Introduction to Least Squares Adjustment for geodetic VLBI

Transcription

1 Introduction to Least Squares Adjustment for geodetic VLBI Matthias Schartner a, David Mayer a a TU Wien, Department of Geodesy and Geoinformation

2 Least Squares Adjustment why? observation is τ (baseline) unknown parameters are: station positions EOP source positions troposphere... How do we link our observations to our unknown parameters? AVN school 1/ 19 (45)

3 A very simple example three sample points data fitting polynome of order 2: y = ax 2 + bx + c observations: (x i, y i ) unknowns: a, b, c AVN school 2/ 19 (45)

4 A very simple example (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) y 1 = ax1 2 + bx 1 + c y 2 = ax2 2 + bx 2 + c y 3 = ax3 2 + bx 3 + c Or in matrix notation: y 1 x y 2 = 1 2 x 1 1 a x2 2 x 2 1 b x3 2 x 3 1 c y AVN school 3/ 19 (45)

5 A very simple example y 1 x y 2 = 1 2 x 1 1 a x2 2 x 2 1 b y 3 x3 2 x 3 1 c solve the equations a x b = 1 2 x y 1 x2 2 x 2 1 y 2 c x3 2 x 3 1 y 3 coefficients for y = ax 2 + bx + c AVN school 4/ 19 (45)

6 A very simple example Warning! Every observation has an error How accurate are our coefficients a, b and c? What can we do? If we have more observations we can average out the error AVN school 5/ 19 (45)

7 A very simple example What happens if I have more observations? y = ax 2 + bx + c three unknowns a, b, c (n unk ) many observations (n obs ) n obs > n unk How do we get our unknown parameters? AVN school 6/ 19 (45)

8 A very simple example simply pick three: (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) y 1 = ax1 2 + bx 1 + c y 2 = ax2 2 + bx 2 + c y 3 = ax3 2 + bx 3 + c Or in matrix notation: y 1 x y 2 = 1 2 x 1 1 a x2 2 x 2 1 b x3 2 x 3 1 c y AVN school 7/ 19 (45)

9 A very simple example y 1 x y 2 = 1 2 x 1 1 a x2 2 x 2 1 b y 3 x3 2 x 3 1 c solve the equations a x b = 1 2 x y 1 x2 2 x 2 1 y 2 c x3 2 x 3 1 y 3 coefficients for y = ax 2 + bx + c AVN school 8/ 19 (45)

10 A very simple example Why not this three? AVN school 9/ 19 (45)

11 A very simple example 50 or this three: Our goal: we want to use every available observations AVN school 10/ 19 (45)

12 A very simple example y 1 x 2 1 x 1 1 y 2 x 2 = 2 x 2 1 a b.... c y n xn 2 x n 1 }{{}}{{}}{{} l A x A R n obs n unk x R n unk l R n obs A x = l (1) AVN school 11/ 19 (45)

13 A very simple example trick and solution A A x = A l (2) ( 1 x = A A) A l (3) residuals v: v i = ax 2 i + bx i + c y i v = A x l (4) minimization v 2 = v v (5) AVN school 12/ 19 (45)

14 A very simple example AVN school 13/ 19 (45)

15 A very simple example conclusion / goal get the most reasonable values for our unknown parameters take all available information more observations then unknown parameters get some error margins How can we do this? Warning! every observation has an error AVN school 14/ 19 (45)

16 Least Squares Methode pros easy while powerful minimize residuals rigorous statistical approach individual weighting (28) accuracy information (32) manipulate geodetic datum (37) combine results of various LSMs (42) cons only for linear functions liniarization via Taylor polynome (22) a priori values iterations (31) we need to model our observations o c outliers requirements for LSM partial derivatives (24) AVN school 15/ 19 (45)

17 What do we want to estimate? station coordinates source coordinates earth orientation parameters tropospheric parameters clock behavior and many more... Many unknown parameters AVN school 16/ 19 (45)

18 What do we want to estimate? Example: Scan with 5 stations = 10 observations Estimates: Troposphere: (15p.) zenith wet delay (5p.) north gradient (5p.) east gradient (5p.) clock (5p.) Parameters Way more unknown parameters then observations Requirement for LSM We need more observations then unknowns! n obs > n unk AVN school 17/ 19 (45)

19 What do we want to estimate? Solution piecewise linear offsets (34) we estimate parameters only every x minutes linear interpolation in between reduces number of estimated parameters! Improvement constraints (35) zwd sta1,03:00 = zwd sta1,04:00 ±1.5 [cm] }{{} constraint AVN school 18/ 19 (45)

20 Conclusion What should you remeber? we calculate our solutions using Least Squares Adjustment we minimize (squared sum of) residuals we need to model our observations! theoretical delay o c we need partial derivatives for our Jacobian matrix A we need a priori values X 0 (usually not a big deal) we estimate additions x to a priori values X 0 we need to eliminate outliers AVN school 19/ 19 (45)

