The classical lognormal PDF of RCS random variable σ is given by^[5]

where σ ₀ and denote the median and mean of σ , respectively; and is the mean-to-median ratio.

For traditional target modeling, the value of ρ is generally assumed to be greater than 1.^[8] While the RCS median σ ₀ of a stealth aircraft is similar to that of a traditional aircraft; its RCS mean , however, is usually smaller than that of a traditional one.^[9] Consequently, in the case of a stealth aircraft, the value of ρ may be smaller than 1, which obviously conflicts with the fitting condition of the lognormal model.

Here, we transform the lognormal problem into a normal problem to avoid calculating parameter ρ directly. Let μ and s denote the mean and standard deviation of ln σ , respectively. Thus, the basic expression of the lognormal PDF may be expressed as^[10]

where ln σ is normally distributed. Equation (2) is convenient to estimate the parameters by Bayesian-MCMC method.

Now we consider the Bayesian derivation of parameter μ as a demonstration. The posterior distribution function of μ is f_{μ | X}(μ | x), where X is a sample set of RCS variables and x is an element of X. According to the Bayesian theory, we have

where

Here, f(μ ) is the prior distribution function of μ , and it is generally assumed to be independent of the observation sample X. Besides, f_X(x) is the standard factor of f_{μ | X}(μ |x), and it is also independent of μ .^[14] Thus equation (3) can be simplified into

where x_i is an element of n-dimensional sample X. Therefore, the estimator of μ is

In estimating μ and s, μ can be estimated with s fixed, and vice versa.

The prior distributions of μ and s are generally specified as a normal distribution and a gamma distribution, respectively, ^[15] i.e., μ ∼ N(η , δ ²), s ∼ Γ (α , β ). The prior distribution of μ is

where η and δ ² are the mean and variance of normal distribution, respectively. The prior distribution of s is

where α is the shape parameter, and β is the scale parameter.

Substituting Eqs. (2), (7), and (8) into Eq. (6), we obtain the posterior estimators

where x_i is an element of X; f_N(x_i; s) and f_N(x_i; μ ) denote the basic lognormal PDF on condition that either s or μ is fixed, respectively.

In the Legendre polynomial model, the probability density of RCS is reconstructed by linear combination of orthogonal polynomials, which is given by^[10]

where ; σ _max and σ _min denote the maximum and minimum of variable σ , respectively; L_m(t) is the m-th Legendre polynomial with ; the coefficients a_m are unknown, but can be determined along with the Legendre polynomials by using random variable σ and order m. However, the computing process is rather complex and the kernel density function of RCS data is required.^[9]

A number of numerical experiments show that satisfied fitting results could be achieved when the first 15 ∼ 25 Legendre polynomials are selected.^[9] Therefore, assuming that the first m (15 ≤ m ≤ 25) polynomials are adopted, we could further obtain the posterior estimates of the polynomial coefficients a₁, a₂, … , a_m with the following procedures.

The prior distribution represents the prior knowledge about the coefficients a₁, a₂, … , a_m before the experimental observation. In practice, the low-information normal distribution is generally chosen as the prior distribution of the coefficients, ^[15] i.e., . The prior distribution expression is

where i = 1, 2, … , m, η _i and are the mean and variance of normal distribution, respectively. Using Eqs. (11), (12), and (6), the posterior estimators could be described as

where f_L(x_i; a₂, a₃, … , a_m), … , f_L(x_i; a₁, … , a_{k− 1}, a_{k + 1}, … , a_m), … , f_L(x_i; a₁, a₂, … , a_{m− 1}) denote respectively the PDF expressions under the condition of fixing all of the parameters except for the estimated one. Note that in practical calculation, the posterior estimates of â ₁, â ₂, … , â _m can be solved once and for all by using Markov chains. Besides, the kernel density function of RCS data is not needed.

In the conventional method, the classical Eqs. (1) and (11) are adopted for curve fitting. Due to the low RCS property of stealth aircrafts, a large error-of-fit may be caused. In contrast, the parameter estimators in Bayesian-MCMC method are reformulated by the prior and posterior distributions of parameters. Then, the parameters estimates can be calculated by the MCMC algorithm. Finally, the estimated parameters are substituted into the basic formulae Eqs. (2) and (11) for curve fitting.

We now consider the parameter estimation procedure by illustrating the principle of the Gibbs sampling algorithm. As shown in Subsection 2.1, the prior distributions of μ and s are μ ∼ N (η , δ ²) and s ∼ Γ (α , β ), respectively. In order to construct a Gibbs sampler for this model, we need to calculate the conditional distributions f(μ |s, X) and f(s|μ , X), and sample sequentially from these distributions. With some mathematical manipulations, ^[15] we could obtain

where , n is the dimension of sample set X, X̄ is the mean of X, and

Using these results, the Gibbs sampling algorithm may be summarized as follows.

