http://section.iaesonline.com/index.php/IJEEI/index Three-Dimensional Modeling of Wave Propagation over Different Types of Terrains and Environments Using the Parabolic Equation Solved by Higher Order Approximation of the Finite Difference

Info 2021 Designing communications and radar systems depends on accurate modeling of ground waves in three-dimensional environment. Propagation of ground waves in the VHF and UHF bands affected by the characteristics of the terrain and the troposphere. Although some three-dimensional modeling of ground waves was found in the literature based on solving the parabolic equation, they were limited to a specific terrain and/or environment. Also, a lot of important factors such as the refractive index of the troposphere were ignored. In this paper, a computational model was developed for predicting the electromagnetic wave propagation over different types of terrains and environments under three-dimensional conditions. The model is based on solving the parabolic equation using higher order approximation of the finite difference method. The model allows specifications of an antenna and the electrical characteristics of the ground. Moreover, the model treats flat and non-flat terrains, mixed path with different electrical characteristics, and forest environment. Furthermore, the model enables calculations to be performed under standard and non-standard refractive conditions of the troposphere that varies in height, width, and range. The results were compared with two-dimensional parabolic equation solved by Fourier split-step and showed excellent


INTRODUCTION
Modeling of ground waves in three-dimensional (3D) environment is a major planning and design problem in communications and radar systems. This is because it gives a complete picture of the field variation in all directions at any desired point. Ground waves depend on characteristics of the terrain, environment, and the atmosphere lower layers.
One important and effective method used to model wave propagation is based on the parabolic equation (PE) [1]- [2]. This is because it provides complete wave solution for the field in the existence of terrain and environment that dependent on range, altitude, and width. Also, the solution of PE exhibit excellent robustness and accuracy for complicated problems involving 3D varying in troposphere refractive index.
A few 3-D implementations based on PE are found in the literature [3]- [11]. These methods have been applied only over restricted specification of an environment. Other environments and/or parameters that affect the propagation such as the refractive index for the troposphere, mixed surfaces environment, and forest environment have been ignored and/or not considered or tested. In this paper, the 3D PE is solved via higher order approximation of finite difference method (FDM) to model wave propagation over different types of terrains and environments under 3D conditions.

RESEARCH METHOD 2.1. The parabolic equation
Figure1 shows the 3D environment to be considered. The 3D PE is defined by [10]: where , y, and z are the range, altitude and width coordinates, 0 = ⁄ is the free space wave number with = 2 is the angular frequency with is the frequency and = 3 × 10 8 m/sec is the speed of light, and n is the refractive index. Incorporating the effect of the troposphere and forest in (1) is done through n. For the troposphere, it is done via the relation between modified refractive index M and the refractive index n as ( , , ) = ( ( , , ) − 1 + ) 10 6 where a is the earth radius taken to be 6371Km and y is the altitude in Km. For standard refractivity ∂ ( , , ) ∂ ⁄ = 0.118 M-units per meter. Incorporating the forest will be discussed in section 2.5.3. Equation (1) is limited to propagation angles less than 15 ∘ above or below the local horizontal.
The transverse boundary condition (BC) over Earth's surface is a mixed BC expressed as where 1 and 2 are constants. For lossy Earth's surface, 1 = 1, 2 = 0 √ − 1 and 1 = 1, 2 = 0 √ − 1/ for horizontal polarization (HP) and vertical polarization (VP), respectively where = + 60 with and are relative permittivity and conductivity of the surface ground at range x and width z, and = ⁄ is the wave length. For the perfectly electrical conductor (PEC) surface, 1 = 0 corresponding to the HP and 2 = 0 corresponding to the VP.

