# Diffraction Loss Prediction of Multiple Edges Using Bullington Method with Neural Network in Mountainous Areas

Authors: Changwon Lee; Sungkwon Park
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

This paper proposes a neural network approach to improve the Bullington method by using parameters obtained from ignored obstacles in mountainous areas. Measurements were performed in mountainous areas to compare the prediction accuracy of propagation loss. And the measured data were used for neural network training. A detailed description of the input parameters of the proposed neural network is presented. The prediction performances were improved by up to 3.20 dB in the average error and 2.11 dB in the standard deviation of errors by the proposed method when compared to traditional diffraction methods.


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## Body

## 1. Introduction

Propagation loss prediction is one of the main problems in planning of radio communication links. And it is also very important to design mobile radio systems [1]. In Korea, the mountainous area is about 70% of the whole land area and the hilly terrain can cause multiple obstacles to the signal path. In hilly terrains, line-of-sight propagation is not possible at all the places and the diffraction becomes dominant on propagation [2]. However, accurate prediction of the diffraction losses is still challenging for realistic propagation environments such as mountainous regions [3]. Practically simplified models are used such as Bullington, Deygout, Causebrook, and Giovanelli for computational efficiency [3, 7]. Those models use knife edges to replace the mountain peaks and ridges for computing diffraction losses [2, 3]. In contrast, more sophisticated techniques such as UTD (uniform theory of diffraction) can be used to improve accuracy but need much more computation time and detailed information of terrains and obstacles such as conductivity of the materials [4].Here, we propose a neural network with Bullington model in the presence of three or more edges. To improve the prediction accuracy of diffraction loss, the proposed method uses ignored edges in the Bullington method to calculate additive diffraction losses. Parameters obtained from terrain elevation information were used as inputs to the neural network. The details are described in Section3.Measurements were conducted in mountainous areas. The measured data were used to evaluate the diffraction loss prediction and train the neural network using training algorithm.The structure of this paper is as follows. Section2 introduces the traditional diffraction models. And the proposed method and training are described in Sections 3 and 4, respectively. Section 5 illustrates the measurement campaign. Section 6 gives the evaluation of the prediction performance. Conclusions are drawn in Section 7.


## 2. Diffraction Methods

### 2.1. Bullington Method

In the Bullington method, the real terrain is reduced to a single equivalent knife edge. The location of the equivalent knife edge is the point at which the extended lines joining the transmitter and receiver to their respective dominant (the greatest angle of elevation as viewed from transmitter or receiver) obstacles meet as shown in Figure1 [5]. Then the diffraction loss is computed using (1) and (2) [4, 6]. This method has the advantage of simplicity but obstacles below the paths of the horizon rays can be ignored. And this may cause large prediction errors. In general, it underestimates path losses [4, 5].Figure 1
Concept of Bullington method.The geometrical parameters are combined in a dimensionless parameter denoted byν as shown in [6]:
(1)ν=h2λ1d1+1d2,where h is the height of the top of the obstacle above (or below) the straight path line and d1 and d2 are the distances of the path ends from the top of the obstacle. If the obstacle is below the straight path line, then h is negative. The diffraction loss as a function of ν is presented in (2) for ν greater than −0.78 [6]. It should be noted that the diffraction loss can be avoided for ν ≤ −0.78 [6].
(2)Jν=6.9+20logν−0.12+1+ν−0.1dB.The approximated diffraction loss,LB, due to the equivalent Bullington edge is equal to (2), and the total path loss using the Bullington method is [4]
(3)Lp=Lfs+LBdB,where Lfs is free space loss [16].


