Page 1

Coherence estimation in synthetic aperture radar data

based on speckle noise modeling

Carlos López-Martínez and Eric Pottier

In the past we proposed a multidimensional speckle noise model to which we now include systematic

phase variation effects. This extension makes it possible to define what is believed to be a novel coherence

model able to identify the different sources of bias when coherence is estimated on multidimensional

synthetic radar aperture (SAR) data. On the one hand, low coherence biases are basically due to the

complex additive speckle noise component of the Hermitian product of two SAR images. On the other

hand, the availability of the coherence model permits us to quantify the bias due to topography when

multilook filtering is considered, permitting us to establish the conditions upon which information may

be estimated independently of topography. Based on the coherence model, two coherence estimation

approaches, aiming to reduce the different biases, are proposed. Results with simulated and experimental

polarimetric and interferometric SAR data illustrate and validate both, the coherence model and the

coherence estimation algorithms.© 2007 Optical Society of America

OCIS codes:

280.6730, 120.3180, 120.5410, 030.6140.

1.

In multidimensional synthetic aperture radar (SAR)

imagery, the complex correlation coefficient has been

revealed as an important source of information. In

particular, the correlation coefficient amplitude, named

coherence, apart from depending on the SAR system

characteristics, is also influenced by the physical

properties of the area under study. The complex cor-

relation coefficient is the most important observable

for SAR interferometry (InSAR).1On the one hand,

and considering the acquisition geometry, it has been

demonstrated that its phase contains information

about the Earth’s surface topography. Therefore,

InSAR phase data are employed to derive digital el-

evation models (DEMs) of the terrain. On the other

hand, despite there not being a complete understand-

ing about the parameters and the physical processes

affecting the interferometric coherence, it has been

shown that this parameter may be successfully em-

ployed to characterize the properties and the dynam-

Introduction

ics of the Earth’s surface. In Ref. 2, interferometric

coherence was employed to detect the descent paths

of pyroclastic flows after the eruption of the Unzen

volcano. Coherence has also been employed for the

study and retrieval of stem volume over forested ar-

eas.3In Ref. 4, techniques based on interferometric

coherence were found to be a good alternative to op-

tical techniques in forestry applications. Diverse

studies have also demonstrated the usefulness of the

interferometric coherence for the study of dry5and

wet6snow covered areas, as well as for the analysis of

ice covered rivers.7The interferometric coherence is

also helpful for the characterization of glaciers, val-

leys, and fjord ice, as shown in Ref. 8. Weydahl8

demonstrated that coherence is important for the de-

tection of spatial details that are not visible in am-

plitude measurements.

The coherence represents an important source of

information when polarimetric SAR (PolSAR) data

are addressed. In particular, the complex correlation

coefficient parameter derived from circularly polar-

ized data has been employed to characterize rough

surfaces,9to study the sea surface,10or to discrimi-

nate sea ice types.11When obtained from linearly

polarized data, the coherence has been employed to

characterize the forest cover in the Colombian Ama-

zon.12In conjunction with polarimetric techniques,

i.e., polarimetric SAR interferometry (PolInSAR), the

interferometric coherence is employed to retrieve the

height of the forest vegetation13or the crop plants.14

The coherence is also important in diverse aspects

Carlos López-Martínez (carlos.lopez@tsc.upc.edu) is with the De-

partment of Signal Theory and Communications, Universitat

Politècnica de Catalunya, Barcelona, Spain. E. Pottier is with the

Institut d’Electronique et de Télécommunications de Rennes, the

SARPolarimétrieHolographieInterférométrieRadaargrammétrie

Team, Université de Rennes 1, 35042 Rennes Cedex, France.

Received 7 July 2006; accepted 28 August 2006; posted 29 Sep-

tember 2006 (Doc. ID 72787); published 17 January 2007.

0003-6935/07/040544-15$15.00/0

© 2007 Optical Society of America

544APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 2

related to the processing of InSAR data such as

phase unwrapping,15DEMs quality assessment,16or

assessment of the SAR system itself.17

All the techniques listed in the previous paragraph

rely on a correct estimation of the coherence param-

eter. The estimated coherence values are overesti-

mated, especially for low coherence values.18Under

the homogeneity hypothesis, the coherence accuracy

and bias depend on the extent of the averaging or

estimation process in such a way that the larger the

number of averaged pixels, the higher the coherence

accuracy and the lower the bias. Therefore since co-

herence accuracy is achieved at the expense of spatial

resolution and spatial details, this point represents a

clear trade-off for coherence estimation. Coherence

estimation techniques also rely on the hypothesis

that all the signals involved in the estimation process

are stationary and, in particular, are locally station-

ary processes. When this is not the case, biased co-

herence values result.18Hence a lack of signal

stationarity can be considered as a second source of

bias for coherence estimation. The departure of the

stationarity condition may be induced by systematic

phase variations mainly due to the terrain topogra-

phy, but also to atmospheric effects or to deformation

gradients. As demonstrated in Ref. 19, the most re-

liable technique to eliminate this bias is to compen-

sate for the topography by means of external DEMs.

Nevertheless, the DEM may not be available for the

scene under study, or its quality may be rather low for

coherence estimation purposes. Alternative coherence

estimation techniques exist that aim to solve these

problems with different levels of success.18–21

Our objective is to employ some recent advances in

speckle noise theory to improve coherence estima-

tion, with special emphasis on InSAR applications. In

Refs. 22 and 23, the authors derived a novel speckle

noise model for characterization of the complex Her-

mitian product of two SAR images. On the basis of

this noise model, coherence estimation by means of

multilook techniques is considered analytically. Here

we establish which conditions are necessary to obtain

an unbiased coherence estimation. In Section 3 we

present two alternatives to estimate interferometric

coherence, which are analyzed in Section 4. Conclu-

sions are presented in Section 5.

