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

## Abstract

**:**

_{22XD}+ T

_{33XD}is quasi-independent of the roughness-induced depolarization, while (T

_{22XD}− T

_{33XD})/(T

_{22XD}+ T

_{33XD}) 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.

## 1. Introduction

## 2. Development of the Extend Fresnel Scattering Model

#### 2.1. Rank 1 Fresnel Scattering

_{Hs}and R

_{Vs}are the horizontal and vertical Fresnel coefficients of the soil scattering plane, which depend on the dielectric constant of the soil ε

_{s}and the local incidence angle ω

_{l}[8]:

_{Ht}and R

_{Vt}are the Fresnel coefficients of the trunk scattering plane, which include the dielectric constant of the trunk ε

_{t}and the local incidence angle ω

_{l}[8]. For modelling reasons, it is assumed that the vertical scattering plane is formed by the plant trunks planted densely in seeding rows.

_{D}] follows as derived in [13] and is parameterized by the complex scattering mechanism ratio α

_{D}and the real backscattering amplitude f

_{D}[14]:

_{D}is represented by the trace of [T

_{D}]: P

_{D}= f

_{D}(1 + |α

_{D}|

^{2}).

_{S}[11,16]:

_{l}is the local incidence angle [12]. The Fresnel coefficients of the soil scatterer (R

_{Hs}, R

_{Vs}) are altered to the so-called modified Fresnel coefficients (RL

_{Hs}, RL

_{Vs}) accordingly [11]:

_{D}is independent of the scattering loss factor L

_{S}, because the multiplicative factor cancels out. Thus, only the backscattering amplitude f

_{D}is affected by the soil roughness loss. However, the roughness-induced depolarization is not included, which would also affect the complex ratio α

_{D}[14].

_{DLoss}] is written as [14]

#### 2.2. Rank 3 Extended Fresnel Scattering

_{Hs}, R

_{Vs}), 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.

_{XD}] is presented as:

_{XD12}= S

_{XD21}). The Pauli scattering vector is formed in Equation (14) to account for distributed targets by calculating in the next step the coherency matrix [T

_{XD}] from the outer product of the scattering vectors [10].

_{1}to θ

_{1}with a uniform probability function (pdf

_{θ}= 1/(2θ

_{1})) representing the rotation limits (±θ

_{1}) and probability distribution (pdf

_{θ}) 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.

_{XD}] is reflection symmetric, and subsequently, the off-diagonal elements T

_{XD13}, T

_{XD23}, T

_{XD31}and T

_{XD32}are zero.

_{Hs}, R

_{Ht}, R

_{Vs}, R

_{Vt}), the rotation limit angle (θ

_{1}) indicating the roughness-induced depolarization, the roughness intensity loss factor (L

_{SXD}) and the differential phase angle (ϕ). From Equation (16), it is clear that the L

_{SXD}-factor is obsolete, if a ratio of the coherency matrix elements is taken for analysis.

_{XD}and a dihedral scattering mechanism α

_{XD}can be defined similar to Equation (6). Furthermore, in the case of extended Fresnel scattering, only the f

_{XD}-component is affected by the L

_{SXD}‑loss factor. However, both variables (α

_{XD}, f

_{XD}) of this extended Fresnel scattering component are influenced by the roughness-induced depolarization, incorporated by the LoS rotation and integration over the limit angle θ

_{1}.

## 3. Sensitivity Analysis of Extended Fresnel Scattering for Distributed Targets

_{SXD}= 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.

_{l}and roughness θ

_{1}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 (ε

_{t}= 10) and a high (ε

_{t}= 30) trunk moisture and a medium soil moisture (ε

_{s}= 20). Figure 2, Figure 3 and Figure 4 show the characteristics of the coherency matrix elements T

_{XD11}, T

_{XD12}, T

_{XD22}and T

_{XD33}with the roughness-induced depolarization (θ

_{1}) for different local incidence angles (ω

_{l}) and dielectric properties of soil (ε

_{s}) and trunk (ε

_{t}). The general trend of the coherency matrix elements persists for different constellations of ω

_{l}, ε

_{s}and ε

_{t}, which means:

- T
_{XD11}, T_{XD12}are decreasing by 3–6 dB with increasing depolarization, while T_{XD11}performs as the more stable component of both coherency matrix elements. - T
_{XD22}decreases by 3–4 dB with increasing depolarization, but stays always higher than T_{XD11}, which is a mandatory condition for the presence of dihedral scattering (compared to surface scattering). - T
_{XD33}increases up to −10 dB from a Rank 1 (T_{XD33}= 0) to a Rank 3 (T_{XD33}> 0) scattering mechanism with increasing depolarization. For weak to medium depolarization (first half of the θ_{1}-range: 0°–45°) T_{XD22}dominates over T_{XD33}, which reverses for the case of strong depolarization (second half of the θ_{1}-range: 45°–90°).

