Coordinate averaging? I need advice! 

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Generic Dude
Posts: 11

Joined: 1/22/2009
Topic  Coordinate averaging? I need advice!       Flag »  Reply »
Ok, maybe "averaging" isn't the correct term, but I don't know what is.

Here is my dilemma:  We need to locate the centers of 4 radio towers, but it's not possible to occupy the centers, or even get near it.  So we set up a series of 4 control points with known coordinates surrounding the tower.  From each of these points we turned angles to the centers (don't ask what they consider the "center"...) of the towers.  You end up with 4 observations on each control point, for a total of 16 angles.

Now, I can easily go into AutoCAD, insert the control points, and intersect the lines from the observed angles.  Problem is, you get 6 possible intersections at each tower center.  They aren't very far apart, but I'm wondering what the best way is to determine the center of the towers based on intersecting angles alone.  I am assuming all the observations are of equal
quality. 

Thanks for any input.
  Thursday, January 22, 2009 at 2:14:44 PM
Isleno
Posts: 42
Location: Gonzales, La USA

Joined: 10/20/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
This is a classic least squares problem.  The equation for each of the intesecting lines had a slope (m) and a y intersect (b), the classical equation of a line being y = mx + b.  A simple soultion requires only two lines (slope m1 and m2) and (intercept b1 and b2) solved by substitution or Gaussian reduction.  A least square solution simultaniously solves for slopes m1, m2,m3.......mi and intercepts b1, b2, b3.........bi.  There are several canned programs that will do this or get a book on practical least squares a write one yourself.  I used to have one I rote in DOS around here somewhere, but I can't find it now.
  Thursday, January 22, 2009 at 3:41:35 PM
Generic Dude
Posts: 11

Joined: 1/22/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Ok, thanks for the advice.  I understand a little about least squares, but it's always done by computer when I deal with it.  Is this something that's even possible/practical to do by hand?  If so, I would like to try it just to learn something new.  I'm not dealing with a large data set...
  Friday, January 23, 2009 at 10:17:23 AM
Isleno
Posts: 42
Location: Gonzales, La USA

Joined: 10/20/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Difficult by hand if the data set is very large, not so bad if the data set is less than 10 or so observations.  It helps if you have some experience with matrix algabra.  There are canned programs out there that can help.  I used to get FORTRAN sub-routines that made it a breeze.  It is very good to learn to use Least Squares by hand.  It is very powerful in applications other than just traverse adjustment or level loops.  The intersection of lines involves two variables, so a 10 set of observations may require 20 or so hand calculations.  Not bad, but the things you will learn about the stastics of data collection will serve you well.  I lost all my custom programs during Katrina, otherwise I'd send you the DOS programs.
  Friday, January 23, 2009 at 10:59:42 AM
Generic Dude
Posts: 11

Joined: 1/22/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
I'm having a tough time finding anything about it on the web.  Most of what I see is for fitting lines, but my problem is that I have a group of x,y coordinates, I just need to find the best-fit center of them.  If you know a good website that could get me started, please let me know.
  Monday, January 26, 2009 at 4:51:03 PM
Dave Lindell
Posts: 16
Location: Pasadena, USA

Joined: 10/20/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Wolfe and Ghilani's "Adjustment Computations" book will explain everything you need to know, and then some!

Their examples are superb!

You will have to know (or be able to grasp) matrices.
  Monday, January 26, 2009 at 5:36:08 PM
Isleno
Posts: 42
Location: Gonzales, La USA

Joined: 10/20/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
E-mail me.  I have an old DOS program for Least Squares fit to a coordiante system.  It is anchient, but then so is math.

                No warranties.
  Tuesday, January 27, 2009 at 5:00:40 PM
Thomas LaCorte
Posts: 62

Joined: 10/17/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Generic Dude, yes least squares will give you the best solution...HOWEVER...if your pulling your hair out on this one simply add all your northings together and divide the total by the number of observations. Same for the Eastings. Example...5000.05....5000.10.....5000.15...now divide by 3 and what do you get? 5000.10..right? Now I know it is an average and only an average but remember your solution is only as good as the measurments that were taken! The real question is how accurate does it have to be? or what is the tolerance of error? I do not dispute Isleno nor Dave on this they are correct just wondering if you over looked a direct averaging of your values. Sometimes we dont see the forest for the tree's!
If your tolerance is very tight by all means go with least squares. But if it's say plus or minus a foot....averaging might be ok. Especially if all of your measured values fall within your tolerance. I believe Isleno and Dave would agree on what I'm trying to say. 
  Tuesday, January 27, 2009 at 11:18:38 PM
Isleno
Posts: 42
Location: Gonzales, La USA

