Terrain Simplification Algorithm in Radio Wave Propagation Prediction

Due to the diversity of landforms, an appropriate threshold should be selected for the simplification of different landforms. The distribution of simplified index was obtained by preprocessing the terrain data, and the distribution was used as a reference for threshold selection. In order to study the influence of terrain simplification on the prediction of complex electromagnetic environment, a visibility algorithm used in the regular model of terrain and a probability-based power propagation model are proposed. The visibility algorithm used in the regular model of terrain is based on the relationship between planes and triangular planes in three-dimensional space. The visible triangular surfaces and the corresponding visible area of the triangular surfaces in the terrain can be calculated with the visibility algorithm. The probability-based power propagation model uses the results of the visibility algorithm to calculate the propagation probability and calculates the change of receiver power in combination with the radio wave propagation characteristics. The spatial visibility algorithm and the probability-based power propagation model can be applied to the complex electromagnetic environment to analyze the influence of terrain simplification on prediction accuracy. The minimum error between the simulation result of the probability-based power propagation model and the Wireless InSite simulation result is 0.5018dB, and the average error is 1.5039dB.


I. INTRODUCTION
Digital Elevation Model (DEM) [1] describes the fluctuation of terrain through the elevation value of terrain, which is the basic element of the radio wave propagation prediction model [2]. However, the large number of triangular surfaces needed to divide the original terrain is a major factor affecting the computational efficiency of the radio wave propagation prediction model [3]- [7] based on the ray-tracing method. There are three ways to improve the simulation speed of ray-tracing algorithm: reducing the number of useless intersecting calculations [8]; improving the speed of intersecting calculation [8]; reducing the grids in dividing terrain [9]. In this paper, terrain data is compressed to reduce the number of triangular surfaces needed for dividing terrain. The prediction results of the model will be affected when the simplified terrain is applied to the complex electromagnetic environment model. To analyze the influence from the perspective of terrain change, this paper proposes a visibility algorithm based on the relationship between planes and tri-The associate editor coordinating the review of this manuscript and approving it for publication was Zhengqing Yun . angular planes in three-dimensional space and a probabilitybased power propagation model.
Researchers carried out lots of related methods in the aspect of terrain simplification. Vertex coordinate was expressed in the form of a two-dimensional matrix with wavelet analysis [10], [11]. Wavelet decomposition and reconstruction algorithm [12], [13] was used to decompose the matrix into lots of low-frequency domain information and high-frequency domain information and the signals of different frequencies are synthesized. However, the parameters of wavelet analysis have no direct correlation with terrain features, which influence the reconstruction effect of DEM greatly. Douglas-Peucker (DP) algorithm [14], [15] was widely used in line element simplification. The three-dimensional DP uses the distance from the point to the base plane as the basis for choosing the value of each discrete point [16], but it is inefficient. In this paper, vertex coordinate is expressed in the form of a two-dimensional matrix. The matrix is divided into blocks in 3 × 3 size. The DP operation in the four directions of the row, column, negative and positive diagonal of each block is considered comprehensively. From the perspective of power propagation, it is necessary to ensure that the triangular surface is visible for the next. The painter's algorithm [17] raster-scans the object from far to near relative to the observation screen to judge the visibility problem between objects. It is applicable when there is no intersection or cyclic overlap between polygons [18]. The depth cache algorithm [19]- [21] uses the depth cache of each pixel to safeguard occlusion information. In addition, [22]- [27] find the two-dimensional or three-dimensional surface visible to the transmitter through computer graphics algorithm. However, the algorithms mentioned above study the visible surface of the transmitter, which can be regarded as a point in the scene rather than a plane. They cannot specifically obtain the area of the visible parts of the observed triangular surfaces. To solve this problem, a visibility algorithm used in the regular model of the terrain is proposed is this paper. By defining the reference point, dividing line, the set of points, and studying the relationship between space planes and triangular planes in threedimensional space, the visible surfaces and their visible areas can be calculated with the visibility algorithm.
