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Article

An Approach to Extended Fresnel Scattering for Modeling of Depolarizing Soil-Trunk Double-Bounce Scattering

German Aerospace Center, Microwaves and Radar Institute, P.O. BOX 1116, 82234 Wessling, Germany
Remote Sens. 2016, 8(10), 818; https://doi.org/10.3390/rs8100818
Received: 15 June 2016 / Revised: 8 September 2016 / Accepted: 26 September 2016 / Published: 1 October 2016

Abstract

:
Focusing on scattering from natural media, dihedral (double bounce) scattering is often characterized as a soil-trunk double Fresnel reflection, like for instance, in most model-based decompositions. As soils are predominantly rough in agriculture, the classical Rank 1 dihedral scattering component has to be extended to account for soil roughness-induced depolarization. Therefore, an azimuthal Line of Sight (LoS) rotation is applied solely on the soil plane of the double-bounce reflection to generate a depolarized dihedral scattering signal in agriculture. The results of the sensitivity analysis are shown for a distributed target in coherency matrix representation. It reveals that the combination of coherency matrix elements T22XD + T33XD is quasi-independent of the roughness-induced depolarization, while (T22XD − T33XD)/(T22XD + T33XD) is quasi-independent of the dielectric properties of the reflecting media. Therefore, a depolarization-independent retrieval of soil moisture or a direct roughness retrieval from the extended dihedral scattering component might be possible in stalk-dominated agriculture under certain conditions (e.g., the influence of a differential phase stays at a low level: ϕ < 15°). The first analyses with L-band airborne-SAR data of DLR’s E-SAR and F-SAR systems in agricultural regions during the AgriSAR, OPAQUE, SARTEO and TERENO project campaigns state the existence and potential of the extended Fresnel scattering mechanism to represent dihedral scattering between a rough (tilled) soil and the stalks of the agricultural plants.

Graphical Abstract

1. Introduction

The increasing availability of fully-polarimetric datasets from space-borne SAR systems like ALOS-1/-2, Radarsat-2, TerraSAR-X and TanDEM-X motivates the investigation of polarimetric scattering mechanisms for natural media [1,2,3,4,5,6,7]. As natural media are mostly distributed scatterers, assuming more than one scatterer within the resolution cell, mathematical notations including second order statistics, like the coherency matrix [T], are needed for a complete description of the polarimetric scattering scenario [8]. As mostly more than one dominant scattering process is occurring within the resolution cell, polarimetric decompositions are one option to analyze the different scattering components.
There are two common ways mainly published in the literature to decompose fully-polarimetric data represented by [T] [8,9] Eigen-based and model-based decompositions. The Eigen-based methods represent a mathematical approach for decomposition, which result in a set of eigenvalues and eigenvectors [9,10]. These parameters are not straightforward to interpret in their physical sense. The model-based decompositions, on the contrary, consist of simple, physically-based electro-magnetic scattering models, which are easy to interpret in terms of scattering physics, but have to be pre-defined by at least some a priori knowledge [11,12]. One of the most popular configurations for a standard three-component, model-based polarimetric decomposition for natural media consists of a surface component, a dihedral component and a volume component [13]. Hajnsek et al. [14] showed how to incorporate Rank 3 extended Bragg surface scattering into the surface component in order to account for roughness-induced depolarization on agricultural ground. This is of special importance for agricultural areas, where the soil roughness is distinct and cannot be neglected. Similarly, a dihedral scattering component will be developed in Section 2, which also accounts for roughness-induced depolarization, because up to now, predominantly smooth double Fresnel scattering is used in standard decompositions [11,13,14]. A sensitivity analysis for the coherency matrix elements on the dielectric content of the soil and the trunk plane, the differential phase, the loss factor, as well as on the local incidence angle is given in Section 3. The appearance of extended Fresnel scattering in agriculture is investigated in Section 4 using fully-polarimetric, airborne SAR data. The potentials and limitations of this novel, depolarizing, scattering component are discussed in Section 5 followed by a summary and the first conclusions in Section 6.

2. Development of the Extend Fresnel Scattering Model

2.1. Rank 1 Fresnel Scattering

The canonical dihedral component is expressed as a double reflection occurring at each of two smooth, orthogonal planes of a dihedral scatterer [8]. Therefore, the dihedral scattering matrix of a soil-trunk reflection (assuming flat ground and stalk-dominated vegetation arranged in dense rows) is modeled as:
[ S D ] = [ R H s R H t ( ω l , ε S , ε T ) 0 0 R V s R V t e i φ ( ω l , ε S , ε T ) ]
RHs and RVs are the horizontal and vertical Fresnel coefficients of the soil scattering plane, which depend on the dielectric constant of the soil εs and the local incidence angle ωl [8]:
R H s = cos ω l ε s sin 2 ω l cos ω l + ε s sin 2 ω l ,   R V s = ε s cos ω l ε s sin 2 ω l ε s cos ω l + ε s sin 2 ω l
RHt and RVt are the Fresnel coefficients of the trunk scattering plane, which include the dielectric constant of the trunk εt and the local incidence angle ωl [8]. For modelling reasons, it is assumed that the vertical scattering plane is formed by the plant trunks planted densely in seeding rows.
R H t = cos ( π 2 ω l ) ε t sin 2 ( π 2 ω l ) cos ( π 2 ω l ) + ε t sin 2 ( π 2 ω l ) ,   R V t = ε t cos ( π 2 ω l ) ε t sin 2 ( π 2 ω l ) ε t cos ( π 2 ω l ) + ε t sin 2 ( π 2 ω l )
The phase angle φ within the factor e i ϕ accounts for the phase difference between the HH- and VV-polarization backscatter incorporated, if propagation through oriented media occurs (further details in [15]). The Pauli scattering vector is obtained as [8,9]:
k D = 1 2 [ R H s R H t R V s R V t e i φ , R H s R H t + R V s R V t e i φ , 0 ] T
[ T D ] = k D k D + = f d [ | α D | 2 α D 0 α D * 1 0 0 0 0 ]
The resulting coherency matrix [TD] follows as derived in [13] and is parameterized by the complex scattering mechanism ratio αD and the real backscattering amplitude fD [14]:
α D = R H s R H t R V s R V t e i φ R H s R H t + R V s R V t e i φ f D = 1 2 | R H s R H t + R V s R V t e i φ | 2
In addition, the power PD is represented by the trace of [TD]: PD = fD (1 + D|2).
As the dihedral scattering model assumes two specular reflections on each of the two orthogonal planes, the model is adapted subsequently for application on natural, respectively lossy, surfaces. Thus, the loss due to scattering at the soil plane is considered by a scattering loss factor LS [11,16]:
Gaussian :   L S = exp ( 2 k 2 σ 2 cos 2 ω l )
Exponential :   L S = exp ( 2 k σ cos ω l )
where k is the wave number, σ the standard deviation of the vertical roughness of the soil and ωl is the local incidence angle [12]. The Fresnel coefficients of the soil scatterer (RHs, RVs) are altered to the so-called modified Fresnel coefficients (RLHs, RLVs) accordingly [11]:
R L H s = R H s × L S R L V s = R V s × L S
The complex ratio αD is independent of the scattering loss factor LS, because the multiplicative factor cancels out. Thus, only the backscattering amplitude fD is affected by the soil roughness loss. However, the roughness-induced depolarization is not included, which would also affect the complex ratio αD [14].
f D l o s s = 1 2 | R H s L s R H t + R V s L s R V t e i φ | 2 = 1 2 | L S ( R H s R H t + R V s R V t e i φ ) | 2 = f D | L S | 2
Finally, the coherency matrix for the modified dihedral scattering contribution [TDLoss] is written as [14]
[ T D L o s s ] = f D | L S | 2 [ | α D | 2 α D 0 α D * 1 0 0 0 0 ]

