## 2 Materials and Methods
### Determination of the Track Axis and Measurement Uncertainty  
The track axis is a theoretical line on the track surface, positioned midway between the rail heads (or, on horizontal curves, referenced to the outer rail) along the track's longitudinal direction. As a designed element, its characteristics directly influence operational parameters--most notably the permissible train speeds. As discussed in Section 1, determining the actual spatial geometry of the track axis within a coordinate system is essential for assessing existing railway lines. The measurements conducted by the authors supply the required data to compute both the track axis coordinates and their associated uncertainties. The measurement method fundamentally relies on synchronous satellite observations acquired by GNSS receivers [10; 15]. Receiver positions are computed in the WGS-84 reference frame using known satellite ephemerides and measured pseudoranges [11], a process known as spatial trilateration. Cartesian Earth-Centered Earth-Fixed (ECEF) coordinates and the receiver clock bias are iteratively estimated via least-squares adjustment. These ECEF coordinates (\(x,y,z\)) are then transformed into curvilinear coordinates--latitude, longitude, and ellipsoidal height (B, L, h) [21]. Ellipsoidal heights are further corrected using a discrete geoid or quasi-geoid model to yield orthometric or normal heights, which reference the mean sea level [22; 23]. The choice of vertical reference system is governed by national legislation; in Poland, the EUVN2007 normal height system, tied to the Amsterdam tide gauge, is employed [24]. Three-dimensional ellipsoidal coordinates must be projected into planar map coordinates to facilitate the computation of linear and angular measurements, as well as areas. Among the various cartographic projections employed in geodesy, navigation, hydrography, and military applications, Gauss-Kruger projection--known in English as the Transverse Mercator projection--is widely used for its conformality and minimal distance distortions [25; 26]. In this method, the convex surface of a segment of the Earth's rotational ellipsoid is mapped onto narrow meridian zones, which are typically 3\({}^{\circ}\) or 6\({}^{\circ}\) wide. A selected zone segment, representing the area of interest, is then used for detailed mapping. In the classical Gauss-Kruger projection, only the central meridian is free of linear distortion; distortion increases progressively toward the zone boundaries. To optimize distortion distribution, a scale factor is applied within each meridian zone [27; 28]. For instance, the PL-2000 coordinate system--used for railway measurements in Poland--employs a 3\({}^{\circ}\) meridian zone with a scale factor of 0.999923. This results in linear distortions of approximately \(-7.7\) cm/km on the central meridian and +7.0 cm/km at the zone edges. Consequently, lines of zero distortion are shifted east and west of the central meridian, occurring at two specific longitudes. This adjustment produces a more uniform distortion distribution, which is crucial for national geodesy and spatial planning. Coordinates in the PL-2000 system are expressed as northings (\(X\)) and eastings (\(\Upsilon\)) [19]. Synchronous satellite measurements from six receivers mounted on the platform are processed either in real time using RTK/RTN corrections or during post-processing with precise SP3 orbit data [11]. In both modes, each receiver's instantaneous position is reported alongside a parameter quantifying its accuracy. First, the partial derivative matrix of the GNSS observation equations is used in a least-squares adjustment to compute the variance-covariance matrix [29]. Its diagonal entries represent the variances of the adjusted estimators for the ECEF coordinates and the receiver clock bias.