2.

An InSAR system acquires two SAR images from

slightly different positions in space, denoted in the

followingasS1andS2,respectively.Theseimagescan

be jointly acquired (single-pass InSAR), or they may

be obtained at different times (repeat-pass InSAR).

Before generating the complex interferogram, it is

necessary to coregister both SAR images and to range

filter the noncommon parts of the spectra of S1and S2

to increase coherence.24Coherence is then defined as

Synthetic Aperture Radar Coherence Model

??????ej?x??

?E?S1S2

?E??S1?

*??

2?E??S2?

2?

, (1)

where E?x? is the expectation value, |z| represents

the absolute value of z, and the processes S1, S2, and

S1S2

To obtain coherence it is necessary to estimate the

values of the different expectation values involved in

Eq. (1) by means of the ensemble average. Neverthe-

less, since there are not multiple realizations of the

SAR images S1and S2, it is also required that S1, S2,

and S1S2

ble averages in the realizations space by the space

averages in the image space. Hence it is possible to

estimate coherence by means of

*are assumed stationary and jointly stationary.18

*be ergodic in mean to substitute the ensem-

??MLT?

?

??m?1

N?S1?m, n??

M?n?1

N

S1?m, n?S2

2?m?1

*?m, n??

M?n?1

??m?1

M?n?1

N?S2?m, n??

2,

(2)

where m and n refer to the image dimensions and M

and N are the number of averaged pixels in each

dimension. |?MLT| receives the name of the multilook

coherence estimator and it corresponds to the maxi-

mum likelihood estimator of |?|.25Under the as-

sumption that the SAR images S1and S2may be

described by circular complex Gaussian probability

density functions (pdfs), the statistics of |?MLT| have

been completely characterized.18,26The main draw-

back of |?MLT| is that it overestimates coherence,

especially for low coherence values and for small val-

ues of MN, that is, small averaging windows. A sec-

ond problem that may appear for large windows is

that the processes S1, S2, and S1S2

ary within the analysis window, resulting in mean-

ingless estimated coherence values.

A meaningful coherence estimation is restricted to

those cases in which S1, S2, and S1S2

processes within the analysis window. Nevertheless,

this hypothesis cannot be fulfilled if the SAR images

S1 and S2 present a systematic phase variation.

InSAR data represent the best example of this prob-

lem, since the SAR images differ by the topographic

induced phase ?x. When the topographic phase com-

ponent ?xis no longer constant within the averaging

window, |?MLT| results in biased coherence values.

Other sources of signal nonhomogeneities can be at-

mospheric effects or terrain deformations. The topo-

graphic induced phase ?xcan be derived by means of

external DEMs making possible its compensation in

Eq. (2):

*can be nonstation-

*are stationary

??PHCDEM?

?

??m?1

M?n?1

M?n?1

N?S1?m, n??

N

S1?m, n?S2

2?m?1

*?m, n?e?j?x?

M?n?1

??m?1

N?S2?m, n??

2.

(3)

ThemaindrawbackofEq.(3)isthatthequalityofthe

external DEM can be rather low, or there may be no

DEM at all, making the phase compensation process

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS545

Page 3

not possible. When such maps do not exist, there is

still an alternative to estimate coherence without the

topographic induced bias. In Ref. 27, the authors pro-

posed to model topography by means of a plane and

then to maximize coherence with respect to this

plane. This methodology was improved in Ref. 19 by

considering more complex phase models

??PHCmodel?

?

max

?x??m?1

??m?1

M?n?1

N

S1?m, n?S2

*?m, n?e?jf??x??

M?n?1

N?S1?m, n??

2?m?1

M?n?1

N?S2?m, n??

2,

(4)

where the function f??x? represents the local phase

model, depending on the parameter vector ?x. Nev-

ertheless, in Ref. 19 it was agreed that the estimator

coherence estimator [Eq. (4)] presents the handicap

that f??x? may not model topography correctly, result-

ing in coherence biases, and concluding that the best

coherence estimator is the one in which phase is com-

pensated by means of external data [Eq. (3)].

An alternative to avoid topographic induced biases

is to derive topography independent coherence esti-

mators. In Ref. 20, an intensity based coherence es-

timator was proposed based on

RINT?

?m?1

N?S1?m, n??

M?n?1

N?S1?m, n??

2?S2?m, n??

4?m?1

2

??m?1

M?n?1

M?n?1

N?S2?m, n??

4,

(5)

where coherence is obtained as

??INT????2RINT?1

0

RINT?1?2

RINT?1?2.

(6)

AsobservedinEq.(5),theestimator|?INT| isbasedon

the SAR images high-order moments. The main dis-

advantage of Eq. (6) is that it presents a reduced sta-

tistical confidence compared with estimators based on

the complex SAR images directly. In Ref. 19, it was

also concluded that coherence estimators based on

high-order moments are characterized by a low statis-

tical confidence.

A.

As observed in Eqs. (1) and (2), the coherence pa-

rameter is basically determined by the complex

Hermitian product of SAR images S1and S2. Con-

sequently, analysis of the coherence value needs to be

performed on the basis of a first study of this complex

Hermitian product. In the frame of PolSAR data,22,23

and under the hypothesis that SAR images may be

characterized by correlated circular, complex, Gauss-

ian pdfs, we have introduced and validated a mul-

tilook data model to describe the effects of speckle

noise in the complex Hermitian product of two com-

Complex Hermitian Product Speckle Noise Model

plex SAR images. The speckle noise model is pre-

sented in the following, whereas details about its

analysis can be found in Refs. 22 and 23.

The complex Hermitian product of SAR images S1

and S2can be written as follows:

S1S2

*??S1S2

*?ej??1??2??zej?,(7)

where ?1and ?2represent the phase of S1and S2,

respectively, z denotes the Hermitian product ampli-

tude, and ? is the phase difference. Under the hypoth-

esis that the SAR images are statistically described by

zero-mean complex Gaussian pdfs, the multilook ex-

pression of Eq. (7) may be modeled as follows:

?S1S2

*?n??nmexp?j?x?

