Everything you did not want to know about least squares and positional tolerance! (in one hour or less) Raymond J. Hintz, PLS, PhD University of Maine

Transcription

1 Everything you did not want to know about least squares and positional tolerance! (in one hour or less) Raymond J. Hintz, PLS, PhD University of Maine Least squares is used in varying degrees in -Conventional survey GPS Photogrammetry GIS Datum transformations 1

2 Most common form of least squares is an observation equation model The model is an equation which defines the measurement in terms of parameters being solved for Conventional survey In plane surveying distances are described by the Pythagoreum theorem, and angles and bearings are described by the tangent inverse. An elevation difference is simply described by a coordinate difference. 2

3 GPS processing raw to vectors (generally the fixed ambiguity solution) is a complex observation! Network adjustments are simply described by coordinate differences derived from raw to vector processing In RTK localizing is a 3-D coordinate transformation Geoid models are derived from least squares collocation Photogrammetry measuring fiducials to set up a photo coordinate system is a 2-D coordinate transformations Absolute orientation is a 3-D coordinate transformation Aerotriangulation, defined by collinearity, is the equivalent of the survey network adjustment and is called the bundle adjustment 3

4 GIS Primarily 2-D coordinate transformations, with polynomials often added to the simply model to try to estimate systematic coordinate errors. Datum transformations Least squares collocation is used to define a generic model which defines based on location what coordinates shifts are estimated. The least squares defines the model! 4

5 Many least squares solutions are equal weighted (often called unweighted) but mixing observation types (distances and angles) forces weighting as units differ. Weight = (1 / error estimate) squared 5

6 If one cannot estimate the errors in one s measurements, one is not surveying (said by a lost and obliterated surveyor many years ago) Individual measurements after least squares have a residual associated with them. A residual is how much the measurement adjusted. 6

7 Example measured distance is , by coordinate inverse it is , the residual is 0.02 You determine is that 0.02 is too much by comparing it to the error estimate 7

8 The error estimate not is a thou shalt not exceed. By the bell shaped curve 33% of your residuals will be larger than the error estimate. But you are 95% sure something is wrong if your residual is more than 3 times its error estimate. 8

9 What can cause residuals more than 3 times the error estimate? Poor measurement, incorrect point id, incorrect control coordinates, etc., etc. etc. its like a puzzle and you have to figure it out. Least squares condition Minimize the sum of the squares of the weighted residuals Sum {[residual * (1/error est.)]2} Note residual/error est. will center around 1.0 if nothing is unusual 9

10 Overall least squares indicators Standard error of unit weight Sqrt (sum of square of weighted residuals / degrees of freedom) Degrees of freedom is how much redundancy you have a loop traverse has 3 degrees of freedom which relate to sum of latitudes =0, Sum of departures = 0, and (n-2)*180 for angular closure. 10

11 Standard error of unit weight should be near 1. You get a larger denominator if you have more checks. More checks allow residuals to be more effectively modeled. Chi squared test at 95% confidence were your error estimates valid based on standard error of unit weight. 11

12 Do you have to pass the chi squared test? No, look at your residuals, and judge for yourself if you can live with the results. You can play with error estimates & pass the chi squared test. Other quality indicators Root mean square (rms) error average residual for a particular type of measurement limited in use if error estimates vary all overall statistics in large adjustments can hide individual outliers 12

13 Some day a big indicator may be post adjustment standard deviations (repeatability) of adjusted quantities such as coordinates positional tolerance normally these values are produced at 95% confidence to ensure a more sure region of reproducibility Positional tolerance is the term being used for post-adjustment coordinate reliabilities It should be obvious the further from control a coordinate is it is harder to reproduce as all survey measurements propagate error 13

14 In other words a point 50 miles from a control point is more difficult to reproduce its coordinates to 0.10 ft. than a point 50 ft. from a control point. Can positional tolerance simplify numerical quality standards? - It is a goal oriented standard instead of a procedure standard. 14

15 Example GPS control standards (1) You must tie to n control points. (n= 2, 3, or 4 probably) Example GPS control standards (2) You must occupy each point twice (or else how do you check setup) 15

