Coherent change detection under a forest canopyP. B. Pincus∗† and M. Preiss∗

∗ Defence Science & Technology Group, Edinburgh, SA, Australia† School of Electrical & Electronic Engineering, University of Adelaide, Adelaide, SA, Australia

Abstract—The possibility of repeat-pass coherent change de-tection of the ground in a forest using synthetic aperture radarimaging is investigated. By vertically beamforming multipleacross-track channels per pass, the canopy interference canbe suppressed. We formulate this as a dual-layer (ground andvolume) multichannel coherence estimation problem and derivethe beamformer that optimally suppresses the volume compo-nent in each pass. Beamformer performance is analysed usingthe random-volume-over-ground model of forest scattering. It isshown that reasonable volume suppression is achievable with justthree channels, and this performance is not overly sensitive tomodel parameters. We also discuss the trade-off between foliagepenetration and change sensitivity for wavelength selection.

I. INTRODUCTION

Coherent change detection (CCD) is an application ofairborne or spaceborne synthetic aperture radar (SAR) thatresolves subtle changes on the ground [1], [2, ch. 5.5]. Bycomputing the complex correlation coefficient (the complexcoherence) between two repeat-pass SAR images as a localaverage in a sliding window, and taking the magnitude, a changemap is generated, with areas of high correlation indicatinglittle change and areas of low correlation indicating substantialchange. The method is sensitive to changes in the complexspeckle pattern of the ground clutter caused by rearrangedscattering elements, as opposed to changes in scattering in-tensity [3], [4]. In typical operation, high-frequency (e.g. Ku-band (λ∼1.8cm) [5]), wideband radar systems generate fine-resolution SAR images of open landscapes for CCD.

We study the possibility of extending the CCD technique todetect ground disturbances on forest-covered terrain. The ideawas first proposed by the authors in [6]. The twin difficultiesthat must be overcome to achieve this are illustrated in Fig. 1:for a given position on the ground, the near-range part of thecanopy will act as a lossy propagation medium, attenuatingthe incident and scattered energy, and the far-range part of thecanopy will layover (i.e. project) onto the ground, such thatthe pixel value at the nominal ground position will actuallycontain a mixture of ground and canopy scattering.

The propagation problem is overcome simply by selecting anoperating wavelength for which the expected foliage attenuationis not too great. Past foliage penetration experiments suggestthat L-band (λ∼23cm) is the highest suitable frequency [7].Whilst lower frequencies offer better penetration, we alsorequire high sensitivity to change, for which higher frequenciesare intuitively better. Here we use a SAR simulation to showhow coherence varies with wavelength, given a fixed resolution.

The layover problem can be overcome by coherently com-bining multiple 2D SAR images acquired at slightly differentgrazing angles to produce a new ‘3D’ image that is steered tothe ground such that scattering contributions from other heights

1.

2.

3.

4.

pulse

transmit

phase centre

receive �

1 2 3

foliage

attenuation

layover along

wavefronts

Fig. 1. Left: a dual-antenna, across-track interferometer that synthesises threeeffective phase centres separated in grazing angle ψ by alternating the transmitantenna pulse-to-pulse and receiving every pulse on both antennas. Right:canopy attenuation and layover.

CCD

3 1 2 , , , ∗ ∗ ∗

3 1 2 , , , ∗ ∗ ∗

Reference pass

Repeat pass

3D SAR

beamforming Complex image

resolved in height

Change map for

ground scattering

SAR images

at different

grazing angles

Fig. 2. The proposed 3D SAR CCD concept. SAR images from each pass ofa multichannel radar system are vertically beamformed using complex weightsw=[w1,w2,...]T, which may vary spatially, to produce a complex ‘3D’ imagesteered to the ground, with canopy scattering suppressed. The magnitude of thecomplex coherence between two 3D images indicates change at ground level.

are suppressed. This is 3D SAR beamforming (a.k.a. SARtomography) [8], [9]. The usual goal is to characterise howscattering intensity varies vertically, for which useful verticalresolution requires many (ten or more) images acquired onseparate passes, which is an impractical collection burden andleads to a difficult phase calibration problem [10]. For the CCDapplication here, the vertical resolution requirement can berelaxed, permitting fewer image channels, but the full complex3D output is needed (rare in the SAR literature [11]), so thebeamforming method must preserve phase. Fig. 1 depicts aradar system that alternates on transmit between two antennasto acquire three channels in a single pass. Adaptive beamform-ing (minimum-variance distortionless response (MVDR) [12,ch. 6.2,6.3,6.6] a.k.a. Capon [9]) using this minimal design hasbeen demonstrated by Intermap [13].