2.2. Rank 3 Extended Fresnel Scattering

In order to account for soil roughness-induced depolarization, a Line of Sight (LoS) rotation of θ, solely around the soil plane (RHs, RVs), is introduced in Equation (12). It is important to note that the rotation is only applied to the soil plane, like in Equation (12), while the trunk plane is assumed static (no trunk roughness) [12]. A conceptual visualization is given in Figure 1. Hence, the orientation of the vegetation volume is not a variable in this approach. Therefore, the assumption is made that the second dihedral scattering event (scattering center at the stalks/stems of the vegetation) is approximated as Fresnel scattering from a vertical plane. The orientation of the vegetation is expected to be located within this plane. Hence, only rotations perpendicular to the soil and to the incidence plane are assumed. Rotations would only matter as far as the vegetation structure forming the second plane reaches outside of this plane [17,18]. The double Fresnel reflection is a distinct assumption within the dihedral scattering component and represents a significant simplification of the scattering scenario. However, if only the soil plane of the dihedral scattering mechanism rotates, whereas the trunk one does not, backscattering is no more at specular direction. This is due to incoherent scattering from rough soil and coherent reflection by the trunk plane. For a non-rotated soil plane, we have coherent reflection from both rough soil, as well as trunk. Hence, the Fresnel coefficients can be used for any terrain roughness. Conversely, for a (only) rotated soil plane, its scattering is incoherent, and Bragg or Fresnel coefficients can be applied according to surface roughness: Bragg for small roughness; Fresnel for distinct roughness. The latter was adopted in the case of extended Fresnel scattering in agriculture. These simplifications are needed to keep the model complexity and the space of variables low. Eventually, the test with experimental SAR data in Section 4 will give the first indications if this novel scattering model can describe the data to a certain extent.
[ S X D ] = L S X D 2 ( ( [ 1 0 0 1 ] [ 1 0 0 e i φ ] [ C o s θ S i n θ S i n θ C o s θ ] [ R H s 0 0 R V s ] [ C o s θ S i n θ S i n θ C o s θ ] T [ R H t 0 0 R V t ] ) + ( [ R H t 0 0 R V t ] [ C o s θ S i n θ S i n θ C o s θ ] [ R H s 0 0 R V s ] [ C o s θ S i n θ S i n θ C o s θ ] T [ 1 0 0 e i φ ] [ 1 0 0 1 ] ) )
Equation (12) contains both possible ray paths to obtain reciprocity (see also Equation (10) in Dahon et al., [19]): soil-trunk reflection and trunk-soil reflection, respectively. Lee et al. published a depolarizing dihedral component, where both the soil and the trunk plane are rotated simultaneously by a LoS rotation of the dihedral coherency matrix [20]. However, the assumption of a trunk roughness (rotation around the trunk plane) in agriculture is physically hardly justifiable, as the stalks forming the trunk plane are supposed to be static during acquisition. In addition, Neumann and Al-Kahachi introduced the rotation solely around the vertical plane (forest trunks/methane bubbles in lake ice) in the dihedral component to model different forest vegetation (height >10 m)/ice components [21,22] at L-band. However, both approaches also increase the entropy of the trunk scattering and are not as closely related to the physical scattering mechanism of a rough soil and an agricultural plant (e.g., wheat stalks) as the proposed one. This might be present in agricultural areas (height < 3 m) at the L-band and will be investigated with real data in Section 4. Here, only a rough and depolarizing soil plane is assumed, modelled by an LoS rotation, and a smooth trunk plane (plant stalks). Hence, the rotated scattering matrix [SXD] is presented as:
[ S X D ] = L S X D [ R H t ( R H s cos 2 θ + R V s sin 2 θ ) 1 4 ( R V s R H s ) ( R V t R H t e i φ ) sin 2 θ 1 4 ( R V s R H s ) ( R V t R H t e i φ ) sin 2 θ R V t e i φ ( R V s cos 2 θ + R H s sin 2 θ ) ] = L S X D [ S X D 11 S X D 12 S X D 21 S X D 22 ]
It can be seen that the model of extended Fresnel scattering fulfills the reciprocity theorem, which is obligatory for monostatic acquisition systems (SXD12 = SXD21). The Pauli scattering vector is formed in Equation (14) to account for distributed targets by calculating in the next step the coherency matrix [TXD] from the outer product of the scattering vectors [10].
k X D = L S X D 2 [ S X D 11 + S X D 22 S X D 11 S X D 22 S X D 12 + S X D 21 ] T
Within the formation of the coherency matrix in Equation (15), the angle accounting for the soil roughness is integrated over a rotation angle range from −θ1 to θ1 with a uniform probability function (pdfθ = 1/(2θ1)) representing the rotation limits (±θ1) and probability distribution (pdfθ) of the rotation angles of the soil plane [11]. Hence, the strength of the soil roughness-induced depolarization is steered by this rotation limit angle. The assumption of a uniform distribution width pays tribute to a randomly-oriented small-scale roughness, which is predominant for stalk-grown agricultural crops, where the seedbed (with more smooth soil conditions) was prepared during planting. Hence, major row structures from ploughing should not be present anymore, and therefore, a uniform distribution (more random distribution of soil roughness) seems likely to represent the true scattering case for seedbed conditions. It is not expected that certain roughness angles are of higher probability to occur than others, especially within the vegetation growing period in agriculture (after seedbed preparation). However, this is an assumption and can be adapted in the future to generalize the approach to different soil roughness cases. Anyway, as the pdf is a factorial term within the calculus of the coherency matrix elements (cf. Equation (15)), the ratios of coherency matrix elements will not be affected by a change of the probability density function.
After integration, the resulting coherency matrix [TXD] is reflection symmetric, and subsequently, the off-diagonal elements TXD13, TXD23, TXD31 and TXD32 are zero.
[ T X D ] = | L S X D | 2 θ 1 θ 1 k X D k X D T * p d f θ d θ
The single, non-zero coherency matrix elements are listed in Equations (16) and (18–21). They show the different dependencies on the Fresnel coefficients (RHs, RHt, RVs, RVt), the rotation limit angle (θ1) indicating the roughness-induced depolarization, the roughness intensity loss factor (LSXD) and the differential phase angle (ϕ). From Equation (16), it is clear that the LSXD-factor is obsolete, if a ratio of the coherency matrix elements is taken for analysis.
[ T X D ] = | L S X D | 2 [ T X D 11 T X D 12 0 T X D 12 * T X D 22 0 0 0 T X D 33 ]
In order to comply with the notation of standard model-based scattering components, a dihedral intensity component fXD and a dihedral scattering mechanism αXD can be defined similar to Equation (6). Furthermore, in the case of extended Fresnel scattering, only the fXD-component is affected by the LSXD‑loss factor. However, both variables (αXD, fXD) of this extended Fresnel scattering component are influenced by the roughness-induced depolarization, incorporated by the LoS rotation and integration over the limit angle θ1.
f X D = | L S X D | 2 T X D 22     α X D = T X D 12 / T X D 22
T X D 11 = 1 16 ( ( R V t 2 e i 2 φ ( 3 R V s 2 + 2 R V s R H s + 3 R H s 2 ) 2 R V t R H t e i φ ( R V s 2 + 6 R V s R H s + R H s 2 ) + R H t 2 ( 3 R V s 2 + 2 R V s R H s + 3 R H s 2 ) ) + ( R V s R H s ) ( R V t e i φ + R H t ) ( 4 ( R V s + R H s ) ( R V t e i φ R H t ) sinc ( 2 θ 1 ) + ( R V s R H s ) ( R V t e i φ + R H t ) sinc ( 4 θ 1 ) ) )
T X D 22 = 1 16 ( ( 2 R V s R H s ( R V t 2 e i 2 φ + 6 R V t R H t e i φ + R H t 2 ) + R V s 2 ( 3 R V t 2 e i 2 φ + 2 R V t e i φ R H t + 3 R H t 2 ) + R H s 2 ( 3 R V t 2 e i 2 φ + 2 R V t e i φ R H t + 3 R H t 2 ) ) + 4 ( ( R V s 2 R H s 2 ) ( R V t 2 e i 2 φ R H t 2 ) sinc ( 2 θ 1 ) ) + ( ( R V s R H s ) 2 ( R V t e i φ R H t ) 2 sinc ( 4 θ 1 ) ) )
T X D 12 = 1 4 ( R V s 2 R H s 2 ) ( R V t 2 e i φ + R H t 2 ) sinc ( 2 θ 1 ) ( R V t 2 e i 2 φ R H t 2 ) ( 1 16 ( 3 R V s 2 + 2 R V s R H s + 3 R H s 2 + ( R V s R H s ) 2 sinc ( 4 θ 1 ) ) )
T X D 33 = 1 16 ( R V s R H s ) 2 ( R V t R H t e i φ ) 2 ( 1 sinc ( 4 θ 1 ) )
Figure 1 is a conceptual visualization of the double-bounce scattering scenario for a depolarizing (rotated) soil and a static trunk plane including the different input variables for extended Fresnel scattering.

