# Extended Geometry and Probability Model for GNSS+ Constellation Performance Evaluation
## 1 Introduction  
Global Navigation Satellite System (GNSS) performance can be evaluated from several aspects, such as satellite signal quality [1; 2], the precision of broadcast ephemerides [3; 4; 5], the multipath effect [6; 7], convergence speed and precision of positioning [8; 9]. Among all of these aspects, the satellite constellation, which defines the observation geometry distribution of navigation systems [10; 11], is of fundamental importance, especially under the progress of constellation design. Satellite visibility and dilution of precision (DOP), which reflect the geometry distribution quality of constellations,is regarded as the metric that defines the mapping factor between the ranging error and positioning error [12; 13]. With the development of the GNSS, the traditional Walker-delta constellation has been transformed into the hybrid constellation. In order to enhance the service in the area of the Asia-Pacific region (60\({}^{\circ}\)S-60\({}^{\circ}\) N, 50\({}^{\circ}\)E-170\({}^{\circ}\) E), the third-generation BeiDou navigation satellite system (BDS-3) constellation comprises a Walker constellation, inclined geosynchronous orbit (IGSO) constellation, and also the geostationary earth orbit (GEO) constellation, which dramatically improves the visible satellite number and DOP in the Asia-Pacific region [14]. Many researchers have analyzed the contribution of IGSO and GEO constellations for the BDS-3 [2; 15]. However, the DOP of current GNSS constellations in high latitude areas is much larger than regions of middle and low latitude [15; 16], which causes unfavorable positioning, navigation, and timing (PNT) performance in these areas. Meanwhile, the current coverage of the satellite-based augment system (SBAS) [17], relies on GEO satellites to broadcast the correction message, also lacks the continuous coverage of the polar regions. Besides, because the Arctic shipping routes are excepted to be the shortest and of considerable economic value from the Atlantic Ocean to Pacific Ocean [18], it is desirable to improve the satellite visibility and geometry distribution quality in high latitude areas, especially in the Arctic region. Therefore, given the facts mentioned above, highly eccentric orbits (HEO) satellites could be used [19]. Among these, the Russian Molniya satellites are the most common, which can be used to provide communication services for polar regions. Conventional methods of satellite visibility and DOPs (GDOP (geometric DOP), HDOP (horizontal DOP), PDOP (position DOP), TDOP (time DOP), and VDOP (vertical DOP) ) assessment are based on real ephemerides [20] or simulated almanacs [21; 22] after complex calculation. [PERSON] et al. [16] proposed the geometry and probability model (GAPM) for the Global Positioning Satellite System (GPS) constellation. The GAPM method applies the concept of statistics to satellite visibility analysis and defines the satellite spatial distribution probability density function based on the satellite angular velocity. This method only requires some constellation parameters and dramatically reduces the calculation burden. This method was later used by [PERSON]. [23] to calculate the GNSS DOP of low earth orbit (LEO) satellites. However, GAPM is only suitable for the medium earth orbit (MEO) or LEO constellations, and to evaluate the contributions of IGSO and GEO satellites of BDS-3, [PERSON] et al. [15] developed the modified GAPM (MGAPM) method. Through the constraint of the domain of a function to the ground track of IGSO or GEO satellites, MGAPM realizes the extension of GAPM. By revisiting their method and considering the GNSS+ constellation with possible non-circular HEO and quasi-zenith orbit (QZO), we present an extended and modified geometry and probability model (EGAPM) with the capability of analyzing the contribution of large eccentricity satellite orbit constellations to navigation performance. EGAPM is validated through the satellite visibility and DOP results compared with that of precise ephemerides of GPS and BDS-3+QZO.