3.1.1. Sensitivity of Height and ZWD Mean Correlation Coefficient to Processing Modeling
The correlation coefficient between height and ZWD has been calculated for each of the GNSS signal processing models in the 200 different tropospheres. The mean height and the ZWD correlation coefficient were then calculated from the 200 resulting correlation coefficients for each processing model. Therefore, there remains a mean correlation coefficient between height and ZWD and its standard deviation for each of the processing models. The standard deviations of the correlation coefficients are all lower than 0.1. In the following, we will focus on the mean correlation coefficient between ZWD and height presented in
Figure 5.
As expected,
Figure 5 shows that the mean correlation coefficient increases with the cut-off angle when using elevation weighting function modelings
sine and
sqrtsin, which corroborates the fact that height and ZWD are better decorrelated when embedding the signals coming from the low elevation satellites above the horizon.
However, we observe the opposite behavior on correlation and height estimation with
uniform and
cos4 elevation weighting functions. As shown in
Figure 4, both
uniform and
cos4 give more weight to measurements of very low elevation angles than
sine and
sqrtsin.
cos4 weighting reaches
sqrtsin around 7° of elevation angle, and the weight of
cos4 is twice as high as
sqrtsin at 3° of elevation angle.
Figure 5 also shows that the mean correlation coefficient between height and ZWD is lower for low random walks. The lowest mean correlation coefficient is 0.16 obtained for 3° of cut-off angle,
sine weighting function, and 1 mm·h
of random walk. The highest mean correlation coefficient is 0.78 for 10° of cut-off angle,
uniform weighting function, and 20 mm·h
. The function
sine is the best weighting for decorrelating height and ZWD, while
uniform appears to be the worst.
sqrtsin appears to decorrelate height and ZWD slightly better than
cos4 at 7° and especially at 3° of cut-off angle, and both at 10° of cut-off angle give the same correlation coefficient. We can notice that the change in random walk value impacts the correlation coefficient more than the cut-off angle. This might be because a tight constraint on ZWD will force this variable to absorb a limited amount of errors coming from the antenna movement, particularly the change height that could be wrongly assimilated to ZWD in the case of loose random walk value. The height part thus wrongly assimilated to the ZWD could be responsible for its correlation being increased with the height estimate.
3.1.2. Sensitivity of Height and ZWD Biases to Processing Modeling
The biases between the estimation and the simulation on height and ZWD have been calculated for each of the GNSS signal processing models over the 200 different tropospheres. The average height and ZWD biases were then calculated from the 200 resulting biases for each processing model, as well as the associated dispersion. From this, the root mean square error (RMSE) of the height bias and ZWD have been calculated for each of the processing models and are presented in
Figure 6.
Figure 6 shows that, despite the large amplitude of the simulated vertical movements, the height and ZWD estimates are still of good quality. Indeed, the height bias RMSE is 3.7 mm at most, with (
) mm of height bias obtained for 10° of cut-off angle, 1 mm·h
of processing random walk and
sine weighting function. This processing modeling also gives the highest ZWD bias RMSE of 1.3 mm, with (
) mm of ZWD bias. The major part of the highest RMSE values on biases is due to the dispersion of the biases being greater than the average. Indeed, the bias averages in the ZWD estimation are less than 0.1 mm, except when using a cut-off angle of 10°, which occurs with an average ZWD bias of at most 0.2 mm for all weighting functions but
sine. Furthermore, average height biases range from 0.0 mm to 0.2 mm when using a cut-off angle of 3° or 7°, except when using
uniform weighting function occurring at a maximum height bias average of 0.4 mm with 3° of cut-off angle. For 10° of cut-off angle, the average height biases range from 0.3 mm to 0.6 mm, depending on the weighting function and random walk modeling.
Figure 6 also shows that increasing the random walk value in the ZWD will decrease the RMSE in the biases for height and ZWD for any given cut-off angle and weighting function modeling. In light of the elements presented in the previous paragraph, this shows that releasing the constraint on the ZWD will allow us to obtain more precision on the biases between simulation and estimation of height and ZWD. This result corroborates the observations made on ground antennas by Young et al. [
76], recommending the use of at least 6 mm·h
random walk on ZWD to process the GNSS data.
