A Discrete Interferometric Model for a Layer of a Random Medium: Effects on InSAR Coherence, Power, and Phase

Abstract

The remote sensing community increasingly demands precise ecosystem monitoring, environmental change detection, and natural resource management, particularly in forestry. Key metrics such as biomass and total area index require accurate estimation, necessitating extensive experiments and reliable scattering models. Recent advances in radar interferometry introduce two essential parameters—interferogram phase and correlation coefficient—containing crucial target information. Understanding their relationship to forest biophysical parameters requires analyzing wave interactions with vegetation particles. This study presents a discrete interferometric model for a random medium layer, establishing the link between radar interferometry and forest biophysical properties. Correlation analysis plays a vital role in estimating one variable based on another, reducing uncertainty in random media. The research introduces a novel modeling approach that enhances theoretical foundations and supports empirical studies in the literature. Bridging theoretical analysis and practical observations, this work enhances the precision and applicability of radar interferometry for vegetation monitoring. The findings contribute to improving remote sensing methodologies and expanding their potential in ecological and environmental research. Ultimately, this study advances the use of interferometric models in extracting critical forest parameters with greater accuracy.

1. Introduction

The extraction of vegetation characteristics from radar data has emerged as a significant concern in remote sensing [1,2,3,4]. The demand for ecosystem monitoring, natural resource management, including forests, and environmental change assessment is rising. A substantial number of experiments are necessary to acquire important data. Diverse models are employed to achieve correlation, phase, and coherence attributes [5,6,7,8,9,10,11,12,13]. Refs. [12,13] employ models to ascertain correlation features. It is claimed that certain models yield consistent outcomes. Additionally, other parameters are thought to influence the features and sensitivity analysis.

Radar functions as a distance measurement instrument, exemplified by altimeters, or for imaging purposes, as seen in Side-Looking Airborne Radar (SLAR) and Synthetic Aperture Radar (SAR). SAR is a radar system that generates a highly focused effective beam through the sophisticated processing of radar data [5,6,7,8]. SAR signal processing involves coherent phase correction that rectifies the particular phase shift history of the signal reflected from a designated location on the ground, accounting for the velocity of the aircraft (radar). The necessary phase adjustment can be inferred from the anticipated phase history of a ground return to an aerial radar [14,15,16].

Interferometric synthetic aperture radar (InSAR) technology is employed in remote sensing and geodesy. This geodetic technique employs variations in wave phases returning to the satellite or aircraft to generate digital elevations or surface alterations from two or more synthetic aperture radar images. Numerous studies and applications exist for InSAR and other radar technologies [16,17,18,19,20,21,22,23,24,25,26]. These studies delineate the derivation of signal statistics, an efficient estimator for the interferometric phase, and the requisite equations for calculating the height error budget. Additionally, they provide a set of design guidelines for enhancing InSAR performance. It was determined that interferometric mapping offers a high-resolution topographic system capable of functioning effectively in challenging topographical conditions. It develops a model that incorporates both radar system characteristics and scene scattering parameters, utilizing it to analyze observations in a forested region with varying coherence. The interferometric height discontinuity exhibits a strong correlation with a dense forest.

The literature review indicates that initial studies have employed mathematics to clarify InSAR observations through the physical characteristics of vegetation, such as scatterer amplitude, extinction coefficient, incident angle, baseline distance, and the height of the ground surface or the depth of the vegetation layer [8,12,13,17,18,19,20,21,22,23]. The comparable frequency decorrelation bandwidth is the only thing that matters when trying to figure out the height of a spread target. The baseline distance has no effect on this. There are references [8,19,20] that describe how to evaluate the extinction, physical height, and height of the scattering phase center in a closed, uniform, semi-infinite canopy using InSAR data.

