Assumes that M and S represent the master and slave images of a single interferogram. The master image can be split into two SLC images, and , following the MAI principle. Similar divisions of the slave image can be made into the two SLC images, and . The standard D-InSAR processing steps, such as registration, multi-look, interference, removal of flat and topographic phases, filtering, and so forth, can then be used to process the data. The forward interferogram is created by using and . The backward interferogram can also be created using . Finally, the phase difference between the two interferograms above creates the MAI interferogram.
The following is the relationship between the azimuth displacement and the MAI interferogram:
where x is along-track displacement, l is adequate antenna length, and n is the faction factor of the full aperture
3D surface displacement
The three-dimensional co-seismic deformations were reconstructed by combining the measurements of LOS and along-track data. The imaging geometry of 3D vectors for surface deformation for the dot P in yellow color, the vector for DInSAR and MAI measurers (Figure 2.6) defined as Equations (5) and (6) (Wang et al., 2018)
where θ represents the incidence angle of radar, α is the azimuth angle, def is the surface deformation observation, with sub-index, los and AZ are the observations of range and azimuth direction, respectively, and e and n are the east and north direction, respectively.
We restructured the 3D surface deformation map based on Equations (5) and (6). Surface deformation and the integration of InSAR and MAI data sets can be expressed as follows (7):
d = A ∗ u. (7)
where d is a matrix containing our data, A is a design matrix, and u is the unknown 3D components.
Expanding Equation (7) using angles, as in Figure 2.6, Equations (8)–(10) can be generated:
. (8)
. (9)
. (10)
where def is deformation observations, sub-indices (asc) and (desc) represent ascending and descending orbit passes, respectively; super-indices represent phase measurements (LOS), azimuth (azi), and range(rng) offsets, respectively; and factor u is the estimated unknown component of the displacement in the east, north, and up
igure (2.6): Geometrical projection relationships between the SAR imaging geometry and the three-dimensional (3D) motion components, up-down deformation (Du), north-south deformation (Dn), and east-west deformation (De), above a specific area (P), with a SAR sensor having a right lateral look (red ascending and blue descending). DAZ correspond on deformation along azimuth, direction incidence angle of radar (θ) and azimuth angle (α)
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