ArticlePDF Available

A New Decorrelation Phase Covariance Model for Noise Reduction in Unwrapped Interferometric Phase Stacks

Authors:

Abstract and Figures

The accuracy of geophysical parameter estimation made with interferometric synthetic aperture radar (InSAR) time-series techniques can be improved with rapidly increasing available data volumes and with the development of noise covariance matrices applicable to joint analysis of networks of interferograms. In this article, we present a new decorrelation phase covariance model and discuss its role in noise reduction in unwrapped interferometric phase stacks. We demonstrate with an example in which we average unwrapped interferogram phase stacks that span over a transient event how a noise covariance model can aid in noise reduction. Our model suggests that, for rapidly decorrelating surfaces (i.e., surfaces with much shorter correlation time than SAR acquisition intervals), it is preferable to incorporate all available interferograms from long observation windows. For slowly decorrelating surfaces (i.e., surfaces with longer correlation time than SAR acquisition intervals), our model suggests that a small subset of interferometric pairs is sufficient. We validate our model and three existing models of decorrelation phase covariance matrices in both Cascadia, a region with heavy vegetation cover, and Death Valley, a desert region with C-band Sentinel-1 A observations. Our proposed model matches observations with the smallest average discrepancy between theory and observations.
Content may be subject to copyright.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1
A New Decorrelation Phase Covariance Model for
Noise Reduction in Unwrapped Interferometric
Phase Stacks
Yujie Zheng, Student Member, IEEE, Howard Zebker, Fellow, IEEE, and Roger
Michaelides, Student Member, IEEE
Abstract—The accuracy of geophysical parameter es-
timation made with Interferometric Synthetic Aperture
Radar (InSAR) time-series techniques can be improved
with rapidly increasing available data volumes, and with
the development of noise covariance matrices applicable
to joint analysis of networks of interferograms. Here
we present a new decorrelation phase covariance model
and discuss its role in noise reduction in unwrapped
interferometric phase stacks. We demonstrate with an
example wherein we average unwrapped interferogram
phase stacks that span over a transient event how a
noise covariance model can aid in noise reduction. Our
model suggests that, for rapidly decorrelating surfaces (i.e.,
surfaces with much shorter correlation time than SAR
acquisition intervals), it is preferable to incorporate all
available interferograms from long observation windows.
For slowly decorrelating surfaces (i.e., surfaces with longer
correlation time than SAR acquisition intervals), our
model suggests that a small subset of interferometric pairs
is sufficient. We validate our model and three existing
models of decorrelation phase covariance matrices in both
Cascadia – a region with heavy vegetation cover, and
Death Valley – a desert region, with C-band Sentinel-1 A
observations. Our proposed model matches observations
with the smallest average discrepancy between theory and
observations.
Index Terms—Decorrelation noise, Covariance matrix,
InSAR noise reduction.
I. INTRODUCTION
INTERFEROMETRIC Synthetic Aperture Radar
(InSAR) is a widely used remote sensing
technique that combines coherent radar images to
form interferograms, which can be used to generate
high-precision measurements of surface topography or
crustal deformation over large areas with meter-level
resolution [1]–[4]. In the past few years, three new
InSAR satellites – Sentinel-1A/B and ALOS-2, became
Y. Zheng and R. Michaelides were with Stanford University for the
majority of this work, now with California Institute of Technology
and Colorado School of Mines, respectively. H. Zebker is with
Stanford University.
operational. In the near future, additional InSAR
satellites such as NISAR and ALOS-4 are anticipated
to come online. Featuring higher temporal sampling
rates and wider spatial coverage, this new generation
of satellites provides large volumes of high-quality
radar measurements and thereby makes observations of
mm-level signals such as very slow, long wavelength
interseismic velocities possible [5].
The accuracy of measured geophysical signals with
InSAR is inherently limited by atmospheric noise
and decorrelation. Atmospheric noise is caused by
fluctuations in wave propagation delays through the
atmosphere due to the presence of water vapor. Since
the revisit time of SAR satellites is on the order of
days, atmospheric noise is essentially uncorrelated in
time but correlated in space and is often modeled as
a long-wavelength artifact in individual interferograms
[6]–[9]. Decorrelation, on the other hand, can be
related to changes between radar measurements in
surface scattering properties, imaging geometries and
thermal noise among others [10], [11]. Temporal
decorrelation due to independent motions of scatterers
in the resolution cell translates to stochastic noise in
interferometric measurements. In contrast, processes
that result in both correlation loss and systematic
phase shifts (e.g., variations in soil moisture [12])
have non-stochastic effects on interferometric stacks
[13]–[15], and are linked with observations of phase
biases in short temporal-span interferograms [16]. In
this paper, we limit our focus on the first category
of decorrelation that results in stochastic noise in
interferometric measurements.
InSAR time-series techniques are methodologies that
exploit interferogram stacks with the aim of retrieving
desired geophysical signals whilst minimizing the effect
of decorrelation. There are two broad categories of
InSAR time-series algorithms. One involves identifying
“persistent scatterer” (PS) pixels with highly stable
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 2
scattering mechanisms [17], [18]. The PS pixels are
minimally affected by decorrelation but are mostly
limited to man-made structures and cities. The other
approach involves exploitation of pixels that are affected
by decorrelation, known as “distributed scatterers”
(DS) pixels. The DS method can be further divided
into two categories. The first category of DS methods
is based on analysis of unwrapped interferometric
phase stacks. For example, the conventional Small
BAseline Subset (SBAS) technique [19] obtains phase
time-series by solving a linear system of equations of
unwrapped phases. The SBAS method limits the effect
of decorrelation by restricting use of interferometric
measurements to a subset of interferograms with small
spatial and temporal baselines. The second category
of DS methods performs time-series estimation based
on analysis of SAR correlation matrices before phase
unwrapping. Accounting for target statistics of SAR
measurements [20], [21], approaches such as SqueeSAR
[22] and its ensuing algorithms [23]–[25] relax the
coherence constraint imposed in SBAS and allow for
use of all interferograms [20], [21]. In this paper, we
focus on the first category of DS methods that operates
on networks of unwrapped interferometric phases.
Hereinafter we refer to the first category of DS methods
as “SBAS-like time-series algorithms”.
Phase statistics of multi-interferometric measurements
play a critical part in improving the performance
and design of SBAS-like time-series algorithms.
Several studies have evaluated covariance between
interferometric phases. In the simple mathematical
framework developed in [6] to describe common noise
sources in InSAR, decorrelation phase is modeled
as an independent noise term, uncorrelated between
interferograms. More recent work such as [13], [14],
[20], [26]–[28] show that interferometric phases are
partially correlated between interferograms and present
analytical models with various forms. With respect
to these works, we propose a new covariance model
for decorrelation phase based on surface scattering
characteristics. We compare and validate our model
with previously published models against Sentinel-1
data collected over both low coherence and medium-
to-high coherence areas. We also demonstrate with an
example how a network noise covariance model can
facilitate decorrelation noise reduction.
This paper is organized as follows. In section II, we
review existing decorrelation phase covariance models
and then present the proposed model. In section III, we
demonstrate how decorrelation covariance models can
be incorporated into SBAS-like time-series algorithms
to great effect and provide theoretical comparisons
between the existing and the proposed models. In
section IV, we validate all models using data collected
by Sentinel-1. We conclude with a summary in section V.
II. DE CO RR EL ATIO N PHA SE COVARIANCE MODELS
Similar to the definition of the correlation coefficient
ρbetween radar measurements, we define γas the
correlation coefficient between interferometric measure-
ments. Let σ2
x,ij denote the variance associated with
φdecor
x,ij (by convention, i<j), the decorrelation phase
component of the interferometric measurement between
SAR acquisitions with indices iand jfor pixel x. The
covariance of φdecor
x,ij and φdecor
x,kl is
cov(φdecor
x,ij , φdecor
x,kl ) = γ(φdecor
x,ij , φdecor
x,kl )·σx,ij σx,kl (1)
Phase variance σ2
x,ij have been comprehensively
studied in literature and can either be derived from a
probability distribution function (PDF) of interferometric
phases under the assumption of a distributed scatterer
mechanism [2], [29]–[32] or be approximated by the
Cramer-Rao bound [33] when the correlation coefficient
ρbetween radar measurements is close to 1. The main
focus of this paper is therefore the derivation of the
correlation coefficient between interferometric phases
γ(φdecor
x,ij , φdecor
x,kl ). In this section, we first review the
existing covariance models. Then we introduce the
proposed new covariance model.
It is worth emphasizing that the focus of this paper
is the covariance models for multi-interferometric phase
measurements of a single pixel. Hereinafter we omit the
subscript xto simplify mathematical notations.
A. Existing Models for γ(φdecor
ij , φdecor
kl )
The first statistical evaluation of temporal decorrela-
tion noise is given by [6], which models decorrelation
as a fully independent noise term in each interferogram
in a network:
γ(φdecor
ij , φdecor
kl ) = δik δjl (2)
where δij = 1, if i=j, and δij = 0 if otherwise.
In contrast, more recent work such as [13], [14], [26]–
[28] argue that decorrelation noise is correlated between
interferograms. [26] and [13] provide estimations under
the simplified assumption that interferometric measure-
ments can be described as circular complex Gaussian
random variables. [28] later notes that interferometric
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 3
phase generally does not follow a circular Gaussian
distribution and provides a closed-form analytical ap-
proximation of the correlation between interferometric
decorrelation phases using the method of nonlinear error
propagation:
cov(φdecor
ij , φdecor
kl )ρik ρjl ρilρjk
2ij ρkl
(3)
where Lis the number of looks. Note that when i=
k, j =l, (3) becomes the Cramer-Rao bound [33], which
is the lower bound for decorrelation phase variance. [14]
also presents a similar equation with (3). We can derive
γ(φdecor
ij , φdecor
kl )from (3):
γ(φdecor
ij , φdecor
kl ) = ρikρj l ρilρj k
q(1 ρ2
ij )(1 ρ2
kl)
(4)
Since (4) has been derived with an approximation that
only holds for a large coherence, (4) is not a complete
covariance model for decorrelation phase noise.
