cloude 2006 radio science

Polarization coherence tomography

Shane R. Cloude1

Received 14 December 2005; revised 31 March 2006; accepted 12 April 2006; published 25 August 2006.

[1] In this paper we introduce a new radar-imaging technique, called polarizationcoherence tomography (PCT), which employs variation of the interferometric coherencewith polarization to reconstruct a vertical profile function in penetrable volume scattering.We first show how this profile function can be efficiently represented as a Fourier-Legendre series, with tomographic reconstruction reducing to estimation of the unknowncoefficients of this series from coherence data. We then show that we can linearize thisinversion by using a priori knowledge of two parameters, namely, volume depth andtopographic phase. We further propose a new algorithm based on polarimetricinterferometry to estimate these two from the data itself. To assess stability, we investigateboth the single- and dual-baseline conditioning of the associated matrix inversion and thenconcentrate on the single-baseline case to demonstrate that for sufficient multilooking(around 50), stable retrievals of profiles can be obtained in the presence of coherencenoise. Finally, we apply the technique to simulated L band coherent radar data todemonstrate its potential for new applications in radar remote sensing.

Citation: Cloude, S. R. (2006), Polarization coherence tomography, Radio Sci., 41, RS4017, doi:10.1029/2005RS003436.

1. Introduction

[2] Polarimetric Interferometric Synthetic ApertureRadar (POLInSAR) is a new radar-imaging technology[Cloude and Papathanassiou, 1998] with importantapplications in the remote measurement of vegetationproperties such as forest height [Papathanassiou andCloude, 2001] and biomass [Mette et al., 2004] as well asemerging applications in agriculture [Sagues et al., 2000;Williams and Cloude, 2005], snow/ice thickness moni-toring [Papathanassiou et al., 2005a; Dall et al., 2003]and urban height and structure applications [Schneider etal., 2005]. Traditionally only simple two-layer randommedia-scattering models have been used to estimate asmall number of layer parameters from POLInSARdata [Cloude and Papathanassiou, 2003], typicallyvegetation height and underlying surface topography[Papathanassiou et al., 2005b]. In this paper wegeneralize this approach to consider estimation of avertical structure function through the layered mediumunder investigation. As this involves estimation of afunction rather than model parameters, but still usesthe variation of interferometric coherence with polariza-tion, we term it polarization coherence tomography (PCT).

[3] POLInSAR differs from conventional radar inter-ferometry in that it involves generation of interferogramsfor different transmit/receive polarization pairs [Cloudeand Papathanassiou, 1998]. It is then possible to usethe change of interferometric phase with polarizationto extract important bio and geophysical parameters[Papathanassiou and Cloude, 2001]. This is especiallyimportant in the case of remote sensing of vegetated landsurface, where polarimetry alone suffers from the inher-ent high-depolarization problem [Cloude and Pottier,1996] while standard single-channel interferometryremains underdetermined [Treuhaft et al., 1996]. Realiz-ing this, several authors have proposed the processing ofmultibaseline data to reconstruct vertical profilesusing a range of signal processing methods [Reigber,2002; Reigber and Moreira, 2000; Reigber et al., 2001;Treuhaft and Siqueria, 2000; Treuhaft et al., 2000, 2002,2004], often based on high-resolution spectral analysistechniques [Lombardini, 2005; Fornaro et al., 2003],although Treuhaft et al. [2004] employed a multipleGaussian parametric model to reconstruct leaf area index(LAI) profiles by combining radar with hyperspectraldata.[4] While these methods often provide excellent

results, they suffer from an inherent requirement forcollection of a large number of operational baselines.One attraction of POLInSAR is that it enables parameterestimation from only single- or dual-baseline sensorsoperating at a single frequency. This minimizes issues

RADIO SCIENCE, VOL. 41, RS4017, doi:10.1029/2005RS003436, 2006

1AEL Consultants, Fife, UK.

Copyright 2006 by the American Geophysical Union.

0048-6604/06/2005RS003436

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related to temporal decorrelation in repeat pass systemsand matches recent technology developments in multi-sensor deployment [Krieger and Moreira, 2005], whichit is envisioned in the future will be able to providesingle-pass interferometry for a limited number of spatialbaselines. With this as motivation, in this paper weconsider the possibility of performing functional recon-struction or tomography using only a single- or dual-baseline polarimetric interferometer.[5] We will show that we can formulate the tomo-

graphic reconstruction problem using a Fourier-Legendreseries, estimation of the parameters of which can belinearized by exploiting a priori knowledge of volumedepth and ground topography. Estimation of the coeffi-cients of this series then permits reconstruction of anestimated ‘‘band-limited’’ profile. For example, we showthat even for single-baseline sensors we can estimate thisseries up to a second-order Legendre series and henceobtain sensitivity to a variety of different volume-scattering environments. We analyze the stability of thistomographic scheme to noise and provide examples of itsapplication to high-fidelity L band SAR simulations ofvegetated terrain.[6] In section 2 we first review current algorithms used

in POLInSAR to estimate the two important parametersrequired for PCT, namely, height and surface topography.We then propose a new algorithm that requires just twopolarized interferograms. In section 3 we then show howknowledge of these two parameters allows us to formu-late and linearize the tomographic problem as a Fourier-Legendre series expansion. In section 4 we analyze thesingle- and dual-baseline tomographic reconstructionproblems based on this series. In section 5 we estimatethe sensitivity of the single-baseline algorithm to statis-tical fluctuations in coherence estimation. In section 6 we

apply the single-baseline algorithm to a full wave prop-agation and scattering simulation of a random canopyabove ground. Finally in section 7 we provide conclu-sions and a perspective on possible future developmentsof this approach.

2. POLInSAR Height and Topography

Retrieval Algorithms

2.1. Models for the Interferometric Coherence ofRandom Media

[7] The basic geometry of radar interferometry isshown in Figure 1. The presence of distributed scattererson the ground at point P leads to a loss of phasecoherence in the interferogram, formed as a phasedifference between signals at either end of the baseline.[8] The complex interferometric coherence observed

for a random vertical distribution of scatterers can beformulated (following range spectral filtering [Bamlerand Hartl, 1998; Gatelli et al., 1994]) as a ratioof integrals as shown in equation (1) [Papathanassiouand Cloude, 2001, Cloude and Papathanassiou, 2003;Treuhaft et al., 1996]

g ¼ eikzzo

Rhvo

f zð Þeikzzdz

Rhv

o

f zð Þdz¼ eifo

Rhv

o

f zð Þeikzzdz

Rhvo

f zð Þdz; ð1Þ

where zo is the position of the bottom of the scatteringlayer, j0 we call the topographic or ground phase andf (z) we call the vertical structure function (see Figure 1).It is the aim of tomography to reconstruct this functionfrom far field scattered data. Note that f (z) may contain a

Figure 1. Geometry of radar interferometry and definition of the vertical structure function f (z) ata point P on the surface.