Fig. 2 shows the proposed 3D SAR CCD concept. In thiswork, CCD under a forest canopy is formulated as a mul-tichannel dual-layer coherence estimation problem, and theoptimal beamforming solution is derived. The random-volume-over-ground (RVOG) model [14, ch. 5.2.4, 7.2-7.4], used toestimate forest height via model inversion given polarimetric-interferometric SAR (PolInSAR) data [15], is repurposed toanalyse the potential performance of a three-channel system.

II. SENSITIVITY TO CHANGE

The complex speckle pattern is the pixel-to-pixel fluctuationin the net coherent response of many scattering elements indistributed clutter [3]. A random rearrangement of scatteringelements (e.g. rearranged pebbles on a gravel road) may causea change in the speckle pattern without changing the overallscattering intensity. CCD aims to detect such subtle changes asdecorrelation (coherence loss) between repeat observations [1].

Fig. 3 indicates how shorter wavelengths are more sensitiveto subtle scene change. The curves were obtained using aSAR imaging simulation. Sets of raw radar pulse data weresynthesised for a homogeneous clutter scene consisting of manyrandomly distributed point scatterers (10/m2). For each dataset,the scatterers were randomly (and independently) shifted rela-tive to their original position, giving rise to a different specklepattern. The shifts were uniformly distributed in angle andnormally distributed (with zero mean) in distance. The standarddeviation of the distance distribution was changed from datasetto dataset; this is the quantity plotted on the x axis. A SARimage was formed from each dataset and its coherence withthe original (zero-shift) image computed (1032 effective looks);this is the quantity plotted on the y axis. To permit comparisonacross wavelengths, the bandwidth (140 MHz) was fixed to givea constant ground-range resolution (1.7 m), and the aperturelengths were varied to give a constant azimuth resolution (1 m).

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

standard deviation of shift (cm)

cohere

nce

2.4 GHz, 12.5 cm

1.2 GHz, 25 cm

800 MHz, 37.5 cm

600 MHz, 50 cm

480 MHz, 62.5 cm

400 MHz, 75 cm

Fig. 3. Coherence magnitude at different wavelengths as a function of the levelof change in a clutter scene. Each curve was obtained by repeated executionof a SAR simulation where change was implemented as random shifts in thepositions of point scatterers. The resolution was fixed.

For the same change, the decorrelation is greater at shorterwavelengths, even with the resolution held constant. Completedecorrelation (say, coherence 0.15) occurs when the shifts’standard deviation is roughly half of the wavelength. The curvesbroadly agree with those in [16,Fig.4] for a different simulation.

Given these results, L-band (λ∼23cm) is favoured over lowerfrequencies, despite the better foliage penetration they offer.

III. MULTICHANNEL COHERENCE

Model a pixel in a radar image of clutter as a zero-meancomplex random variable x with expected power σ2

x = E{xx∗}.Given a registered pair of images acquired on repeat passes

a and b at grazing angles ψa and ψb, the complex coherence[3, p. 38] between corresponding pixels xa and xb is

γab =E{xax∗b}√

E{xax∗a}E{xbx∗b}≈ 〈xax∗b〉√

〈xax∗a〉〈xbx∗b〉(1)

where the ensemble averages E{...} are implemented as localspatial averages 〈...〉 over realisations of the same wide-sensestationary scattering process. The coherence magnitude 0≤|γab|≤1 quantifies the similarity of the two speckle patterns [1].

Coherence is now reformulated to allow for multiple imagechannels per pass, combined as a weighted sum.

Consider a set of M registered radar images acquired atgrazing angles ψ= [ψ1,ψ2,...,ψM ]T . Let x= [x1,x2,...,xM ]T

denote the pixel values at one pixel position. The coher-ence γij between any pair of pixels xi and xj is γij =

E{xix∗j}/√σ2i σ

2j . The covariance matrix is R= E{xxH}=[

E{xix∗j}]

=[√σ2i σ

2j γij

]for i,j = 1,2,...,M . In the special

case of constant channel power σ2 = σ2i ∀i, R= σ2Γ, where

Γ = [γij ] is a matrix of pair-wise coherences, with γii = 1∀i.For a given beamformer w= [w1,w2,...,wM ]T , the weighted

sum is y=wHx with power σ2y = E{yy∗}=wHRw.

