International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 2749035, 10 pages

http://dx.doi.org/10.1155/2016/2749035

## Investigation on the Inversion of the Atmospheric Duct Using the Artificial Bee Colony Algorithm Based on Opposition-Based Learning

^{1}School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China^{2}School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China

Received 14 January 2016; Accepted 21 March 2016

Academic Editor: Sotirios K. Goudos

Copyright © 2016 Chao Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The artificial bee colony (ABC) algorithm is a recently introduced optimization method in the research field of swarm intelligence. This paper presents an improved ABC algorithm named as OGABC based on opposition-based learning (OBL) and global best search equation to overcome the shortcomings of the slow convergence rate and sinking into local optima in the process of inversion of atmospheric duct. Taking the inversion of the surface duct using refractivity from clutter (RFC) technique as an example to validate the performance of the proposed OGABC, the inversion results are compared with those of the modified invasive weed optimization (MIWO) and ABC. The radar sea clutter power calculated by parabolic equation method using the simulated and measured refractivity profile is utilized to carry out the inversion of the surface duct, respectively. The comparative investigation results indicate that the performance of OGABC is superior to that of MIWO and ABC in terms of stability, accuracy, and convergence rate during the process of inversion.

#### 1. Introduction

The lower atmospheric duct commonly encountered in marine boundary layer is an abnormal electromagnetic environment due to the sharp variations of atmospheric temperature and humidity above the sea surface. In the ducting environment, the performance of radar system and communication system can be significantly changed, such as the maximum operation range, creation of radar holes where the radar is practically blind, and strengthened sea surface clutter [1, 2]. Therefore, it is of great importance to infer the atmospheric duct owing to its considerable effect on the radar and communication system that are designed to work under standard atmospheric conditions with a typical slope of 0.118 M-units/s [3].

In general, the atmospheric duct is represented by the modified refractivity profile. The traditional methods of determining the atmospheric duct include radiosondes, rocketsondes, microwave refractometers, and lidar. Nevertheless, the traditional measurement methods have the drawbacks of high cost and containing many restrictive factors. Recently, RFC technique [4, 5] has been a promising method to infer the atmospheric duct. It uses the propagation characteristics of radar sea clutter signal to infer the modified refractivity profile information of atmospheric duct. And the RFC technique has the advantages of simple devices and easy implementation.

Inversion of atmosphere duct from the RFC technique has been an important research subject over the past several decades owing to its important applications in radar system and communication system. More attention is paid to the study of inversion model and optimization model in RFC technique. The detailed procedures of RFC technique are given by Gerstoft et al. [4], and the inversion of the range dependent and independent atmospheric duct using RFC technique is implemented by genetic algorithm. Karimian et al. [5] published a review paper on the latest research developments and the direction of future work to be done about RFC technique. Zhao et al. [6, 7] derive the inversion theoretical framework of adjoint method from parabolic equation model, and the feasibility of the adjoint method is validated by numerical simulations. As is known to all, estimation of atmosphere duct using RFC technique is an inverse problem. Taking the nonlinear relation between the forward propagation model and atmospheric duct parameters into consideration, the investigation on the optimization models with high performance is one of the most important research topics in the field of inversion of the atmospheric duct from radar sea clutter. For instance, the least square support vector machine, the particle swarm optimization (PSO), the simulated annealing algorithm, and the ABC algorithm have been applied to infer the atmospheric duct using RFC technique [8–11].

The ABC algorithm [12] is one of the most recently proposed swarm intelligence algorithms which simulates the intelligent behavior of honeybee swarm. In ABC, the optimization procedures are implemented by simulating the intelligent foraging behavior of a honeybee swarm to share information of bees for the purpose of finding the optimal solution. Currently, the ABC algorithm has been applied to the design of antenna and electromagnetic devices [13–15]. In addition, the ABC algorithm has been used to infer the atmospheric duct from RFC technique [11], and the comparative study results demonstrate that the performance of ABC is superior to that of the PSO for the inversion of atmospheric duct. However, the ABC also has the drawbacks of easily falling into local optima and slower convergence rate. To overcome this issue, the improved ABC has been proposed by updating the search equation to enhance its optimization performance, and the improved ABC is validated by benchmark function [16, 17]. In recent years, the OBL was introduced by Rahnamayan et al. [18] and has been proven to be a useful strategy to enhance the accuracy and convergence rate of the optimization algorithm, such as differential evolution and PSO.

In this paper, the OGABC is proposed by incorporating the OBL strategy and global best search equation into the ABC to enhance the performance of ABC in the inversion of atmospheric duct. In OGABC, the OBL is used to accelerate the convergence rate, and the global best search equation is adopted to balance the local and global search ability.

#### 2. The Propagation Model and Objective Function

##### 2.1. Parabolic Equation Method

Considering that the parabolic equation method has the advantages of high stability and accuracy, it has been extensively utilized to investigate the tropospheric electromagnetic wave propagation. In rectangular coordinates, the parabolic equation can be represented aswhere is the refractive index and is the free space wave number.