The rest of this paper is structured as follows. The simplification method of regular terrain triangular mesh model is introduced in Section II. A visibility algorithm based on the relationship between triangular surface and space plane in three-dimensional space is proposed in Section III, which can eliminate terrain triangular surfaces that are not visible by the observation point and calculate the area of terrain triangular surfaces visible by the observation point. The ''calculation of the set of the visible surfaces'' of the algorithm can be used to derive the probability-based power propagation model in Section IV, which is used to analyze the influence of terrain simplification on electromagnetic prediction accuracy. By comparing and analyzing the simulation results of the probability-based power propagation model with the simulation results of Wireless Insite, the accuracy of the model proposed in this paper is verified in Section V, followed by the conclusion in Section VI.

II. TERRAIN SIMPLIFICATION
The regular model of the terrain, as shown in Fig. 1, is equally divided at spacings of x and y in the x and y directions   respectively. The top view of the regular terrain is shown in Fig. 2.
DP algorithm judges whether to keep the point or not according to the relationship between the threshold and the vertical distance from the point to the line. The method of applying the DP algorithm to terrain simplification in this paper is: the height data of terrain is expressed in the form of two-dimensional numerical matrix, then the matrix is VOLUME 10, 2022 divided into 3 points ×3 points as a terrain block. As shown in Fig. 3, a large terrain is divided into four terrain blocks: A1, A2, A3 and A4. This processing method will reserve some points in advance and avoid the narrow triangular surface after simplification, so this method will not cause a large difference between the simplified terrain and the real terrain.
The advantage of using terrain blocks to process the terrain is that it can eliminate the disadvantage of lacking the influence of adjacent nodes in a certain direction when using DP algorithm to simplify the terrain. Therefore, for the terrain block shown in the Fig. 4, this paper uses the DP operation in the four directions of row, column, positive and negative diagonal to determine whether the point E is retained or deleted. The point sets in four directions are {B, E, H}, {D, E, f}, {A, E, I} and {C, E, G}. Then each point set has a vertical distance value about point E, and the maximum value is taken as the vertical distance of point E. In this way, each terrain block will get a vertical distance value. The statistical vertical distance distribution is used as a reference for selecting terrain simplification threshold. Fig. 5 shows the terrain without simplification and the terrain with simplification.
This method saves the vertical distance of each terrain block to directly compare the vertical distance of each terrain block with the threshold to judge whether this terrain block should be simplified or not. Since the second DP operation is not required.

III. VISIBILITY ALGORITHM BASED ON TRIGONOMETRIC PLANES AND SPATIAL PLANES RELATIONSHIP
In the complex electromagnetic environment, some rays transmitted by the transmitter are propagated to the free space, and others are received by the receiver through the transmission between the surfaces. The transfer of power from one surface to the next involves the judgment of visibility. This paper puts forward a visibility algorithm based on the relationship between the triangular plane and plane in three-dimensional space to calculate the visible surfaces and the corresponding visible area in the regular model of the terrain. These results are applied to the probability-based power propagation model.

A. DEFINITION OF REFERENCE POINT AND DIVIDING LINE
In Fig. 6, the pole of polar system is represented by O and the axis of polar system is represented by Ox. The coordinate of any point M in the polar system is (ρ, θ), in which ρ is the distance between points M and O. This distance is called the polar diameter and θ is the angle between vector −→ OM and the polar axis, which is called the polar angle.
In three-dimensional space, there is a triangular surface T 1 T 2 T 3 out of the observation point p obs . Two requirements need to be met for the observed triangular surface to be visible to the observation point. The normal vector of the observed triangular surface is −→ n b_c . Any point of the observed triangular surface is p. If the observed triangular surface can be seen by the observation point, no other object completely blocks the observation point and the observed triangular surface, and at least one group of (p obs , p) satisfies the following formula The angle between the vector −−→ pp obs from any point of the observed triangular surface to the observation point and the normal vector of the observed triangular surface is within the interval [0, π/2).
