Is it true that some higher quantity of observations done with a so called "X" instrument" can achieve the same confidence level (say a 95% confidence interval), as a fewer number of observations done under identical circumstances with a so called "<X" instrument"? Say 5 vs. 3 or 3 vs. 1.
I've thought of mocking up the two scenarios in Star*net, but you can't very well "create" the data; you could make the outcome anything you want. I've thought of playing with the instrument variables, but that doesn't do it either as the number of redundant observations doesn't vary.
Thoughts?
You want to test one instrument vs. another and see what? Where the 95% confidence interval is? This doesn't make any sense.
A 95% confidence interval is the interval whereby (empirically speaking 2 sigma or 2 standard deviations from the mean), 95% of all observations will lie between the edges (1.965 T Value if you please). This is of course after you've made 30 (don't ask why but it is the industry standard for a sample set) observations, reviewed the IQR (inner quartile range) and reject anything that is greater than 1.5 from the mean (don't ask why because no statistician can give you a good answer other than it's the industry standard) again, and then crunch the numbers using the weights from the DIN specs if you don't want to generate your own specs from tons of measurements.
Only then can you really tell, and ultimately, you will end up with a DIN spec, or close to it.
It almost sounds like you're trying to do this in reverse with Star Net. 95% confidence intervals are generated by the software using the specs fed it. For instance, for a single measurement with a 5" gun and 5mm+5pp, on a 300' line with a 90å¡ single angle turned from the backsight would generate a semi-minor axis that is 0.0145' right and left of the 90å¡ and a semi major axis 0.0334' forward and behind the target, or one axis that is 0.066' across and another that is 0.029' across. This is the edges of the error ellipse for a single shot at that distance. That is to say if you turned that same 90å¡ and measured that distance of 300 feet, 100 times, 95 of them would lie in that ellipse without any blunders and under perfect conditions. Since you have two measurements, (angle and distance) you have to edges to consider which is how the ellipse is drawn. Double the angle, redundant distance, et cetera and you can tighten it up.
Does this make sense?
Kris Morgan, post: 373888, member: 29 wrote: You want to test one instrument vs. another and see what? Where the 95% confidence interval is? This doesn't make any sense.
A 95% confidence interval is the interval whereby (empirically speaking 2 sigma or 2 standard deviations from the mean), 95% of all observations will lie between the edges (1.965 T Value if you please). This is of course after you've made 30 (don't ask why but it is the industry standard for a sample set) observations, reviewed the IQR (inner quartile range) and reject anything that is greater than 1.5 from the mean (don't ask why because no statistician can give you a good answer other than it's the industry standard) again, and then crunch the numbers using the weights from the DIN specs if you don't want to generate your own specs from tons of measurements.
Only then can you really tell, and ultimately, you will end up with a DIN spec, or close to it.
It almost sounds like you're trying to do this in reverse with Star Net. 95% confidence intervals are generated by the software using the specs fed it. For instance, for a single measurement with a 5" gun and 5mm+5pp, on a 300' line with a 90å¡ single angle turned from the backsight would generate a semi-minor axis that is 0.0145' right and left of the 90å¡ and a semi major axis 0.0334' forward and behind the target, or one axis that is 0.066' across and another that is 0.029' across. This is the edges of the error ellipse for a single shot at that distance. That is to say if you turned that same 90å¡ and measured that distance of 300 feet, 100 times, 95 of them would lie in that ellipse without any blunders and under perfect conditions. Since you have two measurements, (angle and distance) you have to edges to consider which is how the ellipse is drawn. Double the angle, redundant distance, et cetera and you can tighten it up.
Does this make sense?
Perhaps I'm asking the question the wrong way.
All other thing being equal except the stated accuracy of the instrument and the number of observations, can a less accurate gun produce as accurate a survey as a more accurate gun, simply by increasing the number of redundant observations?
rfc, post: 373880, member: 8882 wrote: I've thought of mocking up the two scenarios in Star*net, but you can't very well "create" the data; you could make the outcome anything you want. I've thought of playing with the instrument variables, but that doesn't do it either as the number of redundant observations doesn't vary.
Thoughts?
Actually, if you know the standard error of an observation, you can use the "Preanalysis" function of Star*Net to answer the question well enough if the problem is more complicated than can be easily handled with a pocket calculator.
In your case, I think you should be looking at a new statistical function that describes the Riffic Error function* which tends to be distinctly non-Gaussian.
*Also described in the surveying literature of the internet as the "rfc error function".
Suppose you make one observation with an instrument that produces measurements whose mean error is 3 seconds. The error in that measurement might be 1 second, or 3 seconds, or 5 seconds, or some other number of seconds that usually will fit within a range defined by the standard deviation.
If you make 10 observations, some will likely contain more than 3 seconds of error, some less, but the mean error in these observations will be close to 3 seconds. If you make 100 observations, the mean of their errors will be even closer to 3 seconds.
If it is truly a 3-second instrument, no number of observations will turn it into a 1-second instrument.