Decreasing the cut-off angle allows us to decrease the RMSE according to the three graphs for both variables in
Figure 6, which is logical considering the fact that a low cut-off angle allows decorrelating height and ZWD. However, this result is due to the fact that the multipath effect is quite low in this shipborne antenna simulation, requiring a well-placed antenna, for example, at the very top of the vessel. Bosser et al. [
27] showed that some antenna locations are not suitable for retrieving ZWD from shipborne antennas, as the quality of ZWD can be degraded even when using a cut-off angle greater than 3°. Thus, it will be preferred to find the most suitable location as possible for the GNSS antenna onboard the vessel to prevent the multipath effect, and then process the data with a low cut-off angle to well decorrelate height and ZWD.
The weighting function sine appears to give the worst RMSE, while the weighting functions cos4 and sqrtsin have a similar behavior both in the resulting height and ZWD estimates. Indeed, if sine systematically gives the best average bias on both variables, it appears that the dispersion of the ZWD bias is up to five times worse than when using the other weighting functions, while it is up to three times worse in the height bias dispersion. uniform seems to be the best estimation weighting function model, except to estimate the height at 3° of cut-off angle. However, it is worth mentioning that the uniform function provides less accuracy than the other weighting models since it gives a systematically worst average of the bias on both height and ZWD, which can be up to three times higher on ZWD and four times higher on height than with any other weighting function.
3.1.3. Sensitivity of Height and ZWD Standard Deviations to Processing Modeling
The standard deviation of the difference between the estimates and the initial simulation for ZWD and height has been calculated for each of the 200 different tropospheres simulated with each processing modeling. The average of the 200 standard deviations of the difference between the estimation and the simulation of height and ZWD has then been computed for each processing modeling, as well as the associated dispersion.
First,
Table 4 presents some of the dispersion figures of the standard deviations of the differences between the estimation and the simulation of height and ZWD.
The dispersion figures in
Table 4 are very low, most of the time lower than 1 mm, and even lower than 0.5 mm on ZWD. The highest dispersion is always displayed for the weighting function
sine, implying more variability in the accuracy of the height and ZWD estimation when using this elevation weighting modeling, compared to the other weighting functions tested. It appears that using low random walk values or increasing the processing cut-off angle increases the variability of the standard deviations on height and ZWD differences, for any elevation weighting function. In light of this very small dispersion of the standard deviation of the differences, the following will focus on the average of the standard deviation of the differences between the estimation and the simulation of height and ZWD.
The averages of the standard deviations for height and ZWD are presented in
Figure 7.
Figure 7 shows the averages of the standard deviation of the differences in ZWD and height as a function of the processing random walk, for each weighting function (different colors) and cut-off angle (different graphs) tested. In each plot, the use of a
uniform weighting function worsens the average standard deviation compared to the other weighting functions. The only exception is when processing with very low processing random walks, where
uniforms performs better at estimating ZWD, but still provides the worst height estimate of all weighting functions. Then, it appears that setting up a weighting function on the elevation of the observed satellites helps to improve the estimation of height and ZWD during the GNSS PPP signal processing instead of giving the same weighting to all observations.
The weighting function sine systematically provides us with the lowest ZWD average standard deviation for any elevation cut-off angle modeling. The minimum value of the ZWD average standard deviation is obtained for random walks in the range of 10 to 12 mm·h, depending on the cut-off angle. Then it appears that the ZWD average standard deviation is better when using higher random walk values than in the simulation. As an approximation, we can state that the sine function needs to loosen the random walk twice to minimize the standard deviations of the differences.