Solution of 3D PE using higher order FDM
Basically, solving (1) with BC (2) using higher order FDM requires the following steps: Step 1: Dividing the solution region into a grid of nodes as shown in Figure 2. Step 2: Approximate the derivatives in (1) by finite difference [12] as follows: The first derivative of ( , , ) at is given by (3.c) The second derivative of ( , , ) at is given by The second derivative of ( , , ) at is given by From (6), it is clear the present value of f depends on the three previous values of f .
Step 4: Write (6) in matrix form as where [ ]=[ are filled using (4.b) and (5.b) respectively. Also, zero is put in place of any point outside the grid.
Step 5: Factor out [ ] to put (7) in standard form Af=g. Equation (7) becomes 2. Place the boundary value C in the same rows numbers of g. iii. VP BC is mixed BC with = 0. Note that to be able to use (7), the values of 1, , , 2, , , and 3, , have to be known. 1, , is the initial values of the field which obtained via antenna's pattern. To find 2, , , we re-derive (7) by replacing (3.c) by (3.a). The solution for the new equation will be 2, , . In the same manner 3, , is found. We re-derive (7) by replacing (3.c) by (3.b). The solution for the new equation will be 3, , . After that, (7) can be used to solve the problem.

Upper and sides boundaries
Since the propagation problem unbounded when y→∞, z→-∞ and z→∞, an absorption BC is required at specific height and width. Perfectly Matched Layer method is used in this paper for that purpose [13]. sin +√ −cos 2 for VP where is the angle of incidence.

Different types of terrains and environments 2.5.1. Non-flat terrain
The approach used here is presented in [2]. First, a sequence of down and up stair steps approximates the irregular terrain. Then, on each step, the perpendicular field is evaluated in the normal way, forcing the chosen BCs on the ground surface. Finally, zeros are placed inside the field values of the terrain.

Mixed path
In some real environments, the wave propagates over several types of ground sections in such a way that each section has different electrical characteristics. To deal with such environment, update and fix electrical characteristic within the boundary, i.e. (2), of each section in the solution.

Forest environment
Two approaches are used to include forest environment to the PE. The first approach approximates the forest by one dielectric slab as shown in Figure 3.a [14]. This approach is useable for frequency range from 20 to 200MHz. The other approach approximates the forest by three dielectric slabs; ground, trunk, and canopy slab as shown in Figure 3.b. [15]. This model is valid for frequency band from 200MHz to 2GHz. The dielectric slab for both models is given by complex refractive index = √ − 0 ⁄ where and are relative permittivity and conductivity of forest dielectric slab layer respectively.

RESULTS AND DISCUSSION
To validate the proposed method, the mean square error (MSE) is calculated for the proposed approach which is define as where and are the results that have been gotten from higher order FDM and a reference solution of the parabolic equation respectively. The two-dimensional (2D) Fourier split-step (FSS) solution method is used as a reference solution.

Propagation over perfectly flat surfaces under standard and non-standard atmosphere
The propagation of a HP electromagnetic (EM) wave radiated by an omni-directional antenna over PEC flat surface under standard atmosphere (M(x,y,z) = 0.118y) and non-standard atmosphere (∂M(x,y,z) /∂y=0.3 for 0<y<50, ∂M(x,y,z) /∂y=-0.15 for 50<y<150, and ∂M(x,y,z) /∂y=0.118 for y>150) are shown in Figure  4 and at the middle (cross section 0<x<200Km, 0<y<300m, and z=0Km) is shown in Figure 4.a and Figure 5.a respectively. The field as a function of height at ranges 50, 100, 150, and 200Km from the transmitter is shown in Figure 4.b and Figure 5.b respectively. Also, the field as a function of range at heights of 100, 200, 300, and 400m from the surface is shown in Figure 4.c and Figure 5.c respectively. The results are compared with 2D FSS. Excellent agreement appears between the results and MSE = 9.46 × 10 −6 for Figure 4 and MSE= 8.43 × 10 −6 for Figure 5.

