4.1 Typical results

(1)
Partial chaotic data experiment
A partial chaotic series is simulated to validate the SCCS approach. The partial chaotic signal is constructed by a Lorenz chaotic signal and a lowfrequency signal extracted from real data.
A typical waveform obtained via numerical integration of the Lorenz system described in Eq. (2) is illustrated in Fig. 10. The corresponding phasespace projection is shown in Fig. 11.
Here, signal y in the Lorenz system is utilized to construct the partial chaotic signal. Note that the amplitude of y is normalized to the amplitude range level of the real measured sea clutter series. Then the constituent partial chaotic series is
$$ {\mathrm{s}}_{pc}=\mathrm{s}+\mathrm{s}\odot \mathrm{y} $$
(10)
where s represents the lowfrequency signal in a sea clutter signal. In this work, s is a segment of the lowfrequency signal extracted from data S4.
A typical waveform of the constituent partial chaotic signal \( {\mathrm{s}}_{pc} \) is shown in the top subplot of Fig. 12. The lowfrequency signal s is also illustrated for comparison, which can describe the fluctuant trend of \( {\mathrm{s}}_{pc} \). The normalized spectrum of \( {\mathrm{s}}_{pc} \) is given in the bottom subplot of Fig. 12. It can be observed that \( {\mathrm{s}}_{pc} \) has a relatively broad flat spectrum with a highenergy component in the very lowfrequency region, which is similar to the spectrum of real sea clutter shown in Fig. 5a. Note that this result is another proof of the decomposition model introduced in Section 3.2.
Then the proposed SCCS approach is applied to model the partial chaotic signal \( {\mathrm{s}}_{pc} \). Results are given in Fig. 13, where the vertical ordinate is the clutter signal amplitude. Five thousand points are used for training, and the next 2000 points are used for testing. Results show that the SCCS approach can excellently fit the constituent partial chaotic signal.