The methods for extracting vegetation characteristics and the underlying ground surface topography from InSAR data are proposed in both Treuhaft’s and Seker’s Model [12,13,17]. The modeling procedure encompasses the simulation of electromagnetic scattering and radar processing that constitute the InSAR observations. The vegetation and topography parameters are then selected for estimation. Parameter discrepancies are then assessed against the performance of InSAR sensors, and parameter estimates resulting from InSAR data are presented and compared to real-world observations.

While Treuhaft’s model extends radar interferometry applications by representing vegetation as a stochastic medium, enabling mathematical generalization with experimental parameters for improved forest monitoring, this paper departs from Treuhaft’ s work in three important ways: (1) This paper presents a discrete interferometric model utilizing physical characteristics instead of experimental ones, resulting in a more realistic approach. (2) The backscattering coefficients of a vegetation layer over flat ground are provided using a discrete scatterer model. In addition the impact of the ground on bi-static fields is assessed for different types of ground. (3) Unlike Treuhaft’s model, this paper introduces different numerical densities of vegetative scatterers for each multi-layers for the first time, which is essential for power computation. This study theoretically examines and models a discrete interferometric framework for a stochastic medium layer, leading to more accurate forest structure assessments without relying on experimental parameters. As discussed in conclusion, all the outputs of this study align with Treuhaft’s model and successfully validate its results.

The rest of this paper proceeds as follows. Section 1.1 and Section 1.2 encapsulate the methodologies, equations, and propositions employed in the Treuhaft and Seker models, respectively. A comparative analysis of the parameters and methodologies of both models, supplemented by tables that delineate their similarities and differences are introduced in Section 1.3. Section 2 is dedicated to the simulations of theInSAR parameters, power, coherence and phase. Power in relation to tree height with regard to varying concerning different forest densities, and InSAR coherence in relation to tree height with regard to varied angles and ratios are shown. The section proceeds with the outcomes of the horizontal and vertical components of the reflected-direct bi-static field’s influence on coherence as a function of tree height, the impact of forward scattering amplitude on coherence, power, and phase, culminating in the conclusion.

1.1. Treuhaft’s Model

Treuhaft et al. [13] employs a uniform, random volume model of the forest medium to assess the sensitivity of power and interferometric synthetic aperture radar to tree height and vegetation density, considering extinction along with speckle and thermal noise. Treuhaft’s InSAR experiments support a simple model of forest vegetation as a homogeneous, randomly oriented volume [19,20]. The model radar power, phase, coherence, and interferometric complex cross correlation as a function of vegetation height for extinction 0.2 dbm⁻¹, resulting from performing the integrals of Trehauft et al. in 1996 [12], omitting the ground’s contribution for simplicity is given as:

$$\begin{gathered} Power \alpha {\rho }_N {\displaystyle \frac{1-e^{-\gamma h_v}}{\gamma }} \\ Phase=\left[{\displaystyle \frac{\gamma sinsin {\alpha }_z h_v-{\alpha }_{z}coscos {\alpha }_z{h}_v+{\alpha }_z{e}^{-\gamma h_v} }{\gamma coscos {\alpha }_z{h}_v-\gamma e^{-\gamma h_v}+{\alpha }_{z}sinsin {\alpha }_z{h}_v }}\right] \\ Coherence \alpha {\left[{\displaystyle \frac{1+e^{-2\gamma h_v}-2e^{-\gamma h_v}coscos {\alpha }_z{h}_v }{{\gamma }^2+{\alpha }_z^2}}\right]}^{1/2}{\displaystyle \frac{\gamma }{1-e^{-\gamma h_v}}} \end{gathered}$$

(1)

$${\alpha }_z={\displaystyle \frac{\partial Phase}{{\partial }_z}}={\displaystyle \frac{kBcoscos (\theta -{\theta }_B) }{r_{1}sinsin \theta }}$$

where h_v is the forest height, ρ_N is the number density of scatterers (homogeneity of the density), k is the wave number, γ the medium ($\gamma ={\displaystyle \frac{2{\sigma }_x}{cos\theta }}$_, ${\sigma }_x$ is the extinction coefficient), B the baseline length, ${\theta }_B$ the angle the baseline makes with the horizontal, $\theta$ is the incidence angle, r₁ is the path length from a point in the medium defined by the vertical coordinate z to the one end of the interferometer, α_z denotes the partial derivative of the interferometric phase with respect to altitude above the ground surface.

