is true, then you don't need to
calculate the wind except once at the beginning. Can this be stored
for later use each time?
*****

Hey, Kev. We're not going to be adding tracks, we'll just review what
was done with the existing aircraft. The real kicker is that the
operator can effectively take control an aircraft and "fly" it
according to an alternate set of instructions the controller could have
given the pilot. We can then watch the original track of the aircraft
alongside the simulated track and see how the situation would have
developed if the controller had made a different choice.

The playback has no idea what the winds are. If we don't account for
the winds, the simulated track for an aircraft would behave as though
the wind was calm, which is highly unrealistic. Since this is a short
time frame and a relatively small chunk of airspace, a single
calculation for winds would be fine.


Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC

[Stefan]

Kev schrieb:

> I think you were visualizing a triangle, and thought of the two obvious
> solutions. But there are a lot more.

Frankly, I didn't contemplate very much and just posted the obvious.

Stefan

[Stefan]

Jose schrieb:

> Yanno, each airplane that has GPS could transmit its data to Center,
> which could then sort it out. This is possibly much more useful than
> the Mode S ID

Mode S (the flavor used in heavy metal) does exactly that and a lot more.

Stefan

[Jose[_1_]]

>
> Mode S (the flavor used in heavy metal) does exactly that and a lot more.

Thanks. I thought mode S (the flavor used on little airplanes)
basically just transmitted its unique ID.

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.

[Stefan]

Jose schrieb:

>> Mode S (the flavor used in heavy metal) does exactly that and a lot more.

> Thanks. I thought mode S (the flavor used on little airplanes)
> basically just transmitted its unique ID.

You thought correctly. But the OP specifically asked about big planes.

Stefan

[Jose[_1_]]

> You thought correctly. But the OP specifically asked about big planes.

Missed that. It was a while ago, but I don't think I ever realized that.

Jose
--
"There are 3 secrets to the perfect landing. Unfortunately, nobody knows
what they are." - (mike).
for Email, make the obvious change in the address.

[[email protected]]

Chad Speer > wrote:
> I didn't specify in my original post because I didn't expect the
> question to be raised, but we will be using aircraft within a 2000
> foot window. At the higher altitudes, that rarely involves a
> difference of more than a few degrees and maybe six knots of wind.

Sounds reasonable. So far I've only played below 14,000 feet AGL (and
most often under 11,000 feet AGL), and in Oklahoma and Texas, that means
the winds can sometimes shift 90 degrees in a thousand feet.

> If the FAA really gets involved, this will be a useless program.
> Never fails.

I couldn't possibly comment. [0]

> I really like the open discussion of Usenet and would love to make
> this solution an eternal part of rec.aviation. If that doesn't
> happen, I'll bribe some grad students...

Even if you have to get "outside" help, that doesn't mean you can't
share the solution with the group. I figure you (or somebody) will
probably end up writing a short "how this training software works"
document, either for other controllers that want to use it for training,
or to show your boss what you've been fooling around with all this time.
You could probably post that document here, or put it on the web and
provide a link.

Matt Roberds

[0] You ever hear of a little company in Kansas City called Wilcox
Electric? How about a little program called WAAS?

[Kev]

Chad Speer wrote:
> Hey, Kev. We're not going to be adding tracks, we'll just review what
> was done with the existing aircraft. The real kicker is that the
> operator can effectively take control an aircraft and "fly" it [...]
>
> The playback has no idea what the winds are. If we don't account for
> the winds, the simulated track for an aircraft would behave as though
> the wind was calm, which is highly unrealistic. Since this is a short
> time frame and a relatively small chunk of airspace, a single
> calculation for winds would be fine.

Okay then. This is like my earliest days of programming back with 4K
memory and a slow cpu :-) Sometimes the simplest ways are best.