21 Example P2 P3 5 stations 3 with known coordinates 2 with unknown coordinates P5 P4 P AVN school 20/ 19 (45)

22 Example 2 measured distance s ij unknowns: (x4, y4) (x5, y5) n unk = 4 observations: s ij n obs = 7 A R 7 4 x R 4 l R 7 n obs > n unk P2 P3 P5 P4 P AVN school 21/ 19 (45)

23 Some basics about LSM We need to model our system: s ij = (x j x i ) 2 + (y j y i ) 2 linearization not linear Taylor expansion: T,x0 (x) = f (x 0 ) + f (x 0 ) 1! = n=0 f (n) (x 0 ) (x x 0 ) n n! liniarized observation equation What do we need? We need partial derivative We need a priori values x 0 (x x 0 ) + f (x 0 ) (x x 0 ) ! AVN school 22/ 19 (45)

24 Some basics about LSM definitions observations: L (computed) approximated observations: L 0 reduced observation vector (observed minus computed): l l = L L 0 (6) estimated unknowns: ˆX approximated unknowns: X 0 reduced unknowns: x x = ˆX X 0 (7) Designmatrix or Jacobian matrix: A (partials) AVN school 23/ 19 (45)

25 Back to Example 2 unknown: x 4, y 4, x 5, y 5 known: x 1, y 1, x 2, y 2, x 3, y 3 approximate values: x 4,0, y 4,0, x 5,0, y 5,0 s ij = (x j x i ) 2 + (y j y i ) 2 ( ) s14 ( x 4 ) s15 x 4 A = (. ) s45 x 4 ( ) s14 y 4 ( ) s15 y 4 (. ) s45 y 4 ( ) s14 x 5 ( ) s15 x 5 (. ) s45 x 5 ( ) s14 ( y 5 ) s15 y 5 (. ) s45 y 5 s 14 x 4 = x 4,0 x 1 ( x4,0 x1) 2 + ( y 4,0 y 1 ) 2 s 14 y 4 = y 4,0 y 1 ( x4,0 x1) 2 + ( y 4,0 y 1 ) 2 s 14 x 5 = s 14 y 5 = AVN school 24/ 19 (45)

26 Example 2 unknown: x 4, y 4, x 5, y 5 known: x 1, y 1, x 2, y 2, x 3, y 3 approximate values: x 4,0, y 4,0, x 5,0, y 5,0 reduced observation vector (observed minus computed) l = L L 0 s 14 s 14,0 s 15 s 15,0 l =. s 45 s 45,0 s 14,0 = (x 4,0 x 1 ) 2 + (y 4,0 y 1 ) AVN school 25/ 19 (45)

27 Example 2 reduced observation vector (observed minus computed) l = L L 0 Jacobian matrix (partials): A reduced unknowns: x = X X 0 trick Ax = l A Ax = A l ( 1 x = A A) A l (8) What do we estimate? We don t estimate whole unknown parameter ˆX but only additions x to some a priori values X AVN school 26/ 19 (45)

28 Example The coordinates of our unknown points change P5(ˆx 5, ŷ 5 ) P4(ˆx 4, ŷ 4 ) AVN school 27/ 19 (45)

29 Some more basics about LSM usually not every observations is equally accurate: Covariance matrix Q ll R n obs n obs, Weight matrix P: σ 2 l 1 σ l1,l 2... σ l1,l n σ l1,l Q ll = 2 σ 2 l 2... σ l2,l n σ l1,l n σ l2,l n... σl 2 n if observations are independent σ li,l j = 0 σ 2 l σ 2 Q ll = l P = 1. s σl 2 n covariance matrix }{{} usually1 Q 1 ll (9) AVN school 28/ 19 (45)

30 Some more basics about LSM covariance matrix Variance-Covariance matrix Q ll, Weight matrix P R n obs n obs : P = 1 s0 2 Qll 1 = }{{} usually σl σl σln 2 good observation small standard deviation σ li and variance high weight in P matrix σ 2 l i (10) AVN school 29/ 19 (45)

31 Some more basics about LSM Final formulas: A} {{ PA} x = A} {{ Pl } N b equations (11) x = N 1 b (12) Important for LSM Normal equation matrix N = right hand side b = A Pl we minimize v Pv ( ) A PA b R n unk N R n unk n unk ( 1 ˆX = X 0 + A PA) A Pl = X 0 + N 1 b (13) AVN school 30/ 19 (45)

32 Some more basics about LSM iteration Sometimes a iteration is necessary. inaccurate a priori values X 0 numerical reasons You can use ˆX again as new a priori values X 0 and redo everything X 0,n+1 = ˆX n = X 0,n + N 1 n b n (14) AVN school 31/ 19 (45)