According to Treuhaft et al. [13], the InSAR coherence falls as tree height increases and saturates beyond about 40 m. The decrease of the coherence can be seen clearly in Figure 1, which shows the contribution of vegetation at two different heights, with corresponding interferometric phases, ϕ₁, ϕ₂. To account for attenuation at all heights, the lengths of the vectors that represent each vegetation element are proportionate to the vegetation density. When vegetation components are arranged vertically, the coherence is less than one and the overall interferometric amplitude is less than the zero baseline. The highest and lowest vegetation elements correspond to the total interferometric phase. The interferometric phase rises and the interferometric coherence decreases with increasing canopy height [13].

Figure 2 illustrates the two antennas employed in the InSAR technology, wherein transmitted microwave radiation is reflected from the Earth’s surface and received by these antennas. These antennas represent the termini of an interferometric baseline, shown as B in the picture. B is equivalent to the positional difference between receivers 2 and 1, R₂–R₁. For the sake of simplicity, both the baseline and the ground surface are assumed to be horizontal. Two receivers and one transmitter are applicable for usage aboard aircraft. The observed vegetation area extends from z = 0, representing the ground surface height, to z₀ + h_v, where h_v denotes the depth of the vegetation layer. The intricate relationship of the fields obtained at the extremities of the baseline is the primary form of interferometric data.

The arguments of the range resolution functions yield the final expression for the cross-correlation [12]:

$$<E\left(R_1\right)E^{\ast }\left(R_2\right)\ge A^4{\rho }_0<f_b^2>W_{\eta }^2\left(\eta -{\eta }_0\right)\ast e^{{\displaystyle \frac{-8\pi {\rho }_{0 }Im<f_f>}{k_{0}cos\theta }}}\ast e^{-i{k}_0{r}_2}\ast I_1$$

(2)

$$I_1=\int d{r}_1^3{e}^{i{k}_0{r}_1}{W}_r^2\left(x\right)$$

(3)

$$A^4={\displaystyle \frac{{| f_R(\widehat{0}\left)\right|}^2{\left|f_T\right(\widehat{i}) |}^2}{{4\pi R^{2}R}_1{R}_2}}$$

(4)

where A is the distance for spherical waves, ${\displaystyle \frac{1}{|{\overrightarrow{R}}_1-{\overrightarrow{R}}_0 |}}$. ${\overrightarrow{R}}_0$ is the center of the range and azimuth resolutions at z₀. Both ${\overrightarrow{R}}_1$ and ${\overrightarrow{R}}_0$ are shown in Figure 2. W_η is the weight function, η is the azimuthal angle measured in the horizontal plane. The W_η function results from the process of synthesizing the aperture.

$$W_{\eta }=e^{i{k}_0(2s_z-{\tau }_A)}{\displaystyle \frac{sinsin \left[\left(n+1\right)\right]k_0\nu T/2 }{sinsin \left(\frac{k_0\nu T}{2}\right) }}$$

(5)

The range resolution function W_r is given by

$$Ref\left(t\right)\ast {\int }_{-\infty }^{\infty }G\left(\omega \right)e^{-i\omega t}d\omega |\_\tau =\int [{\int }_{-\infty }^{\infty }G\left(\omega \right)e^{-i\omega x}d\omega ] Ref(x-\tau ) dx=W_r\left(\tau \right)e^{-i{\omega }_0\tau }$$

(6)

where the transmitted pulse, with Fourier transform G(ω), is centered at time t = 0, and ω₀ is the center frequency of the transmitted signal. W_r is symmetric about its maximum, which occurs when its argument is zero.

