In a recent thread, there was some discussion of the standard error of a direction measured with an electronic total station with a minimum readout increment of 10" that one poster had bought on eBay to begin his hobby as a rural surveyor without a license. Various posters immediately concluded that the uncertainty that any angle measured with that instrument would have is 10" or more.
Some wiseass suggested that it was easy enough to actually test the standard error of a direction measured with that instrument. I have done some reading on the subject and think that he is correct, both that it can easily be tested and that the least count is a poor indicator of angular accuracy.
Consider these sets of directions taken with a digital total station with a least count of 6" (a Zeiss Elta 46R). There are four sets of directions from the same setup to six different targets, rotating the circle between sets as indicated by the change in the directions to Object 1. To underscore that point: each set was taken on a different part of the instrument's circle.
[pre]
S e t 1
------------------------------------------
Obj F Lt F Rt Mean Reduced
1 0-00-00 00-00 00 0-00-00
2 49-54-48 54-48 48 49-54-48
3 87-20-24 20-24 24 87-20-24
5 123-21-42 21-42 42 123-21-42
6 129-48-06 48-06 06 129-48-06
S e t 2
------------------------------------------
Obj F Lt F Rt Mean Reduced
1 45-09-36 09-36 36 0-00-00
2 95-04-24 04-06 15 49-54-39
3 132-30-00 30-00 00 87-20-24
5 168-31-18 31-12 15 123-21-39
6 174-57-42 57-42 42 129-48-06
S e t 3
------------------------------------------
Obj F Lt F Rt Mean Reduced
1 90-55-12 55-12 12 0-00-00
2 140-50-00 50-00 00 49-54-48
3 178-15-42 15-36 39 87-20-27
5 214-17-00 16-54 57 123-21-45
6 220-43-18 43-24 21 129-48-09
S e t 4
------------------------------------------
Obj F Lt F Rt Mean Reduced
1 136-20-24 20-12 18 0-00-00
2 186-15-06 14-54 00 49-54-42
3 223-40-42 40-42 42 87-20-24
5 259-42-06 42-00 03 123-21-45
6 266-08-30 08-30 30 129-48-12
[/pre]
So, what does the data indicate is the standard error of a direction observed with the instrument, taken as the mean of F Lt and F Rt? Note that one special feature of this situation is that there is nothing magical about the assignment of 0-00-00 to Object 1. Any of the other objects could have been made 0-00-00 and the other directions reduced in relation to it. The reduced directions express the relative angular relationships measured between five targets from one instrument station.
The best estimates of the directions to the five targets are just the means of directions to each target from the four sets (although in this case the directions had also been observed with two different one-second theodolites to verify that there was no anomoly in the answer derived from just the directions observed with the Elta 46R with the 6" least count.
[pre]
Means of Sets 1 - 4
--------------------
Obj Direction
1 0-00-00
2 49-54-44
3 87-20-25
5 123-21-43
6 129-48-08
[/pre]
> [pre]
> Means of Sets 1 - 4
> --------------------
> Obj Direction
>
> 1 0-00-00
> 2 49-54-44
> 3 87-20-25
> 5 123-21-43
> 6 129-48-08
> [/pre]
Okay, so then if the above are the best estimates of the directions to each target, what were the residuals (apparent errors) in each set of observed directions listed in the first post? Given the directions from all four sets, the apparent errors, v', in the directions from a set are taken by subtracting each observed direction from the mean direction from all four sets and then adding a constant to that whole set of residuals so that the sum is 0 (the means coincide).
[pre]
Deviations from
Mean Directions
Set 1 Obj v v' Sum(v')^2
-------------------
1 +0 +0
2 -4 -4
3 +1 +1
5 +1 +1
6 +2 +2
--------------------------
0 22
Set 2 Obj v v'
-------------------
1 +0 -2
2 +5 +3
3 -1 -3
5 +4 +2
6 +2 +0
--------------------------
0 26
Set 3 Obj v v'
-------------------
1 +0 +1.8
2 -4 -2.2
3 -2 -0.2
5 -2 -0.2
6 -1 +0.8
--------------------------
0 8.8
Set 4 Obj v v'
------------------
1 +0 +0.6
2 +2 +2.6
3 +1 +1.6
5 -2 -1.4
6 -4 -3.4
--------------------------
0 23.2
[/pre]
So, the sum of the squares of the residuals (v')^2 = 22 + 26 + 8.8 + 23.2 = 80.0
> So, the sum of the squares of the residuals (v')^2 = 22 + 26 + 8.8 + 23.2 = 80.0
And that means that the best estimate of the standard error of a single direction taken as the mean of F Lt and F Rt is :
s.e. = SQRT [Sum(v'^2)/(Sets-1)(Targets-1)]
s.e. - SQRT [(80.0)/(4-1)(5-1)] = 2.6"
So, the standard error of a direction taken with that Elta 46R with a 6" readout is +/-2.6".
This means that roughly 68% of all directions measured with it under the same conditions as the test will be expected to have instrumental errors of less than 2.6"
This same standard error was also the value obtained by comparing the four sets of directions observed with the Elta 46R to the control values of the directions to the same targets taken as the means of multiple sets independently observed with two one-second instruments. In other words, the standard error estimate was realistic.
> So, the standard error of a direction taken with that Elta 46R with a 6" readout is +/-2.6".
This means that if one were determining the angular relationship of several objects by taking the directions to them as the mean of directions observed on each face of the total station, then the standard error of each direction would be reasonably estimated as +/-2.6"
If one were measuring angles between two objects by taking directions observed on each face of the total station and then subtracting one direction from the other to get the angle between them, then the standard error would be expected to be
SQRT(2) x 2.6" = +/-3.7"
The increase in the standard error is due to the fact that each of the two directions from which the angle was computed had a standard error of +/-2.6" with the root sum of squares being SQRT[(2.6)^2 + (2.6)^2] = SQRT(2) x 2.6 = 3.7"
Well done Kent.
