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 oppositionbased 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.
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 [
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 [
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. [
The ABC algorithm [
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.
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 as
If the initial field is provided, the split step Fourier solution of parabolic equation method at different range can be easily obtained by [
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 [
In this paper, the surface based duct is described by the following fourparameter model [
In the process of inversion, the commonly used least squares objective function is given by [
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 [
Let
Let
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
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
Step
Calculate the probability of each food source according to
Step
Memorize the best solution so far.
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
Repeat Step
The flowchart of the proposed OGABC is shown in Figure
The flowchart of the proposed OGABC algorithm.
In this section, the inversion results are given to validate the optimization performance of the proposed OGABC. In the following, we take the inversion of the fourparameter surface duct with RFC technique as an example to analyze the performance of OGABC and the inversion results are compared with those of the MIWO [
In fact, the essence of the inversion of surface duct is to obtain its corresponding refractivity profile determined by (
The lower and upper search bounds of the parameters.
Parameter  Lower bound  Upper bound  Units 


0.0  0.25  Munits/m 

−3.5  −1.0  Munits/m 

25.0  50.0  m 

10.0  30.0  m 
In numerical simulation, the inversions are implemented by the radar sea clutter power calculated by parabolic equation method using the simulated and measured refractivity profile, respectively. During the inversion, the simulated radar sea clutter power from 10 Km to 50 Km is regarded as the observed radar sea clutter power, and the radar system operates at a frequency of 10 GHz, power of 91.4 dBm, antenna gain of 52.8 dB, antenna height of 7 m, beam width of 0.7°, 600 m range bin, and HH polarization. In addition, the control parameters of OGABC are given as follows: the population size is 60, the number of food sources is 30, the parameter
For the simulated refractivity case, the radar sea clutter power computed by the parameters of the surface duct
Figures
The comparison of the histograms of the inversion results for different algorithms with the noise level of 0 dB.
The comparison of the histograms of the inversion results for different algorithms with the noise level of 1 dB.
The comparison of the histograms of the inversion results for different algorithms with the noise level of 2 dB.
The comparison of the histograms of the inversion results for different algorithms with the noise level of 3 dB.
To study the convergence performance of the OGABC, the comparisons of the convergence curves of different algorithms based on the inversion results given in Figures
The comparison of the convergence curves of different algorithms with the same noise level.
The accuracy of the inversion of atmospheric duct is of crucial importance to exactly predict the marine electromagnetic environment. Hence, the comparisons of the difference between the inverted and actual radar coverage diagram simulated by parabolic equation method for different noise level are shown in Figure
The comparison of the difference between the inverted and actual coverage diagram with different noise level: (a) 0 dB; (b) 1 dB; (c) 2 dB; and (d) 3 dB.
The MAE of the three algorithms for different noise level obtained by (
The comparison of the MAE for different algorithms with different noise level.
Noise level  Algorithms  

MIWO  ABC  OGABC  
0 dB  2.22  1.64 

1 dB  1.88  0.88 

2 dB  3.54  1.84 

3 dB  3.25  0.79 

Then, in order to further test the performance of OGABC, the radar sea clutter power generated by measured refractivity profile [
The comparison of the inverted profile with the measured profile.
In addition, Figure
The comparison of the convergence curves of different algorithms.
In this paper, an improved ABC algorithm called OGABC is presented by simultaneously merging the OBL strategy and global best search equation into the standard ABC algorithm to tackle its deficiency of slow convergence rate and falling into the local best during the process of inversion of atmospheric duct. Taking the inversion of the surface duct using RFC technique as an example, the propagation characteristics of radar sea clutter obtained from the simulated and measured refractivity profile are treated as the observed sea clutter power to examine the performance of OGABC, respectively. For the simulated refractivity profile case, the Gaussian noise is added to the simulated radar sea clutter power to investigate the stability of the proposed OGABC algorithm, and the histograms and the convergence curves are used to analyze the accuracy and convergence rate. Further investigation using the radar sea clutter power generated by the measured refractivity profile is also involved, and the accuracy and convergence rate of the algorithms are discussed by comparing the inverted refractivity profile with the measured one and analyzing their convergence curves. In addition, the inversion results are also analyzed and compared with those of the MIWO and ABC. The obtained results verify that the proposed OGABC algorithm outperforms MIWO and ABC in terms of stability, accuracy, and convergence rate. Future work will focus on the experimental research and the improvement of the inversion model.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Science Fund for Distinguished Young Scholars of China (no. 61225002) and the Young Scientists Fund of the National Natural Science Foundation of China (no. 61302050).