_{l}) for different roughness (θ

_{1}) and moisture conditions of the soil (ε

_{s}) and the trunk (ε

_{t}). However, only the ω

_{l}-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 (ω

_{l}) 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:

- T
_{XD11}decreases until approximately 45° and then increase again to the starting level. - T
_{XD12}shows the same behavior as T_{XD11}, but less pronounced. - T
_{XD22}and T_{XD33}increase until approximately 45° and then decrease to the starting level, while it depends on the roughness depolarization level (θ_{1}), which curve is superior with respect to the other. - The crossing points between T
_{XD11}and T_{XD22}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.

_{s}, ε

_{t}), as well as the local incidence ω

_{l}and the roughness depolarization angle θ

_{1}. 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 [T

_{XD}] [8].

_{XD}] 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]:

_{1}, as shown by Figure 8 and Figure 9. While, Figure 10 in comparison with Figure 8 states that an increase of the dielectric constants (ε

_{s}, ε

_{t}) does not lead to a distinct increase in polarimetric entropy, which is driven by θ

_{1}. 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 H – α scattering plane for a medium local incidence angle (ω

_{l}= 35°), revealing the sensitivity of H and α concerning roughness-induced depolarization θ

_{1}and the dielectric content of both scattering planes (ε

_{s}, ε

_{t}). The dynamics are indicated by black arrows within the figures. The increase in entropy H with θ

_{1}is clearly visible for both cases and expected, because an increase in roughness/depolarization is clearly linked with a rise in disorder/entropy.

_{t}, as well as the soil ε

_{s}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 H − α 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 H − α 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 T

_{XD22}+ T

_{XD33}is investigated with its dependencies on ω

_{l}, θ

_{1}, ε

_{t}and ε

_{s}in Figure 16, Figure 17 and Figure 18. Concentrating on Figure 16, the combination T

_{XD22}+ T

_{XD33}exhibits almost no sensitivity with respect to the roughness-induced depolarization, even for different values of ω

_{l}, ε

_{t}and ε

_{s}. 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 (ε

_{s}) and the dielectric constant of the trunk (ε

_{t}). Here, the sensitivity is strongest for the lower range of dielectric constants from two to approximately 25. Therefore the combination of coherency matrix elements T

_{XD22}+ T

_{XD33}seems to be an appropriate candidate to study the dielectric properties of the reflecting media without influence from roughness-induced depolarization.

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) with respect to the roughness-induced depolarization (θ

_{1}). Since also the function sinc(4θ

_{1}) 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 ω

_{l}and of dielectric constants (ε

_{s}, ε

_{t}) 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 (θ

_{1}) 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 (θ

_{1}), ambiguities occur for strongly depolarizing scenarios with θ

_{1}> 63°.

#### Impact of Differential Phase ϕ and Scattering Loss L_{SXD} on Coherency Matrix Combinations

_{XD22}+ T

_{XD33}and (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}). The influence of ϕ on T

_{XD22}+ T

_{XD33}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 |(T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33})| for phase differences of up to ϕ = 60°. With increasing ϕ the dynamic range of |(T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33})| decreases, and the sensitivity to higher roughness levels (θ

_{1}> 45°) declines strongly until a complete loss of sensitivity in the range of θ

_{1}> 40° for a ϕ-level of 30°. Focusing on the impact of the scattering loss L

_{SXD}, the coherency matrix combination T

_{XD22}+ T

_{XD33}is directly depending on the loss level reducing its power from −7 dB down to −11.5 dB with decreasing of L

_{SXD}from 1.0 (no loss) to 0.6 (60%), as indicated in Figure 25. Fortunately, the matrix combination (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) is a ratio, canceling the L

_{SXD}-factor, and is therefore independent from the influences caused by scattering losses.

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

_{XD}is calculated from the model (see Equation (17)) and from the L-band SAR data:

_{XD}from the X-Fresnel scattering model with the variation of the roughness depolarization angle θ

_{1}for different values of local incidence angle ω

_{l}assuming a medium moisture scenario (ε

_{s}= 25, ε

_{t}= 15). The analysis indicates that α

_{XD}from the X-Fresnel model generally decreases until about 50° local incidence and then strongly rises until ω

_{l}= 70°. This increase is especially strong for roughness angles around 50°. Concerning very rough soils (θ

_{1}> 70°), the behavior is opposed to the previous case, and α

_{XD}diminishes with respect to increasing local incidence.