(2)
Real sea clutter data experiments
In this part, the proposed SCCS approach is applied to model real sea clutter data introduced in Table 1. Three typical results using the dataset S4 are depicted in Figs. 14, 15, and 16, respectively. Five thousand points are used for training, and the next 2000 points are used for testing. The amplitude modulation coefficient λ is fixed to 1.4 in the following experiments discussed in this work.
Results suggest that the SCCS approach can fit the spiky measured sea clutter without divergence phenomenon. Figure 14 shows that the proposed SCCS approach is able to fit measured data with large sea clutter spikes. In Fig. 15, the amplitude of some samples in the SCCS output are larger than that of the measured data. While in Fig. 16, the amplitude of some samples in the SCCS output are smaller than that of the measured data, and the approximation errors usually appear when the signal comes to wave crests. This kind of fitting error is the reason to involve the amplitude modulation coefficient λ in Eq. (9), and these errors can be reduced by adjusting λ to an appropriate value. Since λ is fixed in the experiments discussed in this work, fitting results illustrated here are not optimal performance which the SCCS method can achieve.
4.2 Performance comparison
To further test the effectiveness of the SCCS approach, a longer series is reconstructed by SCCS. As mentioned earlier in Section 2, the predictable samples’ number is also an important performance evaluation indicator in sea clutter modeling. In a longer series test, 5000 points are used in the training process and the next 6000 points are used for testing. In comparison, a same length series is predicted directly by a singleRBF network as described in Section 2, as well as an MMRBF network with two singleRBF models and one gating model (2MRBF).
In the 2MRBF network utilized in this work, two RBF predictors (named as RBF1 and RBF2) are different in their spread coefficients. The spread coefficient is a variance parameter in the nonlinear transfer function in an RBF network and can be set as a multiple of the average distance between data points considered in the fitting process [18]. Therefore, by changing the spread coefficient, the network can be adjusted to fit different segments of sea clutter series with different average data distances. In this work, the spread coefficients in all singleRBF networks are set as σ
_{SRBF} = 1. The spread coefficient of each RBF predictors in 2MRBF network is set as σ
_{MMRBF1} = 1 and σ
_{MMRBF2} = 10, respectively. Note that σ
_{MMRBF1} is set equal to σ
_{SRBF} in order to give a comparative study.
In addition, in order to describe the timedecomposition modeling strategy of the MMRBF network, the output of the gate model, i.e., the model choosing results are obtained from the posterior fitting accuracy. Note that in real applications, the output of the gating model should be trained with a priori information [9]. Let the prediction outputs of the two RBF predictors in 2MRBF be \( {\tilde{\mathbf{c}}}_{M1}\left[n\right] \) and \( {\tilde{\mathbf{c}}}_{M2}\left[n\right] \), respectively. Output signal of the gate model in 2MRBF is given by:
$$ g\left[n+m\right]=\left\{\begin{array}{l}0,\kern1em \left{\tilde{c}}_{M1}\left[n+m\right]c\left[n+m\right]\right\le \left{\tilde{c}}_{M2}\left[n+m\right]c\left[n+m\right]\right\\ {}1,\kern1em \mathrm{otherwise}\end{array}\right. $$
(11)
Here m = 1, 2, …, 6000. Then the final output of 2MRBF is
$$ {\tilde{c}}_{MM}\left[n+m\right]=\left(1g\left[n+m\right]\right)\cdot {\tilde{c}}_{M1}\left[n+m\right]+g\left[n+m\right]\cdot {\tilde{c}}_{M2}\left[n+m\right] $$
(12)
which means that in step m, if the gate model output is 0, the RBF1 predictor should be set active and the output of 2MRBF is equal to \( {\tilde{\mathbf{c}}}_{M1}\left[n\right] \); otherwise, the RBF2 predictor should be set active and the output of 2MRBF is equal to \( {\tilde{\mathbf{c}}}_{M2}\left[n\right] \).
The fitting results are presented in Fig. 17. It shows that waveforms of the SCCS output fit the measured data well, while the predicted signal of the singleRBF approach becomes seriously divergent in some areas. As a suboptimal performance fitting result, the output of 2MRBF is able to fit the real data in some segments, but incurs relatively large fitting errors in other areas.
Detail information of a typical data area without divergence or large fitting errors is presented in Fig. 18. In Fig. 18, waveforms as well as the gate model output g[n] from data point 475 to 520 are depicted. In the data point area from 475 to 485, g[n] is stable at 1, which means that in this data segment, RBF2 in 2MRBF has a better fitting performance and should be set as the active predictor. In this area, 2MRBF has a better fitting performance than singleRBF since RBF2 is active. Meanwhile, in the data point area from 491 to 504, g[n] is stable at 0, which means that in this segment RBF1 has a better performance. As shown in Fig. 18, in this segment the output of 2MRBF is identical to that of singleRBF since the spread coefficient of RBF1 is equal to that of the singleRBF network. This result suggests that the RBF2 model works worse than RBF1 in this data segment, and none of the predictors in 2MRBF is suitable for this data segment. In contrast, the SCCS output fits the real data well in all the data segments, as shown in Fig. 17.
The comparison results show that the performance of singleRBF is unacceptable for its risk to diverge. The 2MRBF network is a better predictor than singleRBF, but still not stable since the number of predictors in MMRBF is limited and cannot be very large in real applications. When none of them is suitable, the fitting performance will degrade. While for the proposed SCCS, since the sea clutter signal is decomposed before prediction, a good and stable fitting performance can be obtained by two simpleRBF networks without any parameter adjustment or gate model training process, which makes it a better choice for sea clutter modeling.
4.3 Detection performance discussion
A typical target detection scene is considered in this subsection to examine the detection performance of the SCCS model. The target echo signal is assumed to exist in the ith pulse, then the echo signal series is
$$ y\left[n\right]=\left\{\begin{array}{l}{s}_t+c\left[n\right],n=i\\ {}c\left[n\right],\kern1.62em n\ne i\end{array}\right. $$
