A network-based approach on detecting dredgers’ illegal behavior of dumping dredged sediments

Abstract

Focusing on detecting the illegal behavior of dumping dredged sediments by dredgers at ports, a sensor network composed of two acoustic sources and several passive sensor arrays is proposed in this article. The approach can be divided into two parts: the local sensor decision model and the fusion center algorithm. Local sensors receive signals emitted from the acoustic sources and extract the corresponding Hilbert–Huang marginal spectrum, which is significantly different between the scenario of illegal dumping dredged sediments and the contrary. Local decision is made based on spectrum feature extraction and classification. In regard to the fusion center, a fusion strategy with a scanning window is adopted to make a system-level decision. With a proper window width and the corresponding Bayes optimum threshold, the proposed approach performs well in simulations, in terms of a low system-level false alarm probability and a low system-level miss alarm probability.

Introduction

Marine transportation, in its long history, has a close relationship with the maritime countries’ development. As a result, the port, which acts as a hinge linking the ocean and the land, is more and more important. Port is a complicated system composed of many facilities, one of which is called disposal site, that is, waste-dumping zone for dredged sediments. Dredged sediments, which are sediments excavated from the bottom of waterway including ports and estuaries, are expected to be dumped to the disposal site by dredgers. The underwater excavation and dumping process is called dredging. Dredging is important to maintain the depth of waterway. If dredging is performed in a perfunctory manner or not done in time, it will bring hidden safety troubles for the navigation, shipping, and ship berthing.

However, dredgers often dump dredged sediments on the road during their transportation instead of to the prescribed disposal site. Such behavior is harmful and illegal. Therefore, we need a stable and reliable method to detect the misbehavior, viz illegal dumping. Insofar, as it can be ascertained, the related schemes can be classified into two types: ship-equipping and ship-equipping-free. Ship-equipping refers to the approaches depending on installing associated apparatus on board to detect the dumping behavior, for instance, the CCTV video monitoring device ¹ and the monitoring recorder for marine wastes dumping. ² By contraries, ship-equipping-free approaches could work without equipping any devices on dredgers. Thought ship-equipping schemes can achieve better judgments, they would give rise to many problems, for example, ambiguity of liability. Thereby ship-equipping-free detecting has drawn the attention of multiple researchers. The most popular approach is to measure the waterline of dredgers with the starting point: sediments dumping lightens a dredger. Luo et al. ³ advanced a method for automatic detection of ships’ waterline based on the image-processing method, using Canny operators to detect edges and Hough transform to detect the locations of water trace and the waterline after geometric correction. Liu et al. ⁴ forwarded a texture spectrum method, using K-means clustering algorithm to segment ship waterline images. Gai et al. ⁵ used a laser rangefinder to get the waterline, but the equipment is expensive, and the method has high requirements for the water turbidity and the measuring cycle period. Hua ⁶ proposes five dynamic measuring methods to detect the waterline. These methods are little affected by wind and waves. Nevertheless, they have respective limitations: very limited detection range, prohibitive cost of necessary equipment, strict requirements for installation, and so on.

Another way is to detect the presence of flowing sediments below the sea surface. The underlying principle is that flowing sediments will cause the suspended sediment concentration higher than usual. Therefore, we can monitor the suspended sediment concentration, or utilize the impacts of the concentration change to detect the presence of flowing sediments. If a dredger is detected having flowing sediments on its way to the disposal site, it is declared to dump dredged sediments illegally. This involves a subject known as underwater target detection. Unfortunately, the problem abovementioned has rarely been discussed. The detecting targets of most relative literatures are fishes or sea bed sediments,^7–10 not the diffusional flowing sediments. Therefore, in order to meet the need of the background, we develop a new approach in the article based on marginal spectrum of Hilbert–Huang Transform (HHT) for local sensors to detect the event.