As is shown in Fig. 7, the observation point is taken as the pole and the p obs x as the polar axis. The corresponding polar angle of the three vertices T 1 , T 2 and T 3 are θ T 1 , θ T 2 and θ T 3 (θ T 1 > θ T 3 > θ T 2 ). The vertex with the second largest polar angle in the observed triangular surface will be used as the reference point. In Fig. 7, T 3 is used as the reference point. If the polar angle of the two vertices in the observed triangular surface is equal, the vertex far from the observation point will be selected as the reference point. As shown in Fig. 8, T 3 is selected as the reference point in the observed triangle surface.
A straight line is constructed of two points in the observed triangular surface except the reference point, which is called the dividing line. In Fig. 7 and Fig. 8, vertex T 3 is the reference point. Except for the reference point T 3 , the two vertices T 1 and T 2 form the dividing line L T 1T 2 . The reference point T 3 and the observation point p obs are on both sides of the dividing line. In Fig. 9, vertex T 3 is the reference point. Except for the reference point T 3 , the two vertices T 1 and T 2 form the dividing line L T 1T 2 . The reference point T 3 and the observation point p obs are on the same side of the dividing line.

B. DEFINITION OF THE SET OF POINTS
A dictionary represented by a symbol map is defined in this paper. The key of dictionary is polar angle intervals and the value is the space plane, which is formed by two vertices of a terrain triangular surface and the observation point. The elements in the dictionary are ordered from the smallest to the largest in accordance with the key. The values of the adjacent elements in the dictionary are on the common edges, namely, the adjacent space planes have common edges.
The dictionary of the observation point p obs is represented by the symbol map obs . In the polar system with the observation point as the pole, the polar angle interval of the observed triangular surface out of the pole can be obtained in the polar system. In Fig. 7, the polar angle interval of the observed triangular surface is [θ T 2 , θ T 1 ). The set of all space planes in map obs containing the polar angle range of the observed triangular plane is denoted as { }. The set of points formed

C. CALCULATION OF THE AREA OF THE VISIBLE SURFACE
According to the fact that whether the observation point and the reference point are on the same side of the dividing line and the relationship between the reference point and the space plane, the area of the triangular surface that can be seen by the observation point is calculated in six cases: Case one: The observation point and the reference point are on the different sides of the dividing line. The reference point is located above the space plane.
The area of the observed triangular surface that can be seen from the observation point is where, y = 1 + 2 . The elements in y are deduplicated and arranged in descending polar order. p m and p n are adjacent points in y . p s is the reference point. λ is VOLUME 10, 2022 expressed as where, p s is the reference point. T is any vertex in the observed triangular surface except the reference point. As shown in Fig. 10, p s is equal to T 3 , which is above the space plane. { } = { 1 , 2 , 3 , 4 }. i 6 is the intersection point of the two adjacent space planes in { } and the plane of T 1 T 2 T 3 . i 1 , i 2 , i 3 , i 4 , i 5 , i 7 are the intersection points of the space plane in { } and the edge of Case two: The observation point and the reference point are on the different sides of the dividing line, and the reference point is located below the space plane.
The area of the observed triangular surface that can be seen from the observation point is where, y = 1 + 2 . The elements in y are deduplicated and arranged in descending polar order. l,y is the set of points in y whose polar angle is greater than the reference point, and r,y is the set of points in y whose polar angle is smaller than the reference point. p m and p n are adjacent points in y . p s is the reference point. The definition of λ is the same as formula .
As shown in Fig. 11, p s is equal to T 3 , which is blow the space plane. { } = { 1 , 2 , 3 , 4 }. i 1 , i 2 , i 3 , i 4 are the intersection points of the space plane in { } and the edge of The polar angle of T 1 , i 1 and i 2 is greater than the polar angle of T 3 . The polar angle of i 3 , i 4 and T 2 is smaller than the polar angle The area of the observed triangular surface that can be seen from the observation point can be calculated by the formula where, n = 1 + 3 , the elements of n are deduplicated and arranged in descending order of polar angle. l, n is the set of points in n whose polar angle is greater than the polar angle of the reference point, and r, n is the set of points in n whose polar angle is smaller than the polar angle of the reference point. p m and p n are adjacent points in n . p s is the reference point. The definition of λ is the same as formula . As is shown in Fig. 12, The polar angle of T 1 , i 1 and i 2 is larger than the polar angle of T 3 . The polar angle of i 3 , i 4 and T 2 is smaller than the polar angle of T 3 in the polar system with the observation point as the pole.