RS4017 CLOUDE: POLARIZATION COHERENCE TOMOGRAPHY

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mixture of surface and volume scattering and will ingeneral depend on the polarization of the incident wave.[9] For the case of a uniform random layer of vegeta-

tion, a common assumption is that the structure functionis polarization-independent and can be modeled as amean wave extinction process as shown in equation (2)[Cloude and Papathanassiou, 2003; Treuhaft et al.,1996, 2000]:

f zð Þ / e2scos qz ) ~g ¼ ~gv hv; sð Þeifo ; ð2Þ

where s is the one-way extinction coefficient, q the angleof incidence and ~gv is called the volume decorrelation.This leads to a commonly used model for the complexinterferometric coherence of a layer of vegetation, whichis independent of polarization and can (in the absence oftemporal and SNR effects) be obtained by inserting (2)into (1) and expressed in the form shown in equation (3)[Papathanassiou and Cloude, 2001; Cloude andPapathanassiou, 2003]:

~gv ¼ eifop

p1

ep1hv � 1

ephv � 1where

p ¼ 2scos q

p1 ¼ pþ ikz

kz ¼4pDql sin q

8>>>>>><>>>>>>:

: ð3Þ

In inverting this equation, the desired unknown vegeta-tion parameters obtained are the vegetation height hv ,ground topography jo and mean wave extinction s,while the known system parameters (see Figure 1) are thewavelength l,q the angle of incidence and Dqthe apparent angular separation of the baseline from thescattering point P. We see that even in this simple case,parameter estimation is ambiguous, as we have threeunknown parameters on the right (hv , s and jo) and onlytwo observables (complex coherence) on the left. For thisreason single-channel interferometry is not well suited toparameter estimation, at least in the absence of externalconstraints (a priori knowledge of one or moreparameters for example). Instead we look to multichannelinterferometry as a more robust way to provide the extrainformation required to enable parameter estimation. Oneimportant way to diversify the observation vector is toobtain interferograms for multiple polarizations [Cloudeand Papathanassiou, 1998]. This is attractive because aswe diversify the polarization some gross structuralparameters (e.g., height and topography) remain constantwhereas others, for example wave extinction [Bessetteand Ayasli, 2001], can change. By exploiting this‘‘selective diversity’’ we can then target the unknownsusing parameter estimation techniques. Before consider-ing such an approach we must however consider oneimportant complicating factor in equation (3), namely,the presence of a ground surface under the vegetation.

[10] In practice equation (3) is found to be too simpleto describe the observed coherence variations of naturalterrain [Papathanassiou and Cloude, 2001; Cloude andPapathanassiou, 2003; Treuhaft and Siqueria, 2000]. Inparticular, it has been suggested that the observed vari-ation of coherence with polarization in real data is dueprimarily to the effects of the surface beneath thevegetation, with polarization variations in extinctionbeing a secondary effect [Papathanassiou and Cloude,2001; Cloude and Papathanassiou, 2003]. In this ran-dom-volume-over-ground (RVOG) case, the model ofequation (3) has to be extended to a two-layer expressionas shown in equation (4) [Papathanassiou and Cloude,2001; Cloude and Papathanassiou, 2003; Treuhaft et al.,1996] where m is the (real) ratio of surface-to-volumescattering and w is a vector representing the choice ofpolarization [Cloude and Pottier, 1996; Cloude andPapathanassiou, 1998]. This model predicts a linearvariation of complex coherence with polarization vectorw [Papathanassiou and Cloude, 2001; Cloude andPapathanassiou, 2003] that has been validated, mainlyat L and P bands, for a wide range of forest types[Mette et al., 2004; Papathanassiou et al., 2005b]. Notethat in equation (4) the phase jo is the underlyingground topography, which is not the same as the ob-served interferometric phase due to the presence of thecomplex volume decorrelation term in the numerator.This is termed vegetation bias and bias removal is animportant challenge in radar interferometry over vege-tated landscapes:

~g wð Þ ¼ eifo~gv hv; sð Þ þ m wð Þ

1þ m wð Þ

¼ eifo ~gv þ L 1� ~gvð Þð Þ ; L ¼ m wð Þ1þ m wð Þ : ð4Þ

This approach can be further extended to three layers tomodel finite crown depth d, that is, to accommodate agap between the ground surface and bottom of thecanopy via a phase offset as shown in equation (5)[Cloude and Papathanassiou, 2003]:

~g wð Þ ¼ eifoeikz hv�dð Þ~gv d; sð Þ þ m wð Þ

1þ m wð Þ¼ eifo eid~gv þ L 1� eid~gv

� �� : ð5Þ

This does not change the linear coherence model but

adds an extra unknown parameter to the problem. For our

purposes inclusion of d allows us to investigate

sensitivity of algorithms to structure in a simple way.


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[11] These models lead to several different strategiesfor algorithm design in vegetation parameter estimationas we now consider.

2.2. Coherence Model Inversion

[12] The basic idea is to invert a model M relatingstructural parameters to observed coherences in multiplepolarization channels. We can then write the estimationproblem formally as shown in equation (6).

p ¼ M�1o: ð6Þ

When we employ equation (4) as the model M andconsider a dual-polarization channel measurement sys-tem, one of which, wv, we assume is dominated byvolume scattering so mv � 0 (often linear crosspolarization HV for example) and the second ws containsan unknown mixture of surface and volume so that ms > 0(often HH for example), then our model can be rewrittenin the explicit form shown in equation (7):

~gwv¼ eifo~gv hv; sð Þ

~gws¼ eifo ~gv hv; sð Þ þ msð Þ

1þ ms

9>>=>>;) p ¼

fo

hv

s

ms

0BBBBBBBBBB@

1CCCCCCCCCCAo ¼

~gwv

~gws

0@

1A:

ð7Þ

Subject to the underlying assumptions (zero m in wv andno polarization dependence of extinction), we are now ina position to consider inversion of the model forparameter estimation, having four unknowns and fourobservables in equation (7). For our purposes, weconsider estimation of the two parameters key to PCT,namely, topographic phase j0 and height hv.

2.3. Estimation of Topographic Phase

[13] The first parameter of interest is the ground phasejo, that is, to remove the vegetation bias. The simplestway to do this is to find a polarization channel for whichms 1, called ws, in which case the ground phase is justthe phase of the interferogram formed in this channel asshown in equation (8) [Cloude and Papathanassiou,1998]:

f ¼ arg ~gwS

� �: ð8Þ

There have been several strategies proposed to findws, including coherence optimization [Cloude andPapathanassiou, 1998; Colin et al., 2003] and super-resolution using the ESPRIT algorithm [Yamada et al.,

2001]. However, one problem with this approach is thehigh signal depolarization encountered in radar polarime-try [Cloude and Pottier, 1996] which implies thatscattering in all polarization channels w contains somevolume component. Hence an alternative strategy is toaccept some finite but unknown ms for ws in equation (7)and then use the second coherence channel wv, this timewith ms � 0, that is, strongly dominated by volumescattering. In this case the phase can be found indirectly byusing the coherences in the two channels in a line fit asshown in equation (9):

~gwv¼ eifo~gv

~gws¼ eifo ~gv þ m2ð Þ

1þ m2

9>>=>>;) eifo ¼

~gwv� ~gws

1� Lwsð Þ

Lws

:

ð9Þ

The phase can then be obtained by solving for Lws, itself

obtained as the appropriate root of a quadratic equation as

shown in equation (10) (the root is chosen to guarantee that

ws has a lower phase center in the volume than wv):

f ¼ arg ~gwV� ~gwS

1� LwSð Þ

� �0 LwS

1

AL2wS

þ BLwSþ C ¼ 0 ) LwS

¼ �B�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � 4AC

p

2A

A ¼ ~gwS

�� 2�1 B ¼ 2Re ~gwV� ~gwS

� �:~g�wS

� �C ¼ ~gwV

� ~gwS

�� 2: ð10Þ

Since this algorithm is more robust against signal

depolarization, we propose use of equation (10) rather

than (8) to calculate the ground phase.