Furthermore, a network, accompanied by a global fusion approach, is developed to monitor the whole ship channel. Detection fusion has drawn significant attention from academia. Niu and Varshney ¹¹ proposed a counting-rule-based approach which used the total number of detections coming from local sensors as the fusion statistic. A local vote decision fusion approach was introduced by Katenka et al. ¹² Applying the principle, whereby the minority is subordinate to the majority, the decision of a local sensor was voted by its entire neighborhood within a fixed distance. Song et al. ¹³ proposed a scan statistic–based fusion approach, using a scanning window to find the most significant detection cluster in a special part of the data set. In this article, we will show that the fusion strategy with scan statistic is suitable to our network-based application, which is detailed in the following section.

The remainder of the article is organized as follows. Section “Related work” discusses the related work. Section “Detection approach for local sensors” presents our approach of detection for local sensors. Section “Global detection fusion” addresses the data fusion problem of our detection network. Section “Experimental results” presents our simulation results. Section “Conclusion and future work” draws the conclusion and discusses the future work.

Related work

There are several researches on sediment-related issues. The issues can be divided into several types: sediment dredge-and-dump strategies, ¹⁴ suspended sediment concentration monitoring, ¹⁵ sediment pollution problem, ¹⁶ and the ecological effect. ¹⁷ And at present, Haimann et al. ¹⁸ propose an ecologically oriented integrated monitoring system which consists of several stages, for example, preliminary assessment stage and post-dumpling monitoring stage.

In order to detect suspended sediment concentration, for both Cutroneo et al. ¹⁵ and Blanco et al., ¹⁶ the dredger needs to be equipped with an acoustic Doppler current profiler (ADCP). ADCP is a kind of equipment which makes use of the Doppler effect of acoustic wave scattered back from particles within the water column to measure water current velocities over a depth range. That is, ADCP emits signal into the water and collects the signal reflected by the particles in the water. Therefore, one can estimate the suspended sediment concentration based on the backscatter intensity.

All the related studies take the detection of the suspended sediment concentration as an indicator to guide the dredgers finishing the dredging operations effectively. The dredgers are assumed to do their duty and follow the instructions well. However, the dredgers in China are employed from third-party companies. And because of the chaos in the management, many dredgers try to find various excuses to remove the monitor facilities equipped on them, and dump dredged sediments on the road during their transportation instead of to the prescribed disposal site to reduce fuel consumption. Therefore, we need a ship-equipping-free approach to detect suspended sediment concentration and take the result as an indicator of whether the illegal dumping happened. Unfortunately, to the best of our knowledge, there is no reference focused on similar application background. Therefore, we strive to develop an effective approach to detect the illegal dumping behavior and take advantage of network structure to extend the detection coverage area.

Detection approach for local sensors

Typically, like Niu and Varshney ¹¹ and Katenka et al., ¹² each sensor obtains an energy reading from the environment. However, the pattern is not applicable in our application, as the suspended sediments would not emit any energy signal. Therefore, we introduce position-fixed acoustic sources (equipped with underwater transducers), and develop a single-input single-output (SISO) detecting model for local sensors (equipped with hydrophones). The detecting signal transmitted by the acoustic sources periodically is composed of two parts: hyperbolic frequency modulated (HFM) signal as synchronization code (denoted by $s_{\mathrm{sy}}(t)$ ) and several evenly spaced chirp signals as detecting codes (denote a certain one by s(t) for convenience). Once the signal synchronization is completed at the receiving end (i.e. the local sensor), discriminating features are extracted from the signal segment corresponding to s(t). Then a specific classifier is adopted to make a judgment whether the dredger is dumping sediments or not.