Now consider a repeat-pass pair of multichannel imagesacquired at grazing angles ψa and ψb. Applying weight vectorswa and wb to corresponding pixel vectors xa and xb generatesbeamformed pixels ya =wH

a xa and yb =wHb xb with coherence

γyab=

E{yay∗b}√E{yay∗a}E{yby∗b}

=wHa Rabwb√

wHa RawawH

b Rbwb

(2)

where Ra = E{xaxHa }, Rb = E{xbxHb } and Rab = E{xaxHb }.The cross-covariance matrix Rab will not be Hermitian.

For repeat-pass change detection, the multichannel grazingangles will typically satisfy the constraints

|ψbi−ψai |. 1◦ ∀i and ∆ψij = ∆ψaij ≈∆ψbij ∀i,j (3)

where ∆ψaij =ψaj−ψai within pass a and similarly for pass b.Given (3), assume that all channels in each (calibrated)

multichannel acquisition receive the same power.

σ2a =σ2

ai∀iσ2b =σ2

bi∀i

}⇒

Ra =σ2aΓa

Rb =σ2bΓb

Rab =√σ2aσ

2b Γab

⇒ γyab=

wHa Γabwb√

wHa ΓawawH

b Γbwb

(4)Note that Γab =

[γaibj

]contains the coherences across passes.

Furthermore, assume that the two sets of coherences observedin the two passes are the same. This is reasonable if, in additionto (3), the scene does not change dramatically.

Γa≈Γb⇒Γ = (Γa+Γb)/2

w=wa =wb

}⇒ γyab

≈ wHΓabw

wHΓw(5)

This multichannel coherence is analogous to that in PolInSAR[14, ch. 6.2.2]; where PolInSAR exploits diversity in polarisa-tion, the 3D processing here relies on diversity in grazing angle.

IV. DUAL-LAYER COHERENCE

Model the forest canopy as a volume containing many scat-tering elements that together provide a macroscopic scatteringresponse but also permit lossy propagation. Model the groundunderneath as a propagation boundary that provides a surfacescattering response. This dual-layer structure is shown in Fig. 4.

ground

volume �

�

Fig. 4. Dual-layer forest model, con-sisting of a volume, which is a lossypropagation medium, above a groundsurface at z = zg , which is a hardpropagation boundary. The model isassumed reasonable at L-band. TheSAR images are focused to z= 0.

Decompose the observed scattering x into independent com-ponents g (signal) and v (interference), where g denotes allscattering mechanisms whose interferometric phase centre islocated at the ground height, notably direct-surface and ground-trunk double-bounce, and v captures all scattering located abovethe ground. Modulate g by a (two-way) volume propagationfactor p, where 0≤ |p| ≤ 1. For simplicity, ignore noise andassume that the radar is calibrated perfectly. Hence,

x= pg+v, (6)

σ2x = |p|2σ2

g +σ2v (power σ2

q=E{qq∗} for source q), (7)

µ= |p|2σ2g/σ

2v (effective ground-volume power ratio). (8)

Given two observations xa and xb at ψa and ψb, split eachcontributing component into a part that is correlated, c, acrossthe observations, and parts that are uncorrelated, ua and ub [16]:

ga = gc+gua , va = vc+vua ,

gb = gc+gub, vb = vc+vub

,

E{gag∗b}=σ2gcejφg , E{vav∗b}=σ2

vcejφv ,

⇒ E{xax∗b}= ejφg (pap∗bσ

2gc +σ2

vcejφvg ), (9)

where φg is the interferometric phase for the ground layer andφvg =φv−φg is the interferometric phase for the volume layerrelative to the ground. The volume manifests interferometricallyas an effective phase centre [17] at height (above the ground)

hv,eff =φvg/kz0 (10)

where kz0 =4π

λ

∆ψ

cosψ0

(ψ0 =

ψa+ψb2

)[2, p. 318] (11)

is the interferometric wavenumber after aperture trimming.Assume that, across observations, the component powers and

the propagation modulation are constant. This is reasonable if,in addition to (3), the scene does not change dramatically.