33 How good are our estimates? a posteriori variance factor σ 2 0 : covariance matrix σ0 2 = v Pv (15) n obs n unk remember our residuals v are: v = Ax l Variance-Covariance Matrix Q xx R n unk n unk σ 2 x 1 σ x1,x 2... σ x1,x n Q xx = σ0n 2 1 σ x1,x = 2 σ 2 x 2... σ x2,x n (16) σ x1,x n σ x2,x n... σ 2 x n σx 2 i σ xi is variance of unknown parameter x i is standard deviation AVN school 32/ 19 (45)

34 Back to example Now we know how accurate our coordinates are P5(ˆx 5, ŷ 5 ) P4(ˆx 4, ŷ 4 ) AVN school 33/ 19 (45)

35 Concepts What are piecewise linear offsets? time dependent parameters Example: Troposphere (mf = mapping function) x 2 x 1 x L w (t) = mf w (t)x 1 + mf w (t) t t 1 (x 2 x 1 ) t 2 t 1 L w = mf w (t) mf w (t) t t 1 x 1 t 2 t 1 L w = mf w (t) t t 1 x 2 t 2 t 1 t 1 t t AVN school 34/ 19 (45)

36 Concepts What are constraints? pseudo observations help us with singularities useful concept A simple example: (think of troposphere between two time epochs) A c R nconst n unk l c R nconst A c = x j x i = 0 ± 5 x j 1 x i 1 = 0 ± 3 (± means standard deviation) ( ) ( ) 0 l c = 0 P c R nconst nconst ( 1 ) P c = AVN school / 19 (45)

37 Concepts What are constraints? N c R n unk n unk b c R nconst Combination with the real observations: N c = A c P c A c (17) b c = A c P c l c (18) N total = A PA + A c P c A c = N + N c (19) b total = A Pl + A c P c l c = b + b c (20) The only thing that changes is: σ 2 0,const = v Pv + v c P c v c n obs + n const n unk (21) AVN school 36/ 19 (45)

38 Concepts What is a geodetic datum? - Example 2 Think of example 2 This time no station has fixed coordinates unknowns: all coordinates observations: s ij we have a priori coordinates P2 P3 P5 P4 P AVN school 37/ 19 (45)

39 Concepts What is a geodetic datum? we know inner geometry very well we can not estimate absolute coordinates N matrix is singular is this correct? P2 P1 P4 P5 P AVN school 38/ 19 (45)

40 Concepts What is a geodetic datum? - Example 2 we know inner geometry very well we can not estimate absolute coordinates N matrix is singular or this? P2 P1 P4 P5 P AVN school 39/ 19 (45)

41 Concepts What is a geodetic datum? - Formulas We need datum stations with additional conditions for example in this case we could say: translation in x translation in y rotation around z scale dxi = 0 dyi = 0 (yi dx i x i dy i ) = 0 (xi dx i y i dy i ) = 0 This would lead to the following matrix: G R n datum n unk G = y 1 x 1... y n x n x 1 y 1... x n y n (22) AVN school 40/ 19 (45)

42 Concepts What is a geodetic datum? - Solution The solution is now: ( ) ( ) ( ) A PA G x A G = Pl 0 k 0 (23) What else is different: σ 2 0 = v Pv n obs + n datum n unk ; Geodetic datum in VLBI ( ) Qxx Q xk = σ0 2 Q xk Q kk ( A PA ) 1 G G 0 (24) usually we use NNT/NNR - No Net Rotation and No Net Translation conditions (3 NNT and 3 NNR) scale is fixed by observations AVN school 41/ 19 (45)

43 Concepts What is a global solution? combination of large number of sessions to estimate accurate common parameters combination is done at the normal equation level you need to make sure N matrix has same structure you want to get rid of some parameters ( ) ( ) ( ) N11 N Nx = b 12 x1 b1 = N 21 N 22 x 2 b 2 ( ) N 11 N 12 N22 1 N 21 }{{} N reduc 1 x 1 = b 1 N 12N22 b 2 (25) }{{} b reduc N REDUC = N reduc1 + N reduc N reducn (26) b REDUC = b reduc1 + b reduc b reducn (27) AVN school 42/ 19 (45)

44 Conclusion What is to remeber? we calculate our solutions using Least Squares Adjustment we minimize v Pv we need to model our observations! theoretical delay we need partial derivatives for our Jacobian matrix A we need a priori values X 0 (usually not a big deal) we estimate additions x to a priori values X AVN school 43/ 19 (45)

45 Link to VLBI AVN school 44/ 19 (45)

46 Matthias Schartner a, David Mayer a, a TU Wien, Department of Geodesy and Geoinformation