1.2. Seker’s Model

Seker’s model introduces a discrete interferometric model of a random medium layer intended for radar interferometry applications. The forest is represented by defining tree trunks, branches, and leaves using randomly oriented, lossy dielectric particles with specified area and orientation. Seker and Lang (2009) examine the issue of scattering of a time-harmonic electromagnetic wave by N discrete identical lossy dielectric scatterers with random positions and orientations within a layer [17]. Scatterers are regarded as autonomous entities; hence, adjacent particles may not be aligned. Initially, an approximation equation for the coherent field and Green’s function is delineated as [17]

$$\underset{\_}{E}\left({\underset{\_}{x}}_t,0\right)={\underset{\_}{f}}_T\left(\widehat{i}\right){\displaystyle \frac{e^{i{k}_{o}r}}{R}}$$

(7)

where $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ is the wave number, ${\underset{\_}{f}}_T\left(\widehat{i}\right)$ is the antenna pattern of the transmitter, and R is the distance. The far field from transmitter A on the interface of the layer of the random medium electric field is shown in Figure 3.

Refs. [21,22] find the correlation of the fluctuating component of the scattering field as

$$<E\left(x_1\right)E^{\ast }\left(x_2\right)>={\int }_{v}ds\rho \left(s\right)\underset{\_}{e_{se}\left(x_1,s\right)e_{se}^{\ast }(x_2,s)}$$

(8)

where e_se(x,s) is the scattered field at x due to a scatterer at s and ρ(s) is the density of particles. The correlation of the field is given as [17]

$$C\left(x_1,x_2\right)=<E\left(x_1\right)E^{\ast }\left(x_2\right)>={\rho }_0\int s{e}_{se}\left(x_1,s\right)e_{se}(x_2,s)$$

(9)

where $\rho \left(s\right)={\rho }_0$ is assumed.

Figure 4 illustrates the four trajectories of the bi-static field. These four options manifest as reflection coefficients during the correlation of fields. The aforementioned equations and Figure 3 and Figure 4 yield the four routes of the bi-static field for a narrow-band signal.

$$C_{\alpha }\left(x_1, x_2\right)={\sum }_{\alpha =1}^N\left[C_{\alpha }^1+C_{\alpha }^2+C_{\alpha }^3+C_{\alpha }^4\right]$$

(10)

where α = 1 for the trunk, α = 2 for the branch, and α = 3 for leaves.

As it is derived in Reference [17] in detail, here, only the final form $C_{\alpha }^1$ is given:

$$C_{\alpha }^1={\rho }_0{\displaystyle \frac{{| f_R(\widehat{0}\left)\right|}^2{| \underset{\_}{f}(0^{+},i^{-}\left)q^0\right|}^2{\left|f_T\right(\widehat{i}) |}^2}{4\pi {\widehat{R}}^2{R}_1{R}_2}}\int {| W_r|}^2{{|W}_{\eta }|}^2{e}^{i{k}_0(R_1-R_2)}{e}^{-{\displaystyle \frac{8\pi \rho }{k_{0}cos\theta }}Im\underset{\_}{f}{s}_z}{d}_{s_z}$$

(11)

1.3. Comparisons Between the Two Models

An analysis of the terminology in Treuhaft’s model and Seker and Lang’s model revealed that both are identical. In Seker’s model, the parameters of the weight and range resolution functions are not displayed. Furthermore, the expression $h_v-\left(z-z_0\right)$ is designated as s_z. The particle density ${\rho }_0$is assumed to be constant in both equations. The symbols (r₁–r₂) and (R₁–R₂) in both equations appear identical, as do the notations $Im\underset{\_}{f}$, $Im<f_f>.$

Without the arguments, correlation equation can be given as

$$<E\left(R_1\right)E^{\ast }\left(R_2\right)>={\rho }_0{A}^4<f^2>\int {{|W}_r|}^2{{|W}_n|}^2{e}^{ik(r_1-r_2)}{e}^{-{\displaystyle \frac{8\pi \rho }{kcos\theta }}Im<f>s_z}d{r}_1^3$$