3. Sensitivity Analysis of Extended Fresnel Scattering for Distributed Targets

In this section, the sensitivity of the extended Fresnel scattering model regarding the different input parameters is investigated in detail. The emphasis was put on the depolarizing components. Therefore, the influence of the differential phase and the roughness loss factor are neglected (ϕ = 0°, LSXD = 1) in the first place, but investigated in the following subsection. Furthermore, combinations of coherency matrix elements will be studied to find candidates, which are at best independent of the roughness-induced depolarization or dielectric properties of the reflecting media.
For Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, the model sensitivity is shown for the full range of incidence ωl and roughness θ1 angles to provide an overview of how the roughness-induced depolarization affects the different coherency matrix elements, as well as the polarimetric entropy and the scattering alpha angle at all possible incidence angles. The fixed dielectric constants are partly deduced from measurements for the soil (see Table 1 in the manuscript), but are selected according to a reasonable choice for a low (εt = 10) and a high (εt = 30) trunk moisture and a medium soil moisture (εs = 20). Figure 2, Figure 3 and Figure 4 show the characteristics of the coherency matrix elements TXD11, TXD12, TXD22 and TXD33 with the roughness-induced depolarization (θ1) for different local incidence angles (ωl) and dielectric properties of soil (εs) and trunk (εt). The general trend of the coherency matrix elements persists for different constellations of ωl, εs and εt, which means:
  • TXD11, TXD12 are decreasing by 3–6 dB with increasing depolarization, while TXD11 performs as the more stable component of both coherency matrix elements.
  • TXD22 decreases by 3–4 dB with increasing depolarization, but stays always higher than TXD11, which is a mandatory condition for the presence of dihedral scattering (compared to surface scattering).
  • TXD33 increases up to −10 dB from a Rank 1 (TXD33 = 0) to a Rank 3 (TXD33 > 0) scattering mechanism with increasing depolarization. For weak to medium depolarization (first half of the θ1-range: 0°–45°) TXD22 dominates over TXD33, which reverses for the case of strong depolarization (second half of the θ1-range: 45°–90°).
Figure 5, Figure 6 and Figure 7 illustrate the behavior of the coherency matrix elements along the local incidence angle (ωl) for different roughness (θ1) and moisture conditions of the soil (εs) and the trunk (εt). However, only the ωl-range from 25°–70° is analyzed, wherein the majority of remote sensing platforms acquire data for non-mountainous regions. In general, the trend with local incidence angle (ωl) has a maximum or a minimum at approximately 45° depending on the coherency matrix element. Thus, one rises from 25°–45° and then decreases from 45° down to 70°, or vice versa. The behavior of the single elements can be explained in more detail:
  • TXD11 decreases until approximately 45° and then increase again to the starting level.
  • TXD12 shows the same behavior as TXD11, but less pronounced.
  • TXD22 and TXD33 increase until approximately 45° and then decrease to the starting level, while it depends on the roughness depolarization level (θ1), which curve is superior with respect to the other.
  • The crossing points between TXD11 and TXD22 represent the Brewster angles of the soil (right crossing) and trunk (left crossing) planes, respectively (see the red points in Figure 5, Figure 6 and Figure 7). The position of the Brewster angles along incidence is related to the soil and trunk dielectric constants. For example, Watanabe et al. analyzed the angular position of the Brewster angles for the potential to retrieve moisture of the soil and the trunks in forested areas [23]. This is an alternative multi-angular method for moisture retrieval, which directly depends on the distinct change of the co-polar phase in dihedral scattering along incidence. The more the covering vegetation canopy is changing the polarization of the penetrating EM waves, the less significant is the phase change and the more biased is the localization of the Brewster angles.
In addition, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 depict the behavior of the polarimetric entropy H and the mean scattering alpha angle α with different soil and trunk dielectric contents (εs, εt), as well as the local incidence ωl and the roughness depolarization angle θ1. The polarimetric entropy H is a measure for the degree of disorder in scattering leading to depolarization, while the mean scattering alpha angle α, ranging from 0–π/2, represents an intrinsic scattering type. Both parameters (H, α) stem from an Eigen-based decomposition of the coherency matrix [TXD] [8].
[ T X D ] = λ 1 e 1 e 1 T * + λ 2 e 2 e 2 T * + λ 3 e 3 e 3 T *
where the coherency-matrix [TXD] is decomposed into its eigenvalues λ and normalized eigenvectors e, in which T* denotes the transpose conjugate. Together with the pseudo-probabilities P and n = 3 (for monostatic systems), the polarimetric entropy H and the mean scattering alpha angle α are formed [8]:
P i = λ i / j = 1 n λ j
H = i = 1 n P i log 3   P i  
α = i = 1 n P i acos ( | e i 1 | )   ,   e i = [ e i 1 e i 2 e i 3 ] T
The derivation of both Eigen-based parameters (H, α), as well as their physical meaning are explained thoroughly in [8].
The polarimetric entropy, which is closely linked to the roughness-induced depolarization [24], increases significantly with θ1, as shown by Figure 8 and Figure 9. While, Figure 10 in comparison with Figure 8 states that an increase of the dielectric constants (εs, εt) does not lead to a distinct increase in polarimetric entropy, which is driven by θ1. Therefore, both plots exhibit approximately the same entropy-level of H = 0.5. Moreover, the mean scattering alpha angle α, displayed in Figure 11, Figure 12 and Figure 13, ranges always above 60° between local incidence angles of 20°–70°, which is the most common angle range for agricultural SAR monitoring. As known from polarimetric scattering theory, this behavior of α is expected for dihedral dominant scattering, which should always range higher than 60° stating anisotropic to isotropic dihedral scattering (α → 90°) [25]. Finally, Figure 14 and Figure 15 present the polarimetric Hα scattering plane for a medium local incidence angle (ωl = 35°), revealing the sensitivity of H and α concerning roughness-induced depolarization θ1 and the dielectric content of both scattering planes (εs, εt). The dynamics are indicated by black arrows within the figures. The increase in entropy H with θ1 is clearly visible for both cases and expected, because an increase in roughness/depolarization is clearly linked with a rise in disorder/entropy.
However, the dynamics concerning the mean scattering alpha angle are related to the variation in the trunk εt, as well as the soil εs dielectric constant, indicating a similar trend (see black arrows in Figure 14 and Figure 15) for both parameters within the scattering planes. Therefore, the discrimination and inversion of soil and trunk moisture cannot be conducted unambiguously within the polarimetric Hα scattering plane, as changes of both moistures trigger similar patterns within the plane. In addition, for entropies higher than 0.6, the different realizations are localized in a dense grid, and an unambiguous inversion becomes impossible. This is a major difference to the X-Bragg model, where an LoS-rotation-invariant inversion of soil moisture and roughness/depolarization in the Hα plane was feasible straightaway [24]. A variety of combinations of coherency matrix elements were tested with respect to their independence on roughness-induced depolarization or on dielectric properties of the reflecting media for future inversion purposes. The coherency matrix combination TXD22 + TXD33 is investigated with its dependencies on ωl, θ1, εt and εs in Figure 16, Figure 17 and Figure 18. Concentrating on Figure 16, the combination TXD22 + TXD33 exhibits almost no sensitivity with respect to the roughness-induced depolarization, even for different values of ωl, εt and εs. This indicates quasi-independence on roughness-induced depolarization. In addition, Figure 17 and Figure 18 present a strong dependency on the dielectric constant of the soil (εs) and the dielectric constant of the trunk (εt). Here, the sensitivity is strongest for the lower range of dielectric constants from two to approximately 25. Therefore the combination of coherency matrix elements TXD22 + TXD33 seems to be an appropriate candidate to study the dielectric properties of the reflecting media without influence from roughness-induced depolarization.
In contrast, Figure 19, Figure 20, Figure 21 and Figure 22 show the combination of coherency matrix elements (TXD22TXD33)/(TXD22 + TXD33) with respect to the roughness-induced depolarization (θ1). Since also the function sinc(4θ1) is plotted as a reference in Figure 19, Figure 20, Figure 21 and Figure 22, the overlap of the two curves for the different ranges of ωl and of dielectric constants (εs, εt) is clearly visible. Hence, this combination of coherency matrix elements is only depending on the roughness-induced depolarization and not on the dielectric properties of the media. Moreover, the trend with soil roughness depolarization (θ1) can be modeled by a simple sinc-function and is equivalent to the behavior of the circular coherence magnitude (for reflection symmetric scattering) [12,24]. However, in the case of an inversion for soil roughness depolarization (θ1), ambiguities occur for strongly depolarizing scenarios with θ1 > 63°.

