# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix D  
The CYGNSS observatory has three antennas: one zenith antenna that is used for direct GPS-to-CYGNSS signal tracking, as well as two nadir science antennas that are used to capture the scattered signal from Earth's surface. Each of the eight spacecraft had all three antennas characterized pre-launch, and values were stored in a look-up table for science processing. Errors in the antenna gain pattern can arise for a variety of reasons. First, the measurement equipment on the ground is essentially a receiver but in controlled conditions. This means that while systematic and correlated errors are likely well-constrained, uncorrelated speckle-type error can still occur. To produce realistic antenna gain patterns, the results of the ground characterization were smoothed with various filters and techniques. Another source of error is the fact that CYGNSS antennas were not characterized while integrated with the spacecraft. This was a cost-saving measure decided by the mission management team. However, the electromagnetic properties of the antenna couple in some fashion with the spacecraft bus, and that will inevitably change the gain patterns. Initial analysis of CYGNSS data shortly after launch showed significant retrieval performance dependence on the observation azimuthal angle with respect to the CYGNSS body frame, which was later hypothesized to originate in errors in the CYGNSS antenna patterns. To compensate for this deviation from measured patterns, the CYGNSS antenna patterns have been updated at several instances over the mission life via empirical calibration. The nadir antenna patterns are updated by comparing a climatology of CYGNSS measurements of \(\sigma^{o}\) (>2 years) with model-generate \(\sigma^{o}_{mod}\) and plotting a scaling factor in the antenna reference coordinate system. \(\sigma^{o}_{mod}\) is generated by using modeled reanalysis winds to generate mean-squared slope with the L-band spectrum extension model, as described in [38]. However, during the generation of these updated patterns, a number of smoothing filters are applied. This prompts a discussion of a conjecture used extensively for this section:  
**Conjecture A1**. : _Uncorrelated errors can become correlated by post-processing with averaging and filters._  
This insight drives much of this section's analysis. Smoothing and filtering will necessarily impose a correlation in error between previously uncorrelated errors. For white noise, that implies that the choice of filter will add color and structure to the noise. In particular, a handy lemma allows us to demonstrate that for white noise, the information required to capture correlated error structure is the filtering kernel itself. We will explore this behavior for a one-dimensional case, but it is generalizable to higher dimensions in our application, as the two-dimensional filters used for antenna smoothing are separable by construction. **Lemma A1**. : _For a filtered signal \(F(t)=K(t)*D(t)\), where \(K(t)\) is a filtering kernel and \(D(t)\) is an arbitrary data signal, the autocorrelation \(F*F\) is the convolution of the autocorrelated kernel \(K*K\) and the autocorrelated signal \(D*D\)._  
Proof of Lemma A1. : Assume convolution and cross-correlation have the standard definitions for two real-valued timeseries \(K(t)\) and \(D(t)\), that is,  
\[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n-t)=\sum_{t=-\infty}^{\infty}K(n -t)D(t) \tag{101}\]  
and  
\[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]  
where \(*\) is the convolution operator, \(\star\) is the cross-correlation operator, and \(n\) is the lag argument.