Robust Centralized CFAR Detection for Multistatic Sonar Systems

Ⅰ.   Introduction

Anti-submarine warfare (ASW) is always a challenging mission^[1]. Meanwhile, it is more difficult to satisfy the surveillance requirement using passive and monostatic active sonar system as the noise reduction and acoustic stealth technologies are widely applied in modern submarines^{[2, 3]}. The multistatic sonar system including several sources and receivers can acquire more target information from different viewing angles, which is regarded as an effective measure against the quiet submarine^{[4, 5]}.

There are two categories of processing architectures for target detection in multistatic sonar system, namely, distributed and centralized detection^[6]. The former makes the first decision in local nodes, and then the fusion center makes the final decision using different fusion rules, such as, "AND", "OR", "MAJORITY"^{[7, 8]}. The latter transmits all the echo data into fusion center, and makes a centralized fusion decision. Obviously, the former needs less communication capacity. But the latter loses less target and background information which is beneficial for subsequent detection processing^[9]. In centralized architecture, the target information from multiple nodes can be accumulated efficiently, which is important for the detection of weak underwater target. Unlike the distributed detection, the relevant studies on the centralized detection are much less.

However, it is still hard to maintain robust Constant false alarm ratio (CFAR) property in heterogeneous underwater environment despite using centralized detection. Especially for the multistatic active sonar echo, there is much more reverberation and kinds of interferences than passive and monostatic active sonar on account of more sources, rolling seabed terrain, underwater artificial facilities, and so on^[10-12]. In order to improve CFAR detection performance in non-homogeneous environments, many specific detectors have been introduced^[13-15], some of which are applied in centralized detection, such as SO-CFAR, OS-CFAR, and ACMLD^[16]. In Ref.[17], several more complex ordered CFAR detectors are derived for Multi input multi output (MIMO) radars. However, these specific detection algorithms only perform well in specific application scenarios. They are not always suited in various undersea heterogeneous environments.

We propose a robust centralized CFAR detection algorithm to mitigate the influence of changeable heterogeneous background. The algorithm takes advantage of the idea of S-CFAR^[18] and Ⅵ-CFAR^[19]. They compute some background statistics firstly, and then choose an appropriate CFAR detector. According to the homogeneity judgment of the leading and lagging reference windows, the Ⅵ-CFAR selects the detector from the CA-CFAR, GO-CFAR, and SO-CFAR. Some researchers applied the Ⅵ-CFAR to monostatic sonar system, which performs good detection property in reverberation edge and single interference environments^{[20, 21]}. Likely, we fully assess the homogeneity and consistency of the echo data from all the transmit-receive nodes. Then the matched detector from the CCA-CFAR, CGO-CFAR, COS-CFAR is chosen in terms of comprehensive background assessment.

The rest of this paper is organized as follows. Section Ⅱ introduces some classical CFAR detectors. In Section Ⅲ, we introduce the CMVI-CFAR detection algorithm in detail. Besides, the CGO-CFAR detector is derived. Section Ⅳ demonstrates the detection performance of the CMVI-CFAR detection algorithm on simulated and measured sonar data in kinds of underwater environments. Finally, conclusion is given in Section Ⅴ.

For ease of reading, we list all the abbreviations of this paper in Table 1.

Table  1.  Comparison table of acronyms and full names

Acronym	Full name
ASW	Anti-submarine warfare
CFAR	Constant false alarm ratio
SO-CFAR	Smallest of CFAR
OS-CFAR	Order statics CFAR
ACMLD	Automatic censored mean level detector
MIMO	Multi input multi output
S-CFAR	Switching CFAR
Ⅵ-CFAR	Variability index CFAR
CA-CFAR	Cell averaging CFAR
GO-CFAR	Greatest of CFAR
CCA-CFAR	Centralized CA-CFAR
CGO-CFAR	Centralized GO-CFAR
COS-CFAR	Centralized OS-CFAR
CMVI-CFAR	Centralized modified Ⅵ-CFAR
CUT	Cell under test
SNR	Signal-to-noise ratio
INR	Interference-to-noise ratio
RNR	Reverberation-to-noise ratio
MR	Mean ratio
ROC	Receiver-operating-characteristic
PDF	Probability density function
MGF	Momentum generating function