### 2.2. Deygout Method

The Deygout method is drawn in Figure2 for a path with three obstacles. The first step is to compute ν parameter using (1) for each edge alone, as if all other edges were absent, that is, all ν parameters are calculated for the paths Tx-A-Rx, Tx-B-Rx, and Tx-C-Rx in Figure 2. If edge B is the main edge in Figure 2, then the diffraction losses, which are J(νD_A) and J(νD_C), for edge A and edge C are found with respect to a line joining the main edge to the Tx and Rx. And those are added to the main edge loss (J(νD_B)) to obtain a total approximated diffraction loss (LD), by Deygout method [4, 7]. The diffraction loss at each edge is calculated using (2). The total path loss using the Deygout method is presented in (4) [4]. This procedure can be repeated until all the edges have been considered for more than three edges [4, 8]. However, it is common to compute the total loss as the sum of one main edge and two subsidiary main edges on either side in practical applications [4, 8].
(4)Lp=Lfs+LD=JvD_B+JvD_A+JvD_CdB,where Lfs is free space loss [16].Figure 2
Concept of Deygout method.


### 2.3. Causebrook Method (Correction)

To reduce an overestimation problem of the Deygout method, Causebrook proposed an approximate correction derived from the exact analysis of the two-edge solution [24]. The corrected form is given in
(5)LCorrected=Lp−C1−C2dB,where LD is the diffraction loss from (4). The correction factors C1 and C2 are presented below [24]:
(6)C1=6−L2+L1cosα1,C2=6−L2+L3cosα2,where cosα1=d1d3+d4/d1+d2d2+d3+d4, cosα3=d1+d2d4/d1+d2+d3d3+d4, and L1 and L3 are the losses due to edges A and C, as if they existed on their own between the Tx and Rx. And d1, d2, d3, and d4 are shown in Figure 2.


### 2.4. Giovanelli Method

Another development of the Deygout method has been proposed [24, 25]. The concept for the Giovanelli method is shown in Figure 3. It is assumed that edge A is the main edge in this case. Then a reference field point F′ is found by projecting AB onto F′F″ and h1″ is defined in [24]
(7)h1''=h1−d1Hd1+d2+d3,where H = h2 + md3 and m = (h2 − h1)/d2. The effective height for the secondary edge B is then given by [24]
(8)h2''=h2−d3h1d2+d3.Figure 3
Concept of Giovanelli method.The total diffraction loss is now given in [24]
(9)LGV=Jvd1,d2+d3,h1''+Jvd2,d3,h2''dB,where J(·) is the loss from (2). This method is conveniently extended to more than two edges by recursively applying the above procedure [24].


## 2.1. Bullington Method

In the Bullington method, the real terrain is reduced to a single equivalent knife edge. The location of the equivalent knife edge is the point at which the extended lines joining the transmitter and receiver to their respective dominant (the greatest angle of elevation as viewed from transmitter or receiver) obstacles meet as shown in Figure1 [5]. Then the diffraction loss is computed using (1) and (2) [4, 6]. This method has the advantage of simplicity but obstacles below the paths of the horizon rays can be ignored. And this may cause large prediction errors. In general, it underestimates path losses [4, 5].Figure 1
Concept of Bullington method.The geometrical parameters are combined in a dimensionless parameter denoted byν as shown in [6]:
(1)ν=h2λ1d1+1d2,where h is the height of the top of the obstacle above (or below) the straight path line and d1 and d2 are the distances of the path ends from the top of the obstacle. If the obstacle is below the straight path line, then h is negative. The diffraction loss as a function of ν is presented in (2) for ν greater than −0.78 [6]. It should be noted that the diffraction loss can be avoided for ν ≤ −0.78 [6].
(2)Jν=6.9+20logν−0.12+1+ν−0.1dB.The approximated diffraction loss,LB, due to the equivalent Bullington edge is equal to (2), and the total path loss using the Bullington method is [4]
(3)Lp=Lfs+LBdB,where Lfs is free space loss [16].


## 2.2. Deygout Method

The Deygout method is drawn in Figure2 for a path with three obstacles. The first step is to compute ν parameter using (1) for each edge alone, as if all other edges were absent, that is, all ν parameters are calculated for the paths Tx-A-Rx, Tx-B-Rx, and Tx-C-Rx in Figure 2. If edge B is the main edge in Figure 2, then the diffraction losses, which are J(νD_A) and J(νD_C), for edge A and edge C are found with respect to a line joining the main edge to the Tx and Rx. And those are added to the main edge loss (J(νD_B)) to obtain a total approximated diffraction loss (LD), by Deygout method [4, 7]. The diffraction loss at each edge is calculated using (2). The total path loss using the Deygout method is presented in (4) [4]. This procedure can be repeated until all the edges have been considered for more than three edges [4, 8]. However, it is common to compute the total loss as the sum of one main edge and two subsidiary main edges on either side in practical applications [4, 8].
(4)Lp=Lfs+LD=JvD_B+JvD_A+JvD_CdB,where Lfs is free space loss [16].Figure 2
Concept of Deygout method.


