1 SVY207: Lecture 18 Network Solutions Given many GPS solutions for vectors between pairs of observed stations Compute a unique network solution (for many stations)

2 Supposing: –you have 2 GPS receivers but 4 stations to survey –1 station has known coordinates –other 3 stations are to be positioned precisely So: –you survey all possible (6) vectors in separate sessions –data processing gives coordinates for all 6 vectors But you need: –a unique solution for the 3 unknown stations –which is free of obvious blunders Motivation

3 Triangular Network –3 stations P, Q, R, 3 sessions, observe a vector each session: Q  P, Q  R, R  P –solve for position of Q and R (holding P fixed) Step 1: (e.g., Geogenius) –Apply GPS processing software to each observed vector –Produces coordinates and covariance for each vector Step 2: (e.g., Geogenius) –Apply “network adjustment” software to all GPS solutions –Produces coordinates and covariance for Q and R positions Network Solution Example P Q R

4 Network Computation: Where to start? Write down the observation equations: x Q-P  x Q  x P  v 1 y Q-P  y Q  y P  v 2 z Q-P  z Q  z P  v 3 x Q-R  x Q  x R  v 4 y Q-R  y Q  y R  v 5 z Q-R  z Q  z R  v 6 x R-P  x R  x P  v 7 y R-P  y R  y P  v 8 z R-P  z R  z P  v 9 –on the left side are the GPS relative coordinates for the observed vectors » given by the GPS software » input to the network computation » these are treated as observations –on the right side is the model » similar to levelling - but in 3D » position coordinates of Q and R (in bold and italics) are treated as parameters to be estimated » arbitrary coordinates chosen for P

5 Preparation for Least Squares Linearize functional model, and put into matrix form: –That was easy – because equations are already linear –As usual, interpret each term as a “correction” to provisional values

6 Design Matrix, A Dimensions columns = parameters = 6 (coords of Q and R) rows = observations= 9 (coords of Q  P, Q  R, R  P) Example: –Easy to figure out A by inspection

7 Weight Matrix, W W is the inverse covariance for observations Here, the “observations” are GPS solutions for the relative coordinates of each observed vector –3x3 covariance matrix for each vector from GPS software –e.g., for vector Q-P Construct 9x9 covariance for all 9 “observations” –invert this to form the weight matrix

8 Weighted Least Squares Computation –observation eqn:Ax = b + v –WLS solution : x  (A T WA)  1 A T W b –Covariance of estimates:C x  (A T WA)  1 Notes on fixed station: –one station (P) is not estimated in network solution –can use any value you like for coordinates of P –estimated positions of Q and R should be interpreted as being dependent on the choice of coordinates for P –can fix its value to the pseudorange point position, but take care not to over-interpret results: a point position might be in error by up to m

9 Weighted Least Squares Notes on computed errors –Covariance C x (for Q and R coords) should be interpreted in as position errors relative to fixed station (P) –C x is determined »network geometry (i.e. which vectors are observed?) »number of vector solutions –Network geometry (GPS contrasted with classical) »vector observations are geometrically far more robust compared to distance or angle observations »no problem with “long/thin” networks »with GPS, long distances are estimated precisely, so better to include direct observation of the longer vectors in the network –Data redundancy important for blunder detection »each station should be in at least 3 sessions

10 Error Assessment Internal (Precision): –Vector coordinate residuals from network solution »should behave as expected for precision GPS –Unit variance »is the scatter of residuals as low as expected? –Goodness of fit »are these residuals normally distributed? –Internal Reliability »theoretical detection level for badly-fitting vectors »good surveys have high internal reliability »requires high redundancy External (Accuracy) –External Reliability »Effect of an undetected blunder on final coordinates –External comparison of solution with another method »Problem: cannot rely on OSGB36 for accuracy