## Research Activities

The routine data processing and analyses are enlarged by several additional research activities. The current subjects of interest are shortly described in following sub-sections

### Advanced Analyses of the post fit observation residuals

DORIS residuals are usually presented as range-rate residuals (in mm/s), but they can be expressed as phase count residuals (mm) as well. The basic relation between the range-rate and the count residuals is simple: the range-rate residual multiplied by the length of the observation time interval is equal to the count residual. The situation is more complicated due to the frequently switched chained/unchained observation mode. The observations with different time intervals are usually acquired (and processed) together. It is thus desirable to analyze the residuals of both types of observations separately, i.e., to distinguish the mixed “short” 7 s or 9 s observation intervals (unchained mode) and the “long” 10 s observation intervals (chained mode), or at least to pay attention to their ratio. Note that if observations with different time intervals are processed together, there is a difference between the minimization of the sum of squared range-rate residuals and the minimization of the sum of squared count residuals. In the GOP solution, the sum of squared count residuals is minimized. All results described in this chapter are derived from the daily GOP single-satellite solutions with adjusted orbits, station coordinates, ZTD and beacon frequency offsets but with fixed ERP.

A profound and detailed comparison of the “short” and “long” measurement accuracy based on the analysis of post-fit residuals is difficult, since both types of observations are processed together. The amount of the short observation intervals is usually too low to be processed separately. The presented approach can thus give only an approximate estimation of the relation between the measurement precision and the length of the observation interval.
The approximate computation of the post fit residual RMS of the chained and unchained observations σ_{c} , σ_{u} (eq. 4)

(4a) σ_{u} = √( (Σe^{2} / n_{u})*(n / n - u))

(4b) σ_{c} = √( (Σe^{2} / n_{c})*(n / n - u))

was used to compare the residuals of both types of the measurement, where 'e' are post-fit residuals referring to unchained or chained observations, n is the total number of observations, n_{u} the number of unchained observations, n_{c} the number of chained observations, and u the total number of estimated parameters. The fact, that the short (unchained) observations have higher range-rate residuals but lower count residuals than the long ones (chained), makes it clear that the measurement noise partly depends on the length of the observation time interval. We may, as a simple approximation, explain the accuracy of the count measurements as the result of a combination of two uncorrelated processes, where the first one is independent and the second one dependent on the count interval. More specifically, we may write the variance of the count measurements in the form (eq. 5)

(5) σ^{2}
= σ_{1}^{2}
+ (Tσ_{2})^{2}

where σ_{1} is related to the phase measurement noise while σ_{2} relates to the frequency dependent noise and T is the count interval. We may thus write (eq. 6)

(6a) σ_{u}^{2}
= σ_{1}^{2}
+ T_{u}^{2} σ_{2}^{2}

(6b) σ_{c}^{2}
= σ_{1}^{2}
+ T_{c}^{2} σ_{2}^{2}

where T_{u} (T_{c}) is the time interval of the unchained (chained) observation. We may then use the post-fit residual RMS computed using eq. 1 as the estimate of the observation standard deviation for the short and long observation intervals. The system of the two equations (eq.6) can be then solved for the two noise components σ_{1} and σ_{2}. The values obtained by this procedure for the time period 2002-2008 are similar for all satellites except for SPOT-2. The unchained observations of SPOT-2 are based on a 9 s time interval, while unchained observations of the other satellites have 7 s time interval. The standard deviation σ_{2} ranges between 0.225 and 0.257 mm/s for all satellites except for SPOT-2, where σ_{2} is 0.307 mm/s. On the other hand, σ_{1} is lower in the case of SPOT-2 (2.57 mm) than for the other satellites (2.77-3.16 mm). Fig. 9 shows the corresponding average values for each year of the analyzed time interval. The estimated relations between constant and time-dependent term are very similar, except for SPOT-2.

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