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Ⅱ.   Classical CFAR Detection Algorithms

In this section we just quickly review some classical CFAR detection algorithms, such as CA-CFAR, GO-CFAR, and OS-CFAR, which are the basis of subsequent proposed algorithm. With reference to one sonar node and one sweep ping, suppose that $x_{0}$ and $x_{j}$, $j = 1, \ldots, K$ denote the square-law detected samples from the CUT and the adjacent range cells, respectively.

1   CA-CFAR

The CA-CFAR estimates the background noise power through the averaging of the adjacent reference samples^[13]. Its decision procedure is shown as

where $T_{\rm C A}$ denotes the detection threshold. The performance of the CA-CFAR is good when the CUT has enough echo power and the K reference cells are identical.

2   GO-CFAR

The reference window including $K$ cells is divided into two parts, namely the leading (F) and lagging (B) windows. The GO-CFAR firstly estimates the background power of the F and B windows and then selects the maximum one as the final estimation^[22]. Its decision procedure is shown as

where $T_{\rm GO}$ denotes the detection threshold. The GO-CFAR can effectively suppress the potential false alarm point located at reverberation edge.

3   OS-CFAR

The OS-CFAR estimates the background noise power through selecting the suitable sample from the sorted reference cells in increasing order. Its decision procedure is shown as

where $T_{\rm OS}$ is the detection threshold, $x_{k}$ denotes the kth sample after the sorting action. The OS-CFAR can improve the detection performance when there exist several strong outliers in leading or lagging windows. But it has a little loss of detection performance than the CA-CFAR when the $K$ reference cells are identical.

Although these classical detectors have different advantages in the corresponding environments, they still exist some problems. For example, they are difficult to hold the detection performance when the echo from the target is weak or environment is heterogeneous and changing.

Ⅲ.   Centralized CFAR Detection Algorithm

This section is devoted to the proposed centralized CFAR detection algorithm, which offers a solution to reduce or eliminate the impact of the weak target and heterogeneous environments. On the one hand, under the centralized architecture the fusion center could accumulate the target power from multiple sensors to increase the SNR of weak target. On the other hand, we employ the idea of the Ⅵ-CFAR to select different detectors for different heterogeneous environments in order to improve the robustness.

Consider a centralized multistatic sonar system with M transmit-receive combinations. In Fig. 1, the system consists of one source and two towed array receivers. The distributed platforms can acquire different target sound scattering intensity at different transmit-receive viewing angles. Besides, different node has different adjacent reference cells. These facts offer the potential of enhanced and robust detection for the undersea target. We utilize the centralized processing architecture. All the raw echo data should be transmitted into the fusion center.

Figure  1.  The schematic diagram of centralized multistatic sonar detection

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Suppose that each node includes K reference cells $x_{i, j}, i = 1, \ldots, M, j = 1, \ldots, K$ and one CUT $x_{i, j}, i = 1, \ldots, M, j = 0$. All the reference cells and CUTs in different nodes are independent. Ignoring the potential strong reverberation and interferences, the distribution of all the background elements are modeled as^[16]

where $\mu$ denotes the average background noise power.

The structure of the CMVI-CFAR detector is shown in Fig. 2. Firstly, compute the Ⅵ and MR statistics of the leading (F) and lagging (B) windows as

Figure  2.  The structure chart of the CMVI-CFAR detector

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and

respectively, where

The VI statistic is a function of the estimated population mean $\hat{\mu}$ and estimated population variance $\hat{\sigma}^{2}$, which is closely related to an estimate of the shape parameter. The Ⅵ statistic indicates whether the leading or lagging window cells are variable or non-variable. The MR statistic is the ratio of the mean values of the leading and lagging reference window cells, which indicates the consistency of two reference window samples.

