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To study the DOP reduction due to the fusion of cellular and GNSS signals, consider an environment comprising a receiver making pseudorange measurements on M GNSS SVs and N terrestrial cellular BTSs. The pseudorange measurements are fused through a WNLS estimator to estimate the states of the receiver , where and δt_r are the 3D position and clock bias of the receiver, respectively, and c is the speed of light. To simplify the discussion, assume that the measurement noise is independent and identically distributed across all channels with variance σ². If the measurement noise was not independent and identically distributed, the weighted DOP factors must be considered [84]. The estimator produces an estimate and an associated estimation error covariance matrix P = σ²(H^TH)⁻¹, where H is the measurement Jacobian matrix. Without loss of generality, assume an east, north, up (ENU) coordinate frame to be centered at . Then, the Jacobian in this ENU frame can be expressed as

where c(·) and s(·) are the cosine and sine functions, respectively; and are the elevation and azimuth angles, respectively, of the m‐th GNSS SV; and and are the elevation and azimuth angles, respectively, of the n‐th cellular tower as observed from the receiver. Therefore, G ≜ (H^TH)⁻¹ is completely determined by the receiver‐to‐SV and receiver‐to‐BTS geometry. The diagonal elements of G, denoted g_ii, are the DOP factors: geometric DOP (GDOP), horizontal DOP (HDOP), and vertical DOP (VDOP)

With the exception of GNSS receivers mounted on high‐flying aerial vehicles and SVs, all GNSS SVs are typically above the receiver [85]; that is, the elevation angles in H_sv are theoretically limited between 0° and 90°. Moreover, GNSS receivers typically ignore signals arriving from GNSS SVs below a certain elevation mask (typically 0° to 20°), since such signals are heavily degraded due to the ionosphere, troposphere, and multipath. When using GNSS together with cellular signals for navigation, the elevation angle span may effectively double to be between −90° and 90°. For ground vehicles, useful measurements can be made on cellular towers at elevation angles of . For aerial vehicles, cellular BTSs can reside at elevation angle as low as , for example, if the vehicle is flying directly above the BTS.

To compare the DOP of a GNSS‐only navigation solution with a GNSS + cellular navigation solution, a receiver position expressed in an Earth‐Centered Earth‐Fixed (ECEF) coordinate frame was set to . The elevation and azimuth angles of the GPS SV constellation above the receiver over a 24 hour period was computed using GPS SV ephemeris files from the Garner GPS Archive [86]. The elevation mask was set to el_{sv, min} ≡ 20^°. The azimuth and elevation angles of three towers, which were calculated from surveyed terrestrial cellular CDMA tower positions in the receiver’s vicinity, were and . The resulting VDOP, HDOP, GDOP, and associated number of available GPS SVs for a 24 hour period starting from midnight, September 1, 2015, are plotted in Figure 38.61. These results were consistent for different receiver locations and the corresponding GPS SV configurations. The following can be concluded from these plots for using N ≥ 1 cellular towers. First, the resulting VDOP using GPS + N cellular towers is always less than the resulting VDOP using GPS alone. Second, using GPS + N cellular towers prevents large spikes in VDOP when the number of GPS SVs drops. Third, using GPS + N cellular towers also reduces both HDOP and GDOP. Additional analysis is given in [7, 8].

Figure 38.61 Figure (a) represents the number of GPS SVs with an elevation angle > 20° as a function of time. Figures (b)–(d) correspond to the resulting VDOP, HDOP, and GDOP, respectively, of the navigation solution using GPS only, GPS + 1 cellular tower, GPS + 2 towers, and GPS + 3 towers (Morales et al. [7]).

Source: Reproduced with permission of Z. Kassas (International Technical Meeting Conference).