Joined: 10/20/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
I did a little memory search.  The first step is to convert the lines measured in radial terms to the cartisian terms.  The cartisian equation of a line is y=mx + b.  If you have a line of azimuth "A" measured from point X1, Y1 then m=1/tan(A) and b = Y1- (1/tan(A))X1 will give you the values for the cartisian definition.  Convert all of your observations to this format.  Then you need to create the "normal" equations from the list of observed equations.  Refer to page 48 of "A text-book on the method of least squares"  by M. Merriman.  It is all just multiplication and addition.  The problem with most least squares books is that they don't explain how to convert lines defined by azimuth and point-on-line to y=mx + b.
  Wednesday, January 28, 2009 at 8:18:01 AM
Generic Dude
Posts: 11

Joined: 1/22/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Thomas, To tell the truth, I have already averaged the northings and eastings to get my coordinates.  They were all within a 0.20 foot radius of each other and the method by which the angles were measured is dubious at best.  The averaging worked fine for this one.

Still, it's something I want to learn anyway in case sometime down the road averaging/guessing is not good enough.  I know computers can do it, but I hate just accept stuff like that without knowing any of the theory behind it.  It goes on way too much these days, in my opinon. 

I've ordered the book Dave suggested, and with the great advice I've got from this forum, I'm going to learn it anyway.  I just think it would be a good skill to have.
  Wednesday, January 28, 2009 at 1:39:27 PM
Dave Lindell
Posts: 16
Location: Pasadena, USA

Joined: 10/20/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Tom, least squares adjustments give you the MOST PROBABLE value of an adjustment, not always the "best".  As stated, that depends on the data input.
  Thursday, January 29, 2009 at 9:26:19 PM
Thomas LaCorte
Posts: 62

Joined: 10/17/2008
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
I stand corrected "most probable" not "best". Thanks Dave.
  Friday, January 30, 2009 at 10:24:36 AM
jwahl
Posts: 8

Joined: 1/31/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
Another advantage of least squares is that it might show you how good your measurements are and thus find any outliers which may be affecting your results as long as you have some redundancy.

- jlw
  Friday, February 06, 2009 at 6:28:16 PM
Amberftw
Posts: 11

Joined: 1/2/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
I am still in school for surveying but this is my idea. Some how I think maybe you could use resection to figure it out the northing and eastings to the center of the towers? Resections I have done in class are used to get cordinates of point by using alpha beta angles. We had an occupying point, known cordinate points, and a cirlce and a few triangles. Would this be used in this type of a situation?
  Friday, February 06, 2009 at 11:43:06 PM
Generic Dude
Posts: 11

Joined: 1/22/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
The resection would be fine if you had one set of observations and it was a good one.  The problem here is that I have multiple low-quality observations of the same point, which ends up being like four resections at once.
  Monday, February 09, 2009 at 9:27:33 AM
k-bob
Posts: 15
Location: MI USA

Joined: 2/6/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
two things:
1. how good are your 'control points'?

2. least squares is not magic; the old adage garbage in = garbage out rings just as true with it as with any adjustment procedure.

Otherwise, kudos for pursuing least squares methodology, it is a highly beneficial procedure to understand, and typically provides the most robust 'adjustment'. Good luck!
  Tuesday, February 10, 2009 at 8:59:47 PM
Generic Dude
Posts: 11

Joined: 1/22/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
The control points are gold, so the measurements from them are not entirely horrible.  I understand what you're saying about garbage in, garbage out.  And for this case, I'd be just as well off averaging the northings and eastings.

Just for the sake of learning, I'm taking this scenario and pretending that what if guessing wasn't good enough, or what if the measurements were all high precision and a more accurate solution was needed.

I'm just using it as a training exercise to learn least squares.  I've got the book mentioned previously by Dave.  A little bit of a steep learning curve, but if I figure it out, it should be extremely useful.

 
  Wednesday, February 11, 2009 at 9:34:10 AM
Jack Chiles
Posts: 3

Joined: 2/13/2009
Reply  Re: Coordinate averaging? I need advice! Flag »  Reply »
The method I have always used was to tie the corners (4) of the towers and intersect the cross tie lines. Why estimate where one thinks the center is? Use the legs to determine that.
We alwasy tied the outside corner, by the way.
  Friday, February 13, 2009 at 2:50:31 PM

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