## 3 Coherent Pointing Estimation using Notch and Boresight Pattern  
### Difference Pattern Approach  
The main drawback of the conventional pointing estimation technique as described in the previous section is the requirement for homogeneous distributed targets, like the Amazon rainforest. This kind of targets are unfortunately sparsely spread around the globe. Hence, they cannot be used to track pointing variations along the orbit which may arise due to thermal and gravity induced changes in the surface shape of large deployable reflector (LDR)-based antenna systems proposed for various future SAR missions [18]. Inhomogeneous distributed targets could be employed for more continuous elevation pointing estimation if the variations in the backscatter from ground could be compensated for. The proposed technique combines two coherent SAR datasets to mitigate the effect of the underlying scene. With the target scene vanishing, the differences in the antenna pattern become apparent (neglecting noise for the moment). From the images of the two recorded coherent SAR datasets, a difference dataset can be generated which is modeled as follows:  
\[\Delta_{Pattern}(t,\tau)=\frac{S_{Target}(t,\tau)\cdot A_{Pat,1}(t,\tau)\cdot A_ {Prop}(t,\tau)+n_{1}(t,\tau)}{S_{Target}(t,\tau)\cdot A_{Pat,2}(t,\tau)\cdot A_ {Prop}(t,\tau)+n_{2}(t,\tau)}. \tag{2}\]  
The two level 1 SAR images used to form the difference image \(\Delta_{Pattern}(t,\tau)\) are described as the combination of the underlying scene \(S_{Target}(t,\tau)\), propagation effects \(A_{Prop}(t,\tau)\), the antenna pattern \(A_{Pat,1}(t,\tau)\), \(A_{Pat,2}(t,\tau)\), and noise \(n_{1}(t,\tau)\), \(n_{2}(t,\tau)\), respectively. The variables \(t\) and \(\tau\) account for the two-dimensional nature of the scene in slow and fast time. All terms in the equation are complex-valued and \(S_{Target}(t,\tau)\) has the form of scattering coefficients. The variations in the underlying scene \(S_{Target}(t,\tau)\) and the propagation effects \(A_{Prop}(t,\tau)\) can be neglected if the two datasets are acquired coherently at the same instant of time, i.e., in the same overpass. Hence, only the difference in the antenna pattern \(A_{Pat,X}(t,\tau)\) remains, if noise and ambiguity contributions can be neglected:  
\[\Delta_{Pattern}(t,\tau)\approx\frac{A_{Pat,1}(t,\tau)}{A_{Pat,2}(t,\tau)}. \tag{3}\]  
This concept is valid for any combination of antenna patterns and not limited to notch beams. By selecting characteristic patterns, their differences can be analyzed to retrieve information about the antenna pointing. One promising selection of antenna patterns is the combination of a notch/mono-pulse pattern with its steep null in elevation direction and a nominal boresight beam pattern with its nearly flat gain response along elevation. This selection yields a difference datasetwith characteristic properties which can be used to estimate the antenna pointing in elevation without disturbances by the underlying scene. Hence, even the \(\gamma\)-correction step required in the conventional method is not necessary. It should also be stressed that no assumption upon the recorded scene is necessary, not even its homogeneity has to be known for deriving the elevation pointing as long as sufficient energy is scattered back to the sensor (i.e., noise term in Equation (2) is negligible). A major advantage of the difference technique proposed here in comparison to the conventional technique is the availability of both amplitude and phase information for further analysis. In the conventional elevation pointing estimation technique only the power information, i.e., the gain drop of the notch pattern could be used because the amplitude behavior of the scene (homogeneous distributed target) is known while the (backscatter) phase is random. The estimation of the elevation antenna pointing using the coherent difference method as proposed here allows the extraction of both, amplitude and phase, from the difference dataset.