Set θ = (μ , s) for the lognormal model or

for the Legendre polynomial model, and set the initial values θ ⁽⁰⁾. Then, for t = 1, … , T,

Step 1 Let θ = θ ^{(t− 1)}.

Step 2 For j = 1, … , J, update θ _j from θ _j ∼ f(θ _j| θ _{∖ j}, X), where θ _{∖ j} = (θ ₁, … , θ _{j− 1}, θ _{j + 1}, … , θ _J).

Step 3 Set θ ^(t) = θ for saving the generated set of values at t + 1 iteration of the algorithm.

Hence, given a particular state of the Markov chain θ ^(t), we generate the new parameter values in Step 2 by

So far, an iteration of the Gibbs algorithm has been completed, and the parameter set θ ^{(t− 1)} has been updated as . The first t randomized samples can be removed by a large number of iterations T, and consequently a Gibbs system sample θ ^{(t+ 1)}, θ ^{(t+ 2)}, … , θ ^(T) can be established. The Gibbs system sample depends on the actual distribution f(θ ), which can be used to calculate the parameter estimates.

In the analysis of the MCMC output, the Monte Carlo error (MC error) is adopted to measure the performance of each Markov chain. A small MC error generally corresponds to a parameter estimate with relatively high estimation precision.

The MCMC output provides us with a random sample of the type θ ⁽¹⁾, θ ⁽²⁾, … , θ ^(t), … , θ ^(T). For any function G(θ ) of the interesting parameter θ , we can obtain a sample by simply considering G(θ ⁽¹⁾), G(θ ⁽²⁾), … , G(θ ^(t)), … , G(θ ^(T)). Using these functions, we adopt the batch mean method to estimate MC error.^[15] Firstly, partition the output sample into K batches (usually K = 30 or K = 50). Thus, each batch mean G̅ _b(θ ) is given by

where v denotes the sample size of each batch, ; t is the iteration time, and t = 1, … , T. Thus, the overall sample mean is

on the assumption that we keep θ ⁽¹⁾, … , θ ^(T) observations. Then an estimate of the MC error is simply acquired by the standard deviation of the batch mean estimates G̅ _b(θ ), ^[15]

The MC error changes with the batch value K (or iteration time t). When the maximum value of the MC error is less than a designated threshold ε , the sampling results can be considered to be convergent. The procedure of the Bayesian-MCMC method is summarized in Fig. 1.

Fig. 1.

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	Fig. 1. Flow chart of the Bayesian-MCMC method.

We consider two typical stealth aircraft models, namely type I and type II. Due to the complex absorbing material and its unknown electrical parameters, ^[17] we only consider the effect on RCS caused by the shape stealth technology. The aircraft models shown in Fig. 2 are constructed in accordance with their real size in computer aided design (CAD), and the coordinate system of stealth aircraft is also illustrated in Fig. 2(b), where ϕ and Θ denote the azimuth angle and pitch angle, respectively. Besides, the RCS data are simulated by the multilevel fast multipole method (MLFMM) in the FEKO (FEldberechnung bei Korpern mit beliebiger Oberflache). Considering the computational complexity and accuracy, this experiment is carried out at 1.3 GHz with horizontal polarization (HH), and the azimuth angle sampling interval is 0.1° (leveled off, pitch angle 0° ). The RCS sampling data of both types are shown in Fig. 3.

Fig. 2.

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	Fig. 2. Models of stealth aircraft (a) type I and (b) type II and the coordinate system, where the head direction of an aircraft is set to be azimuth angle 0° and pitch angle 0° .

Fig. 3.

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	Fig. 3. RCS data of (a) type I and (b) type II for different azimuth angles. The RCS data of both aircrafts are concentrated in − 20∼ 0 dB.

Notice that the three peaks of RCS data curves correspond to the wings and tails of the aircrafts, respectively. Further, the RCS data of both aircrafts concentrate in the range of − 20 dB ∼ 0 dB, and the data of head direction gather in the range of − 20 dB ∼ 10 dB, which agree with the public data. The RCS changing trend of type II is similar to that of type I, but the mean value of RCS data of type II is less than that of type I due to its smaller physical dimension. With some direct calculations, we could find that the number of RCS data below 0 dB is up to 62.2% and 74.4% for type I and type II, respectively.

In this section, we first obtain the parameter estimates by the Bayesian-MCMC method, and then fit curves by substituting the estimated parameters into the basic functions of models. Moreover, comparisons are made between the curves fitted by the proposed method and the conventional method.