Model establishment

Denote r(t) as the signal segment received corresponding to s(t) ( $S(\omega )$ in the frequency domain), and its frequency domain form is given by

$$R(\omega )=\sum _{\ell =0}^{L-1}{A}_{\ell }(\omega )S(\omega )e^{-j\omega {\tau }_{\ell }}+W(\omega ),$$

(1)

that is, we consider a multipath channel, where L is the number of paths decided by the Zielinski N-path model, ¹⁹ $A_{\ell }(\omega )$ is the attenuation coefficient of the $\ell$ -th path, ${\tau }_{\ell }$ is the delay of the $\ell$ -th path, and $W(\omega )$ is the noise. The noise is not white but generated according to the typical mean spectrum level diagram of marine noise. ²⁰

In the shallow sea, $A_{\ell }(\omega )$ is mainly caused by the propagation loss, which can be expressed as ²¹

$$A_{\ell }(\omega )=\frac{R_{\ell }}{x_{\ell }^{\frac{n}{2}}}·{10}^{-{\alpha }_{\ell }(\omega )x_{\ell }/20}$$

(2)

where $x_{\ell }$ is the propagation distance of path $\ell$ , n is propagation index (we choose 1 here), and $R_{\ell }$ is the corresponding reflection loss coefficient decided by

$$R_{\ell }={\tilde{r}}_s^n{\tilde{r}}_b^m$$

(3)

where ${\tilde{r}}_s$ ²² and ${\tilde{r}}_b$ ²³ represent the reflection coefficients of sea surface and sea bed, respectively, and n and m are the corresponding occurrences of reflections.

The acoustical absorption coefficient ${\alpha }_{\ell }(\omega )$ under turbid sea-water conditions is given by

$${\alpha }_{\ell }(\omega )={\alpha }_{\ell ,w}+{\alpha }_{\ell ,v}+{\alpha }_{\ell ,s}$$

(4)

where ${\alpha }_{\ell ,w}$ is the acoustical absorption coefficient of pure sea-water (dB/m), ${\alpha }_{\ell ,v}$ is the viscous attenuation coefficient (dB/m), and ${\alpha }_{\ell ,s}$ is the scattering loss coefficient (dB/m), which can be neglected, since it is far less than the other two. The exist of flowing dredged sediments will affect the value of ${\alpha }_{\ell }(\omega )$ by bringing a big change to the value of ${\alpha }_{\ell ,v}$ , which will further affect the suspended sediment concentration.

Flowing dredged sediments under the sea surface are assumed to exist in forms of particle clouds. Figure 1 is a schematic illustration when a single acoustic ray travels through a particle cloud. Let us take the example of Xiamen Port in China. The concentration of the suspended sediments in the sea at Xiamen Port has consistently kept at a low level less than 0.1%, whereas it will locally exceed 1% in a short time when there is a dredger dumping sediments, which leads ${\alpha }_{\ell ,v}$ in equation (4) to be about 10 times larger than normal.

Figure 1.

Schematic illustration of the acoustic signal passing through the particle cloud. $\Delta x_{\ell }$ is the distance of the signal traveling through the particle cloud.

The semiempirical formula of ${\alpha }_{\ell ,w}$ is expressed as ²¹

$${\alpha }_{\ell ,w}=(20\log e)(\frac{\mathrm{AS}{f}_T{f}^2}{f_T^2+f^2}+\frac{B{f}^2}{f_T})(1-6.67\times {10}^{-5}d)$$

(5)

where e is the base of the natural logarithm, A and B are constant with the value 2.03 × 10⁻⁶ and 3.38 × 10⁻⁶, respectively, f is the frequency of the acoustic signal (kHz), d is the depth of the transducer, T is the temperature of the sea surface, we take it as 21.5°C at Xiamen Port, f_T is the relaxation frequency with the value $f_T=21.9\times {10}^{6-1520/(T+273)}\approx 151\mathrm{kHz}$ , and S is the salinity $(\mathrm{‰})$ . When the seawater density ${\rho }_w=1025\mathrm{kg}/{\mathrm{m}}^3$ , $S=1.305({\rho }_w-1000)+(T-17.5)\times 0.3=33.825‰$ .

