Principles of Modern Radar - Volume 2 1891121537

8.5 InSAR Processing Steps 3670.1 0.2 0.3Δ1 Δ4Δ2 Δ3–0.1 –0.2 –0.40.10.2 0.3Δ1 Δ4Δ2 Δ3–0.1 –0.2 –0.4–0.2 –0.2(a)–0.3–0.2 –0.2 –0.3(b)FIGURE 8-17 Illustration of path dependence and residues in two-dimensional phaseunwrapping. (a) Path with no residue. (b) Path with residue of −1. See text for details.(From Ghiglia and Pritt [37]. With permission.)practical matter, aliasing can be very difficult to avoid: large and sudden changes in actualterrain elevation, say, at cliffs or building sides, can cause large changes in the actual IPD.Path-dependent data can be recognized by a simple test. Consider the idealized 3 × 3segment of wrapped phase data in Figure 8-17. The values shown are in cycles; thus, a valueof 0.1 represents a wrapped phase value of 0.2π radians. Because wrapped phases are inthe range [−π, +π], the values in cycles are in the range [−0.5, +0.5]. Path dependencecan be tested by integrating the wrapped phase difference around a closed path. Becausewe start and end at the same pixel, the phase values at the beginning and end of the pathshould be the same. Consequently, the integral of the phase differences around such a pathshould be zero. In Figure 8-17a, the sum of the differences of the wrapped phase aroundthe path shown is1 + 2 + 3 + 4 = (−0.2) + (−0.1) + (+0.4) + (−0.1) = 0 (8.34)However, the path in Figure 8-17b has the sum1 + 2 + 3 + 4 = (−0.4) + (−0.2) + (−0.3) + (−0.1) =−1 (8.35)This occurs because 3 =+0.7 is outside of the principal value range of [−0.5, +0.5]and therefore wraps to 0.7 − 1.0 =−0.3. In this second case, the closed-path summationdoes not equal zero, indicating an inconsistency in the phase data. A point in the wrappedIPD map where this occurs is called a residue. The particular residue of Figure 8-17b issaid to have a negative charge or polarity; positive residues also occur. Conducting this testfor each 2 × 2 pixel closed path is a simple way to identify all residues in the wrapped IPDmap. If residues exist, then the unwrapped phase can depend on the path taken throughthe data, an undesirable condition.A solution to the residue problem is to connect residues of opposite polarity bypaths called branch cuts and then to prohibit integration paths that cross branch cuts. Theremaining legal integration paths are guaranteed to contain no pixel-to-pixel phase jumpsof more than π radians, so integration yields consistent unwrapping results. In a real dataset, there may be many residues and many possible ways to connect them with branchcuts. Thus, the selection of branch cuts becomes the major problem in implementing pathfollowing. One of the limitations of path-following methods is that portions of the wrappedphase map having high residue densities can become inaccessible, so no unwrapped phaseestimate is generated for these areas and holes are left in the unwrapped phase map [38].The most widely known path-following approach is the Goldstein-Zebker-Werner(GZW) algorithm [45], which is reasonably fast and works in many cases. Many practicalsystems begin with the GZW algorithm. For instance, the SRTM mission uses a GZW