# Correcting for Mobile X-Band Weather Radar Tilt Using Solar Interference
## 3 Solar Interference Assessment Methodology
### Application of a Multi-Parameter 2D Gaussian (Bullseye) Fit  
To analyse the raw solar hits, a five-parameter model is fitted to their position offsets and recorded solar power. This models the two-dimensional Gaussian form of the combined sun, antenna beam shape and azimuthal beam averaging as first proposed by [PERSON] and [PERSON] [12]. This approach is commonly referred to as the \"bullseye\" method, as the solar power decreases from the centre of the disk, producing a bullseye target. The [PERSON] and [PERSON] [12] method is set out here for completeness. The form of this model is:  
\[p(x,y)=a_{1}x^{2}+a_{2}y^{2}+b_{1}x+b_{2}y+c \tag{1}\]  
where \(p(x,y)\) is the received solar power, \(x\) is the difference between the radar azimuth and the solar azimuth (read-sun), and \(y\) is the difference between the radar elevation and the solar elevation (read-sun). The equation is described by the linear parameters \(a_{1,2}\), \(b_{1,2}\) and \(c\), allowing the solar hit data to be fit to it using least squares regression. From these linear parameters, five physical parameters: azimuth width (\(\Delta_{\phi}\)), elevation width (\(\Delta_{\phi}\)), azimuth offset (\(x_{0}\)), elevation offset (\(y_{0}\)) and peak solar power (\(p_{0}\)) can be calculated through the following equations:  
\[\Delta_{\phi}=\sqrt{-\frac{40\log_{10}2}{a_{1}}}, \tag{2}\]  
\[\Delta_{\theta}=\sqrt{-\frac{40\log_{10}2}{a_{2}}}, \tag{3}\]  
\[x_{0}=-\frac{b_{1}}{2a_{1}}, \tag{4}\]  
\[y_{0}=-\frac{b_{2}}{2a_{2}}, \tag{5}\]  
and  
\[p_{0}=c-\frac{b_{1}^{2}}{4a_{1}}-\frac{b_{2}^{2}}{4a_{2}}. \tag{6}\]  
Here it is important to note that the azimuth and elevation widths are a function of the convolution of the antenna beam widths and the apparent width of the solar radiation rather than the absolute beam widths of the weather radar, which are already known. To further improve the quality of the fit, outliers are removed based on a comparison to the initial modelled parameters. In this case, any solar hits that deviate by more than 1 dBsfu from the modelled Gaussian form are removed before refitting the model. In the case of both the initial and second (final) fits, the root mean square error (RMSE) is calculated to assess the bullseye model. The RMSE is calculated as follows:  
\[\text{RMSE}=\sqrt{\frac{\Sigma_{n=1}^{N}(p_{n}-\hat{p}_{n})^{2}}{N}} \tag{7}\]  
where \(p_{n}\) is the observed relative solar flux, \(\hat{p}_{n}\) is the relative solar flux predicted by Equation (1), and N is the total number of solar observations in the analysis window. Calculating the RMSE also allows comparison of different time windows. During their work, [PERSON] and [PERSON] [12] used a daily time interval for fitting their data, whereby they were able to obtain 20 to 30 solar hits per day across a range of elevations. In contrast, the NXPol-1's RAiN-E dataset typically contains fewer than ten hits per day. To mitigate this reduced number of hits and increase the stability of the resulting fit, two modifications are made to their original methodology. Firstly the azimuth and elevation widths (\(\Delta_{az}\) and \(\Delta_{el}\)) are fixed to \(1.29^{\circ}\) and \(1.11^{\circ}\) respectively, for the RAiN-E campaign, and secondly, the time windows analysed are lengthened. The model azimuth and elevation widths used here (\(1.29^{\circ}\) and \(1.11^{\circ}\)) were derived using the five-parameter fitted model using data from a previous NXPol-1 deployment at Chilbolton Observatory where, during a 7-week period, decreased SNR requirements led to a greater number of low-intensity (estimated solar power below \(-110\) dBm) observations being recorded across a wider elevation and azimuth range (off-axis hits). The additional observations enable a more precise derivation of the 2D power Gaussian as expected considering NXPol-1's beam width of \(0.98^{\circ}\), the antenna rotation speed and the apparent width of the solar radiation source (Figure 2).