complex interferometric coherence loci for...
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COMPLEX INTERFEROMETRIC COHERENCE LOCI FOR THE RVOGAND OVOG MODELS WITH A DOMINANT DOUBLE-BOUNCESCATTERING IN A SINGLE-TRANSMIT ACQUISITION MODE
J. David Ballester-Berman Juan M. Lopez-Sanchez Yolanda Marquez
Departamento de Fısica, Ing. Sistemas y Teorıa de la Senal (DFISTS), University of AlacantP.O. Box 99, E-03080 Alacant, Spain, Phone / Fax: +34965909597 / +34965909750,
E-mail: [email protected], [email protected], [email protected]
ABSTRACT
Mathematical expressions for the interferometric co-herence of homogeneous-volume-over-ground models(RVoG and OVoG) for the case of dominant specular scat-tering and considering a single-tx interferometer are de-rived. The analysis of the positions of the coherences onthe complex plane, as a function of the polarization chan-nel, yields important differences for the single-tx and thealternate-tx acquisition modes when the ground return isdominated by the double-bounce contributions. An ad-ditional volume decorrelation term appears when the in-terferometer is operated in a single-tx mode. This fea-ture makes more difficult the parameter inversion fromPolInSAR observables, since the topographic phase is notrepresented anymore by the crossing between the lineformed by the coherences and the unit circumference.The effect of this bistatic volume decorrelation is evenmore noticeable for the cross-polar channel in the ori-ented volume case, since the phase of the coherence withan infinite ground-to-volume ratio does not correspond tothe topographic phase, as observed for the copolar chan-nels. Additionally, a preliminary study of the influenceof the bistatic volume decorrelation as a function of thebistatic angle has been carried out.
1 INTRODUCTION
Remote sensing observations on agricultural crops withradar systems have demonstrated that for many crop typesand at many frequency bands the main contribution ofthe ground to the backscattered signal comes from thedouble-bounce interaction between the stems and theground, and not from the direct backscattering from theground surface. Consequently, direct electromagneticmodels and inversion algorithms must describe correctlythis type of ground response.In addition, the presence of the stems and other orientedstructures of agricultural plants produces an anisotropicpropagation when the electromagnetic waves travelthrough the vegetation volume. As a consequence, theattenuation (or extinction) of the signal depends on polar-ization. In general, the vertical polarization suffers froma stronger attenuation when compared to the horizontalone. This oriented nature of the volume must be also in-corporated in the model of the scene.A third important characteristic of radar observables ac-quired on agricultural crops is the presence of impor-
tant changes between measurements carried out at dif-ferent instants. The fast temporal evolution of this typeof scene constrains the use of interferometric analysesto single-pass systems, or repeat-pass systems with ex-tremely short revisiting cycles. When implementing asingle-pass interferometer, one can choose between a sys-tem with one or two transmitters. In the first case thesystem is called single-transmit (single-tx) or bistatic. Incontrast, if both antennas are used for transmitting and re-ceiving the radar signal the system is known as alternate-transmit (alternate-tx) or ping-pong.
The work presented in this paper is focused on the de-velopment of a mathematical expression for describingthe complex interferometric coherence of a simple modelnamed oriented volume over ground (OVoG) model whenthe ground contribution is dominated by the double-bounce terms and the interferometer is operated in single-tx mode. In addition, the formulation for the random vol-ume over ground (RVoG) model developed in [1] is revis-ited in order to compare also the coherence loci for bothsingle-tx and alternate-tx modes.
All mathematical derivations rely completely on theframework presented in the papers by Treuhaft et al. [1, 2,3], and hence the reader is referenced to them as a start-ing point for finding all parameter and variable definitionsand notation criteria (see Table 3 in [2, page 150]). Onlythe main steps and some definitions will be explainedagain in this work for the sake of clarity.
Once the expressions are derived, the associated posi-tions on the complex plane (also known as coherenceloci [5, 6]) are analyzed in detail and compared with theones corresponding to the alternate-tx mode. This analy-sis is useful for the interpretation of the model physicsand for obtaining important insights about the inverseproblem, i.e. the estimation of biophysical parametersfrom PolInSAR observables.