Attempting to extend the covariance model from high
to moderate coherence level, [27] starts with a pseu-
docovariance matrix ˜
ifg that is derived from a SAR
coherence matrix sar and the InSAR incidence matrix
A(Definition of Acan be found in [19]): ˜
ifg =
1
2AsarAT. The pseudocovariance of φdecor
ij and φdecor
kl
is then expressed as
fcov(φdecor
ij , φdecor
kl ) = ρik +ρj l ρil ρjk
2(5)
where ρij is the correlation coefficient between radar
measurements siand sj. The pseudocovariance is then
used to approximate the true covariance of φdecor
ij and
φdecor
kl :
cov(φdecor
ij , φdecor
kl ) =
σij
p1ρij ·fcov(φdecor
ij , φdecor
kl )·σkl
1ρkl
(6)
where the scaling factor σij /p1ρij implies that
cov(φdecor
ij , φdecor
ij ) = σ2
ij . We can also derive
γ(φdecor
ij , φdecor
kl )from (6):
γ(φdecor
ij , φdecor
kl ) = ρik +ρj l ρil ρjk
2p1ρij 1ρkl
(7)
For ease of discussion, hereinafter we refer to the model
suggested by [6] as the ”Hanssen model”, the model
suggested by [27] as the ”Agram-Simons model”, and
the model suggested by [28] as the ”Samiei-Esfahany
model”. We omit discussions of models suggested in
[13], [14], [26] to avoid redundancy as these models
are well represented by the aforementioned three models.
B. A New Model for γ(φdecor
ij , φdecor
kl )
In the following, we present a new covariance model
based on surface scattering characteristics. Consider a
series of radar signals s1, s2, ...snacquired at different
times observing the same target. We adopt the style of
analysis presented in [10]:
si= (ρC+p1ρDi)ei(8)
where ψirepresents propagation phases (e.g., deforma-
tion signal, atmospheric delay) at time ti,Crepresents
persistent scatterers that remain coherent, and Direp-
resents distributed scatterers at time tithat decorrelate
gradually over time. For the sake of simplicity and
without loss of generality, we let C= 1,E[Di] = 0
and E[|Di|2]=1, so that the expected intensity of
siis unity. ρacknowledges the contribution from
persistent scatterers [34]. Examining the properties of
the distributed scatterers in detail,
DiDj
=ρd
ij +Rij (9)
E[DiDj
] = ρd
ij (10)
where ρd
ij describes the correlation between distributed
scatterers Dat times tiand tj.Rij describes the re-
maining uncorrelated part of DiDj
and has an expected
value of zero. Let zij represent the interferometric mea-
surement between signals siand sj:
zij =sisj
=ρ+ (1 ρ)DiDj
+pρ(1 ρ)(Di+Dj
)ei(ψiψj).(11)
The correlation ρij between signals siand sjis
ρij =E[zij ]
pE[|si|2]E[|sj|2]
=ρ+ (1 ρ)ρd
ij .(12)
Note that ρd
ij can be of any generic form that de-
scribes temporal decorrelation of distributed scatterers.
For example, with an exponential decay model ρd
ij =
exp−|titj|, (12) takes the form of the generic
decorrelation model presented in [34], [35].
ρ(t) = ρ+ (1 ρ)et
τ(13)
where τis the characteristic correlation time of the
surface.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 4
Substitute (9) and (12) into (11),
zij =ρij + (1 ρ)Rij
+pρ(1 ρ)(Di+Dj
)ei(ψiψj)(14)
The first term of (14) represents coherent signals between
time tiand tj, which is the expected value of zij, the
second and third terms are associated with decorrelation
during this time period:
zij E[zij ] = (1 ρ)Rij
+pρ(1 ρ)(Di+Dj
)ei(ψiψj)
(15)
where zij E[zij ]represents the zero-mean complex
decorrelation noise component in the interferometric
measurement zij .
The covariance of the complex decorrelation noise
components in zij and zkl is then
cov (zij ,zkl) = E(zij E[zij])(zkl E[zkl])
= (ρikρj l ρ2)ei(ψiψjψk+ψl)(16)
Detailed derivation of (16) is shown in the Appendix A.
The correlation coefficient between complex interfero-
metric measurements is therefore:
γ(zij , zkl) = cov (zij , zkl)
qE|zij E[zij ]|2E|zkl E[zkl ]|2
=ρikρj l ρ2
1ρ2(17)
We find via numerical simulation that the relation be-
tween γ(φdecor
ij , φdecor
kl )and γ(zij , zkl )exhibits a power-
law behavior (See Appendix B)
1γ(φdecor
ij , φdecor
kl )1γ(zij , zkl )1
2(18)
Combining (17)) and (18), we have
γ(φdecor
ij , φdecor
kl )=1s1ρikρjl
1ρ2
(19)
Note that since we need ρij in (19), this model is only
practically applicable to multi-looked interferograms in
which estimates of correlation coefficients ρij can be
obtained.
III. DECORRELATION REDUCTION IN UNWRAPPED
IN TE RF ERO ME TR IC P HA SE S TACK S:A STACKING
EXAMPLE
Typical SBAS-like time-series algorithms estimate de-
sired geophysical parameters by linearly combining a set
of unwrapped interferometric phase measurements over
the same resolution unit on the ground:
P=WΦ(20)
where Prepresents the estimated geophysical
parameters, Wis the weighting or inversion matrix,
and Φ = {φij }is the set of unwrapped interferometric
phases involved in the estimation.
Each φij can be represented as the sum of deforma-
tion, atmospheric noise and decorrelation noise [4], [18],
[19]
φij =φdef
ij +φatm
ij +φdecor
ij (21)
Note that we have not included phase noise terms such
as digital elevation model (DEM) error or thermal noise
because they are either deterministic terms that can
be reasonably well modeled and removed, or typically
negligible uncorrelated noise terms. We have also omit-
ted an integer ambiguity term that accounts for phase
unwrapping errors. We assume that phase unwrapping
is performed consistently and accurately in space and
time. Modeling of phase unwrapping errors requires
detailed mapping of terrain-dependent back-scatterers
and is beyond the scope of this manuscript. Deforma-
tion is a deterministic process. Atmospheric noise and
decorrelation noise, on the other hand, are stochastic
variables with assumed zero means and are independent
of each other because they represent unrelated physical
processes. Therefore, the covariance matrix for a set
of interferometric measurements Φ = {φij }can be
expressed as:
Cov(Φ) = Covatm) + C ovdecor )(22)
The covariance matrix for atmospheric noise is well
researched [7], [9], [36]. In this paper, we focus on the
covariance matrix for decorrelation phase Covdecor).
Assuming no atmospheric noise is present, we can prop-
agate measurement uncertainties to uncertainties in the
estimated parameters:
Cov(P) = W Covdecor)WT(23)
So far we have described four models for
Covdecor): the Hanssen model, the Agram-Simons
model, the Samiei-Esfahany model and the proposed
model. In this section, we illustrate their differences
by feeding each model into (23) and then comparing
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 5
their respective predictions of Cov(P). To achieve that,
we need to first specify Pand W. While there are
many different deformation scenarios and time-series
processing schemes in real applications, we find that
a simple stacking exercise over a transient event is
sufficient for our purposes.
A. Simulation Setting: stacking over a transient event
Consider a radar target that decorrelates over time.
Assume that a transient deformation event took place and
we have Mconsecutive radar measurements acquired
over the surface before the event s1, ..., sMand M
measurements acquired after the event sM+1, ..., s2M.
Therefore, we have a total of M2interferograms that
span the transient event:
{φ1,M+1, ..., φ1,2M, ...φM,M +1, ...φM,2M}
Out of these M2interferograms, there is a maximum
of Minterferograms formed using unique pairs of radar
measurements. One such combination is
{φ1,M+1, φ2,M +2, ..., φM,2M}.
The Minterferograms with unique pairs of radar mea-
surements are often referred to as “independent inter-
ferograms” in literature in the context of atmospheric
noise mitigation [7], [36], [37]. Since these interfero-
grams are not necessarily truly independent, we adopt
the term ”non-repeating stack” instead. In contrast, we
refer to the previous group of measurements with all
possible measurements as a “repeating stack”. Note that
in practice, a repeating stack does not have to include
all available interferograms. Assuming no atmospheric
noise is present,
φij =φdef +φdecor
ij (24)
where φdef corresponds to phase change caused by the
transient event and is present in every interferogram
that span the event.
To retrieve signals associated with the transient de-
formation event φdef , we can average either stack. Both
stacking strategies are widely adopted in literature [37]–
[41]. By stacking a non-repeating stack, we have
Φnrp = [φ1,M +1, φ2,M +2, ..., φM ,2M]0
W= [ 1
M
1
M... 1
M](25)
P=WΦnrp
0 5 10 15 20 25 30 35 40 45
/ t
0
2
4
6
8
10
12
14
Predicted phase noise 2(P), [rad]
2
Hanssen
Agram-Simons
Samiei-Esfahany
Proposed
Non-repeating
Repeating
Fig. 1. Predicted uncertainty associated with decorrelation noise
after non-repeating stacking and repeating stacking using the Hanssen
model, the Agram-Simons model, the Samiei-Esfahany model and the
proposed model.
