Application of the Finite Difference Parabolic Equation Model in Forestry Remote Telemetry

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Introduction
With the development of communication network, remote telemetry technology is becoming more and more perfect. However, for the terrains that are complicated and inaccessible, traditional telemetry technology is unable to meet the needs of measurement, especially in the complex forest environment. Te electromagnetic waves (EWs) can create diferent propagation properties near the surface of the diferent ground [1]. How to efciently predict propagation loss (PL) distribution of EWs for a large-scale forest environment is the most important part [2].
Tamir frst found that the forest can be equivalent to a lossy dielectric medium layer for radio propagation prediction [3]. When the transmitting and receiving antennas are situated within the vegetation, the direct EWs and ground-refected EWs are dominant; the lateral EWs are tied to the forest-air interface. Also, a four-layer forest model can be used to predict the propagation loss at frequencies up to 2 GHz [4]. However, it is very hard to accurately calculate the PL in an actual forest because the PL values in the above models only depend on the frequency and propagation distance. In fact, the water content and volume rate of the plant have a signifcant infuence on the PL distribution. Without taking vegetation's own biophysical factors into consideration, the above traditional models are not accurate for EWs propagation prediction in the forest environment.
In the current research, most propagation prediction models were just empirical models, which were applied in the wireless sensor network of precision agriculture. Te commonly used empirical models are LITU-R, FITU-R, Cost 235, and log-normal [5]. It should be noted that these models were used in areas with foliage depths less than 1 km and were only explored the infuence of the volume density of plants, ignoring the other factors, such as water content. Tus, the above empirical models used in the forest are limited. It is necessary to fnd an electromagnetic numerical model, which is more suitable for the large-scale complicated forest environment.
As is known to all, if we regard the forest as a loss dielectric layer, the PL values of EWs vary with the biophysical parameters of vegetation [6]. Te efective permittivity is the main factor afecting PL, which is directly related to the volume content and moisture content parameters of plants. Te Debye-Cole dual dispersion model was used to simulate the efective permittivity of forest [7]. Te models regarded the forest as a mixture of air, water, and plants. It is found that the efective permittivity is a function of EWs frequency, moisture content, and volume content of plant. Te more the moisture content and volume content, the higher the efective permittivity values of forest. With the increase in frequency, the efective permittivity is decreasing. Te results show that the diference in efective permittivity will produce diferent EWs scattering, refection, refraction, and difraction efect. Te research frst reveals the linear relationship between the efective permittivity of plants and PL values of EWs.
However, it is essential to accurately predict the EWs distribution characteristics in the large-scale forest environment. Te common electromagnetic numerical methods are geometrical optics (GO) and fnite-diference timedomain (FDTD) [8,9], which are suitable for simple scenarios and take longer computing time. Tey are inefcient and not appropriate for the forest environment. Te PE method is an iterative electromagnetic numerical algorithm, which greatly reduces the calculation and improves the applicability in complicated environment. In recent years, the parabolic equation (PE) method has been widely used to model radio propagation in the troposphere [10][11][12][13].
In this paper, we present the fnite diference parabolic equation (FDPE) method to predict the PL of EWs in the forest environment [14]. It has higher accuracy than the commonly used split-step Fourier transform (SSFT) parabolic equation method [15]. Te FDPE is more convenient to solve the radio propagation problem in an in-homogeneous atmosphere.Te results of FDPE model are compared with the advanced refractive efects prediction system (AREPS) [16] and measured data in literature [17], which verify its efciency and superiority.
Based on the above results, the article will ofer an innovative way for forest remote telemetry using large-scale EWs propagation. It is more convenient, efcient, and cost-efective than the traditional telemetry way. It involves just setting up the transmitter and receiver antennas in the woods. Te polynomial ftting method is adopted to process a large number of PL sampled data obtained from the receiver. Ten, the statistical model is established, and the volume content and moisture content parameters can be estimated according to the polynomial ftting function. Te coefcient of determination R 2 is almost equal to 1. It proves that the ftting function has a great ft characteristic. Te study provides a novel and efcient theoretical method for large-scale forestry remote telemetry.