Case five: The observation point and the reference point are on the same side of the dividing line, and the reference point is below the space plane.
The area of the observed triangular surface that can be seen from the observation point is calculated by the formula  where, n = 1 + 3 , the elements of n are deduplicated and arranged in descending polar order. p m and p n are adjacent points in n , p s is the reference point. The definition of λ is the same as formula . As is shown in Fig. 13, Case six: The observation point and the reference point are on the same side of the dividing line, and the reference point is in the space plane. This case is computed as case four.   The normal vector of the observation plane is −→ n obs . Any point of the observation plane is p i . The normal vector of the observed triangular surface is −→ n b_c . Any point of the observed triangular surface is p j . If the observed triangular surface can be seen from the observation point, there is at least one pair of combinational (p i , p j ) that satisfies the formula.

D. CALCULATION OF THE SET OF THE VISIBLE SURFACES
There is at least one point p i on the observation plane and one point p j on the observed triangular surface. The angle between the vector − → p i p j and the normal vector of the observed triangular surface and the angle between the vector − → p j p i and the normal vector of the observation plane are all within the interval [0, π/2).
Steps for calculating the set of visible triangular surfaces are as follows: VOLUME 10, 2022 (1) Initialize the observation point and corresponding dictionary map obs . Then process triangular surfaces within the smallest rectangle that contains the observation point and judge whether the triangular surface is visible through the formula (3)(4)(5)(6)(7)(8). The area of the visible part of the triangular surface will be calculated if it is visible. The dictionary of observation point map obs is updated with the observed triangular surface.
(2) Calculate the area of the triangular surface that can be seen from the observation point by the ring. As shown in Fig. 14, the blue region is the ring region closest to the observation point, which is called the first ring. In the ring, the area formed by the four adjacent vertices p i,j , p i+1,j , p i,j+1 , p i+1,j+1 is the minimum processing unit. p i,j , p i+1,j , p i,j+1 , p i+1,j+1 can be divided by two triangular surfaces. The triangular surface close to the observation point is first processed, and then the triangular surface far from the observation point is processed. If they are equidistant from the observation point, they can be processed in any order. After processing all the triangular surfaces in the ring, the dictionary of observation point map obs will be updated with the triangular surfaces in the ring. init }, and init is formed by p 2,2 , p 2,3 , p 3,3 . Then the visible area of p 2,2 p 3,3 p 3,2 is calculated. As shown in Fig. 14, if the observation point is on hypotenuse p 2,2 , p 3,3 of triangular surface excluding the vertices of triangular surface, map obs = {I 1 : 1 , I 2 : 2 . In this equation, I 1 is the polar interval of p 2,2 p 2,3 p 3,3 , I 2 is the polar interval of p 2,2 p 3,3 p 3,2 , 1 is the plane of p 2,2 p 2,3 p 3,3 and 2 is that of p 2,2 p 3,3 p 3,2 . As shown in Fig. 15, if the observation point is on the right-angled side excluding vertices of the triangular surface, the observation point is on line segment p 3,2 p 3,3 excluding vertices p 3,2 and p 3,3 , map obs = {I 1 : 1 , I 2 : 2 }. In this equation, I 1 is the polar interval of p 2,2 p 3,3 p 3,2 , I 2 is the polar interval of p 3,2 p 3,3 p 4,3 , 1 is the plane of 2,2 p 3,3 p 3,2 and 2 is that of p 3,2 p 3,3 p 4,3 . Then the visible area of triangular surface in the red region is calculated. If the observation point is vertex of triangular surface, as shown in Fig. 16, it is necessary to initialize the dictionary with the polar angle interval of the triangular element in the blue region and the corresponding plane and calculate the visible area of the triangular surface in red region.
