Bill93, post: 384106, member: 87 wrote: In a specific instance of one actual error thats true. In the probability though the rms of NE error is the same as E or N rms error because often the N error and E error are not large at the same time. That's what a circular "ellipse" means.
Except that the semi-major axis of even a circular error ellipse around a coordinate with standard errors of 5.8mm in its N and E components won't be 2 x 5.8mm, but will be 2.445 x 5.8mm = 14.2mm, right? So if the 2-sigma error ellipse has a semi-major axis of 14.2mm, the 1-sigma error ellipse has a semi-major axis of 14.2 / 2 = 7.1mm, doesn't it?
Kent McMillan, post: 384111, member: 3 wrote: Except that the semi-major axis of even a circular error ellipse around a coordinate with standard errors of 5.8mm in its N and E components won't be 2 x 5.8mm, but will be 2.445 x 5.8mm = 14.2mm, right? So if the 2-sigma error ellipse has a semi-major axis of 14.2mm, the 1-sigma error ellipse has a semi-major axis of 14.2 / 2 = 7.1mm, doesn't it?
2-sigma is a common rule of thumb that comes from rounding off 1.96 sigma, which is the 95% confidence value for a 1-dimensional distribution.
We've got a 2-dimensional distribution here (bivariate Gaussian, also called Rayleigh), and for that, 95% confidence is at 2.445 sigma. So 2-sigma is irrelevant. You've multiplied by 2.445 correctly. Going back to the component in one direction, you'd need to divide by that 2.445..
Bill93, post: 384132, member: 87 wrote: We've got a 2-dimensional distribution here (bivariate Gaussian, also called Rayleigh), and for that, 95% confidence is at 2.445 sigma. So 2-sigma is irrelevant. You've multiplied by 2.445 correctly. Going back to the component in one direction, you'd need to divide by that 2.445..
But in the case of a circular error ellipse, about a point X,Y such that the errors in x and y are random, uncorrelated, and normally distributed with the same standard error, s
then the length of the semi-major axes of the error ellipse of probability, p is
k*s, where
k = SQRT[-2*ln(1-p)]
p=0.50 k= 1.177
p=0.68 k= 1.510
p=0.95 k= 2.447
So, where s = 5.8mm, the 68% confidence error ellipse would have a semi-major axis of 1.510 x 5.8 = 8.7mm, not 5.8mm, right?
So, if the maximum allowable size of the semi-major axis of the 95%-confidence relative error ellipse is 2cm, and both points compared are positioned with the same standard errors in N and E components,
then the maximum allowable semi-major axis of the 95%-confidence relative error ellipse of each point would be 2cm / 1.414 = 1.4cm
and, assuming that the N and E components of the coordinates of the points compared have the same standard error, that means that the maximum permissible standard errors of N and E would be
1.4cm / 2.447 = 5.7mm (as I believe you suggested).
Kent McMillan, post: 384142, member: 3 wrote: So, where s = 5.8mm, the 68% confidence error ellipse would have a semi-major axis of 1.510 x 5.8 = 8.7mm, not 5.8mm, right?
Yes that's the 68% value but for a bivariate 68% is not the rms.
I guess no one else noticed that a nationally recognized ALTA expert posted about components of an RTK system being stolen, in another thread.
Bill93, post: 384148, member: 87 wrote: Yes that's the 68% value but for a bivariate 68% is not the rms.
Yes, thanks for the clarification. It leads me to wonder exactly what the RTK manufacturers are specifying as positioning accuracy with the tag of "RMS" as in
Accuracy: Horizontal 8mm + 1ppm RMS
If, as you suggest, standard errors of 8mm in N and E components produce a radial RMS error of 8mm, then the spec would be equivalent to a position with a 95% confidence error ellipse with a semi-major axis not to exceed 2cm + 2.4ppm, which is pretty poor and easily explains why RTK surveys tend to be so sloppy.
Dave Karoly, post: 384150, member: 94 wrote: I guess no one else noticed that a nationally recognized ALTA expert posted about components of an RTK system being stolen, in another thread.
It looks as if they were using the RTK equipment to locate utilities (note the underground locators also taken). That sounds about right to me. How accurately do valve boxes and manholes really need to be located?
Dave Karoly, post: 384150, member: 94 wrote: I guess no one else noticed that a nationally recognized ALTA expert posted about components of an RTK system being stolen, in another thread.