Finite Difference Parabolic Equation Method for Radio Propagation
A brief description of the fnite diference parabolic equation (FDPE) will be presented in this section [18].
In the two-dimensional Cartesian coordinates (x, z), we assume that the felds are independent of the y direction, where x and z correspond to the distance and height, respectively, and exp −iωt is the time-dependence of the felds.
For horizontal polarization, the electric feld E only has one nonzero component E y , while for vertical polarization, the magnetic feld H only has one nonzero component H y . We work with the appropriate feld component defned by for horizontal polarization and for vertical polarization. Also, the feld component satisfes the following two-dimensional scalar wave equation: where k is the free space wave number and n is the refractive index. We introduce the reduced functions associated with the paraxial direction x as follows: Ten, the scalar wave equation is given as follows: where u represents a scalar component of the electric feld for horizontal polarization or the magnetic feld for vertical polarization, and m is the modifed refractive index defned as follows: where a e is the radius of the earth. Te equation (5) can be formally written as follows: Here the wave equation is split into two terms, the u + and u − , which represent, respectively, the forward and back propagating waves.
Ten, the one-way PE is given as follows: Te pseudo-diferential operator Q is given as follows: Te Q operator can be approximated by using the Greene form as follows: 2 International Journal of Antennas and Propagation Here, χ 1 � 0.99987, χ 2 � 0.79624, χ 3 � 1.00, and χ 4 � 0.30102. Te Greene approximation FDPE can certainly give better results for irregular terrain [14].
Using the approximation in (10) for (8), the forward Greene approximation PE model can be written as follows: In order to numerically solve (11), we need to transform the form of diferential into diference. Te resulting system to be solved by diference scheme is the pentadiagonal matrices as follows: where j � 1, . . ., Z. Ten, (12) is expressed by a matrix given as follows: Te matrix elements ϑ m j , θ m j , α m j , φ m j , and ϕ m j in (13) can be written as follows: Respectively, where the symbol T, D, and d are defned by the following equation: We use the Leontovich boundary condition on the surface as follows: Te boundary feld u (x m , 0) is found from Ten, the propagation loss (PL) of electric or magnetic felds u are defned by [18] PL � −20 log 10 |u| + 10 log 10 d − 20 log 10 λ + 20 log 10 (4π), (18) where d is the horizontal propagation distance in km and λ is the wavelength in km. In the following experiments of Section 4, the PL values of EWs are calculated using equation (18).

Forest Propagation Environment Modeling
Te calculation region of the forest propagation environment is divided into three layers: the absorbing layer, the air layer, and the loss dielectric layer, as shown in Figure 1.
Te top absorbing layer is designed to eliminate the refected waves coming from the upper boundary. Here, we will set the Tukey window function. Te thickness of the absorbing layers is chosen to be 30% of the maximum height.
Te air layer is in the middle of the calculation region, and the modifed refractive index of the air layer is calculated using the equation (6).

International Journal of Antennas and Propagation
Te lower layer is defned by the forest medium. When the EWs propagate in the forest environment, the vegetation can be considered as the lossy dielectric layer. Te two-phase mixture refraction model regards the forest as a mixture of air and vegetation. Te efective permittivity ε e of forest is as follows [19]: where V represents the volume content of vegetation and ε v the represents permittivity of vegetation.
According to the Debye-Cole dual dispersion model, the vegetation is considered as a mixture of plant, free water, and bound water. Te permittivity of vegetation ε v is defned by [7] ε Here, v f and v b represent the volume content of free water and bound water, respectively. v f is defned as follows: Te plant permittivity is ε p , and the free water permittivity ε f and the bound water permittivity ε b are defned as follows: where the unit of frequency f is GHz, and σ indicates the conductivity of the free water. It can be represented by the salinity S as follows: Here, salinity S is 8.5%. Te volumetric moisture content M v can be represented by weight moisture content M g as follows: In Figures 2 and 3, the relations between the efective permittivity and frequency f and volume content V and moisture content M g are shown. Te efective permittivity in Figures 2 and 3 is correlated with the equation (19). It shows that the efective permittivity is a function of the frequency, weight moisture content, and volume content of plant. Te moisture contents are from 40% to 80%, the volume content is from 0.1% to 1%, and the frequency is 200 MHz and 300 MHz, respectively. It is found that the higher the moisture content and volume content, the higher is the efective permittivity and the efective permittivity increases with decreasing frequency.