Then judge the homogeneity and consistency of background statistics of each node

where $c_{\rm Ⅵ}$ and $c_{\rm M R}$ are the constant threshold coefficients of the Ⅵ and MR statistics, respectively^[19]. If the Ⅵ value of the leading or lagging window cells exceeds $c_{\rm Ⅵ}$, those samples of the window are variable or non-homogeneous. Similarly, if the MR value exceeds $c_{\rm M R}$ or is less than $c_{\rm M R}^{-1}$, the power level of the two windows has an enough gap and is not consistent.

Secondly, to quantify the multiple homogeneous and heterogeneous environments, we analyze and calculate the values of statistical classification $h(m), m = 1, \ldots, M$ according to the marked-choice criterion as shown in Table 2. For example, $h(m) = 2$ when the F and B windows are non-variable (see Eq.(8)) and they are not consistent (see Eq.(9)). It means that the samples from both windows tend to be homogeneous independently, but the gap of their mean statistics is too large.

Table  2.  The marked-choice criterion of statistics for different situations^[19-21]

F non-variable	B non-variable	Consistent	h
Yes	Yes	Yes	1
Yes	Yes	No	2
Yes	No	-	3
No	Yes	-	3
No	No	-	4

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Fuse the values of statistical classification to obtain the comprehensive background evaluation

where $\operatorname{NUM}(h(m) \in B), m = [1, M]$ denotes the number of h in M nodes which belongs to the set B. For example, when $M = 5$ and the values of statistical classification are $\{1, 2, 4, 4, 4\}, B_{\max } = 4$. It is worth noting that the value of $B_{\max}$ has uniqueness. If there are more than one maximum value, we utilize the following priority rules to determine the only value

The above rules mean that the comprehensive background statistics tend to be non-homogeneous when there is the same amount of homogenous and non-homogeneous nodes. For example, when the numbers of the combinations of both $h = 2$ and $h = 1$ are 2, so the value of $B_{\max}$ may be 1 or 2 according to Eq.(10). But we choose $B_{\max } = 2$ according to the priority rules in Eq.(11).

Thirdly, select the appropriate centralized detector to compute the background noise level statistics and corresponding threshold factor. From $B_{\max} = 1$ to $B_{\max } = 4$, choose the CCA-CFAR, CGO-CFAR, $K/2$ CCA-CFAR, and COS-CFAR in turn.

Obviously, the CCA-CFAR is a reasonable choice when $B_{\max } = 1$ which means all the reference cells are homogeneous. In order to suppress the potential false alarms in reverberation edges, select the CGO-CFAR when $B_{\max } = 2$. $B_{\max } = 3$ implies that there may be discrete strong interferences located at the leading or lagging windows for the majority of nodes. When $B_{\max } = 4$, several interferences or strong reverberation located at both windows are the possible scenario. Herein we choose the COS-CFAR detector. However, the Ⅵ-CFAR uses the SO-CFAR for monostatic sonar system in this scene. Apparently, it cannot avoid the strong interference and results in target shielding effect^[8].

The background noise level statistics are given by

where $x_{i, k}$ denotes the kth greatest reference value of the ith node. In general, the value of k is set to be $3 K / 4$ or $5 K / 6$. If $B_{\max} = 3$, the statistic x will select the mean values of those non-variable reference window cells as the background statistic^[19].

