The GNSS integer ambiguities: estimation and validation

the propagation delay does not depend on the frequency of the signal. Moreover, the
tropospheric delays are equal for carrier phase and code observations.
Usually a distinction is made between a wet delay caused by water vapor, and a hydrostatic
delay caused by a mixture of dry air and water vapor that is considered to be in a
hydrostatic equilibrium. The wet delay can be 0-40 cm in the zenith direction, and can
only be determined with an accuracy of 2-5 cm based on semi-empirical models. The
hydrostatic delay is much larger, 2.2-2.4 m in the zenith direction. However, it can be
predicted with high accuracy based on surface pressure observations.
The tropospheric delay parameter will be denoted as T s r . A priori models can be used
to correct for the tropospheric delays, see (Hopfield 1969; Saastamoinen 1973). It is
necessary to estimate the tropospheric delay parameters if it is expected that one cannot
fully rely on these models, see (Kleijer 2004).
Ionospheric delays
The ionized part of the atmosphere ranges approximately from 80 to 1000 km altitude.
This atmospheric layer contains both uncharged and charged particles. The charged
particles are created when neutral gas molecules are heated and electrons are liberated
from them. This effect is called ionization. The rate of ionization depends on the density
of the gas molecules and the intensity of the radiation. Because of the varying intensity
of solar radiation, the free electron density is very variable in space and time.
The free electrons in the ionosphere affect the code and phase measurements differently.
That is because the phase velocity is advanced, whereas the group velocity is delayed.
So, the ionospheric effects on code and phase observations have opposite signs. Furthermore,
the ionosphere is a dispersive medium, which means that the ionospheric effect is
dependent on the frequency of the signal. The effects are inversely proportional to the
square of the frequency of the signal. This allows for estimating the first order ionospheric
effects by collecting measurements on different frequencies. Under worst-case
conditions the first order term may be tens of meters. According to Odijk (2002) the
higher order effects may be neglected in the case of relative positioning (see section
2.3.2) with inter-receiver distances up to 400 km.
The ionospheric effects on code and phase observations, respectively, are given by:
with
dI s r,j = µjI s r ,
δI s r,j = −µjI s r ,
µj = f 2 1
f 2 j
(2.14)
= λ2j λ2 . (2.15)
1
The atmospheric effects in equations (2.7) and (2.12) can now be written as:
da s r,j = T s r + µjI s r
δa s r,j = T s r − µjI s r
(2.16)
GNSS observation equations 11