The text is organized as follows. Section 2 is devoted torevisit briefly the formulation of the RVoG model and toanalyze in detail the associated coherence loci. In Sec-tion 3 the main steps for the derivation of the complexinterferometric expression for the OVoG model are pre-sented, and the position of possible coherences is studied.Finally, the conclusions of this work are summarized inSection 4.
2 COHERENCE LOCI FOR THE RVOG
The formulation derived in [2] for the RVoG model fordominant double-bounce returns in a single-tx acquisi-tion mode is reviewed in this section. The interferometriccross product or correlation is defined as:
〈p1 · ~Et1(~R1) · p∗2 · ~E∗
t2(~R2)〉 =⟨
M∑j=1
p1 · ~Et1(~Rj)
M∑k=1
p∗2 · ~E∗t2
(~Rk)
⟩(1)
where the angle brackets perform an ensemble averageover the scattering characteristics and spatial locationsof scatterers. The received signal at ends 1 and 2 are~Et1
(~R1) and ~Et2(~R2), respectively, where subscriptst1
andt2 indicate the transmitted polarizations. Vectors~R1
and ~R2 denote the ends of the baseline. Unitary vectorsp1 andp2 represent the received polarization at each endof the interferometer, andM is the total number of con-tributing signals backscattered from elements located at~Rj .
θ0
R1
R
θspec
R1
Rspec
z=z0+hv
z=z0
z=0
z=z0+hv
z=z0
z=0
θspec
R1
Rspecz=0
R2 R2 R2
Fig 1. Volume and double-bounce scattering contributions forsingle-tx acquisition mode. Blue and red lines correspond tofields received at~R1 and ~R2, respectively.
Fig. 1 illustrates the contributions to the total receivedfield when the radar response from the ground is dom-inated by the double-bounce scattering [2]. The totalbackscattered field at endi of the interferometer can beobtained as:
〈 ~Eti(~Ri, ω0; ~R)〉 = 〈 ~EV
ti(~Ri)〉+〈 ~EGV
ti(~Ri)〉+〈 ~EV G
ti(~Ri)〉(2)
The path followed by the field for each contribution, fromthe transmitter to the receiver, depends on the interfero-metric mode.In this paper we are interested in interferometric prod-ucts obtained by combining two images acquired with thesame polarization combination, but not in polarimetric ra-tios. Then, a coherence expression can be obtained [2]by assuming a reciprocal medium and considering thatthe transmitted and received polarizations,t1 = t2 andp1 = p2, are denoted by arbitrary polarizationst and pfor both ends of the baseline. The resulting expressionfor the coherence of the RVoG model for an arbitrary po-
larization channelpt is [2, Eq.14]:
γpt = ejφ0 ·∫ hv
0e
2σxzcos θ0
+jαzzdz + µpt ·sin(κzhv)
κzhv
cos θ02σx
·(e
2σxhvcos θ0 − 1
)+ µpt
= ejφ0 ·γv + 1
I0· µpt ·
sin(κzhv)κzhv
1 + 1I0
· µpt
(3)
where all terms are defined in [2]. Note that the verticalwavenumberkz used in [4] corresponds exactly to thedefinition ofαz, so it should not be confused with theκz
term employed in this work.The distributions of possible coherences on the complexplane as a function of the ground-to-volume ratio for aRVoG model are plotted in Fig. 2 for a typical agricul-tural configuration with several combinations of extinc-tions and vegetation heights. Figs. 2a and 2c correspondto the alternate-tx configuration, which is geometricallyequivalent to a repeat-pass interferometer. Therefore, theloci occupied by the coherences coincide with previousreferences [5, 4] where they have been successfully em-ployed for estimation of biophysical parameters of vege-tation covers from airborne radar data. In Figs. 2b and 2dthe coherence loci for the single-tx mode is depicted.Note that for comparison purposes we have used the samewavenumberαz for both configurations, so the physi-cal baseline in the alternate-tx case is half the one in thesingle-tx case. In these plots, each solid line correspondsto a different scene, and one can move along the line bychanging the polarization channel, which in turn modifiesthe ground-to-volume