σ2g =σ2

ga =σ2gb

σ2v =σ2

va =σ2vb

p= pa = pb

⇒ σ2x =σ2

xa=σ2

xb= |p|2σ2

g +σ2v

E{xax∗b}= ejφg (|p|2σ2gc +σ2

vcejφvg )(12)

Using (7), (8) and (12), the observed coherence γab in (1) is

γab = ejφg|p|2σ2

gc+σ2vcejφvg

|p|2σ2g +σ2

v

= ejφg

(µ

1+µγg+

1

1+µγv

)(13)

= ejφgγg+γv/µ

1+1/µ(14)

where

γg =σ2gc/σ

2g (ground-only coherence; real) (15)

γv = (σ2vc/σ

2v)ejφvg (volume-only coherence; complex). (16)

From (13), ground and volume scattering contribute to the totalcoherence as a weighted sum, where the weights depend onjust the ground-volume power ratio µ. (14) matches (7.38) in[14], except that there it was assumed that γg = 1.

The goal is to estimate γg, the change parameter for theground layer. If |γab| from (13) was used as an estimator for γg ,then the estimate would be biased by the volume, to the extentdetermined by µ. If γg = 1 (no change), then any decorrelationexhibited by the volume (|γv|< 1) would bias the estimatedownwards. If γg = 0 (complete change), then any correlationexhibited by the volume (|γv|> 0) would bias the estimateupwards. Vertically beamforming multiple channels offers away to increase µ and therefore decrease the volume bias.

V. MULTICHANNEL DUAL-LAYER COHERENCE

Model the coherence for every channel pair in (4) by the dual-layer expression in (13). For pairs within a pass, the groundcoherence will be unity (ignoring noise and assuming aperturetrimming [2, ch. 5.2]). Across passes, all pairs observe the sametemporal decorrelation (i.e. change); denote the common groundcoherence as γg . Given (12), µ will be constant for all channels.Given (3) and the weak dependence of kz0 on ψ0, the groundinterferometric phases for the pairs within one pass will beapproximately equal to the phases for the corresponding pairswithin the other pass i.e. φgij =φgaij

≈φgbij ∀i,j, and similarlyfor the volume coherences (assuming also that the volume doesnot change dramatically) i.e. γvij = γvaij

≈ γvbij ∀i,j. Hence,

Γ = Γa = Φg ◦(

µ

1+µ1+

1

1+µΓv

)≈Γb, (17)

Γab = Φgab◦(

µ

1+µγg1+

1

1+µΓvab

), (18)

where Φg =[ejφgij

]is a matrix of ground interferometric

phasors for pairs within a pass (Φgabis the same for pairs

across passes), Γv is a matrix of volume coherences for pairswithin a pass (Γvab

is the same for pairs across passes), 1 is anM×M matrix of ones, and ◦ indicates the Hadamard product.

Substituting (17) and (18) into (5), the output coherence is

γyab≈wHΓabw

wHΓw=µwHΦgab

wγg+wHΦgab◦Γvab

w

µwHΦgw+wHΦg ◦Γvw=δγg+βvαv/µ

1+αv/µ(19)

whereαv =

wHΦg◦Γvw

wHΦgw, βv =

wHΦgab◦Γvab

w

wHΦg ◦Γvw, δ=

wHΦgabw

wHΦgw.

(20)αv is a real non-negative number because its constituent matri-ces are Hermitian positive semi-definite. Moreover, comparingthe numerator and denominator matrices, 0≤αv,|βv|,|δ| ≤ 1.

Compare (19) with (14). The multichannel volume attenuationfactor αv captures the effect of the beamformer w: the effectiveground-volume power ratio within a pass is increased from µ to

µ′=µ/αv (0≤αv ≤ 1). (21)

If the topography of the forested terrain is known (or can beestimated using multichannel/PolInSAR data [13], [15]), thenthe focal surface for the SAR images can be made to matchthe true ground surface (the ideal for CCD of the ground), so

zg = 0 ⇒ φgij = 0 ∀i,j ⇒ αv =wHΓvw

wH1w. (22)

If the repeat-pass geometries are identical i.e. ψai =ψbi∀i(the ideal for change detection), and the volume is constant,then Γvab

=Γv and βv=1. If either the collection geometriesare identical or the focal and ground surfaces match, then δ=1.