But that didn't exactly answer your question. Just as people do win lotteries with odds approaching 300 million to 1, it is possible that a surveyor could use a 5-second instrument, take more measurements than he did with a 1-second instrument, and produce a better result with the 5-second instrument. It would, however, be due to chance rather than the number of observations.
MathTeacher, post: 373923, member: 7674 wrote: If it is truly a 3-second instrument, no number of observations will turn it into a 1-second instrument.
Except that isn't entirely true if the characteristic of a 1-second instrument is that horizontal direction angles with a standard error of +/-1" can be measured with it from just the mean of F1 and F2 pointings.
Nine rounds of directions with an instrument that gives direction angles with a standard error of +/-3" (mean of F1 and F2) should in fact give mean direction angles with standard errors of +/-1", assuming that a proper program is followed for observations (including rotating the circle between rounds).
Inverse law of errors. Yes an instrument of lower precision can by repetitive measurement achieve higher accuracy.
Even a T16 (a 20" gun) can return 1" accuracy. But you have to wrap the angle many times. Like 40 - 60 times.
I meant inverse square law of errors.
To get back to the original question, yes you can get angles that are better than the "least count" of an instrument. The real question is how good are the optics (including how thin is the crosshair) in terms of arc-angle. I started out surveying with an instrument that had much better optics (and crosshair) than it had angle readout. It was a three button 20 second "theodolite", with a magnifying glass for reading the vernier. Really, it was just a theodolite shaped transit, but with pretty good glass. The way it worked was, I would turn an angle (lets say 80 degrees), and record it. I was trained (when using this method) to ignore anything between the lines of the vernier. So, I might notice that the angle was 80 degrees plus a fuzz, but the fuzz part being unmeasured, it was not to be guessed at. I would then leave that angle in the gun (not rezero!) and point the instrument at the backsight again, and turn to the foresight again. I would then read that angle (lets say 160-00-20) and record that (as raw data). I would then divide that angle by two (because it is the SECOND angle of the set), and record the result (80-00-10). This process (winding up) would then be repeated yet again, with an example value of 240-00-40, and, divided by three this would yield a third angle of 80-00-13. I would do about five of these. I could then invert the scope, rezero, and turn another five wound angles. The key point is that the angles get more "accurate" (though not more precise) as they accumulate. The fuzzy part of the measurement (the part you cannot guess at) accumulates until it is readable. Each additional angle has its own fuzz, but that fuzz gets divided by larger and larger numbers, so it affects the computed result less and less. Each successive angle is less different from its predecessor than the previous pair. For example, leaving out the raw angles, and showing only the values computed by dividing, this example might look something like this:
First angle (raw) 80-00-00
Difference 10 seconds
Second (reduced) 80-00-10
Difference 3 seconds
Third (reduced) 80-00-13
Difference 1 seconds
Fourth (reduced) 80-00-12 THESE ARE NOT REAL NUMBERS!
That would be an example of "winding up" four pairs of pointings with the instrument in direct mode. This could be repeated with the instrument reversed/inverted. That would give you another set of four angles, exhibiting the same pattern of increasing agreement, and four more pairs of pointings.
The key thing here is, you would use the LAST angle, since it is the result of averaging all four pairs of pointings. The numerical angle 80-00-12 comes from four foresights and four backsights, mechanically added together on the verneir plate. If you rezeroed, and did another set of four pairs of pointings with the instrument inverted, you would also use THAT last angle. You would also look to see that they were in reasonable agreement, based on your expectation from previous experience with that instrument. I would typically put both of the "last angles" one direct, one inverted, into a starnet adjustment also containing redundant distance measurements.
The point of all this blather is that, with that ancient 20 second instrument, I could get direct/reversed pairs of angles that agreed down to five or six seconds more than 95 percent of the time. Further winding up of the angles would NOT get me tighter than that, because I ran into the limit imposed by the telescope glass and the crosshair width. I learned about "winding up" angles as a way to get around the 20 second limit of the verniers I was reading back in the 1970s. I have used this method ever since, and it allows me to take a 5 second total station and get 2 second results out of it.
This procedure can be used on any instrument, though it is pointless on the really good one or two second instruments. You can take any total station, and wind up the angles. The more you turn, the more they will tend to converge on a result, and then get no tighter, but wander around a value. Most of the gear used by most surveyors can be "wound up" to get results in the two to three second range, regardless of the least count of the instrument. This most probably means that the crosshairs most of us are looking at are roughly that wide, angle wise.
To try winding angles for yourself, take your instrument, zero it, point it at any backsight, turn to a foresight, and record the angle. Repeat that process (without re zeroing!) so that the first angle is still in the gun or on the readout when you again point at the backsight. Turn again to to the foresight to complete the second pair of pointings. Record THAT angle, but also divide it by two, and record that computed angle. Repeat again, and divide that angle by three, and the next one by four etc. The computed angles will tend to converge on a result, and then not change much once you are within a few seconds of the true angle. The better your telescope and the thinner your crosshair, the tighter the angles will get before they start wandering around a mean and getting no better.
