# Validating CYGNSS Wind Speeds with Surface-Based Observations and Triple Collocation Analysis
## 2 Materials and Methods
### Triple Collocation Analysis Methodology  
Assumptions of stationarity and linearity between the datasets were checked as in [41], using scatterplots and an Augmented Dickey-Fuller T-test. The assumption of independence was tested by using another reference dataset, ERA5 [42], and observing any correlations in the errors. Due to the large ERA5 file size, a comparison was performed over the 3-month period from August to October 2023, as it contained the highest number of coincident measurements in TCA. Appendix A Figures A1 and A2 display the results, with the highest correlation of errors being 0.56 between SMAP and SAR when used with Merged. This is due to SAR wind algorithm being trained on SMAP data [43], and ERA5 being known to underestimate extreme winds [44]. As SMAP measurements are not used for ongoing calibration in the SAR algorithm and low correlation of errors in the independence test, we considered it acceptable to include in the analysis. ERA5 winds improve the results for NOAA. SAR and SMAP are not ingested to ERA5, which enables independence for TCA [42]. Before performing the TCA, an optimal time threshold to aggregate CYGNSS measurements was required. The R and RMSD were calculated using coincidental measurements for SAR and CYGNSS NOAA. SAR was chosen to identify the integration time as it only measured during storm events, whereas SMAP and CYGNSS provide continuous monitoring. NOAA v1.2 was the chosen CYGNSS product, instead of Merged, as it provided continuous monitoring. Collocated measurements within the sampling window were aggregated to a 0.25\({}^{\circ}\) grid for all datasets, and the average for each cell is compared. Only grid cells with an average SAR value above 15 m/s were included because including all SAR values resulted in ambiguous results, and this study's focus is GNSS-R performance in TC environments. Values of R and RMSD at each time threshold were calculated to see if an optimal threshold was present. Once the optimal timing threshold was determined, TCA was performed using both the additive and multiplicative error models. The additive model considers error to be the difference between the measurement and the truth, whereas multiplicative errors are the ratio of the difference [45]. The additive error model is represented in Equation (1) as:  
\[X_{i}=\alpha_{i}+\beta_{i}T+\epsilon_{i} \tag{1}\]  
where \(X\) is the measurement for instrument \(i\), \(T\) is the unknown true value being measured, \(\alpha\) and \(\beta\) represent the error coefficients, and \(\epsilon\) is the random error. The multiplicative error model is similar in Equation (2):  
\[X_{i}=\alpha_{i}T^{\beta_{i}}\,e^{\epsilon_{i}}, \tag{2}\]  
and when arranged in logarithmic form becomes Equation (3):  
\[ln(X_{i})=\alpha_{i}+\beta_{i}\,ln(T)+\epsilon_{i}. \tag{3}\]  
The multiplicative error model was found to be more applicable to rainfall due to the variability of daily values and the separation of systematic errors [45]. There are no concerns of taking the log of zero-values, unlike precipitation datasets [46]. Previous TCA of ocean winds have used the additive model [47; 48; 49; 50]. This analysis will use both error models and will compare their findings, as in [46]. The TCA yielded values for approximate cross-correlation and root mean square error (RSME). Using the additive error model, cross correlation (\(CC\)) is calculated in Equation (4) and standard error in Equation (5) as:  
\[CC=\sqrt{\frac{C_{X,Y}C_{X,Z}}{C_{X,X}C_{Y,Z}}} \tag{4}\]  
\[RMSE=\sqrt{C_{X,X}-\frac{C_{X,Y}C_{X,Z}}{C_{Y,Z}}} \tag{5}\]  
where \(C\) is the covariance matrix of the three datasets, \(X\) is either NOAA or Merged CYGNSS wind speed product, \(Y\) is SAR, and \(Z\) is SMAP. The formulas look similar in the multiplicative error model, except that the logarithm is taken of the wind speed estimates before calculating the covariance matrix.