According to the setting probability of false alarm ($P_{\rm f a}$), we can compute the corresponding threshold factor. The $P_{\rm f a}^{\rm C C A}, P_{\rm f a}^{\rm C G O}, P_{\rm f a}^{K / 2\; {\rm C C A}}$, and $P_{\rm f a}^{\rm C O S}$ of the CCA-CFAR^[16], CGO-CFAR, $K / 2$ CCA-CFAR, and COS-CFAR^[16] detectors are given by Eqs.(13)–(16).

where $\alpha_{\rm C C A}, \alpha_{\rm C G O}, \alpha_{K / 2\; {\rm C C A}}$, and $\alpha_{\rm COS}$ denote the corresponding threshold factors. The symbols $(\begin{array}{l}m \\ n\end{array})$ and $\{f(x)\}_{x \rightarrow -1}$ denote the combination operation and limit operation, respectively. The derivation of the CGO-CFAR false alarm Eq.(14) is shown in Appendix $A$.

Finally, make a decision to judge whether there exists target in the CUT:

The test statistic $x_{0}$ in Eq.(17) suggests that the centralized detection has the advantage of accumulating the power of target when compared with the distributed. It means that more target echoes from more nodes maybe offer the greater SNR. Besides, the CMVI-CFAR could select the corresponding centralized detection methods for different values of $B_{\max }$ according to Eq.(12). Therefore, the CMVI-CFAR has better adaptation for the changing heterogeneous environments.

Ⅳ.   Performance Assessment

This section is devoted to the performance assessment of the CMVI-CFAR detector in terms of controlling false alarms and target detection. Especially, we compare it with the CCA-CFAR in multiple homogeneous and heterogeneous environments.

1   Numerical simulations

As in Ref.[19], the values of $c_{\rm Ⅵ}$ and $c_{\rm M R}$ are set to 4.76 and 1.806, respectively. Unless stated, we set the theoretical $P_{\rm f a} = 10^{-3}$ and the number of reference cells of each node to be 32. According to Eqs.(13)–(16), the threshold factors of all the detectors are computed in Table 3.

Table  3.  The threshold factors of the different detectors

$M$	1	2	3	4
CCA-CFAR	0.24	0.15	0.12	0.11
$K$/2 CCA-CFAR	0.54	0.33	0.26	0.22
CGO-CFAR	0.43	0.27	0.22	0.19
COS-CFAR	5.05	3.16	2.50	2.15

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Consider a homogeneous environment where the distribution of background is the Rayleigh with parameter 3. The amplitude of the added target follows the Swerling Ⅰ model. The number of Monte-Carlo is 10⁵.

The probability of detection ($P_{\rm d}$) versus SNR of the CCA-CFAR and CMVI-CFAR in homogeneous environment is shown in Fig. 3. Both the detectors have the same detection performance under the same number of nodes. It also means the fact that the CMVI-CFAR selects the CCA-CFAR detector in homogeneous environment. Moreover, the CMVI-CFAR has higher $P_{\rm d}$ as the number of nodes increases. The gain of the CMVI-CFAR with $M = 4$ is about 7.5dB over single node situation when $P_{\rm d} = 0.5$.

Figure  3.  P_d versus SNR of the CMVI-CFAR and CCA-CFAR in homogeneous environment

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To analyze the performance of controlling false alarms, we design the scene of reverberation edges. The theoretical $P_{\rm f a}$ and the number of Monte-Carlo are set to $\{10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}\}$ and $100 / P_{\rm f a}$, respectively. The results are shown in Fig. 4. The CMVI-CFAR achieves lower real $P_{\rm f a}$ than the CCA-CFAR and has better performance of controlling false alarms.

Figure  4.  Theoretical P_fa versus real P_fa of the CMVI-CFAR and CCA-CFAR in reverberation edges

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Consider the typical heterogeneous environments in shallow sea where the reference windows of each node include single or several interferences. Unless stated, we set INR as 20dB, following the Swerling Ⅰ model. The detection curves of the CCA-CFAR and CMVI-CFAR for single and multiple interferences are shown in Fig. 5(a)and (b), respectively. Moreover, we give the ROC curves of the CMVI-CFAR and CCA-CFAR in Fig. 6, where the target SNR is set to be 10dB.