If the initial field is provided, the split step Fourier solution of parabolic equation method at different range can be easily obtained by [19]where and are the Fourier transform and inverse Fourier transform, respectively, is the transform variable, is the distance interval, and is the initial field. It should be pointed out that this research mainly focuses on the inversion of atmospheric duct; more detailed information on the propagation problem with parabolic equation method can be found in [19].

##### 2.2. Radar Sea Clutter Power

In RFC technique, the objective function is described by the radar sea clutter power at different propagation distances. Taking the influence of atmosphere condition into account, the received radar sea clutter power based on radar equation can be expressed in dB by [4] where is the propagation loss calculated by the parabolic equation method, is the radar cross section obtained by the GIT sea clutter model [20], is the propagation distance, is a constant that includes wavelength, transmitter power, and antenna gain, and is the parameter vector of the atmospheric duct.

In this paper, the surface based duct is described by the following four-parameter model [2]:where is the base refractivity and and stand for the slope and thickness of the base layer, whereas and represent the slope and thickness of the inversion layer, respectively.

##### 2.3. The Objective Function

In the process of inversion, the commonly used least squares objective function is given by [4]where and stand for the observed and received sea clutter power at different ranges and and denote the average power of and , respectively.

#### 3. The Proposed OGABC Algorithm

##### 3.1. The OBL

The OBL strategy can improve the convergence rate and accuracy of optimization algorithm by simultaneously evaluating the initial solution and opposite solution for the population initialization and for the generation jumping. The probability theory indicates that the opposite solution can increase the opportunity of approaching the global best solution in the search process. The definitions of opposite number and opposite solution are given by [18].

*Definition 1. *Let be a real number. Its corresponding opposite number is defined by

*Definition 2. *Let be a solution in -dimensional space, where and , are lower and upper bounds of the dimension. The corresponding opposite solution is defined by

In this paper, the inversion of atmospheric duct is a minimization problem. With the help of the definition of opposite solution, the OBL in the inversion of atmospheric duct can be described by the following: if , then random solution can be replaced with ; otherwise, we continue with . Additionally, according to a jumping rate, the better population for the next iteration can be obtained by the generation jumping using the current and their corresponding opposite population. Evidently, the random solution and opposite solution are simultaneously evaluated to select the better solution in the search process.

##### 3.2. The Proposed OGABC and Its Implementation Steps

The ABC is one of the most recent swarm intelligence optimization algorithms proposed by Karaboga under the inspiration of the intelligent foraging behavior of honeybee swarm. In ABC, there are three types of honeybees: employed bees, onlooker bees, and scouts. The position of a food source stands for a possible solution of the optimization problem and the nectar amount of a food source is employed to evaluate the quality of the solution. The number of employed bees is equal to the number of food sources and the half of the population size. The employed bees undertake the responsibility of searching for food sources and share the effective information with onlooker bees. The onlooker bees try to make a further selection of the excellent food sources based on the information provided by employed bees. If the quality of food source cannot be improved through a predetermined condition, the corresponding food source becomes a scout. Then, the scout begins to randomly generate a new food source at the neighborhood of the hive.

In order to enhance the performance of ABC in the inversion of atmospheric duct, the OGABC is presented by incorporating the OBL strategy and global best search equation into ABC algorithm. The main steps of OGABC are summarized as follows.

*Step 1 (opposition-based population initialization). *Step 1 contains the following.

*Step 1.1*. Randomly produce -dimensional population of solutions by

*Step 1.2*. Generate the opposite population of by

*Step 1.3*. Choose the best solutions from according to the fitness value to produce the initial population.

*Step 2 (in employed bees stage). *Step 2 contains the following.

*Step 2.1*. Update the position of food sources using the global best search (10) [16] and evaluate the quality of the new position of food sources:where the subscripts , , and are randomly selected and satisfy , is the element of the global best solution, and and are uniform random number in and , respectively.

*Step 2.2*. Apply the greedy selection mechanism to choose the better food source between the old and new food source.

*Step 3. *Calculate the probability of each food source according towhere represents the fitness value of the food source computed in employed bees stage.

*Step 4 (in onlooker bees stage). *Step 4 contains the following.

*Step 4.1*. Update the position of food sources using (10) according to the probability computed in Step 3.

*Step 4.2*. Apply the greedy selection mechanism again to choose the better food source.

*Step 5. *Memorize the best solution so far.

*Step 6. *In scouts stage, decide whether a food source becomes a scout or not; if it exists, the food source is replaced by a new random solution.

*Step 7 (opposition-based generation jumping). *Step 7 contains the following.

*Step 7.1*. According to the jumping rate, decide whether opposition-based generation jumping appears or not; if it appears, the new opposite population of are produced bywhere and are the minimum and maximum value of the dimension in the current population.

*Step 7.2*. Choose the best solutions from according to the fitness value to generate the population for the next iteration.

*Step 8. *Repeat Step 2 to Step 7 until a terminating condition is met.

The flowchart of the proposed OGABC is shown in Figure 1.