Looking for a math wiz!

"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

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

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.

"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

[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/pro...p?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

"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.

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

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

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

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/ca...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