The update rule of the dictionary is to take the plane located above the space within the same polar angle. In one ring, the area formed by the four adjacent vertices p i,j , p i+1,j , p i,j+1 , p i+1,j+1 is the minimum processing unit. p i,j , p i+1,j , p i,j+1 , p i+1,j+1 can be divided by two triangular surfaces. The processes of calculating the area visible from the observation point are as follows: (1) In the polar system with the observation point as the pole, p i,j , p i+1,j , p i,j+1 , and p i+1,j+1 form a polar angle interval I four . Then find the dictionary map 1 corresponding to I four in map obs .
(2) Among p i,j , p i+1,j , p i,j+1 , p i+1,j+1 , the point closest to the observation point is denoted as p closest , and the point adjacent to the hypotenuse of p closest is denoted as p a . If p a is a point outside the current ring and does not belong to any of p i,j , p i+1,j , p i,j+1 , p i+1,j+1 , we will consider its impact on the calculation of the visible area, otherwise it will not be considered. As shown in Fig. 14, the blue area which is denoted as R is the first ring closest to the observation point. p 4,1 is the point of the outer circle of R, and p 3,2 is the point of the inner circle of R. In the polar system with the observation point as the pole, the polar angle interval formed by p a and p closest is I papclosest . If there is an intersection between I papclosest and I four , we will consider its impact on the calculation of the visible area is necessary, otherwise the impact will not be considered. Then we discuss the case where I papclosest and I four have the intersection. In the three-dimensional space, the plane papclosest is formed by p a , p closest and the observation point. I papclosest and papclosest form the dictionary map 2 , where map 2 = {I papclosest : papclosest }. Using map 2 to update map 1 . If there are two vertices closest to the observation point, they are both need to be considered. As shown in Fig. 14, the plane formed by p 2,1 , p 3,2 , and p obs and the plane formed by p 4,3 , p 3,2 and p obs will affect the calculation of the visible area of the triangular surfaces formed by the points p 3,1 , p 4,1 , p 3,2 and p 4,2 , which are visible from the observation point.
(3) p i,j , p i+1,j , p i,j+1 , and p i+1,j+1 can form two triangular surfaces A and B. First calculating the area of the triangular surface A that is close to the observation point with map 1 . Then updating map 1 with triangular surface A, and calculating the area of triangular surface B with map 1 .

IV. PROBABILITY-BASED POWER PROPAGATION MODEL
In order to study the influence of terrain simplification on predicting complex electromagnetic environment, a probabilitybased power propagation model is proposed. The triangular surfaces of the terrain fitted by the digital elevation model are mirror-smooth. Since the surface of terrain is rough and the terrain is composed of infinite number of micro-surfaces, the directions of micro-surfaces are random, and they are not always the same as that of macro triangular surface. Therefore, the process of power arrives at the ground from the transmitter is a deterministic event, which is transmitted between surfaces and finally reaches the receiving terminal as a probability-based event.  In Fig. 17, the transmitter is represented by Tx, and the receiver is Rx. The visible triangular surface of Tx is f 1 . The visible object of f 1 is f 2 and Rx. The visible object of f 2 is Rx. A certain probability of power is radiated from Tx to Rx through f 1 and f 2 . A certain probability of power is radiated from Tx to Rx through f 1 .
In Fig. 18, the power is transferred from triangular surface F 1 to its visible triangular surface F 2 . The distance between the two visible triangular surfaces is where, O 1 and O 2 are the gravity of surfaces F 1 and F 2 , respectively. The reflection angle [28], [29] is where, − → n 2 is the normal vector of micro-surface in F 2 , and − −− → O 2 O 1 is the vector from O 2 to O 1 . As shown in Fig. 18, the reflection angle is between the vector − −− → O 2 O 1 and the normal vector of the triangular surface F 2 , and the reflection angle is defined on the triangular surface F 2 .
