Hypothesis test

A fundamental approach for the integer ambiguity validation problem can be based on general hypothesis test theory (Teunissen, 2007). Similarly, the analysis can be performed in terms of

62 Methodology for integer ambiguity resolution Table A.1 Statistics, configuration parameters and validation thresholds for integer ambiguity resolution in the test case using GRACE data from October 2011.

Float ambiguity estimation

Priors GNSS observations

State Value Type Value

Relative position σ_x = 0.01 m Pseudorange σ_P = 0.50 m

DD ionospheric delay σı = 1.00 m Carrier phase σ_Φ = 0.01 m

DD ambiguities L1, L2 σn→ ∞ Integer ambiguity resolution

Validation Partial resolution

Test/Parameter Standard

configuration

Criterion Standard method Success rate ILS (§A.3) Ps,ILS = 0.80 Selection of

ambiguity set^*d

Analysis of ambiguity formal errors

Ellipsoidal integer aperture^*a (Eq.(A.5))

µ²_EIA = 100

Ratio test integer aperture^*b (Eq.(A.6))

FixedPf = 0.001

Widelane ambiguity residuals^*c (§A.3.3)

TW L= 0.25cy

Formal error MW ambiguities (§A.3.3)

¯

σM W = 0.10cy

Interval to search for integer MW ambiguities (§A.3.3)

ˆ

aM W ±3¯σM W

Hamming distance between best ILS solutions (Pub. 1, Eq.(2))

D_h= 0

* Variable configuration tests: ^a§A.3.1. ^b §A.3.2. ^c §A.3.3. ^d§A.3.4.

model selection/comparison. In this way, the two hypothesis (or models) to compare can be expressed as

M_f :y =Aˆa+Bˆb+e,ˆa∈R^q,ˆb ∈R^p, e∈R^q (A.2a) M_i :y =Aˆn+Bˆb+e,nˆ∈Z^q,ˆb∈R^p, e∈R^q (A.2b) In this case, the null hypothesis is given by the float model M_f and implies that ambiguities aˆ are real numbers. The alternative hypothesis is given by the integer/fixed model M_i which

A.3 Integer ambiguity estimation and validation 63 considers ambiguities nˆ as integer values. Other non-ambiguity parameters are contained in vectorb(e.g. epochwise relative position vector and differential ionospheric delays in the models introduced in Publication 1).

Given that the integer ambiguity solutionnˆ is directly dependent on the float solution ˆa, it is not possible to perform a model comparison by means of a direct evaluation of M_f and M_i. Instead, the probabilistic analysis of the ambiguity acceptance test must be done in terms of the ambiguity residuals εˆ= ˆa−nˆ and/or their distribution fˆε(x) (Wang, 2015). Such an analysis leads to the definition of the so-called acceptance and rejection regions which determine which ambiguity solutions should be discriminated. Acceptance regions are also called aperture (denoted as Ω) and they are always a subset of the ILS pull-in regions (Verhagen, 2005). In this way, ambiguity acceptance tests based on this principle can be generalized into the so-called integer aperture estimators (IAEs), which represent a unified framework for integer ambiguity estimation and validation (Teunissen, 2003).

One interesting feature of IAEs is their hybrid nature, i.e. the output of the estimator can be defined as

ˆ

aIAE = ˆa+ X

n∈Z^q

(n−ˆa)ω_n(ˆa) (A.3)

with the function ω_n(ˆa) given by

ωn(ˆa) =

1 if ˆa∈Sn

0 if ˆa̸∈S_n (A.4)

where S_n is (generally) the ILS pull-in region. Thus, the IAE will have as output the float solution ˆa if ˆa̸∈Ω_n or the fixed solution n if ˆa∈Ω_n.

One of the simplest estimators in this class is the ellipsoidal IAE, whose principles derive directly from the hypothesis test and the analysis of ambiguity residuals. The ellipsoidal integer aperture (EIA) region is defined as (Teunissen, 2003)

ΩEIA,n= ΩEIA,0+n ={a ∈Sn| ||a−n||²_Qˆa ≤µ²_EIA} (A.5)

which is an ellipsoid defined by the distance between the float and integer ambiguity solution vectors in the space with metricQ_ˆa (variance-covariance matrix of the estimated float ambiguity vector). The size of the aperture is defined by the parameter µEIA. Figure A.4 depicts an example of Ω_EIA,n shown as a subset of the ILS pull-in region.

The parameterµEIA can be used for the computation of the success and failure rate of the EIA estimator (Verhagen, 2005; Wang, 2015). However, in the present work, the EIA estimator has been mainly used as a hypothesis (complementary) test within a multi-step validation scheme (see Fig. A.1). In this way, the selection of the aperture parameter µEIA is done in function of an expected confidence in the float solution. The larger the value selected for µEIA, the larger the confidence in the integer solution, which effectively reduces the probability of false alarm but increases the probability of failure (see e.g. Wang (2015)). The selection of µEIA is performed in a heuristic way and a more formal probabilistic characterization of the overall validation test scheme is left to the analysis of the ILS success rate and a subsequent test using a second integer aperture estimator (see §A.3.2).

64 Methodology for integer ambiguity resolution

−2 −1 0 1 2

ˆa₁ [ cy]

−2

−1 0 1 2

ˆa²

[cy]

Fig. A.4 Aperture region of the ellipsoidal integer aperture estimator (blue) as a subset of the ILS pull-in region (black contour).

In order to observe the impact of different possible threshold values for µEIA, various tests have been run, analyzing as key indicators the integer ambiguity fixing rate and the final baseline precision. Figure A.5 depicts the results obtained with valuesµ²_EIA = 25, 50, 200, 300 in comparison with a value µ²_EIA = 100 corresponding to a standard configuration. Values roughly lying on the diagonal line indicate a similar performance for both values under comparison, whereas off-diagonal values show an improved performance in favor of either configuration.

As depicted, threshold values below 100 appear more stringent in terms of the confidence on the resolved integer ambiguities. Although the probability of failure is effectively reduced for such configuration values, the ambiguity fixing rate is decreased and, in consequence, the final baseline precision is slightly reduced. In comparison, the achieved performance with configuration values above 100 is very similar, which indicates that although the probability of failure is increased, the precision of baseline solutions is not degraded for most of the days. In terms of the so-called integer test (Verhagen, 2005), threshold values above 50-100 are obtained in case of very low levels of significance and/or a large number of degrees of freedom n of an F(n,∞) distribution for a test statistic ||a−n||²_Q

ˆ

a/n. In this sense, by using high threshold values, the test gives a large tolerance margin to the evaluated integer ambiguity solution. In the proposed validation scheme, the implementation of this hypothesis test in terms of the ellipsoidal integer aperture estimator allows a more prompt interpretation of thresholds values and facilitates its selection according to any given scenario under analysis.