ratio. The dashed line denotes thetopographic phaseφ0.The most important feature is the additional volumedecorrelation term (sinc term in (3)) generated by thedouble-bounce contributions in single-tx mode, as wasfirstly indicated in [1, 2]. This term produces two impor-tant effects: 1) even when the ground-to-volume ratio isinfinite, one can not reach a unit coherence, and 2) the to-pographic phase is not represented anymore by the cross-ing between the line formed by the coherences and theunit circumference. For more details and a deeper analy-sis of this scenario consult [8].This new feature leads to practical problems for the im-plementation of a geometrical approach for parameterinversion, since at least one channel with a ground-to-volume ratio as high as possible is required. Note that, forthe example in Fig. 2, changing the maximum ratio from30 to 40 dB introduces more than 10 degrees of phasechange in the estimation. Consequently, any future inver-sion approach derived from this model should take thischange into account. In particular, the requirement of atleast one channel with highµ is often met in agriculturalscenarios, but not in forestry (even at L band), so this factmay limit the use of the single-tx configuration to shortvegetation monitoring applications.On the other hand, note that together with the coherencedecrease, a potential inversion approach based only on
0.4
0.6
0.8
270
300
330
hv=1m
hv=1.5m
hv=2m
φ0
Increasing µ
0.4
0.6
0.8
270
300
330
hv=1m
hv=1.5m
hv=2m
φ0
Increasing µ
(a) (b)
0.4
0.6
0.8
270
300
330
σx=0 dB/m
σx=1 dB/m
σx=4 dB/m
φ0
Increasing µ
0.4
0.6
0.8
270
300
330
σx=0 dB/m
σx=1 dB/m
σx=4 dB/m
φ0
Increasing µ
(c) (d)
Fig 2. Loci of possible coherences of the RVoG model foralternate-tx ((a) and (c)) and single-tx ((b) and (d)) modes. Pa-rameters:θ0 = 45◦, αz = 1, κz = 0.5, φ0 = −40◦, -40 dB≤ µpt ≤ +40 dB. Cases: (a–b)σx=1 dB/m,hv= 1, 1.5 and 2 m;(c–d)σx=0, 1 and 4 dB/m,hv= 1.5 m.
the geometry of the coherence loci as proposed in [4] maynot result straightforward. However, this would not be thecase if a numerical method is used, since the number ofunknowns does not increase and a regularization shouldbe performed in order to obtain unambiguous solutionsas in the alternate-tx case.
3 COHERENCE LOCI FOR THE OVOG
In contrast with the random volume model, if an ori-ented volume is present, extinction becomes polarizationdependent. In this case, the propagation of the electro-magnetic wave can be expressed in terms of two orthog-onal polarizations calledeigenpolarizations[7]. In manycases, the vegetation layer is basically composed by ver-tically oriented elements, such as the stems in agriculturalfields. Therefore, the eigenpolarizations corresponding tothis scenario are vertical and horizontal polarizations.Note that the interferometric cross product of an orientedvolume, without an underlying ground surface, was al-ready treated in [2] and [3]. As in that case, the studyof the OVoG model needs an eigendecomposition of theaverage forward scattering matrix [7],〈Ff 〉, which is nota multiple of the identity matrix as for a random volume,and whose eigenvectors,pa andpb, are the already men-tioned eigenpolarizations that describe the propagatedwave through the oriented volume.The cross correlation of electric fields for the volumecontribution alone [2, Eq.27] is calculated by projecting
the transmitted and received fields on the eigenpolariza-tions. Then, following the indications in [2], an expres-sion of the interferometric cross correlation for an ori-ented volume with a double-bounce dominant return forthe single-tx acquisition mode will be derived. Note thata zero slope terrain is assumed. The main step is thederivation of the new expressions for the electric fields ofthe double–bounce scattering terms:〈 ~EGV
t1〉 and〈 ~EV G
t1〉.
