Class 25: Even More Corrections and Survey Networks Project Planning 21 April 2008

Deadlines Reading assignments (2) due 30 April 2008 –Sample of what I expect is posted to class wiki page Extra credit due 23 April 2008 No lab assignments/homework will be accepted after today.

Textbook EM , Chapters 4-10 Geodesy text, Chapter 11

Does distance matter in GPS? Article available on NGS web site

Plumb line Indicates the average direction of gravity between the point of suspension and the plumb bob. It is perpendicular to every each level surface it intersects. –Equipotential surfaces are NOT parallel Effected by mass of the earth –Oceans are a mass deficiency –Mountains a mass concentration

Astro-Geodetic

Deflection of the Vertical Angle between the plumb line and a normal to the ellipsoid is the deflection of the vertical. City of Corpus Christi, TX Ft. Davis, TX Denali, AK

Computation of Deflection Has a North South component ξ (xi) and East West component η (eta) Computed with respect to astronomical azimuth (A) –ξ = θ cos A and η = θ sin A Positive ξ indicates astronomic latitude will fall north of the geodetic latitude Positive η indicates astronomic longitude will fall east of geodetic longitude

Mason-Dixon Line Monument Latitude (  ) Xi ( ξ) Lat. Seconds ( Φ) 50N N N N Near Warfordsburg N MARPENNN  = Φ - ξ

Distance Corrections Refraction – may be overcome by reciprocal vertical/zenith angles –Otherwise: k is coefficient of atmospheric refraction R is mean earth radius h A is ellipsoid height of instrument station

Curvature Correction Accounts for the fact that plumb lines are not parallel at different locations on earth’s surface.

Geodetic Distance Computed from level distance (LD) –LD * [ R / (R + h) ] –LD * [ R / (R + H + N) ] Use R = 6,371,000 m –H = Orthometric Height –h = ellipsoid height –N = geoid height

Planning GPS Control Surveys Your plan will be developed in accordance with your client’s goals and the relevant standards. EM describes work for the USACE. –These differ from FGCS standards and specifications.

Issues in Any Plan Project Datum –Mostly will be NAD 83 (but which version?) Intended accuracy –Horizontal, Vertical or both Which height system (ellipsoidal or NAVD 88) More accuracy = More $$$$ Monumentation Equipment needed (and available)

Project Planning/Standards and Specifications See class page for links to relevant documents.

Meeting standards We must have redundant observations in order to evaluate their precision. We must have ties to fixed control (more than one) in order to attach our observations to this framework as well as to verify the relationship of the fixed points. When our observations are more accurate than the fixed network, we degrade them to fit the network.

Level Data Reduction and Adjustment 5

Leveling summary Difference in elevation for section (edge) Section length kilometers A to D = A to E = C to D = B to E = E to D =

Populating the design matrix Height difference (i to j) is equal to the observed difference and its residual/variance. Line 1. D – A = variance Line 2. E – A = v Line 3. D – C = v Line 4. E – B = v Line 5. D – E = v

Design Matrix (Free) Nodes ->ABCDE Section Section Section Section Section “From” station gets a “-1” “To” station gets a “1”

Note diagonal elements!

Observations/Distances Section Height Diff. (m) Distances (KM)

Account for Known BMs Review diagram Solve for heights where direct connection to unknown 5

Updated Matrices DE HeightD.ESum

Rearrange to put BMs on right side D = v E = v D = v E = v D – E = v

Why not just mean heights? DE Mean: Mean: How do we account for the difference in elevation from E to D? How accurate is our result?

Matlab Un-weighted Result Calculate heights Calculate residuals

Weight Matrix We assume that error will accumulate as a function of distance. Weights assigned as 1/dist KM. SectionLengthWeight A to D A to E C to D B to E E to D

Weight Matrix P used for weight matrix (also W) Diagonal Matrix assumes no correlations Applied to both design matrix and observation matrix X = A T PA -1 *A T PL

D = E =

Differences in Results Simple MeanLeast Squares Un- weighted Least Squares Weighted D = E = ResidualResidual (weighted)