## 2 Radar Techniques for Space Observation and Products  
### Generation of High-Quality Observation Vectors for Orbit Determination  
The tracking radar of the TIRA system measures for each transmitted pulse the range of a space object as well as its range rate, azimuth angle, and elevation angle. The estimation accuracy of each measured parameter depends on its specific resolution and on the signal-to-noise ratio (SNR). It is well known that averaging over several measurements improves the estimation accuracy of a random variable. Indeed, the estimation accuracy is improved by a factor \(\sqrt{N}\) when averaging over \(N\) realizations. The latter assumes that the process of the random variable is wide-sense stationary, i.e., the variance is finite, the expected value is time-invariant, and the autocovariance is translationally invariant.  
However, for LEO objects, the different parameters strongly change over time due to the fast motion of these objects relative to the radar. Averaging over a large number of pulses is counterproductive as it deteriorates the estimates due to the underlying dynamics. This effect is examined in the framework of orbit determination [8] where a processing time of about 1 s is derived as a good compromise between improvement in the estimation accuracy and process stationarity.  
As introduced in [6], a coherent processing over \(N\) subsequent pulses can be applied to increase the estimation accuracy of the range rate and to derive a new parameter, the range-rate rate (Figure 3c). The latter is determined independently of the aforementioned observables. This parameter is particularly interesting for initial orbit determination (IOD) [6]. Currently, the tracking radar uses a correlator quartet to determine the range and range rate of the object. This method works similarly to a monopulse system by comparing the outputs of the matched filter at different delays and Doppler frequencies. The major advantage of this method is its rapidity as it avoids a grid search to estimate the Doppler frequency. The price to pay is that the estimated values are slightly biased (Figure 3b, red dots). The bias is much more important than the center of the correlator quartet being shifted from the true object position in the range/Doppler plane. This solution was chosen in the past as the best tradeoff for a real-time implementation. It will be replaced by the realization of a matched filter bank in the future development stage of TIRA to obtain unbiased estimates and better control the residuals. The range rate bias can be corrected by the coherent integration filter [6] (Figure 3b, black dots). The refined range rate and the range-rate rate can be used to improve the range estimate (Figure 3a, black dots).  
If the object was also observed with the imaging radar, a more accurate range could be determined from the imaging radar data. Indeed, the used bandwidth of the imaging radar is much larger compared to the one of the tracking radar, yielding a much finer range resolution. Since the imaging radar transmits chirp waveforms, a coupling between the range and the Doppler exists. This ambiguity has to be first solved in order to estimate accurately the range of the object [6] (Figure 3a, blue dots).  
A high-quality observation vector at a given epoch can be derived, which regroups the new range estimate with the refined range rate, the range rate, and the angular parameters. The benefit and longevity of these highly accurate measurements for orbit determination and cataloging are currently under evaluation.