(13)
where \( {s}_t=\sqrt{\mathrm{SCNR}\cdot {P}_{CN}} \); SCNR refers to the signal to clutterandnoise power ratio and P
_{
CN
} refers to the power of clutter and noise. Note that since the real sea clutter datasets include clutter echo and noise, c[n] is actually the clutterandnoise signal.
For the nth pulse, the output of the detector is
$$ d\left[n\right]=\left\{\begin{array}{l}1,\kern1em if\;\left\tilde{c}\left[n\right]y\left[n\right]\right\ge \varepsilon \\ {}0,\kern1em \mathrm{otherwise}\end{array}\right. $$
(14)
where \( \tilde{c}\left[n\right] \) is the predicted output of SCCS and \( \varepsilon \) is the detector threshold. The same 6000point series shown in Fig. 17 is used for the detection experiment. Let i changes from 1 to 6000, then the detection probability is calculated as
$$ {P}_d={\displaystyle \sum_{i=1}^{6000}d\left[i\right]}/6000 $$
(15)
Figure 19 shows the P
_{
d
} results under different SCNR with three ε values:
$$ {\varepsilon}_1=0.05\cdot \operatorname{var}\left(\mathbf{c}\left[n\right]\right),\kern1em {\varepsilon}_2=0.1\cdot \operatorname{var}\left(\mathbf{c}\left[n\right]\right),\kern1em {\varepsilon}_3=0.15\cdot \operatorname{var}\left(\mathbf{c}\left[n\right]\right) $$
(16)
where var(c[n]) indicates the variance of c[n]. In a low SCNR area (less than −3 dB) illustrated in Fig. 19, three curves are stable at values of 0.21, 0.05, and 0.01 separately. These values are equal to the false alarm probabilities under their ε values. Since a false alarm will occur when the prediction error becomes larger than ε, for a determined threshold ε, the false alarm probability is proportional to the prediction error level. On the other hand, for a determined prediction error level, the larger the threshold, the less the false alarm probability as well as the less the sensitivity of the detector. In real applications, if the threshold is set as ε
_{3}, the false alarm probability is 0.01, and the existing target in a single pulse can be detected if its power is comparable to the clutterandnoise power.
Note that under this detection scenario, a target signal in a single pulse can hardly be detected from the waveform or spectrum, especially when the target is near or overlapped by the sea spikes.
4.4 Applicability tests
To further evaluate the applicability of the SCCS approach in different situations, we consider applying SCCS to fit the five real measured datasets introduced in Table 1. The rootmeansquared error (RMSE) is used as the performance measurement:
$$ {\mathrm{RMSE}}_{d,l}=\frac{\sqrt{{\displaystyle \sum_{m=1}^M{\left(\tilde{\mathbf{c}}\left[n+m\right]\mathbf{c}\left[n+m\right]\right)}^2}}}{M} $$
(17)
where M refers to the number of testing samples, d refers to the dataset number, and l refers to the data start point in each dataset. Five thousand samples are used for training, and the next 2000 samples are used for testing (M = 2000).
To get a fair and comprehensive performance evaluation, seven series from each dataset are used for training and testing. The start points of the seven series in each dataset are 1, 20,000, 40,000, 60,000, 80,000, 100,000, and 120,000, respectively. So we have d = [1, 2, 3, 4, 5], l = [1, 20, 000, 40, 000, 60, 000, 80, 000, 100, 000, 120, 000]. The RMSE of 35 tests (5 datasets, 7 series in each dataset) are shown in Table 2.
Results show that for all the 35 tests, the fitting RMSE remains under 0.1, which means that the SCCS approach is able to reconstruct measured sea clutter series under different sea state situations.
To evaluate the relationship between the fitting performance and sea clutter characteristics, mean value ε
_{
d
} and variance σ
_{
d
} of the fitting RMSE of each dataset are calculated as:
$$ {\varepsilon}_d={\displaystyle \sum_l{\mathrm{RMSE}}_{d,l}}/7,\kern1em {\sigma}_d={\displaystyle \sum_l{\left({\mathrm{RMSE}}_{d,l}{\varepsilon}_d\right)}^2}/7 $$
(18)
Results of ε
_{1} ∼ ε
_{5} and σ
_{1} ∼ σ
_{5} are shown in Fig. 20, where the x axis is the wave height of each dataset. As mentioned above, the wave height can be used as a measure of sea state.
It can be observed that as the wave height increases, the mean values of RMSE are monotonously increasing, and the variances have a certain upward trend. This indicates that the fitting performance of SCCS has a kind of proportional relationship with the sea state. The reason might lie in the fact that under a high sea state circumstance, the increased degree of freedom and fluctuation in sea clutter makes it more difficult to approximate the deAM signal with a structure fixed neural network. However, comparing to the prediction outputs of the singleRBF method, the SCCS outputs remain convergent in all the experiments. Improved neural networks can be used to deal with this sea state related performance deterioration.
Based on the real data involved experiment results, the following observations could be made about the SCCS modeling approach.

(1)
Free of divergence problem
The SCCS approach can fit sea clutter with different waveform properties and barely encounter divergence problem, which obviously outperforms the singleRBF approach since the divergence problem can cause unacceptable prediction errors.

(2)
Stable performance when long testing series is needed
Keeping the training samples’ number unchanged, with the increase of the testing samples’ number, the SCCS approach consistently maintains a stable fitting performance. This indicates that the trained SCCS model successfully approximates the compound underlying process within the target sea clutter.

(3)
Applicable under different sea states
Sea clutter collected in various sea states can be quite different. For most of the clutter datasets, we have investigated, the SCCS approach has shown a good overall fitting performance.
These observations confirm the effectiveness of the SCCS approach. Most of them come from the utilization of the SCD model proposed in this paper, in which a more suitable component for chaotic modeling is found. Again, we emphasize that the role of the SCD model is a preprocessing approach, which can be used to reduce the difficulty of sea clutter modeling. So the SCDSCCS approach is not exclusive. RBF predictors in SCCS can be replaced by other modeling approach (such as improved intelligent algorithms, fractal models, and statistics models) as an alternative choice in different application situations.