2.4. Height Estimation

[14] Turning now to estimation of height hv, there areagain two main strategies involved. In the simplest case,as an extension of equation (8), we can propose heightretrieval as a phase difference between interferograms.This DEM differencing approach was first proposed byCloude and Papathanassiou [1998] and is shown inequation (11):

hv ¼arg gwV

� �� f

kz; kz ¼

4pDql sin q

; ð11Þ

where now we make use of polarization wv, dominatedby volume scattering (ms � 0). The problem with this isthat even with ms = 0, the phase centre does not lie at thetop of the vegetation unless the extinction tends toinfinity (or the crown depth d in (5) tends to zero). In


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general therefore this approach tends to underestimateheight [Yamada et al., 2001].[15] A second approach is to ignore phase completely

and parameter match the observed coherence amplitudeto that expected for pure volume scattering. With theassumption that d = hv, that is, a full canopy, this leads toa solution as shown in equation (12):

minhv

F ¼ ~gwv

�� p

p1

ep1hv � 1

ephv � 1

��

��

wherep ¼ 2s

cos qp1 ¼ pþ ikz

8<: : ð12Þ

There are two problems with this approach. First we needto know or estimate the mean extinction s in the canopyand secondly this approach is sensitive to verticalstructure variations in crown depth d, which makes itunsuitable for PCT. For shallow crowns (d small) itunderestimates vegetation height, as there is reduceddecorrelation by the limited random volume scattering.[16] To further improve the robustness of the height

estimate to structure, we can instead try and match boththe phase and coherence of the volume-dominated chan-nel to the model [Papathanassiou and Cloude, 2001;Cloude and Papathanassiou, 2003]. This approach leadsto a least squares inversion, where the extinction and

height are varied until the function F in equation (13) isminimized, which we note makes use of the estimate ofground phase from equation (10):

minhv;s

F ¼ ~gwV� eif

p

p1

ep1hv � 1

ephv � 1

��: ð13Þ

This approach requires a computationally expensive 2-Dsearch procedure for each pixel [Papathanassiou andCloude, 2001; Cloude and Papathanassiou, 2003] and isalso susceptible to two main types of error. In the firstcase, if the selected volume channel wv is contaminatedby some surface components and so mv > 0 then (13) nolonger provides a unique solution, but a family ofsolutions given by a parameter 0 k 1 as shown in(14). In case mv > 0 then assuming k = 0, as in equation(13), can lead to an underestimation of height [Cloudeand Papathanassiou, 2003]:

minhv;s

F kð Þ ¼ ~gwvþ k eif2 � ~gwv

� �� eif

p

p1

ep1hv � 1

ephv � 1

��

where

p ¼ 2scos q

p1 ¼ pþ ikz

f2 ¼ arg ~gws� ~gwv

1� LwVð Þ

� �: ð14Þ

8>><>>:

Figure 2. Illustrating height estimation from coherence using an inverse sinc function as proposedin equation (15) (for kz = 0.12).


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The other potential problem we face is the presence ofstructure (finite crown depth d in (5)). In case d tends tozero, application of equation (13) leads to error mainly inthe extinction estimate. In effect the two-parametermodel fit tries to compensate the high phase centre andhigh coherence associated with small crowns by using avery high extinction. Fortunately the height estimateobtained from (13) seems not very sensitive to suchstructure variations and remains a fairly robust parameter,showing residual errors of the order of 10% or so[Papathanassiou and Cloude, 2001; Papathanassiou etal., 2005b].[17] We have seen that the most robust strategy for

height estimation is to employ both the phase andcoherence amplitude in the inversion process. However,full model-based inversion, although robust to structurevariations, still has some disadvantages related to com-putational complexity [Cloude and Papathanassiou,2003]. With this in mind, we can use the relativeinsensitivity of coherence to extinction to devise asimpler yet robust phase and coherence algorithm. Ournew proposed algorithm is shown in equation (15). Theheight estimate in equation (15) comprises two compo-nents. The first is just a phase difference between thevolume channel and the ground phase estimate jo. Asmentioned earlier, this by itself generally underestimatesheight and so needs augmenting by the second term,

which is based on inversion of the coherence amplitude.This second stage involves inversion of the coherenceamplitude using the zero extinction version of equation(12), which results in a ‘‘sinc’’ function. Figure 2demonstrates graphically how the sinc inversion stageis implemented.

hv ¼arg ~gwv

� �� f

kzþ e

2 sin c�1 ~gwv

�� kz

where

f ¼ arg ~gwV� ~gwS

1� LwSð Þ

� �0 LwS

1

AL2wS

þ BLwSþ C ¼ 0 ) LwS

¼ �B�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � 4AC

p

2A

A ¼ ~gwS

�� 2�1 B ¼ 2Re ~gwV� ~gwS

� �:~g�wS

� �C ¼ ~gwV

� ~gwS

�� 2: ð15ÞVariations in structure or extinction in the volume cannow both be accommodated in this model by correctselection of the parameter e as we now demonstrate. Thebasic idea is to choose e so as to minimize the error inequation (15) across the expected range of extinctionsand structure variations. In the simple case of zeroextinction, e = 0.5 is the correct choice for compensatingfor arbitrary variations in crown depth d. At the oppositeextreme of infinite extinction then e = 0 is the correct

Figure 3. Height estimation versus extinction for various e values in equation (15) (true height =20 m) demonstrating that choice of e = 0.4 keeps height error within 10% bounds for a wide rangeof extinction values.


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choice, as the phase term itself will then give the top ofthe vegetation. However, in practice the extinction ofmicrowaves by forest vegetation seldom exceeds 1 dB/m[Bessette and Ayasli, 2001], depending on frequency anddensity. Over this range we suggest it is possible tochoose a single value of e that keeps the maximuminversion error within the 10% bounds. As an example,we show in Figure 3 plots of the estimated height usingthe full random volume-over-ground or RVOG model asinput with extinction varying from 0 to 1 dB/m. We showthe worst-case scenario with 100% crown occupancy(which emphasizes the importance of extinction) andtake 20 m as the vegetation height with a kz value of 0.12at 45� angle of incidence, corresponding to typicalbaselines required for forest height estimation usingPOLInSAR.[18] We note that the e = 0.5 case causes errors beyond

the 10% point for moderate extinctions, while the e = 0.3case shows a consistent underestimation of height untilrelatively high extinction values of 0.5 dB/m. This leadsus to propose 0.4 as a suitable compromise for heightestimation in varying forest density and structure envi-ronments. Just to reiterate, this inversion scheme gen-erates estimates of the two parameters required for PCTfrom four observables. Importantly, this scheme remainsrobust to structure and density variations (but is conse-quently unable to determine any structure parameters).It does however rely on two important underlyingassumptions:[19] 1. It assumes that ms = 0 in the wv channel. If this

is not true then height will be underestimated by equation(15).[20] 2. It assumes that the volume decorrelation is

independent of polarization (the random volume assump-tion). If there are polarization changes in the volume[Treuhaft and Cloude, 1999] then the ground phaseestimate jo will be in error, as the straight-line hypoth-esis will be incorrect. This again will lead to height errorsin (15).[21] Some of these issues have been dealt with further

by Cloude and Papathanassiou [2003] and Treuhaft andCloude [1999] but here we note only that our estimatesof the two parameters based on any POLInSAR approachare likely to contain errors because of limitations of theunderlying assumptions. The key issue is the magnitudeof these errors and their impact on structure estimation aswe now consider.