Impact of Differential Phase ϕ and Scattering Loss LSXD on Coherency Matrix Combinations

As seen in Equation (13), the differential phase ϕ between the two co-polarizations (HH, VV) accounts for the propagation phase difference occurring during the pass through the vegetation volume. Vegetation effects due to preferential orientations have an individual influence on the different polarizations. In Figure 23 and Figure 24, the effect of an increased propagation difference affecting the two co-polarizations is shown for the two T‑matrix combinations TXD22 + TXD33 and (TXD22TXD33)/(TXD22 + TXD33). The influence of ϕ on TXD22 + TXD33 for phase differences of up to 30° stays below 1 dB of change in power for all roughness levels (0 < ks < 1), which are conventionally occurring due to soil cultivation. Therefore, the differential phase indicates a low impact on this coherency matrix combination. Figure 24 indicates the influence of ϕ on |(TXD22TXD33)/(TXD22 + TXD33)| for phase differences of up to ϕ = 60°. With increasing ϕ the dynamic range of |(TXD22TXD33)/(TXD22 + TXD33)| decreases, and the sensitivity to higher roughness levels (θ1 > 45°) declines strongly until a complete loss of sensitivity in the range of θ1 > 40° for a ϕ-level of 30°. Focusing on the impact of the scattering loss LSXD, the coherency matrix combination TXD22 + TXD33 is directly depending on the loss level reducing its power from −7 dB down to −11.5 dB with decreasing of LSXD from 1.0 (no loss) to 0.6 (60%), as indicated in Figure 25. Fortunately, the matrix combination (TXD22TXD33)/(TXD22 + TXD33) is a ratio, canceling the LSXD-factor, and is therefore independent from the influences caused by scattering losses.

