lecture 17: geodetic datums and a little linear regression
DESCRIPTION
Lecture 17: Geodetic Datums and a little Linear Regression. GISC-3325 24 March 2008. Update. Reading for next few classes Chapters Eight and Nine Start reading GPS Positioning Guide Chapters 1-3. Exam results and answers are posted to class web page. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/1.jpg)
Lecture 17: Geodetic Datums and a little Linear Regression
GISC-3325
24 March 2008
![Page 2: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/2.jpg)
Update
• Reading for next few classes– Chapters Eight and Nine– Start reading GPS Positioning Guide
Chapters 1-3.
• Exam results and answers are posted to class web page.
• Extra credit opportunities are available and should be discussed with Instructor
![Page 3: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/3.jpg)
Azimuth for DRISCOLL to DRISCOLL RM 1 is taken from the datasheet.
![Page 4: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/4.jpg)
Datum
• “A datum is any quantity or set of quantities that may serve as a referent or basis for calculation of other quantities.”source: NGS Glossary
• Historically there were separate bases for horizontal and vertical datum.
![Page 5: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/5.jpg)
Vertical Datum
• A set of fundamental elevations to which other elevations are referred.
• National Geodetic Vertical Datum of 1929 (NGVD 29) formerly known as Mean Sea Level Datum of 1929.– Because mean sea level varies too much!
• North American Vertical Datum of 1988– Readjustment not referenced to mean sea
level.
![Page 6: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/6.jpg)
Global Sea Level
![Page 7: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/7.jpg)
NGVD 29• Define by heights at 26 tide stations in the
US and Canada.
• Gages connected to vertical network by leveling
• Water-level transfers to connect leveling across the Great Lakes
• Used normal orthometric heights– scaled geopotential numbers using normal
gravity
![Page 8: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/8.jpg)
![Page 9: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/9.jpg)
Problems with NGVD 29
![Page 10: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/10.jpg)
NAVD 88
• Datum based on an equipotential surface
• Minimally constrained at one point: Father Point/Rimouski on St. Lawrence Seaway
• 1.3 million kilometers of level data
• Heights of 585,000 permanent monuments
![Page 11: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/11.jpg)
Father Point/Rimouski
![Page 12: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/12.jpg)
Elements of NAVD 88
• Detected and removed height errors due to blunders
• Minimized effects of systematic errors in leveling data– improved procedures better modeling
• Re-monumentation and new leveling
• Removal of height discrepancies caused by inconsistent constraints.
![Page 13: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/13.jpg)
![Page 14: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/14.jpg)
![Page 15: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/15.jpg)
![Page 16: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/16.jpg)
Height Relations
h – H – N = zero + errors
![Page 17: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/17.jpg)
![Page 18: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/18.jpg)
h – H – N ≠ 0 WHY?
H
h
N
![Page 19: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/19.jpg)
NAVD 88 height by GPS = 1.83 m
NAVD 88 height adjusted = 1.973 m
Difference = 0.14 m
![Page 20: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/20.jpg)
New vertical datum to be based on h (ellipsoid heights) and N (gravimetric geoid model). Remember: h – H – N = 0 plus errors
![Page 21: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/21.jpg)
Vertical Datum Transformations
• First choice: Estimate heights using original leveling data in least squares
• Second choice: Rigorous transformation using datum conversion correctors estimated by adjustment constraints and differences
• Third option: VERTCON
![Page 22: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/22.jpg)
Linear Regression
• Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data.
• A linear regression line has an equation of the form Y = mX + b, where X is the explanatory variable and Y is the dependent variable. The slope of the line is m, and b is the intercept (the value of y when x = 0).
![Page 23: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/23.jpg)
![Page 24: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/24.jpg)
Results in Excel
http://phoenix.phys.clemson.edu/tutorials/excel/regression.html
![Page 25: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/25.jpg)
Why not Matlab?
![Page 26: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/26.jpg)
Matlab to the rescue!
![Page 27: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/27.jpg)
![Page 28: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/28.jpg)
![Page 29: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/29.jpg)
Rod Calibration
![Page 30: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/30.jpg)
![Page 31: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/31.jpg)
Two-Plane Method of Interpolating Heights (Problem 8.3)
• We can approximate the shift at an unknown point (when observations are unavailable) using least squares methods.– Need minimum of four points with known
elevations in both vertical datums.– Need plane coordinates for all points.– Calculates rotation angles in both planes (N-S
and E-W) as well as the vertical shift.
![Page 32: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/32.jpg)
Problem 8.3 in text
Benchmark NGVD 29 Height ft.
NAVD 88 m
Northing Easting
Q 547 4088.82 1247.360 60,320 1,395,020
A 15 4181.56 1275.636 60,560 1,399,870
AIRPORT 2 4085.32 1246.314 56,300 1,397,560
NORTH BASE 4191.80 1278.748 57,867 1,401,028
T 547 4104.04 Unknown 58,670 1,397,840
![Page 33: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/33.jpg)
Function model
• (NAVD88i-NGVD29i)=αE(Ni-N0)+ αN(Ei-E0)+tZ
• Where we compute the following (all values in meters):– NAVD88i-NGVD29i = difference in heights
– Ni-N0 = is difference of each North coordinate of known points from centroid
– Ei-E0 = is difference of each East coordinate of known points from centroid
![Page 34: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/34.jpg)
Solving Problem
• Determine the mean value (centroid) for N and E coordinates (use known points only)– N0: 58762 E0: 1398370 (wrong in text)
• Determine NAVD 88 - NGVD 29 for points with values in both systems. Note signs!
Δ Q 547 = 1.085
Δ A 15 = 1.094
Δ AIRPORT 2: = 1.106
Δ NORTH BASE = 1.085
![Page 35: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/35.jpg)
Compute differences from centroid
Station Difference in N Difference in E
Q 547 1558 -3350
A 15 1798 1500
AIRPORT 2 -2462 -810
NORTH BASE -895 2658
![Page 36: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/36.jpg)
Compute parameters
• B the design matrix consists of three columns:– Col.1: difference in Northings from centroid– Col.2: difference in Eastings from centroid– Col.3: all ones
• F the observation matrix– Vector of height differences
• Parameters are computed by least squares: (BTB)-1BTf
![Page 37: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/37.jpg)
![Page 38: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/38.jpg)
Applying parameters
• Our matrix inversion solved for rotations in E and N as well as shift in height.
• Compute the shift at our location using our functional model: αE(Ni-N0)+ αN(Ei-E0)+tZ – Result is the magnitude of the shift.
• We calculate the new height by adding the shift to the height in the old system.
![Page 39: Lecture 17: Geodetic Datums and a little Linear Regression](https://reader035.vdocuments.site/reader035/viewer/2022062518/5681495c550346895db6aed6/html5/thumbnails/39.jpg)
We validate the accuracy of our result by computing the variances.