3. Generalized Structure Estimation

3.1. Introduction

[22] In the previous section we showed how dualpolarized interferograms can be combined to provideestimates of underlying ground topography and mean

vegetation height (equation (15)). While these two param-eters are of importance in themselves [Mette et al., 2004;Papathanassiou et al., 2005b], it is interesting to speculateif we can extend this approach to estimate structuralparameters of the vegetation cover. To investigate further,we turn again to the basic definition of coherence inequation (1) and, assuming we can now obtain estimatesof the two parameters hv and jo, consider techniques forthe reconstruction of f (z), the vertical structure functionitself. To do thiswe first remove the topography phase termand normalize the range of the integral by a change ofvariable as shown in (16):

Zhv0

f zð Þeikzdz ��z0¼2zhv�1�!

Z1�1

f z 0ð Þeikz 0dz0: ð16Þ

We also rescale variation of the real nonnegative functionf (z) so that if 0 f (z)1 then f (z0) = f (z)� 1 and�1f (z0)1. Critically we can now develop f (z0) in a Fourier-Legendre series on [�1,1] as

f z0ð Þ ¼Xn

anPn z0ð Þ

an ¼ 2nþ 1

2

Z1�1

f z0ð ÞPn z0ð Þdz0;ð17Þ

where the first few Legendre polynomials of interest to us

are given explicitly as

P0 zð Þ ¼ 1

P1 zð Þ ¼ z

P2 zð Þ ¼ 1

23z2 � 1� �

P3 zð Þ ¼ 1

25z3 � 3z� �

P4 zð Þ ¼ 1

835z4 � 30z2 þ 3� �

:

ð18Þ

[23] The numerator and denominator of the generalexpression for coherence can now be written fromequation (16) as

Zhv0

f zð Þeikzzdz ¼ hv

2ei

kzhv2

Z1�1

1þ f z0ð Þð Þeikzhv2 z0dz0

Zhv0

f zð Þdz ¼ hv

2

Z1�1

1þ f z0ð Þð Þdz0;

ð19Þ


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from which it follows that the coherence can be writtenas

~g ¼ eikzhv2

R1�1

1þ f z0ð Þð Þeikzhv2 z0dz0

R1�1

1þ f z0ð Þð Þdz0

¼ eikv

R1�1

1þPnanPn z0ð Þ

� �eikvz

0dz0

R1�1

1þPnanPn z0ð Þ

� �dz0

: ð20Þ

By expanding the series and collecting terms, this can be

rewritten in simplified form as

~g ¼ eikv1þ a0ð Þ

R1�1

eikvz0dz0 þ a1

R1�1

P1 z0ð Þeikvz0dz0 þ a2R1�1

P2 z0ð Þeikvz0dz0 þ a3R1�1

P3 z0ð Þeikvz0dz0 þ . . .

1þ a0ð ÞR1�1

dz0 þ a1R1�1

P1 z0ð Þdz0 þ a2R1�1

P2 z0ð Þdz0 þ a3R1�1

P3 z0ð Þdz0 þ . . .

¼ eikv1þ a0ð Þf0 þ a1f1 þ a2f2 þ ::anfn

1þ a0ð Þ ;

where we note that evaluation of the denominator is

simplified by using the orthogonality of the Legendre

polynomials. Evaluation of the numerator involves

determination of the functions fn, which is straightfor-

ward but involves some algebra and repeated use of the

following identity:

Zznebzdz ¼ ebz

bzn � nzn�1

b

�þn n� 1ð Þzn�2

b2. . .

�1ð Þnn!bn

�:

ð22Þ

For reference, we give the explicit form of these

functions up to fourth-order in equation (23). We note

the following important points:

ð21Þ

Figure 4. Plot of coherence basis functions from the Fourier-Legendre expansion (equation (23)).Functions are grouped in two classes: in blue we show the real functions and in green theimaginary. The solid curves show the single-baseline functions, and the dashed curves show theextra functions obtained for dual-baseline operation.


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[24] 1. The even index functions are real while the oddare pure imaginary. We note also that the unknowncoefficients an are all real.[25] 2. The functions vary only with the single param-

eter kv, itself defined from the product of two parameters,vegetation height hv and the vertical component ofinterferometric wave number kz.[26] Graphs of these functions are shown in Figure 4.[27] We see that the first function is a sinc relation

between coherence and increasing height-baseline prod-uct. This is the expected functional relationship forscattering by a uniform vegetation layer in the zeroextinction limit, as used in equation (15):

fo ¼sin kvkv

f1 ¼ isin kvk2v

� cos kvkv

� �

f2 ¼3 cos kv

k2v� 6� 3k2v

2k3vþ 1

2kv

� �sin kv

f3 ¼ i30� 5k2v

2k3vþ 3

2kv

� �cos kv

�

� 30� 15k2v2k4v

þ 3

2k2v

� �sin kvÞ

f4 ¼35 k2v � 6� �2k4v

� 15

2k2v

� �cos kv

þ35 k4v � 12k2v þ 24� �

8k5vþ30 2� k2v� �8k3v

þ 3

8kv

� �sin kv:

ð23Þ

The other functions are arranged in pairs in anticipationof their use for tomographic reconstruction. In blue inFigure 4 we show the functions f1 and f3. The first we seeis a strong function of height-baseline product, while thesecond shows poor sensitivity below height-baselineproducts of 1.5. In green we show the functions f2 and f4.These again are split, with f2 showing greater sensitivityto variation in height-baseline product. The relativebehavior of these functions is important in determiningthe optimum baseline to use in coherence tomography aswe show later. These functions can be considered thebasis functions for a generalized treatment of informationextraction in multibaseline interferometry as we nowdemonstrate.

3.2. Coherence Tomography

[28] We showed in the last section that the coherencefor an arbitrary vertical structure function f (z) can bewritten in the form

~ge�ikv ¼ ~gk ¼ f0 þ a10f1 þ a20f2 þ ::an0fn; ð24Þ

where the unknown coefficients have been normalizedby the zeroth-order term so that

an0 ¼an

1þ a0: ð25Þ

If we can estimate the set of unknown coefficients an0 by

inverting this relation, then we can use them to generate

an estimate of the unknown vertical structure profile,f (z0), as

f z0ð Þ ¼ 1þ a10P1 z0ð Þ þ a20P2 z0ð Þ þ . . . an0Pn z0ð Þ;ð26Þ

which determines the unknown structure function up to

an arbitrary scaling factor (1 + ao); that is, we can only

retrieve a relative structure function using interferometry.