4. Investigation of Experimental SAR Data for Extended Fresnel Scattering in Agriculture

A rough dihedral scattering component, exhibiting extended Fresnel (X-Fresnel) scattering, requires a distinct vertically-oriented scattering medium, e.g., plant stalks, as well as a horizontal scattering medium with a rough surface, like for instance, a rough soil surface created by agricultural field cultivation.
In order to study extended Fresnel scattering, fully polarimetric SAR data at the L-band of the AgriSAR [26], OPAQUE [27], SARTEO [28] and TERENO [29] project campaigns, including a big variety of crop types in different phenological stages were used. Initially, the data were acquired by DLR’s E-SAR and F-SAR sensor for polarimetric scattering analyses over several agricultural sites [26,27,28,29]. L-band was selected as the appropriate wavelength for the investigation of X-Fresnel scattering due to its higher penetration capability into agricultural vegetation compared to shorter wavelength (X- and C-band) and due to its sufficient signal-to-noise ratio compared to even longer wavelengths, like P-band. Winter crop fields, like for instance winter wheat or winter barley, and farming grassland, which have a distinct stalk component in May, June and July exhibiting potentially X-Fresnel scattering, are investigated for their scattering behavior.
In order to compare the model with data, the scattering mechanism parameter αXD is calculated from the model (see Equation (17)) and from the L-band SAR data:
α X D D a t a = | T 12 | / T 22
Figure 26 shows the modeled αXD from the X-Fresnel scattering model with the variation of the roughness depolarization angle θ1 for different values of local incidence angle ωl assuming a medium moisture scenario (εs = 25, εt = 15). The analysis indicates that αXD from the X-Fresnel model generally decreases until about 50° local incidence and then strongly rises until ωl = 70°. This increase is especially strong for roughness angles around 50°. Concerning very rough soils (θ1 > 70°), the behavior is opposed to the previous case, and αXD diminishes with respect to increasing local incidence.
Moving to the analysis of αXD in the data, Figure 27 displays exemplarily the main scattering mechanisms indicated by the normalized scattering components of the Pauli decomposition for two campaigns (AgriSAR 2006, OPAQUE 2007). The focus of analysis concerning the different scattering mechanisms is on the red color in Figure 27a–c indicating the even-bounce/dihedral scattering as dominant, which is clearly visible on several agricultural fields. The land use of the AgriSAR and OPAQUE campaigns (Figure 27d,h) states that predominantly winter crops and grassland exhibit dihedral/Fresnel and potentially X-Fresnel scattering as the dominating scattering mechanism compared to surface or volume scattering in May–July.
For a first comparison, the αXD-parameter is calculated from the OPAQUE and AgriSAR data and presented in Figure 27e–g. Regions with strong surface scattering, which appear blue in Figure 27a–c, exhibit high αXD-values (αXD > 1.0) in Figure 27e–g, while regions with distinct dihedral scattering indicate low αXD-values (αXD < 0.8). This implies a domination of the correlation term T12 over the even-bounce/dihedral term T22 in the case of surface Bragg scattering compared to dihedral Fresnel scattering and vice versa. In Figure 28, the αXD-values from the Fresnel scattering model (red dashed line) and from the extended Fresnel (black lines) scattering model are compared with mean of field values from the agricultural regions within the AgriSAR, OPAQUE, SARTEO and TERENO project data [26,27,28,29]. Black lines with varying symbols indicate different roughness (θ1) (depolarization) cases in Figure 28. The single signs with error bars represent αXD-values from the different campaigns (AgriSAR: plus = June 2006, triangle up = July 2006; OPAQUE: square = May 2007; SARTEO: diamond = May 2008; TERENO: triangle down = May 2012 at the Bode test site, triangle right = May 2012 at the Demmin test site E-W track, triangle left = May 2012 at the Demmin test site N-S track); and the colors assigned to different stalk-dominated crop types from field grass, winter barley, winter triticale, grassland to winter wheat. Despite the distinct standard deviation (gray bars in Figure 28), quantifying around αXD = 0.1 for all of the means of field values, the match of the mean of the field values with the validity region, spanned by the modeled αXD-curves for different soil roughness scenarios, is clearly recognizable. While, for instance, the winter barley field (yellow color in Figure 28) seems to be more or less close to the red dashed line, revealing more Fresnel scattering, especially the field grass and winter triticale fields (orchid and green color in Figure 28) appear far from the Fresnel scattering line, showing extended Fresnel scattering with a distinct soil roughness component.
The different plots within Figure 28 reveal the dynamics of the modelled scattering mechanism αXD with local incidence angle ωl for different dielectric constants of the soil and the stalks (εs, εt). While the level of modelled αXD for low incidence angles (ωl < 35°) is strongly depending on εt (stalks) and minor on εs (soil), the situation is reversed for high incidence angles (ωl > 55°). Therefore, the best modelled αXD-case is taken in Figure 29 by adjusting the level of εs according to available field measurements of soil moisture from the different in situ measurement campaigns. A mean εs-level of 10 was found as the most representative approximated dielectric constant value of the soil concerning all campaigns (see Table 1). Unfortunately, only sparse measurements on the vegetation water content were conducted during the field campaigns. Thus, a meaningful and representative εt-level from field measurements could not be derived for modelling of the X-Fresnel scattering mechanism αXD. However, the area of modelled αXD-values (using the εs-level from the measurements) in Figure 29 contains most of the data-derived αXD-values compared to the non-adapted modelling cases (see Figure 28). This trend is important to signify the logic correctness of the model, as input of in situ conditions during modelling should improve the modelled αXD-predictions. In Figure 28, more αXD-values from the data are located outside the modelled αXD-curves. This again strengthens the indication that the extended Fresnel formalism might be an appropriate model to explain the occurring scattering mechanism in agriculture. However, the analyses just reveal first insights into a potential extended Fresnel scattering mechanism present at L-band for certain crop types in agriculture. Especially the mismatch of the black, modelled curves compared to the extent of the gray bars, representing the standard deviation of the SAR-derived αXD-values for each of the agricultural fields, in Figure 28 and Figure 29, indicates that by far, not all double-bounce scattering scenarios in agriculture can be represented by a depolarizing double-bounce mechanism using extended Fresnel scattering, as it still represents a significantly simplified scattering scenario.

