# An Improved Pigeon-Inspired Optimisation Algorithm and Its Application in Parameter Inversion

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## Abstract

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## 1. Introduction

- An improved PIO algorithm (IPIO) is proposed by introducing the particle swarm optimisation (PSO) algorithm, an inverse factor, and a Gaussian factor into the PIO algorithm. In order to verify the effectiveness of the improvement, the benchmark functions are used to do the experiments and compare the experimental results with other algorithms.
- The improved algorithm for the pre-stack AVO elastic parameter inversion is used. The experimental comparison results with other optimisation algorithms indicate that the proposed IPIO algorithm can achieve better inversion results.

## 2. PIO Algorithm

#### 2.1. PIO Algorithm and Its Improvement

_{gbest}is the global optimal position obtained by comparing the positions of all the pigeons after Nc-1 iteration cycles. Once the required number of iterations is reached, the operation of the map and compass operator is stopped and the landmark operator is activated.

_{center}, which is used as a landmark, i.e., as a reference flight direction. Accordingly, the position of the ith pigeon (X

_{i}) is updated using the Formulas (3)–(5).

^{2}. The parameter μ is the mean or expected value (which is also the mid-value and the mode) of the distribution and the parameter σ is the standard deviation (and its variance is σ

^{2}).

_{1}and R

_{2}are two random numbers between 0 and 1, R

_{n}is a Gaussian random number between 0 and 1, and T

_{2}is the maximum number of iterations.

- Initialise the parameters, initialise the population, calculate the fitness value of each individual in the population, and select the optimal position ${X}_{gbest}$ of the population.
- Update the position and speed of each individual according to the PSO algorithm, and calculate the position of the individual’s corresponding reverse individual.
- Calculate the fitness value, compare the fitness values of the individual and its reverse individual, retain the better one of the two, remove the poor one, and update the global optimum and the historical optimum of each individual.
- Determine whether the maximum number of iterations of the particle swarm operator is reached. If yes, proceed to the next step, otherwise return to Step 2.
- Calculate the centre position of the population using the landmark operator of the PIO algorithm.
- Update the position of each individual based on the improved landmark operator.
- Calculate the fitness value, and update the global optimum.
- Determine whether the maximum number of iterations of the landmark operator is reached. If yes, terminate the operation, otherwise return to Step 6 to operate until the iteration is stopped.

#### 2.2. Experimental Results and Analysis

## 3. Pre-stack AVO Elastic Parameter Inversion Problem

#### 3.1. Inversion Model

#### 3.2. Inversion Results Evaluation

## 4. Experimental Simulation and Analysis

#### 4.1. Parameter Setting

#### 4.2. Simulation Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Logging curves and seismic records of inversion generated by basic genetic algorithm (GA).

**Figure 5.**Logging curves and seismic records of inversion generated by particle swarm optimisation (PSO).

**Figure 7.**Logging curves and seismic records of inversion generated by pigeon-inspired optimisation (PIO).

**Figure 9.**Comparison of the mean correlation coefficients of the three parameters for the GA, PSO, DE, PIO and IPIO algorithms.

Function | Range of Independent Variable | Dimension | Optimal Value | Category |
---|---|---|---|---|

f(1) | [−100,100] | 5 | 0 | Single solution |

f(2) | [−100,100] | 5 | 0 | Single solution |

f(3) | [−100,100] | 6 | 0 | Single solution |

f(4) | [−100,100] | 5 | 0 | Single solution |

f(5) | [−1,4] | 5 | 0 | Single solution |

f(6) | [0,10] | 5 | 0 | Single solution |

f(7) | [−100,100] | 5 | 0 | Multiple solutions |

f(8) | [−100,100] | 5 | 0 | Multiple solutions |

f(9) | [−100,100] | 6 | 0 | Multiple solutions |

Function | Algorithm | Minimum Value | Maximum Value | Mean |
---|---|---|---|---|