(12)

Comparing both models, the only parameters needed to equate are the terms ${\displaystyle \frac{{| f_R(\widehat{0}\left)\right|}^2{| \underset{\_}{f}(0^{+},i^{-}\left)q^0\right|}^2{\left|f_T\right(\widehat{i}) |}^2}{4\pi {\widehat{R}}^2{R}_1{R}_2}}$ and $A^4$ that makes the Equations (2) and (12) equal. The list of the terms used both in the Treuhaft and Seker model is given in

Table 1. The terms used in the models.

Terms in the Treuhaft Model	The Corresponding Terms in the Seker Model
$h_v-\left(z-z_0\right)$	sz
${\rho }_0$	${\rho }_0$
$Im<f_f>$	$Im\underset{\_}{f}$
$<f^2>$	${| \underset{\_}{f}(0^{+},i^{-}\left)q^0\right|}^2$
$A^4$	${\displaystyle \frac{{| f_R(\widehat{0}\left)\right|}^2{\left|f_T\right(\widehat{i}) |}^2}{{4\pi R^{2}R}_1{R}_2}}$

.

Treuhaft’s model, characterized by a homogeneous randomly oriented volume, serves as a valuable foundation for considering “interferometry” (INSAR), “polarimetry” (POLSAR) and “polarimetric interferometry” (POLINSAR). Each model scenario presented in [13] includes a summary of its parameters and observations. The decrease in specular amplitude caused by roughness is represented by the term Γ_rough.

$${ \Gamma }_{rough}\equiv expexp \left[-2k^2{\sigma }_H^{2}coscos {\theta }_{sp1, R} \right]$$

(13)

where σ_H is the expected Gaussian-distributed ground heights standardly derivable. Although it always multiplies the reflection coefficient, the ground roughness term is added for completeness; therefore, by itself, σ_H will not appear as a parameter. The whole path length is included in the equal phases of the ground–volume and volume–ground components. The length of this path, ${\widehat{R}}_1\to {\widehat{R}}_{sp1,\widehat{R}}\to \widehat{R}(x,y,z)\to {\widehat{R}}_1$, is approximately equivalent to $2|{\widehat{R}}_1-\widehat{R}(x,y,z_0\left)\right|$. The cross-correlation is obtained by utilizing the equivalence of these path lengths.

Seker’s discrete model demonstrates that the interferometric cross-correlation yields four fundamental terms, as elucidated in [17]. The various combinations of ground–volume and volume–ground that are interrelated are illustrated by the four ground terms in the cross-correlation. Due to attenuation in the vegetation, components related to ground–volume–ground returns (two specular reflections) are omitted as they are often negligible. Bistatic fields in the Seker model and the corresponding scattering mechanisms used in the Treuhaft model are listed in

Table 2. Scattering mechanism comparison of [13,17,20].

Bistatic Fields in the Seker Model [17]	Corresponding Scattering Mechanisms in the Treuhaft Model [13,20]
Direct incidence–direct scattered field	Direct ground
Reflected–direct field	Specular ground–volume
Direct–reflected field	Specular volume–ground

. Besides these bistatic fields, Seker’s model addresses an alternative scattering mechanism, specifically the reflected–reflected bistatic field which is not addressed in Treuhaft’s model.

2. Simulation of InSAR Parameters, Power, Coherence and Phase

Treuhaft et al., in 1996 and 2004, determined InSAR power, phase, and coherence using an extensive derivation process of which quantitative values are also presented with a sequence of simulations [12,13]. This research employs the parameters of the Treuhaft model, as enumerated in

Table 3. Parameters and the values used in the simulations.