_{XD}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.

_{XD}-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 α

_{XD}-values (α

_{XD}> 1.0) in Figure 27e–g, while regions with distinct dihedral scattering indicate low α

_{XD}-values (α

_{XD}< 0.8). This implies a domination of the correlation term T

_{12}over the even-bounce/dihedral term T

_{22}in the case of surface Bragg scattering compared to dihedral Fresnel scattering and vice versa. In Figure 28, the α

_{XD}-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 (θ

_{1}) (depolarization) cases in Figure 28. The single signs with error bars represent α

_{XD}-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 α

_{XD}= 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 α

_{XD}-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.

_{XD}with local incidence angle ω

_{l}for different dielectric constants of the soil and the stalks (ε

_{s}, ε

_{t}). While the level of modelled α

_{XD}for low incidence angles (ω

_{l}< 35°) is strongly depending on ε

_{t}(stalks) and minor on ε

_{s}(soil), the situation is reversed for high incidence angles (ω

_{l}> 55°). Therefore, the best modelled α

_{XD}-case is taken in Figure 29 by adjusting the level of ε

_{s}according to available field measurements of soil moisture from the different in situ measurement campaigns. A mean ε

_{s}-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 ε

_{t}-level from field measurements could not be derived for modelling of the X-Fresnel scattering mechanism α

_{XD}. However, the area of modelled α

_{XD}-values (using the ε

_{s}-level from the measurements) in Figure 29 contains most of the data-derived α

_{XD}-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 α

_{XD}-predictions. In Figure 28, more α

_{XD}-values from the data are located outside the modelled α

_{XD}-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 α

_{XD}-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

_{XD22}+ T

_{XD33}and (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) showed special sensitivities concerning only soil roughness-induced depolarization or concerning only the dielectric properties of the reflecting media.

_{XD22}+ T

_{XD33}appears solely dependent on the dielectric properties and not on roughness-induced depolarization, (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) has reversed sensitivities. However, it is important to note that the combination (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) is also free from the roughness scattering loss factor (L

_{SXD}), whereas the combination T

_{XD22}+ T

_{XD33}is still depending on this factor (see Figure 25). Hence, the L

_{SXD}-dependence of T

_{XD22}+ T

_{XD33}requires an exact modelling of the intensity of this component, compared to L

_{SXD}-independent ratios, like α

_{XD}and (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}). However, the exact formulation of the L

_{SXD}-factor is complicated to elaborate rigorously in the case of a depolarizing ground plane with dependence on the angle θ

_{l}. 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., α

_{XD}and (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33})). The model serves to a minor extent for obtaining predictions about absolute backscattering intensities of depolarizing dihedral scattering.

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) and T

_{XD22}+ T

_{XD33}, 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 T

_{XD22}+ T

_{XD33}is quasi-insensitive of differential extinction up to ϕ = 30°, whereas (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) 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.

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}), ambiguities occur for high to very high roughness ranges, where θ

_{1}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 $\ll $ 0.7). In contrast, the fields are heavily ploughed at non-vegetated times, but then dihedral scattering should not emerge.

## 6. Summary and First Conclusions

_{θ}) and orientation angle width (2θ

_{1}). The results of the sensitivity analysis are shown for the elements of the coherency matrix [T

_{XD}].

_{22XD}+ T

_{33XD}is quasi-independent of the surface roughness-induced depolarization, while the combination (T

_{22XD}− T

_{33XD})/(T

_{22XD}+ T

_{33XD}) 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.

_{XD}-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

## Conflicts 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: ω

_{l}= local incidence angle, θ

_{1}= rotation limit angle of soil plane, L

_{SXD}= roughness intensity loss factor.

**Figure 2.**Sensitivity of coherency matrix elements T

_{XD11}(blue), T

_{XD12}(purple), T

_{XD22}(red) and T

_{XD33}(green) (dB) with roughness angle θ

_{1}(ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 30°, L

_{SXD}= 1, ϕ = 0°).

**Figure 3.**Sensitivity of coherency matrix elements T

_{XD11}(blue), T

_{XD12}(purple), T

_{XD22}(red) and T

_{XD33}(green) (dB) with roughness angle θ

_{1}(ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 50°, L

_{SXD}= 1, ϕ = 0°).