The solid angle [30]- [32] of the area visible to the triangular surface F2 seen from the triangular surface F 1 with O 1 as the centre of sphere is where, − → R is the vector from O 1 to the any point on F 2 , and S F2,y is the area when F 2 can be seen from O 1 the gravity of surfaces F 1 , S F2 is the area of F 2 . The solid angle is an area on the unit sphere. Therefore, the solid angle of the center of the sphere is equal to 4π for a unit sphere.
As shown in Fig. 19, the gravity of the triangular surface F 3 is O 3 , the center of the sphere is O r , and the solid angle of the sphere of radius r relative to the point O 3 is The proportion of power radiated from the transmitter to the triangular surface is In the formula, Tx,f is the solid angle of the visible surface of the transmitter on the unit ball with the transmitter position as the center of the sphere. It is the proportion of power radiated from the transmitter to the triangular surface. The rays are emitted in all directions by the transmitter, and is not blocked by terrain. Therefore, the denominator of the formula is 4π. The probability of power transferring from one triangular surface to the next is In the formula, f,f is the solid angle of the triangular surface on the unit sphere with the gravity of the surface where the power is radiated as the center of sphere. The probability of power transfer from one triangular surface to the receiving sphere is where, f,Rx is the solid angle of the receiving end on the unit ball whose center is the gravity of the surface where the power is radiated. Formulas (4-6) and (4)(5)(6)(7) show that the ray is transmitted from the triangular surface to the next triangular surface and the ray is transmitted from the triangular surface to the receiving ball. Since the ray can only be reflected from above the terrain surface rather than emitted in all directions like the transmitting ball, the denominator of the formula is 2π. The upper half unit ball divided by the terrain surface.
The power is transmitted from one surface through reflection, and the probability of one of the paths received by the receiver through N − 1 reflection is The power is radiated from the transmitter, and the probability of one of the paths received by the receiver after N reflections is The power radiated by the transmitter and eventually received by the receiver after being reflected for N times is where, M is the number of effective channels through which the power transmitted by the transmitter is received by the receiver through the surfaces; P pow is the transmitter power; g i is the product of the transmit and receive antenna gain; N i is the reflection number of the ith channel; d i is the path length of the ith path; R N i i is the product of the reflection coefficient; P pow_Rx is the power of the receiver; λ is the wavelength.
In the prediction of complex electromagnetic environment based on ray-tracing technology, the terrain simplification may change the path that is used between receiver and transmitter, accordingly affect the calculation results. The change of power of the receiver caused by the terrain simplification with the probability-based power propagation model is where, P pow_Rx is the power received by the receiver in the original terrain. P pow_Rx is the power received by the receiver when the terrain is simplified.

V. RESULTS AND ANALYSES A. RESULT AND ANALYSIS ON ACCELERATION OF RADIO WAVE PROPAGATION PREDICTION BY TERRAIN SIMPLIFICATION
In order to study the radio wave propagation prediction time acceleration and result accuracy using the simplified terrain relative to the original terrain, this paper merely changes the threshold of terrain in the simulation, and other settings remain the same. The original accuracy of the terrain is 12m, and the width and the length of the terrain are both 4320m. The whole terrain consists of 260642 terrain triangles. And the whole terrain scene includes flat terrain and complex rugged terrain. The relative permittivity of the material is 25 and the electrical conductivity is 0.02S/m. The power of each transmitting antenna is 46dBm and the frequency of each transmitting antenna is 1000MHz. It is located in a relatively flat area. The height of the receiving antenna in the simulation is 10m, and the height of transmitting antenna is 30m, 40m and 50m, respectively. The transmitting antennas are located  in half way up the mountain. The simulation software is Wireless InSite. The terrain simplification method can retain the basic framework of terrain to the greatest extent, and construct flat and rugged terrain with triangles with different accuracy, which can reduce the number of triangles required to construct terrain model and represent the multi-precision terrain. In this paper, the vertical distance distribution is used as a reference for threshold selection. As shown in Fig. 20, the number of vertical distances that is less than or equal to 0.3 accounts for 35.45% of the total vertical distance, the number of vertical distances that is less than or equal to 0.5 accounts for 64.46% of the total vertical distance, and the number of vertical distances that is greater than 0.5 accounts for 35.54% of the total vertical distance. Therefore, 0.3 and 0.5 are selected as the simplified threshold of the terrain. A terrain block is simplified, the number of triangular surfaces of the terrain block will be changed from 8 to 2, and the number of triangular surfaces is reduced by 6. The proportion of vertical distance increases by 35.45% from the original accuracy terrain to the terrain with threshold value of 0.3. The proportion of vertical distance increased by 29.01% from the threshold value of 0.3 to the threshold value of 0.5. The increment of the vertical distance from the original terrain to the threshold of 0.3 is greater than that from 0.3 to 0.5.
