Reliability of GPS cycle slip and outlier detection

The theory and application of statistical quality control is well established in surveying and geodesy. In recent years it is also gaining momentum in applications of marine positioning. The set of quality control recommendations of the United Kingdom Offshore Operators’ Association (UKOOA) is a good example in this respect. Quality control is made up of various contributing factors, one of which is the concept of reliability. By means of the Minimal Detectable Biases (MDBs), the concept of reliability provides diagnostic tools to infer the strength with which positioning models can be validated. In this contribution we will present in analytical form the MDBs of singleand dual-frequency GPS code data (pseudoranges) and carrier phases. These MDBs describe the size of outliers in the code data and the size of slips in the phase data which can just be detected by the appropriate test statistics. These MDBs will be given for three different GPS models, the geometry-free model and two variants of the geometry-based model. INTRODUCTION Minimal Detectable Biases (MDBs) as introduced by Baarda (1967, 1968) are important diagnostic tools for inferring the strength of model validation. As such they can also be used to study the strength of the various GPS positioning models, in particular with respect to the detectability of outliers in the code data or slips in the carrier phase data. The fact that a whole suit of different GPS models exists, implies that there are different stages at which quality control can be excercized. Roughly speaking, one can discriminate between the following four levels: • Receiver-level: in principle it is already possible to validate the time-series of undifferenced data of a single GPS receiver. Single-receiver quality control is very useful for reference receivers that are used in active GPS control networks or in DGPS. • Baseline-level: in this case a pair of receivers is used. When the observation equations are parametrized in terms of the baseline vector, the strength of the model primarily stems from the presence in the design matrix of the relative receiversatellite geometry. Additional redundancy enters when the baseline is considered stationary instead of moving. 1 Proceedings INSMAP98, Melbourne, FL, USA, Nov. 30-Dec. 4, 1998 • Network-level: when sufficient (independent) baselines are used to form a network, redundancy enters by enforcing the closure of the ‘baseline loops’, similar as the loops of a traditional levelling network. • Connection-level: additional redundancy enters again when a free GPS network is connected to points of an existing control. In this case the redundancy stems from the fact that the shape of the free network is compared with the shape of the existing control. In this contribution we will consider the baseline level. Three different such models will be considered, the geometry-free model and two variants of the geometry-based model, the rovingand stationary variant. Attention will be restricted to the ‘short’ baseline case. The term ‘short’ refers to the assumption that double-differenced (DD) observables are sufficient insensitive to orbital uncertainties in the fixed orbits and to residual ionospheric and tropospheric delays. INTERNAL RELIABILITY In this section a brief review is given of internal reliability. For more details see e.g. (Baarda, 1968) or (Teunissen, 1985). Reliability in the context of GPS is also treated in (Teunissen and Kleusberg, 1998), (Tiberius, 1998) and (de Jong, 1998). Internal reliability as represented by the MDBs describes the size of the model errors which can just be detected using appropriate test statistics. Consider the following nullhypothesis Ho and alternative hypothesis Ha: b Ax y E H and Ax y E H a o + = = } { : } { : with E{.} the expectation operator, y the m-vector of normally distributed observables, A the m×n design matrix, x the n-vector of unknown parameters, and b the unknown bias vector. The bias vector is assumed to describe the model error. Hence it is absent under the null-hypothesis, but present under the alternative hypothesis. It is further assumed that the bias vector can be parameterized as unknown known c with c b = ∇ = ∇ = , The vector c specifies the type of model error, while ∇ describes its unknown size. The (uniformly) most powerful test statistic for testing Ho against Ha is given as