Consider using |γyab| from (19) as an estimator for γg . In the

ideal case βv = δ= 1, the coherence estimation error would be

εγg= |γyab|−γg=

1

1+µ/αv(1−γg)=

1

1+µ/αvγg=0

0 γg=1

(23)

Limiting the coherence error to, say, 0.1, sets the required αv:

εγg < 0.1 ⇒ αv <µ/9 (α(dB)v <µ(dB)−9.5dB). (24)

VI. 3D SAR BEAMFORMING

Scattering elements above or below the focal surface of aSAR image will layover into the image and contribute residualpropagation phase; both effects depend on scatterer heightand radar grazing angle. Multiple images acquired at differentgrazing angles will collectively be sensitive to scatterer heightvia the differential layover and phase. SAR beamformingexploits this by applying a height-selective complex weightvector w [9]. For CCD, the complex reflectivity of the groundis sought, so only phase-preserving methods are considered.

After accurate motion compensation and removal of flat-earth phase, scatterers on the focal surface will contribute noresidual propagation phase. Therefore, the height (z= 0) of thefocal surface is the SAR-equivalent of the ‘broadside direction’,for which the steering vector is a vector of ones, denoted v0.

The gain variation of a beamformer w is specified by itsbeampattern b(z)= |wHv(z)|2. In the SAR case, the ith elementof the propagation vector v(z) = [ejφi(z)] contains the interfer-ometric phase φi(z) = kziz (see (11)) that would be observedif the first and ith channels illuminated a surface at height z.

Two standard types of beamformers are the conventionalwconv and the adaptive (a.k.a. minimum-variance distortionlessresponse (MVDR) [12, ch. 6.2,6.3,6.6] or Capon [9]) wmvdr:

wconv =1

Nchv0 (25) wmvdr =

R−1v0

vH0 R−1v0

, (26)

where the sample covariance matrix R requires a spatial average.For approximately uniformly spaced channels, the beampatternof wconv will be sinc-like with mainlobe peak-to-null width ρzand a grating-lobe ambiguity at hamb:

ρz =λ

2M∆ψcosψ (27) hamb =Mρz (28)

where ψ is the mean grazing angle and ∆ψ the mean spacing.wmvdr optimally minimises the power from all heights withoutdistorting the signal from the steered height; up to M−1 nullsare steered onto interferences. Compared to wconv, wmvdr offersimproved performance but greater sensitivity to errors [12].

We seek the optimal beamforming weight vector wopt thatminimises αv in (22) and thereby maximises µ′ in (21).

First note that Γv is Hermitian positive semi-definite, sinceit is a scaled covariance matrix. It will almost certainly bepositive definite in practice, so assume that Γ−1v exists.

Observe from (22) that

1

αv=

wHAw

wHBw=

wHCC−1AC−HCHw

wHCCHw=

w′HDw′

w′Hw′(29)

where A=1 is Hermitian, B=Γv is Hermitian positive definite,B=CCH is the Cholesky factorisation of B [18, pp. 90,441],D=C−1AC−H, and w′=CHw is the transformed weight vector.The right-hand side of (29) fits the general form of a Rayleighquotient, whose bounds are the minimum and maximumeigenvalues of D, achieved when w′ equals the correspondingeigenvectors [18, pp. 234–235]. The eigenvalue problem is

Dw′=λw′ ⇒ C−1AC−HCHw=λCHw ⇒ Ew=λw (30)whereE=C−HC−1AC−HCH=(CCH)−1A(CC−1)H=B−1A. (31)

In this case, E=B−1A= Γ−1v 1 has unit rank. The only non-trivial eigenvalue λmax> 0 and corresponding eigenvector are

λmax =vT0 Γ−1v v0 wmax = cΓ−1v v0 (32)

for any c∈C (N.B. 1=v0vT0 ). Hence, the optimal solution is

αv,min = 1/λmax wopt =wmax/λmax (33)

which matches the usual form for optimal beamforming [12].Note that wopt is optimal for αv, but not necessarily for

εγg= |γyab|−γg , unless βv=1 (requires identical geometries).

VII. RANDOM VOLUME COHERENCE

It would be useful to compare the performance of differentradar designs (M,∆ψ) for different model scenes. We use therandom-volume-over-ground (RVOG) model, commonly usedin PolInSAR for forest height inversion [17], because it hasbeen shown to be widely applicable to different forest types andoffers a simple analytical expression for the volume coherencethat we can use to populate Γv . The canopy is modelled as just ahomogeneous random volume of scattering elements extendingfrom the ground to a height hv . For energy incident at ψ, thetwo-way penetration length through the volume to the groundand back is lp = 2hv/sinψ. Propagation through the volumein direction ψ is subject to exponential decay, parameterisedby an extinction coefficient σe (units: m−1). The subsequenttwo-way attenuation of scattered energy is given by

fv(z′) = e−2σe(hv−z′)/sinψ (34)

where z′= z−zg is referenced to the ground surface, so fv(hv)= 1 and |p|2 = fv(0) = e−2σehv/sinψ = e−lpσe > 0. Figure 5sketches the effective vertical structure of a random volume.