Ç

Multiplicative term

???????Ncz ?n?exp?j?x????nar?jnai?

Ç

Additive term

,

(8)

where the symbol ? ?nstands for the n sample average

or multilook. The following list details the different

parameters of Eq. (8).

? denotes the average power in the two chan-

nels,

Y

???E??Si?

2?E??Sj?

2?.(9)

Y

Y

?xcorresponds to the average phase of ?.

Nctakes the expression

???2F1?

Nc???n?1⁄2???3⁄2?

??n?

3

2?n,

1

2; 2;???

2?. (10)

As has been demonstrated in Ref. 28, parameter Nc

contains the same information as the coherence.

Y

Considering Eq. (7), z ?nrepresents the normal-

ized Hermitian product amplitude, that is, z ?n ?

E?z???.

Y

The first speckle noise component is given by

nm, which is characterized by the following mo-

ments:

E?nm??Ncz ?n,(11)

?nm

2?Nc

2?1????

2?

2n

. (12)

This noise component presents a multiplicative

nature that is dominant for high coherence data.

Consequently, this term is referred to as the mul-

tiplicative term of the model presented by Eq. (8). If

the Hermitian product is constructed with the same

image, i.e., SS*, this component reduces to the clas-

sical multiplicative noise model for the SAR image

intensity.

546APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 4

Y

The second speckle noise component corre-

sponds to the complex additive noise nar? jnai, which

is characterized by the moments

E?nar??E?nai??0,(13)

?nar

2??nai

2?

1

2n?1????

2?1.32?n. (14)

In contrast with the multiplicative term indicated in

the previous point of this list, the additive term of the

model presented by Eq. (8) makes reference to that

part of Eq. (8) not affected by the speckle noise com-

ponent nm.

For further analysis, it must be taken into account

that the components nmand nar? jnaiare not com-

pletely uncorrelated.22Nevertheless, this correlation

can be neglected due to its small value. We direct

readers interested in the details of Eq. (8), as well as

in the different consequences this model has in the

estimation of information, to Refs. 22 and 23. In

short, one can affirm that in the case of low coher-

ences, the stochastic behavior of the Hermitian prod-

uct [Eq. (8)] is determined by the complex additive

speckle noise component. On the contrary, speckle in

high coherence areas is determined by the multipli-

cative noise component nm. In this situation, it must

also be considered that the multiplicative speckle

component is multiplied by the complex phase term

exp?j?x?. This modulation implies that the final

speckle noise nature, in case of high coherences, will

also depend on the phase information, in such a way

that in some cases, high coherence areas may present

a strong additive speckle noise behavior.

B.

Equation (8) has been derived on the basis that the

true phase component ?xis constant. Consequently,

the model is not sensitive to systematic phase varia-

tions, as for instance the topographic phase compo-

nentpresentinInSARdata.Thissectionisdevotedto

analyzing and including these effects in the speckle

noise model [Eq. (8)]. As mentioned in Section 1, a

systematic phase variation can be due to different

natural processes. Nevertheless, if data are con-

sidered locally, these variations may be modeled

accurately by a linear phase term. For 2D data,

systematic phase variations are modeled in what

follows as a separable model in which phase is lin-

ear in each dimension:

Introduction of Systematic Phase Variations

?x?m, n??2?

sx

m?2?

sy

n??x0, (15)

where sxand syrepresent the spatial phase periods in

each orthogonal dimension and ?x0 is a constant

phase term. Consequently, the orthogonal space fre-

quencies are defined as

?x?2?

sx, (16)

?y?2?

sy. (17)

For single-look data, the only task to perform is to

substitute the constant phase term of Eq. (8), partic-

ularized to the single-look case, by the model pre-

sented in Eq. (15). However, the key study here is to

determine the effects of the linear phase components

when data are filtered, that is, when the Hermitian

product of SAR images, modeled as in Eq. (8), is fil-

tered by Eq. (2).

A multilook filter, as employed in Eq. (2), presents

the following impulse response:

h?m, n??

1

MN?

p?1

M

?

q?1

N

??p?m???q?n?, (18)

with a Fourier transform equal to

H??x, ?y??

1

M

sin?

sin?

M

2?x?

?x

2?

1

N

sin?

sin?

N

2?y?

?y

2?

, (19)

where ??k? is the Kronecker ? function and ?xand ?y

indicate the spatial frequencies. To determine the

effects of the multilook filter, it is necessary to calcu-

late the convolution of the single-look speckle noise

model [Eq. (8)], with the impulse response [Eq. (18)].

Since Eq. (8) contains random signal terms, this anal-

ysis must be performed in the frequency domain, con-

sidering the product of the spectral density function

of Eq. (8) with Eq. (19). Consequently, it is first nec-

essary to derive the spectral density function of Eq.

(8), where the correlation between the first and the

second additive terms is neglected, due to its small

value, to simplify the analysis. As a result, the anal-

ysis expressed above may be done separately for ev-

ery additive term of Eq. (8). Additionally, and without

loss of generality, the constant phase term ?x0can be

considered equal to zero.

The autocorrelation function of the first additive

term of Eq. (8),

u1?x, y???nmexp?j?x?x, y??, (20)

under the hypothesis of spatially uncorrelated speckle

is equal to

ru1u1?k, l???2Nc

2?z ?n

2??1????

2?

2n

??k, l??

?ej??2??sx?k??2??sy?l?,(21)

giving as a result the following spectral density func-

tion:

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS547

Page 5

Su1u1??x, ?y???2Nc

2?1????

2?

2n

sx????y?2?

??2Nc

2z ?n

sy?.