Similarly, by stacking a repeating stack, we have
Φrp = [φ1,M +1, ..., φ1,2M, ...φM,M+1 , ...φM,2M]0
W= [ 1
M2
1
M2... 1
M2](26)
P=WΦrp
B. Construction of Covdecor)
We are interested in determining σ2(P)for both
stacks. Adopting the Cramer-Rao bound [33]
σ2(φij ) = 1ρ2
ij
2ρ2
ij
as phase variances, we construct Covdecor)using (1)
with γ(φdecor
ij , φdecor
kl )given by each of the four models.
We also need to specify a temporal decorrelation
model ρ(t). Here we choose the generic decorrelation
model (13). The parameters in (13) are highly dependent
on land covers, wavelength and climate [34], [35]. In
this simulation, we examine the impacts of correlation
time constant τand persistent correlation ρon
Covdecor). We also examine the impacts of different
numbers of radar measurements (2M), and acquisition
intervals on the performance of decorrelation noise
reduction.
C. Comparison Between Existing And The Proposed
Decorrelation Covariance Models
Fig. 1 depicts the predictions of σ2(P)from both
the existing and the proposed models for surfaces
with varying decorrelation rates and fixed persistent
coherence (ρ= 0.1). We use the ratio between
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 6
characteristic correlation time τand SAR acquisition
interval tto represent apparent surface correlation
time – higher ratio means slower apparent decorrelation.
We set M= 25. The Hanssen model (green lines)
predicts larger phase variance with non-repeating
stacking regardless of surface decorrelation rates; the
Agram-Simons model (blue lines) predicts zero or
small differences between two stacking strategies with
either rapidly or very slowly decorrelating surfaces;
and both the Samiei-Esfahany model (magenta lines)
and the proposed model (red lines) predict significantly
higher noise reduction from repeating stacking when
τ/t < 15 and similar noise reduction performances
from either stacking strategy when τ/t > 15.
Additionally, we observe that:
1. The Agram-Simons model predicts the highest
phase noise (i.e., the least reduction in decorrelation
noise) for both non-repeating and repeating stacking.
This means that the Agram-Simons model suggests high
correlation between decorrelation noise components.
In contrast, the Hanssen model, which assumes
independence between decorrelation noise components,
predicts the lowest uncertainty. The Samiei-Esfahany
model suggests higher correlation than that suggested
by the proposed model, but lower than that suggested
by the Agram-Simons model (within the range of τ/t
plotted in Fig.1).
2. Predictions from the Samiei-Esfahany, the Agram-
Simons model and the proposed model model all show
a peak around τ/t6while predictions from the
Hanssen model shows a steady decrease with increasing
apparent correlation time τ/t. Two factors influence
the predicted phase variance: a) the average phase noise
level in interferograms and b) the degree of correlation
between decorrelation noise components. The higher
the average phase noise, the higher the predicted phase
variance; the higher the degree of correlation between
decorrelation noise terms, the higher the predicted phase
variance. When τ/tis small, correlation between
decorrelation noise terms are negligible in all three
models, and when τ/tis large, the average phase noise
level is low. Therefore the predicted phase variances
in all three models are low at both ends of τ/tbut
high in the middle. Since there is no correlation in the
Hanssen model, the predicted phase variance decreases
monotonically with τ/t.
Fig. 2 depicts how varying ρor the number of
SAR measurements (2M) impacts predicted phase
noises with respect to different apparent decorrelation
0 10 20 30 40 50
/ t
0
1
2
3
4
5
6
Predicted phase noise 2(P), [rad]
2
= 0.1
= 0.3
=0.5
Non-repeating
Repeating
0 20 40 60 80 100
Number of SAR measurements
0
5
10
15
20
25
Predicted phase noise 2(P), [rad]
2
/ t= 0.1
/ t= 1
/ t= 10
Non-repeating
Repeating
(a) (b)
Fig. 2. The influence of (a) ρand (b) the number of SAR
measurements with respect to different apparent decorrelation rates
on predicted uncertainty using the proposed model.
rates τ/t. Fig. 2 (a) shows that increasing ρ
decreases the dependency of predicted uncertainty
on surface decorrelation rate as contributions from
persistent scatterer increases. Fig. 2 (b) illustrates
that for a slowly decorrelating surface (τ/t= 10),
increasing Mincreases predicted uncertainties while for
a rapidly decorrelating surface (τ/t= 0.1), increasing
Mreduces predicted uncertainty. Increasing Mis
equivalent to increasing the observation window length.
For rapidly decorrelating surfaces, longer observational
window provides additional measurements that are at
a comparable noise level (determined by ρ) with the
original stack. Since these measurements are almost
independent with the original stack, increasing M
reduces noise. In contrast, for slowly decorrelating
surfaces, long temporal span interferograms provide
highly correlated measurements at a higher noise level.
Therefore increasing Mincreases noise. The contrasting
impacts of Mwith respect to surface decorrelation rates
again reflect the two factors that influence the predicted
phase variance: 1) the average phase noise level in
interferograms and 2) the degree of correlation between
decorrelation noise terms. Increasing Mraises average
noise level in interferograms but reduces the degree of
correlation between decorrelation noise components.
Therefore, when SAR measurements are sampled at a
comparable interval with the surface correlation time
(τ/t=1 ), the predicted uncertainty first increases
then decreases when increasing M.
D. Implication for stacking strategies
Ultimately our goal is to effectively reduce
decorrelation noise. Knowledge of Covdecor)
can help us determine optimal processing strategies.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 7
In our simple example of stacking over a transient
event, we compared the performances of two common
stacking practices: non-repeating stacking and repeating
stacking. If the target decorrelates exponentially over
time, Fig.1 provides predictions of performance of
either stack on decorrelation reduction with the given
configuration of M= 25 and ρ= 0.1. For example, if
radar measurements are not sampled 10 times the rate of
surface decorrelation (τ/t < 10), the proposed model
suggests use of repeating stack because the performance
differences between the two stacks are significant.
On the other hand, if τ/t > 10, the performance
differences between the two stacks is insignificant,
non-repeating stack becomes a better choice because it
involves computation of fewer interferograms and hence
is more efficient. Fig. 2 (b) demonstrates that for rapidly
decorrelating surfaces, it is preferable to have long
observation windows, though the rate of noise reduction
declines with increasing M. In comparison, for surfaces
that exhibit slow decorrelation rates, a small subset of
interferometric pairs is sufficient to achieve satisfactory
results.
In real applications, the weighting matrix Wcan take
on more complicated forms than the ones shown in (25)
and (26). Moreover, there are often constraints on M.
Nevertheless the same procedure as described above
can be followed to guide the choice of Wand the set
of interferometric pairs to be used.
IV. VALIDATI ON WI TH RE AL DATA
In this section, we compare and assess all four
decorrelation phase covariance models – the Hanssen
model, the Samiei-Esfahany model, the Agram-Simons
model, and the proposed model – with C-band Sentinel-
1 data collected in both Cascadia and Death Valley.
Similar to Section III, we construct Cov(φdecor
ij , φdecor
kl )
using (1) with γ(φdecor
ij , φdecor
kl )given by each of the
four models. Note that we adopt the correlation forms
of the models to avoid estimation bias caused by phase
variance estimation – the Samiei-Esfahany model uses
the Cramer-Rao bound to estimate phase variances
while all three other models use observed phase
variances. Since the Cramer-Rao bound only holds
for high coherence targets and hence underestimate
phase variances of the stacks, we modify the original
Samiei-Esfahany model (3) by adopting observed phase
variances instead. We estimate observed phase variances
of individual interferograms σ2(φij), and of averaged
stacks σ2(φdecor
nrp ),σ2(φdecor
rp )by calculating phase
0 200 400 600 800
0.2
0.4
0.6
0.8
1
0 500 1000 1500
0.2
0.4
0.6
0.8
1
0 500 1000 1500
0.2
0.4
0.6
0.8
1
D
C
D
A
A
B
C
B
(a) (b) (c)
Time span, days
Coherence
<latexit sha1_base64="6eS2BO7AuLYOi7WOAzXi0icXhGQ=">AAAB63icbVDLSgNBEOz1GeMr6tHLYBA8hd0Y0GPAi8cI5gHJEmYns8mQeSwzs0JY8gtePCji1R/y5t84m+xBEwsaiqpuuruihDNjff/b29jc2t7ZLe2V9w8Oj44rJ6cdo1JNaJsornQvwoZyJmnbMstpL9EUi4jTbjS9y/3uE9WGKfloZwkNBR5LFjOCbS4N9EQNK1W/5i+A1klQkCoUaA0rX4ORIqmg0hKOjekHfmLDDGvLCKfz8iA1NMFkise076jEgpowW9w6R5dOGaFYaVfSooX6eyLDwpiZiFynwHZiVr1c/M/rpza+DTMmk9RSSZaL4pQjq1D+OBoxTYnlM0cw0czdisgEa0ysi6fsQghWX14nnXotuK7VHxrVZqOIowTncAFXEMANNOEeWtAGAhN4hld484T34r17H8vWDa+YOYM/8D5/ABzEjj0=</latexit>
Fig. 3. Examples of observed temporal variations of correlation
in the Cascadia and the Death Valley regions. (a) Scatter plots of
correlation vs time in locations A, B, C and D. (b) A and B are located
in the heavily vegetated Cascadia region, (c) C and D are located in
the desert of Death Valley. A and B both exhibit a characteristic
decorrelation time of roughly 30 days and an asymptotic coherence
ρof 0.1. C and D, on the other hand, show much longer decor-
relation time of approximately 400 days and 200 days and higher
asymptotic coherence ρof 0.2 and 0.6, respectively.
variances inside 50 pixel by 50 pixel boxes (15 by
15 km2in area), assuming the spatial variances of
atmospheric noise and phase unwrapping errors are
negligible in the box. It is reasonable to assume that
areas with moderate to high correlations are reliably
unwrapped. Low correlation areas, on the other hand,
are prone to phase unwrapping errors. We assume that
phase unwrapping errors are either uniform or sparsely
distributed inside a small estimation window and hence
contribute little to the overall phase variance.
