# Accurate Collaborative Globally-Referenced Digital Mapping with Standard GNSS
## 3 GNSS Error Analysis
### Navigation Filter  
At each epoch, \(n_{z}\) multipath-free, ionosphere-free, and troposphere-free simulated pseudorange measurements were combined with corresponding simulated multipath tracking delay errors and fed to a navigation filter that estimates the receiver state. The navigation filter implemented in this paper is an extended Kalman filter (EKF) with a nearly constant velocity motion model following [53]. The standard details of the EKF are omitted for brevity. The effect of multipath tracking on the navigation solution is strongly dependent on the receiver's multipath rejection scheme. Two schemes are explored here. The first is a hypothetical ideal multipath rejection scheme that excludes all signals for which the LOS signal has a smaller-than-10-dB advantage over its multipath echoes. The second scheme implements a normalized innovation squared (NIS) test to reject multipath signals based on measurement innovations [53]. At the \((k+1)\)th measurement update step, the difference between the predicted and observed measurement vector, called the innovation and denoted \(\
u(k+1)\), is squared and normalized by its covariance, which is the sum of the measurement covariance matrix, \(R(k+1)\), and the propagated state covariance transformed through the measurement sensitivity matrix, \(H(k+1)P(k+1|k)H(k+1)^{T}\). In the absence of multipath tracking errors, the resulting NIS statistic is chi-squared distributed with \(n_{z}\) degrees of freedom. If the NIS statistic exceeds a chosen threshold, then the signal with the largest normalized innovation is dropped. This continues until the NIS statistic falls below the threshold or the number of remaining signals drops to a preset minimum number of required signals. Simulation Results  
Figure 5 shows the mean position error in the east, north, and up directions over 1000 sessions for the two multipath rejection schemes mentioned previously. From Figure 5a, it can be seen that sub-20 cm average error is achievable with hypothetical ideal multipath exclusion. A closer look at Figures 4 and 5a reveals that the decimeter-level sinusoidal position error trend, initially in the north direction and later in the east direction, in fact corresponds with the along-track accelerations of the vehicle that were not adequately tracked by the nearly-constant-velocity-model-based navigation filter. Figure 5: Mean position error in the east-north-up (ENU) frame over 1000 sessions due to multipath. **(a)** Ideal multipath exclusion. **(b)** Normalized innovation statistic (NIS)-based multipath exclusion. The black, gray, and dashed-black lines represent the error in the east, north, and up directions, respectively. The up error in the bottom panel reached a maximum magnitude of 1.75 m.  
Figure 5b shows that the NIS test based exclusion of signals was able to approach the performance of ideal exclusion in the horizontal plane, save for the first stationary period where the vehicle was moving at low speed between buildings on both sides. The average vertical position error was much worse, growing as large as 1.75 m in magnitude. To determine whether the average errors shown in Figure 5 are in fact persistent biases, a study of the standard deviation of position errors was conducted. The standard deviation of the average errors in east, north, and up directions was computed for disjoint averaging ensembles of size 1, 2, 4, 8, 16, 32, 50, and 100 sessions taken from the total of 1000 simulated sessions. For instance, 125 disjoint ensembles of eight sessions were selected, and the position errors were averaged over the eight sessions in each set. The standard deviation of the eight-session-averaged errors was then computed across the 125 ensembles. In the case of an averaging ensemble with only a single session (i.e., no averaging), the computed standard deviation is simply the measured standard deviation of the position error across all 1000 simulated runs.