Figure  5.  P_d versus SNR of the CMVI-CFAR and CCA-CFAR for (a) Single interference; (b) Multiple interferences

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Seen from Fig. 5, the performance of $P_{\rm d}$ for the CMVI-CFAR is better than the CA-CFAR and CCA-CFAR in both single and multiple interference scenes. In Fig. 5(a), the gain of the CMVI-CFAR with $M = 4, M = 3$, and $M = 2$ is about 9.5dB, 8dB, and 7.8dB over the CCA-CFAR when $P_{\rm d} = 0.5$. Likely, in Fig. 5(b), the gain of the CMVI-CFAR with $M = 4, M = 3$, and $M = 2$ is about 11.5dB, 11dB, and 10dB over the CCA-CFAR when $P_{\rm d} = 0.5$. In Fig. 6, the ROC curves show that the CMVI-CFAR has better performance of $P_{\rm d}$ under the same $P_{\rm f a}$ than the CCA-CFAR in single interference environment. So the CMVI-CFAR can effectively avoid the target shielding effect in discrete interference environments. However, it is difficult to avoid target shielding effect for the Ⅵ-CFAR because it often chooses the SO-CFAR in multiple-interference environment. But the CMVI-CFAR uses the COS-CFAR which can tolerate limited number of interferences whichever windows they are located at.

Figure  6.  ROC curves of the CMVI-CFAR and CCA-CFAR for single interference

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Different nodes often have different reference cells due to different transmit-receive angles. Therefore, we consider two kinds of mixed multistatic detection scenes in the following: one consists of a node with interference and another three nodes being homogeneous; the other scene consists of two nodes with single interference and two nodes with multiple interferences. The settings of all the interferences are the same as in Fig. 5.

Fig. 7 compares the target detection performance of the CMVI-CFAR and CCA-CFAR detectors with kinds of node combinations in mixed heterogeneous environments. In Fig. 7(a), the gain of the CMVI-CFAR with $M = 3, 2, 1$ is about 4.5dB, 5dB, and 6.5dB over the CCA-CFAR when $P_{\rm d} = 0.5$. However, the fact that both the detectors have the same performance when $M = 4$ implies that the CMVI-CFAR selects the CCA-CFAR in this situation. In Fig. 7(b), the CMVI-CFAR which selects the $K/2$ CCA-CFAR or COS-CFAR also achieves better $P_{\rm d}$ than the CCA-CFAR. Especially, the relative gain of the CMVI-CFAR with $M = 3, 4$ has 9dB.

Figure  7.  P_d versus SNR of the CMVI-CFAR and CCA-CFAR for (a) Mixed interference and homogeneous environments; (b) Mixed single and multiple interferences

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Fig. 8 compares the ROC performance of the CMVI-CFAR and CCA-CFAR detectors in mixed heterogeneous environments. Apparently, the CMVI-CFAR has better ROC performance than the CCA-CFAR.

Figure  8.  ROC curves of the CMVI-CFAR and CCA-CFAR for mixed single and multiple interferences

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In conclusion, the CMVI-CFAR is more robust than the CCA-CFAR in various heterogeneous environments in terms of target detection and false alarm controlling. Besides, centralized CFAR detection performs a better $P_{\rm d}$ than monostatic sonar no matter what detector and no matter what environment.

2   Performance assessment of measured sonar data

In this subsection, we analyze the performance of the DMVI-CFAR using the measured sonar data with simulated interference or strong reverberation. The measured data was collected in East China Sea, which is typical shallow sea environment. The number of the total nodes are 4. Fig. 9 shows the collected measured data of one node. The CUT is the 61th range cell. The numbers of protection units and reference cells are set to 40 and 32, respectively.