5. Discussion on Potentials and Limitations

Section 3 and Section 4 introduced an extended Fresnel scattering model and first analysis of the existence of this scattering mechanism in fully-polarimetric, longer wavelength (L-band) SAR data. Shorter wavelengths (X- and C-band) were not analyzed due to their reduced penetration depth in stalk-dominated, maturing winter crops, unlikely to induce a double Fresnel reflection (rough-soil to plant stalks). Hence, the model is only qualified for lower frequency analysis of properties, like soil conditions (roughness, moisture) in agriculture.
The sensitivity analysis of the model on soil roughness-induced depolarization and dielectric properties of the media revealed the dependencies of the coherency matrix elements. The two coherency matrix combinations TXD22 + TXD33 and (TXD22TXD33)/(TXD22 + TXD33) showed special sensitivities concerning only soil roughness-induced depolarization or concerning only the dielectric properties of the reflecting media.
While TXD22 + TXD33 appears solely dependent on the dielectric properties and not on roughness-induced depolarization, (TXD22TXD33)/(TXD22 + TXD33) has reversed sensitivities. However, it is important to note that the combination (TXD22TXD33)/(TXD22 + TXD33) is also free from the roughness scattering loss factor (LSXD), whereas the combination TXD22 + TXD33 is still depending on this factor (see Figure 25). Hence, the LSXD-dependence of TXD22 + TXD33 requires an exact modelling of the intensity of this component, compared to LSXD-independent ratios, like αXD and (TXD22TXD33)/(TXD22 + TXD33). However, the exact formulation of the LSXD-factor is complicated to elaborate rigorously in the case of a depolarizing ground plane with dependence on the angle θl. Therefore, the proposed model for extended Fresnel scattering can be mainly utilized to understand a depolarizing dihedral scattering mechanism and its ratio terms (e.g., αXD and (TXD22TXD33)/(TXD22 + TXD33)). The model serves to a minor extent for obtaining predictions about absolute backscattering intensities of depolarizing dihedral scattering.
In addition to the loss influence, the combinations of coherency matrix elements, (TXD22TXD33)/(TXD22 + TXD33) and TXD22 + TXD33, depend on the differential phase angle (ϕ), which was investigated theoretically for its influence in Figure 23 and Figure 24. This sensitivity study revealed that TXD22 + TXD33 is quasi-insensitive of differential extinction up to ϕ = 30°, whereas (TXD22TXD33)/(TXD22 + TXD33) correlates clearly with the change of ϕ. For scattering scenarios of strongly polarizing, meaning strongly-oriented media (ϕ > 30°), the contribution of the differential phase can hardly be neglected. Future analysis for the dependency on ϕ with experimental SAR data should provide deeper insight into this relation and the impact, as well as the level of ϕ in real agricultural SAR data. It is anticipated, that in the case of inversion, the retrieval of soil parameters will be constrained until impossible in media causing high (ϕ > 30°) to very high (ϕ > 60°) differential phases.
However, in Section 4, the analyses with E-SAR and F-SAR L-band data over agricultural regions reveal first insights into the extended Fresnel scattering mechanism, occurring only on different winter crops (winter wheat/barley/triticale and winter rape) and grassland fields for the AgriSAR, OPAQUE, SARTEO and TERENO project campaigns. As indicated by Figure 28, several agricultural fields with different crop types can only be modeled with the extended Fresnel (black lines in Figure 28 and Figure 29) instead of the Fresnel scattering mechanism (red dashed line in Figure 28 and Figure 29). Nevertheless, the data analysis based on these selected fields just represents a first approach to X-Fresnel scattering, whose occurrence is confirmed, but will be investigated in more detail within upcoming agricultural SAR campaigns. Especially the lack of soil roughness and plant moisture measurements for the analyzed vegetated winter crop fields hampers a more detailed analysis of the potential to invert for instance soil roughness. Moreover, in situ measurements of soil roughness will lead to a clear distinguishing of which scattering type should be used for modelling the scattering at the soil plane: Bragg or Fresnel scattering. At the moment, Fresnel scattering is applied assuming rough (tilled) soils in agriculture.
Nonetheless, it can be already anticipated from the model analysis in Section 3 that in the case of inversion for soil roughness and soil moisture, constraints exist using the respective coherency matrix elements. For the inversion of surface roughness with (TXD22TXD33)/(TXD22 + TXD33), ambiguities occur for high to very high roughness ranges, where θ1 is bigger than 63° (ks = 0.7). Therefore very high roughness conditions cannot be inverted unambiguously. This might be of minor concern in agriculture, when dihedral scattering is present. Then, the fields are normally vegetated by crops, and the soil was prepared before for seeding, leading to small roughness conditions (ks 0.7). In contrast, the fields are heavily ploughed at non-vegetated times, but then dihedral scattering should not emerge.
In the end, the data analyses for extended Fresnel scattering imply that this particular scattering case is rather a special type of scattering mechanism in stalk-dominated, agricultural crop fields than a universal type of mechanism leading to wide-area inversion capabilities of soil and plant properties.

6. Summary and First Conclusions

The classical Rank 1 dihedral scattering, often included in model-based decompositions, was extended to account for soil roughness-induced depolarization appearing in the scattering of natural media. Therefore, an azimuthal Line of Sight (LoS) rotation on the soil plane of the double-bounce reflection was used to generate a depolarized dihedral scattering signal after integration over the chosen orientation angle distribution (pdfθ) and orientation angle width (2θ1). The results of the sensitivity analysis are shown for the elements of the coherency matrix [TXD].
It reveals that the combination of coherency matrix elements T22XD + T33XD is quasi-independent of the surface roughness-induced depolarization, while the combination (T22XDT33XD)/(T22XD + T33XD) is quasi-independent of the dielectric properties of the reflecting media and only depends on the roughness-induced depolarization. Therefore, a depolarization independent retrieval of soil moisture or a direct roughness retrieval from an extended dihedral scattering component might be possible under certain vegetation and soil conditions. The influence of the differential phase should be negligible or stay at least at a low level (ϕ < 15°), and the soil roughness range should range below ks < 0.7 to avoid ambiguities in a later inversion for soil roughness.
Future studies with polarimetric SAR data will investigate the capability of the suggested soil moisture and soil roughness dependencies for inversion purposes from regions with dominant dihedral scattering mechanism in agriculture. Up to now, the first investigations for agricultural fields (showing dominant dihedral scattering) within the AgriSAR, OPAQUE, SARTEO and TERENO project campaigns, detailed in Section 4, already indicate the benefit of modeling the scattering scenario for stalk-dominated winter crops and grassland with an extended Fresnel instead of a classical Fresnel scattering mechanism, since a wider spectrum of αXD-values can be represented by the novel model, and inversion for geo-physical parameters, like soil moisture or soil roughness, might be enabled under the discussed constraints (see Section 5). This can be also interesting for novel types of polarimetric decompositions using, for instance, multi-angular, polarimetric SAR data, when the dihedral component is non-dominant (unlike in the presented case) and superimposed by volume scattering [30,31].

Acknowledgments

The author gratefully acknowledges Ludovic Villard, Irena Hajnsek and Kostas Papathanassiou for encouraging the study, helpful discussions and recommendations. The author also thanks Eric Pottier for reading and commenting on the manuscript. The German Federal Ministry of Education and Research (BMBF), Helmholtz initiative ‘TERENO’ and the European Space Agency (ESA) are acknowledged for data provision and support with the airborne and ground-based research activities. This study was partly done under the funding of the Helmholtz Alliance (HA)-310 ‘Remote Sensing and Earth System Dynamics’.

Conflicts of Interest

The author declares no conflict of interest.

