6.4 Local Refractivity Effects
6.4.3 Single Baseline Studies
In order to investigate refractivity variations on a local scale, three WHISP sessions (WHISP1-3, see Tab. 6.2) consisting of only a single baseline between the 20 m diameter radio telescope (Wz) and the north antenna of the twin telescopes (Wn) are used. In a first solution, differential zenith wet delay parameters are estimated for Wz relative to Wn. The results for the zenith wet delays vary mostly between ±2 mm (red dots in Fig. 6.7). In a second independent least squares adjustment, an offset parameter valid for the whole 22h session (red line) is estimated to identify remaining systematics. Please note, that these are only relative changes of the zenith wet delays.
The behavior of the hydrostatic calibrations reduced by a constant term (about 2.15 m, blue dots) gives some indication of the overall weather fluctuations within the sessions which is quite severe in the last one (see Fig. 6.7(c)).
The temporal variations of the estimated relative atmospheric parameters over each 22h period with only a few millimeters appear to be realistic estimates. This is supported by the fact that they are not (anti-) correlated with the hydrostatic calibrations. Since the two stations are only separated by about 120 m and both telescopes always point in the same direction, it can be assumed that any systematic effect not stemming from the atmosphere is canceled out in the differential mode.
Tests with variations in the clock parametrization support this assumption (not shown here). The non-zero estimates of the average ZWD offsets are presumably caused by differential paraboloid deformation effects which have invariant sin (elevation) characteristics (Nothnagel et al. 2017).
As a consequence of these interpretations, the majority of the fluctuations within each hourly segment are then attributed to the stochastic character of the neutral part of the atmosphere.
In order to validate this assumption, the effect of refractivity fluctuations due to turbulent motions in the atmosphere are investigated. Here, the atmospheric turbulence model presented in Sec. 5.1 is applied to describe the stochastic behavior of the small-scale refractivity variations and introduce physical correlations between the observations. Again, differential zenith wet delays are estimated for Wz relative to Wn (see red dots in Fig. 6.8), which, compared to the results in Fig. 6.7, only differ in the application of the turbulence model. The 22h session offset parameter (red line) and the reduced hydrostatic calibrations (blue dots) are plotted as well. Comparing these results to the standard case of VLBI data analysis (Fig. 6.7), it is apparent, that, the scatter of the differen-tial zenith wet delay estimates are considerably smoother, particularly for WHISP1 and WHISP3 (Fig. 6.8, (a) and (c)). Additionally, the standard deviations of the ZWD estimates become clearly larger and more realistic, remembering the fact that the standard deviations of space-geodetic
88 6. Case Study: The WHISP Project
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Figure 6.7: Differential zenith wet delay estimates (CPWFL, red) between the VLBI stations Wz and Wn for WHISP1-3 (a-c). An offset parameter estimated over the 22h session length is plotted as red line. The modeled zenith hydrostatic delays are represented in blue (reduced by a constant offset).
6.4. Local Refractivity Effects 89
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Figure 6.8: Differential zenith wet delay estimates (CPWFL, red) between the VLBI stations Wz and Wn for WHISP1-3 (a-c) using the atmospheric turbulence model described in Sec. 5.1. An offset parameter estimated over the 22h session length is plotted as red line.The modeled zenith hydrostatic delays are represented in blue (reduced by a constant offset).
90 6. Case Study: The WHISP Project
techniques are generally too optimistic (see, e.g.,Halsiget al.2016a). Except for two to three data points, the estimates for all differential zenith wet delays are not different from zero considering their 1σ standard deviations. In the standard analysis, many more data points exceed this criterion.
Consequently, the atmospheric turbulence model is able to improve the solution and the assumption that the majority of the variations from hour to hour can be assigned to the stochastic character of the atmosphere is hereby confirmed.
Second, the results of the three WHISP sessions are examined whether there exists a dependency on the observation geometry. It is generally known, that the extra signal path length through the neutral atmosphere with respect to a theoretical path in vacuum depends on the elevation angle ε of the observation: the lower the elevation angle, the larger the path length. Generally, mapping functions are used to map an observation from an arbitrary elevation to the zenith direction (cf.
Sec. 3.3.4).
Figure 6.9: Separation distance (dashed red line) of the signal rays of two radio telescopes for (a) an observation perpendicular to the baseline, and for (b) an observation in the direction of the baseline.