Parameter	Value
Extinction coefficient (σx)	0.2 db/m
Baseline distance	5 m
Incidence angle (θ)	30°
Wavelength (λ)	5.7 cm (C-Band)
Radar altitude	8 km
B/r1	10−4

. In contrast to Treuhaft’s model, this study offers the numerical density of vegetative scatterers (ρ_N), which is essential for power computation. The estimated parameter values for the simulations are displayed in

Table 4. Forest density parameters.

θ: Incidence Angle	σx: Extinction Coefficient	γ	k: Wavenumber	B/r1	αz: Vertical Wavenumber
300	0.2 db/m	0.462	110.23	10−4	1.909 × 10−2

.

A simulation evaluating the influence of forest density on InSAR power, concerning tree height, was performed with differing vegetation numerical densities, as illustrated in Figure 5. The power demonstrates clear saturation after around 10 m of tree height, supporting the conclusions of Treuhaft’s model [13].

As forest density increases, the maximum power value correspondingly ascends, exhibiting evident saturation beyond a tree height of 10 m. For a specific numerical density, InSAR power increases with tree height; however, tree height has little effect on power beyond 10 m. The fluctuation in numerical density is assumed to exert no influence on γ, given that the extinction coefficient and incidence angle stay constant. When the baseline, polarization, and frequency are specified, the interferometric phase and coherence function as the two InSAR observations employed in parameter estimation.

Figure 6 presents the simulation results of InSAR coherence relative to tree height. InSAR coherence decreases with increasing tree height, exhibiting apparent saturation beyond 10 m, thereby corroborating the findings presented in Treuhaft’s model [13]. The vertical wavenumber α_z is influenced by various parameters, including the wavenumber, incidence angle, the angle of the baseline relative to the horizontal, and B/r₁. The known incidence angle and wavenumber from the provided data allow for an investigation into the effects of varying θ_B and B/r₁ on coherence. Figure 6 illustrates that coherence exhibits periodicity at specific angles due to the characteristics of the cosine function. At θ_B of 120°, the height of the tree does not influence coherence. At a θ_B of 30°, InSAR coherence diminishes as tree height increases. As the angle increases, both the maximum value and the rate of decrease in coherence diminish, demonstrating clear saturation beyond 10 m of tree height for all angles.

Altering the ratio of B/r₁ influences the computed InSAR coherence. A significant alteration in the ratio, on the order of tens, leads to a rapid increase in coherence, rendering the effects of lower ratios negligible, as illustrated in Figure 7. The maximum ratio of B/r₁ in Figure 7 is 0.1, while the minimum is 10⁻⁶. A change in the order of ten significantly enhances coherence, rendering the previous values negligible.

The extinction coefficient correlates with biomass, reflecting changes in density. Additionally, InSAR coherence increases with extinction, as higher extinction allows for lower vegetation layers to contribute, resulting in a more compact appearance of the vegetation. This discovery substantiates the arguments articulated in [13,20].

2.1. Effect of Bistatic Fields

According to the discussion in Seker’s model [17], InSAR coherence is influenced by the four paths of the bistatic field. The Fresnel coefficients utilized in calculating bistatic field occurrences are as follows:

$$\begin{gathered} {\Gamma }_{sq}={\Gamma }_{gq}{e}^{2i{k}_{sq}d} , qϵ\left\{\underset{\_}{\widehat{h}},{\widehat{\underset{\_}{v}}}_{-}\right\} \\ {\Gamma }_{sp}={\Gamma }_{gp}{e}^{2i{k}_{ip}d} , pϵ\left\{\underset{\_}{\widehat{h}},{\widehat{\underset{\_}{v}}}_{+}\right\} \end{gathered}$$

(14)

vertical fields:

$${\Gamma }_g^{\left(h\right)}={\displaystyle \frac{k_{z0}-k_{zg}}{k_{z0}+k_{zg}}}, {\Gamma }_g^{\left(v\right)}={\displaystyle \frac{{\epsilon }_g{k}_{z0}-k_{zg}}{{\epsilon }_g{k}_{z0}+k_{zg}}}$$

(15)

where ${\epsilon }_g$ is the permittivity of the ground and $k_{z0}$ and $k_{zg}$ are given as $k_{z0}=k_{0}cos{\theta }_0$ and $k_{zg}=\sqrt{k_0^2{\epsilon }_g-k_{x0}^2}$, $k_{x0}=k_{0}sin{\theta }_0$, where ${\theta }_0$ is the incidence angle.