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Figure 1. Conceptual visualization of the depolarizing soil-trunk double-bounce scattering including the different input variables for extended Fresnel scattering: ωl = local incidence angle, θ1 = rotation limit angle of soil plane, LSXD = roughness intensity loss factor.
Figure 1. Conceptual visualization of the depolarizing soil-trunk double-bounce scattering including the different input variables for extended Fresnel scattering: ωl = local incidence angle, θ1 = rotation limit angle of soil plane, LSXD = roughness intensity loss factor.
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Figure 2. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with roughness angle θ1s = 20, εt = 10, ωl = 30°, LSXD = 1, ϕ = 0°).
Figure 2. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with roughness angle θ1s = 20, εt = 10, ωl = 30°, LSXD = 1, ϕ = 0°).
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Figure 3. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with roughness angle θ1 (εs = 20, εt = 10, ω l = 50°, LSXD = 1, ϕ = 0°).
Figure 3. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with roughness angle θ1 (εs = 20, εt = 10, ω l = 50°, LSXD = 1, ϕ = 0°).
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Figure 4. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with roughness angle θ1 (εs = 20, εt = 30, ωl = 30°, LSXD = 1, ϕ = 0°).
Figure 4. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with roughness angle θ1 (εs = 20, εt = 30, ωl = 30°, LSXD = 1, ϕ = 0°).
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Figure 5. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with local incidence angle ωl (εs = 20, εt = 10, θ1 = 30°, LSXD = 1, ϕ = 0°). Red points indicate the Brewster angle locations.
Figure 5. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with local incidence angle ωl (εs = 20, εt = 10, θ1 = 30°, LSXD = 1, ϕ = 0°). Red points indicate the Brewster angle locations.
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Figure 6. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with local incidence angle ωl (εs = 20, εt = 10, θ1 = 50°, LSXD = 1, ϕ = 0°). Red points indicate the Brewster angle locations.
Figure 6. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with local incidence angle ωl (εs = 20, εt = 10, θ1 = 50°, LSXD = 1, ϕ = 0°). Red points indicate the Brewster angle locations.
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Figure 7. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with local incidence angle ωl (εs = 20, εt = 30, θ1 = 30°, LSXD = 1, ϕ = 0°). Red points indicate the Brewster angle locations.
Figure 7. Sensitivity of coherency matrix elements TXD11 (blue), TXD12 (purple), TXD22 (red) and TXD33 (green) (dB) with local incidence angle ωl (εs = 20, εt = 30, θ1 = 30°, LSXD = 1, ϕ = 0°). Red points indicate the Brewster angle locations.
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Figure 8. Sensitivity of polarimetric entropy H with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 30°).
Figure 8. Sensitivity of polarimetric entropy H with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 30°).
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Figure 9. Sensitivity of polarimetric entropy H with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 60°).
Figure 9. Sensitivity of polarimetric entropy H with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 60°).
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Figure 10. Sensitivity of polarimetric entropy H with local incidence angle ω1 (εs = 20, εt = 30, LSXD = 1, ϕ = 0°, θ1 = 30°).
Figure 10. Sensitivity of polarimetric entropy H with local incidence angle ω1 (εs = 20, εt = 30, LSXD = 1, ϕ = 0°, θ1 = 30°).
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Figure 11. Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 30°).
Figure 11. Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 30°).
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Figure 12. Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 60°).
Figure 12. Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω1 (εs = 20, εt = 10, LSXD = 1, ϕ = 0°, θ1 = 60°).
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Figure 13. Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω1 (εs = 20, εt = 30, LSXD = 1, ϕ = 0°, θ1 = 30°).
Figure 13. Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω1 (εs = 20, εt = 30, LSXD = 1, ϕ = 0°, θ1 = 30°).
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Figure 14. Sensitivity of polarimetric Hα scattering plane with roughness angle θ1 (εs = (2,42), εt = 10, LSXD = 1, ϕ = 0°, ω1 = 35°). Color changes with increasing θ1.
Figure 14. Sensitivity of polarimetric Hα scattering plane with roughness angle θ1 (εs = (2,42), εt = 10, LSXD = 1, ϕ = 0°, ω1 = 35°). Color changes with increasing θ1.
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Figure 15. Sensitivity of polarimetric Hα scattering plane with roughness angle θ1 (εs = 10, εt = (2,42), LSXD = 1, ϕ = 0°, ω1 = 35°). Color changes with increasing θ1.
Figure 15. Sensitivity of polarimetric Hα scattering plane with roughness angle θ1 (εs = 10, εt = (2,42), LSXD = 1, ϕ = 0°, ω1 = 35°). Color changes with increasing θ1.
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Figure 16. Sensitivity of coherency matrix combination TXD22 + TXD33 (blue) (dB) with roughness angle θ1; blue: εs = 20, εt = 10, ωl = 30°; green: εs = 20, εt = 30, ωl = 30°; red: εs = 20, εt = 10, ωl = 50° and LSXD = 1, ϕ = 0°.
Figure 16. Sensitivity of coherency matrix combination TXD22 + TXD33 (blue) (dB) with roughness angle θ1; blue: εs = 20, εt = 10, ωl = 30°; green: εs = 20, εt = 30, ωl = 30°; red: εs = 20, εt = 10, ωl = 50° and LSXD = 1, ϕ = 0°.
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Figure 17. Sensitivity of coherency matrix combination TXD22 + TXD33 (blue) (dB) with the dielectric of the soil εs; blue: θ1 = 30°, εt = 10; red: θ1 = 50°, εt = 10 and ωl = 30°, LSXD = 1, ϕ = 0°.
Figure 17. Sensitivity of coherency matrix combination TXD22 + TXD33 (blue) (dB) with the dielectric of the soil εs; blue: θ1 = 30°, εt = 10; red: θ1 = 50°, εt = 10 and ωl = 30°, LSXD = 1, ϕ = 0°.
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Figure 18. Sensitivity of coherency matrix combination TXD22 + TXD33 (blue) (dB) with the dielectric of the trunk εt (θ1 = 30°, εs = 20, ωl = 30°, LSXD = 1, ϕ = 0°).
Figure 18. Sensitivity of coherency matrix combination TXD22 + TXD33 (blue) (dB) with the dielectric of the trunk εt (θ1 = 30°, εs = 20, ωl = 30°, LSXD = 1, ϕ = 0°).
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Figure 19. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 20, εt = 10, ωl = 30°, ϕ = 0°) and comparison with sinc(4θ1) (red).
Figure 19. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 20, εt = 10, ωl = 30°, ϕ = 0°) and comparison with sinc(4θ1) (red).
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Figure 20. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 20, εt = 10, ωl = 60°, ϕ = 0°) and comparison with sinc(4θ1) (red).
Figure 20. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 20, εt = 10, ωl = 60°, ϕ = 0°) and comparison with sinc(4θ1) (red).
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Figure 21. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 20, εt = 30, ωl = 30°, ϕ = 0°) and comparison with sinc(4θ1) (red).
Figure 21. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 20, εt = 30, ωl = 30°, ϕ = 0°) and comparison with sinc(4θ1) (red).
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Figure 22. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 30, εt = 30, ωl = 30°, ϕ = 0°) and comparison with sinc(4θ1) (red).
Figure 22. Sensitivity of (TXD22TXD33)/(TXD22 + TXD33) with roughness angle θ1 (blue) (εs = 30, εt = 30, ωl = 30°, ϕ = 0°) and comparison with sinc(4θ1) (red).
