In the previous section were computed the standard deviations (1 σ) of P , T and e on each pressure level (see Table 4.9). Those standard deviations are now input to an error propagation in order to assess the impact on the determination of the total refractivity N. The parameter e was assumed to be independent from T so that the variance covariance matrix of the parameters X (CXX) has null non-diagonal elements and (assuming perfect refractivity constants as well) can be written as:
and the transition matrix F written as:
FT = · ∂N
To compute the results, average pressure in each pressure layer was used. Results for the total refractivity are presented in Figure 4.27. The value used for P is the average measured pressure on the pressure level over years 2005 and 2006 for
all sites. The range of the plots were limited to possible situations only, but it is emphasized the pressure used for various combination of T and e was fixed to the average one computed as mentioned above. Those plots indicate that the error on the total refractivity can easily reach several ppm. The worst case scenario is found for pressure layer 850–700 hPa with a propagated error of 7 ppm. No major difference could be observed when using the pressure of the bottom pressure level instead. It is also important to note that the thickness of the pressure layer might play a significant role in the error found in the refractivity (see Table 4.10).
0
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 123.2 [0.07, 1.10, 0.00]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 84.1 [0.05, 0.98, 0.00]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 49.2 [0.04, 1.00, 0.00]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 274.2 [0.11, 0.64, 0.00]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 223.9 [0.13, 1.17, 0.00]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 173.4 [0.10, 1.44, 0.00]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 595.1 [0.50, 0.81, 0.46]
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 448.3 [0.19, 0.49, 0.11]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 347.9 [0.27, 0.69, 0.03]
0.00 0.25 0.50 0.75 1.00 N
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 974.4 [0.10, 0.70, 0.65]
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 899.0 [0.10, 0.71, 0.78]
180 200 220 240 260 280 300 Temperature [K]
Water vapour pressure [hPa]
P= 772.3 [0.23, 0.99, 0.94]
3 4 5 6 7
N
Figure 4.27: Error propagated on the total refractivity N (ppm) for the different pressure levels. A priori sigma for P , T , and e (assumed independent) were previously determined (Table 4.9) and are reproduced in brackets on each plot.
4.7 Zenith Delay Determination: High Resolu-tion Radiosonde versus NWM, a comparison
The UK Met Office high resolution mesoscale NWM (MESO) was ray-traced at the location of the five hiRes RS sites (using exactly the same implementation as for the OMF derivation) and the zenith delays obtained compared. This com-parison, after the assessment of the models themselves based solely on radiosonde data, includes not only the models’ inherent limitations, but also the errors in the NWM itself. Radiosondes launched at around 11 (23) UTC, were compared to ZD ray-traced using the 12 (00 of the following day) UTC NWM output. The ascent takes about 30-45 minutes, so the approximation made here on the timing will be considered acceptable. Above it was already shown that Methods 1 and 2, based on standard data, were producing results in a relative disagreement, al-though found to be in a very close agreement when used on the hiRes RS data.
Therefore, results are exposed for the two methods for completeness.
Table 4.13 shows that a much better agreement on the ZTD derivation is found for Method 1 between the NWM and the reference solution (either Method 1 or 2 on hiRes RS data) than found for Method 2. However, the ZHD is in a closer agreement for Method 2 than 1, that is, the difference is really made on the ZWD where biases are between 0.3 and 4.2 mm for Method 1 and between 5.2 and 10.0 mm for Method 2. Overall, the level of noise is of comparable magnitude.
Table 4.13: Statistics (mean and associated standard deviation) of the difference between zenith delays derived from the NWM and the hiRes RS data.
RS ZTD Difference ZHD Difference ZWD Difference Sample
Site (mm) (mm) (mm) size
Method 1
ALBE -0.2 ± 10.9 −1.5 ± 1.9 1.2 ± 10.6 2244
CAMB 2.5 ± 12.8 −1.7 ± 1.9 4.2 ± 12.6 2318
HERS -2.3 ± 11.4 −3.2 ± 1.8 0.9 ± 11.1 2093
LERW -0.6 ± 9.9 −0.9 ± 2.2 0.3 ± 9.4 2310
WATN -0.5 ± 10.9 −2.8 ± 1.8 2.3 ± 10.8 2213
Method 2
ALBE 6.2 ± 10.7 −0.2 ± 1.5 6.4 ± 10.5 2243
CAMB 9.6 ± 12.7 −0.4 ± 1.6 10.0 ± 12.6 2317
HERS 4.6 ± 11.4 −1.8 ± 1.4 6.4 ± 11.3 2093
LERW 5.7 ± 9.8 0.6 ± 1.9 5.2 ± 9.4 2310
WATN 6.3 ± 11.1 −1.5 ± 1.5 7.8 ± 11.1 2213
4.8 Summary
The quality of the models used for interpolating meteorological states with height, based on pressure levels data, were studied using several years of high resolu-tion radiosonde data. Running the models on extracted standard pressure levels allowed for a direct comparison between interpolated values for pressure, tem-perature and humidity to be quantified. It was found that the expected total refractivity error (assuming true data on pressure levels being used in the inter-polation) to be about 0.5%, with annual variations, mostly induced by error in the humidity. Error in interpolating the pressure was found to be small in comparison, negative, but systematic. The error made on the absolute temperature averages to zero and appeared to be random. The models’ performances was detailed on a pressure basis. The accuracy of the models was quantified in terms of total zenith delays (using two integration approaches), taking the high resolution RS results as the truth. Various models were tested for interpolating the humidity information, out of which, assuming a linear variation of the relative humidity with height as
a model for humidity, appeared to be the best.
Chapter 5
Results: Azimuthally Symmetric Mapping Functions
This chapter presents the results of a comparison of seven different modelling techniques of the troposphere delay in the processing of GPS data, all of which assume a symmetrical atmosphere. 3.8 years long time series for thirty stations in the UK are examined. All the processing was carried out using the Bernese GPS Software [Dach et al., 2007], and all the solutions were obtained using the precise point positioning technique. The seven solutions are the results of the combination of different a priori information on the zenith delay (Saastamoinen based or numerical weather model ray traced) and different mapping functions.
The main objective is to compare and validate the azimuthally symmetric OMF against the best mapping functions currently available before examining in the next chapter the performance of the azimuth dependent OMF.
5.1 Introduction
The principal objective of this chapter is to validate/invalidate the concept of the OMF via the processing of GPS data. It was a necessary step to investigate the azimuthally symmetric version of the OMF before introducing the azimuthal dependency (Chapter 6). The performances of the azimuthally symmetric OMF are compared against those of the global mapping functions currently available, in particular the (gridded) VMF1, considered the best mapping function available, and that is recommended by the IGS.
The data processing described in the first section (5.2) covers the essential parts of how all results presented in this thesis (this chapter, and Chapters 6 and 7, and the appendices) were generated, although the scope of the present is limited to symmetric mapping functions. In particular, the atmospheric pressure loading models tested are examined and compared here for consistency but their impact on positioning is presented in Subsection 5.3.2 and Section 5.4.