1. SVY207: Lecture 10 Computation of Relative Position from Carrier Phase u Observation Equation u Linear dependence of observations u Baseline solution –Weighted least squares u Statistical dependence of observations –Double difference weight matrix

2. Observation Equation u Double differenced carrier phase (metres)   L AB jk   AB jk  AB jk    N AB jk  where:  AB jk  (  A j   B j  (  A k   B k )  A j  ((x j  x A ) 2  (y j  y A ) 2  (z j  z A ) 2 )  AB jk   atmospheric delay  N AB jk   phase ambiguity –For simplicity, from now on:  Assume  AB jk is negligible for short baselines  Drop the  notation –Observation equation for our purposes: L AB jk   AB jk   N AB jk

3. Double Differencing Data:L AB jk  L A j  L B j   L A k  L B k  Model:L AB jk  AB jk   N AB jk AjAj BjBj AkAk BkBk Station A records L A j and L A k at epochs 1, 2, 3,... Station B records L B j and L B k at epochs 1, 2, 3,... Satellite j Satellite k

4. Linear Dependence of Observations  3 Combinations:L AB jk  L A j  L B j   L A k  L B k  L AB jl  L A j  L B j   L A l  L B l  L AB lk  L A l  L B l   L A k  L B k  Station A records L A j, L A k, L A l at epochs 1, 2, 3,... AjAj BjBj AkAk BkBk Station B records L B j, L B k, L B l at epochs 1, 2, 3,... Satellite j Satellite k Satellite l AlAl BlBl

5. Linear Dependence of Observations F Double differences can be linearly dependent –Note that:L AB lk = L AB jk  L AB jl –Therefore, data from L AB lk provides no new information –Similarly:L AB jk = L AB lk  L AB jl L AB jl = L AB jk  L AB lk –The set  L AB jk, L AB jl, L AB lk  is linearly dependent –A linearly independent set is needed for least squares –Examples of linearly independent sets: set  j = {L AB ab | a=j, b  a} = (L AB jk, L AB jl ) set  k = {L AB ab | a=k, b  a}= (L AB kj, L AB kl ) set  l = {L AB ab | a=l, b  a}= (L AB lj, L AB lk )

6. Linear Dependence of Observations F Reference satellite concept –One method of ensuring linear independence of data for baseline solution (between stations A and B) –Select a reference satellite. –For example, satellites 1, 2, 3, 4, and 5 are tracked. –Select reference satellite 3:  3 =  L AB jk | j=3, k  3  =  L AB 31, L AB 32, L AB 34, L AB 35  –All possible sets are valid F Reference satellite selection –Select satellite which has longest data span –Better: select the best reference satellite every epoch

7. Linear Dependence of Observations F Reference station/satellite concept F For network solutions –between stations A, B, C,... –Select a reference satellite and a reference station –For example, satellites 1, 2, 3, 4, and stations A, B, C –Select reference satellite 4 and reference station A  A 4 =  L ab jk | j=4, k  4, a=A, b  A  =  L AB 41, L AB 42, L AB 43, L AC 41, L AC 42, L AC 43   Number of independent observations at each epoch N   (N stations  1)(N satellites  1) –Example: 3 stations and 4 satellites give 6 observations

8. –Linearised model: b  Ax    noise  b  column of observation residuals  observed (L AB jk )  computed (  AB jk   N AB jk ) –select a reference satellite j, and only include  j –initially use provisional coordinates and N   x  column of parameters  adjustment to provisional values x, y, z, and N –If our estimate of N leads to an obvious integer, then fix this integer in computed model (“fixed” solution)  A  design matrix (partial derivatives  L/  u, etc.) –Weighted least squares solution: Baseline Solution

9. Statistical Dependence of Observations F Double differences are also statistically dependent –L AB 12 and L AB 13 are correlated due to data in common L A 1 –An error in L A 1 will affect both L AB 12 and L AB 13 –Positive error in L AB 12 usually => positive error in L AB 13 F Weighted least squares is appropriate –“Stochastic model” is needed to derive weight matrix –Weight matrix is the inverse of the covariance matrix for the data: W = C  1  What is the covariance matrix for double differenced data, C  ?

10. Rule of propagation of errors F In general, consider a vector X with covariance C X  What is the covariance for Y  KX ? (K is a matrix)  Solution: C X   X X T   means “expected value” C Y   Y Y T  define:Y  KX   KX (KX) T  recall: (AB) T  B T A T   KX X T K T  K  K (constant)  K  XX T  K T C Y  K C X K T

11. Double Difference Weight Matrix F Apply propagation of errors to double differencing –We can write  L = D L –Where D is a matrix with elements of  or  Number of columns = number of observations Number of rows = number of independent double differences –Double differenced data covariance: C  = D C D T –Hence, weight matrix for double differenced data is: W  = (D C D T )  –For C, use a diagonal matrix, assuming a value for the standard deviation of an observation Assumes observation errors are uncorrelated Realistic value for observation error ~ few mm