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Figure 23. Sensitivity of TXD22 + TXD33 along roughness angle θ1 (εs = 20, εt = 10, ωl = 30°, LSXD = 1) with differential phase ϕ = 0° (blue), ϕ = 15° (red, dashed), ϕ = 30° (green) and ϕ = 60° (purple).
Figure 23. Sensitivity of TXD22 + TXD33 along roughness angle θ1 (εs = 20, εt = 10, ωl = 30°, LSXD = 1) with differential phase ϕ = 0° (blue), ϕ = 15° (red, dashed), ϕ = 30° (green) and ϕ = 60° (purple).
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Figure 24. Sensitivity of |(TXD22 - TXD33)/(TXD22 + TXD33)| along roughness angle θ1 (red) (εs = 20, εt = 10, ωl = 30°, LSXD = 1) with differential phases ϕ = 0° (a), ϕ = 15° (b), ϕ = 30° (c), ϕ = 60° (d) and comparison with |sinc(4θ1)| (blue).
Figure 24. Sensitivity of |(TXD22 - TXD33)/(TXD22 + TXD33)| along roughness angle θ1 (red) (εs = 20, εt = 10, ωl = 30°, LSXD = 1) with differential phases ϕ = 0° (a), ϕ = 15° (b), ϕ = 30° (c), ϕ = 60° (d) and comparison with |sinc(4θ1)| (blue).
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Figure 25. Sensitivity of TXD22 + TXD33 along roughness angle θ1 (εs = 20, εt = 10, ωl = 30°, ϕ = 0°) with scattering loss LSXD = 1.0 (blue), LSXD = 0.8 (red) and LsXD = 0.6 (green).
Figure 25. Sensitivity of TXD22 + TXD33 along roughness angle θ1 (εs = 20, εt = 10, ωl = 30°, ϕ = 0°) with scattering loss LSXD = 1.0 (blue), LSXD = 0.8 (red) and LsXD = 0.6 (green).
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Figure 26. Sensitivity of αXD along the roughness angle θ1; αXD is derived from the X-Fresnel scattering model (εs = 25, εt = 15, ϕ = 0°) for different angles of local incidence ωl. Gray dashed arrow indicates dynamics with ωl.
Figure 26. Sensitivity of αXD along the roughness angle θ1; αXD is derived from the X-Fresnel scattering model (εs = 25, εt = 15, ϕ = 0°) for different angles of local incidence ωl. Gray dashed arrow indicates dynamics with ωl.
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Figure 27. First row: RGB-composite of normalized Pauli decomposition-based scattering components (R: 1/2|SHH − SVV|² even-bounce/dihedral scattering, G: 2|SXX|² volume/vegetation scattering, B: 1/2|SHH + SVV|² odd-bounce/surface scattering) for the May acquisition of the OPAQUE 2007 (a), the June acquisition (b) and the July acquisition (c) of the AgriSAR 2006 campaign; (d) the land use of the AgriSAR 2006 campaign; second row: comparison of αXD of the data for the OPAQUE 2007 campaign (e) and the AgriSAR campaign (f) June acquisition and (g) July acquisition); (h) the land use of the OPAQUE 2007 campaign.
Figure 27. First row: RGB-composite of normalized Pauli decomposition-based scattering components (R: 1/2|SHH − SVV|² even-bounce/dihedral scattering, G: 2|SXX|² volume/vegetation scattering, B: 1/2|SHH + SVV|² odd-bounce/surface scattering) for the May acquisition of the OPAQUE 2007 (a), the June acquisition (b) and the July acquisition (c) of the AgriSAR 2006 campaign; (d) the land use of the AgriSAR 2006 campaign; second row: comparison of αXD of the data for the OPAQUE 2007 campaign (e) and the AgriSAR campaign (f) June acquisition and (g) July acquisition); (h) the land use of the OPAQUE 2007 campaign.
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Figure 28. Comparison of αXD from extended Fresnel scattering (black lines) and from standard Fresnel scattering (red dashed line) along the local incidence angle ωl; αXD is derived from the X-Fresnel model (εs = 25, εt = 15, ϕ = 0° (a); εs = 15, εt = 25, ϕ = 0° (b); εs = 25, εt = 25; ϕ = 0° (c)) for different roughness angles θ1 and from the AgriSAR (plus (June) and triangle up (July) sign), OPAQUE (square sign), SARTEO (diamond sign) and TERENO (triangle down = Bode site, triangle right Demmin site E-W track, triangle left Demmin site N-S track) L-band data for several agricultural fields with a stalk-dominated phenology (yellow = winter barley, green = field grass, gold = winter wheat, pink = winter wheat (field 221), cyan = grassland, red = undefined land use, orchid = winter triticale); gray bars indicate the standard deviation of αXD for each of the agricultural fields.
Figure 28. Comparison of αXD from extended Fresnel scattering (black lines) and from standard Fresnel scattering (red dashed line) along the local incidence angle ωl; αXD is derived from the X-Fresnel model (εs = 25, εt = 15, ϕ = 0° (a); εs = 15, εt = 25, ϕ = 0° (b); εs = 25, εt = 25; ϕ = 0° (c)) for different roughness angles θ1 and from the AgriSAR (plus (June) and triangle up (July) sign), OPAQUE (square sign), SARTEO (diamond sign) and TERENO (triangle down = Bode site, triangle right Demmin site E-W track, triangle left Demmin site N-S track) L-band data for several agricultural fields with a stalk-dominated phenology (yellow = winter barley, green = field grass, gold = winter wheat, pink = winter wheat (field 221), cyan = grassland, red = undefined land use, orchid = winter triticale); gray bars indicate the standard deviation of αXD for each of the agricultural fields.
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Figure 29. Comparison of αXD from extended Fresnel scattering (black lines) and from standard Fresnel scattering (red dashed line) along the local incidence angle ωl; αXD is derived from the X-Fresnel model with εs-input from local field measurements (εs = 10, εt = 25, ϕ = 0°) for different roughness angles θ1 and from the AgriSAR (plus (June) and triangle up (July) sign), OPAQUE (square sign), SARTEO (diamond sign) and TERENO (triangle down = Bode site, triangle right Demmin site E-W track, triangle left Demmin site N-S track) L-band data for several agricultural fields with a stalk-dominated phenology (yellow = winter barley, green = field grass, gold = winter wheat, pink = winter wheat (field 221), cyan = grassland, red = undefined land use, orchid = winter triticale); gray bars indicate the standard deviation of αXD for each of the agricultural fields.
Figure 29. Comparison of αXD from extended Fresnel scattering (black lines) and from standard Fresnel scattering (red dashed line) along the local incidence angle ωl; αXD is derived from the X-Fresnel model with εs-input from local field measurements (εs = 10, εt = 25, ϕ = 0°) for different roughness angles θ1 and from the AgriSAR (plus (June) and triangle up (July) sign), OPAQUE (square sign), SARTEO (diamond sign) and TERENO (triangle down = Bode site, triangle right Demmin site E-W track, triangle left Demmin site N-S track) L-band data for several agricultural fields with a stalk-dominated phenology (yellow = winter barley, green = field grass, gold = winter wheat, pink = winter wheat (field 221), cyan = grassland, red = undefined land use, orchid = winter triticale); gray bars indicate the standard deviation of αXD for each of the agricultural fields.
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Table 1. Approximated mean dielectric level of soil from in situ measurements of the different campaigns.
Table 1. Approximated mean dielectric level of soil from in situ measurements of the different campaigns.
CampaignDateApprox. Mean
εs-Level (-)
AgriSAR7 June 20069
AgriSAR5 July 20065
OPAQUE31 May 200717
SARTEO27 May 200811
TERENO
Bode
22 May 20128
TERENO
Demmin
23 May 20129

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Jagdhuber, T. An Approach to Extended Fresnel Scattering for Modeling of Depolarizing Soil-Trunk Double-Bounce Scattering. Remote Sens. 2016, 8, 818. https://doi.org/10.3390/rs8100818

AMA Style

Jagdhuber T. An Approach to Extended Fresnel Scattering for Modeling of Depolarizing Soil-Trunk Double-Bounce Scattering. Remote Sensing. 2016; 8(10):818. https://doi.org/10.3390/rs8100818

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Jagdhuber, Thomas. 2016. "An Approach to Extended Fresnel Scattering for Modeling of Depolarizing Soil-Trunk Double-Bounce Scattering" Remote Sensing 8, no. 10: 818. https://doi.org/10.3390/rs8100818

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