# Ambiguity-Resolved Model Tests for Carrier-Phase GNSS
## 1 Introduction  
GNSS model validation and monitoring constitutes an essential part of any GNSS data processing scheme [1, 2, 3]. Statistical tests are then employed to test for the occurrence of model misspecifications--e.g., pseudorange outliers, carrier-phase slips, or neglected atmospheric delays [4, 5, 6]--or, in the case of monitoring, to test for the stability of estimated parameters--for instance, in GNSS time-series for displacement, deformation, or landslide studies [7, 8, 9, 10]. To achieve the highest accuracy in GNSS parameter estimation, carrier-phase measurements are generally used, as they have a two orders of magnitude better precision than their pseudorange (code) counterparts [11, 12, 13]. However, since the interferometric carrier-phase observables are integer-ambiguous, their GNSS models consist of both real-valued and integer-valued parameters; therefore, they are of the mixed-integer type. Although the theory of mixed-integer inference is well developed for GNSS parameter estimation [1, 2, 3], such is not yet the case for the validation and monitoring of the mixed-integer GNSS carrier-phase models. It is the goal of this study to contribute to this field by introducing a class of ambiguity-resolved model tests for carrier-phase GNSS. This contribution is organized as follows. In Section 2, we formulate the to-be-tested mixed-integer null and alternative hypotheses of carrier-phase GNSS and introduce our class of mixed-integer model (MIM) tests for their testing. The hypotheses are formulated in a general form such that they apply to any carrier-phase GNSS system of observation equations, whether single-epoch or multi-epoch, single-frequency or multi-frequency, undifferenced, single-differenced or double-differenced, or whether for use with single-GNSS or multi-GNSS. Our MIM test statistic is a function of the least-squares residualsof both null and alternative hypotheses, as well as of their integer admissible ambiguity estimators. As different forms of the test statistic are useful for different applications, we provide various useful relationships between these normed residuals. To obtain a first insight into the expected performance of the MIM test, we consider two of its limiting cases in Section 3. In the first case, the integerness of the ambiguities is not taken into account, while in the second case, the integer ambiguities are assumed to be known. The first is referred to as ambiguity-float (AF), while the second is referred to as ambiguity-known (AK). In both these limiting cases, the MIM test becomes equal to a \(\chi^{2}\)-test with well-known power functions (i.e., functions that provide, for a given false-alarm probability, the detection probability as a function of the biases). It is shown that for carrier-phase GNSS, the AK power significantly outperforms the AF power, i.e., when the ambiguities are assumed known, the carrier-phase observables start to act as very precise pseudoranges, thereby contributing significantly to the power improvement of the test. The AK test, however, is not an operational test, since in the case of carrier-phase GNSS, the ambiguities can never be assumed as known, i.e., despite the fact that the ambiguities are known to be integers, they still remain unknown. Although the AK test cannot be used operationally, its large power difference with the AF test does indicate the great potential of our MIM test, a test that operates in between the AF and AK tests. This is shown in Section 4, where, for different circumstances, different forms of the MIM test statistic are presented together with their associated probability distributions. The MIM test is worked out in Section 4.1 for the case when the null hypothesis is tested against the most relaxed alternative hypothesis. This case of detection requires, in contrast to the general MIM test, only integer ambiguity resolution under the null hypothesis.