IEEE Geoscience and Remote Sensing Magazine - March 2020 - 100

norm, that is, A, B = tr ^ A H Bh, being tr ^ $ h and H, the
matrix trace and the Hermitian (transpose and conjugate)
operators, respectively. Finally, F is a proper normalization factor used to provide a coherence indicator, that is,
K ! 60, 1@, and $ is the operator that extracts the absolute
value of a complex number.
As for the low spatial frequency analysis in Figure 2, an
initial grid of reliable points/areas is generally identified.
This can be implemented by either exploiting only the amplitude information, that is, for PSI (i.e., the PS candidates)
or performing a contextual analysis in terms of spatial coherences (as in the SCI methods) measured on the stacked
interferograms by means of spatial multilook [5], [82]. It is
evident that the latter case presumes an implicit assumption on the local stationarity.
For some applications, such as the modeling of earthquakes, for which geophysical models may already be pertinent and sufficient to describe the main signal component,
an analysis in the complex signal (i.e., an analysis with
restricted phase values) may be adequate. In these limited
cases, a spatial coherence obtained by relating, in (5), A to
the geophysical model and f ^Ch to the complex data and
evaluating the scalar product over different pixels may be
sufficient [117]. For other uses, including landslide monitoring, analysis limited only to the complex (i.e., phaserestricted) domain is inadequate, and the separation and
filtering of the atmospheric phase component is necessary.
PU procedures aimed at estimating the absolute phase
variability on a larger (i.e., unrestricted) interval of variation
is mandatory: this processing step may heavily affect the accuracy of the final product, regardless of the resolution with
which it is provided. Because PU is a highly nonlinear and, often, a critical step, it is difficult to provide a direct quantitative

index exemplifying the measurement accuracy. Nevertheless,
a consistency index measuring the spatial reliability in a normalized (i.e., [0,1]) interval of the PU process can be provided
[5], [118]. This is carried out in (5) by selecting as f ^Ch the
function that extracts the complex phase of the covariance
matrix in case the acquisition indexes p and q identify a pair
belonging to the set S of the chosen interferograms:
f ^C h : c pq " e j+cpq if pq ! S, 0 elsewhere, (6)

where + is the operator that extracts the phase of a complex number, with c pq = " C ,pq, and " $ ,pq is the operator
extracting the pq element of a matrix.
The consistency index c (downstream of the unwrapping
process) considers the complex phase residuals between the
starting interferograms and corresponding phase beatings
evaluated from the unwrapped phase signals after the LS
inversion of (4), that is, D @ . The elements of matrix A in (5)
are specified as
a pq = e j" D $ D $ D{ ,pq, (7)
@

where F = S , that is, the number of selected interferograms. Larger values of the unwrapping consistency index
identify areas in which unwrapping is likely more reliable.
Generally, the farther the pixel is from the tie point/area
(used as a starting reference point for PU), the lower the
consistency index. Such a behavior of the index agrees with
the results of a comparison between A-DInSAR deformation measurements and external independent measurements (e.g., leveling or GPS) published in the literature
[118]. On one hand, as in the case of products provided at
the low-spatial-frequency analysis, the consistency index

Low Spatial Frequencies

High Spatial Frequencies

+
Unmodeled/Model-Supported
Phase Analysis

Model-Based Analysis
-
+

DEM

Atmospheric
Disturbance

Residual Deformation
+

(Small-Scale) Nonlinear
Deformation

Residual
Topography

Large-Scale Nonlinear
Deformation

FIGURE 2. A general data analysis two-step scheme based on low and high spatial frequencies. All processing outputs are in red.

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IEEE Geoscience and Remote Sensing Magazine - March 2020

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