Accuracy assessment of position, velocity and acceleration

GPS not only provides the precise position of aircraft but also the velocity and acceleration for data processing. Here, accuracy assessments of position, velocity and acceleration using the GPS data collected in the multiple altitudes airborne gravity surveys are presented. The positions of aircraft were determined by the Bernese 5.0 software (Beutler et al., 2004; Rolf et. al., 2007). Ten sessions of GPS data from the Kuroshio and Taiwan Strait surveys at 1500 m, and another ten from the Taiwan Island survey at 5000 m were selected for the computation of position, velocity and acceleration and for accuracy assessments. Each session of about 4 hours in data length was divided into two independently processed sub-sessions with a 30-minute overlap. The 20 overlapping sessions were selected so that the tracks of the overlaps cover coastal plains, high mountains, and oceans.

Figure (3.6) shows a typical example of position differences in an overlapping

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session, which vary slowly and contain spikes and discontinuities. Figure (3.7) shows the PSD of position differences for the three components. The spectral index, i.e., the value of the exponentxinS(f)  f ^x , where S is PSD and f is frequency, is -0.7,-0.8 and -1.4 for the north, east, and vertical components, respectively. Therefore, the PSD approximately indicates flicker noise (x1) for the north and east components, and a combination of flicker noise and a random-walk process (x2) for the vertical component. Thus, the overlapping differences of position estimates are dominated by the long-wavelength errors in GPS positioning. These results are consistent with other studies, e.g., Verdun et al. (2003) represent the combined effect of long and short wavelength GPS errors. Furthermore, values of the PSD of the vertical component are 0.5 to 1.5 orders of magnitude smaller than the PSD of the north component, but similar to the PSD of the east component at high frequencies.

Values of the PSD of the north component are 2 to 4 smaller than the east component at all frequencies.

The overlapping difference can be regarded as the internal accuracy of GPS positioning. It was found that spikes and discontinuities were associated with phase ambiguity changes, i.e., change of visible satellites and/or cycle slips in one or more of the baselines. The position differences in Figure (3.6) are mainly caused by the differences in the estimated common parameters associated with the overlapping sub-sessions, e.g., phase ambiguities and tropospheric parameters. Since these common parameters will remain unchanged within a certain time span, their effects will be reduced upon differentiation. That is, we would expect that the differences in position-derived velocity and acceleration (by numerical differentiation) are less affected by the differences in common parameters.

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Figure 3.6: Differences of position at an overlapping session in the north, east, and vertical components. The left vertical scale is for the north and east components, and the right vertical scale for the vertical component. Discontinuities occur at seconds 615 and 699 (Hwang et al., 2007).

Figure 3.7: Power spectral densities of the differences (Hwang et al., 2007).

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Table 3.2: Averaged RMS differences of 10 overlapping sessions of GPS kinematic solution from the Kuroshio and Taiwan Strait surveys at the altitude of 1500 m.

North East Vertical

Position (m) 0.137 0.268 0.421

Velocity (m·s^-1) 0.0064 0.0081 0.0108

Acceleration (mgal) 93.88 74.53 906.28

Table 3.3: Averaged RMS differences of 10 overlapping sessions of GPS kinematic solution from the Taiwan Island survey at the altitude of 5000 m.

North East Vertical

Position (m) 0.073 0.208 0.293

Velocity (m·s^-1) 0.0003 0.0003 0.0013

Acceleration (mgal) 23.56 26.56 104.85

Table (3.2) and (3.3) lists the average RMS differences in position, velocity, and acceleration from the 10 overlapping sessions at 1500 m and 5000 m, respectively.

The overall positioning accuracy is of the order of decimeter, with the vertical component being the largest. The RMS velocities show that the long wavelength positioning errors does not propagate to the errors in velocity and acceleration. The statistics of differences in velocity and acceleration at the altitude of 1500 m are larger than those at the altitude of 5000 m. The reasons are (1) the measuring environment at lower altitude contains more turbulence and (2) the GPS signals at 1500 m contain more atmospheric influence such as tropospheric errors than the case at 5000 m.

Assume that the RMS differences in Table (3.3) are equal to the standard errors of the respective quantities. In theory, a standard error of 0.293 m in vertical position

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0   m·s^-2 in vertical acceleration. However, the standard errors in Table (3.3) are much smaller than these two numbers. This implies that differentiation of positions does reduce the effects of long-wavelength errors of positions on velocity and acceleration (Verdun et al., 2003).

Table 3.4: Standard deviations of differences at an overlapping session of 5000 m in the north and east velocity components and the vertical acceleration with different filter window size (Hwang et al., 2007).

Window

120 5000 0.000030 0.000042 0.000015

180 7500 0.000027 0.000037 0.000012

240 10000 0.000026 0.000034 0.000010

Furthermore, to see the performance of the Gaussian filter and the effect of filter window size on the GPS-derived velocities and accelerations we performed the following tests using the 20 overlapping sessions. The results in Table (3.4) are based on an overlapping session. We assume that the aircraft velocity is 300 km/hour, and the equivalent spatial resolution of airborne gravity data is just the multiplication of the aircraft velocity and the half-window size. In general, with a window size greater than 60 seconds, the standard deviations of velocities are at the 0.005 m·s^-1 level and

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the standard deviations of vertical acceleration are at the mgal level. A window size larger than 200 seconds yields a sub-mgal standard deviation. Through Table (3.4) we can initially understand how to select an adequate window size for the filtering of airborne data. In an actual flight, the aircraft may experience turbulence and GPS data may contain cycle slips, so the window size to be used will vary from one case to another.