IEEE Geoscience and Remote Sensing Magazine - March 2020 - 80

In summary, a large number of deformation models
can be selected for InSAR-based mechanical parameter
inversion. Among these, the PIM is a classical model that
has been widely used and is preferred over mechanical
parameter inversion in the case of critical extraction or
supercritical extraction (i.e., spatial prediction) because
of its good performance in these cases. However, for a
subcritical extraction or spatiotemporal deformation description, a generalized or temporal PIM is preferred. Note
that both the PIM and temporal and generalized PIMs still
have some typical limitations, such as poor performance
in mountainous areas [120]. Thus, testing additional mining deformation models is necessary for optimizing InSAR-based mechanical parameter inversion.
FUNCTION MODEL CONSTRUCTION
InSAR-derived deformations are along the LOS and/or
azimuth directions, but existing deformation models
(including the previously discussed PIM and temporal
and generalized PIMs) can describe only mining surface
deformation patterns in vertical and horizontal motions
[see, e.g., (10)]. Therefore, a function model must be constructed to relate the model parameters to the InSAR-derived deformation observations. To this end, in 2014, Fan
et al. [112], [113] first considered InSAR LOS deformation
as subsidence observations directly, without taking the
strict SAR geometry into account, and then constructed
a functional relationship for the subsidence-related mechanical model parameters of the PIM [i.e., P in (10)
except for horizontal motion constant b]. This work, to
some extent, promotes the application of InSAR on mining deformation model parameter inversion; however,
such a function model cannot consider the joint contribution of 3D displacement components to the LOS direction, potentially causing large errors in the subsequent
model parameter inversion. Moreover, it cannot invert
the model parameters of the PIM related only to horizontal movements (e.g., horizontal motion constant b)
from InSAR measurements due to their absence in the
constructed function model.
To overcome this, in 2016, a new functional relationship
between the full model parameters of the PIM and InSAR
LOS displacement measurements was constructed [111], in
which the formulas of 3D mining displacements were first
expressed using the following PIM:

	

*

W (x, y) = f1(x, y, P, G)
E(x, y) = f2(x, y, { E, P, G) , (11)
N (x, y) = f2(x, y, { N, P, G)

where { E and { N are the angles from the advancing direction to the eastern and northern directions, respectively.
According to the projection relationship of the 3D displacements onto the LOS direction [see (3)], the new function
model relating InSAR LOS deformation to all the model
parameters of the PIM was then constructed:
80

	

R
VT R f (x, y, P, G) V
cos i
S
W S 1
W
S
LOS(x, y) = sin i cos a h W S f2(x, y, { N, P, G)W, (12)
S- sin i sin a W S f (x, y, { , P, G) W
E
h
2
T
X T
X

Moreover, in the same year, Diao et al. [114] followed
the same core idea presented in [111] to construct a function
model that was similar to that of (12) but was a radar coordinate system between the model parameters of the PIM and
InSAR measurements. Compared with the function model
constructed in [112] and [113], the model of (12) is mathematically and theoretically rigorous. In addition, (12) offers
the potential to invert all the PIM model parameters with InSAR measurements; therefore, the function model is preferable for InSAR-based model parameter inversion. In addition
to the PIM (spatial model), the functional model between the
spatiotemporal deformation prediction models (e.g., the temporal or generalized PIMs) and InSAR measurements can be
constructed by following the processes discussed previously.
MODEL PARAMETER INVERSION
Due to the mathematical complexity and nonlinearity of
PIM-based deformation description formulas, InSAR model parameter inversion based on InSAR measurements has
thus far generally been conducted by nonlinear searching.
In 2014, Fan et al. [113] inverted the subsidence-related
mechanical parameters of the PIM from DInSAR measurements using a classical genetic algorithm (GA). Following
this, Yang et al. [111] inverted all of the model parameters
related to both subsidence and horizontal motions from
InSAR measurements based on (12), using an improved
GA termed the GA with gross error elimination (GAGEE). The
GAGEE algorithm first obtains a large number of GA estimates of the PIM model's parameters. Two times the standard variance of each parameter's estimate is then used
to eliminate those estimates with gross errors. Finally, a
fitness-based weighting scheme is used to determine the
final model parameters of the PIM from the remaining estimates. In addition to the GA and GAGEE algorithms, other
nonlinear search algorithms, such as simulated annealing
(SA) [121] and hybrid GA/SA algorithms [122], have been
applied to InSAR-based model parameter inversion.
Among these nonlinear search algorithms, the classical
GA has a powerful global optimization capability without
requiring initial parameter values, but its estimates are
generally polluted by different-magnitude random errors
due to random processes in the GA [123]. To a large extent,
the GAGEE algorithm can overcome the aforementioned
limitation of the classical GA but does so at the expense of
increased time consumption. The SA algorithm generally
has a lower time cost for searching, but the accuracy of its
estimates depends on the initial values of the model's parameters. Thus, the hybrid GA/SA algorithm combines the
advantages of GAs and SA so that it can rapidly obtain an
acceptable accuracy of model parameter estimates. Consequently, today, the hybrid GA/SA algorithm is preferred for
InSAR-based mining model parameter inversion.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

MARCH 2020
IEEE Geoscience and Remote Sensing Magazine - March 2020

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