Next up a pointing/sighting error estimate exercise? (ie how well can an individual actually acquire the target with a given instrument)
This is Great Stuff! Thanks!
So, are you saying that the dumbass really
wasn't a dumbass?
Excuse me, I meant wiseass, lol
I have gathered (without looking up the thread to which you refer) that you meant someone other than yourself, but knowing your self-deprecating humor, I may be wrong.
Pointing Accuracy of Total Station
> Next up a pointing/sighting error estimate exercise? (ie how well can an individual actually acquire the target with a given instrument)
That is pretty much just a function of the telescope magnification.
For distant targets with poor contrast, the empirical estimate is:
s.e. = 20"/M
where M = telescope magification
For close range targets with sharp contrast :
s.e. = 12"/M
So, for a total station with a 28X telescope, a pointing error of 12"/28 = +/-0.42" s.e. would be the expected value to sharp, clear targets.
Naturally, along lines of sight near the ground, target blurring and shimmer can hugely degrade pointing errors unless proper attention is given to things.
Note that the standard error of a direction as obtained by the test procedure given above contains the pointing error to the targets as a component. If for some reason a surveyor will have to measure angles to targets that are blurred or subject to scintillation, then just setting up targets that duplicate those poor conditions and running the test will capture the pointing error component.
I deal with the question in my own practice by using excellent, high contrast targets and by not trying to measure angles under marginal seeing. If you use prism poles in tripods to support targets, you have the option of raising the target a couple of feet to increase ground clearance for the line of sight and to improve seeing. If that doesn't do it, then shortening the line of sight or returning at a different time of day is often the better option.
The Element of Drama in Error Analysis
> I have gathered (without looking up the thread to which you refer) that you meant someone other than yourself, but knowing your self-deprecating humor, I may be wrong.
Every good thread needs an element of human drama. In this case the wiseass is one of these know-it-all types that we deal with so frequently. He shares an office with me. The problem with know-it-alls is that some of them are usually right, particularly about obvious stuff like testing the standard error of a direction taken with some particular total station.
framed on the wall...
behind the owner's desk at a firm I worked for years ago:
"People who think they know what they are talking about are a hindrance to those of us that do."
Haven't thought about it for years until you mentioned the wiseacre with whom you share an office.
framed on the wall...
> "People who think they know what they are talking about are a hindrance to those of us that do."
You would think that somewhere in cyberspace somewhere there would be a message board for surveyors organized along those lines. :>
BTW, this character in my office who originally mentioned the testing of standard errors of angles measured with a total station is looking for either a BMW K75 or a pre-2000 Kawasaki Drifter 800 that hasn't been Shrinered up.
Pointing Accuracy of Total Station
Closely related issue that always got me upset. Pet Peeve.
That is, what target to sight first.
Typically in multi target traverse setups.
People always mindlessly would say "sight the farthest first, you'll turn better angles"
Am I wrong in saying that if you are using relative angles i.e. difference in plate directions (like a Wild T2), that it makes no difference which is "first".
The simplest example is just one angle, with one BS target and one FS target.
Makes no difference which one you sight first.
Where Plate Zero is, is just academic.
I can see where if you were raying out points you would want a strong (long) BS.
Not what I'm talking about.
I think these people are confusing measuring angles and using angles in calculations where you of course want the strongest direction (usually farthest point?) to orient your survey. But that was not what I was pointing out to them.
And I explained that many times and could not get through.
Many old timers were stuck on this.
Pointing Accuracy of Total Station
> Am I wrong in saying that if you are using relative angles i.e. difference in plate directions (like a Wild T2), that it makes no difference which is "first".
Yes, that's right. It doesn't matter. The uncertainty in the angle taken as the difference in directions between two targets doesn't depend at all upon the order in which they were taken. For example, you could be measuring the angle from a statiion to a survey target on a control point 600 ft. away and the center of a 24" bearing tree 30 ft. away. The answer is qualitatively the same regardless of which is the foresight object.
Pointing Accuracy of Total Station
I am sure that Jered McGrath probably remembers this lab as well.
Pointing and Reading (filetered for outliers@95% confidence level in excel)...
Reading only (filtered for outliers@95% confidence level in excel)...
Wild T2 specs...
I wonder if my technique has gotten better to allow better results...
Pointing Accuracy of Total Station
> I wonder if my technique has gotten better to allow better results...
That's interesting, but a surprising large standard error for a direction taken as the mean of F Lt and F Rt. The Zeiss Th2 (a one-second theodolite similar to the Wild T-2, but better) that I tested extensively gave directions with a standard error of +/-0.88". Obviously, the pointing error was significantly less than that.
Should one assume that by "reading" what was done was to measure the scatter of repeated micrometer settings and estimate the standard error from them?
Pointing Accuracy of Total Station
> That's interesting, but a surprising large standard error for a direction taken as the mean of F Lt and F Rt. The Zeiss Th2 (a one-second theodolite similar to the Wild T-2, but better) that I tested extensively gave directions with a standard error of +/-0.88". Obviously, the pointing error was significantly less than that.
Granted... I am still a "young feller" in the survey community, but I was just a punk when this lab was conducted.
> Should one assume that by "reading" what was done was to measure the scatter of repeated micrometer settings and estimate the standard error from them?
Yes, that is exactly what we did.
Pointing Accuracy of Total Station
Full disclosure... that lab may have occurred on a Friday:-D
Pointing Accuracy of Total Station
> Full disclosure... that lab may have occurred on a Friday
Well, Monday and Friday errors are well known and difficult to model. :>