## 3 Procedures of InSAR Image Co-registration
### Measurement for Quality of an Interferogram  
The quality of an interferogram is an indicator for the reliability of SAR image co-registration. So far, the root mean square error (RMSE) of the residuals at tie points after least-squares adjustment was adopted as a measure for the quality of an interferogram. It implies that the smaller the RMSE, the better the interferogram. However, it is not always the case. In some cases, when the RMS is small, the resultant interferogram is not so good [68].  
From the literature, it can be found that visual appearance of the fringes is widely used to indicate the quality of interferograms [68, 78], that is, clear fringes imply good quality of an interferogram. In other words, if the interferometric fringes can be observed clearly, the quality of interferogram must be good. On the other hand, the blurry fringes will be found in bad interferogram. Visual inspection is a kind of manual interpretation.  
The coherence of interferometric SAR is another indicator for the quality of interferogram [16, 56]. The coherence between two SAR images can be theoretically defined on a pixel basis as the correlation coefficient and varies in the range of 0 to 1. It significantly influences the accuracy of phase in interferogram. So the degree of coherence can be used as a quality measure for the interferogram. The stronger the coherence, the better the quality. The coherence of an image varies from area to area or even from pixel to pixel. Therefore coherence image is produced to show the  
Figure 12: A pair of SAR images. (a) the master image; (b) the slave image.  
Figure 13: The corresponding coherence image.  
quality variation of an interferogram over the entire area. Figure 12 is a pair of SAR image and its corresponding coherence image is shown in Figure 13[79]. In this image, some areas are brighter while some areas are darker. The brighter areas correspond to good coherence and the dark areas correspond to bad coherence.  
To make a coherence measure quantitative, histogram of the coherence image is used. More precisely, the mean of the distribution and the standard deviation (STD) from the mean are the two quantitative parameters for the coherence. However, it is found even though the coherence images are quite different and the distributions of the pixel number in different interval of coherence are quite different, the actual values for the means and STD can be quite similar[79]. Therefore, such statistical parameters are not sufficient to differentiate the quality of interferograms. Based on the above, [PERSON] _et al_.[79] has proposed a more reliable quantitative measure for the InSAR interferogram based on the \"sum of phase differences\" (SPD).  
The SPD method is based on the characteristics of the interferogram. Because an interferogram is generated by means of the product of the master image and the complex conjugate of the registered (slave) image, it contains an interferometric phase. Different colour fringes express different phase values in the interferogram. In an ideal case, the phase value changes gradually from 0 to \(2\pi\) radians. The value of the phase difference between two pixels that are in neighboring fringes should be infinitesimal. If the two pixels are in the same colour fringe, the phase difference between them should be zero. Therefore, the smaller the SPD, the better the quality of the interferogram.  
The \(SPD_{local}\) of one pixel is calculated as follows[79]:  
\[SPD_{local}\ (x,y)=\frac{1}{8}\sum_{l=-1}^{1}\ \ \sum_{k=-1}^{1}\left|\varphi_{ \textit{flattened}}\ (x,y)-\varphi_{\textit{flattened}}\ (x+l,y+k)\right| \tag{16}\]  
where \(x=0,1,2,\cdots,\)\(p-1;\ y=0,1,2,\cdots,\)\(q-1\) and the interferogram image size is \(p\times q\).  
The \(SPD_{whole}\) of the whole interferogram is obtained by summarizing the \(SPD_{local}\) of all pixels as follows:  
\[SPD_{whole}=\sum_{x=0}^{p-1}\ \ \sum_{y=0}^{q-1}SPD_{local}(x,y) \tag{17}\]  
The indicator is adopted to measure the quality of the interferogram by [PERSON][71, 80].