The semiempirical formula of ${\alpha }_{\ell ,v}$ is expressed as ²⁴

$${\alpha }_{\ell ,v}=\frac{C{(1-C)}^2{(1-\sigma )}^2\varsigma ·\omega ·{\rho }_w·\sqrt{\frac{k_c}{\rho }}}{\sqrt{2({\varsigma }^2+Q^2)}·\sqrt{{\varsigma }^2+\mathrm{PQ}+\sqrt{({\varsigma }^2+P^2)({\varsigma }^2+Q^2)}}}$$

(6)

where C is the volume fraction of the solid particulate; $\omega$ is the angular frequency of the signal (rad/s); k_c is the volume compressibility of the solid–liquid mixture; ${\rho }_w$ is the density of pure sea-water (kg/m³); $\rho$ is the density of the mixed liquid (kg/m³); and P, Q, $\lambda$ , and $\varsigma$ are dimensionless parameters. Detailed parameter’s determination is specified in Table 1, ²⁴ where ${\rho }_s$ is the density of the solid particulate (kg/m³), a_s is the mean radius of the solid particulate (m), $\mu$ is the kinematic viscosity of seawater, ${\mu }_0$ is the kinematic viscosity of pure water, T is the sea surface temperature (°C), Cl_v is the chlorosity of seawater (kg/m³), Cl is the chlorinity of seawater $(\mathrm{‰})$ , K_s is the bulk modulus of the solid particulate (set to 7 × 10⁹ Pa), and K_w is the bulk modulus of seawater (set to 2.43 × 10⁹ Pa).

Table 1.

Parameter calculating formulas for ${\alpha }_{\ell ,v}$ .

Parameter	Calculating formula	Parameter	Calculating formula
P	${\rho }_s·(1-C)/\rho +\lambda$	Q	$\delta (1-C{)}^2+C(1-C)+\lambda$
$\delta$	${\rho }_s/{\rho }_{\omega }$	$\lambda$	$1/2+9/(4\beta a_s)$
$\rho$	${\rho }_s·C+{\rho }_{\omega }·(1-C)$	$\varsigma$	$9(1+(\beta a_s{)}^{-1})/(4\beta a_s)$
$\beta$	$\sqrt{\omega /(2\mu )}$	$\mu$	${\mu }_0(1+q\sqrt{C{l}_V}+p·C{l}_V)$
${\mu }_0$	$\{21.482[(T-8.435)+\sqrt{8078.4+{(T-8.435)}^2}]-1200{\}}^{-1}$	Clv	$\mathrm{Cl}·{\rho }_{\omega }·1000$
q	$(0.5185T+1.0675)/10000$	p	$(0.33T-25.91)/10000$
kc	$C·K_s+(1-C)·K_{\omega }$

As shown in Figure 1, we denote ${\alpha }_{\ell ,v}$ and $({\alpha }_{\ell ,v}+\Delta {\alpha }_{\ell ,v})$ as the viscous attenuation coefficients in the no-dredged-sediments area and the dredged-sediments area respectively, where $\Delta {\alpha }_{\ell ,v}$ is caused by the influence of the flowing sediments. Therefore, equation (2) can be written as

$$A_{\ell }(\omega )=\frac{R_{\ell }}{\sqrt{x_{\ell }}}·{10}^{-\frac{[({\alpha }_{\ell ,w}+{\alpha }_{\ell ,v})x_{\ell }+\Delta {\alpha }_{\ell ,v}·\Delta x_{\ell }]}{20}.}$$

(7)

Local sensor node decision

By performing HHT on r(t), we obtain the corresponding Hilbert marginal spectrum. HHT is a time–frequency analysis method advanced in 1998 by Norden E. Huang et al. Compared with the classical signal analysis methods (e.g. the Fourier transform), HHT breaks out the restrains of linearity and stationarity and overcomes the restriction of Heisenberg uncertainty principle.