2?2??2

????x?2?

(22)

The details of the process resulting in Eqs. (21) and

(22) are given in Appendix A. The spectral density

function of the filtered Hermitian product first term,

v1?x, y?, is obtained by multiplying Eq. (22) by the

square of the amplitude of Eq. (19):

Sv1v1??x, ?y???2Nc

2?1????

2?

2n

??

1

M

sin?

sin?

2?2??2?2???x?

M

2?x?

?x

2?

1

N

sin?

sin?

N

2?y?

?y

2??

sx????y?

2

??2Nc

2z ?n

2?

2?

sy?,

(23)

where

???

1

M

sin?

sin?

M?

sx?

?

sx?

1

N

sin?

sin?

N?

sy??

sy?

?

. (24)

The inverse Fourier transform of Eq. (23) gives as a

result the autocorrelation function

rv1v1?k, l???2Nc

2z ?n

2?2ej??2??sx?k??2??sy?l?

2?1????

2n

1

N?

??2Nc

2?

1

M?

k???M?1?,

?M?1??

l???N?1?, ?N?1??, (25)

where ?

tween ?K and K. If Eq. (25) is compared with Eq.

(21), it can be observed that both equations are con-

ceptually the same except for the spatial correlation

introduced by the multilook processing, given by the

triangle function, and the presence of the coefficient

?. Consequently, the first term of the filtered Hermi-

tian product can be considered as

k??K, K? denotes the triangle function be-

v1?x, y???? exp?j?x?x, y??n?m?x, y?,(26)

where the multiplicative speckle term n?mpresents

the statistical moments

E?n?m??Ncz ?n, (27)

?n?m

2?Nc

2?1????

2nMN

2?

.(28)

Analysis of the second additive term of Eq. (8),

u2?x, y?, is straightforward as it corresponds to the

convolution of a constant value with the impulse re-

sponse [Eq. (18)]. Hence the signal

u2?x, y????????Ncz ?n?exp?j?x?x, y??, (29)

filtered by Eq. (18) results in

v2?x, y?????????Ncz ?n?exp?j?x?x, y??. (30)

The last step in this analysis corresponds to the

study of the multilook filtering of the third additive

term of Eq. (8):

u3?x, y????nar?x, y??jnai?x, y??.(31)

In this case, only the analysis of the real part of Eq.

(31) will be considered, since the imaginary part be-

haves in the same way. Again, under the hypothesis

that speckle noise is spatially uncorrelated, the real

part of Eq. (31) presents the autocorrelation function

r??u3???u3??k, l???

1

2n?1????

2?1.32?n??k, l?, (32)

which spectral density function is

S??u3???u3???x, ?y???

1

2n?1????

2?1.32?n.(33)

Employing the same procedure as was done for the

first additive term of Eq. (8), the spectral density

function of the filtered third additive term is

S??v3???v3???x, ?y??

1

2nMN?1????

??

2?1.32?nNM

2?x?

sin?

1

M

sin?

M

?x

2?

1

N

sin?

sin?

N

2?y?

?y

2??

2

,

(34)

resulting in a filtered term presenting the autocorre-

lation function

r??v3???v3??k, l??

1

2nNM?1????

2?1.32?nNM1

M?

k???M?1?,

?M?1??

1

N?

l???N?1?, ?N?1??.

(35)

The autocorrelation function for the imaginary part

of signal u3?x, y? presents the same expression as Eq.

(35). Hence the filtered signal v3?x, y? can be supposed

to be

548APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 6

v3?x, y????n?ar?x, y??jn?ai?x, y??, (36)

where the additive noise components are character-

ized by

E?n?ar??E?n?ai??0,(37)

?n?ar

2??n?ai

2?

1

2nMN?1????

2?1.32?nMN. (38)

Considering the previous results, one can write the

filtered Hermitian product of two SAR images as

?S1S2

*?nMN???n?mexp?j?x?????????Ncz ?n?exp?j?x?

???n?ar?jn?ai?.(39)

The multiplicative speckle component n?mis now char-

acterized by Eqs. (27) and (28) and it presents a spa-

tial correlation proportional to the dimensions of the

multilooking. In the same way, the real and imagi-

nary parts of the speckle complex additive component

are characterized by Eqs. (37) and (38), and they also

present the same spatial correlation due to the mul-

tilook filter. It is important to consider, at this point,

the consequences of parameter ?, which expression is

given in Eq. (24). The expected value of Eq. (39) is

E??S1S2

*?nMN????E?n?m?exp?j?x?????????Ncz ?n?

?exp?j?x?

???Ncz ?nexp?j?x?????????Ncz ?n?

?exp?j?x?

??????exp?j?x?. (40)

As ? ? 1, this parameter introduces a bias in the

amplitude of the filtered Hermitian product, in such

a way that the amplitude is underestimated with

respect to its actual value. The physical origin of this

bias can be explained as follows. Since the frequency

response of the multilook filter [Eq. (19)] is not equal

to one in all its bandwidth, the modulation intro-

duced by the phase term exp?j?x? produces the

Hermitian product amplitude to be multiplied by ?,

which introduces the bias.

Two possible forms exist to eliminate this bias due

to systematic phase variations. The first method is

based on the use of an estimation of the complex

phase term exp?j?x?, which can be derived from the

own Hermitian product or by means of external

methods. This estimation can be employed, on the

one hand, to compensate for ?xor, on the other hand,

it can be considered in Eq. (24) to eliminate the bias.

As can be deduced from Eqs. (19) and (24) bias ? is

filter dependent. Consequently, the second method to

eliminate bias ? could be to find a filter presenting a

flat frequency response of value one in the bandwidth

of interest. Since it is of importance to have this be-

havior for any value of ?x, the only possibility to

achieve this property is by means of filter bank tech-

niques. The validity of this filtering option has been

already demonstrated in Ref. 28, where a wavelet

filter bank has been considered. The final conclusion

that is possible to extract here is that it is possible to

estimate the Hermitian product amplitude and the

coherence value independently of systematic phase

variation. This is of importance since it implies that it

may be possible to estimate the information of inter-

est, in case of InSAR data, independently of topogra-

phy.