Similar to section III, we form both non-repeating
and repeating stacks that span over a given time interval
in both regions, and predict residual decorrelation noise
after averaging each stack using both the proposed
and existing decorrelation phase covariance models.
We estimate ρij using sampled local averages over
300 meter by 300 meter windows. We then compare
predicted residual decorrelation noise with observed
residual decorrelation noise.
Both Cascadia and Death Valley exhibit exponential
decay in correlation with time, but with contrasting
decorrelation rates (Fig. 3). For the Cascadia region,
we form a non-repeating stack consisting of 40
interferograms and a repeating stack consisting of 1600
interferograms. For the Death Valley region, we form a
non-repeating stack consisting of 28 interferograms and
a repeating stack consisting of 784 interferograms. By
averaging the respective stacks, we obtained φnrp and
φrp for both regions.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 8
0 100 200 300 400
Data Index
0
0.1
0.2
0.3
0.4
0.5
0.6
Prediction Error, [rad]2
0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Data
Agram-Simons
Hanssen
Samiei-Esfahany
Proposed
0 0.1 0.2 0.3 0.4 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Data
Agram-Simons
Hanssen
Samiei-Esfahany
Proposed
Cascadia Death Valley
2(rp),[rad]2
<latexit sha1_base64="x19PuoYpEV26unaVVm3HIUb3ihk=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahgpSkFh+7ghuXFewDkjRMJpN26EwSZiZCCd248VfcuFDErf/gzr8xSYv4OnDhcM693HuPFzMqlWF8aKWFxaXllfJqZW19Y3NL397pyigRmHRwxCLR95AkjIako6hipB8LgrjHSM8bX+Z+75YISaPwRk1i4nA0DGlAMVKZ5Or7tqRDjgaNmh2PqJuKeHp0DC2BfGfQcPWqUTcKwL/EnJMqmKPt6u+2H+GEk1BhhqS0TCNWToqEopiRacVOJIkRHqMhsTIaIk6kkxZfTOFhpvgwiERWoYKF+n0iRVzKCfeyTo7USP72cvE/z0pUcO6kNIwTRUI8WxQkDKoI5pFAnwqCFZtkBGFBs1shHiGBsMqCqxQhXOQ4/Xr5L+k26uZJvXndrLaa8zjKYA8cgBowwRlogSvQBh2AwR14AE/gWbvXHrUX7XXWWtLmM7vgB7S3Tz+Yl9w=</latexit>
2(rp),[rad]2
<latexit sha1_base64="x19PuoYpEV26unaVVm3HIUb3ihk=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBahgpSkFh+7ghuXFewDkjRMJpN26EwSZiZCCd248VfcuFDErf/gzr8xSYv4OnDhcM693HuPFzMqlWF8aKWFxaXllfJqZW19Y3NL397pyigRmHRwxCLR95AkjIako6hipB8LgrjHSM8bX+Z+75YISaPwRk1i4nA0DGlAMVKZ5Or7tqRDjgaNmh2PqJuKeHp0DC2BfGfQcPWqUTcKwL/EnJMqmKPt6u+2H+GEk1BhhqS0TCNWToqEopiRacVOJIkRHqMhsTIaIk6kkxZfTOFhpvgwiERWoYKF+n0iRVzKCfeyTo7USP72cvE/z0pUcO6kNIwTRUI8WxQkDKoI5pFAnwqCFZtkBGFBs1shHiGBsMqCqxQhXOQ4/Xr5L+k26uZJvXndrLaa8zjKYA8cgBowwRlogSvQBh2AwR14AE/gWbvXHrUX7XXWWtLmM7vgB7S3Tz+Yl9w=</latexit>
2(nrp),[rad]2
<latexit sha1_base64="4TmyjBy/VzZk0qkeGvGJsf+JUkE=">AAACBnicbVDLSsNAFJ3UV62vqEsRBotQQUpSi49dwY3LCvYBSRomk0k7dDIJMxOhhK7c+CtuXCji1m9w59+YpEV8HbhwOOde7r3HixmVyjA+tNLC4tLySnm1sra+sbmlb+90ZZQITDo4YpHoe0gSRjnpKKoY6ceCoNBjpOeNL3O/d0uEpBG/UZOYOCEachpQjFQmufq+LekwRINGzY5H1E25iKdHx9ASyHcGDVevGnWjAPxLzDmpgjnarv5u+xFOQsIVZkhKyzRi5aRIKIoZmVbsRJIY4TEaEiujHIVEOmnxxhQeZooPg0hkxRUs1O8TKQqlnIRe1hkiNZK/vVz8z7MSFZw7KeVxogjHs0VBwqCKYJ4J9KkgWLFJRhAWNLsV4hESCKssuUoRwkWO06+X/5Juo26e1JvXzWqrOY+jDPbAAagBE5yBFrgCbdABGNyBB/AEnrV77VF70V5nrSVtPrMLfkB7+wQWyphU</latexit>
2(nrp),[rad]2
<latexit sha1_base64="4TmyjBy/VzZk0qkeGvGJsf+JUkE=">AAACBnicbVDLSsNAFJ3UV62vqEsRBotQQUpSi49dwY3LCvYBSRomk0k7dDIJMxOhhK7c+CtuXCji1m9w59+YpEV8HbhwOOde7r3HixmVyjA+tNLC4tLySnm1sra+sbmlb+90ZZQITDo4YpHoe0gSRjnpKKoY6ceCoNBjpOeNL3O/d0uEpBG/UZOYOCEachpQjFQmufq+LekwRINGzY5H1E25iKdHx9ASyHcGDVevGnWjAPxLzDmpgjnarv5u+xFOQsIVZkhKyzRi5aRIKIoZmVbsRJIY4TEaEiujHIVEOmnxxhQeZooPg0hkxRUs1O8TKQqlnIRe1hkiNZK/vVz8z7MSFZw7KeVxogjHs0VBwqCKYJ4J9KkgWLFJRhAWNLsV4hESCKssuUoRwkWO06+X/5Juo26e1JvXzWqrOY+jDPbAAagBE5yBFrgCbdABGNyBB/AEnrV77VF70V5nrSVtPrMLfkB7+wQWyphU</latexit>
0 20 40 60 80 100
Data Index
0
0.1
0.2
0.3
0.4
0.5
0.6
Prediction Error, [rad]2
(a) (b)
(c) (d)
Fig. 4. Predicted phase variance and prediction errors after non-
repeating stacking vs repeating stacking in (a) (c) the Cascadia region
and (b) (d) the Death Valley region. Data (circles) show that for the
Cascadia region, repeating stacking produces smaller phase variances
and for the Death Valley region, similar phase variances, as compared
to non-repeating stacking. Model predictions are shown in (a) and (b)
while their respective errors are shown in (c) and (d) with the average
prediction errors marked by solid lines. Predictions from the proposed
model (stars) match the data best in both cases. The accuracy of each
model prediction is summarized in Table.I
TABLE I
AVERA GE PREDICTION ERROR FO R EAC H MOD EL OV ER T WO
SU RFACE S. WE DEFINE P REDIC TION ER ROR S AS THE DI STAN CES
(UN IT:[rad]2)B ETWEE N POSIT IONS OF DATA PO IN TS (BEIGE
CIRCLES)AND POSITIONS OF THEIR CORRESPONDING
PREDICTION POINTS (PU RPL E STAR S,O RAN GE S QUARES ,CYAN
DI AMO ND S OR MA GEN TA INV ERT ED TR IA NGL ES )IN FIG . 4.
Models Cascadia Death Valley
Agram-Simons 0.300 0.368
Hanssen 0.097 0.114
Samiei-Esfahany (Modified) 0.096 0.231
Proposed 0.092 0.105
Comparisons between phase variances associated
with φnrp and φrp are shown in Fig. 4 (a) and (b).
We obtained σ2(φdecor
nrp )and σ2(φdecor
rp )at evenly
distributed image grid points. For the Cascadia region
(τ/t1or 2), the repeating stack yields smaller
uncertainties – on average 0.037 rad2) than the
non-repeating stack – on average 0.284 rad2(Fig.
4 (a), beige circles), confirming predictions from the
proposed model. For the Death Valley region where
the apparent correlation time τ/t > 20, non-repeating
and repeating stacking produce comparable results – on
average 0.105 rad2and 0.102 rad2, respectively (Fig.
4 (b), beige circles), which are also consistent with
predictions from the proposed model.
Predictions from both existing and the proposed
models are shown in Fig. 4 and Table. I. Predictions
from the proposed model (purple stars) match best
with actual observations (beige circles). Predictions
from existing models are consistent with the simulation
results in Section III. For example, in Cascadia where
the decorrelation rate is rapid, the Agram-Simons
model (cyan diamonds) have the highest estimation
errors (Fig. 4 (a)(c)). In the Death Valley region where
the decorrelation rate is on the order of years, the
proposed model and the Hanssen model have the lowest
estimation errors (Fig. 4 (d)). However, predictions from
the Hanssen model systematically deviate from actual
observations (Fig.4 (b)) .