Once these two contributions have been calculated, thetotal fields at~R1,2 will be obtained by adding them to thedirect contribution from the volume, whose expressionwas derived in [2, Eq. D6]. To proceed with the calcu-lation of the first double-bounce term (ground–volume),we have to start with the expression of the field incidenton the volume scatterer at~R, coming from the ground re-flection, which can be obtained by replicating the sameprocedure as in [2] for the RVoG case:
〈 ~Et1(~R, ω0; ~Rspec)〉 =
A · 〈R(θspec)〉 · Γrough · ejk0[|~R1−~Rspec|+|~R−~Rspec|] ·
·[(t1 · pa) · pa · e
j2πρ0λa(hv+z)k0 cos θ0 +
(t1 · pb) · pb · ej2πρ0λb(hv+z)
k0 cos θ0
](4)
where there appear the following set of parameters: com-plex eigenvalues (λa andλb), refractivity values (χa andχb), extinction coefficients (σa andσb), wavenumber infree spacek0, and density of scatterersρ0 [2, Sect. 2.3.1].The exponential terms with(hv + z) arise as a conse-quence of the propagation of the eigenpolarizations in-side the volume from the transmitter to the specular pointon the ground and then to the volume scatterer.
Next, the incident field at~R is multiplied by the volumespecular matrix,Fspec→~R1
and a free-space propagationterm, and, finally, the effect of the volume is added. Thus,the received field at~R1 due to the ground-volume inter-action is
〈 ~EGVt1
(~R1)〉 =
ejk0·|~R1−~R|
|~R1 − ~R|· Fspec→~R1
· 〈 ~Et1(~R, ω0; ~Rspec)〉+
+∫
vol
ρ0 ·ejk0·|~R1−~R′|
|~R1 − ~R′|· 〈F~R→~R′→~R1
〉·
〈 ~Et1(~R′, ω0; ~R)〉d3R′ (5)
where〈F~R→~R′→~R1〉 is the scattering matrix for a wave
coming from~R, impinging on a scatterer at~R′ and scat-tered towards~R1, and〈 ~Et1
(~R′, ω0; ~R)〉 represents the av-
erage field at~R′ from a volume scatterer at~R.
The main contribution in the volume integral comes fromthe forward scattering matrix, so it can be expressed in
terms of the eigenpolarizations and its eigenvalues:
〈 ~EGVt1
(~R1)〉 =
A2 · ejk0·[|~R1−~Rspec|+|~R−~Rspec|+|~R1−~R|]·Fspec→~R1
· 〈R(θspec)〉 · Γrough·[(t1 · pa) · pa · e
j2πρ0λa(hv+z)k0 cos θ0 +
(t1 · pb) · pb · ej2πρ0λb(hv+z)
k0 cos θ0
]+
+∫
vol
ρ0 ·ejk0·|~R1−~R′|
|~R1 − ~R′|·∑
l
λl · (pl · 〈 ~Et1(~R′, ω0; ~R)〉) · pld
3R′ (6)
Finally, following the same manipulations as for the ori-ented volume alone [2, Eqs. D2-D6], it yields:
〈 ~EGVt1
(~R1)〉 =
A2 · ejk0·[|~R1−~Rspec|+|~R−~Rspec|+|~R1−~R|] · Γrough·∑i
∑l
pl · (t1 · pi) · (pl · Fspec→~R1· 〈R(θspec)〉 · pi)·
ejk0[(χi+χl)hv−(χi−χl)z]
cos θ0 · e−(σi+σl)hv−(σi−σl)z
2 cos θ0 (7)
The field due to the volume-ground contribution can beobtained in the same way as the ground-volume term buttaking into account that the order of occurrence of scat-tering mechanisms must be interchanged and that nowthe volume specular matrix isF~R1→spec = FT
spec→~R1.