For very short baselines such as the one used here, the separation distance of the two ray paths of the radio signal (dashed red line, Fig. 6.9), referred to as ray distance in the following, also depends on the azimuth and elevation angle of the observation. Assuming both radio telescopes pointing into zenith direction, the separation distance would be maximal and equal to the baseline length separating the telescopes, which is about 123 m in the Wettzell case. When moving to a radio source with an arbitrary azimuth and elevation angle, the separation distance becomes smaller:
the lower the elevation angle, the lower the distance between the two rays. Fig. 6.9 shows two different pointing directions and the corresponding separation distances illustrated as dashed red lines. The colored triangles are supposed to emphasize, that for the determination of the separation distance a two-dimensional problem (in azimuth and elevation) is taken into account instead of a one-dimensional one only relating to the elevation angle. Mathematically, the ray distance can be calculated following Eqs. (5.6) to (5.11) in Sec. 5.1. The separation distance is evaluated at a certain height H = 2000 m representing the effective tropospheric height, which, however, results from implementing the turbulence model and is not important for the determination of the separation distance at this point. The ray distances are determined for each observation and a residual analysis is performed with respect to the separation distance.
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Figure 6.10: Post-fit residuals of the WHISP1 (a), WHISP2 (b) and WHISP3 (c) experiment as a function of the distance separating the ray path of the radio signals.
92 6. Case Study: The WHISP Project
Under the conditions set above that any remaining variations can be attributed to the wet atmo-sphere, it is assumed that there should be a dependency of scatter of the post-fit residuals on the ray distances. In Fig. 6.10, the post-fit residuals of the three WHISP-sessions are shown as a function of the separation distance of the ray path of the radio signals. To eliminate the elevation-dependent multiplication of the refraction errors implemented as mapping functions, the post-fit residuals have to be scaled by an inverse mapping function, simply approximated by sin (ε). This ensures that only the spatial dependency of the residuals due to the separation distance is analyzed, and not blended by other effects necessary for the data analysis.
The assumption, that the post-fit residuals become larger for an increasing separation is doubtlessly verified, particularly for WHISP2 and WHISP3 (Fig. 6.10, (b) and (c)). It may be assumed that this is due to the fact that the turbulent refractivity variations lose their spatial correlations with increasing ray separation. In order to see whether the turbulence model had any effect, two solutions with and without applying the turbulence model are compared, but there is no a clear evidence, that the post-fit residuals would become more randomly when introducing spatial correlations into the data analysis.
Figure 6.11: The spatial structure function of the post-fit residuals Dr(d), illustrated as a typical log-log-plot with respect to the separation distanced(the x-axis label is given in linear scale). The green and red dotted line correspond to the typical 2/3 and 5/3 power law exponents according to turbulence theory.
This issue is further investigated by calculating the spatial structure function of the post-fit residuals Dr(d), which is plotted with respect to the separation distance d. As an example, the result for WHISP1 is shown in Fig. 6.11. The findings for WHISP1 are consistent with the results of WHISP2 and WHISP3 (not shown here). As already described in Sec. 4.2, the structure functions can generally be represented as straight lines with different slopes, which are equal to the specific exponents of the power law processes. According to turbulence theory, power law processes with
6.4. Local Refractivity Effects 93
two characteristic exponents of2/3(2D turbulence; green line) and5/3 (3D turbulence; red line) are predicted. A more detailed description on the power law relations for atmospheric turbulence can be found in Sec. 4.2 or, e.g., Thompson et al. (2001). Compared to Fig. 4.3 in Sec. 4.2, which describes the idealized general behavior of a structure function, the computed slopes from the post-fit residuals do not clearly reflect the theoretical2/3and5/3power law exponents, but are still close to the expected values. The initial slopes for short distances of 15 to 20 m follows approximately the power law exponent of5/3and quickly decreases to a value close to, but slightly higher than the expected 2/3. One explanation for the difference between the computed and the theoretical slopes might the be fact, that the post-fit residuals are superimposed by other unmodeled effects. It is also conceivable that the calculated slope would become even closer to the theoretical one when further increasing the separation distance, at least up to a certain distance, where the structure functions becomes flat again (with a power law exponent close of zero) and the post-fit residuals are uncorrelated. Nonetheless, the separation distance is limited by the baseline length between the two telescopes, and therefore the maximum distance is about 123 m when looking in the zenith direction.
Finally, the solutions with shorter piece-wise linear segments of 15 or 30 minutes show only minor effects in the dependency of the post-fit residuals on the separation distance, although in both cases the solution itself improves (compare Tables 6.3 and 6.4).