The simulation time in Fig. 22 includes the time to simplify the terrain. With the increase of threshold, the number of triangular surfaces is reduced and the simulation time is shortened. In the radio wave propagation prediction based on ray-tracing technology, the most time-consuming  part is calculating the intersection of ray and triangular surfaces. The tree structure is used to organize and store the data structure of triangular surfaces, and the time complexity is logarithmic with the number of triangular surfaces. As shown in Fig. 21 to Fig. 22, the reduction of simulation time is related to the reduction of the number of triangular surfaces.
To analyze the power error between using the simplified terrain and using the original terrain, power simulation under different terrain simplified thresholds is carried out in this paper. Fourteen simulations are carried out under each threshold. One receiver and one transmitter are set for each simulation. Only the coordinates and height of the transmitter and the coordinates of the receiver are changed for each simulation. In Fig. 23, the ordinate is the power received by the transmitter and the power unit is dBm. The abscissa is the serial number of each simulation. The solid blue line represents the simulated power under the original terrain. The red dotted line represents the simulated power under the terrain with the simplified threshold of 0.3; The green double solid line represents the simulated power under the terrain with the simplified threshold of 0.5. When the threshold is 0.3 and 0.5, the average error of power is 1.05dB and 1.19 dB, respectively. The formula of power error between using the simplified terrain and using the original terrain is In this formula, P j i is the power received by the receiver on the terrain simplified by threshold j in ith simulation. P j 0 is the power received by the receiver on the original terrain in ith simulation. n is the number of simulations.
In this formula, T 0 is the simulation time under the original terrain, T i is the simulation time under the simplified terrain with the threshold i.
E is the power deviation when the unit simulation time is shortened, which is used as the indicator for the selection of terrain simplified threshold. When the terrain simplification threshold is 0.3, E is 0.3684. When the terrain simplified threshold is 0.5, E is 0.3343. By comparing E under different thresholds, when the threshold is 0.5, the value of E is the smallest, that is, the shortening of simulation time has the least impact on the power prediction results. Therefore, this threshold is used as the threshold of the terrain simplification. From the above analysis, it can be seen that within a certain allowable power error range, compared with the original terrain, simplifying the terrain can shorten the simulation time. For the terrain with large size and high modeling accuracy, terrain simplification has an obvious impact on the simulation time. And the main reason for power error is that the effective ray paths in the original terrain are different between those in the simplified terrain.

B. RESULT AND ANALYSIS OF THE IMPACT OF TERRAIN SIMPLIFICATION ON COMPLEX ELECTROMAGNETIC ENVIRONMENT BY THE PROBABILITY-BASED POWER MODEL
The simulation result of the probability-based power propagation model is compared with the simulation result of Wireless InSite to verify the accuracy of the probability-based power model. The original accuracy of the terrain is 12m, the width is 1008m and the length is 1008m. The whole terrain consists of 14112 terrain triangles. And the whole terrain scene includes flat terrain and complex rugged terrain. The relative permittivity of the material is 25 and the electrical conductivity is 0.02S/m. The power for each transmitting antenna is 46dBm and the frequency of each transmitting antenna is 1000MHz, and it is located in a relatively flat area. The height of the receiving antenna in the simulation is 10m and the height of transmitting antenna is 30m, 40m and 50m. All the transmitting antennas are located in half way up the mountain.