However, this relative profile is still important, indicating

as it does the variation of density of microwave scattering

throughout the volume, allowing for example the

identification of layering. As we are reconstructing a

function from a set of ‘‘projections,’’ that is, the

coherence, we can formulate this problem as an example

of tomography.[29] Importantly, if we assume we know the vegetation

height and location of the ground topography, then theestimation of the unknown terms becomes much simpler,involving just the solution of a set of linear equations.We have seen that single-baseline POLInSAR can pro-vide an estimate of these parameters by combininginterferograms at two polarizations (equation (15)). Toderive a tomographic scheme we use the fact that thebasis functions are either real or imaginary and theunknown coefficients are all real to formulate a set ofequations for each baseline as shown in equation (27):

Re ~gkð Þ � f0 ¼ a20f2 þ a40f4 þ . . .

Im ~gkð Þ ¼ �i a10f1 þ a30f3 þ . . .ð Þ¼ a10f1i þ a30f3i þ . . . ; ð27Þ

where the subscript i has been added to the odd functionsto indicate that the imaginary part is to be taken. Here weconsider the special cases of single- and dual-baselineinterferometry so that for the same height hv we allow kzto take on two values and obtain the complex coherencefor each. The approach can easily be generalized tomultibaseline interferometry, providing the potential foreven higher-resolution reconstructions. However, wenote that practical limitations mean that currently,multibaseline measurements are expensive or difficultto obtain for a large number of baselines and so it is


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important to first assess how much information can beextracted for a small number of baselines. Our formula-tion allows us to adopt a parsimonious approach to thisissue, extracting full information from each additionalbaseline.

4. Single- and Dual-Baseline Tomography

4.1. Introduction

[30] Starting with a single baseline, we note that wewill always be limited in the resolution we can achieve inany reconstruction, as a single baseline provides only asingle complex coherence. Hence only two unknownparameters can be obtained in the Fourier-Legendreseries, namely, a10 and a20 so that we have the followingalgorithm to estimate the structure function in single-baseline problems (again assuming we have a prioriknowledge of the height and ground topography):

a20 ¼Re ~gkð Þ � fo

f2

a10 ¼Im ~gkð Þf1i

9>>=>>;) f zð Þ ¼ 1þ a10z

þ a202

3z2 � 1� �

� 1 z 1:

ð28Þ

To illustrate the resolution capabilities of this algorithm,we show in Figure 5 the three basis functions availablefor single-baseline analysis. We see that we can model anarbitrary mixture of a uniform, linear and quadraticvariation across the height.[31] To obtain higher resolution, we clearly need to

include higher-order Legendre polynomials in the recon-struction and this is only possible by adding baselines tothe interferometer. For example, the addition of a secondbaseline leads to estimation of up to four unknowncoefficients. If we denote the two baselines by super-scripts x and y then we can write the expansion in matrixform as shown in equation (29):

fx1 0 fx

3 00 fx

2 0 fx4

fy1 0 fy3 0

0 fy2 0 fy4

2664

3775:

a10a20a30a40

2664

3775 ¼

Im ~gxk� �

Re ~gxk� �

� fx0

Im ~gyk� �

Re ~gyk� �

� fy0

2666666664

3777777775) F½ �:a ¼ g;

ð29Þfrom which we can obtain an estimate of the unknowncoefficients as

a ¼ F½ ��1g: ð30Þ

In this case we can estimate profile functions obtained asarbitrary mixtures of the set of five Legendre functions

Figure 5. Plot of the Legendre polynomial basis functions available from single-baselineinterferometry (see equation (18)).


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Figure 6. Plot of the Legendre polynomial basis functions available from dual-baselineinterferometry (see equation (18)).

Figure 7. Condition number of dual-baseline [F] matrix versus baseline ratio (equation (29)) forthree different starting baselines, 0.25, 0.5, and 1, showing the high condition number for dual-baseline PCT.


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shown in Figure 6. Extension of this idea to three andmore baselines follows immediately, with two extraLegendre polynomials being added for each baseline,steadily increasing the resolution of the reconstruction.[32] One important consequence of this matrix inver-

sion formulation is that we can assess the accuracy of theestimation and its susceptibility to noise in the measure-ment data. It follows that the stability of the inversiondepends on the condition number of the matrix [F],which in turn depends only on the functional form ofthe elements F (see Figure 4). These can be calculatedindependently of the actual profile and depend only onthe baseline ratio, By/Bx, and baseline-height product kz

x.Clearly if the baseline ratio approaches unity then theinversion becomes unstable as det([F]) tends to zero. Onthe other hand, a large baseline ratio will cause thecoherence to fall to zero at the longer baseline and thiswill generate severe bias and noise issues in the measure-ments. It is therefore of interest to examine the variationof condition number with baseline (for single-baselinesensors) and baseline ratio (for dual baseline). Figure 7shows the dual-baseline variation of condition number of[F] with baseline ratio for three different starting base-lines. If the baseline-height product is small (kz

x = 0.25 inFigure 7) then the baseline ratio is allowed to be large(>10) but the condition number remains high. At the

other extreme, when the smaller baseline is chosen tohave a higher value (kz

x = 1.0) then although the range ofbaseline ratios is now more limited, the condition num-ber falls off much faster. For example it is often easy togenerate a factor of 2 in baselines by employing the‘‘ping-pong’’ interferometric mode (transmitting andreceiving from either end of the baseline compared tojust transmitting from one and receiving on both).Figure 7 shows that this factor of 2 in baselines wouldbe best employed by using a kv value around 1, in whichcase the condition number can be reduced below 100.[33] However, we see from Figure 7 that dual-baseline

inversion is generally poorly conditioned, requiring inthe worst case extremely high precision in the measure-ment of complex coherence (of order of 10�4) in order toachieve accuracies of a few percent in the estimation ofthe Legendre parameters. However, it can also be envis-aged that dual baseline could be used as part of anoverdetermined system to either estimate a subset ofparameters or help compensate for secondary errorsources such as temporal decorrelation in repeat passsensors. Such dual-baseline/multibaseline issues will bethe subject of a separate study and here we return toconsider the conditioning of single-baseline tomography,which as we have seen in equation (28) is possible up toa second-order Legendre approximation.

Figure 8. Variation of single-baseline Legendre matrix condition number with baseline-heightproduct (equation (32)) showing a generally lower condition number than dual baseline (seeFigure 7).


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4.2. Single-Baseline Tomography

[34] We start by reformulating equation (28) as amatrix equation as shown in equation (31):

1 0 00 f1i 00 0 f2

24

35: a00

a01a02

24

35

¼1

Im ~gkð ÞRe ~gkð Þ � f0

24

35) F½ �a ¼ b ) a ¼ F½ ��1

b;

ð31Þ

where the functions f0, f1 and f2 are given in equation(23). Using these relationships we can then obtain anexplicit expression for the condition number of thematrix [F] as

CN ¼ � 1

f2¼ � k2v

3 cos kv � 3� k2v� �

sin kvkv

: ð32Þ

Figure 8 plots this function versus normalized wave

number kv.[35] We note that for small baseline/height products the

inversion is poorly conditioned. For baseline/heightproducts around unity, the condition number is around

10–20. Since F is diagonal we can also identify theworst-case scenario, when the system becomes mostsensitive to errors. From (31) this arises for perturbationsof a true solution of the form

b ¼100

2435) bþ db ¼

10b

2435; ð33Þ

which physically corresponds to uniform, zero extinctionvolumes. In this worst case the error in the Legendrecoefficient vector is of the order of CN.b. The coefficientb can be related to the coherence and effective number oflooks L by using the Cramer-Rao bound [Touzi et al.,1999; Seymour and Cumming, 1994] and considering thelimiting case of zero extinction volume scattering.Considering the worst case from equation (33), it followsthat the largest error contribution is from the real part ofthe phase corrected coherence ~gk. For a uniform zeroextinction volume the real part error is then dominatedby the Cramer-Rao variance on coherence rather thanphase estimation [Seymour and Cumming, 1994]. Takingthe standard deviation as a measure of the coherenceerror we can then write

b �1� g2v� �

ffiffiffiffiffiffi2L

p � 1ffiffiffiffiffiffi2L

p 1� sin kvkv

� �2 !

