## 2 Quality Control Algorithms for GPS Observations
### Satellite Elevation and Dilution of Precision (DOP)  
GPS errors are mainly affected by satellite arrangements [9]. It is possible to identify these arrangements by calculating the azimuth and altitude of all of the satellites during the periods of observations. The observables of the 3-D geodetic model are the azimuth (\(\alpha\)), the vertical angle (\(\beta\)) and the slant distance (\(s\)). The forces of gravitational and non-gravitational origin perturb the motion of GPS satellites, causing the orbits to deviate from a Keplerian ellipse in inertial space defined by the six elements, semi-major axis a, eccentricity \(e\), inclination \(i\), longitude of the ascending node \(\Omega\), angle of perigee \(\omega\), and true anomaly \(f\) or eccentric anomaly \(E\). The perturbations are characterized by periodic and secular components, and must be continually determined through the analysis of tracking data. In order to adequately describe the GPS orbits for the interval of time during which the ephemeris information is transmitted, a representation based on Keplerian elements plus perturbations is used. The satellite ephemerides are broadcasted two hours in advance of the epoch for which they were calculated. Generally, the update occurs daily, though is sometimes more frequent. Therefore, the portions of ephemeris data for the second through fourteenth days are not normally transmitted, except when the upload is not possible. At the same time, each satellite's clock state is estimated and then extrapolated into the future, and the information is finally formatted into the navigation message. It is easy to calculate the earth-centered and earth-fixed (_ECEF_) Cartesian coordinates of a satellite (\(X_{S}\), \(Y_{S}\), \(Z_{S}\)) on a specific epoch using satellite ephemeris information and WGS84 constants. The \(\alpha\) and \(\beta\) between the satellite and receiver were derived from Equations (1) and (2), respectively. The _ECEF_ Cartesian coordinate of the receiver was calculated from the approximate coordinate recorded in a header part of the GPS observation data. The earth-fixed reference system is presently based on the WGS84 system [9],  
\[\alpha=\tan^{-1}\left(\frac{-\sin\lambda_{\mathrm{g}}\Delta X+\cos\lambda_{ \mathrm{g}}\Delta Y}{-\sin\varphi_{\mathrm{R}}\cos\lambda_{\mathrm{R}}\Delta X -\sin\varphi_{\mathrm{R}}\sin\lambda_{\mathrm{R}}\Delta Y+\cos\varphi_{ \mathrm{R}}\Delta Z}\right) \tag{1}\]  
\[\beta=\sin^{-1}\left(\frac{\cos\varphi_{\mathrm{R}}\cos\lambda_{\mathrm{R}} \Delta X+\cos\varphi_{\mathrm{R}}\sin\lambda_{\mathrm{R}}\Delta Y+\sin\lambda _{\mathrm{R}}\Delta Z}{\sqrt{\Delta X^{2}+\Delta Y^{2}+\Delta Z^{2}}}\right) \tag{2}\]  
where, \(\varphi_{\mathrm{R}}\) and \(\lambda_{\mathrm{R}}\) are the geodetic coordinates of the receiver based on the WGS84 ellipsoid, which is calculated by the direct coordinate conversion from _ECEF_ to geodetic [10]. This conversion is not exact and provides centimeter accuracy for heights below 1,000 km. \(\Delta X\), \(\Delta Y\) and \(\Delta Z\) are the geometrical differences in the _ECEF_ coordinate system between the receiver (\(X_{R}\), \(Y_{R}\), \(Z_{R}\)) and GPS satellite (\(X_{S}\), \(Y_{S}\), \(Z_{S}\)). Dilution of precision (_DOP_) values are commonly used to indicate the quality of satellite arrangements. _DOP_ is the ratio of positioning accuracy to the observation accuracy. _DOP_ is always a number greater than unity when there are no redundant observations. The _DOP_ values vary from epoch to epoch according to the change of a satellite's geometric distribution. The satellite's geometric distribution does not cause inaccuracies in the position determination that can be measured in meters. In fact, the _DOP_ values amplify other inaccuracies. High _DOP_ values merely amplify other GPS errors to a greater extent than do low _DOP_ values. A number of different _DOP_ factors are possible, depending on the coordinate component, or combination of coordinate components of the local geodetic coordinate system, usually expressed as _N, E_ and \(H\).