# Space Object and Background Polarization Models and Detectability Analysis
## 2 Basic Theory of the Polarization Bidirectional Reflection Distribution Function Model
### Parametric Genetic Algorithm Inversion Verification  
In this paper, a genetic algorithm is selected to invert five key unknown parameters in the model, denoted as \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\) and \(a_{5}\). The basic parameters and operations of the genetic algorithm are set and selected for the needs of this paper as follows. Step 1: Create an initial population of chromosomes encoded by genes at random. We use universal binary coding in this paper and assume that the range of individual \(a\) values to be inverted is \([a_{\text{min}},a_{\text{max}}]\), so the binary number of nnn bits \(b\) is  
\[b=\frac{(2^{n}-1)(a-a_{\text{min}})}{(a_{\text{max}}-a_{\text{min}})} \tag{24}\]  
The first-generation initial population consists of all the individuals compiled using the adopted coding at the very beginning, where the genes on the individuals in the initial population are usually generated randomly within their range of values. In this paper, we set the initial population size to 100 randomly generated individuals. Step 2: Calculate the fitness function value for each chromosome, end the loop iteration by satisfying the output condition, and output the optimal five covariates after decoding;if the output condition is not satisfied, continue to run down. In this paper, the fitness function is chosen as the standard mean squared deviation model of the simulated value \(k\) and the experimentally measured polarization value \(n\), to ensure the trend of selecting individuals during the selection process. \[f(n,k,\sigma,k_{s},\rho)=\frac{\sum\limits_{\theta_{i}}{(P^{\prime}-P)^{2}}}{ \frac{\sum\limits_{\theta_{i}}{(P)^{2}}}} \tag{25}\]  
Step 3: The \(100-N\) individuals with relatively good fitness functions are chosen to form a new population. In the new population, the set crossover probability and method operation are run In the selected population, there is a chance that two individuals are randomly chosen to exchange the same type of inverse covariates, which is known as the crossover operation in the genetic algorithm. In other words, two new chromosomes are formed by swapping alleles from a randomly paired pair of chromosomes. The crossover chance is set to 0.02, and single-point, multi-point, and uniform crossover methods can all occur at the same time. Step 4: The set mutation probability and method operations are run in the new population; the inverse covariates in the population's individuals are all given a certain probability to revalue in the range of values to produce a new individual, which is called the mutation operation in the genetic algorithm, i.e., the genes on the chromosome mutate to other alleles with a certain probability to produce a new chromosome. The mutation probability is 0.02, and uniform, boundary, and basic position mutations can all occur at the same time. Step 5: The population is run again in step 2 after selection, crossover, and mutation. When the output condition is met, the population genetic iteration is terminated. The output condition is that the difference in fitness function between the largest individual in the population and the average fitness function of all individuals in the population is less than \(10^{-3}\), and the algorithm's maximum number of inheritance \(N\) cannot exceed 100. In order to verify the reliability of the theoretical model, the polarization measurements in the literature [18] are used to invert the parameters of complex refractive index N, specularity coefficient \(k_{s}\) and reflectivity parameter \(\rho_{d}\) of aluminum and black lacquer materials, and the inversion results are obtained and compared with the reference literature to obtain the error values. The complex refractive index of the material is \(N=n+ki\), where \(n\) is the refractive index and \(k\) is the absorption coefficient, both of which are wavelength-dependent.