00 1

hv

z′

fv(z′)

00

hv

z′

fv(z′) (dB)

Fig. 5. Effective verti-cal structure of a randomvolume with propagationloss given by fv(z′) in(34). The ground is atz′= 0.

One aspect that is not clearly explained in the SAR literature(compare to [14, p. 230]) is that the propagation loss can beexpressed as a one-way attenuation rate σdB

e (units: dB/m),which is equivalent to the intuitive concept of cable loss. Givenfv(0) = e−lpσe , the decibel loss per unit length in direction ψ is

σdBe =−10log10e−lpσe

lp= (10log10e)σe =

10

ln10σe. (35)

When observed interferometrically, the coherent superposi-tion of scattering elements from different heights causes volumedecorrelation. This is separate from any change in the volume.The random volume coherence is given by [17], [14, ch. 5.2.4.2]

γv =E{vav∗b}√

E{vav∗a}E{vbv∗b}e−jφg =

p1(ep2hv−1)

p2(ep1hv−1)(36)

where p1 = 2σe/sinψ and p2 = p1 + jkz0. Note that γv iscomplex. If σe = 0, then fv(z′) = 1, and by L’Hopital’s rule,γv = ejkz0hv/2sinc kz0hv/2π (so hv,eff =∠γv/kz0=hv/2), withγv = 0 when hamb = 2π/kz0 = hv. As σe increases, both |γv|and hv,eff increase—the volume tends towards a surface at hv .

VIII. RVOG BEAMFORMER PERFORMANCE

Define the RVOG beamformer wrvog as the optimal beam-former in (33) obtained when Γv is populated with volumecoherences predicted by (36) according to the random volumemodel. Implementation requires estimated or assumed valuesof the random volume parameters (hv and σdB

e ) together withthe known multichannel collection geometry.

The performance of the RVOG beamformer depends on howclosely the modelled volume matches the actual canopy. Theassumptions are two-fold: firstly, that the canopy can be well-modelled as a random volume, and secondly, that the canopyis well-modelled by the specified random volume.

Compare wrvog = Γ−1v v0 from (33) with wmvdr = R−1v0

in (26), neglecting scale factors. The former is clairvoyantand the latter is data-driven. In cases where it is difficult toobtain a reliable estimate of R, say due to a limited numberof resolution cells to average over, it may be safer to insteademploy wrvog. A mixed approach, whereby the adaptivity isconstrained within reasonable RVOG bounds, may be possible.

Fig. 6 shows the best-case performance of the RVOGbeamformer for different collection parameters, given a typicalRVOG scene. Note that, as the angular spacing ∆ψ decreases,the optimal performance improves despite the resolution ρz in(27) of the conventional beamformer becoming coarser. Evenwith only three channels, choosing ∆ψ∼0.05◦ (matching theIntermap design [13]) would support volume attenuation around−12 dB, which from (24) is sufficient to permit accurate groundcoherence estimates when µ>−2.5dB (assuming zg = 0).

The beampatterns in Fig. 7 show how different beamformerssuppress the volume. The optimal RVOG beamformer (red)steers two nulls inside the mainlobe of the conventionalbeamformer (blue), which encompasses the whole volumeextent (hv = 20m vs ρz = 36m). The trade-off for this highperformance is large gain for any scattering received fromoutside the expected scene extent.

0 0.05 0.1 0.15 0.2 0.25 0.3−25

−20

−15

−10

−5

0

grazing angle spacing (deg)

volu

me a

ttenuation facto

r (d

B)

Nch

3579

zzzM Fig. 6. RVOG beamformerperformance: the optimalmultichannel volume attenu-ation factor αv,min as a func-tion of grazing angle spacing∆ψ for different numbersof channels M (ψ = 35◦,λ=23cm). Fixed scene pa-rameters: hv=20m, σdB

e =0.1dB/m. Each point oneach curve was obtained bypopulating Γv with volumecoherences from (36) for allchannel pairs in the givenconfiguration and then eval-uating (33).