I noticed it. He used to post on the old forum years ago. I think he was driven away by threats and insults of the forum's notorious incarcerated Indy fence surveyor. That being said...it would have been Interesting in to hear his take on RTK boundary tolerances.
Kent McMillan, post: 384157, member: 3 wrote: It looks as if they were using the RTK equipment to locate utilities (note the underground locators also taken). That sounds about right to me. How accurately do valve boxes and manholes really need to be located?
+/-0.07' between any two manholes.
Dave Karoly, post: 384165, member: 94 wrote: +/-0.07' between any two manholes.
... if used as boundary control. :>
Kent McMillan, post: 384166, member: 3 wrote: ... if used as boundary control. :>
I have a measuring wheel that I inherited from my Father.
(Here is the original post revised to take into account the statistical detail that Bill93. pointed out)
The current MINIMUM STANDARD DETAIL REQUIREMENTS FOR ALTA/NSPS LAND TITLE SURVEYS (Effective 02/23/2016) specify that :
"The maximum allowable Relative Positional Precision for an ALTA/NSPS Land Title Survey is 2cm (0.07 feet) plus 50 parts per million (based on the direct distance between the two corners being tested)."
and it defines Relative Positional Precision as:
"the length of the semi-major axis, expressed in feet or meters, of the error ellipse representing the uncertainty due to random errors in measurements in the location of the monument, or witness, marking any corner of the surveyed property relative to the monument, or witness, marking any other corner of the surveyed property at the 95 percent confidence level."
So, how does one compute relative positional precision? The ALTA specification states that:
"Relative Positional Precision is estimated by the results of a correctly weighted least squares adjustment of the survey."
If you are independently determining the coordinates of two points by some method such as, say, RTK to pass the ALTA/NSPS accuracy standard, you want to be certain that the length of the semi-major axis of the relative error ellipse between two corners is no more than 2cm at the 95%-confidence level. By "independent" what is meant is that the random errors in present in the coordinates of one point are not related to those present in the coordinates of the other. In the situation where the uncertainties in the North and East components of both points are all the same (the error ellipses of both points are circular), what is the maximum uncertainty that is allowable for the position of each point?
The answer is that a RMS uncertainty greater than 5.8mm for both of the points will fail the ALTA/NSPS specification for their relative positional precision when they are close enough together that the magnitude of 50ppm x D is neglible, where D = Distance between points. By RMS, what is meant is the standard error of the N and E components of the coordinates of the points, which for the purposes of the general case may be assumed to be equal in magnitude, i.e. the error ellipse around the point is circular. The RMS error of the N and E components is also the RMS error of the radial error of the position of the coordinates as measured from the point's actual position.
In other words, to be certain that one is meeting the ALTA/NSPS specification, the points need to be positioned with an RMS uncertainty significantly less than +/-5.8mm. If you are using a positioning technique that on a good day, under ideal conditions, just barely has an RMS uncertainty of 5.8mm, then the odds are very good that the results are substandard.
Note that the manufacturers of RTK GNSS equipment quote uncertainties of 8mm + 1ppm RMS for their equipment. That would only yield a relative positional uncertainty greater than 2.8cm between two points positioned at those same uncertainties. That is, the length of the semi-diameter of the relative error ellipse between the two points would be 8mm x 2.447 x 1.414 = 2.8cm = 0.09 ft.
As distances between corners increase, the 50ppm x D portion of the minimum positional uncertainty specification becomes significant and the tolerances loosen significantly. Where D = 160m (= 525 ft.), the maximum allowable length of the 95%-confidence relative error ellipse would increase to 2cm + 50ppm x 160m = 2.8cm, which the RTK coordinates measured with an uncertainty of 8mm + 1ppm might barely meet if the distance to the base was negligible.
The cure is simple. When the points positioned are close togther, use a better positioning technique than one that only delivers 8mm RMS. A total station should be able to most reliably do this.
Kent McMillan, post: 384187, member: 3 wrote: (Here is the original post revised to take into account the statistical detail that Bill93. pointed out)
The current MINIMUM STANDARD DETAIL REQUIREMENTS FOR ALTA/NSPS LAND TITLE SURVEYS (Effective 02/23/2016) specify that :
"The maximum allowable Relative Positional Precision for an ALTA/NSPS Land Title Survey is 2cm (0.07 feet) plus 50 parts per million (based on the direct distance between the two corners being tested)."
and it defines Relative Positional Precision as:
"the length of the semi-major axis, expressed in feet or meters, of the error ellipse representing the uncertainty due to random errors in measurements in the location of the monument, or witness, marking any corner of the surveyed property relative to the monument, or witness, marking any other corner of the surveyed property at the 95 percent confidence level."
