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SVY207: Lecture 18Network Solutions

• Given many GPS solutions for vectors between pairs of observed stations

• Compute a unique network solution (for many stations)

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• 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

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• Triangular Network– 3 stations P, Q, R,

3 sessions, observe a vector each session: QP, QR, RP

– 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

QR

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Network Computation:Where to start?

• Write down the observation equations:xQ-P xQxPv1

yQ-P yQyPv2

zQ-P zQzPv3

xQ-R xQxRv4

yQ-R yQyRv5

zQ-R zQzRv6

xR-P xRxPv7

yR-P yRyPv8

zR-P zRzPv9

–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

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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

b Ax

x

y

z

x

y

z

x

y

z

x

y

z

x

y

z

Q P

Q P

Q P

Q R

Q R

Q R

R P

R P

R P

Q

Q

Q

R

R

R

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

1 0 0 1 0 0

0 1 0 0 1 0

0 0 1 0 0 1

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

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Design Matrix, A• Dimensions

columns = parameters = 6 (coords of Q and R)

rows = observations = 9 (coords of QP, QR, RP)

• Example:

– Easy to figure out A by inspection

Ax

x x

Axx x

Ayx x

QQ R

RQ R

RQ R

414

1

444

4

454

5

1

1

0

b

x

b

x

b

x

A

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0

0 1 0 0 1 0

0 0 1 0 0 1

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

1 1 0

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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

W C

C

C

C

C

C

C

Q P

Q R

R P

Q P

Q R

R P

1

1 1

1

1

0 0

0 0

0 0

0 0

0 0

0 0

CQ P

x xy xz

xy y yz

xy yz z Q P

2

2

2

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Weighted Least Squares• Computation

– observation eqn: Ax = b + v

– WLS solution : x (ATWA)1ATW b

– Covariance of estimates: Cx (ATWA)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 10-20 m

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Weighted Least Squares• Notes on computed errors

– Covariance Cx (for Q and R coords) should be interpreted in as position errors relative to fixed station (P)

– Cx 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

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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