C.

As defined in Eqs. (1) and (2), an exact model for the

coherence parameter needs to consider both the nu-

merator and the denominator jointly. To overcome

the complexity derived from considering the correla-

tion between both components, in what follows they

will be modeled separately to derive a coherence sig-

nal model. The analysis of the numerator is straight-

forward from the previous results. Hence

Coherence Signal Model

?S1S2

*?MN???n?mexp?j?x?????????Ncz ?n?exp?j?x?

???n?ar?jn?ai?. (41)

In Eq. (41), original data are assumed to be single-

look, i.e., n ? 1. The modelization of the denominator

is performed under a criterion of simplicity. This con-

dition is necessary as it permits us to derive an in-

vertible coherence signal noise model allowing the

retrieval of its actual value. Due to the finite averag-

ing of the multilook process, the denominator will

present a certain degree of variance. This component

could be modeled through an additional noise compo-

nent, but to do it would force the necessity to consider

its correlation with the noise components of Eq. (41).

Hence only the mean value of the argument of the

square root is considered. This value equals

E??|S1|2?MN?|S2|2?MN????1????

2

MN?.(42)

The denominator of Eq. (2) is then considered as

??|S1|2?MN?|S2|2?MN? ???1?

1

MN?,(43)

where the dependence on the coherence is neglected

as it has a small influence, even when MN is small.

Introducing Eq. (41) and expression (43) into Eq. (2)

results in

?MLT

?

?n?mexp?j?x????????Ncz ?n?exp?j?x???n?ar?jn?ai?

??1?

1

MN?

.

(44)

Equation (44) has been obtained by considering the

hypothesis MN ? 1 [expression (43)]. If one keeps

this hypothesis, Eq. (44) can be simplified as follows:

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS 549

Page 7

?MLT????? exp?j?x???1?

1

MN?

1?2

?n?ar?jn?ai?,

(45)

wheredetailscanbefoundinAppendixB.Expression

(45) represents the signal model for the complex cor-

relation coefficient characterizing a pair of complex

SAR images. This model has been derived under the

premise that a multilook process has been employed

to filter the signal. Nevertheless, the previous model

can be generalized to consider any type of filter just

taking into account that parameter ? will depend on

the particular filter and the number of looks MN

must be substituted by the effective number of looks

of the resulting data.

Expression (45) is now analyzed to derive an ex-

pression for the coherence characterizing a pair of

complex SAR images. To take just the amplitude of

expression (45) is not possible as it would make it

necessary to consider the amplitude of the complex

additive speckle noise component. However, this pa-

per is simply considering the intensity of expression

(45). Then, the expected value of the intensity of ex-

pression (45) is

MN?

E???MLT?

2?????

2?2??1?

1

?11

MN?1????

2?1.32?MN.

(46)

Expression (46) is able to identify the different

sources of bias when coherence is estimated. The first

source of bias, introduced by parameter ?, has its

origin on the terrain topography and its interaction

with the filtering method employed to estimate co-

herence. As deduced in Eq. (24), parameter ?, in the

case of the multilook filter, always presents a value

less than or equal to 1. This implies that this bias, in

the case of multilooking, induces an underestimation

of the coherence. Nevertheless, this fact does not pre-

vent us from inducing an overestimation in the case

of different filtering approaches. As deduced, this

source of bias does not depend on the coherence value

itself. This is not the case for the second source of

bias, which is due to the additive speckle noise com-

ponent nar? jnai. As one can observe in expression

(46), this source of bias is important for low coher-

ences, even though its effect decreases with the num-

ber of looks. The effect of the additive speckle noise

component is to introduce an overestimation of the

coherence values. Figure 1 shows the effect of this

second source of bias and compares it with the exact

coherence bias.18,26As observed, the bias predicted by

the coherence model is close to this actual value. One

must take into account that the differences may be

explained by the fact that several approximations

have been necessary to derive a simple coherence

model. In the case of high coherences, the proposed

model underestimates the coherence bias helping to

maintain these values unaltered. On the contrary,

the coherence bias is slightly overestimated for low

coherences. As a concluding remark of this subsec-

tion, it is worth mentioning that the coherence model

presented in expression (45) is able to identify and

quantify the different sources of bias found in the

related literature.18

3.

The availability of the expressions given by expres-

sions (45) and (46) makes it possible to define an

algorithm for coherence estimation in multidimen-

sional SAR imagery. Consequently, it is necessary to

compensate for two bias sources. The first source is

the overestimation for low coherence values. From

expression (46), this bias presents the expression

??1?

Coherence Estimation

?speckle

2

1

MN?

?11

MN?1????

2?1.32?MN, (47)

which, as has been demonstrated, is due to the addi-

tive speckle noise component nar? jnai. As has been

presented in Section 2, Eq. (47) slightly overesti-

mates the speckle bias for low coherences. This effect

could be reduced by considering a weighting of Eq.

Fig. 1.True and estimated speckle biases of coherence.

550APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 8

(47) into expression (46). However no weighting has

been considered in what follows to assess the prop-

erties of expression (46).