Numerical Results and Discussion
In this section, the FDPE model is applied to a radio propagation simulation experiment in the forest environment. Te results of FDPE model are compared with the advanced refractive efects prediction system (AREPS) [16] and measured data in literature [17], which verify its efciency and superiority. Te simulation conditions are set as follows: In the standard atmosphere, the transmitter is a Gaussian horizontally polarized (HP) antenna at a height of 10 m. Te frequency is 300 MHz, and the half-power beam width is 3°w ith an inclination angle of 0°. Te maximum distance is 10 km, and the maximum height is 3 km. Te horizontal step ∆x and vertical step ∆z are normally set to be one-half of the wavelength. Suppose the type of ground is moderately dry. Te relative permittivity is ε g � 20, and the conductivity is σ g � 0.01 S/m. Te thickness of the forest dielectric layer is 15 m.
First, the FDPE model is compared with the AREPS to prove the validity in Figure 4. Suppose the surface types of bare ground and forest cover are both moderately dry ground. Te volume content is set to 0.1%, the moisture content is set to 40%, and the efective permittivity is 1.0059 + 4.36e − 5i. Te results of the FDPE model are compared with those of AREPS, and a good agreement is observed. It is shown that forward EWs mainly propagate in the form of lateral waves in the forest environment, and the PL of lateral waves increases with the horizontal distance. Due to the efect of the lateral waves, the PL values in forest-covered environment are signifcantly reduced compared to the PL values in bare ground environment.
Ten, the FDPE model is compared with measurements in Figures 5 and 6. Te measured data used were carried out by Holm in a fr forest environment [17]. Te frequency is 1355.5 MHz, and the forest is distributed near the ground and its thickness is about 18 m. Te efective permittivity is 1.003 + 0.16e − 3i in the reference. Assuming the efective permittivity is set to be 1.003 + 0.16e − 3i, 1.003 + 0.03e − 3i, and 1.4 + 0.16e − 3i, respectively, in our studies. Te results show that PL values of EWs vary with the diferent efective permittivity. Te diference in PL is very high below the height of 18 m. Tis is because the diferent characteristics of the forest will signifcantly afect the PL distribution. With increasing real and decreasing imaginary parts of efective permittivity, the PL decreased. However, the PL curves are found to be in excellent agreement beyond the height of 18 m. Because there is no forest and radio waves propagate in the air medium, the PL values are almost same over the top of the forest.
Second, the infuence of moisture content of vegetation on the PL distribution is discussed in Figure 7. Te thickness of    International Journal of Antennas and Propagation the forest is 15 m, and the height of the transmitter antenna is 10 m. Te volume content is set to be 0.1%. Assuming the moisture content is 40%, 43%, 47%, and 50%, then the effective permittivity is 1.0059 + 4.36e − 5i, 1.0062 + 13.34e − 5i, 1.0067 + 27.70e − 5i, and 1.0070 + 40.04e − 5i, respectively. Te results show that EWs mainly propagate in the form of lateral waves in the forest. It is also found that when the moisture content of vegetation changed, the action range of lateral waves changed. Because the higher the moisture content, the faster the attenuation speed of lateral waves, and the smaller the action scope of lateral waves.
Tird, the infuence of volume content of vegetation on the PL distribution is discussed in Figure 8. Other simulation conditions are the same as above. Te moisture content is set at 40%. Assuming the volume content is set to be 0.1%, 0.3%, 0.5%, and 0.7%, then the efective permittivity is 1.0059 + 4.36e − 5i, 1.0177 + 13.18e − 5i, 1.0295 + 22.09e − 5i, and 1.0415 + 31.11e − 5i, respectively. Te real and imaginary parts of efective permittivity both increased with increasing volume content. It is shown that the higher the volume content, the larger the propagation loss.
In Figures 9 and 10, the PL curves along with distance are shown for diferent moisture content and volume content. Te intense oscillation appears at the boundary of lateral waves' distribution areas. With increasing moisture content, the action scope of lateral waves decreased (Figure 9). It is found that as the volume content gets larger, the whole PL curves move upward to a higher level and oscillation becomes more intense in Figure 10. Tis is because there are larger real parts of efective permittivity for volume content values of 0.5% and 0.7%. Te results show that the larger real parts of efective permittivity have greater impact on the superposition efect of refection, refraction, and lateral waves.