**Figure 4.**Sensitivity of coherency matrix elements T

_{XD11}(blue), T

_{XD12}(purple), T

_{XD22}(red) and T

_{XD33}(green) (dB) with roughness angle θ

_{1}(ε

_{s}= 20, ε

_{t}= 30, ω

_{l}= 30°, L

_{SXD}= 1, ϕ = 0°).

**Figure 5.**Sensitivity of coherency matrix elements T

_{XD11}(blue), T

_{XD12}(purple), T

_{XD22}(red) and T

_{XD33}(green) (dB) with local incidence angle ω

_{l}(ε

_{s}= 20, ε

_{t}= 10, θ

_{1}= 30°, L

_{SXD}= 1, ϕ = 0°). Red points indicate the Brewster angle locations.

**Figure 6.**Sensitivity of coherency matrix elements T

_{XD11}(blue), T

_{XD12}(purple), T

_{XD22}(red) and T

_{XD33}(green) (dB) with local incidence angle ω

_{l}(ε

_{s}= 20, ε

_{t}= 10, θ

_{1}= 50°, L

_{SXD}= 1, ϕ = 0°). Red points indicate the Brewster angle locations.

**Figure 7.**Sensitivity of coherency matrix elements T

_{XD11}(blue), T

_{XD12}(purple), T

_{XD22}(red) and T

_{XD33}(green) (dB) with local incidence angle ω

_{l}(ε

_{s}= 20, ε

_{t}= 30, θ

_{1}= 30°, L

_{SXD}= 1, ϕ = 0°). Red points indicate the Brewster angle locations.

**Figure 8.**Sensitivity of polarimetric entropy H with local incidence angle ω

_{1}(ε

_{s}= 20, ε

_{t}= 10, L

_{SXD}= 1, ϕ = 0°, θ

_{1}= 30°).

**Figure 9.**Sensitivity of polarimetric entropy H with local incidence angle ω

_{1}(ε

_{s}= 20, ε

_{t}= 10, L

_{SXD}= 1, ϕ = 0°, θ

_{1}= 60°).

**Figure 10.**Sensitivity of polarimetric entropy H with local incidence angle ω

_{1}(ε

_{s}= 20, ε

_{t}= 30, L

_{SXD}= 1, ϕ = 0°, θ

_{1}= 30°).

**Figure 11.**Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω

_{1}(ε

_{s}= 20, ε

_{t}= 10, L

_{SXD}= 1, ϕ = 0°, θ

_{1}= 30°).

**Figure 12.**Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω

_{1}(ε

_{s}= 20, ε

_{t}= 10, L

_{SXD}= 1, ϕ = 0°, θ

_{1}= 60°).

**Figure 13.**Sensitivity of polarimetric mean scattering alpha angle α with local incidence angle ω

_{1}(ε

_{s}= 20, ε

_{t}= 30, L

_{SXD}= 1, ϕ = 0°, θ

_{1}= 30°).

**Figure 14.**Sensitivity of polarimetric H – α scattering plane with roughness angle θ

_{1}(ε

_{s}= (2,42), ε

_{t}= 10, L

_{SXD}= 1, ϕ = 0°, ω

_{1}= 35°). Color changes with increasing θ

_{1}.

**Figure 15.**Sensitivity of polarimetric H − α scattering plane with roughness angle θ

_{1}(ε

_{s}= 10, ε

_{t}= (2,42), L

_{SXD}= 1, ϕ = 0°, ω

_{1}= 35°). Color changes with increasing θ

_{1}.

**Figure 16.**Sensitivity of coherency matrix combination T

_{XD22}+ T

_{XD33}(blue) (dB) with roughness angle θ

_{1}; blue: ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 30°; green: ε

_{s}= 20, ε

_{t}= 30, ω

_{l}= 30°; red: ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 50° and L

_{SXD}= 1, ϕ = 0°.

**Figure 17.**Sensitivity of coherency matrix combination T

_{XD22}+ T

_{XD33}(blue) (dB) with the dielectric of the soil ε

_{s}; blue: θ

_{1}= 30°, ε

_{t}= 10; red: θ

_{1}= 50°, ε

_{t}= 10 and ω

_{l}= 30°, L

_{SXD}= 1, ϕ = 0°.

**Figure 18.**Sensitivity of coherency matrix combination T

_{XD22}+ T

_{XD33}(blue) (dB) with the dielectric of the trunk ε

_{t}(θ

_{1}= 30°, ε

_{s}= 20, ω

_{l}= 30°, L

_{SXD}= 1, ϕ = 0°).