The second source of bias is the underestimation

for the whole coherence range due to the topographic

bias ?. Before we can detail such an algorithm, it is

necessary to comment on a few aspects about expres-

sions (45) and (46). As observed, the algorithm needs

to operate with |?|2instead of |?| to take advantage

of Eq. (47), as the additive speckle noise source pre-

sents an expected value equal to zero [Eq. (37)]. In

addition, and due to the data stochastic nature, the

process leading to an estimated value of coherence,

which implies the use of the square root function,

needs to control that the estimated value of the

square of coherence is within the dynamic range

?0, 1?. Finally, the bias [Eq. (47)] depends on a previ-

ous estimation of the coherence value. Since this first

coherence is biased, the bias derived from it will be

underestimated. This fact suggests, in a very direct

way, that the algorithm to be proposed must present

an iterative nature. For simplicity reasons, two co-

herence estimation algorithms are considered. The

first one considers only the bias due to speckle,

whereas the second one considers the combined re-

duction of the two bias sources.

A.

The scheme of the algorithm focused on the reduction

of the speckle bias of coherence is depicted in Fig. 2.

In this case, the input coherence corresponds to the

value derived by means of the multilook approach

[Eq. (2)], but squared. From this value, it is straight-

forwardtoestimatethebias[Eq.(47)]consideringthe

dimensions of the multilook filter that is M and N. To

reduce the variance of the estimated nonbiased co-

herence, it is helpful to filter the estimated bias.

There is not a special reason to consider a particular

filter to perform this task, although the use of the

same multilook filter employed to estimate the initial

coherence produces good results, as will be observed

in Section 4. The estimated coherence is derived by

subtracting this bias from the squared coherence.

This estimated value is controlled in such a way that

itsvalueiswithintherange?0, 1?.Therearedifferent

ways to deal with values outside this range. In what

follows, values outside this range are made equal to

the extremes accordingly. The estimated coherence is

finally derived through the square root function. This

process is iterated, as given in Fig. 2, to improve the

estimation of the bias component. Although this al-

gorithm has been considered under the hypothesis

Algorithm for Speckle Bias Reduction

that coherence was obtained with a multilook ap-

proach, other estimation approaches are possible.

The only difference is that when Eq. (47) is used, the

product MN should be substituted by the equivalent

number of looks.

B.

Biases Reduction

Thesecondofthealgorithmspresentedinthissection

considers the combined reduction of biases due to

speckle and due to systematic phase variations. This

new algorithm consists of a variation of the previous

one in order to consider the systematic phase varia-

tion bias component. As shown in Section 2, since this

bias component does not depend on coherence, it

must not be reduced in the iterative loop of Fig. 2.

Therefore,thisnewalgorithmconsistsofsubstituting

the dashed square in Fig. 2 by the processing steps in

Fig. 3.

The first step to take into account is to estimate the

systematic phase variations on the data, that is, the

topographic phase component in the case of InSAR

data. This estimation process can be performed by

any of the alternatives proposed in the literature. In

what follows, the particular approach proposed by

Trouvé et al.,29based on the multiple signal classifi-

cation algorithm, is considered due to its good perfor-

mance. The retrieved information about systematic

phase variations is next employed to calculate the

bias component given by Eq. (24). At this point it

would be possible to use expression (46) to invert the

effects of topography and speckle jointly, but this

process may be complex due to the complex nature of

the equation itself. Hence the following simplification

is considered:

Algorithm for a Combined Speckle and Topographic

??MLT?

2???NBS?

2?2?

1

MN?1???NBS?

2?. (48)

The previous simplifications allow a simple process to

eliminate both bias sources jointly as follows:

??NBS?

2????MLT?

2?

1

MN??

MN

MN?2?1?.(49)

The retrieved coherence value is checked to ensure

that it is within the range ?0, 1?. After this combined

reduction, the speckle bias is reduced iteratively as

depicted in Fig. 2.

Fig. 2.

mation.

Algorithm for speckle bias reduction in coherence esti-

Fig. 3.

phase variations.

Variation for the reduction of biases due to systematic

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS551

Page 9

4.

This section contains, first, a validation process of

the theory and algorithms presented previously. This

process is based on simulated data, granting access to

thetheoreticalcoherence,andphasevaluesemployed

to simulate data. In a second process, the algorithms

are applied to experimental interferometric and po-

larimetric SAR data.

Results

A.

On the basis of the technique presented in Ref. 30,

512 ? 512 pixel correlated SAR images have been

simulated for the whole coherence range. Initial co-

herences have been estimated by using a multilook

approach [Eq. (2)], considering square windows of

side: 3, 5, 7, and 9 pixels. Figure 4 details the esti-

mated coherence values retrieved by the first algo-

rithm, referred to as |?NBS|, of Section 3, as well as

the mean square error (MSE). Values corresponding

to the multilook estimation process are included for

comparison, referred to as biased |?MLT|. As ob-

served, the proposed algorithm is able to reduce the

bias due to speckle for low coherence values in all the

cases. It is clear that, despite the proposed algorithm

being able to reduce the bias, it is not able to cancel it

Simulated Synthetic Aperture Radar Data Results

completely. A complete cancellation of this bias com-

ponent would mean that the input data must present

a standard deviation equal to zero. This is only pos-

sible if the initial coherence is estimated by consid-

ering a multilook process with an infinite number of

samples. From a practical point of view, this would

translate into using large averaging windows, which

are not possible as the homogeneity condition neces-

sary to estimate coherence would not be fulfilled. In

the case of real SAR data, neither the terrain topog-

raphy nor the surface properties can be considered as

homogeneous in large areas. Figure 4 also contains

the MSE of both estimators obtained as

MSE??? ˆ???E???? ˆ??????2??var??? ˆ???b2??? ˆ??,

(50)

where |? ˆ| and |?| indicate the estimated and true

coherence values, respectively, and b?x? represents

the estimation bias. The MSE is better adapted than

the standard deviation measure to study the perfor-

mance of biased estimators as it also takes into

consideration this biased nature.31As expected, no

differences in the MSE are observed for high coher-

ences as the estimators are nonbiased. However, dif-

Fig. 4.

for the speckle bias reduction ???NBS??.