## 2.3. Causebrook Method (Correction)

To reduce an overestimation problem of the Deygout method, Causebrook proposed an approximate correction derived from the exact analysis of the two-edge solution [24]. The corrected form is given in
(5)LCorrected=Lp−C1−C2dB,where LD is the diffraction loss from (4). The correction factors C1 and C2 are presented below [24]:
(6)C1=6−L2+L1cosα1,C2=6−L2+L3cosα2,where cosα1=d1d3+d4/d1+d2d2+d3+d4, cosα3=d1+d2d4/d1+d2+d3d3+d4, and L1 and L3 are the losses due to edges A and C, as if they existed on their own between the Tx and Rx. And d1, d2, d3, and d4 are shown in Figure 2.


## 2.4. Giovanelli Method

Another development of the Deygout method has been proposed [24, 25]. The concept for the Giovanelli method is shown in Figure 3. It is assumed that edge A is the main edge in this case. Then a reference field point F′ is found by projecting AB onto F′F″ and h1″ is defined in [24]
(7)h1''=h1−d1Hd1+d2+d3,where H = h2 + md3 and m = (h2 − h1)/d2. The effective height for the secondary edge B is then given by [24]
(8)h2''=h2−d3h1d2+d3.Figure 3
Concept of Giovanelli method.The total diffraction loss is now given in [24]
(9)LGV=Jvd1,d2+d3,h1''+Jvd2,d3,h2''dB,where J(·) is the loss from (2). This method is conveniently extended to more than two edges by recursively applying the above procedure [24].


## 3. Proposed Method

As mentioned in Section2, there may be ignored edges in the Bullington method, which generally cause underestimation. Therefore, for multiple edges on the path, we propose a method to obtain the total diffraction loss by adding the loss from the Bullington method and the loss from the ignored edges. Where the diffraction loss from the ignored edges is calculated using a neural network as shown in Figure 4. There are 15 inputs of the proposed neural network, all of which can be obtained from DTM (digital terrain model). Inputs 1 through 9 are consisted of the heights of Tx and Rx, the three largest ν of ignored obstacles, the distances between Tx and three ignored obstacles, and the distance of the total path. Inputs 10 through 15 are approximate front and rear slopes of ignored edges. An approximate slope is calculated as the height difference between the edge and the point spaced by 30 m from the edge as shown in Figure 5. The use of slopes is intended to improve the prediction performance by considering the obstacle radius of curvature [9, 10]. This is an approximate application for the obstacle radius of curvature in the diffraction loss calculation. The proposed method is applied when there are at least three edges to include an ignored edge on the paths. For paths with three or more edges, the prediction of the proposed method is compared with the two traditional diffraction methods in Section 2, and the results are presented in Section 6.Figure 4
Architecture of the proposed method.Figure 5
Slopes of an ignored edge.