−10 −5 0 5 10 15 20 25 30 35 40−40−35−30−25−20−15−10

−505

10

height (m)

gain

(dB

)

convnullrvognotch

Fig. 7. Vertical beampatterns for conventional (αv=−1.8dB), null-steer(αv=−7.4dB) and optimal RVOG (αv=−12.1dB) beamformers. Collectionparameters: M = 3, ∆ψ= 0.05◦, ψ= 35◦, λ= 23cm. Scene parameters:hv=20m, σdB

e =0.1dB/m. The null-steer beamformer (green) was designedto steer a single null at hv,eff ≈ 13m (the height of the volume’s effectivephase centre is slightly different for different channel pairs). The notch lines(black) indicate the ideal filter.

How sensitive is the RVOG beamformer to deviations be-tween the assumed and actual values of the random volumeparameters? Fig. 8 shows the ground coherence estimationperformance of wrvog designed for the collection scenario in Fig.7 when applied to different random volumes. Broadly, blue (red)indicates good (poor) performance. The black loop marks theacceptable operating region where the estimation error is lessthan 0.1. The beamformer is moderately sensitive to deviationsin the model parameters. Near the design point in this example,the volume height should be known to within about threemetres of its true value. Estimating canopy height to this levelof accuracy is achievable using PolInSAR [15], [14, ch. 9.5.5].

attenuation rate (dB/m)

volu

me h

eig

ht (m

)

0 0.05 0.1 0.15 0.2

5

10

15

20

25

30

00.020.040.060.080.10.120.140.160.180.2

Fig. 8. RVOG beamformer sensitivity:the coherence estimation error εγg in(23) for γg=0 when the RVOG beam-former from Fig. 7 is applied to differentrandom volumes. For each (hv , σdB

e )combination, the volume attenuation αvachieved by wrvog was computed via(22) (with Γv generated using (36)), andthe ground-volume power ratio µ wasmodified for the new propagation factor|p|2=fv(0) from (34). The black crossindicates the design point: hv=20m,σdBe =0.1dB/m, µ=0dB. The black

line marks the region where εγg<0.1.

Note that the random volume parameters determine both thevertical structure for beamforming (captured in αv) and thepropagation loss (captured in µ). For example, as the attenuationrate increases, the volume becomes more localised in height, so

it can be more effectively suppressed by the beamformer, but thepropagation loss increases, so the ground signal will be weaker;the latter effect may dominate, as in the top-right of Fig. 8.

IX. SIMULATION

The 3D SAR CCD concept was tested by simulating RVOGclutter using many point scatterers. Fig. 9 shows the results fora three-channel radar configuration and non-identical passes.For the repeat-pass pair formed by the middle channels only,γv=0.241∠339◦ from (36) and |γab|=0.614 from (14) forγg=1, which matches the single-channel coherence observed in(c) in the no-change areas. Combining all channels from eachpass using the adaptive beamformer, the resulting coherenceacross passes is |γyab

|= 0.872 because the volume, whichcauses decorrelation, has been suppressed. Thus, the true scenechanges on the ground in (a) are visible in the 3D CCD in (e).

X. CONCLUSION

3D SAR CCD of the ground in a forest would seem to be fea-sible by applying optimal beamforming techniques to just threeacross-track image channels. A framework has been establishedfor suitable multichannel radar design and performance analysisusing the RVOG model. Of the numerous practical challengesto implementation, obtaining accurate knowledge of the groundheight for SAR image formation may be the most difficult.

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(a) CCD (|γab|) ground-truth (no volume): RADAR2018 BRISVEGAS [19].

(b) SAR image (dB) of full scene (ground + volume). Consists of speckle only.

(c) Single-channel CCD of full scene. ψb−ψa = 0.3◦, |γab|= 0.61

(d) Multichannel CCD using the conventional beamformer. |γyab |= 0.63

(e) Multichannel CCD using the adaptive MVDR beamformer. |γyab |= 0.87

Fig. 9. 3D SAR CCD simulation results. The scene consists of ∼2e6 pointscatterers that satisfy the RVOG model (µ=0dB, hv=20m, σdB

e =0.1dB/m).Raw echo data were synthesised at L-band (λ=22.7cm, M=3, ∆ψ= 0.05◦,ψa=35◦, ψb=35.3◦) and focused into SAR images (res. 1×1.2m no window).Between passes, those ground scatterers under a text mask (line-width 1.9 m)were shifted randomly (std deviation 12 cm), causing the decorrelation in (a).