# Error Sources of Interferometric Synthetic Aperture Radar Satellites
## 4 Random Errors  
Phase noises are represented as \(\Delta\varphi_{\text{noise}}\) in Equation (1). Phase noise, also known as speckle noise, is the drawback of coherent SAR systems since the functionality of SAR systems relies on coherent scattered signals [45]. Figure 6 demonstrates the InSAR imaging geometry and the SAR signal in detail, where \(\Delta\)R is the range difference between two acquisitions, \(H\) is satellite altitude, and \(h\) is topographic height. If the returning waves of both acquisitions oscillate \"in phase\", the resulting signal is coherent. If the waves oscillate \"out of phase\", the resulting signal is incoherent [127]. Therefore, the returned scattering information (s) recorded in each pixel from the first and second interferometric SAR image is composed of the coherent part (\(c\)) and the incoherent part (\(n\)), as expressed in Equation (20) [39; 128]. Subscript 1 or 2 symbolizes the information acquired during the first and second acquisition. \[ \begin{cases}s_{1}=c_{1}+n_{1}\\ s_{2}=c_{2}+n_{2}\end{cases}  \tag{20}\]  
To quantify phase noise and evaluate the quality of an interferogram, the complex correlation coefficient (\(\gamma\)) is defined as Equation (21) [28; 36; 39; 128]. The magnitude of the correlation, i.e., \(|\gamma|\), is often referred to as \"coherence\", which can be used as a measure of phase noise [38]. In Equation (21), \(s^{*}\) is the complex conjugate of s, and \(\langle\cdot\rangle\) means ensemble average. For simplicity, \(\gamma\) is termed coherence hereafter. \[\gamma=\frac{|\langle s_{1}s_{2}^{*}\rangle|}{\sqrt{\big{\langle}s_{1}s_{1}^{* }\big{\rangle}\big{\langle}s_{2}s_{2}^{*}\big{\rangle}}} \tag{21}\]  
Coherence is between 0 and 1. A coherence value equal to 0 indicates that the wave oscillation is fully out of phase; hence, there is no correlation at the given pixel between two SAR images. A coherence value equal to 1 indicates that the condition is completely  
Figure 6: Imaging geometry of interferometry SAR, revised from [129]. coherent, and the signals are completely in phase. If the interferogram is covered by low-coherence pixels, they will obscure the useful information in the interferogram, by which the interferogram will start to be grainy, and the fringes will become unrecognizable [35; 46]. This not only impedes the interpretation of interferograms but also presents difficulties in retrieving useful phase information and imposes adverse effects on phase unwrapping (PU). Phase unwrapping is one of the most important steps in InSAR processing as it calculates the absolute phase values from the original wrapped phase [2; 27; 69; 130]. The consequence of PU is decisive for the quality of the final InSAR results. However, PU is an error-prone process mainly due to the existence of phase noise and steep terrain slopes [28; 131]. Current PU processing methods involve the path of integration, so the local errors caused by phase noise can easily propagate along the path of integration. If the InSAR processing continues after an erroneous PU calculation, the errors will again propagate throughout the rest of the InSAR workflow and will ultimately undermine the precision of the InSAR measurements. Coherence should be maximized in order to reduce phase noise and prevent PU errors. Coherence is determined by several different correlation components. The temporal correlation (\(\gamma_{temp}\)) describes the correlation induced by the temporal interval between acquisitions. The baseline correlation (\(\gamma_{B}\)) describes the correlation affected by different observing geometries. The volumetric correlation (\(\gamma_{volume}\)) is the correlation influenced by the vertical extent of the scatterers. The noise correlation (\(\gamma_{SNR}\)) is the correlation influenced by receiver thermal noise inside of the radar system. The processing-induced correlation (\(\gamma_{processing}\)) is the correlation influenced by the InSAR processing procedures.