As shown in Fig. 24 and Table 2, the vertical distance less than or equal to 0.3 accounts for 45.39% of the total   vertical distance. In comparison with the original terrain, the number of triangular surfaces decreases by 15.26%. The vertical distance less than or equal to 0.5 accounts for 71.05% of the total vertical distance, and the number of triangular surfaces decreases by 23.89%. The percentage of vertical distance greater than 0.5 is extremely small. The simplification of terrain should retain the basic skeleton of the terrain in the radio wave propagation prediction model based on ray-tracing, and the terrain should not undergo drastic changes. Therefore, the thresholds for simplified terrain are 0.3 and 0.5.
To verify the accuracy of the probability-based power propagation model, comparisons of the simulation results of Wireless InSite and the model under different thresholds are carried out in this paper. At each threshold, the simulation results of Wireless InSite are compared with the model six times. One transmitter and one receiver are set for each simulation, and the coordinates and height of the transmitter are changed for each comparison. In Fig. 25 and Fig. 26, the ordinate is comparison of the simulation results of Wireless InSite and the model under different thresholds. And the abscissa is the serial number of each comparison. The blue curve and the red curve represent the power difference between power obtained under the original terrain and the power obtained after the terrain is simplified under the thresholds. The calculation method of ordinate ''power difference'' is shown in formula .
P pow_Rx is the power received by the receiver in the original terrain; P pow_Rx is the power received by the receiver after the terrain is simplified. Since the ordinate is the power difference rather than the power value, the results in Fig. 21 and 22 are always negative values.
The minimum error between the simulation result of the probability-based power propagation model and the WI simulation result is 0.5018dB. The average error is 1.5039dB. It can be verified that the probability-based power propagation model can better analyze the influence of terrain simplification on the receiver power. The formula of error between the simulation results of the probability-based propagation model and the simulation results of WI is The unit is dB, Pp is the difference between the power of receiver under a given threshold and that in the original terrain predicted with the probability-based power propagation model. Pw is the difference between the power of receiver obtained by WI simulation and that in the original terrain under a given threshold.

VI. CONCLUSION
As for the simplification of terrain, the numerical matrix of the original terrain is divided into 3 × 3 blocks. The results of Douglas-Peucker algorithm in four directions of row, column, positive and negative diagonal are considered compre- hensively in each small terrain, by which the shortcoming of neglecting the influence of adjacent nodes in a certain direction can be made up. The vertical distance distribution obtained in the pretreated terrain is used as a reasonable reference for the selection of the threshold of terrain simplification.
In order to analyze the influence of terrain simplification on the prediction accuracy of complex electromagnetic environment, two algorithms are proposed, which are the visibility algorithm based on the relationship between triangular plane and plane in space and the probability-based power propagation model. The areas of visible and visible surfaces are calculated with the visibility algorithm. In reality, the terrain is composed of a large number of micro-surfaces. Not all the normal vectors of micro-surfaces are the same as the macro normal vectors. Therefore, the process of power transfer from the transmitting source to the receiving end through the surfaces is regarded as a probabilistic process. The probability-based power propagation model uses the results of the visibility algorithm to calculate the probability of power propagation, and is combined with the characteristics of radio wave propagation to calculate the variation in the prediction results caused by terrain simplification. Table 3. His current research interests include electromagnetic compatibility, ultra-wideband wireless communication, and mobile internet.

See
FENGSHUO WEI is currently pursuing the master's degree with the Beijing University of Posts and Telecommunications, Beijing, China. His current research interests include electromagnetic compatibility, ultra-wideband wireless communication, and mobile internet.
CHENG LIAN received the bachelor's degree in electronic engineering from the Chongqing University of Posts and Telecommunications, Chongqing, China, in 2020. He is currently pursuing the master's degree in electronic engineering with the Beijing University of Posts and Telecommunications, Beijing, China. His research interests include electromagnetic compatibility and machine learning.