# Accuracy of Code GNSS Receivers under Various Conditions
## 3 Materials and Methods
### Methods  
All survey instruments were located on a cart and stabilized in polystyrene in a vertical position at a distance of \(\sim\)10-15 cm (Figure 3 right), which allowed them to be in a fixed position relative to each other and at a constant height relative to the terrain (Figure 3 left).  
Figure 4 shows a panorama of each site where the measurement was performed and Figure 5 shows sky visibility conditions. The experiment was conducted on 3 test objects. These were a circle with a diameter of 35.5 m forming a surface object (yellow color, Figure 2), a line with a length of 100 m forming a linear object (orange color, Figure 2), and a point (red color, Figure 2), and each measurement lasted 20 min with an interval of 1 s. The circle was measured 10 times, the cart was driven along the line 20 times, while the measurement on the point lasted over 1200 epochs.  
Figure 4: _Cont_.  
Figure 3: Survey equipment design and GNSS devices used in the study.  
Figure 4: Panorama of objects: (**a**) surface, (**b**) line, (**c**) point, (**d**) static session.  
Figure 5: _Cont._  
The results obtained from the RTK measurement were taken as a reference for further analyses. However, due to significant obscuration of the horizon, only some of the solutions were precise. In the case of the point, the obtained precise solutions were averaged, while for the line and the circle, on the basis of precise solutions, the geometric shape of these figures was determined, and the accuracy of coded solutions was defined as deviations from the line and the circle.  
Measurement data from handheld receivers were obtained in _gpx_ format and from phones in _csv_ format. For each measurement object, a list of geographic coordinates was created in decimal format and in degrees/minutes/seconds format, which enabled faster implementation of further stages of work. The Javad receiver recorded geodetic coordinates in the WGS-84 system, while the other receivers recorded \(\varphi\), \(\lambda\) geodetic coordinates in WGS-84 and elevation with respect to the EGM96 global geoid [37]. All receivers measured coordinates at epochs of measurement. First, geoid heights (\(H\)) from code receivers were converted to ellipsoidal heights (\(h\)) using the global undulation (\(N\)) calculator [38]:  
\[h\ =\ H\ +\ N. \tag{1}\]  
The next stage of the work was the transformation of geodetic coordinates to coordinates in the \(XYZ\) system [39]:  
\[\begin{array}{l}X\ =\ (R_{N}\ +\ h)\cdot cos\varphi\cdot cos\lambda.\\ Y\ =\ (R_{N}\ +\ h)\cdot cos\varphi\cdot cos\lambda.\\ Z=\left(\left[1\ -\ e^{2}\right]R_{N}\ +\ h\right)\cdot sin\varphi.\end{array} \tag{2}\]  
where \(R_{N}\) is the radius of curvature in the prime vertical and \(e\). is eccentricity. For this purpose, the calculation program Transpol 2.06 was used. Then, the obtained Cartesian coordinates were converted into topocentric NEU according to the formula [40] for the \(i\)th number of observations (in this paper, this is ~1200--20 min with a 1 s interval):  
\[\begin{bmatrix}N\\ E\\ U\end{bmatrix}\ =\ \begin{bmatrix}-sin\varphi_{i}cos\lambda_{i}&-sin\varphi_{i} sin\lambda_{i}&cos\varphi_{i}\\ -sin\lambda_{i}&cos\lambda_{i}&0\\ cos\varphi_{i}cos\lambda_{i}&cos\varphi_{i}sin\lambda_{i}&sin\varphi_{i} \end{bmatrix}\begin{bmatrix}dX_{i}\\ dY_{i}\\ dZ_{i}\end{bmatrix} \tag{3}\]  
where \(dX_{i}\ =\ X_{i}\ -\ \overline{X}\), \(dY_{i}\ =\ Y_{i}\ -\ \overline{Y}\), \(dZ_{i}\ =\ Z_{i}\ -\ \overline{Z}\); \(\overline{X}\), \(\overline{Y}\), and \(\overline{Z}\) are reference coordinates from RTK measurement; and \(X_{i},\ Y_{i},\ Z_{i}\) are consecutive coordinate components generated by smartphones or handheld receivers.  
Figure 5: Sky visibility conditions during each part of the experiment: (**a**) surface, (**b**) line, (**c**) point, (**d**) static session.