In the Bayesian-MCMC experiment, the prior distribution of each parameter is given in Section 2 and its initial value is selected according to Ref. [15]. Each Markov chain runs for 10000 iterations, and the results of the first 2000 iterations are discarded when calculating the posterior distributions. Besides, the first 20 Legendre polynomials are selected, and the threshold of the convergence criterion is 0.05. It should be noted that the RCS data hereafter are converted from logarithmic space to linear space, hence the following results are based on linear data.^[5]

The MC errors of Markov chains and the posterior distributions of parameters for both aircrafts are illustrated in Figs. 4 and 5, respectively. It is noteworthy that only the first two parameters of the Legendre model are depicted due to a similar converging tendency of the other parameters.

Fig. 4.

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	Fig. 4. MC errors of (a) lognormal model and (b) Legendre polynomial model for both aircrafts. The number of iterations is 10000 and the MC error denotes the Monte Carlo error.

Figure 4 shows that the MC error of Markov chains decreases rapidly as the iteration increases. The MC errors of the parameter estimates of the lognormal model converge after about 2000 iterations, while those of the Legendre polynomial model do not converge until 8000 iterations, which implies that the former estimates approximate the actual value more rapidly. In addition, the MC errors of different parameters in one model are close to the same value gradually, and the errors of the lognormal model after convergence are smaller than that of the Legendre polynomial model. It is illustrated in Fig. 5 that each posterior distribution converges towards its prior distribution (the normal distribution is the limiting distribution of the gamma distribution), and the mean value of a posterior distribution is the estimator of a parameter.

Fig. 5.

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	Fig. 5. (a) ∼ (d) Posterior distributions of parameters for type I, (e) ∼ (h) posterior distribution of parameters for type II, where panels (a), (b), (e), and (f) belong to lognormal model, panels (c), (d), (g), and (h) belong to Legendre model.

The RCS parameter estimation results from different fluctuation models for both aircrafts are summarized in Table 1.

Table 1.

Table 1.

Table 1. Parameter estimates and MC errors. μ σ a1 a2 Type Value (MC error) Value (MC error) Value (MC error) Value (MC error) Type I 1.18 1.31 1.03 1.33 (0.00033) (0.00022) (0.019) (0.018) Type II 0.89 1.39 1.61 2.17 (0.00035) (0.00024) (0.020) (0.017)

Table 1. Parameter estimates and MC errors.

As is easily observed, the estimation accuracy of the lognormal model is almost two orders of magnitude higher than that of the Legendre polynomial model with the same number of iterations. This may result from the fact that the Legendre polynomial model is essentially nonparametric, and its coefficients are not completely independent.

Here, the estimated parameters in Table 1 are substituted into the basic fluctuation models for curve fitting of Bayesian-MCMC method. Then, the fitting curves of conventional method are obtained according to Section 2. The comparisons among the fitting results are illustrated in Fig. 6.

Fig. 6.

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	Fig. 6. Comparison of fitting curves for (a) type I and (b) type II, where ‘ con-method’ denotes the conventional method.

Now, we summarize the characteristics of the RCS distributions and the fitting curves.

Further comparative analyses of the fitting accuracy are given below by goodness-of-fit test.

The commonly used methods of goodness-of-fit test are chi-square test, K-S test, AD (Anderson-Daring) test, etc., ^[18] which are mainly applicable to parametric models. While in this paper, the Legendre polynomial model is a nonparametric one so that the nonparametric test in Ref. [10] is adopted. The expression of error-of-fit is

where p_i is the real probability distribution of RCS, p̂ _i is the estimated probability distribution of each fluctuation model (for both methods), N₀ is the number of the segments partitioned, and . The improvement ratio r is defined as (r = (e_c − e_m)/e_c, where e_c and e_m denote the values of error-of-fit of conventional method and MCMC method, respectively. The results of the test are listed in Table 2.

Table 2.

Table 2.

Table 2. Errors-of-fit for both fluctuation models. Type Models Conventional method (ec) MCMC method (em) Improvement ratio (r) Type I log-normal 15.76% 1.19% 92.45% Legendre 1.29% 0.94% 27.13% Type II log-normal 16.07% 1.91% 88.11% Legendre 2.29% 1.63% 28.82%

Table 2. Errors-of-fit for both fluctuation models.

Due to less parameters, the capability of curve fitting of the log-normal model is obviously inferior to that of the Legendre polynomial model, although the single parameter estimation accuracy of the former is superior as demonstrated in Table 1.

In terms of fitting methods, the value of error-of-fit of Bayesian-MCMC method is less than that of the conventional method for the corresponding fluctuation model. The obtained fitting accuracy amelioration is more than 25% by using the proposed method, and the most noticeable improvement is achieved for the lognormal model.