# GPS Multipath Analysis Using Fresnel Zones
## 3 Identification of Diffracted Satellite Signals Using Fresnel Zones and OAEMs
### Algorithm Assessment  
The algorithm assessment was based on the analysis of the coordinates from a kinematic baseline solution. For this purpose, the original data set was modified in two different ways:  
1. All satellite signals with an elevation angle lower than the respective OAEM values were excluded (hereafter denoted as _classical OAEM_). 2. In addition to the signals detected by the OAEM, all signals that were identified by the proposed algorithm as being potentially influenced by diffraction effects were also excluded (hereafter denoted as _OAEM with Fresnel zones_). Figure 14: (**a**) Orthophoto of the test area from a UAV flight. The red point marks the location of the GPS antenna on the cinder pitch. (**b**) Obstruction adaptive elevation mask (black dashed line) of test area determined from a georeferenced terrestrial laser scanner (TLS) point cloud. The green lines show the GPS satellite tracks during the 24 h observation period. For all data sets, a kinematic baseline solution was carried out. In each case, the VRS served as the master station. The estimation of the baseline parameters is performed in the three steps: (1) float solution, (2) integer ambiguity fixing and (3) fixed solution. In the float solution step, the ambiguities were estimated as real numbers. Afterwards, the float ambiguities and their covariance matrix were used to fix the the ambiguities to integer values using the LAMBDA method [39]. The result of the fixing process was validated using the ratio test [40]. Herein, the squared norm of the residuals of the best set of ambiguities (\(R_{1}\)) and the second best set of ambiguities (\(R_{2}\)) were compared. If the quotient \(R_{2}/R_{1}\) exceeded a predefined threshold of 3, the ambiguity fixing was accepted. In the final step, the fixed solution, the baseline parameters were estimated after the fixed ambiguities had been added to the observations and were eliminated from the parameter vector. Afterwards, for better visualization purposes, the resulting baseline vectors were transformed to a local topocentric coordinate system (East, North, Up), and the coordinates were reduced by the results of a static batch solution from a commercial software package (Leica Infinity ([[https://leica-geosystems.com/en-US/products/gnss-systems/software/leica-infinity](https://leica-geosystems.com/en-US/products/gnss-systems/software/leica-infinity)]([https://leica-geosystems.com/en-US/products/gnss-systems/software/leica-infinity](https://leica-geosystems.com/en-US/products/gnss-systems/software/leica-infinity)))). In Figure 15, the coordinate differences of the up-component and the respective 3\(\sigma\) boundaries are shown exemplarily for each data set. Additionally, Table 2 lists the minimum and maximum coordinate differences and the root-mean-squared error (RMS) of the time series. Furthermore, the percentage of outliers with respect to the 3\(\sigma\) boundary, as well as the percentage of observation epochs in which the carrier phase ambiguities could be fixed to integer values are listed. Figure 15: Differences in the Up-component for the original data set (blue), the data set modified by the classical OAEM approach (red) and the data set modified by the newly proposed algorithm (green). For better visualization, the blue and green curves are shown with offsets of 10 and \(-\)10 cm respectively. Differences exceeding the axis limits are truncated. The black dashed line represents the 3\(\sigma\) boundary of each time series. Both the coordinate differences shown in Figure 15 and the related values listed in Table 2 indicate that using the Fresnel zone concept for detecting and excluding diffracted signals can lead to an improvement of the coordinate accuracy. After the classical OAEM approach has been applied to the data set, the percentage of epochs with successfully fixed ambiguities increases. Although in two cases (cf. 12:00 and 16:00 in Figure 15) the highest differences in the original data set can be reduced, one significant peak remains (cf. 08:00 in Figure 15).