Its expression is:
〈 ~EV Gt1
(~R1)〉 =
A2 · ejk0·[|~R1−~R|+|~R−~Rspec|+|~R1−~Rspec|] · Γrough·∑i
∑l
pl · (t1 · pi) · (pl · 〈R(θspec)〉 · F~R1→spec · pi)·
ejk0[(χi+χl)hv+(χi−χl)z]
cos θ0 · e−(σi+σl)hv+(σi−σl)z
2 cos θ0 (8)
Finally, the total field received at~R1 is given by the sumin (2).A similar expression can be derived to calculate〈 ~Et2
(~R2, ω0; ~R)〉 where the ground scattering matrix ,〈R(θspec)〉, and the volume scattering matrix,Fspec→~R1
,are assumed to be equal to the corresponding ground re-flection point for the end 2. Then, its expression is ob-tained by changing to2 the subscript of unitary and posi-tion vectors.Differences appear on the summatory terms accountingfor the propagation through the oriented volume. Thesesummatories generate four terms in the most generalcase, which involve sixteen cross terms. However, sinceone must use the eigenpolarizations as transmitted andreceived polarizations, the cross terms are simplified.Indeed, the only non-zero terms correspond to indices
i, l = a, b. More explicitly, there is a simple final expres-sion only for the four combinations of eigenpolarizations:aa, ab, ba andbb. If we denote the arbitrary eigenpolar-izations aste andpe, the correlation is calculated with thefollowing expression:
〈pe · ~Ete(~R1) · p∗e · ~E∗
te(~R2)〉 =
A4 · ejφ0 ·∫ 2π
0
W 2η dη·∫ +∞
−∞W 2
r r0ejαrrdr · ρ0 · e−
(σa+σb)hvcos θ0 ·
·[|Fbpete
|2·∫ hv
0
e(σa+σbcos θ0
+jαz)zdz+
Γ2rough · (F ? R(θspec))
∫ hv
0
e
“σa−σbcos θ0
+jκz
”zdz+
Γ2rough · (F ? R(θspec))
∫ hv
0
e−j
“2k0(χa−χb)
cos θ0+κz
”zdz+
Γ2rough · (F ? R(θspec))
∫ hv
0
ej
“2k0(χa−χb)
cos θ0+κz
”zdz+
Γ2rough · (F ? R(θspec))
∫ hv
0
e−
“σa−σbcos θ0
+jκz
”zdz
](9)
Then, simplifying the double-bounce terms and calculat-ing the integrals, expression (9) is rewritten as:
〈pe · ~Ete(~R1) · p∗e · ~E∗
te(~R2)〉 = K · ejφ0 ·[
|Fbpete|2 · 1
σa+σb
cos θ0+ jαz
·(
e
“σa+σbcos θ0
+jαz
”hv − 1
)+ Γ2
rough · (F ? R(θspec))·{2
2k0(χa−χb)cos θ0
+ κz
· sin[(
2k0(χa − χb)cos θ0
+ κz
)hv
]
+2
σa−σb
cos θ0+ jκz
· sinh[(
σa − σb
cos θ0+ jκz
)hv
]}](10)
where the ensemble averages considering the volumeand the ground scattering matrices are represented byF ? R(θspec).
The normalizing factor of the coherence is obtained fromthe computation of the cross correlation in (10) with nointerferometric information considered, i.e.φ0 = 0,αz = 0 and κz = 0. Therefore, for each end of the
baseline one obtains
〈|pe · ~Ete(~R1,2)|2〉 =
K ·
[|Fbpete
|2 · 1σa+σb
cos θ0
·(
e
“σa+σbcos θ0
”hv − 1
)+
Γ2rough · (F ? R(θspec))·{
22k0(χa−χb)
cos θ0
sin[(
2k0(χa − χb)cos θ0
)hv
]+
2σa−σb
cos θ0
sinh[(
σa − σb
cos θ0
)hv
]}](11)
Then, redefining the ground-to-volume ratio as
µpe te=
2 · Γ2rough · (F ? R(θspec))
|Fbpete|
(12)
the coherence function for an oriented volume over thedouble-bounce effect, considering a single-tx interferom-eter, yields
γ = ejφ0 ·
I + µpe te·
sin[(
2k0(χa−χb)cos θ0
+ κz
)hv
]2k0(χa−χb)
cos θ0+ κz
+
sinh[(
σa−σb
cos θ0+ jκz
)hv
]σa−σb
cos θ0+ jκz
/
I0 + µpe te·
sin[(
2k0(χa−χb)cos θ0
)hv
]2k0(χa−χb)
cos θ0
+
sinh[(
σa−σb
cos θ0
)hv
]σa−σb
cos θ0
(13)
where functionsI andI0 have the same form as in theRVoG case (3) but substituting2σx by (σa + σb).Due to the short vegetation depth of agricultural crops,refractivity indices can be assumed to be equal,χa = χb
(see discussion after [1, Eq. 20]). Hence, a more con-ventional form for the complex interferometric coherencedefined for any of the four combinations of eigenpolariza-tionspa andpb in transmission and reception is obtained:
γ = ejφ0
γv + µpete
I0
[sin(κz·hv)
κz+
sinhh“
σa−σbcos θ0
+jκz
”hv
iσa−σbcos θ0
+jκz
]1 + µpete
I0
[hv +
sinhh“
σa−σbcos θ0
”hv
iσa−σbcos θ0
](14)
where numerator and denominator of (13) were dividedby I0.Previous expressions describe a single-tx interferometer.When an alternate-tx system is used, the considerationsstated in the RVoG case apply again, and a multiplying
0.8
270
300
330
0.4
0.6 φ0
Increasing µ
Pol. aaPol. ab=ba
Pol. bbx
x
x
(a)
0.8
270
300
330
0.4
0.6 φ0
Increasing µ
Pol. aa
Pol. ab
Pol. bbxx
x
x
Pol. ba
(b)
Fig 3. Loci of possible coherences of the OVoG model foralternate-tx (a) and single-tx (b) modes. Parameters:θ0 = 45◦,αz = 1, κz = 0.5, φ0 = −40◦, -40 dB≤ µpe te
≤ +40 dB,σa
= 3 dB/m,σb = 0 dB/m,hv= 2 m.
factor modifies theµpe teparameter at both numerator
and denominator of (14). Hence, the coherence for thealternate-tx mode has the same form as in (3), but with-out the sinc term (due to the interferometer mode) andredefining parameterµpt as
µpt → µpe te·
[hv +
1σa−σb
cos θ0
· sinh[(
σa − σb
cos θ0
)hv
]](15)
The coherence loci for the OVoG model corresponding toboth single-tx and alternate-tx modes, for a single typi-cal interferometric configuration, is shown in Fig. 3. Inthis plot, each solid line shows the possible positions ofeach different polarization channel, i.e.aa, bb, ab andba. Lines are defined by considering different vegetatedmedia, which translates into varying ground-to-volumeratios. Evidently, in a real experiment we obtain onlyfour coherence points, each of them lying on each line, as
depicted with crosses on the same plot.In the single-tx case (see Fig. 3b), the lines correspondingto the copolar channelsaa andbb are similar to those ofthe RVoG case when we study different extinctions (seeFig. 2d). Again, the single-tx configuration entails a de-crease in the coherence amplitude, even for an infiniteground-to-volume ratio, as a consequence of the volumedecorrelation produced by the change in the propagationpaths to both ends of the baseline. Phase topographyφ0
can be identified by the phase of the copolar coherenceswith infinite ground-to-volume ratios.The most interesting feature of this formulation, whichis illustrated in Fig. 3b, is the distribution of the com-plex coherence for the cross-polar channels. First, bothcrosspolar channels,ab and ba, do not show the sameloci (except for a null ground contribution) because ofthe different extinctions within the vegetation volume andthe bistatic configuration. Second, the crosspolar linesare not confined in the region defined by the two copo-lar cases, in contrast to the alternate-tx case. Finally,and more importantly, the phase of the crosspolar coher-ences with an infinite ground-to-volume ratio does notcorrespond to the topographic phase. Mathematically,this phase shift can be identified in (14), because whenσa 6= σb the total phase of the coherence is modified.From the physical point of view, the phase change is pro-duced by the different extinctions affecting the double-bounce contributions,〈 ~EGV
t1〉 and 〈 ~EV G
t1〉, when propa-
gating to end 1 of the baseline in contrast to end 2. Thedifferent paths inside the vegetation volume, with differ-ent extinctions in the OVoG case, produce different mag-nitudes of the received fields at both antennas. Conse-quently, the resulting phase of the interferometric crossproducts is not cancelled and a shift proportional to thevegetation height appears.