: ð34Þ

Figure 9. Upper bound on fractional error in Legendre coefficients (see equation (35)) versusnormalized wave number and number of looks for single-baseline PCT.


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Figure 10. Summary of proposed algorithm for single-baseline polarization coherencetomography (PCT).


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Figure 11. Extinction profiles (solid curves) and their single-baseline Legendre seriesapproximations (dashed curves) for high- and low-extinction values.

Figure 12. Two-layer volume plus surface-scattering profile (solid curve) and its Legendre seriesapproximation (dashed curve).


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where L is the number of looks. In this way we can

estimate an upper bound on the fractional error in the

estimate of the Legendre coefficients as a function of just

two parameters, kv and the number of looks L as shown

in equation (35):

k da kk a k sin2 kv � k2vffiffiffiffiffiffi

2Lp

3 cos kv � 3� k2v� �

sin kvkv

� � : ð35Þ

Figure 9 shows how this bound varies as a function of kvfor various number of looks L.[36] We see that we need a large number of looks,

generally L > 50 and to work at moderate normalizedwave numbers (kv = 1) in order to obtain reasonablebounds on the error. We must point out however that thisrepresents a worst-case scenario. In practice the extinc-tion through the vegetation layer will not be zero andthere will be some tapering to the structure function withheight. In this case, as we show in section 5, the

inversion becomes better conditioned and we can obtainerrors much below this upper bound.

5. PCT Algorithm and Its Sensitivity to

Noise

[37] In this section we first summarize our proposedalgorithm for single-baseline polarization coherence to-mography and then validate its performance using sim-ulated data. We employ simulated profiles on the basis ofvarious two-layer models to investigate the basic sensi-tivity issues related to realistic profiles and statisticalfluctuations in coherence estimation. In addition to thedesired volume coherence, there exist several othersources of decorrelation in interferometry [Zebker andVillasenor, 1992]. Key amongst these are signal-to-noiseratio (SNR) and temporal decorrelation. In addition,there is the issue of coherence bias [Touzi et al., 1999],whereby with limited data samples there is a tendency tooverestimate the coherence magnitude. All of these act aspotential noise sources in PCT and need to be considered

Figure 13. (left) Reconstructed versus (right) estimated profiles for the (top) volume-only caseand (bottom) mixed surface plus volume case (known height and ground phase).


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in a full error analysis. However, here we assume thatsufficient multilooking is being performed to minimizebias issues and that SNR and temporal effects can for themoment be ignored. Future studies will address theseauxiliary noise and bias issues on an application specificbasis.[38] Figure 10 shows a summary of the key steps

involved in the proposed PCT algorithm. Here we pulltogether all the relevant strands of the previous sectionsto illustrate the main flow of the procedure. We start byselecting two polarizations wv and ws, the former weassume has dominant volume scattering while the lattersignificant surface scattering. We then estimate thecomplex coherence using a window with an effectivenumber of looks of L. This dual-channel coherence datais then used to estimate two parameters, the groundtopography and height of the vegetation layer. Theseestimates are then used to normalize the coherence in theselected channel via phase correction as shown in stage 2in Figure 10. Note that the choice of polarization for thischannel is arbitrary, so we can use PCT to explore thevariation of structure with polarization by choosingvarious w vectors in stage 2.

[39] The real and imaginary parts of this normalizedcoherence are then used in stage 3 to solve a matrixequation for the unknown structure coefficients a. At thesame time we can obtain an estimate of the error in a asshown in the side box. Finally in stage 4 we use theseestimates to derive a profile function. This function isguaranteed to be bounded by the height estimate and tobe zero outside this range.[40] We now demonstrate application of the algorithm

to simulated data sets. As a first step in assessing thecapability of single-baseline tomography for estimatingprofile functions, we consider the set of true densityprofiles built from combinations of exponential andGaussian functions, where in general we can write

f zð Þ ¼ m1e2scos qz þm2e

�g2 z�zoð Þ2 : ð36Þ

We begin with the simplest case of an exponential profile(m2 = 0, m1 = 1) with varying extinction s. Figure 11shows two examples for low and high extinction rates of0.1 and 0.3 dB/m, respectively, for a tree height of 20 mand 45� angle of incidence. Shown in dash are the single-baseline Legendre series approximations of the true

Figure 14. (left) Reconstructed versus (right) estimated profiles for the (top) volume-only caseand (bottom) mixed surface plus volume case (estimated height and ground phase).


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profile. We see that for low extinction the single-baselineapproximation is very good, whereas for higher extinc-tion the profile match is poorer.[41] A key feature of POLInSAR is that the above

exponential profiles are modified by the presence of apolarization-dependent surface contribution. We can in-vestigate the effect of such a change by including aGaussian function in equation (36) with an offset zo = 0,that is, on the surface. Figure 12 shows an example fors = 0.3 dB/m, hv = 20 m, q = 45� and g = 0.01hv.[42] The parameters m1 and m2 are chosen so that the

ratio of integrals of the two components is unity, whichthen corresponds to a value of m = 0 dB in equation (4).By comparing the dash curves in Figures 11 and 12 wenote that while the single-baseline approximation cannotmatch profiles exactly, it is able to track changes in therelative level of the surface component and hence give anindication of the relative levels of surface and volumescattering in a channel.[43] Having established the ability of the single-base-

line Legendre series to respond to changes in the ratio ofsurface and volume scattering, we now turn to considerthe ability of the algorithm in Figure 10 to estimate the

dashed profiles in Figures 11 and 12. To investigate therobustness of the algorithm, we start by employing thetwo solid line profiles in Figures 11 and 12 (for s =0.3 dB/m). These profiles can then be used to estimatethe coherence by direct numerical evaluation ofequation (1), using a baseline corresponding to kz =0.1. The volume-only channel (Figure 11) we then useto model the coherence in the wv cross polarized or HVchannel. The mixed surface-and-volume case (Figure 12)we use to model the coherence in the ws, copolarizedHH-VV channel [Cloude and Papathanassiou, 2003]. Togenerate realistic fluctuation noise on these coherencesamples we adopt a multivariate Gaussian assumption[Touzi et al., 1999] and use the coherences to generatenumerical samples from a normal distribution with thesame underlying coherence and a specified number oflooks L = 50. We then employ 64 pairs of samplecoherences (HV, HH-VV) as input to the tomographicalgorithm in Figure 10. Finally we stack the 64 estimatedheight profiles to generate an image. By looking atstructure in the image we can then quickly assess theaccuracy of the estimation technique. By taking the normof the difference between the estimated Legendre coef-

Figure 15. (left) Reconstructed versus (right) estimated profiles for the low extinction volume(0.03 dB/m) case (known height and ground phase).