So, how does one compute relative positional precision? The ALTA specification states that:
"Relative Positional Precision is estimated by the results of a correctly weighted least squares adjustment of the survey."
If you are independently determining the coordinates of two points by some method such as, say, RTK to pass the ALTA/NSPS accuracy standard, you want to be certain that the length of the semi-major axis of the relative error ellipse between two corners is no more than 2cm at the 95%-confidence level. By "independent" what is meant is that the random errors in present in the coordinates of one point are not related to those present in the coordinates of the other. In the situation where the uncertainties in the North and East components of both points are all the same (the error ellipses of both points are circular), what is the maximum uncertainty that is allowable for the position of each point?
The answer is that a RMS uncertainty greater than 5.8mm for both of the points will fail the ALTA/NSPS specification for their relative positional precision when they are close enough together that the magnitude of 50ppm x D is neglible, where D = Distance between points. By RMS, what is meant is the standard error of the N and E components of the coordinates of the points, which for the purposes of the general case may be assumed to be equal in magnitude, i.e. the error ellipse around the point is circular. The RMS error of the N and E components is also the RMS error of the radial error of the position of the coordinates as measured from the point's actual position.
In other words, to be certain that one is meeting the ALTA/NSPS specification, the points need to be positioned with an RMS uncertainty significantly less than +/-5.8mm. If you are using a positioning technique that on a good day, under ideal conditions, just barely has an RMS uncertainty of 5.8mm, then the odds are very good that the results are substandard.
Note that the manufacturers of RTK GNSS equipment quote uncertainties of 8mm + 1ppm RMS for their equipment. That would only yield a relative positional uncertainty greater than 2.8cm between two points positioned at those same uncertainties. That is, the length of the semi-diameter of the relative error ellipse between the two points would be 8mm x 2.447 x 1.414 = 2.8cm = 0.09 ft.
As distances between corners increase, the 50ppm x D portion of the minimum positional uncertainty specification becomes significant and the tolerances loosen significantly. Where D = 160m (= 525 ft.), the maximum allowable length of the 95%-confidence relative error ellipse would increase to 2cm + 50ppm x 160m = 2.8cm, which the RTK coordinates measured with an uncertainty of 8mm + 1ppm might barely meet if the distance to the base was negligible.
The cure is simple. When the points positioned are close togther, use a better positioning technique than one that only delivers 8mm RMS. A total station should be able to most reliably do this.
I wonder what my measuring wheel can do?
I'm not so sure about all of that. Please refer to page 13 of the linked pdf:
http://www.eng.usf.edu/~tdavis/resume/confidence%20regions.pdf
2.44 isn't the multiplier from 1 Sigma to 2 Sigma for 2D. According to the chart 1 Sigma is already 1.515 in 2D. So to go from 1.515 to 2.44 you multiply by 1.6.
I'm still thinking 9mm RMS (radial/2D) at each point will satisfy the requirement.
1.6*SQRT(9mm^2+9mm^2) = 20mm
Shawn Billings, post: 384193, member: 6521 wrote: 2.44 isn't the multiplier from 1 Sigma to 2 Sigma for 2D. According to the chart 1 Sigma is already 1.515 in 2D. So to go from 1.515 to 2.44 you multiply by 1.6.
2.447 actually is the multiplier for the semi-major axis of the 95%-confidence error ellipse where the standard errors of the N and E components are identical. As Bill93 correctly reminded me above, 2-sigma is not the limit of the 95% confidence interval for a bivariate distribution such as that of the radial error of a point position that is the root sum of squares of two variables (the N and E components) with a normal or Gaussian distribution.
Page 4 would seem to agree with what you and Bill93 are saying.
Shawn Billings, post: 384217, member: 6521 wrote: Page 4 would seem to agree with what you and Bill93 are saying.
I think that the conclusion to be drawn is that in testing the positioning uncertainty of RTK, it's useful to compute the standard errors of N and E (and their covariances).
I'll see if I can revisit my data.
