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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

2Lρij 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 ˜

Ωifg that is derived from a SAR

coherence matrix Ωsar and the InSAR incidence matrix

A(Deﬁnition of Acan be found in [19]): ˜

Ωifg =

1

2AΩsarAT. 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

p1−ρij ·fcov(φdecor

ij , φdecor

kl )·σkl

√1−ρkl

(6)

where the scaling factor σij /p1−ρij 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

2p1−ρij √1−ρkl

(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= (√ρ∞C+p1−ρ∞Di)eiψi(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. ρ∞acknowledges 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(ψi−ψj).(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−|ti−tj|/τ, (12) takes the form of the generic

decorrelation model presented in [34], [35].

ρ(t) = ρ∞+ (1 −ρ∞)e−t

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(ψi−ψj)(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(ψi−ψj)

(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 −ρ∞2)ei(ψi−ψj−ψk+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 −ρ∞2

1−ρ∞2(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)

1−γ(φdecor

ij , φdecor

kl )≈1−γ(zij , zkl )1

2(18)

Combining (17)) and (18), we have

γ(φdecor

ij , φdecor

kl )=1−s1−ρikρjl

1−ρ2

∞

(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) = Cov(Φatm) + C ov(Φdecor )(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 Cov(Φdecor).

Assuming no atmospheric noise is present, we can prop-

agate measurement uncertainties to uncertainties in the

estimated parameters:

Cov(P) = W Cov(Φdecor)WT(23)

So far we have described four models for

Cov(Φdecor): 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 Cov(Φdecor)

We are interested in determining σ2(P)for both

stacks. Adopting the Cramer-Rao bound [33]

σ2(φij ) = 1−ρ2

ij

2ρ2

ij

as phase variances, we construct Cov(Φdecor)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 ρ∞on

Cov(Φdecor). 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/∆t≈6while 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 ρ∞or 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) ρ∞and (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 Cov(Φdecor)

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

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0.2

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1

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0.2

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0 500 1000 1500

0.2

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D

C

D

A

A

B

C

B

(a) (b) (c)

Time span, days

Coherence

⇢

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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

ρ∞of 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 ρ∞of 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

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0.5

0.6

Prediction Error, [rad]2

0 0.1 0.2 0.3 0.4 0.5 0.6

0

0.1

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0.3

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Data

Agram-Simons

Hanssen

Samiei-Esfahany

Proposed

0 0.1 0.2 0.3 0.4 0.5 0.6

0

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Data

Agram-Simons

Hanssen

Samiei-Esfahany

Proposed

Cascadia Death Valley

2(rp),[rad]2

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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

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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

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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/∆t≈1or 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

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-2

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Phase, rad

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-1

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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

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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. Di’s can be represented as

Di=ρd

i−1,iDi−1+q1−ρd

i−1,i

2ni(27)

where niis a complex Gaussian random variables with

an expected intensity of unity, and that is uncorrelated

with D’s and n’s 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(ψi−ψj−ψk+ψl)

=(1 −ρ∞)2E(Rij R∗

kl)

+ρ∞(1 −ρ∞)(ρd

ik +ρd

jl )ei(ψi−ψj−ψk+ψl)

=(1 −ρ∞)2ρd

ikρd

jl

+ρ∞(1 −ρ∞)(ρd

ik +ρd

jl )ei(ψi−ψj−ψk+ψl)

= (ρikρj l −ρ2

∞)ei(ψi−ψj−ψk+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 1−γ(φdecor

ij , φdecor

kl )≈1−γ(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

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42.5°N

-5

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124.5°W 123.5°W 122.5°W 121.5°W

47.5°N

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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.

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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.