f(1) | GA | 2.56 × 10^{−3} | 5.14 × 10^{−1} | 1.01 × 10^{−1} |

DE | 3.94 × 10^{−2} | 45.08599 | 27.59336 | |

PSO | 3.60 × 10^{−1} | 4.224161 | 1.934762 | |

PIO | 1.32 × 10^{−7} | 3.55 × 10^{−2} | 8.44 × 10^{−3} | |

IPIO | 5.69 × 10^{−8} | 3.71 × 10^{−5} | 5.42 × 10^{−6} | |

f(2) | GA | 1.19 × 10^{−1} | 1.604513 | 5.17 × 10^{−1} |

DE | 4.63 × 10^{−7} | 11.1002 | 1.684983 | |

PSO | 2.213867 | 4.607223 | 3.56288 | |

PIO | 8.84 × 10^{−5} | 3.376968 | 1.703053 | |

IPIO | 3.82 × 10^{−14} | 2.15 × 10^{−1} | 1.69 × 10^{−2} | |

f(3) | GA | 5.81 × 10^{−2} | 2.706744 | 1.060794 |

DE | 2.44 × 10^{−8} | 55.00046 | 7.248065 | |

PSO | 7.609825 | 20.19414 | 15.15247 | |

PIO | 8.81 × 10^{−9} | 8.328467 | 1.403083 | |

IPIO | 1.07 × 10^{−14} | 1.15 × 10^{−1} | 5.73 × 10^{−3} | |

f(4) | GA | 1.90 × 10^{−2} | 4659.428 | 594.2779 |

DE | 9.68 × 10^{−2} | 22078.02 | 3841.383 | |

PSO | 11827.93 | 3459811 | 362612.9 | |

PIO | 5.26E + 02 | 114849.4 | 23962.32 | |

IPIO | 3.88 × 10^{−8} | 318704.3 | 1.63E+04 | |

f(5) | GA | 1.67 × 10^{−11} | 9.68 × 10^{−6} | 1.81 × 10^{−6} |

DE | 2.29 × 10^{−6} | 4.11 × 10^{−1} | 3.09 × 10^{−2} | |

PSO | 1.21 × 10^{−4} | 8.53 × 10^{−4} | 4.84 × 10^{−4} | |

PIO | 1.62 × 10^{−7} | 2.32 × 10^{−3} | 3.23 × 10^{−4} | |

IPIO | 3.87 × 10^{−24} | 5.45 × 10^{−19} | 7.82 × 10^{−20} | |

f(6) | GA | 6.56 × 10^{−4} | 7.93 × 10^{−3} | 3.40 × 10^{−3} |

DE | 3.35 × 10^{−6} | 30.11047 | 7.679355 | |

PSO | 4.68 × 10^{−4} | 1.67966 | 8.87 × 10^{−2} | |

PIO | 4.02 × 10^{−1} | 3.06216 | 1.59 | |

IPIO | 9.86 × 10^{−32} | 6.99 × 10^{−15} | 6.79 × 10^{−16} | |

f(7) | GA | 2.99 × 10^{−3} | 2.68 × 10^{−2} | 1.09 × 10^{−2} |

DE | 7.31 × 10^{−2} | 5.163348 | 1.671903 | |

PSO | 3.07 × 10^{−1} | 4.484861 | 1.868947 | |

PIO | 0.00 | 86.51133 | 33.82977 | |

IPIO | 7.45 × 10^{−9} | 6.22 × 10^{−5} | 4.70 × 10^{−6} | |

f(8) | GA | 2.82 × 10^{−4} | 2.18 × 10^{−1} | 3.22 × 10^{−2} |

DE | 8.06 × 10^{−6} | 8.353849 | 9.78 × 10^{−1} | |

PSO | 1.772018 | 3.991754 | 3.012487 | |

PIO | 3.02 × 10^{−5} | 3.41 × 10^{−2} | 1.66 × 10^{−4} | |

IPIO | 3.55 × 10^{−15} | 3.73 × 10^{−3} | 2.26 × 10^{−4} | |

f(9) | GA | 3.30 × 10^{−4} | 1.908191 | 4.45 × 10^{−1} |

DE | 4.88 × 10^{−3} | 295.5104 | 21.82682 | |

PSO | 9.761999 | 47.82482 | 24.92847 | |

PIO | 3.06 × 10^{−4} | 2.236064 | 1.41 × 10^{−1} | |

IPIO | 5.50 × 10^{−12} | 0.058537 | 2.93 × 10^{−3} |

N | ω | C1 | C2 | p | Max_Iteration1 | Max_Iteration2 |
---|---|---|---|---|---|---|

40 | 0.5 | 2 | 2 | 0.3 | 4000 | 1000 |

Experimental Environment | Parameter Description |
---|---|

JAVA version | 1.8.0_111-b14 |

Compiler environment | Eclipse-jee-luna-SR1a-win32-x86_64 |

Processor | Intel(R) Core (TM) i5-6500 CPU @ 3.10 GHZ |

Installed Memory (RAM) | 8.00 GB |

System type | 64-bit operating system |

**Table 5.**Comparison of the mean correlation coefficients of the three parameters for the five algorithms.

GA | PSO | DE | PIO | IPIO | |
---|---|---|---|---|---|

${V}_{p}$ | 0.556364 | 0.765481 | 0.701792 | 0.980645 | 0.937951 |

${V}_{s}$ | 0.650805 | 0.832094 | 0.787196 | 0.894879 | 0.908213 |

$\rho $ | 0.462346 | 0.649089 | 0.606702 | 0.537096 | 0.925746 |

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**MDPI and ACS Style**

Liu, H.; Yan, X.; Wu, Q.
An Improved Pigeon-Inspired Optimisation Algorithm and Its Application in Parameter Inversion. *Symmetry* **2019**, *11*, 1291.
https://doi.org/10.3390/sym11101291

**AMA Style**

Liu H, Yan X, Wu Q.
An Improved Pigeon-Inspired Optimisation Algorithm and Its Application in Parameter Inversion. *Symmetry*. 2019; 11(10):1291.
https://doi.org/10.3390/sym11101291

**Chicago/Turabian Style**

Liu, Hanmin, Xuesong Yan, and Qinghua Wu.
2019. "An Improved Pigeon-Inspired Optimisation Algorithm and Its Application in Parameter Inversion" *Symmetry* 11, no. 10: 1291.
https://doi.org/10.3390/sym11101291