102 6. Case Study: The WHISP Project
Using piece-wise linear segments of 30 minutes instead of 60 minutes leads to an improvement, but the effect due to the different phase calibration methods is even more relevant in this analysis. As expected, the WRMS of post-fit residuals increases sharply when applying manual phase calibration instead of scan-by-scan phase calibration. However, it is striking that particularly the short baseline gets considerably worse: in case of manual phase calibration the WRMS of post-fit residuals is more than twice as high as for the normal scan-by-scan system calibration. But also the degradation for the longer baselines is still considerable. When only considering the solutions with 30 minute piece-wise linear intervals, the solution degrades at least for about 4 ps up to almost 25 ps in quadrature (degradation of about 4 and 13 ps in quadrature for baseline On-Wn, and about 23 and 24 ps in quadrature for baseline On-Wz, in each case for WHISP5 and WHISP7, respectively). Somehow surprising, the effect of degradation is always higher for the baseline On-Wz compared to On-Wn, but, of course, still much lower compared to the short baseline in Wettzell.
6.5. Discussion 103
Second, two independent baselines between the two Wettzell antennas and a telescope in Onsala have been observed to estimate (absolute) zenith wet delays for the two Wettzell stations. The correlation between the zenith wet delay parameters of the two adjacent telescopes has been found to be in the order of 0.94 to 0.99, e.g., when modeling small scale refractivity fluctuations with the turbulence model. Although this should have been expected, it is the first proof that the VLBI systems are capable to measure these effects reliably.
For WHISP5 and WHISP7, the differences between the ZWDs of both stations generally vary only in the range of 1-3 millimeters, which fits very well the conclusions obtained for the single baseline WHISP sessions. The situation looks worse for WHISP6, where the ZWD differences almost reach the centimeter level. It can be assumed that the reason is due to the fact that “manual” phase calibration had to be applied for both radio telescopes in Wettzell. This assumption could be unequivocally established by analyzing the post-fit residuals of two further experiments, WHISP5-M and WHISP7-WHISP5-M, where the fringe fitting process was repeated, but manual phase calibration was applied for both Wettzell stations. For the first time, the effect of applying manual phase calibration instead of scan-by-scan system calibration has been identified and quantified, which is in the order of about 20 ps.
In all these investigations the large number of observations is necessary to guarantee a very stable estimation of the parameters and warrants a reliable interpretation of the residuals.
Concluding, the individual components of the observing system, particularly the hydrogen maser clocks feeding the local oscillators and other necessary electronics, the uncertainties emerging from the VLBI correlation process, and the effect of phase calibration, have been quantified, in part for the first time. Additionally, atmospheric refraction effects have been found to be in the range of 1-3 millimeters, which has been validated with two different observing strategies pursued by specially designed experiments. The investigations have also benefit from the turbulence model developed in this thesis to characterize refractivity fluctuations in the neutral atmosphere. Finally, the WHISP project has laid the basis for an improved characterization of atmospheric refraction effects, particularly on a local scale (objective 2).
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7. Alternative Strategies for Modeling Atmospheric Refraction
As has been explained in Sec. 3.3.6, certain deficiencies exist in treating the atmosphere in the current tropospheric model of VLBI observations defined in Eq. (3.65). For instance, the pseudo-stochastic character of the piece-wise linear representation, which is used to parametrize the zenith wet delay parameters, is generally not optimal to model the highly dynamic nature of the atmo-sphere. Additionally, soft constraints in the form of pseudo observations are often needed to stabilize the solution due to missing observations in some piece-wise linear segments. A similar situation ap-plies to the atmospheric gradients modeling azimuthal asymmetries of the neutral atmosphere.
Since the estimation of the model coefficients heavily depends on observations at low elevation angles, soft constraints are again necessary to stabilize the solution. Also the mapping functions, which are used to map the zenith delays to an arbitrary elevation angle, are also not optimal, even in case of the Vienna mapping functions 1 as the currently most accurate mapping function. Here, numerical weather models are necessary which are rather coarse with a temporal resolution of only six hours (Böhm et al. 2006b). In order to avoid the mapping function as additional uncertainty source, it would also be desirable to obtain atmospheric delays directly in slant direction, which, however, is only possible, if the number of observations is large enough. Finally, several parameter groups, such as atmospheric and clock parameters as well as the vertical component of the station coordinates, are assumed to be correlated and mutually influence each other, in particular if the stochastic model of the observations is not complete. Consequently, the ZWD estimates do not reflect meteorological and physical conditions in a plausible way in many cases.
In this chapter, two of these issues will be addressed in more detail, leading to two alternative modeling and adjustment strategies which help to estimate atmospheric parameters in a more reliable way. First, an inequality constrained least squares approach (ICLS) has been developed to overcome the deficiency, that occasionally zenith wet delay (ZWD) estimates become negative, which, of course, does not reflect meteorological conditions in a plausible way (Sec. 7.1). Second, the pseudo-stochastic behavior of the piece-wise linear representation for the zenith wet delays, which only models the stochastic character of the atmosphere to a limited extent, was replaced by a least squares collocation method and suitable covariance functions to describe the stochastic properties of the troposphere (Sec. 7.2).