Figure  9.  Measured active sonar data and simulated interference for (a) Reverberation edge scene; (b) Discrete interference scene

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We employ the data in Fig. 9(a) to evaluate the performance of false alarms controlling where the left part is strong reverberation region. The CUT is located at the reverberation edge. Fig. 10 shows the performance curves of $P_{\rm f a}$ versus RNR. As the RNR increases, $P_{\rm f a}$ of the CCA-CFAR tends to 100%. However, the CMVI-CFAR still maintains extremely low $P_{\rm f a}$ because it selects the CGO-CFAR.

Figure  10.  P_fa versus RNR of the CMVI-CFAR and CCA-CFAR using measured sonar data in reverberation edges

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To assess the performance of the target detection we use the data in Fig. 9(b) where the SNR of CUT or target is enough high to be detected ignoring the potential interferences. We add the simulated interferences with Swerling Ⅰ model into reference windows. The INR is set from 10 to 40dB. The number of Monte-Carlo for interferences is 10⁵. Fig. 11 shows the compared performance results of the CMVI-CFAR and CCA-CFAR. As the INR increases, the CMVI-CFAR keeps almost 100% probability of detection no matter single or multiple interferences and no matter what INR equals to. However, $P_{\rm d}$ of the CCA-CFAR goes down as increasing INR.

Figure  11.  P_d versus INR of the CMVI-CFAR and CCA-CFAR using measured sonar data for (a) Single interference; (b) Multiple interferences

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As mentioned in Section Ⅳ.1, consider two similar mixed heterogeneous detection scenes using measured data in Fig. 9(b). Besides, the first scene consists of two nodes with single interference and two homogeneous nodes.

The performance curves of target detection of the CCA-CFAR and CMVI-CFAR using the measured data for complex mixed heterogeneous environments are shown in Fig. 12. Seen from Fig. 12, the proposed detector keeps almost 100% $P_{\rm d}$ no matter what mixed environments. Although $P_{\rm d}$ of the CCA-CFAR can increase as more nodes, it still tends to 0 as INR is added up to 40dB.

Figure  12.  P_d versus INR of the CMVI-CFAR and CCA-CFAR using measured sonar data for (a) Mixed interferences and homogeneous environments; (b) Mixed single and multiple interferences

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Ⅴ.   Conclusions

This paper has developed a novel centralized CFAR detector, namely CMVI-CFAR, for multistatic sonar systems. The CMVI-CFAR detector comprehensively evaluates the statistic properties of reference background from all the nodes. Then it employs the scene-matching centralized detector from the CCA-CFAR, CGO-CFAR, $K/2$ CCA-CFAR and COS-CFAR. The results of numerical simulations and using measured data show that the detection performance of the proposed detector is more robust than the CCA-CFAR in various undersea environments. Besides, the performance of target detection also increases as the number of nodes increases. But the CMVI-CFAR detector still exists some shortcomings. It is difficult to give the accurate values of $c_{\rm Ⅵ}$ and $c_{\rm M R}$ for complex and changing undersea environment. The inaccurate coefficients will result in false statistic classification and disturb the detection performance. Therefore, we will study the influence of these constant coefficients using more measured data in the future.

Appendix.   False Alarm Equation of the CGO-CFAR

Suppose that the statistics $u_{j}$ and $v_{j}$ denote the sum of leading and lagging cells in the jth node, respectively. According to the idea of the GO-CFAR, the total noise power of M nodes is estimated by

where $x_{j} = \max \left(u_{j}, v_{j}\right)$.

The PDF of $x_{j}$ can be written as

where the distributions of u_j and v_j are given by

Then $f_{x_{j}}(x)$ is found to be

The MGF of $f_{x_{j}}(x)$ is given by

So the MGF of $f(x)$ can be expressed as

According to the contour integral^[23], $P_{\rm f a}^{\rm C G O}$ can be computed as

where the symbol $\operatorname{Res}(\cdot)$ denotes residue. The MGF of the CUT under the H₀ hypothesis is given by

which has a pole of order M at $t_{0} = -1 / \mu$.

According to the residue definition at a pole^[24], Eq.(A-7) can be simplified as