Observation vector , containing the code position solution
X(k) = 8X(k)+X(k) and delta position solution X(kk-1)
=5X(k,k-1)+X°(k,k-1).The matrix ® is a 6x6 state transition
matrix and À is the 6x6 observation matrix; w(k) is the 6x1
dynamic noise vector and e(k) is the noise vectors of X(k)
and X(k,k-1). The Kalman filtering solutions to this system
can be written as
Z(k)-(kk-1) 2(k-1)* K()[y()-A(K) d(kk-1) 2-1)]
Q4)
where
K(k)- D(k) A'(k) P (k) Q5)
D(k)- [D" (k.k-1) - A'(k)P' (k) A(k) ]' (26)
D(kk-1)= d(kk-1)D(k-1)'(kk-1) - Q(k-1) (27)
P(k) and Q(k) are the variance matrices for observation
noise and dynamic noise. If the dynamic noise and phase
delta position noise are negligible, comparing to the code
position noise, the Kalman filtering solution of the 3
dimensional position state vector x can be simplified to
$0)-D.()(D, (k-D[ 3(k-1)*- X(k-1)]* P, (k) X(9))
(28)
D,(k)=[D;"(k-1) - P! (9]' (29)
where P, stands for the variance matrix of code position
solution X(k). and D, denotes the filtering variance matrix
of the position state vector; D, ! (0) 20 is assumed for k=1.
If the errors of different position components are assumed to
be mutually independent and identical in accuracy, the
equations (28) and (29) partition into the three estimates of
the individual coordinate components:
X, &)-((k-D[ S, (k-I) ^ X,(kk-D)] -X, (k))k (30)
where the subscript m denotes three position components
latitude, longitude and height or x, y and z when m=1,2, 3.
3.3 Optimal Smoothing of Phase Delta-Position and Code
Position Solutions
Optimal Kalman filtering uses all the to-date measurements
to create the position state vector solutions in real time
mode. Kalman smoothing, however allows us to utilise all
the measurements for achieving uniformly accurate
solutions in postprocessing mode. As the optimal smoothing
equations for the system equations (22) and (23) can be
found in related textbooks, only a simplified formula is
68
given here, based on the assumption that the dynamic noise
and delta-position noise can be ignored with respect to code
position noise. Therefore, the smoothing position vector is
written as
36) - D^ Z^ [X()* XJ] P!) (31)
where
D? 7 [Z4" P, ()]' (32)
X(,k)7 X(k+1,k)+ X(k+2,k+1) + ... XGj-1) (33)
where X(kk)-0 and X(kj)= -X(jk) are assumed.
Similarly, if the errors of different position components are
assumed to be mutually independent and identical in
accuracy, the equations (31) and (32) partition into the three
estimates of the individual coordinate components:
Xa) 7 Xa [X40) * X,GJ)] /n (34)
where n is the cumulative tracking time in epochs, and Fl
25. n
3.4 Asymptotical Performance of Dynamic Filtering and
Smoothing Solutions
Because the linear system of (22) and (23) is uniformly,
completely observable and uniformly, completely
controllable, its Kalman filtering solutions are uniformly,
asymptotically stable (Feng & Kubik, 1994). In other
words, the position solution converges to an asymptotic
level of accuracy with increasing tracking time: The longer
the continuous tracking time, the more accurate the state
filtering solution will be. According to the equations (30)
and (34), and assuming random noise, the variance of the
filtering solution X(k) and the smoothing solution (j) can
be approximately estimated as
Oa (K) * o? /k (35)
Ow () » o? /n (36)
where o? denotes the variance of code position solution X,
and o^, (k) and os, (j) denote the variances of filtering
solution &,(k). and smoothing solution $,,(j). It can be seen
that the smoothing solutions are uniformly accurate while
the filtering solutions are asymptotically stable. This
asymptotical nature allows for the direct comparison
between known baseline vector and the smoothing
positions. Once the smoothing solutions are corrected to the
known baseline, centimetre accuracy may be achievable.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B1. Vienna 1996
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