Try this methodology. It is proof that you can turn better angles than the "least count" specifications of an instrument suggest as the limits of the instrument. The method is useless on really good instruments, but only because the least count is perfectly matched to the glass (the optics) and the crosshair. On the really good instruments, you don't get any benefit by "winding up" but you don't lose anything either. So, I always do my control sets that way. I have never gotten bad results. Blunders are immediately obvious because the reduced angles do not converge on a value range.
John Wetzel, post: 373968, member: 11747 wrote:
To try winding angles for yourself, take your instrument, zero it, point it at any backsight, turn to a foresight, and record the angle. Repeat that process (without re zeroing!) so that the first angle is still in the gun or on the readout when you again point at the backsight.
I understand the principle of wrapping angles, but if I remember correctly, a modern total station is different than a two plate theodolite, and simply "holding" the angle and turning it again on one is very very different than on the theodolite because you're using the same part of the glass; you've just electronically "held" the angle. You will get the advantage of the average of many observations being much closer than if you just observed once though.
To actually use all sectors of the glass on a total station, you must turn the tribrach 120 degrees (or whatever), and do F&R multiple times, with different sections of the glass. (I believe it was Kent or Conrad that advised this).
Mathematically and statistically, yes, accumulating angles will get you better than a theodolite's least count. Accumulating angles was a technique for theodolites with upper and lower motions.
Bear in mind that there will likely be a point of diminishing returns; I doubt you could (reasonably) get one arc second results from a twenty second theodolite within a worthwhile amount of time.
Accuracy improvement is a power curve. So to improve the accuracy by a factor of 2 (half) you wrap 4x, to cut that in half 16x wraps, another one half improvement 64x. And so on.
1" angles were achievable long before a 1" gun was technically achievable.
No statistics here. Yes you can.
MathTeacher, post: 373925, member: 7674 wrote: It would, however, be due to chance rather than the number of observations.
Every textbook, white paper, I've seen on the subject, rejects chance. Repetition is to remove chance. (Chance being random error.) There so much technical information and formulae on std deviation, std err of the measurement, std err of the mean, that a measurement's accuracy and the probable error associated, always improves with repetition predictably within bounds. Assuming, of course, blunders are identified, std dev is maintained, procedure adhered to, and environmental conditions are stable. Not just with survey measurements.
rfc, post: 373916, member: 8882 wrote: Perhaps I'm asking the question the wrong way.
All other thing being equal except the stated accuracy of the instrument and the number of observations, can a less accurate gun produce as accurate a survey as a more accurate gun, simply by increasing the number of redundant observations?
To a point yes. At some point, you run into a point of diminishing returns whereby a 1' mountain gun would require a ridiculous amount of observations to become "kinda" close to a single angle from a 5" gun.
However, wrapping multiple angles with an old GTS 2B (20" gun) can and will produce the same results as a 1" VX robotic station, but you're gonna have to work at it, a lot.
Correct, everyone, but let me ask you this. If an instrument is rated at 20" but consistently produces 12" results, is it a 20" instrument or is it a 12" instrument?
rfc, post: 373973, member: 8882 wrote: I understand the principle of wrapping angles, but if I remember correctly, a modern total station is different than a two plate theodolite, and simply "holding" the angle and turning it again on one is very very different than on the theodolite because you're using the same part of the glass; you've just electronically "held" the angle. You will get the advantage of the average of many observations being much closer than if you just observed once though.
To actually use all sectors of the glass on a total station, you must turn the tribrach 120 degrees (or whatever), and do F&R multiple times, with different sections of the glass. (I believe it was Kent or Conrad that advised this).
Yes to a point again. We didn't have dual compensator instruments until sometime in the late 90's. Not that they weren't available but that we didn't have them. With a single compensator gun, you need to flip the barrel to see the error in the plate as that axis was not compensated for.
With dual compensators, if everything is in adjustment, this is no longer necessary; however, we still flip the barrel because an instrument left to sit can and will go out of level and you won't know it unless you're instrument man is on his game, or you flip the barrel.
MathTeacher, post: 374010, member: 7674 wrote: Correct, everyone, but let me ask you this. If an instrument is rated at 20" but consistently produces 12" results, is it a 20" instrument or is it a 12" instrument?[/QUOTE]
It's a 20" gun because that was what it was rated at when purchased. Whether it is better than the spec is irrelevant.
A 20" second describes the least count on the plate. Not the angle it'll produce.
And that determines the procedure that would be required to "better" that. It's a power curve. And to get down to control grade, long distance shots will take an untenable amount of work. Not impossible.
A 5" gun (least count) repeating instrument will require fewer wraps.
A 20" gun isn't restricted to only being capable of a 20" angle, or 10", or 5" angle measurement. But you'll know upfront that it'll take a lot of work if you need higher accuracy. So, you pick your weapon of choice for the task.