I think someone else noted that since you know the filed TAS and the
radar-derived GS, then the Headwind values will be super easy to
derive. (TAS-GS)

So what I would do is try to find aircraft facing the four or eight
main compass directions. Calculate and store the head or tail wind for
each base direction in a lookup array. Extrapolate if wished for more
compass points. You can do this once at the beginning, or do it at
various intervals if more aircraft will turn and give finer data.

Now just apply that simple wind info (by array lookup of the rough
course) to the TAS of the plane you are taking control of. It should
be quite sufficient since you really just need the basic wind effect on
its TAS as it turns... the crosswinds and airplane headings don't
matter in your case.

Regards, Kev

[Chad Speer]

Kev wrote:
*****
> I think someone else noted that since you know the filed TAS and the
> radar-derived GS, then the Headwind values will be super easy to
> derive. (TAS-GS)
*****

This is interesting. I hadn't considered breaking them up into
headwind and crosswind components. I see the simplicity in that, but
I'll have to run that through the brain a few times to see how it fits.
I certainly can't dismiss it offhand!


Chad Speer
PP-ASEL, IA
ATCS, Kansas City ARTCC

[Kev]

Chad Speer wrote:
> This is interesting. I hadn't considered breaking them up into
> headwind and crosswind components. I see the simplicity in that, but
> I'll have to run that through the brain a few times to see how it fits.
> I certainly can't dismiss it offhand!

No crosswind calcs needed. No trigonometry needed. Just addition and
subtraction.

In your case, aircraft always follow a specified course. Headings
don't matter. Crosswinds don't matter. The only thing that changes is
their speed, and that's simply TAS +/- wind speed for a particular
direction.

So if you calculate at startup the head/tail wind (TAS-GS) for two
aircraft flying basically North or South, and East or West, then you
have enough rough information to extrapolate the speed of the trainee's
airliner, as it turns to face different directions. It's not precise,
but it's good enough to make things more realistic by at least having
the speed change appropriately for each direction.

Kev

[d&tm]

"d&tm" > wrote in message
...
>
> "Chad Speer" > wrote in message
> oups.com...
> > Terry - thanks for the reply, but heading is not known.
> >
> > Stefan - we need to be able to plug these known values into a formula
> > and kick out a result. Assuming the original responder was on the
> > right track, I still don't know what to do with his suggestion. Any
> > ideas on that? I'm not lazy, this just went over my head a long time
> > ago. :-)
> >
> Chad, sorry misread the question. But I like a challenge so I had another
> go.
> Firstly I dont believe the equations given by the original poster are
> correct. if you apply the cosine rule to the wind triangle for 2 aircraft
> you get the following 2 equations.
>
> TAS1^2=WS^2+GS1^2-2WSGS1COS(180-ABS(WD-TR1)
> TAS2^2=WS^2+GS2^2-2WSGS2COS(180-ABS(WD-TR2)
>
> WHERE TAS1 AND TAS2 ARE TRUE AIRSPEEDS FOR AIRCRAFT 1 AND 2
> WS = WIND SPEED
> WD =WIND DIRECTION IN DEGREES MAG
> TR1 AND TR2 ARE TRACKS MAGNETIC FOR AIRCRAFT 1 AND 2
> GS1 AND GS2 ARE GROUND SPEEDS FOR AIRCRAFT 1 AND 2.
> (180-ABS(WD-TR) WILL GIVE YOU THE ACTUAL ANGLE BETWEEN WIND DIRECTION AND
> TRACK.
>
Further to this post ,using the above 2 equations , with the following
correction
cos( abs( abs(wd -tr)-180)), the problem was solved graphically by solving
the 2 quadratics , or 3 for a 3 aircraft case for wind speed, for each of
the 360 degrees of possible wind direction. this gives you a parabola for
each aircraft and the overlap of the parablolas gives you the required
answer. For 2 aircraft there can be either 1 or 2 solutions. having a
3rd aircraft will give one unique solution..
This solution is available as an Excel spreadsheet if anybody is interested.
terry