Finally, we use an example to illustrate the significant
reduction in decorrelation noise offered by including
repeating interferograms in the stack over rapidly decor-
relating areas. Fig. 5(a) and (b) depicts stacks over a
slow slip deformation event [42], [43] that occurred in
February 2016 in the Cascadia region. As a comparison,
non-repeating and repeating stacking results in Death
Valley (Fig.5 (c) and (d)) show minimal differences.
Slow slip events are usually hard to capture in the
Cascadia region with InSAR due to extremely low signal
to noise ratios. Atmospheric noise and decorrelation are
the two main limiting factors. With 10 Sentinel-1 SAR
acquisitions before the slow slip event and 10 Sentinel-
1 SAR acquisitions after, we formed a non-repeating
interferogram stack consisting of 10 interferograms and
a repeating stack consisting of 100 interferograms. It is
apparent that the result from using repeating stacking
yields a much cleaner signal pattern. Fig. 6 compares
phase measurements along four profile lines between
non-repeating and repeating stacking. Again, it is clear
that while both stacking strategies produce measure-
ments of the same expected signal, repeating stacking
produces measurements with 80% less phase variance
– the uncertainty associated with decorrelation reduces
from 0.33 cm to 0.13 cm in the line-of-sight direction.
With the same number of independent acquisitions, at-
mospheric noise is reduced to the same extent after either
non-repeating stacking or repeating stacking. Therefore,
the significant reduction in phase noise reflects reduced
decorrelation noise in the repeating stack. It is worth
noting that repeating stacking may also contribute to re-
duction of phase unwrapping errors. Fig. 8 (Appendix C)
shows that phase unwrapping errors are more prevalent
in longer temporal baseline interferograms. The problem
of phase unwrapping error reduction is analogous to
that of decorrelation noise reduction. On one hand, a
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 9
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
LOS displacement, cm
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
LOS displacement, cm
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Phase, rad
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Phase, rad
(a) (b) (c)
(d)
Fig. 5. Results from (a) non-repeating and (b) repeating stacking
over the February 2016 slow slip event in the Cascadia region. The
result from repeating stacking suffers from less decorrelation noise
than the result from non-repeating stacking. Areas with less coverage
than 20 acquisitions are masked. Phase measurements are converted
to radar line-of-sight (LOS) measurements. Results from (c) non-
repeating and (d) repeating stacking in the Death Valley region. The
differences between the two stacking results are minimal.
repeating stack may help reduce phase unwrapping errors
because of its superior number of measurements. On the
other hand, a repeating stack includes more long tempo-
ral baseline interferograms that are susceptible to phase
unwrapping errors. While detailed statistical models for
phase unwrapping errors are beyond the scope of this
work, a reasonable assumption for phase unwrapping
errors in rapidly decorrelating areas are to treat them as
random variables, i.e., the amount of phase unwrapping
errors in a particular pixel vary independently in different
interferograms. Since decorrelation noise can be treated
as an independent noise term for rapidly decorrelating
surfaces, the addition of another random noise term
does not alter the statistical description for overall phase
noise. Therefore, in the Cascadia case, repeating stacking
performs better in reducing overall phase noise than non-
repeating stacking, as expected from the proposed model.
V. DISCUSSIONS AND CONCLUSIONS
We have described a new decorrelation phase
covariance model for unwrapped interferogram phase
stacks and compared the proposed model with three
existing models – the Hanssen model, the Samiei-
Esfahany model and the Agram-Simons model.
Validations with Sentinel-1 data collected in both the
Cascadia region and Death Valley region show that our
model best captures decorrelation noise propagation in
interferogram stacks. We demonstrated with a simple
stacking exercise that the proposed decorrelation phase
0 500 1000
-4
-2
0
2
4
0 500 1000
-4
-2
0
2
4
0 500 1000
-4
-2
0
2
4
0 500 1000
-4
-2
0
2
4
A
B
C
D
A’
B’
C’
D’
Pixels
Phase, rad
A-A’ B-B’
C-C’ D-D’
Non-repeating st acking
Repeating stacking
Fig. 6. Comparisons between non-repeating and repeating stacking
results over profile lines A-A’, B-B’, C-C’ and D-D’. Repeating
stacking (red dots) produces similar mean measurements as with
non-repeating stacking (blue dots), but with much 83% less phase
variances. Equivalently, repeating stacking produces measurements
with 0.33 cm line-of-sight uncertainties compared to 0.13 cm after
non-repeating stacking.
covariance model can facilitate choice of SBAS-like
time-series processing strategies. The workflow applied
to the stacking example can be easily adapted to general
SBAS-like time-series algorithms in three steps:
1 Obtain the temporal decorrelation characteristics
ρ(t)in the area of interest. In some cases we may
have a prior knowledge or a good estimation of ρ(t).
Otherwise, estimate ρ(t)from the processed stack.
2 Construct the decorrelation phase covariance matrix
with (1) and (19) for the entire interferometric phase
stack Φ. Phase variances can be estimated using the
PDF of interferometric phases [2], [29]–[32].
3 Estimate residual decorrelation noise COV (P)
using (23) with different weighting matrices W.
Determine the optimal Wand subset of interferometric
pairs Φin terms of effective noise reduction and
efficiency.
As we head into an era with an ever-growing SAR
archive, understanding and quantifying uncertainties as-
sociated with decorrelation noise is of critical im-
portance. A rigorous, comprehensive noise covariance
model allows the InSAR community to better assess
uncertainties with InSAR measurements, and to extend
InSAR applications from mid-to-high correlation areas
to low-correlation areas.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 10
APPENDIX A
DERIVATION OF (16)
We model the distributed scatterers Dias complex
Gaussian random variables. Dis can be represented as
Di=ρd
i1,iDi1+q1ρd
i1,i
2ni(27)
where niis a complex Gaussian random variables with
an expected intensity of unity, and that is uncorrelated
with D’s and ns other than itself. With (27), we can
derive that
ρd
ij ρd
jk =ρd
ik (28)
E(DiD
j) = ρd
ij (29)
E(|Di|2|Dj|2) = 1 + ρd
ij
2(30)
Now we can derive (16):
cov (zij E[zij ], zkl E[zkl ])
=E(zij E[zij ])(zkl E[zkl ])
=E(1 ρ)2Rij R
kl
+ρ(1 ρ)(DiD
k+DiDl+D
jD
k+D
jDl)
+ (1 ρ)pρ(1 ρ)Rij (D
k+Dl)
+R
kl(Di+D
j)ei(ψiψjψk+ψl)
=(1 ρ)2E(Rij R
kl)
+ρ(1 ρ)(ρd
ik +ρd
jl )ei(ψiψjψk+ψl)
=(1 ρ)2ρd
ikρd
jl
+ρ(1 ρ)(ρd
ik +ρd
jl )ei(ψiψjψk+ψl)
= (ρikρj l ρ2
)ei(ψiψjψk+ψl)(31)
APPENDIX B
NUMERICAL SIMULATION OF (18)
To find the relation between γ(zij , zkl)and φij, φk l,
we simulate a series of complex Gaussian random vari-
ables with decreasing correlation from 1 to 0. We then
compute the corresponding correlations between phases
of these complex Gaussian random variables (Fig. 7).
APPENDIX C
SAM PL E IN TE RF ERO GR AM S OF T HE CA SC AD IA S TACK
The non-repeating stack of Cascadia consists of 10
interferograms that have similar temporal baselines
(on average 272 days). Fig. 8 (b) and (e) show
wrapped and unwrapped phase from a non-repeating
0 0.2 0.4 0.6 0.8 1
(zij,zkl)
0
0.2
0.4
0.6
0.8
1
( ij,kl)
Simulation
Power-law
Fig. 7. The relation between γ(zij , zkl)and γ(φij , φk l)exhibits a
power-law behavior 1γ(φdecor
ij , φdecor
kl )1γ(zij , zkl )1
2.
stack interferogram with a temporal baseline of 264
days. The repeating stack of Cascadia consists of 100
interferograms, which includes interferograms with
temporal baseline ranging from 48 days to 480 days.
Fig. 8 (a) and (d) show wrapped and unwrapped phases
from an interferogram with a temporal baseline of 48
days. Fig. 8 (c) and (f) show wrapped and unwrapped
phases from an interferogram with a temporal baseline
of 480 days. Phase unwrapping errors manifest as jumps
of 2πradians. Fig. 8 shows that phase unwrapping
errors are more prevalent in longer temporal baseline
interferograms.
ACK NOWLEDG ME NT
We would like to thank the anonymous reviewers for
their constructive criticisms that greatly improved the
quality of this manuscript. This work was supported by
NASA Earth Science and Interior Grant NNX 17AE036.
Copernicus Sentinel-1 data 2015-2018 was retrieved
from ASF DAAC 7 May 2018, processed by ESA,
https://www.asf.alaska.edu.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 11
124.5°W 123.5°W 122.5°W 121.5°W
47.5°N
46.5°N
45.5°N
44.5°N
43.5°N
42.5°N
-2
0
2
4
6
8
10
12
14
rad
124.5°W 123.5°W 122.5°W 121.5°W
47.5°N
46.5°N
45.5°N
44.5°N
43.5°N
42.5°N
-5
0
5
10
rad
124.5°W 123.5°W 122.5°W 121.5°W
47.5°N
46.5°N
45.5°N
44.5°N
43.5°N
42.5°N
-3
-2
-1
0
1
2
3
rad
124.5°W 123.5°W 122.5°W 121.5°W
47.5°N
46.5°N
45.5°N
44.5°N
43.5°N
42.5°N
-3
-2
-1
0
1
2
3
rad
124.5°W 123.5°W 122.5°W 121.5°W
47.5°N
46.5°N
45.5°N
44.5°N
43.5°N
42.5°N
-3
-2
-1
0
1
2
3
rad
124.5°W 123.5°W 122.5°W 121.5°W
47.5°N
46.5°N
45.5°N
44.5°N
43.5°N
42.5°N
-5
0
5
10
rad
Wrapped phaseUnwrapped phase
(a) (b) (c)
(d) (e) (f)
Fig. 8. Three sample interferograms before and after unwrapping.
