## 3 Impact of external errors
### Impact of roll angle knowledge
#### Sensors 2006  
6  
Applying Equation 15, a one arc second error in the knowledge of the roll angle results in a Far Range height error about 50 cm. It then comes that a 0.1 arcsecond maximum error is acceptable, given the 5 cm requirement corresponding to the overall error budget of the height parameter. An a priori knowledge of the roll angle spectral characteristics is provided by the PAD system onboard the satellite. However, the required accuracy of less than a tenth of an arc second is so strong that ground post processing procedures using crossover data need to be done. A crossover is an area where an ascending track crosses a descending track. In the case of WSOA, swath/swath, nadir/swath and nadir/nadir crosses shall be considered as calibration points (see Figure 15). Two successive crossover locations along a track are only 80 seconds apart as a maximum. Thus, assuming we get rid of the high frequency part of the roll spectrum in the PAD, an appropriate interpolation scheme between crossovers may finally give an overall a posteriori knowledge of the roll angle that lies within the specifications. [PERSON] and al ([PERSON] and al (2010)) have computed a crossover maximum likelihood estimation of the roll angle in the WSOA/Jason-2 case, considering a 14 km intrinsic range resolution. The goal of the current study is to determine the influence of external factors on the quality of the roll estimator: the ground pixel size after incoherent onboard processing, the repeat cycle of the orbit and finally the presence of nadir altimeter data. The main idea of the crossover estimator is that the over flying time of the crossovers zones are small enough to consider a linear evolution of the roll angle with time (see Figure 16 (a)) as followed:  
\[\alpha=\alpha_{0}+\dot{\alpha} \tag{16}\]  
Four parameters (two for the ascending track, two for the descending track) are then to be estimated:  
\[\dot{\theta}=\left[\alpha_{0\ -A}\ \ \ \dot{\alpha}_{A}\ \ \ \ \alpha_{0\ -D}\ \ \ \dot{\alpha}_{D}\ \ \right] \tag{17}\]  
Figure 14: (a) Spatial differences in wet tropospheric delays over a large atmospheric feature (typically a few hundred kilometers wide) in mm (b) Statistical distributions of the tropospheric errors over the feature. The height differences at the crossovers are dependent on several factors, which are respectively the roll parameters associated to the geometry of the crossovers (see Figure 15), the instrumental noises (interferometer and nadir altimeter), the ocean variability and the external noises (tropospheric and ionospheric), and can be expressed as followed (all variables are defined in the table of notations):  
\[ \begin{array}{l}\Delta h_{s\,wath/\,swath}=\left[X_{A}\alpha_{{}_{0\_A}}-X_{D} \alpha_{{}_{0\_D}}\right]+\left[X_{A}Y_{A}\dot{\alpha}_{A}-X_{D}Y_{D}\dot{ \alpha}_{D}\right]+2N_{\,\text{int\,}erferometer}+N_{\,ocean}+N_{\,\text{ tropo\_{-\,}iano}}\\ \Delta h_{s\,wath/\,nadir}=\left[X_{A}\right]\alpha_{{}_{0\_A}}+\left[X_{A}Y_{A} \right]\dot{\alpha}_{A}+N_{\,\text{int\,}erferometer}+N_{\,\text{nadir\_{- \,}alimeter}}+N_{\,\text{ocean}}+N_{\,\text{ tropo\_{-\,}iano}}\end{array}  \tag{18}\]  
The interferometer noise and the external noises are derived from previous sections. The nadir altimeter noise is assumed to be 2 cm (see [24]). The covariance of the SLA (Sea Level Anomaly) is computed from [25] as a function of latitude. A maximum time difference of 5 days between ascending and descending tracks has been used in the selection of crossovers to avoid high temporal decorrelation. Thus, for Jason-2, all crossovers can then be considered (see Figure 16 (b)). \"A-D\" (ascending minus descending) differences in Equation (18) are the observations, which we will call \(\text{d}_{i}\), which can be represented using the following:  
\[d_{i}=m\big{(}\theta\big{)}+N_{i}\,,\text{d}_{i}\in\Re^{K} \tag{19}\]  
where \(m\big{(}\theta\big{)}=H\theta\), and (Ni) have covariance matrices \(R\) independent of \(\theta\).