Principles of Modern Radar - Volume 2 1891121537

370 CHAPTER 8 Interferometric SAR and Coherent ExploitationFIGURE 8-21Phase unwrappingusing a leastsquares method.(a) Wrapped phaseof hill with squarenoise patch.(b) Unwrappedphase using theunweighted leastsquares Ghiglia-Prittalgorithm.Along-Track (m)−6−4−2024696979899100101102Range (m)(a)3210−1−2−3Wrapped IPD (radians)Along-Track (m)−6−4−20243020100−10−20−306 −401031049697These are then combined into a driving function d[l,m]:9899100101102103Range (m)(b)104Unwrapped IPD (radians)d[l,m] = ( ˜ y [l,m] − ˜ y [l − 1,m] ) + ( ˜ x [l,m] − ˜ x [l,m − 1] ) (8.37)Let D[q, r]betheM × N two-dimensional DCT 2 of the driving function. 12 The estimateof the unwrapped phase is then obtained as the inverse DCT 2 of a filtered DCT 2 spectrum:ˆφ difab⎧⎨[l,m] = DCT−12 ⎩{2 cos( πqMD[q, r])+ cos( πrN)− 2⎫}⎬⎭ − 1 (8.38)This phase function is then used to estimate the terrain elevation map h[l,m]. Note that theDCT domain filter transfer function is undefined for the “DC value” q = r = 0, emphasizingagain that the overall phase offset of the estimated map ˆφ difab is indeterminate. Noticethat the least squares IPD estimate is offset from the GZW estimate by about 30 radians.Figure 8-21a is the wrapped phase for the hill example but with a larger square noisepatch added to simulate a low reflectivity or degraded area. Straightforward applicationof equations (8.36)–(8.38) produces the unwrapped interferometric phase map estimateof Figure 8-21b. The phase noise remains in the unwrapped map. While it appears to haveremained localized, in fact it tends to have a somewhat regional influence. This can be seenby closely comparing Figure 8-21b to Figure 8-20b, particularly in the northeast corner.The general smoothing behavior of least squares methods, coupled with their inabilityto ignore outliers and other corrupted data, means that data errors tend to have a globalinfluence. It can also be shown that these methods tend to underestimate large-scale phaseslopes [38]. On the other hand, they require no branch cut or path computations andconsequently produce an unwrapped phase estimate everywhere in the map.The least squares approach lends itself naturally to an extension that incorporatesweights on the data. Generally, the weights are related to an estimate of data quality, sohigh-quality data regions have more influence on the solution than low-quality regions.12 There are multiple forms of the DCT in common use. The notation DCT 2 refers to the specific versionidentified as the DCT-2 in [49].