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ficients and the known ‘‘true’’ Legendre coefficients wecan further quantify the fractional error in the estimationprocess.[44] We employ two different approximations. In the

first we assume that the height and ground topography areknown a priori, and we are seeking only to reconstruct thevertical profile from the coherence. In the second weadopt the extreme case where we assume zero a prioriknowledge and must estimate every parameter fromthe data itself (using the full algorithm in Figure 10).Figures 13 and 14 show sample results of this process.[45] In the case of known height and topography

(Figure 13) we clearly obtain the best results, withfluctuation in estimates due to coherence variationsfollowing the analysis of equation (35). By using thisapproximation we estimate a fractional error bound of25%. Across the 64 samples shown in Figure 13 weobtain a mean fractional error of 18%. This discrepancyis consistent with our use of equation (35) as an upperbound, the lower actual error being caused by themoderate extinction in the volume (0.3 dB/m). To verifythis, we show in Figure 15 the case when we reduce the

extinction to 0.03 dB/m. Here we see a much noisierestimation, with 47% predicted error and 45% observedfor this 64 block sample.[46] Returning now to Figure 14 we see that when both

ground phase and height have to be estimated then errorsin these two do occur. The fluctuations in height liearound the 10% level (typical for the POLInSAR tech-nique) with matching errors in ground phase. In this case,even with the high extinction, the fractional errorincreases to around 42%, in excess of the 18% observedin Figure 13. Clearly the approximation of equation 35 isonly a starting point in a fuller error analysis accountingfor height, topography and coherence estimation noisein the fully unconstrained profile estimation problem.Such an analysis will be the focus of a future study,but the results of Figure 14 demonstrate that theestimated profiles, while noisy, are not completelydestroyed by typical height and phase errors appropriateto POLInSAR data.[47] Finally we point out one other important feature of

the PCT approach. It has the ability to map profiles thatdo not strictly satisfy the assumptions of the random-

Figure 16. (left) Reconstructed versus (right) estimated profiles for a Gaussian profile function(known height and ground phase).


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volume-over-ground model. To illustrate this, we con-sider a profile made from a single-offset Gaussian m1 =0, m2 = 1 and z0 = hv/2 in equation (36). This Gaussianprofile does not match the propagation extinction as-sumption inherent in equation (3). Figure 16 shows thereconstruction of such a profile using again L = 50.[48] The Legendre coefficients for this type of profile

are clearly separated from those corresponding to thevolume extinction case. This ability of PCT to check theassumptions underlying models such as RVOG is auseful feature. We shall see later (Figure 20) that suchnon-RVOG profiles arise in real data in regions wherethe propagation path is poorly formed. With this basicconfidence now established in the PCT technique, weturn to consider its application to more realistic imagingscenarios.

6. Application of PCT to POLInSAR Data

[49] In order to illustrate application of the abovealgorithm to synthetic aperture radar, we employ datafrom a 3-D coherent SAR simulator, as detailed byCloude and Papathanassiou [2003] and Cloude et al.[2004]. Note that the model used in this simulator is

more general than the assumptions used here for algo-rithm development, being based on a full Maxwellequation wave propagation and scattering approach tiedto a detailed description of branch, leaf and trunkdistributions [Cloude et al., 2004]. Penetration andscattering are calculated as a function of wavelengthand polarization for each voxel. The underlying surfacescattering is modeled as a tilted Bragg surface andmutual interactions up to third order, such as ground-trunk, canopy-ground and ground-canopy-ground areincorporated in the simulation. The technique isfully coherent and so can be used to model volumedecorrelation effects as required for tomography. Forcontinuity, we employ the same simulations as used todemonstrate POLInSAR height retrieval by Cloude andPapathanassiou [2003]. To achieve this, the SAR sim-ulator was initialized using a 2-D point spread functionmatched to the airborne DLR E-SAR system, with 0.69 mazimuth and 1.38 m ground range resolution. Simulationswere carried out at L band (23 cm wavelength) and at45� angle of incidence from 3 km altitude with 10 mand 20 m horizontal baselines. A full random canopyof 10 m height placed above a flat Bragg surface ischosen as the test configuration. The branches used to

Figure 17. L band POLInSAR simulations of 10 m vegetation layer above a rough surface.


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populate the random canopy have dimensions chosen fromGaussian distributions. Branch length has mean 1.5 m andstandard deviation 0.2 m, branch radius, mean 1.5 cm andstandard deviation 0.2 cm. The branch length distributionwas truncated at 2.0 m and 1.0 m, and radii at 1.0 cm and2.0 cm. There were 59,153 branches in the canopy withdimensions 56 m by 56 m by 10 m, corresponding to amean volume fraction of 0.2%. In the simulation branchesare further subdivided into smaller elements dependingupon the resolution cell size, so that over 200,000 elementsexisted in the simulation. Cloude and Papathanassiou[2003] investigated the ability of single-baselinePOLInSAR to retrieve height and surface topographyfor this scenario. Here we extend the analysis toinvestigate the ability of PCT to estimate variationsin vertical profile in the medium. Note that althoughthe height and topography are constant, the canopy isfilled randomly and so will have density variations notapparent from a simple height estimation. Here weattempt to use PCT to extract these variations.[50] Figure 17 shows simulated SAR images of the

canopy scene. Note that the canopy depolarizes, with

scattering appearing in all polarization channels, includ-ing the cross-polar channel HV. The surface scatteringhas zero cross polarization (HV) but has different scat-tering components in HH and VV as consistent with theBragg model. The underlying surface is flat, with aconstant topographic phase, which can be arranged bysuitable flat earth removal processing, to have zerophase.[51] The kz values for the 10 m and 20 m baseline are

0.128 and 0.256, respectively. For a 10 m canopy heightthis translates into normalized wave numbers of 0.64 and1.28. From Figure 8 the condition number for the 10 mbaseline is therefore around 40 while for 20 m it falls to10. For this reason we adopt the longer baseline (20 m)for tomographic analysis. To further reduce the varianceof the estimate, we employ a large but realistic averagingwindow with an equivalent number of looks L = 60,closely matching the averaging used in Figure 13. Thisleads to a bounding fractional error from equation (35)of around 41%. We then consider only those changesin profile with variations greater than this level assignificant.

Figure 18. (left) Complex coherence pair inside the unit circle and (right) corresponding verticalprofile estimates for dual-polarization 20 m baseline data (point P in Figure 17).


