Principles of Modern Radar - Volume 2 1891121537

8.3 Estimating Elevation Profiles Using Radar Echo Phase 355differential elevationĥ dif (x,y) ≡ ĥ(x,y) − h ref (x,y)= α IF[φ ab (x,y) − φ refab (x,y) ]= α IF φ difab (x,y) (8.29)It is straightforward to compute the IPD φ refab (x,y) that would be observed by our radarwhen viewing the profile h ref (x,y). First, compute the depression angle ψ ref (x,y) to eachpixel using equation (8.2). Next, use that result in equation (8.12) to predict the unwrappedreference IPD φ refab (x,y).Now suppose our radar is used to newly measure a wrapped IPD ˜φ ab (x,y) of thesame scene. Assuming that ˜φ ab (x,y) could be unwrapped to obtain φ ab (x,y), an elevationestimate h ⌢ dif (x,y) could be formed using equation (8.29). However, consider the wrappedversion of φ difab (x,y):〈〉˜φ difab (x,y) = φ ab (x,y) − φ refab (x,y)=〈˜φ ab (x,y) − ˜φ refab (x,y) 〉2π = 〈˜φ ab (x,y) −The last equality indicates that the wrapped differential IPD ˜φ difab2π〈φ refab (x,y) 〉2π〉2π(8.30)can be computed bywrapping the difference between the measured IPD, which is inherently wrapped, and thewrapped reference IPD obtained from the reference DEM.The process of subtracting the wrapped reference DEM IPD from the measured IPDis called interferogram flattening. It removes the gross phase variation, leaving only thedifferential relative to the reference profile. This tends to greatly reduce the number offringes (wrapping cycles) in the data, which in turn tends to simplify phase unwrapping.The reference profile is ideally an existing DEM, but the interferogram can be significantlyflattened simply by referencing it to a nominal profile such as a flat earth surface (forairborne systems) or a standard Earth ellipsoid such as the World Geodetic System (WGS)84 model (for spaceborne systems) [19]. In the flat earth case, the required reference DEMIPD is the flat earth IPD of equation (8.19).Figure 8-8 illustrates the effect of interferogram flattening on the complexity of anIPD. The measured wrapped IPD ˜φ ab of Figure 8-8a shows approximately 70 rapidlyvarying fringes. This function could be unwrapped to produce an estimate of φ ab, whichcould then be used in algorithm 1 or 2 to estimate ĥ. However, Figure 8-8b shows theresult of estimating instead the wrapped differential IPD ˜φ difab by referencing the measuredwrapped IPD ˜φ ab to the WGS 84 smooth surface earth model using equation (8.30). ˜φ difabmust still be unwrapped to estimate ˆφ difab and then ĥ dif and ĥ, but now the number offringes that must be unwrapped is reduced to about 9 or 10. Figure 8-8c references theIPD to an existing DEM of the same area, and the number of fringes is further reduced toabout four. Because of this reduction in the number of fringes, it is common to flatten theinterferogram before phase unwrapping.8.3.7 Range Foreshortening and LayoverStandard SAR image formation processing is designed to assume that all echoes originatefrom a two-dimensional flat surface. Equivalently, the three-dimensional world is projected