The GNSS integer ambiguities: estimation and validation

So, λ0 = λ(αq, γ0, q). From this value one can then compute the corresponding model
error C∇ from equation (2.72).
The vector ∇y = C∇ describes the internal reliability of hypothesis H0 with respect to
Ha. The internal reliability is thus a measure of the model error that can be detected
with a probability γ = γ0 by test (2.73). This measure can be computed by considering
certain types of model errors, as specified by the matrix C. In GNSS applications
common error types are for example carrier cycle slips and code outliers. In those cases,
q = 1, so that C reduces to a vector c, and ∇ to a scalar. From equation (2.72) follows
then that:
|∇| =
λ0
c T Q −1
y QêQ −1
y c
(2.76)
|∇| is called the minimal detectable bias (MDB). With |∇| known, it is possible to
determine the corresponding impact of the model error on the estimates of the unknown
parameters, referred to as the external reliability. This impact can be computed as:
∇ˆx = (A T Q −1
y A) −1 A T Q −1
y ∇y (2.77)
by using ∇y = c|∇|. ∇ˆx is referred to as the minimal detectable effect (MDE).
Note that the measures of the internal reliability and the external reliability can both be
computed without the need for actual observations, and hence can be used for planning
purposes before the measurements are carried out.
2.5.2 Detection, Identification and Adaptation
Once the observations are collected there will be several alternative hypotheses that need
to be tested against the null hypothesis. The DIA-procedure (detection, identification,
adaptation) can be used in order to have a structured testing procedure (Teunissen
1990).
The first step in DIA is detection, for which an overall model test is performed to diagnose
whether an unspecified model error has occurred. In this step the null hypothesis is tested
against the most relaxed alternative hypothesis:
Ha : E{y} ∈ R m
The appropriate test statistic for detection is then:
T m−n = êT Q −1
y ê
m − n
(2.78)
(2.79)
The null hypothesis will be rejected if T m−n > Fαm−n (m − n, ∞, 0). In that case, the
next step will be the identification of the model error.
The most likely alternative hypothesis is the one for which
T qi
Fαq i (qi, ∞, 0)
(2.80)
24 GNSS observation model and quality control