In Figures 11 and 12, the PL curves along with height are shown for diferent moisture content and volume content. At the receiver location of 10 km, it is shown that the lower the moisture content and volume content, the smaller the PL values are. Furthermore, due to the fact that real part of efective permittivity varied in a larger range, the amplitude oscillation of PL curves became more intense with increasing volume content (Figure 12). It provides a theoretical basis for remote telemetry of moisture content and volume content values of vegetation for a large-scale forest environment.
In view of the above experiment results, it is shown that the vegetation's biophysical factors have a direct infuence on the PL of EWs. It is necessary to establish a statistical model to determine some relations between PL values and moisture content or volume content.     International Journal of Antennas and Propagation Te polynomial ftting curve is the perfect linear evaluation method. It guarantees both the accuracy and reliability of the experimental results. Also, it is a simple and convenient empirical way which is given as follows: Here, the polynomial ftting order N � 3, and m represents the number of sampling points. Te other simulation conditions are unchanged. Suppose the receiving point is located at a distance of 10 km and a height of 10 m.
In Figure 13, the moisture contents are set to be 40%, 41%, 42%, and 43%, and the three-order polynomial ftting curves of PL along with the volume content are obtained, respectively. Te corresponding ftting parameters are given in Table 1. Te results show that the higher the moisture content, the higher the efective permittivity. Furthermore, it is found that the greater the volume content, the larger the PL values. In addition, it is shown that with increasing moisture content, the PL curves showed increasing trends. Te absolute values of polynomial ftting coefcients ω 0 , ω 1 , and ω 3 decreased and ω 2 increased with increasing moisture content. Te coefcient of determination (R 2 ) values tend to rise with increasing moisture content.
In Figure 14, the volume content ranges from 0.05% to 0.25% at intervals of 0.05%, and the corresponding PL curves along with the moisture content are shown, and the corresponding ftting parameters are given in Table 2. It is found that the higher the volume content, the higher is the efective permittivity. With increasing volume content, the PL curves showed increasing trends. Te absolute value of polynomial ftting coefcients ω 0 , ω 1 , ω 2 , and ω 3 increased with volume content increase. Te coefcient of determination (R 2 ) values tend to descend with increasing volume content. Te MAE and MAPE values tend to rise with increasing volume content. However, the coefcient of determination (R 2 ) values are almost equal to 1, which have great ftting characteristics.
According to the above two polynomial ftting curves, the volume content and moisture content parameters can be determined. Te results show that the FDPE method is specifcally suitable for propagation prediction in large-scale inaccessible regions with serious environment. It provides novel and efcient theoretical method for forestry remote telemetry.

Conclusions
Tis article presents the fnite diference parabolic equation method to calculate the propagation loss for electromagnetic waves in the forest environment. Te results were compared with those of the AREPS and measurements, and good agreement was observed. It is found that propagation loss of EWs varies with the efective permittivity of vegetation, which is directly related to the biophysical parameters, including the volume content and moisture content. Te polynomial ftting method is adopted to establish a statistical model to determine some relations between them. Ten the volume content and moisture content of vegetation can be determined according to the polynomial ftting function. Tese studies provide a novel theoretical method for remote telemetry in a large-scale complicated forest environment.

Data Availability
No underlying data were collected or produced in this study.

Conflicts of Interest
Te authors declare that there are no conficts of interest.