Propagation over finitely mixed flat surfaces
The propagation of a HP EM wave radiated by Gaussian antenna over mixed path; sea-land-sea, with electrical characteristics, = 80, = 5 siemens/m, = 10, and = 0.002 siemens/m, is shown Figure 6. Each segment has a length of 5Km. The antenna is located at a height of 100 m in the ydirection and 0Km in the z-direction, and it operates at 30 MHz with beamwidth and elevation angles in both y and z directions are equal to 1 o and 0 o respectively, with refractive index of 1 (free space). The coverage diagram of field at the middle (cross section 0<x<15Km, 0<y<500m, and z=0Km) is shown in Figure 6.a. The field as a function of height at ranges 2.5, 7.5, and 12.5Km from the transmitter is shown in Figure 6.b and compared with 2D FSS. Also, the field as a function of range at heights of 100, 200, 300, and 400m from the surface is shown in Figure 6.c and compared with 2D FSS. The MSE = 2.95 × 10 −6 .

Propagation over non-flat terrain
The propagation of a HP EM wave radiated by Gaussian antenna over PEC non-flat terrain surface is shown Figure 7. The antenna is located at a height of 100m in the y-direction and 0Km in the z-direction, and it operates at 300 MHz with beamwidth and elevation angles in y and z directions are 1 o and 0 o respectively, with refractive index of n=1 (free space). The maximum range and width are 50Km. The coverage diagram of field at the middle (cross section 0<x<50KM, 0<y<500m, and z=0Km) is shown in Figure 7.a. The field as a function of height at ranges 10, 20, 30, 40, and 50Km from the transmitter is shown in Figure 7.b and compared with 2D FSS. Also, the field as a function of range at heights of 100, 200, 300, and 400m from the surface is shown in Figure 7.c and compared with 2D FSS. The MSE = 6.37 × 10 −6 .

Forest environment
The propagation loss (PL) (defined as the ratio between the power radiated by the transmitting antenna and the power which would be available at the receiving antenna) of HP EM wave radiated by omni-directional antenna operating at 50MHz located at a height of 5m and 0Km in the y and z directions respectively, with M(x,y,z) = 0.118y outside the forest is shown in Figure 8. The maximum range is 40Km and the forest extends from 0 to 40Km in x and z directions. The altitude of the trees is 18m. The relative permittivity for the forest is = 1.5 and the conductivity is = 1 × 10 −3 siemens/m. The coverage diagram of the PL at the middle (cross section 0<x<40Km, 0<y<100m, and z=0Km) is shown in Figure 8.a. The PL as a function of height at ranges 10, 20, 30, and 40Km from the transmitter is shown in Figure 8.b and compared with 2D FSS. Also, the  PL as a function of range at heights of 20, 40, 60, and 80m from the surface is shown in Figure 8.c and compared with 2D FSS. The MSE = 4.03 × 10 −6 .
Next, the field of HP EM wave radiated by omni-directional antenna operating at 1GHz located at a height of 30m and 0Km in the y and z direction respectively, with M(x,y,z) = 0.118y outside the forest is shown in Figure 9. The maximum range and width are 5Km. The forest extends form 0Km to 5Km in x and z directions respectively. The altitude of the trees is 25m. The electrical characteristics for the forest are assumed to be = 20, = 10 × 10 −3 siemens/m for the ground, = 1.03, = 0.03 × 10 −3 siemens/m for the trunk with 15m height, and = 1.12, = 0.12 × 10 −3 siemens/m for the canopy with 10m height. The coverage diagram of PL at the middle (cross section 0<x<5Km, 0<y<100m, and z=0Km) is shown in Figure 9.a. The PL as a function of height at ranges 1, 2, 3, and 4Km from the transmitter is shown in Figure 9.b and compared with 2D FSS. Also, the PL as a function of range at heights of 20, 40, 60, and 80m from the surface is shown in Figure 9.c and compared with 2D FSS. The MSE = 1.22 × 10 −6 .

CONCLUSION
In this paper, modeling of the HP and VP EM waves radiated by an antenna with specific characteristics; frequency, radiation pattern, altitude, beamwidth and elevation angles, through standard and non-standard refractive index of troposphere; over perfectly and finitely conductive mixed surfaces, flat and non-flat surfaces and forest environment was performed using higher order FDM solution of the 3D PE.