## 4. Training

Figure6 shows a simple neuron. The neuron is presented with inputs as shown in
(10)x=x1x2⋯xnT,and an output value of a neuron is produced as shown in [11]
(11)net=wTx,where (·)T denotes the transpose, and the weights of a neuron w are presented in
(12)w=w1w2⋯wnbT.Figure 6
Neuron model.The activation function has been selected to be the commonly used hyperbolic tangent sigmoid transfer function [11]. Figure 7 shows the architecture of the neural network with inputs of the proposed method. The inputs in Figure 7 correspond to those shown in Figure 5. Prior to training, data with three or more obstacles should be extracted from all of the measurement data. And it is necessary to divide all the measurement data with three or more edges into each of the two disjoint sets which are training and testing sets. The training sets are only used in the training procedure, and the testing sets are used for the evaluation of prediction performances. The training data are randomly chosen to be about 50% of the measurement data with three or more edges in this paper. The input vectors shown in Figures 5 and 7 can be obtained from the DTM corresponding to the transmitting and the receiving positions as described in Section 3. The receiving power may be expressed in (13) in wireless environments [12, 13].
(13)Pr=Pt+Gt+Gr−LpdBm,where Pt is the transmitting power, Gt and Gr are the gains of the Tx and Rx antennas. Lp is the path loss. Bullington proposed that the diffraction loss has to be added to the free space loss when the path is obstructed [4, 14, 15]. In order to find target values, (14) can be obtained from (13) by using the Bullington method as shown in (3).
(14)Pr=Pt+Gt+Gr−Lfs−LB−LtdB,where Lfs is the free space loss and LB is the Bullington diffraction loss. Lt is the target value, which is desired loss due to ignored edges for training. The target value can be obtained using (14) where Pr is measured power in dBm, Gt and Gr are 2.15 dBi gain, and Pt is 36 dBm as drawn in Figure 8. And the free space loss Lfs is computed using [16]
(15)Lfs=32.4+20logf+20logddB,where f is the frequency (MHz) and d is distance (km).Figure 7
Diagram of the proposed method.Figure 8
Measurement configuration diagram.Here, the error is defined for the feed-forward neural network as shown in(16)eNNj=oj−LtjdB,where o(j) is the output of the neural network corresponding to an input vector, Lt(j) is a target value, and j is the index number of training data. In the backpropagation procedure, the weights of the neural network are adjusted by minimizing the mean square error as follows.
(17)MSE=1N∑j=1NeNN2j.The Levenberg-Marquardt algorithm [17] is used to update the weights of the proposed neural network. The algorithm has been introduced to training neural networks faster than the standard gradient descent backpropagation algorithm by more than 10 times [18, 19]. In the trials of many cases, the numbers of neurons in the hidden layers with the best results are N = 20 and M = 12. In the procedure of the backpropagation training, the epochs are as follows: 109 epochs and 128 epochs for the west side of Mt. Maebong and Mt. Ongma, respectively, in Figures 9 and 10.Figure 9
Measurements in the west side of Mt. Maebong.Figure 10
Measurements in Mt. Ongma.


## 5. Measurement Campaign

Measurements have been performed in the three mountain areas as shown in Figures9–11. The heights of the selected mountains for measurement purposes are as follows: 746.7 m, 533.1 m, and 1420.2 m for the west side of Mt. Maebong in Inje, Mt. Ongma in Boryeong, and Mt. Hwaak in Gapyeong, respectively, in Figures 9–11. The selected mountains are typical hilly mountains in the Republic of Korea. In these mountains, it was convenient and safe to conduct measurements using a vehicle. The red triangle marks in Figures 9–11 are the transmitting positions, and the circles are measuring points. The total number of measurement data in Figures 9–11 is as follows: 80,163 for the west side of Mt. Maebong, 77,215 for Mt. Ongma, and 35,057 for Mt. Hwaak.Figure 11
Measurements in Mt. Hwaak.The transmitting signal is 1399 MHz continuous wave. This frequency was temporarily allowed in UHF band for the measurement campaign. The transmitting antenna is fixed on tripod of 6 m height, and the receiving antenna is placed on the roof of a car at a height of 1.8 meter above ground. The power of the transmitter is 36 dBm with a 2.15 dBi gain antenna and a cable of 2.71 dB loss. The measurement data are obtained from a spectrum analyzer connected to a 2.15 dBi gain antenna and a 40 dB gain low-noise-amplifier. A measurement configuration diagram is shown in Figure8. The measurement data with three or more obstacles from the real mountain areas are used to produce train sets for the proposed method and later evaluate the performances of Bullington, Deygout, and the proposed method. The numbers of data with three or more obstacles are as follows: 1676, 5863, and 1147 for the west side of Mt. Maebong, Mt. Ongma, and Mt. Hwaak, respectively. And the numbers of training data are as follows: 838 and 2932 for the west side of Mt. Maebong and Mt. Ongma. And all training data consist of 70% for training and 30% for validation.


