# Analysis of 'Pre-Fit' Datasets of gLAB by Robust Statistical Techniques
## 1 Introduction
In this regard, the pseudorange measurements \(P_{rec}^{sat}\) can be written according to [4] as \[P_{rec}^{sat}=\rho_{rec}^{sat}+c(\delta t_{rec}-\delta t^{sat})+ Trop_{rec}^{sat}+Ion_{rec}^{sat}+IFB^{sat}+\epsilon_{rec}^{sat}\ \ ;\ \ sat=1,\ldots,n\] (1) where \(\rho_{rec}^{sat}\) is the geometric distance between the receiver and the satellite, \(c\) stands for the velocity of light, \(\delta t_{rec}\) and \(\delta t^{sat}\) are the receiver and satellite clock offsets, \(Trop_{rec}^{sat}\) and \(Ion_{rec}^{sat}\) are the propagation delays occurring at the troposphere and ionosphere respectively, \(IFB^{sat}\) is the satellite inter-frequency biases (IFB), and \(\epsilon_{rec}^{sat}\) accounts for the measurement noise. The output of the MODELLING module are the pre-fit residuals, i.e., the difference between the preprocessed measurements and the sum of the model terms: \[prefit_{rec}^{sat}=P_{rec}^{sat}-Pmodel_{rec}^{sat}\ \ ;\ \ sat=1,\ldots,n\] (2) Considering the model terms described in the standard point positioning (SPP) [7], the pre-fit residuals result in: \[prefit_{rec}^{sat}=\rho_{rec}^{sat}+c\delta t_{rec}+\epsilon_{rec}^{sat}\ \ ;\ \ sat=1,\ldots,n\] (3) where Equation (3) contains four unknowns: three receiver coordinates in the geometric distance \(\rho_{rec}^{sat}=\sqrt{\left((x_{rec}-x^{sat})^{2}+(y_{rec}-y^{sat})^{2}+(z_{ rec}-z^{sat})^{2}\right)}\) and the time offset of the receiver clock offset \(\delta t_{rec}\), with respect to GNSS time. * The FILTER module: implements an extended Kalman filter to obtain the estimations of the receiver PVT from the pre-fit residuals. The filter outputs the values of the estimated unknowns together with its co-variance, as a measure of the uncertainty in the estimation process. * The OUTPUT module: is in charge of printing intermediate and final results in a structured manner. This allows extracting useful and complete information at any point in the GNSS data processing chain. Both in gLAB and in commercial software packages, PVT estimation can be deteriorated by _outliers_. _Outliers_ can be intuitively defined as elements of a sample qualitatively distinct as 'bad' compared to the majority of the sample considered as 'good'. Note that it is not said that outliers are 'wrong' data; they are data needing a 'careful look', since their 'being distinct' from the majority could lead to unreliable estimates. _Outlier detection_, by a _comparative_ or _quantitative_ evaluation of their quality of 'being distinct', is a central component of PVT estimation. The aim of the present work was to improve the outlier detection capabilities of gLAB by working at the intermediate step between the MODELLING and the FILTER modules. To this aim, we considered as basic dataset the _pre-fit residuals_ as in Equation (2) above. In this GNSS context, outliers can be more precisely defined as pre-fit residual values from a particular satellite that are too different from the pre-fit residuals median (computed including all satellites). Note that by _pre-fit residuals_ we mean something distinct from _residuals_ in regression theory; the different use of the word _residuals_ should be clear from the context. The most common reason for the presence of outliers in the pre-fit residuals is the multipath effect, when GNSS signals from low-elevation satellites reach the antenna along multiple paths, following a reflection in the ground or surrounding structure. Other outlier sources are the malfunctioning of the receiver or satellite hardware. For the latter, the ground segment (GS) of each constellation constantly monitors the status of its satellites using a worldwide distribution of permanent stations. Thus, the GS can inform users of satellite anomalies by means of the healthy flag broadcast in the navigation message that is typically refreshed every two hours. In contrast, anomalies at one particular isolated receiver are more difficult to detect, as there is no redundancy of measurements. In this case, receiver autonomous integrity monitoring (RAIM) must be applied, following [8].