16 Example GPS control standards (3) Error estimates on all vectors will be x meters plus y ppm (the error estimate model - example meters plus 10 ppm) is defined. 16

17 deg. Of freedom F statistic multiplier

18 Example GPS control standards (4) Coordinate positional tolerance at 95% confidence will not exceed x meters plus y ppm (based on distance to closest control station) ALTA Measurement Standards - The following measurement standards address Relative Positional Precision for the monuments or witnesses marking the corners of the surveyed property. i. Relative Positional Precision means 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. Relative Positional Precision is estimated by the results of a correctly weighted least squares adjustment of the survey. 18

19 ii. Any boundary lines and corners established or retraced may have uncertainties in location resulting from (1) the availability, condition, history and integrity of reference or controlling monuments, (2) ambiguities in the record descriptions or plats of the surveyed property or its adjoiners, (3) occupation or possession lines as they may differ from the written title lines, and (4) Relative Positional Precision. Of these four sources of uncertainty, only Relative Positional Precision is controllable, although due to the inherent errors in any measurement, it cannot be eliminated. The magnitude of the first three uncertainties can be projected based on evidence; Relative Positional Precision is estimated using statistical means (see Section 3.E.i. above and Section 3.E.v. below). iii. The first three of these sources of uncertainty must be weighed as part of the evidence in the determination of where, in the surveyor s opinion, the boundary lines and corners of the surveyed property should be located (see Section 3.D. above). Relative Positional Precision is a measure of how precisely the surveyor is able to monument and report those positions; it is not a substitute for the application of proper boundary law principles. A boundary corner or line may have a small Relative Positional Precision because the survey measurements were precise, yet still be in the wrong position (i.e. inaccurate) if it was established or retraced using faulty or improper application of boundary law principles. 19

20 iv. For any measurement technology or procedure used on an ALTA Land Title Survey, the surveyor shall (1) use appropriately trained personnel, (2) compensate for systematic errors, including those associated with instrument calibration, and (3) use appropriate error propagation and measurement design theory (selecting the proper instruments, geometric layouts, and field and computational procedures) to control random errors such that the maximum allowable Relative Positional Precision outlined in Section 3.E.v. below is not exceeded. v. The maximum allowable Relative Positional Precision for an ALTA Land Title Survey is 2 cm (0.07 feet) plus 50 parts per million (based on the direct distance between the two corners being tested). It is recognized that in certain circumstances, the size or configuration of the surveyed property, or the relief, vegetation or improvements on the surveyed property will result in survey measurements for which the maximum allowable Relative Positional Precision may be exceeded. If the maximum allowable Relative Positional Precision is exceeded, the surveyor shall note the reason as explained in Section 6.B.ix below. 20

21 Improvements in ALTA (1) It used to say must be derived from a minimally constrained adjustment which means only one control point can be used GONE!! (2) Used to say 95% confidence was 2 standard deviations which is only correct at a high degree of freedom (if you used F statistic multiplier) and if you round to one significant figure GONE!! It is very impressive these issues in ALTA standards have been taken care of thanks! Are we into absolute or relative positional tolerance? Boundary obviously the bearing and distance between monumented corners has been the ultimate product. But note for the last 30 years bearings and distances are always derived from coordinates. We thus need to know relative coordinate change between boundary corners and today don t need the absolute coordinates BUT TODAY THE EASIEST THING TO RELOCATE ARE THE ABSOLUTE COORDINATES!!!!! 21

22 (1) Absolute error ellipses Relative to nearest control station Example - we have a semi-major axis of 0.10 ft. and are 1000, 2000, and 3000 ft. from 3 control stations. You use the closest 1000 ft. (straight line not traverse distance) Allowable by ALTA (1000/20000) = actual < 0.12 allowable you pass!!! (2) Relative error ellipses The ellipse can be computed between any two points in your survey (independent of absolute coordinates) I don t like this one because if you are checking 10 stations you have to check lines. So in a big survey it seems like an amazing way to proceed Obviously you are allowed more error on a longer inversed line 22

23 (3) No one considers this one BUT Post adjustment standard deviations in final bearings and distances on a survey product I guess we don t use this one because people think what we show is perfect! Lets look at real data!!! 23