(Left) Mean and (right) MSE values of the estimated coherence values with the multilook algorithm ???MLT?? and the algorithm

552 APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 10

ferences in the MSE are noticed for medium and low

coherences. In the latter case, the bias reduction re-

sults in a clear decrease of the MSE.

If one considers Fig. 4, it can be observed that for

all window sizes, the bias is basically reduced by a

factor equal to 2. This can be interpreted in the fol-

lowing way. For a given level of bias, it is possible to

use the nonbiased coherence estimation algorithm

with a multilook in which the dimensions of the win-

dow may be reduced by a factor of 2. This property is

important as it allows a better preservation of the

image resolution and details. Of course, it must also

be mentioned that the retrieved coherence values

would present the standard deviation values corre-

sponding to the window with reduced dimensions.

Hence the coherence estimation process presents a

clear trade-off between bias reduction and standard

deviation.

In the previous tests, data were simulated consid-

ering a homogeneous phase component in the whole

image. In what follows, a nonhomogeneous phase

component will be introduced into the data to dem-

onstrate the capabilities of the algorithm designed for

a combined speckle and topographic biases reduction.

The true topography corresponds to a phase ramp

that, when wrapped, results in 15 pixel fringes. In all

the cases and for the complete coherence range, to-

pography is estimated from data. Figure 5 details

Fig. 5.

for the speckle bias reduction ???NBS??.

(Left) Mean and (right) MSE values of the estimated coherence values with the multilook algorithm ???MLT?? and the algorithm

Fig. 6.

for the speckle bias reduction ???NBS?? with a 7 ? 7 looks window. Topography is not estimated from data, but offered as an input to the

algorithm.

Mean and standard deviation values of the estimated coherence values with the multilook algorithm ???MLT?? and the algorithm

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS553

Page 11

the retrieved coherence values. When employing a

3 ? 3 pixel window to estimate coherence, it can be

observed as the algorithm is able to reduce the bias

due to topography while it also reduces the speckle

bias for low coherences. As the window size increases,

the algorithm is still able to eliminate completely the

bias due to topography. It is worth noticing that even

when employing a 9 ? 9 pixel window, where the

topographic phase presents an excursion of 6??5 rad

within the window, the algorithm is able to retrieve

the correct coherence. Nevertheless, one can also ob-

serve that for very low coherences, coherence is un-

derestimated. This underestimation is due to the

impossibility of having a reliable estimated topogra-

phy for such low coherences. This limitation is not

duetothecoherenceestimationalgorithmspresented

in Section 3, but to the algorithm in charge of the

topographic phase estimation process. To demon-

strate this, a variation of the algorithm, in which the

true topographic phase is not estimated from the

data, but introduced in the algorithm as an input,

is considered. Figure 6 presents the results when a

7 ? 7 pixel window is considered. As observed, the

proposed algorithm is able to eliminate completely

the bias due to topography and to reduce the bias due

to speckle.

B.

The algorithms presented previously to remove the

different sources of biases when coherence is esti-

mated have also been tested on real SAR data sets.

The algorithm reducing the bias due to speckle for

low coherences has been tested on an experimental

small baseline, 1024 ? 2048 pixel, L-band PolSAR

Experimental Synthetic Aperture Radar Data Results

Fig. 7.Histograms of coherence for the Traunstein image.

Fig. 8.Original and nonbiased estimated coherences based on 3 ? 3 pixel windows.

554APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 12

data set. These data have been acquired by the

Experimental Synthetic Aperture Radar sensor, op-

erated by the Deutsches Zentrum für Luft- und

Raumfahrt, over the Traunstein area located in

southern Germany. Within these data, agricultural,

forested, and urban zones can be identified. To test

the algorithm, only the Hermitian product ShhSvv

been considered. In the previous product, Shhstands

for the scattering coefficient relating the electromag-

netic fields transmitted and received in horizontal

polarization, whereas Svvindicates the same quan-

tity but for electromagnetic fields transmitted and

received with vertical polarization. Figure 7 presents

the histograms of the coherence values obtained with

the multilook approach [Eq. (2)], and the histograms

of the coherence values obtained with the proposed

algorithm, for different sizes of the analysis window.

In all the cases, it may be observed that the histo-

*has

grams are equal for high coherence values, proving

that the proposed algorithm does not alter high co-

herence values, as designed. Nevertheless, these his-

togramspresentcleardifferencesformediumandlow

coherences, which decrease with the dimensions of

the analysis windows as is expected. It is important

to note the peak appearing for a coherence equal to

zero. This peak appears as a consequence of the con-

trol performed on the estimated coherences to avoid

negative values. Since the estimated coherence is it-

self a random variable, it presents a certain degree of

variance, which induces an overestimation of the bi-

ases for those coherences below the mean coherence

value. Since this effect is intrinsic to the estimation

process, it can be reduced or eliminated only by in-

creasing the size of the analysis window. Finally, Fig.

8 presents a detailed image of the coherence term

corresponding to the Hermitian product ShhSvv

the original Traunstein data set of 328 ? 164 pixels.

In this case, coherence has been estimated with a

3 ? 3 multilook approach [Eq. (2)], and the algorithm

proposed to reduce the speckle bias. Table 1 contains

the mean coherence values for three different areas

detailed in Fig. 8, covering all the coherence dynamic

*

of

Fig. 9.

with 7 ? 7 pixel averaging windows.

ERS-1 and ERS-2 tandem interferogram: (a) phase ?, (b) coherence histograms, (c) |?MLT|, (d) |?NBS|. Coherences are obtained

Table 1.Mean Coherence Values for the Areas of Fig. 8

High (H) Medium (M)Low (L)

|?MLT|

|?NBS|

0.8084

0.8014

0.6708

0.6412

0.4631

0.3712

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS555

Page 13

range. As observed, only medium and low coherences

are corrected by the algorithm. This correction, as it

may be observed in Fig. 8, is performed without a loss

of spatial resolution or spatial details.