3.1 Effect of the bistatic angle
In order to quantify the impact of the additional decor-relation term as a function of the bistatic angle, a num-ber of additional simulations have been performed. Pa-rameter values are assumed to be those from the futureTanDEM-X space-borne system [9], with the maximumbaseline of 2 km. Fig. 4a shows the interferometric geom-etry, and Figs. 4b-c illustrate the effect of baseline vari-ation on the coherence loci for incidence angles of 25◦
and 45◦, respectively. It is observed thatαz should bemaintained below 1.5 in order to avoid low coherences,which would degrade a potential inversion application.This upper limit ofαz corresponds to baselines of 889.5m and 1907.4 m, respectively. Note that, for simplicity,we have assumed that the angle formed by the baselineand the horizontal is equal to the incidence angle. Thecorresponding values ofκz are 0.27 and 0.75 for eachincidence angle.Additionally, the spectral shift due to baseline decorre-lation must be also considered [10]. In these cases, thebaselines for incidences of 25◦ and 45◦ produce spectral
A1
A2
B
∆θ
θ1
θ2Bn
r1
r2
H
h
δ
(a)
0.2 0.4
0.6
0.8
210
240
90
270
120
300
150
330
180 0
αz = 2
B = 1186 m
αz = 1.5
B = 889.5 m
αz = 0.5
B = 296.5 m
αz = 1
B = 593 m
αz = 0.75
B = 444.75 m
(b)
0.2
0.4
210
240
90
270
120
300
150
330
180 0
αz = 1.5
B = 1907.4 m
αz = 0.5
B = 635.8 m
αz = 1
B = 1271.6 m
αz = 0.75
B = 953.7 m
0.6
0.8
(c)
Fig 4. Loci of possible coherences of the OVoG model forsingle-tx as a function of bistatic angle: a) InSAR geometry;b) θ0 = 25◦; c) θ0 = 45◦. Parameters:f = 9.65 GHz,δ = θ0,H = 514 km, z0 = −0.7 m, -40 dB≤ µpe te
≤ +40 dB (-10 dB and +10 dB values are marked with bigger dots),σa = 3dB/m,σb = 0.5 dB/m,hv= 2 m.
shifts of 32.5 MHz and 25.3 MHz, respectively, which areacceptable considering that the TanDEM-X system pro-vides a 150 MHz nominal bandwidth.
4 CONCLUSIONS
This study is presented as an extension of the works byTreuhaft et al. [1, 2, 3] and is aimed to derive an ex-pression of the complex interferometric coherence of theOVoG model in a specific situation: a ground responsedominated by the double-bounce contribution.As stated in [2] for the RVoG, an additional volumedecorrelation term appears when the interferometer is op-erated in single-tx mode, which would add complexity ina potential inversion of the model parameters, speciallyby geometric approaches. This effect becomes more im-portant for the cross-polar channels in the OVoG case,since the phase with an infinite ground-to-volume ratiodoes not correspond to the topographic phase, as ob-served for the copolar channels. In addition, copolarchannels do not show the same loci except for a nullground contribution because of the different extinctionswithin the vegetation volume and the bistatic configura-tion.Some results of this work could provide useful informa-tion in the definition of vegetation monitoring applica-tions based on PolInSAR data. Together with the rec-ommendation of a single-pass configuration to avoid theextreme temporal decorrelation caused by the fast growthof crop plants, it is necessary to take into account the op-erational mode (single-tx or alternate-tx) from the view-points of the system design and of the parameter retrievalalgorithms.Complementarily, the formulation for a wide range ofmixtures of ground responses (from pure surface returnto the already treated double-bounce interaction) has tobe addressed for generalizing this technique to a widerrange of agricultural crop types and soil situations.Finally, recent investigations have demonstrated that thevertical distribution of backscattering properties of manycrop types does not fit the exponential distribution as-sumed in this model, leading to other functions whichshould be incorporated in the formulation to obtain moreprecise expressions for the POLInSAR observables.
ACKNOWLEDGMENT
This work has been supported by the Spanish Ministeryof Education and Science (MEC) and EU FEDER, underProject TEC2005-06863-C02-02.
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