The GNSS integer ambiguities: estimation and validation

with ep a p-vector with ones, Ip an identity matrix of order p, and:
⎛
µ = T ⎜
µ1 · · · µf , Λ = ⎝
λ1
. ..
λf
⎞
⎟
⎠ (2.36)
In the same way the geometry-based model is obtained, but the observations collected in
the vector y are now the observed-minus-computed observations from equation (2.22).
The model is given by:
µ
E{y} = (e2f ⊗ G) ∆rqr + (e2f ⊗ Im) T + ⊗ Im I
−µ
(2.37)
0
+ (I2f ⊗ em) dtqr + ⊗ Im a
Λ
with
G = −u 1 qr · · · −u m qr
T
and
−(u s r) T ∆rr + (u s r) T ∆r s − −(u s q) T ∆rq + (u s q) T ∆r s
= −(u s qr) T ∆rqr − (u s qr) T ∆rq + (u s qr) T ∆r s
≈ −(u s qr) T ∆rqr,
(2.38)
(2.39)
since the orbital uncertainty ∆r s can be ignored if the baseline length rqr is small
compared to the satellite altitude (Teunissen and Kleusberg 1998). The term with the
receiver position error ∆rq is zero when the position of receiver q is assumed known. The
parameters in ∆rqr are referred to as the baseline increments; if the receiver position rq
is known, the unknown receiver position rq can be determined from the estimated ∆rqr.
Both single difference models (2.35) and (2.37) are rank deficient. The rank deficiencies
can be resolved by lumping some of the parameters and then estimate these lumped
parameters instead of the original ones. The parameters are lumped by performing
a parameter transformation, which corresponds to the so-called S-basis technique as
developed by Baarda (1973) and Teunissen (1984).
A first rank deficiency is caused by the inclusion of the tropospheric parameters. This
can be solved for the geometry-free model by lumping the tropospheric parameters with
the ranges:
ρ = ρ1 qr + T 1 qr · · · ρm qr + T m T qr
(2.40)
In the geometry-based case, the tropospheric delay parameter is first reparameterized
in a dry and wet component. The dry component is responsible for about 90% of the
tropospheric delay and can be reasonably well corrected for by using a priori tropospheric
models, e.g. (Saastamoinen 1973). The much smaller wet component is quite variable
16 GNSS observation model and quality control