(a)(d) A sample interferogram with 48 days temporal baseline
(b)(e) A sample interferogram with 264 days temporal baseline
(c)(f) A sample interferogram with 480 days temporal baseline.
Phase unwrapping errors manifest as jumps/spikes of 2πrad. Phase
unwrapping errors are more prevalent in longer temporal baseline
interferograms.
REFERENCES
[1] H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, and
C. L. Werner, “On the derivation of coseismic displacement
fields using differential radar interferometry: The landers earth-
quake,” Journal of Geophysical Research: Solid Earth, vol. 99,
no. B10, pp. 19 617–19 634, 1994.
[2] R. Bamler and P. Hartl, “Synthetic aperture radar interferome-
try,Inverse problems, vol. 14, no. 4, p. R1, 1998.
[3] P. A. Rosen, S. Hensley, I. R. Joughin, F. Li, S. N. Madsen,
E. Rodriguez, and R. M. Goldstein, “Synthetic aperture radar
interferometry,” 1998.
[4] R. B¨
urgmann, P. A. Rosen, and E. J. Fielding, “Synthetic aper-
ture radar interferometry to measure earth’s surface topography
and its deformation,” Annual review of earth and planetary
sciences, vol. 28, no. 1, pp. 169–209, 2000.
[5] J. Elliott, R. Walters, and T. Wright, “The role of space-based
observation in understanding and responding to active tectonics
and earthquakes,” Nature communications, vol. 7, p. 13844,
2016.
[6] R. F. Hanssen, Radar interferometry: data interpretation and
error analysis. Springer Science & Business Media, 2001,
vol. 2.
[7] T. Emardson, M. Simons, and F. Webb, “Neutral atmospheric
delay in interferometric synthetic aperture radar applications:
Statistical description and mitigation,” Journal of Geophysical
Research: Solid Earth, vol. 108, no. B5, 2003.
[8] R. B. Lohman and M. Simons, “Some thoughts on the use of
insar data to constrain models of surface deformation: Noise
structure and data downsampling,Geochemistry, Geophysics,
Geosystems, vol. 6, no. 1, 2005.
[9] F. Onn and H. Zebker, “Correction for interferometric synthetic
aperture radar atmospheric phase artifacts using time series of
zenith wet delay observations from a gps network,Journal of
Geophysical Research: Solid Earth, vol. 111, no. B9, 2006.
[10] H. A. Zebker and J. Villasenor, “Decorrelation in interferometric
radar echoes,” IEEE Transactions on geoscience and remote
sensing, vol. 30, no. 5, pp. 950–959, 1992.
[11] D. Just and R. Bamler, “Phase statistics of interferograms with
applications to synthetic aperture radar,Applied optics, vol. 33,
no. 20, pp. 4361–4368, 1994.
[12] F. De Zan, A. Parizzi, P. Prats-Iraola, and P. L´
opez-Dekker, “A
sar interferometric model for soil moisture,” IEEE Transactions
on Geoscience and Remote Sensing, vol. 52, no. 1, pp. 418–425,
2013.
[13] F. De Zan, M. Zonno, and P. L´
opez-Dekker, “Phase incon-
sistencies and multiple scattering in sar interferometry,IEEE
Transactions on Geoscience and Remote Sensing, vol. 53,
no. 12, pp. 6608–6616, 2015.
[14] S. Zwieback, X. Liu, S. Antonova, B. Heim, A. Bartsch,
J. Boike, and I. Hajnsek, “A statistical test of phase closure
to detect influences on dinsar deformation estimates besides
displacements and decorrelation noise: Two case studies in
high-latitude regions,IEEE Transactions on Geoscience and
Remote Sensing, vol. 54, no. 9, pp. 5588–5601, 2016.
[15] S. Zwieback and F. J. Meyer, “Repeat-pass interferometric
speckle,” IEEE Transactions on Geoscience and Remote Sens-
ing, pp. 1–15, 2020.
[16] H. Ansari, F. De Zan, and A. Parizzi, “Study of systematic
bias in measuring surface deformation with sar interferometry,
IEEE Transactions on Geoscience and Remote Sensing, pp. 1–1,
2020.
[17] A. Ferretti, C. Prati, and F. Rocca, “Permanent scatterers in sar
interferometry,IEEE Transactions on geoscience and remote
sensing, vol. 39, no. 1, pp. 8–20, 2001.
[18] A. Hooper, H. Zebker, P. Segall, and B. Kampes, “A new
method for measuring deformation on volcanoes and other
natural terrains using insar persistent scatterers,” Geophysical
research letters, vol. 31, no. 23, 2004.
[19] P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti, “A new
algorithm for surface deformation monitoring based on small
baseline differential sar interferograms,IEEE transactions on
geoscience and remote sensing, vol. 40, no. 11, pp. 2375–2383,
2002.
[20] A. M. Guarnieri and S. Tebaldini, “Hybrid cram´
er–rao bounds
for crustal displacement field estimators in sar interferometry,
IEEE signal processing letters, vol. 14, no. 12, pp. 1012–1015,
2007.
[21] A. M. Guarnieri and S. Tebaldini, “On the exploitation of target
statistics for sar interferometry applications,” IEEE Transactions
on Geoscience and Remote Sensing, vol. 46, no. 11, pp. 3436–
3443, 2008.
[22] A. Ferretti, A. Fumagalli, F. Novali, C. Prati, F. Rocca, and
A. Rucci, “A new algorithm for processing interferometric
data-stacks: Squeesar,IEEE Transactions on Geoscience and
Remote Sensing, vol. 49, no. 9, pp. 3460–3470, 2011.
[23] G. Fornaro, S. Verde, D. Reale, and A. Pauciullo, “Caesar:
An approach based on covariance matrix decomposition to im-
prove multibaseline–multitemporal interferometric sar process-
ing,” IEEE Transactions on Geoscience and Remote Sensing,
vol. 53, no. 4, pp. 2050–2065, 2014.
[24] H. Ansari, F. De Zan, and R. Bamler, “Sequential estimator:
Toward efficient insar time series analysis,IEEE Transactions
on Geoscience and Remote Sensing, vol. 55, no. 10, pp. 5637–
5652, 2017.
[25] H. Ansari, F. De Zan, and R. Bamler, “Efficient phase estimation
for interferogram stacks,” IEEE Transactions on Geoscience
and Remote Sensing, vol. 56, no. 7, pp. 4109–4125, 2018.
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 12
[26] F. Rocca, “Modeling interferogram stacks,IEEE Transactions
on Geoscience and Remote Sensing, vol. 45, no. 10, pp. 3289–
3299, 2007.
[27] P. Agram and M. Simons, “A noise model for insar time series,”
Journal of Geophysical Research: Solid Earth, vol. 120, no. 4,
pp. 2752–2771, 2015.
[28] S. Samiei-Esfahany and R. Hanssen, “On the evaluation of
second order phase statistics in sar interferogram stacks,” Earth
Observation and Geomatics Engineering, vol. 1, no. 1, pp. 1–
15, 2017.
[29] B. Barber, “The phase statistics of a multichannel radar inter-
ferometer,Waves in random media, vol. 3, no. 4, pp. 257–266,
1993.
[30] J.-S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller,
“Intensity and phase statistics of multilook polarimetric and
interferometric sar imagery,IEEE Transactions on Geoscience
and Remote Sensing, vol. 32, no. 5, pp. 1017–1028, 1994.
[31] L. Joughin and D. Winebrenner, “Effective number of looks for
a multilook interferometric phase distribution,” in Proceedings
of IGARSS’94-1994 IEEE International Geoscience and Remote
Sensing Symposium, vol. 4. IEEE, 1994, pp. 2276–2278.
[32] J. Tough, D. Blacknell, and S. Quegan, “A statistical description
of polarimetric and interferometric synthetic aperture radar
data,” Proceedings of the Royal Society of London. Series A:
Mathematical and Physical Sciences, vol. 449, no. 1937, pp.
567–589, 1995.
[33] E. Rodriguez and J. Martin, “Theory and design of interfer-
ometric synthetic aperture radars,” IEE Proceedings F (Radar
and Signal Processing), vol. 139, no. 2, pp. 147–159, 1992.
[34] A. Parizzi, X. Cong, and M. Eineder, “First results from
multifrequency interferometry. a comparison of different decor-
relation time constants at l, c, and x band,” ESA Scientific
Publications, no. SP-677, pp. 1–5, 2009.
[35] Y. Morishita and R. F. Hanssen, “Temporal decorrelation in
l-, c-, and x-band satellite radar interferometry for pasture
on drained peat soils,” IEEE Transactions on Geoscience and
Remote Sensing, vol. 53, no. 2, pp. 1096–1104, 2014.
[36] H. A. Zebker, P. A. Rosen, and S. Hensley, “Atmospheric effects
in interferometric synthetic aperture radar surface deformation
and topographic maps,” Journal of geophysical research: solid
earth, vol. 102, no. B4, pp. 7547–7563, 1997.