The study in [17] identified the effects of four paths of the bi-static field: direct incidence–direct scattered field, reflected–direct field, direct–reflected field, and reflected–reflected field. The reflection coefficients were utilized solely in terms of their magnitudes; thus, when the absolute value is computed, it will equal unity. It is necessary to determine the horizontal and vertical reflection coefficients. The various types of wavenumbers and the permittivity of the ground are essential for these considerations. The impact of the ground on bistatic fields is assessed for two types of grounds, as indicated in references [21,22,23]. The backscattering coefficients of a vegetation layer over flat ground are provided using a discrete scatterer model. The value ε_g = 12 + j3 indicates a volumetric water content of 0.3, signifying dry ground conditions. The wavenumber k₀ was determined to be 110.23 in prior research, and the incidence angle is 30°. The wavenumbers are derived from this information. The magnitudes of the Fresnel reflection coefficients are computed using horizontal and vertical formulas. The Fresnel reflection coefficients are derived, and utilizing these values while disregarding variations in other parameters, the influence of the bistatic fields on coherence can be determined. Previously, the coherence of the direct incidence and direct scattered field was examined, excluding Fresnel reflection coefficients. Figure 8 illustrates the coherence between the reflected and direct fields. Compared to Figure 9, the new coherence has decreased due to the Fresnel reflection coefficients Γ_sh for the horizontal component and Γ_sv for the vertical component, as illustrated in Figure 8 and Figure 9.

Moreover, the reflected–reflected field reduces coherence due to the multiplication of the squares of two Fresnel reflection coefficients. This indicates a notable reduction in coherence, as illustrated in Figure 8 for horizontal components and Figure 9 for vertical components. This important issue is also highlighted in [20]. The terms related to ground–volume–ground reflections, which involve two specular reflections, may have been omitted due to their typically small magnitude resulting from attenuation in vegetation.

2.2. Effect of Vegetation Forward Scattering Amplitude on Coherence

γ is defined as follows:

$$\gamma \equiv {\displaystyle \frac{2{\sigma }_x}{cos\theta }}\equiv {\displaystyle \frac{8\pi {\rho }_{N}Im<f>}{k cos\theta }}$$

(16)

The extinction coefficient (σ_x) and incidence angle (θ) yield a value of γ equal to 0.462 for the simulations conducted. This value is contingent upon the numerical density of vegetation scatterers (ρ_N), wave number (k), and the imaginary component of the vegetation forward scattering amplitude (Im). Given that ρ_N is defined as a constant, it was assigned a value of one for the simulations of power, phase, and coherence. Using the same ρ_N and γ, Im is determined to be 1.7548 from Equation (15).

The sensitivity analysis of scattering amplitude involves calculating and plotting coherence for various imaginary components. Coarse comparisons for Im values ranging from 0.175 to 2.65 are illustrated in Figure 10 and Figure 11. Various observations can be made based on these figures. Figure 10 indicates that, as the amplitude of the imaginary component ranges from 0.175 to 1.7548, InSAR coherence appears to decline with decreasing imaginary component values. Furthermore, as the component diminishes, saturation occurs beyond the point at which the actual component saturates (exceeding 10 m). The forward scattering amplitude influences both the saturation point and the coherence magnitude. The imaginary component ranges from 1.7548 to 2.65 in Figure 11. The saturation point stabilizes at approximately 15 m of tree height.