**Figure 19.**Sensitivity of (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) with roughness angle θ

_{1}(blue) (ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 30°, ϕ = 0°) and comparison with sinc(4θ

_{1}) (red).

**Figure 20.**Sensitivity of (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) with roughness angle θ

_{1}(blue) (ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 60°, ϕ = 0°) and comparison with sinc(4θ

_{1}) (red).

**Figure 21.**Sensitivity of (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) with roughness angle θ

_{1}(blue) (ε

_{s}= 20, ε

_{t}= 30, ω

_{l}= 30°, ϕ = 0°) and comparison with sinc(4θ

_{1}) (red).

**Figure 22.**Sensitivity of (T

_{XD22}− T

_{XD33})/(T

_{XD22}+ T

_{XD33}) with roughness angle θ

_{1}(blue) (ε

_{s}= 30, ε

_{t}= 30, ω

_{l}= 30°, ϕ = 0°) and comparison with sinc(4θ

_{1}) (red).

**Figure 23.**Sensitivity of T

_{XD22}+ T

_{XD33}along roughness angle θ

_{1}(ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 30°, L

_{SXD}= 1) with differential phase ϕ = 0° (blue), ϕ = 15° (red, dashed), ϕ = 30° (green) and ϕ = 60° (purple).

**Figure 24.**Sensitivity of |(T

_{XD22}- T

_{XD33})/(T

_{XD22}+ T

_{XD33})| along roughness angle θ

_{1}(red) (ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 30°, L

_{SXD}= 1) with differential phases ϕ = 0° (

**a**), ϕ = 15° (

**b**), ϕ = 30° (

**c**), ϕ = 60° (

**d**) and comparison with |sinc(4θ

_{1})| (blue).

**Figure 25.**Sensitivity of T

_{XD22}+ T

_{XD33}along roughness angle θ

_{1}(ε

_{s}= 20, ε

_{t}= 10, ω

_{l}= 30°, ϕ = 0°) with scattering loss L

_{SXD}= 1.0 (blue), L

_{SXD}= 0.8 (red) and Ls

_{XD}= 0.6 (green).

**Figure 26.**Sensitivity of α

_{XD}along the roughness angle θ

_{1}; α

_{XD}is derived from the X-Fresnel scattering model (ε

_{s}= 25, ε

_{t}= 15, ϕ = 0°) for different angles of local incidence ω

_{l}. Gray dashed arrow indicates dynamics with ω

_{l}.

**Figure 27.**First row: RGB-composite of normalized Pauli decomposition-based scattering components (R: 1/2|S

_{HH}− S

_{VV}|² even-bounce/dihedral scattering, G: 2|S

_{XX}|² volume/vegetation scattering, B: 1/2|S

_{HH}+ S

_{VV}|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 α

_{XD}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 28.**Comparison of α

_{XD}from extended Fresnel scattering (black lines) and from standard Fresnel scattering (red dashed line) along the local incidence angle ω

_{l}; α

_{XD}is derived from the X-Fresnel model (ε

_{s}= 25, ε

_{t}= 15, ϕ = 0° (

**a**); ε

_{s}= 15, ε

_{t}= 25, ϕ = 0° (

**b**); ε

_{s}= 25, ε

_{t}= 25; ϕ = 0° (

**c**)) for different roughness angles θ

_{1}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 α

_{XD}for each of the agricultural fields.

**Figure 29.**Comparison of α

_{XD}from extended Fresnel scattering (black lines) and from standard Fresnel scattering (red dashed line) along the local incidence angle ω

_{l}; α

_{XD}is derived from the X-Fresnel model with ε

_{s}-input from local field measurements (ε

_{s}= 10, ε

_{t}= 25, ϕ = 0°) for different roughness angles θ

_{1}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 α

_{XD}for each of the agricultural fields.

**Table 1.**Approximated mean dielectric level of soil from in situ measurements of the different campaigns.

Campaign | Date | Approx. Mean ε _{s}-Level (-) |
---|---|---|

AgriSAR | 7 June 2006 | 9 |

AgriSAR | 5 July 2006 | 5 |

OPAQUE | 31 May 2007 | 17 |

SARTEO | 27 May 2008 | 11 |

TERENO Bode | 22 May 2012 | 8 |

TERENO Demmin | 23 May 2012 | 9 |

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**MDPI and ACS Style**

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

**Chicago/Turabian Style**

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