The weighting functions
cos4 and
sqrtsin exhibit behavior similar to each other, especially as the cut-off angle increases. However, the use of the weighting function
cos4 always induces a higher standard deviation than
sqrtsin for height estimates. Furthermore, only in the case of a random walk lower than 3 mm·h
the weighting function
cos4 achieved a better estimation of ZWD than
sqrtsin. Finally, the graphs of the average standard deviation on the height difference processed with the weighting functions
uniform and
cos4 in
Figure 4 show that increasing the cut-off angle improves the height estimation with both weighting models. This might be explained by the multipath effect, as discussed in
Section 3.1.4.2. Therefore, we chose to focus on the impact of
sqrtsin and
sine.
Table 5 shows the minimum values of the average standard deviation for ZWD and height differences for the three tested cut-off angles and the
sqrtsin and
sine weighting functions.
It is noticeable that all the standard deviation minima presented in
Table 5 are not obtained with the same random walk value on the ZWD. Therefore, the last column of
Table 5 presents the random walk values that provide the minimum standard deviation on ZWD and height for each pair of modeling on the cut-off angle and weighting function.
The average standard deviation shown in the fourth column of
Table 5 shows that the configuration that gives the closest ZWD and height estimates to the reference is 3° of cut-off angle and weighting function of elevation
sine with a random walk of 11 mm·h
. The rate of change shows that the increase in the cut-off angle induces an increase in the standard deviation of the ZWD differences up to +20.3% for 7° and +38.2% for 10°. The height results also show that increasing the cut-off angle decreases the resulting minimum in the average standard deviation of the height difference, with a change rate of +3.7% for 7° and +7.4% for 10°.
At a given cut-off angle, the minimum average standard deviation in ZWD is obtained using the weighting function
sine instead of
sqrtsin. However, the differences remain very low, in the 0.1 mm range, and tend to decrease when the cut-off angle increases. For a cut-off angle of 3°, the minimum standard deviation obtained by using the
sqrtsin weighting function is +5.3% higher than when using the function
sine. At 7°, the rate of change of
sqrtsin is +3.8% higher than
sine and decreases to +0.9% at a cut-off angle of 10°. The same conclusion is reached when considering the height estimates. A +6.6% increase in the average standard deviation of the difference is obtained when using
sqrtsin compared to
sine for a cut-off angle of 3° and +2.5% at 7°. The function
sqrtsin even provides a +0.3% better average standard deviation minima than
sine for a cut-off angle of 10°. It appears that the choice of weighting functions between
sine and
sqrtsin mainly affects the low-elevation satellite signals; then it decreases with increasing cut-off angle [
46].
Table 6 further details the ZWD estimate for each of these random walks. The three last columns of
Table 6 show that the
sqrtsin function is 9.8% better at estimating ZWD than the function
sine at a cut-off angle of 3°, with the 5 mm·h
random walk corresponding to the simulated ZWD. At a minimum, obtained with a cut-off angle of 3°,
sine weighting function, and 11 mm·h
of random walk,
sqrtsin degrades the estimate of ZWD by +28.2%. In general, the rate of change between
sqrtsin and
sine decreases when the cut-off angle increases, but the ZWD standard deviations also increase compared to a cut-off angle of 3°.
The use of the weighting function
sine implies the need to double the random walk value for a better estimation compared to the simulation. The weighting function
sqrtsin appears to work well with lower random walks, such as 5 mm·h
for 3° of cut-off angle and 7 mm·h
for higher cut-off angles, instead of 11 to 12 mm·h
for obtaining the minimum standard deviation with the weighting function
sine. As ZWD was simulated with a 5 mm·h
random walk,
sqrtsin seems to remain consistent with the physics of atmospheric water vapor compared to the function
sine. Indeed,
Table 7 shows that for the three cut-off angle values tested, the
sqrtsin function performs around +10% better in estimating the ZWD than
sine weighting function when using 5 mm·h
of random walk modeling. The height estimation with
sqrtsin at 5 mm·h
does not change with the elevation cut-off angle. However, the function
sine performs better at a very low elevation cut-off angle, showing an improvement of +5.9% at most for 3°. Both weighting functions give similar results to the rate
for the other two cut-off angles.