2.3. Effect on Power

Figure 12 and Figure 13 illustrate that the InSAR power variations exhibit unique trends as the forward scattering amplitude increases. Employing the previously specified parameters, these figures offer a concise yet illuminating examination of power dynamics in relation to variations in the imaginary component. In Figure 12, as the imaginary component rises from 0.175 to 1.7548, both power and the saturation point demonstrate a steady decline. This suggests that elevated imaginary values lead to a decrease in power, potentially owing to increased absorption or dispersion effects. Outside this range, when the imaginary component escalates from 1.7548 to 2.65, power persists in diminishing, albeit at a decelerated rate. Furthermore, the saturation point stabilizes, indicating that the system attains a threshold beyond which additional fluctuations in the imaginary component exert negligible impact on power levels.

Figure 13 further elucidates this phenomenon by broadening the spectrum of forward scattering amplitudes. The observations validate that the rate of power reduction decreases at elevated levels, corroborating the notion that saturation effects become increasingly significant as the imaginary component exceeds 1.7548, supporting Treuhaft’s model [20]. This stabilization may be associated with the intrinsic scattering characteristics of the medium or certain interaction mechanisms within the interferometric system.

These findings underscore the nonlinear correlation between forward scattering amplitude and InSAR power, highlighting the importance of the imaginary component in regulating power attenuation and saturation effects.

2.4. Effect on Phase

The influence of the forward scattering amplitude on phase, as depicted in Figure 14, shows that an increase in the forward scattering amplitude from 0.175 to 1.7548 corresponds with a rise in phase. Nonetheless, beyond the amount of 1.7548 (Phreal), additional increments in amplitude yield little alterations in phase.

A comparison of this behavior with the Treuhaft model, commonly employed to elucidate the interaction between scattered electromagnetic waves and the medium’s properties, reveals a similar trend. The Treuhaft model emphasizes the relationship between phase changes and scattering amplitude, especially in applications like radar interferometry and remote sensing [13,20]. In this model, phase transitions are more pronounced at lower scattering amplitudes, whereas at elevated amplitudes (akin to the crucial value identified in the present study), the phase fluctuations stabilize. This comparison underscores the uniformity of the phase–amplitude relationship among various models, affirming that upon attaining a specific threshold (1.7548), the influence of augmenting the forward scattering amplitude on phase diminishes significantly, as anticipated by both empirical data and the Treuhaft model [20].

3. Conclusions

This study emphasizes the importance of utilizing models to identify mathematical relationships for generalizations in radar interferometry. Coherence, phase, correlation, and power are directly linked to various forest parameters, including the vegetation extinction coefficient, tree height, wavelength, radar altitude, biomass, topography, incident angle, scattering mechanisms, and ground surface dielectric properties. By comparing the discrete interferometric model with Treuhaft’s model for a layer on a random medium, this work demonstrates that our model generalizes Treuhaft’s empirical approach, providing more accurate and reliable forest structure assessments.

The findings show that power increases with tree height, and higher forest density results in greater maximum power, though both reach saturation beyond 10 m of tree height. Coherence improves as the B/r₁ ratio increases, and as θ_B increases, both the maximum value and rate of coherence decline exhibit clear saturation effects at tree heights greater than 10 m. The impact of reflection coefficients and bistatic fields on vegetation properties is evident, with two specular reflections causing a notable decrease in coherence, as confirmed by Treuhaft’s model.

Dielectric properties of different ground types influence vegetation parameters, and the optimal coherence is achieved on dry ground. The imaginary component of the forward scattering amplitude affects coherence, though not proportionally, and power shows more compact results in relation to changes in the forward scattering amplitude. As the forward scattering amplitude increases, both power and the saturation point decrease. The mathematical relationships between coherence, phase, and power provide useful indicators for predicting parameter changes and their effects.

Overall, this study advances the application of interferometric models for more accurate forest data retrieval, contributing to better vegetation monitoring. Future research should focus on validating these simulation results with empirical radar measurements to further refine the model’s accuracy and applicability in ecological and environmental monitoring.