Looking for a math wiz!

[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.

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

-----Original Message-----
From: Everett M. Greene ]
Posted At: Sunday, December 17, 2006 11:05 PM
Posted To: rec.aviation.ifr
Conversation: Looking for a math wiz!
Subject: Looking for a math wiz!

....
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.

The problem given one aircraft has many solutions because neither
heading nor wind vector can be determined by radar. Two unknowns for the
3 factor problem. Heading will change as a function of wind vector and
vice-versa.

As a sidebar: is this just an exercise or is the goal to empower the
controllers to provide accurate real-time winds aloft?

"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

"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

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

Travis wrote:
*****
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.
*****

This is for enroute training, Travis. It is not the crab angle we are
after, but the winds aloft. If we could pull the actual (or even just
forecast) winds aloft into this system, we'd be set. Unfortunately,
I'm told that data cannot be used for this purpose. I don't know if it
is inaccessible or if there is a contractual reason we can't get to it.
The bottom line is that we have only radar data to estimate the winds.

I see no other way of estimating the winds with this data. I hope that
this eventually provides us a workable solution. I've seen too many
trainees make bad decisions because they didn't consider the effects of
the wind to discount the usefulness of this information. A fair number
of operational errors occur because controllers don't use the wind to
their advantage.

By the way, my wife fell in love with Chicago on our trip last weekend,
so I'll be in your neck of the woods much more often. Atlantic
Aviation at Midway made us feel very welcome. :-)


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

Jim wrote:
*****
As a sidebar: is this just an exercise or is the goal to empower the
controllers to provide accurate real-time winds aloft?
*****

This is just a training tool, Jim. Basically a replay system that
allows us to take control of an aircraft an demonstrate how a situation
could have been handled differently by a trainee. It should be really
cool, but none of this will be used on live traffic.


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

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