International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
4.2 Two stage Kalman filter
The state equation and the observation equation can be modified
for the presence of a constant bias vector by
oi + Bb, 3 T€, Q4)
ym 4x, tC, t1, (25)
x, = F,x,
A new state vector can be defined as
whe (26)
E b, |$r
so the state equation and the observation equation become
z mZG TOE, (27)
y, 7 Lz, tnl, (28)
where
nir
, [A ! & en em
Z EL odit
calo (30)
0|tr
Hd (31)
The observation equation also can be trasformed. The optimal
estimator and its covariance matrix are given by
She 7 p] 7 7 2
mA aO tE, (X, Ld) (32)
um f T
n eZ IH TES Oi GC. G (33)
AK
The variance equation can be modified partitioning the variance-
covariance matrix
mna !
Q, k Qu. In (34)
a =
e. o. Ox Tr
This leads to decouple the estimation of the unbiased state
vector and the bias vector, involving smaller matrices. Moreover,
in many cases the bias vector can be estimated at lower
frequency than the state vector, and this can be applied in GPS
biases estimation to avoid excessive processor loading. Carrier
phase ambiguities represent a special case of constant bias in
GPS signal, but two stage Kalman filter can also handle slow
varying biases and noisy biases, although this implies more
complex equations. Discontinuous biases, such as carrier phase
ambiguities that are discontinued by cycle slips, can be handled
integrating quality control procedures in the two stage Kalman
filter, as will be underlined in the following.
4.3 Quality control
The main error sources in GPS observations are clock jumps in
the receiver clock, cycle slips, outlier and quasi random errors
(mainly multipath, diffraction, ionospheric scintillation). Real time
state space estimation requires robust quality control
procedures, to be applied both to observation space and to
parameter space, testing the correctness of the a priori stochastic
model. The appropriate test statistic can be formulated in terms
of predicted residuals, that have been already defined by the
equations (17) and (18). To handle the various alternative
hypothesis we make use of three steps, as suggested by
(Teunissen, 1998): detection, identification and adaptation.
e Detection: a global model test is performed on the
whole observation set at a given epoch. In case of
global model test failures, the identification step is
performed.
e Identification: is used to identify the potential error
source.
e Adaptation: after identification is possible to cancel
the detected and identified bias, correcting its effects
in the state estimation.
The DIA procedure can be designed for batch and for recursive
solutions too. The recursive (epoch by epoch) form can be
integrated with recursive estimators as Kalman filter. To test a
null hypothesis against an alternative hypothesis, the detection
test value with y^ distribution is given by
7, — V, Qr, (35)
and depends on predicted residuals and their covariance matrix.
In local identification the test value is
where s— 0.0, dsl) Is a flag vector used to identify the
observation to test. The Minimum Detectable Bias is given by
óc
MDB, = (37)
Fi
where r is the redundance, or the trace of the redundance
matrix
R= Qe. (38)
After identification, the detected bias is compared with the MDB
value, that is a treshold value used to identify meaninigful biases.
If the bias candidate has a value less than the MDB, the
observation is accepted. Otherwise the procedure continues with
the adaptation step. In adaptation the state vector and its
covariance matrix are corrected with
an __ 20 TL C
il ee K,s P, (39)
Q^ =() (K S0. s KT (40)
dA = kk kk By Á k
where B, is the least square estimated bias vector and 2 its
C B,
variance, given by
Th
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Se
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Th
inte
Ka
sta
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