[37] T. Wright, B. Parsons, and E. Fielding, “Measurement of
interseismic strain accumulation across the north anatolian fault
by satellite radar interferometry,Geophysical Research Letters,
vol. 28, no. 10, pp. 2117–2120, 2001.
[38] T. J. Wright, B. Parsons, P. C. England, and E. J. Fielding, “Insar
observations of low slip rates on the major faults of western
tibet,” Science, vol. 305, no. 5681, pp. 236–239, 2004.
[39] O. Cavali´
e, C. Lasserre, M.-P. Doin, G. Peltzer, J. Sun, X. Xu,
and Z.-K. Shen, “Measurement of interseismic strain across the
haiyuan fault (gansu, china), by insar,Earth and Planetary
Science Letters, vol. 275, no. 3-4, pp. 246–257, 2008.
[40] D. Schmidt, R. B¨
urgmann, R. Nadeau, and M. d’Alessio, “Dis-
tribution of aseismic slip rate on the hayward fault inferred from
seismic and geodetic data,” Journal of Geophysical Research:
Solid Earth, vol. 110, no. B8, 2005.
[41] M. Furuya and S. Satyabala, “Slow earthquake in afghanistan
detected by insar,Geophysical Research Letters, vol. 35, no. 6,
2008.
[42] H. Dragert, K. Wang, and T. S. James, “A silent slip event on
the deeper cascadia subduction interface,” Science, vol. 292, no.
5521, pp. 1525–1528, 2001.
[43] M. M. Miller, T. Melbourne, D. J. Johnson, and W. Q. Sumner,
“Periodic slow earthquakes from the cascadia subduction zone,
Science, vol. 295, no. 5564, pp. 2423–2423, 2002.
Yujie Zheng received the B.S. degree from
Peking University, Beijing, China, in 2014, and
the Ph.D. degree in Geophysics from Stanford
University, CA, USA, in 2019. She is currently
a Postdoctoral Scholar in the Seismological
Laboratory, California Institute of Technology.
Her main research interests are Interferometric
synthetic aperture radar (InSAR) processing
and InSAR applications in crustal deformation
studies – earthquakes, volcanoes, anthropogenic signals of deforma-
tion such as water management or oil and gas resources.
Howard Zebker (M’87–SM’89–F’99) re-
ceived the B.S. degree from the California
Institute of Technology, Pasadena, CA, USA,
in 1976, the M.S. degree from the University
of California at Los Angeles, Los Angeles,
CA, USA, in 1979, and the Ph.D. degree in
electrical engineering from Stanford Univer-
sity, Stanford, CA, USA, in 1984. He is cur-
rently a Professor of geophysics and electrical
engineering with Stanford University, Stanford, CA, USA, where his
research group specializes in interferometric radar remote sensing. He
was a Microwave Engineer with the NASA Jet Propulsion Laboratory
(JPL), Pasadena, where he built support equipment for the SEASAT
satellite synthetic aperture radar and designed airborne radar systems.
He later developed imaging radar polarimetry, a technique for mea-
surement of the radar scattering matrix of a surface. He is best known
for the development of radar interferometry, leading to spaceborne
and airborne sensors capable of measuring topography to meter scale
accuracy and surface deformation to millimeter scale. More recently,
he has been participating in the NASA Cassini Mission to Saturn,
concentrating on analysis of data acquired by the radar/radiometer
instrument.
Roger Michaelides received the B.A. degree
in physics and science of earth systems from
Cornell University, Ithaca, NY, USA, in 2015,
the Ph.D. degree in geophysics from Stanford
University, Stanford, CA, USA in 2020. He
is currently a Postdoctoral Researcher in the
Department of Geophysics at the Colorado
School of Mines. His research interests include
radar remote sensing applications for terrestrial
and planetary geophysics, especially InSAR algorithm development,
quantifying uncertainties in InSAR retrieval algorithms, and InSAR
applications for studying hydrologic, cryologic, environmental pro-
cesses in periglacial and vegetated landscapes, radar remote sensing
for planetary science applications, and altimtetry and imaging radar
instrument onboard the Cassini mission to Saturn.
... InSAR has been used to study a variety of phenomena including volcanic inflation and deflation, landslides, tectonic deformation across faults, seismic and aseismic fault slip and groundwater-induced deformation [1]- [8]. InSAR time-series algorithms exploit interferogram stacks to reduce the impact of uncorrelated or partially correlated noise in individual InSAR measurements, such as signal decorrelation [9], [10] and atmospheric delay [11], [12]. ...
Article
Full-text available
In this article, we investigate the link between the closure phase and the observed systematic bias in deformation modeling with multilooked SAR interferometry. Multilooking or spatial averaging is commonly used to reduce stochastic noise over a neighborhood of distributed scatterers in interferometric synthetic aperture radar (InSAR) measurements. However, multilooking may break consistency among a triplet of interferometric phases formed from three acquisitions leading to a residual phase error called closure phase. Understanding the cause of closure phase in multilooked InSAR measurements and the impact of closure phase errors on the performance of InSAR time-series algorithms is crucial for quantifying the uncertainty of ground displacement time series derived from InSAR measurements. We develop a model that consistently explains both closure phase and systematic bias in multilooked interferometric measurements. We show that nonzero closure phase can be an indicator of temporally inconsistent physical processes that alter both phase and amplitude of interferometric measurements. We propose a method to estimate the systematic bias in the InSAR time series with generalized closure phase measurements. We validate our model with a case study in Barstow-Bristol Trough, CA, USA. We find systematic differences on the order of cm/year between InSAR time-series results using subsets of varying maximum temporal baselines. We show that these biases can be identified and accounted for.
... Phase history uncertainty assessment: UAVSAR, California −π 0 π −10 0 10 Its evaluation requires information from interferograms not included in the subset, calling for heuristics [18], [31]. ...
Article
Full-text available
Deformation estimation from radar interferometric stacks has to confront speckle over decorrelating distributed targets. Inferring the speckle-induced uncertainty in the estimated phase history is challenging. Previously published estimates based on the Fisher Information can underestimate the errors by an order of magnitude. Here, we introduce three improvements to mitigate the bias. We 1) account for uncertainty in the magnitudes of the interferometric covariance matrix elements; 2) penalize the likelihood to reduce the impact of coherence biases on the phase history uncertainty estimates; 3) constrain the covariance magnitudes to stabilize the estimation. In simulations, these improvements substantially reduced the bias in the uncertainty estimates. The bias reduction was due to an increase in the predicted uncertainty (improvements 1–3) and a decrease in the actual error (improvements 2–3). Temporal correlations – crucial for model fitting and testing – were also estimated more accurately. In observations, the underestimation relative to the observed spatial variability was largely eliminated. In contrast to the alternative estimates based on spatial variability, the improved Fisher Information uncertainty estimates are applicable to small-scale phenomena such as sinkholes. They can serve as foundation for reliable uncertainty estimates of the deformation derived in subsequent interferometric processing steps, thus bolstering model testing and data fusion.
... In keeping with common practice, we only use the diagonal form of Σ t p , , to weight the SBAS inversion. We note that a more accurate form of temporal covariance model accounting for off-diagonal components has been recently proposed by Zheng et al. (2021). The more accurate form would result in higher uncertainty estimates for the SBAS time series, but would not change the static inversion results, as discussed in the next section. ...
Article
Full-text available
From August 2018 to May 2019, Kurn:x-wiley:21699313:media:jgrb54955:jgrb54955-math-0003lauea’s summit exhibited unique, simultaneous, inflation and deflation, apparent in both GPS time series and cumulative InSAR displacement maps. This deformation pattern provides clear evidence that the Halema‘uma‘u (HMM) and South Caldera (SC) reservoirs are distinct. Post-collapse inflation of the East Rift Zone (ERZ), as captured by InSAR, indicates concurrent magma transfer from the summit reservoirs to the ERZ. We present a physics-based model that couples pressure-driven flow between these magma reservoirs to simulate time dependent summit deformation. We take a two-step approach to quantitatively constrain Kurn:x-wiley:21699313:media:jgrb54955:jgrb54955-math-0004lauea’s magmatic plumbing system. First, we jointly invert the InSAR displacement maps and GPS offsets for the location and geometry of the summit reservoirs, approximated as spheroidal chambers. We find that HMM reservoir has an aspect ratio of ∼ 1.8 (prolate) and a depth of ∼ 2.2 km (below surface). The SC reservoir has an aspect ratio of ∼ 0.14 (oblate) and a depth of ∼ 3.6 km. Second, we utilize the flux model to invert GPS time series from 8 summit stations. Results favor a shallow HMM-ERZ pathway an order of magnitude more hydraulically conductive than the deep SC-ERZ pathway. Further analysis shows that the HMM-ERZ pathway is required to explain the deformation time series. Given high-quality geodetic data, such an approach promises to quantify the connectivity of magmatic pathways between reservoirs in other similar volcanic systems.