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[52] We consider two different scenarios. In the firstwe assume that height and ground phase are both knownand we seek to map variations in vertical profile. In thesecond, more challenging case, we assume no a prioriknowledge and must estimate height, topography andvertical structure from the data itself.[53] To demonstrate the nature of this retrieval pro-

cess, we show in Figure 18 a sample coherencediagram [Papathanassiou and Cloude, 2001; Cloudeand Papathanassiou, 2003] and vertical profile esti-mates for two polarizations channels, HV in green andHH-VV in red. The retrievals correspond to a point inthe centre of the canopy, shown as P in Figure 17.[54] Note that the true ground phase is everywhere

zero. We see that equation (15), which involves a line fitthrough the two coherence values and estimation oftopography as the unit circle intersection, predicts atopography estimate close to the true value. We alsosee that the height of the POLInSAR estimate is close tothe true value of 10 m. These issues were described inmore detail by Cloude and Papathanassiou [2003].Here we are more interested in the profile estimationresults. First we note that the profiles are not simple

pulse functions, which would correspond to a sinccoherence function, and this in turn indicates the pres-ence of important structural information in the fullspectrum of Legendre coefficients. We see also thatthe profiles are different in the two polarizations (wehave normalized their values at the top of the volumefor comparison). The HV channel is dominated byvolume scattering, whereas the HH-VV channel has astrong surface component, clearly discernible in theprofile shape. This latter response is due to the strongsurface-canopy dihedral response that, as indicated byCloude and Papathanassiou [2003], has a phase centreclose to the surface. These results give some support tothe ability of PCT to extract useful information onphysical profiles, even in the most challenging case ofunknown height and topography. However, this is still asimulated scenario and questions remain as to how wellPCT will perform on real SAR data from real forestenvironments. The prospects for undertaking such astudy in the near future are good however, as therenow exists an extensive database of single-baselinePOLInSAR data covering European, boreal and tropicalforest environments [Papathanassiou et al., 2005b],

Figure 19. Polarimetric tomographic reconstructions along azimuth line AA0 in Figure 17, usingknown height and ground topography for (top) HV and (bottom) HH-VV.


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many with supporting in situ data suitable for testingPCT.[55] To take a closer look at the stability and resolution

issues of the PCT scheme, we show in Figure 19 avertical ‘‘tomogram’’ obtained by stacking profiles esti-mated along the azimuth line AA0 as shown in Figure 17.[56] In Figure 19 we show the case of profile estima-

tion for known height and phase. The HH-VV channelconfirms that the dihedral response is spatially stableacross the scene, giving a dominant response in thatchannel close to the surface. In the HV channel bycontrast we see a very different profile. Here we seedominant scattering from the top of the vegetation layer.The only exception to this occurs at the edges of thecanopy, but here we have errors due to the spatialaveraging employed in the multilooking, which essen-tially mixes pixels from the canopy with the externalsurface returns. Advanced spatial averaging techniquessuch as the refined Lee filter [Lee et al., 2003] could beused in future to reduce such edge effects. These resultsconfirm the stability of the PCT technique and supportapplication of POLInSAR height retrieval for this vol-ume, as suggested by Cloude and Papathanassiou

[2003], with two well separated phase centers in thetwo polarization channels. This ideal situation is notalways the case however, even in simulated data. Muchmore interesting is to consider a tomogram along a rangeline. This is shown in Figure 20.[57] Here we start to see some variation in structure

with range. In the near range we see clear evidence oflayover. In the HV channel we see the increasingeffective vegetation layer due to layover (45� angle ofincidence). More interestingly we then see an interme-diate region where the propagation extinction has not yetfully developed and there is a response from the fullvolume. We then move into a stable region where themean extinction is evident and scattering occurs predom-inantly from the upper layer. In the HH-VV channel wesee a simpler transition, with strong front surface/volumedihedral return on the ground followed by a layoverregion and then the stable dominant surface response. Inthe far range we again see the effects of layover with asurface response separated from its corresponding vol-ume return because of the latter’s elevation.[58] The above results demonstrate the ability of sin-

gle-baseline PCT to resolve variations in vertical struc-

Figure 20. Polarimetric tomographic reconstructions along range line BB0 in Figure 17, usingknown height and ground topography for (top) HV and (bottom) HH-VV.


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ture when height and topography are known. While onecan envisage some applications where this informationwill be known in advance, a much more challengingscenario for remote sensing is the case of zero a prioriknowledge, that is, when we need to estimate height andtopography as well as vertical structure. We now turn toconsider this problem. This time we apply the fullprocessing chain as shown in Figure 10. We then usethe local height estimate and topographic phase as aninput into the structure estimation algorithm. Figures 21and 22 show results for two sample azimuth and rangelines.[59] We note the following points:[60] 1. The azimuth slice again shows dominant vol-

ume scattering in the HV channel and strong surfacescattering in the HH-VV channel. Closer comparisonwith Figure 18 however indicates that the predictedextinction in the volume is smaller for the height-esti-mated case than in the known height inversion.[61] 2. The range line shows an initial overestimation

of height due to an edge effect but then shows underes-timation of height in the transition region (around azi-muth position of 20 m) where the wave extinction

channel is not yet fully formed. This is consistent witha breakdown of the assumptions behind the RVOGmodel. Into the main region of the canopy the estimatedheight improves in accuracy and the profile is stable,falling off again the far range due to layover effects.

7. Conclusions

[62] In this paper we have introduced a new radar-imaging technique, polarization coherence tomography(PCT). The method extends conventional polarimetricinterferometry by allowing reconstruction of a verticalprofile function, physically representing the variation ofbackscatter as a function of height. We suggest that thishas potential new applications in radar remote sensing,such as improved vegetation species classification, im-proved biomass estimation and subcanopy surface pa-rameter mapping. Detailed analysis of these applicationswill be the subject of future studies and here we haverestricted attention to developing the basic techniqueitself.[63] The technique relies on knowledge of two key

structure parameters of the volume under investigation,

Figure 21. Polarimetric tomographic reconstructions along azimuth line AA0 in Figure 17 usingPOLInSAR estimated height and ground topography (see Figure 10).


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namely, height and ground phase. These two can beobtained by a variety of methods, such as field measure-ments, other sensors (such as lidar) or can themselves beestimated from the radar data in advance of applicationof PCT by using POLInSAR algorithms. With theseparameters in place, the tomographic reconstructionproblem can then be reformulated as a Fourier-Legendreseries expansion and the unknown expansion coefficientsestimated from coherence data by a simple linear matrixinversion. Perhaps surprisingly, we have shown that evenwith a single-baseline, single-frequency interferometer,tomography is still possible, albeit limited in resolutionup to a second-order polynomial expansion. We haveanalyzed in detail the conditioning of the matrix inver-sion involved in tomographic reconstruction and shownthat for dual-baseline applications the conditioning canbe very poor. Future analysis is required to investigatethe dual-baseline and multibaseline situation in moredetail. Here we have concentrated instead on the sin-gle-baseline tomographic case. For single-baseline stud-ies we have established a framework for assessing thenumber of looks required of the interferometer to reduceerrors in the estimated parameters below a prescribed

level. Although only a first-order analysis of such errorsis presented here, we believe that the formulation permitsstraightforward extension to consider other error sources,such as those in the two key structure parameters heightand topography, as well as signal to noise and temporaldecorrelation. Such analyses will be the focus of futurestudies. Finally, although we have focused on applicationof PCT to vegetation mapping using radar imaging, webelieve this technique has a wider range of applicationsin any EM-scattering scenario where there is a combi-nation of surface and volume-scattering effects. Impor-tant examples for future study include snow hydrology,agriculture and ground penetration for moisture estima-tion at low frequencies.

[64] Acknowledgments. Thanks to Mark Williams ofDSTO, Australia, for providing the POLInSAR test simulationsused in this study and to the anonymous referees for their usefulsuggestions for improving the clarity of the paper.

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Figure 22. Polarimetric tomographic reconstructions range line BB0 in Figure 17 usingPOLInSAR estimated height and ground topography (see Figure 10).


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