The second algorithm proposed in Section 3, for a

combined reduction of the speckle and the topographic

biases in coherence estimation, has been tested on a

1024 ? 1024 pixel tandem interferogram acquired by

the ERS-1 and the ERS-2 systems on 30 and 31 July

1999 at the region of Murcia, in eastern Spain. The

interferogram presents a spatial baseline of 249 m.

As observed in the phase image in Fig. 9, the upper

part contains rough topography with low coherence,

whereas the rest of the image corresponds to relatively

smooth topography. The topography has been esti-

mated by the algorithm detailed in Ref. 29. This esti-

mation of the topographic phase has been employed to

correct coherences as detailed in Figs. 2 and 3. In this

case, coherence has been estimated considering an av-

eraging window of 7 ? 7 samples. Figure 9 contains

the original coherence value |?MLT|, the nonbiased

estimation |?NBS|, and the corresponding histograms.

Theaveragevalueof|?MLT|isequalto0.499,whereas

|?NBS| presents an average value of 0.555. As shown,

the proposed algorithm is able to compensate for the

topographic bias.

5.

Speckle noise may be considered as one of the most

important drawbacks when multidimensional SAR

data are considered to study and to analyze the

Earth’s surface. In the past, we introduced and vali-

dated a novel, to our knowledge, multilook, multidi-

mensional speckle noise model. This model has been

able to identify that speckle presents multiplicative

and additive noise components when multidimen-

sional SAR data are considered. Here we considered

an additional generalization of such a model by in-

troducing the effects due to systematic phase varia-

tions. The consequences of these effects depend on

the method employed to estimate data, that is, the

process to remove speckle noise. The multilook ap-

proach has been considered as its simplicity allows

the analytical study of the problem.

The speckle noise model for multidimensional SAR

data has also been employed to perform an in-depth

study of the coherence estimation problem, resulting

in a novel and simple coherence model. This model is

able to identify, in an accurate way, the origin of the

different bias sources when coherence estimation is

considered. As demonstrated, the coherence bias at

low coherence values is due to the complex, additive

speckle noise component. The bias itself has its origin

on the variance of the real and imaginary parts of this

noise component. In addition, as the model takes into

account the effects of systematic phase variations, it

has allowed a quantitative analysis of this second

source of bias in the particular case of the multilook

filter. Thus it has been possible to identify that filters

presenting a flat frequency response equal to one

make it possible to estimate information indepen-

dently of topography.

Conclusions

On the basis of the coherence noise model, two new

algorithms for unbiased coherence estimation have

been introduced. The first method focuses on the re-

duction of the low coherences bias. The algorithm has

been tested on simulated and real SAR data, proving

that it is possible to reduce this bias source without

the loss of spatial resolution or spatial details. Nev-

ertheless, this bias source cannot be completely elim-

inated as it would imply the use of extremely large

analysis windows, with the consequent loss of spatial

resolution. The second algorithm performs a com-

bined reduction of the speckle and the systematic

phase biases without the necessity of external data.

The tests over simulated and experimental data have

demonstrated the performances of the algorithm,

showing also the limitations to estimate very low

coherences. This limitation is not due to the proposed

algorithm but to the method employed to estimate

this systematic phase variation for very low coher-

ence areas.

Appendix A:

Given the first additive term of Eq. (8)

Spectral Density Functions

u1?x, y???nmexp?j?x?,(A1)

and considering the phase model [Eq. (15)], its auto-

correlation function is

ru1u1?k, l?? ?

m???

?

?

n???

?

u1

2?

sx

*?m, n?u1?m?k, n?l?

2?

sy

?nm?m?k, n?l?

??2exp?j?

sx

Under the hypothesis of nonspatially correlated

speckle, the autocorrelation of the multiplicative

speckle noise term nmis

2?z ?n

which can be approximated for single-look data, that

is n ? 1, as

??2exp?j?

k?

l?? ?

m???

?

?

n???

?

nm

*?m, n?

2?

k?

2?

sy

l??rnmnm?k, l?.(A2)

rnmnm?k, l??Nc

2??1????

2?

2n

??k, l??, (A3)

rnmnm?k, l??n?1?Nc

2z ?n

2?1???k, l??.(A4)

The spectral density function of Eq. (A2) is ob-

tained as

Su1u1??x, ?y?? ?

k???

?

?

l???

?

ru1u1?k, l?exp??j?xk?exp??j?yl?

??2?

k???

?

?

l???

?

Nc

2?1????

2?

2n

exp??j???x?2?

??k, l?

??2Nc

???y?2?

2z ?n

2?

k???

sy?l??,

?

?

l???

?

sx?k

(A5)

which reduces to the expression given by Eq. (22).

556APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 14

Appendix B:

Equation (44) can be rewritten as

Coherence Model Simplification

?MLT??n?mexp?j?x?

??1?

??n?ar?jn?ai?

??1?

1

MN?

???????Ncz ?n?exp?j?x?

??1?

1

MN?

1

MN?

.(B1)

For multilook data it is possible to simplify Eq. (B1)

considering MN ? 1. The second additive term of Eq.

(B1) can be neglected under this hypothesis as ob-

served in Fig. 10. Considering Eq. (27) and Fig. 10 it

is possible to prove

?n?mexp?j?x?

??1?

1

MN?

→

MN?1

????exp?j?x?,(B2)

where the variance associated with the multiplicative

speckle noise term is neglected. Introducing the pre-

vious results into Eq. (B1), the simplified model for

coherence presented by expression (45) results.

The authors thank the Deutsches Zentrum für Luft-

und Raumfahrt and the European Space Agency for

providing the experimental data employed in this

study and Pablo Blanco and David Navarrete for their

helpinprocessingtheInSARdata.Thisworkhasbeen

financed by the Spanish Ministry of Education and

Science.

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