In this fourth step of the methodology, the ray-tracing algorithm used for numerical integration is applied as described in section 5.3.3. As well as introducing the elevation dependency, the distance between ground station and aircraft is also extended beyond the fixed distances of the predefined ten segments. Finally, its unknown along which portion of the path is the tropospheric delay decorrelated between the segment end points. For this reason, when using elevations lower than 90 degrees, an additional search parameter is introduced, by translating the ground-to-aircraft plane by a series of distances Dw. as shown in the following figure. Indeed,
without translation the worst condition (represented by the cloud) could be missed, therefore it is needed to move the segment for finding the worst differential range tropospheric delay.
Figure 80-Segment translation
At the end of the loop, the worst case over Dw is used. Figure 81 shows how the perpendicular translation is
performed, shifting the midpoint M to a new midpoint M’.
Figure 81-Finding the worst case for Differential Range Tropospheric Delay
It is important to remind here in this report that, the differential range tropospheric delay varies with distance between aircraft and ground as shown in the following figure obtained by using the MHM described in 5.3.2 for a satellite at 5° of elevation:
Figure 82-Differential range Tropospheric Delay
Therefore, in this section the distance D was fixed as the sum of the ground distance of the aircraft to the runway threshold (LTP, Landing Threshold Point) at the 200ft Cat I decision height which is considered as the furthest point where Cat II/III performance requirements are applicable and a nominal of 5km between the LTP and ground station according SARPs [117]. In the previous Figure 82, it is interesting to remark that the first part of the curve grows up with distance and then the second part decreases with distance. This could be explained by the fact that the first part shows the spatial decorrelation between aircraft and ground station. Then, this study assumes a glide path angle at 2.5° with an “infinite” approach therefore the second part decreases because the horizontal variation of the troposphere decrease with the height of aircraft. This approach is not realistic but is ok as the critical part is the closest to the ground station and therefore the first part of the curve.
A fitting curve has been derived using the Wall model parameters defined in [7] with the Modified Hopfield Model (MHM) explained in 5.3.1 to compute the differential range tropospheric delay and plotted in blue denoted by the Curve A in Figure 83. Then in order to compare this “Ohio” parameterization 5.2.1.2 with the “European” parameterization, the same approach as explained in 5.2.1 was derived by using the “Harmonie” data and “Arome” data. Indeed, by examining these data (5.2.2.3), worst case weather conditions appear for “Harmonie” to be Tw at 37.01°C, Pw set at 1003.2hPa, RHw set at 100% and for “Arome” to be Tw at 36.9°C, Pw
set at 1009.8hPa, RHw set at 100%. Then, the red dashed Curve represents the Wall model derived with Harmonie
Figure 83-Wall model with different parameterizations
Differences in the curves can be explained due to the means the parameters were derived and ways the worst conditions were found and the environment of the region of data employed. Indeed, for analyzing the environment impact, it is possible to compare the “Wall Model” curves in Figure 83 , then it can be noticed an important difference between the blue curve representing the Ohio parameterization derived from a restricted set of data (GPS data only at one location) and the dashed curves which represent the “European” parameterization derived from a large amount of data (Arome and Harmonie). Therefore, this figure suggests that gradients in Europe could be more important than for this set of data examined at Ohio because differences on maximum differential range tropospheric delay appear to be about 0.75m at 5° of elevation and 0.15m at 90°. But it should be added that theses differences could be also explained by the fact that worst conditions in Harmonie and Arome data were found by using a conservative methodology. Indeed the search was realized for the worst conditions on all data and compared to standard values even if in the same set of data standard conditions and worst conditions are not present at the same time and at “approach” distance (up to 12km).
Then, Figure 84plots the worst cases found with the methodology explained in this section 5.4 for each of the 4945 sample points of Harmonie NWM over 2 years. A red curve denoted Curve B was fitted to worst case over time for each five degrees of elevation.
Figure 84-Max differential range tropo delay for Harmonie
Also the Figure 85 plots these worst cases found with the methodology explained in this section 5.4 for each of 2920 sample points of Arome NWM over 1 year, then a green curve denoted Curve C was fitted to worst case over time for each five degrees of elevation.
Figure 85-Max differential range tropo delay for Arome
It is relevant to notice the sharped shape of the Curve C at low elevations of 5°. This phenomena can be explain by the fact that in this Alpines area, gradients in atmospherical parameters are higher and they change more frequently as in Netherlands area particularly closer to the surface (which is the case for low elevations case in
this research). Also, it is important to notice that for this Arome Case, the dash line is below the green line for elevations angle smaller than 6°,therefore the Wall Model methodology appear conservative only for angle superior to 6° of elevation. Furthermore, for Alpines data realy conservative assumptions about the way to find the worst case were taken (as possible approaches closed to the mountain) which are no realistic these can explain the highest values obtained for low elevation angles. Then it could be interesting to only plot results just for elevation angle from 10° to 90° as presented in the following figure and to fit them by the Curve D.
Figure 86-Max differential range tropo delay for Arome from 10°
Finally, the bounding/fitting curves obtained with each model and data (Ohio, Arome and Harmonie) is represented starting at 5° elevations angle (Curve A,B and C) in Figure 87 and at 10° elevation angle (Curve A,B and D) in Figure 88.
Figure 87-Bounding Curves for each model stating at 5°
Figure 88-Bounding Curves for each model stating at 10°
The differential range tropospheric delay follows an expected trend with respect to elevation. Equations of these curves A, B, C and D are detailed in section 6.3.1.2. Also according to some previous analysis [7] [128], differential range tropospheric delays appeared as large as 0.4m in Ohio for a particular set obtained with GPS data, in the Figure 87, maximum differential range tropospheric delay appear to be about 0.8m by seeing the red Curve B and about 1.1m regarding the Curve C at 5° of Elevation (0.55m at 10° regarding the Curve D) so gradient in Europe seems again (as thought by seeing Figure 83) more important than in the data set examined by Ohio
particularly in Alpines area at low elevation. This could be explained by the fact that abnormal changings in atmospherical parameters should happen more frequently and should be more localized close to the surface (i.e at low elevation) in Alpines area.
Also, it is relevant to notice that the dashed red/green lines (representing the Wall Model curves)are really above the red/green lines that means that using the “Wall model” as a methodology for finding the maximum differential range tropospheric delay is a too conservative methodology therefore the Wall Model methodology appear really conservative.