Article
Full-text available
The Gaussian speckle model for homogeneous distributed targets is commonly assumed to apply in repeat-pass radar interferometric analyses, for instance in deformation estimation. This is despite widespread evidence from snapshot intensity observations indicating deviations from Gaussianity, as many natural land surfaces are intrinsically heterogeneous. The concern is that neglecting heterogeneity will deteriorate the phase estimates and induce underestimation of the uncertainty. Here, we introduce and theoretically characterize compound models that extend the Gaussian speckle model for repeat-pass stacks by representing heterogeneity in intensity and phase. In two L-band repeat-pass data sets, we find pervasive deviations from Gaus-sianity. Our estimates suggest that the heterogeneity in intensity is largely due to time-invariant, rather than dynamic, texture. Deviations from Gaussianity associated with phase heterogeneity are generally less pronounced. One notable exception with large estimated phase heterogeneity occurs over a permafrost wetland, where degrading ice wedges induce subsidence that is variable on the resolution scale. For deformation analyses, accounting for heterogeneity has, on average, a moderate impact on the phase estimates and on the estimated phase uncertainty, which increases by 10% on average. However, in intrinsically heterogeneous areas such as the permafrost wetland, the accuracy of the phase estimate can realistically improve by up to 20%, and the predicted phase uncertainty increase by 30%. The improvements in phase estimation accuracy and in the quality of the uncertainty estimates when accounting for heterogeneous speckle can on occasion make a notable difference for subtle or small-scale deformation.
Article
Full-text available
This article investigates the presence of a new interferometric signal in multilooked synthetic aperture radar (SAR) interferograms that cannot be attributed to the atmospheric or Earth-surface topography changes. The observed signal is short-lived and decays with the temporal baseline; however, it is distinct from the stochastic noise attributed to temporal decorrelation. The presence of such a fading signal introduces a systematic phase component, particularly in short temporal baseline interferograms. If unattended, it biases the estimation of Earth surface deformation from SAR time series. Here, the contribution of the mentioned phase component is quantitatively assessed. The biasing impact on the deformation-signal retrieval is further evaluated. A quality measure is introduced to allow the prediction of the associated error with the fading signals. Moreover, a practical solution for the mitigation of this physical signal is discussed; special attention is paid to the efficient processing of Big Data from modern SAR missions such as Sentinel-1 and NISAR. Adopting the proposed solution, the deformation bias is shown to decrease significantly. Based on these analyses, we put forward our recommendations for efficient and accurate deformation-signal retrieval from large stacks of multilooked interferograms.
Article
Full-text available
Signal decorrelation poses a limitation to multipass SAR interferometry. In pursuit of overcoming this limitation to achieve high-precision deformation estimates, different techniques have been developed, with short baseline subset, SqueeSAR, and CAESAR as the overarching schemes. These different analysis approaches raise the question of their efficiency and limitation in phase and consequently deformation estimation. This contribution first addresses this question and then proposes a new estimator with improved performance, called Eigendecomposition-based Maximum-likelihood-estimator of Interferometric phase (EMI). The proposed estimator combines the advantages of the state-of-the-art techniques. Identical to CAESAR, EMI is solved using eigendecomposition; it is therefore computationally efficient and straightforward in implementation. Similar to SqueeSAR, EMI is a maximum-likelihood-estimator; hence, it retains estimation efficiency. The computational and estimation efficiency of EMI renders it as an optimum choice for phase estimation. A further marriage of EMI with the proposed Sequential Estimator by Ansari et al. provides an efficient processing scheme tailored to the analysis of Big InSAR Data. EMI is formulated and verified in relation to the state-of-the-art approaches via mathematical formulation, simulation analysis, and experiments with time series of Sentinel-1 data over the volcanic island of Vulcano, Italy.
Article
Full-text available
Wide-swath Synthetic Aperture Radar (SAR) missions with short revisit times, such as Sentinel-1 and the planned NISAR and Tandem-L, provide an unprecedented wealth of Interferometric SAR (InSAR) time series. The processing of the emerging Big-Data is however challenging for the state-of-the-art InSAR analysis techniques. This contribution introduces a novel approach, named Sequential Estimator, for efficient estimation of the interferometric phase from long InSAR time series. The algorithm uses recursive estimation and analysis of the data covariance matrix via division of the data into small batches, followed by the compression of the data batches. From each compressed data batch artificial interferograms are formed, resulting in a strong data reduction. Such interferograms are used to link the “older” data batches with the most recent acquisitions and thus to reconstruct the phase time series. This scheme avoids the necessity of re-processing the entire data stack at the face of each new acquisition. It is shown that the proposed estimator introduces only negligible degradation compared to the Cramér-Rao-Lower-Bound. The estimator may therefore be adapted for high-precision Near-Real-Time processing of InSAR and accommodate the conversion of InSAR from an off-line to a monitoring geodetic tool. The performance of the Sequential Estimator is compared to the state-of-the-art techniques via simulations and application to Sentinel-1 data.
Article
Full-text available
Displacements of the Earth's surface can be estimated using differential interferometric synthetic aperture radar. The estimates are derived from the phase difference between two radar acquisitions. When at least three such acquisitions are available, one can compute the displacement between the first and the third acquisition and compare it with the sum of the two intermediate displacements. These two are expected to be equal for a piston-like spatially uniform deformation. However, this is not necessarily the case in measured data. Such lack of phase closure can be due to decorrelation noise alone. It has also been attributed to complex scattering processes such as soil moisture changes or multiple scattering sources. However, the nature of these nonrandom effects is only poorly understood in cold regions, as the role of snow and freeze/thaw processes has not been studied to date. To distinguish the noise-like and the systematic effects, an asymptotic Wald significance test is proposed. It detects situations when the observed closure error cannot solely be explained by noise. Such situations with p < 0.05 are observed at the Ku-band during snow metamorphism and melt and following a summer precipitation event in Sodankylä, Finland. They can also be prevalent (> 25%) in the X-band observations of ice-rich permafrost regions in the Lena Delta, Russia, indicating the presence of processes that can have systematic and deleterious impacts on the estimation of surface movements. Satellite-based monitoring of these displacements is thus possibly subject to complex error sources in high-latitude regions.
Article
Full-text available
Synthetic aperture radar (SAR) tomography has been strongly developed in the last years for the analysis at fine scale of data acquired by high-resolution interferometric SAR sensors as a technique alternative to classical persistent scatterer interferometry and able to resolve also multiple scatterers. SqueeSAR is a recently proposed solution which, in the context of SAR interferometry at the coarse scale analysis stage, allows taking advantage of the multilook operation to filter interferometic stacks by extracting, pixel by pixel, equivalent scattering mechanisms from the set of all available interferometric measurement collected in the data covariance matrix. In this paper, we investigate the possibilities to extend SqueeSAR by allowing the identification of multiple scattering mechanisms from the analysis of the covariance matrix. In particular, we present a new approach, named “Component extrAction and sElection SAR” algorithm, that allows taking advantage of the principal component analysis to filter interferograms relevant to the decorrelating scatterer, i.e., scatterers that may exhibit coherence losses depending on the spatial and temporal baseline distributions, and to detect and separate scattering mechanisms possibly interfering in the same pixel due to layover directly at the interferogram generation stage. The proposed module allows providing options useful for classical interferometric processing to monitor ground deformations at lower resolution (coarse scale), as well as for possibly aiding the data calibration preliminary for the subsequent full-resolution interferometric/tomographic (fine scale) analysis. Results achieved by processing high-resolution Cosmo-SkyMed data, characterized by the favorable features of a large baseline span, are presented to explain the advantages and validate this new interferometric processing solution.
Article
The quantity and quality of satellite-geodetic measurements of tectonic deformation have increased dramatically over the past two decades improving our ability to observe active tectonic processes. We now routinely respond to earthquakes using satellites, mapping surface ruptures and estimating the distribution of slip on faults at depth for most continental earthquakes. Studies directly link earthquakes to their causative faults allowing us to calculate how resulting changes in crustal stress can influence future seismic hazard. This revolution in space-based observation is driving advances in models that can explain the time-dependent surface deformation and the long-term evolution of fault zones and tectonic landscapes.
Article
Repeat-pass Interferometric Synthetic Aperture Radar (InSAR) provides spatially dense maps of surface deformation with potentially tens of millions of data points. Here we estimate the actual covariance structure of noise in InSAR data. We compare the results for several independent interferograms with a large ensemble of GPS observations of tropospheric delay and discuss how the common approaches used during processing of InSAR data affects the inferred covariance structure. Motivated by computational concerns associated with numerical modeling of deformation sources, we then combine the data-covariance information with the inherent resolution of an assumed source model to develop an efficient algorithm for spatially variable data resampling (or averaging). We illustrate these technical developments with two earthquake scenarios at different ends of the earthquake magnitude spectrum. For the larger events, our goal is to invert for the coseismic fault slip distribution. For smaller events, we infer the hypocenter location and moment. We compare the results of inversions using several different resampling algorithms, and we assess the importance of using the full noise covariance matrix.
Article
With three coherent synthetic aperture radar images, it is possible to form three interferograms. In some cases, the phases of the three averaged interferograms will be significantly inconsistent and indicate a sort of phase excess or deficit (which we call lack of triangularity or inconsistency). In this paper, we illustrate theoretically which models can explain such phenomenon and provide some real-data examples. It is also shown that two or more independent scattering mechanisms are necessary to explain phase inconsistencies. The observation of lack of consistency might be useful to derive information on the target and as a warning that the scatterer presents a temporal covariance matrix, which is not intrinsically real, with consequences for the processing of interferometric stacks.
Article
Interferometric Synthetic Aperture Radar (InSAR) time-series methods estimate the spatio-temporal evolution of surface deformation by incorporating information from multiple SAR interferograms. While various models have been developed to describe the interferometric phase and correlation statistics in individual interferograms, efforts to model the generalized covariance matrix that is directly applicable to joint analysis of networks of interferograms have been limited in scope. In this work, we build on existing decorrelation and atmospheric phase screen models, and develop a covariance model for interferometric phase noise over space and time. We present arguments to show that the exploitation of the full 3D covariance structure within conventional time-series inversion techniques is computationally challenging. However, the presented covariance model can aid in designing new inversion techniques that can at least mitigate the impact of spatial correlated nature of InSAR observations.