## 6. Comparison of Prediction Performance

The proposed method is applied to real mountainous environments. Two scenarios are discussed for the comparisons of diffraction methods in mountainous areas as shown in Figures12–14. The details of the scenarios are drawn in Sections 6.1 and 6.2.Figure 12
Scenario 1 (in Mt. Maebong).Figure 13
Scenario 1 (in Mt. Ongma).Figure 14
Scenario 2 (in Mt. Hwaak).In Section6.1, the area-specific performance of the proposed method is investigated as shown in Figures 12 and 14. In Section 6.2, we try to find out the performance of the proposed method in another area as shown in Figure 14.


### 6.1. Prediction Results of the Proposed Method with Traditional Methods

The proposed method is evaluated by comparing with the four diffraction methods described in Section2. As described in Section 4, the testing sets are applied to the proposed method, Bullington method, Deygout method, Causebrook method, and Giovanelli method for the evaluation of prediction performances. The error is defined as shown in [2, 20, 21]
(18)ei=pi−midB,where p is the prediction of received power, m is the measured value, and i is the point number of measurement data. The prediction values are obtained by using (14) with outputs of the neural network. The prediction values are calculated for each method as follows:
(1)
Prediction values for the proposed method using (14)
(19)pi=Pt+Gt+Gr−Lfsi−LBi−oidB
(2)
Prediction values for the Bullington method using (3) and (13)
(20)pi=Pt+Gt+Gr−Lfsi−LBidB
(3)
Prediction values for the Deygout method using (4) and (13)
(21)pi=Pt+Gt+Gr−Lfsi−LDidB
(4)
Prediction values for the Causebrook method using (5) and (13)
(22)pi=Pt+Gt+Gr−Lfsi−LCorrectedidB
(5)
Prediction values for the Giovanelli method using (9) and (13)
(23)pi=Pt+Gt+Gr−Lfsi−LGVidBThe average error and the standard deviation of errors from (18) are considered to represent the prediction performance [22, 23].As described in Table1 and Figures 15 and 16, the results show that prediction performances are improved when using the proposed method in mountain areas with three or more obstacles on the paths.Table 1
Comparison of prediction performance.
Area
Method
Average error (dB)
Standard deviation of errors (dB)
Westside of Maebong mountain
Bullington
8.50
6.32
Deygout
4.92
5.27
Causebrook
3.53
5.19
Giovanelli
4.86
5.25
Proposed
2.75
4.66
Ongma mountain
Bullington
4.12
7.49
Deygout
6.51
7.41
Causebrook
7.17
7.51
Giovanelli
6.45
7.42
Proposed
0.92
5.30Figure 15
Comparison of the measurements and predictions in the west side of Mt. Maebong.Figure 16
Comparison of the measurements and predictions in Mt. Ongma.


### 6.2. Prediction Results of the Trained Neural Network in the Area Where Data Are Not Used in Training

The proposed method which has been trained with measured data from Mt. Maebong and Mt. Ongma is applied to Mt. Hwaak as illustrated in Figure14. To validate the pretrained neural network, the prediction accuracy is compared with Bullington, Deygout, Causebrook, and Giovanelli methods in the area where data are not used in training.For the paths with three or more obstacles, neural network has been trained with all of the training data from the two areas in the previous subsection. The calculation for the prediction of each method uses (19), (20), (21), (22), and (23) as in Section 6.1. As shown in Table 2 and Figure 17, the comparison results show that prediction performance is improved by using the proposed method (pretrained neural network) in the area where data are not used in training.Table 2
Comparison in the area where data are not used in training (Mt. Hwaak).
Area
Method
Average error (dB)
Standard deviation of errors (dB)
Hwaak mountain
Bullington
6.58
6.55
Deygout
7.73
7.43
Causebrook
9.45
7.45
Giovanelli
8.11
6.83
Proposed
0.71
5. 39Figure 17
Comparison in the area where data are not used in training (Mt. Hwaak).


## 6.1. Prediction Results of the Proposed Method with Traditional Methods

The proposed method is evaluated by comparing with the four diffraction methods described in Section2. As described in Section 4, the testing sets are applied to the proposed method, Bullington method, Deygout method, Causebrook method, and Giovanelli method for the evaluation of prediction performances. The error is defined as shown in [2, 20, 21]
(18)ei=pi−midB,where p is the prediction of received power, m is the measured value, and i is the point number of measurement data. The prediction values are obtained by using (14) with outputs of the neural network. The prediction values are calculated for each method as follows:
(1)
Prediction values for the proposed method using (14)
(19)pi=Pt+Gt+Gr−Lfsi−LBi−oidB
(2)
Prediction values for the Bullington method using (3) and (13)
(20)pi=Pt+Gt+Gr−Lfsi−LBidB
(3)
Prediction values for the Deygout method using (4) and (13)
(21)pi=Pt+Gt+Gr−Lfsi−LDidB
(4)
Prediction values for the Causebrook method using (5) and (13)
(22)pi=Pt+Gt+Gr−Lfsi−LCorrectedidB
(5)
Prediction values for the Giovanelli method using (9) and (13)
(23)pi=Pt+Gt+Gr−Lfsi−LGVidBThe average error and the standard deviation of errors from (18) are considered to represent the prediction performance [22, 23].As described in Table1 and Figures 15 and 16, the results show that prediction performances are improved when using the proposed method in mountain areas with three or more obstacles on the paths.Table 1
Comparison of prediction performance.
Area
Method
Average error (dB)
Standard deviation of errors (dB)
Westside of Maebong mountain
Bullington
8.50
6.32
Deygout
4.92
5.27
Causebrook
3.53
5.19
Giovanelli
4.86
5.25
Proposed
2.75
4.66
Ongma mountain
Bullington
4.12
7.49
Deygout
6.51
7.41
Causebrook
7.17
7.51
Giovanelli
6.45
7.42
Proposed
0.92
5.30Figure 15
Comparison of the measurements and predictions in the west side of Mt. Maebong.Figure 16
Comparison of the measurements and predictions in Mt. Ongma.


## 6.2. Prediction Results of the Trained Neural Network in the Area Where Data Are Not Used in Training

The proposed method which has been trained with measured data from Mt. Maebong and Mt. Ongma is applied to Mt. Hwaak as illustrated in Figure14. To validate the pretrained neural network, the prediction accuracy is compared with Bullington, Deygout, Causebrook, and Giovanelli methods in the area where data are not used in training.For the paths with three or more obstacles, neural network has been trained with all of the training data from the two areas in the previous subsection. The calculation for the prediction of each method uses (19), (20), (21), (22), and (23) as in Section 6.1. As shown in Table 2 and Figure 17, the comparison results show that prediction performance is improved by using the proposed method (pretrained neural network) in the area where data are not used in training.Table 2
Comparison in the area where data are not used in training (Mt. Hwaak).
Area
Method
Average error (dB)
Standard deviation of errors (dB)
Hwaak mountain
Bullington
6.58
6.55
Deygout
7.73
7.43
Causebrook
9.45
7.45
Giovanelli
8.11
6.83
Proposed
0.71
5. 39Figure 17
Comparison in the area where data are not used in training (Mt. Hwaak).


## 7. Conclusion

A neural network-based diffraction method is proposed, and the method can be applied with the DTM in mountainous areas which have three or more obstacles on the paths. With a set of measured data of real mountain areas, the proposed method was evaluated with Bullington, Deygout, Causebrook, and Giovanelli methods and the results showed that the prediction performances were improved by up to 3.20 dB in the average error and 2.11 dB in the standard deviation of errors by the proposed method. The remaining results are shown in Table1.In addition, the prediction performance of the proposed method is validated by applying the proposed method which was trained with data from two areas to the area where data are not used in training. The results show that prediction performance is improved as described in Table2. We used 838 data as minimum train set within about 10 km between Tx and Rx. To use the proposed method at longer distances, it is recommended to train the neural network with data from corresponding distances.It is expected that the proposed method based on neural network would show better prediction results with more measurement data in hilly mountainous areas with multiple obstacles.



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