Denote $R_H(\omega )$ as the Hilbert marginal spectrum of r(t). $R_H(\omega )$ curves under different situations are shown in Figure 2, wherein sharp contrasts can be seen in terms of the concentration degree and the amplitude. In order to distinguish different situations, three features are chosen: the peak energy (denoted as E_p), the ratio between 3 dB bandwidth and 50 kHz (denoted as $R{B}_{3\mathrm{dB}}$ ), and the ratio between the energy within the frequency range from 20 to 25 kHz and the total energy (denoted as $R{B}_{\mathrm{se}}$ ), all of which are described by

$$\{\begin{array}{ccc}{E}_p\hfill & =\hfill & max\{R_H(\omega )\}\hfill \\ \multicolumn{1}{c}{R{B}_{3\mathrm{dB}}}& =\hfill & \frac{B_{3\mathrm{dB}}}{50\mathrm{kHz}}\hfill \\ \multicolumn{1}{c}{R{B}_{\mathrm{se}}}& =\hfill & \frac{E_0}{E_{\mathrm{total}}}\hfill \end{array}$$

(8)

where E₀ is the energy within the frequency range from 20 to 25 kHz, and $E_{\mathrm{total}}$ is the total energy.

Figure 2.

Hilbert marginal spectrums of the signals received under different situations.

Once the feature selection is done, a specific classifier is used to discern the scenario of illegal dumping sediments and the contrary. Thereafter, the local sensor declares its decision accordingly, that is, declares the value 1 when an illegal dumping behavior happened, and declares the value 0 for the opposite case.

Global detection fusion

Optimal detection fusion rule with scan statistic

As shown in Figure 3, the sensor network is composed of two acoustic sources and M sensor arrays. Each sensor array contains K passive sensors arranged in the vertical direction. The first $\lfloor M/2\rfloor$ sensor arrays and source 2 are deployed on the same bank, and the rest sensor arrays and source 1 are deployed on the other bank. $\lfloor ·\rfloor$ denotes the floor function that gives the greatest integer less than or equal to the corresponding variable. The sensor arrays are spaced evenly along the banks, and the sources are deployed at the 1/4 and 3/4 locations of their corresponding banks, respectively. Assume that each sensor array can only receive signals from its corresponding source, that is, the first $\lfloor M/2\rfloor$ arrays are related to source 1, and the rest are related to source 2. Each sensor array has a cluster head (CH), which is responsible for collecting data from the sensors within the array through wire transmissions, and then relay the merged data to the fusion center using electromagnetic waves. That is, each sensor node is composed of a hydrophone for passive signal receiving, digital signal processing (DSP) units for feature selection and classification, and wire communication units to send data to its CH. And all of the CHs are equipped with wireless communication units, wire signal receiving units, and the necessary DSP units.

Figure 3.

The sensor network is composed of two acoustic sources and M sensor arrays, each of which contains K passive sensors, in order to detect dredgers’ misbehavior of illegal dumping sediments.

Denote the binary data from the kth sensor of sensor array m (i.e. sensor (m,k)) as $x(m,k)=\{0,1\}$ and $x_{\mathrm{mk}}$ in brief. $x_{\mathrm{mk}}$ takes the value 1 when there is a positive detection; otherwise, it takes the value 0. The mth CH counts the number of positive decisions made by the sensors $\{(m,k){\}}_{k=1}^K$ , and sends it to the fusion center periodically. When the fusion center receives local decisions transmitted from CHs (transmission time is negligible), it will make a final decision about whether there is a dredger dumping sediments illegitimately or not, and it will also know the approximate place where the event happens (we call a dumping-sediments event an event in brief in the article).

The null hypothesis of no event happens is ${\mathcal{H}}_0$ : $x_{\mathrm{mk}}=0$ for all mk combinations, while the alternative is simply $H_1$ : $x_{\mathrm{mk}}=1$ for some mk combinations.

If all sensor decisions are counted to get a system-level decision, it is obvious that we can get an extreme low system-level false alarm probability (denoted by $P_f$ ) and, unfortunately, a very low system-level detection probability (denoted by $P_d$ ). This is because only a part of the sensors would be affected by the event. Thereby, we introduce a scanning window by which we can sacrifice the performance of $P_f$ for higher $P_d$ , that is, make a tradeoff between $P_f$ and $P_d$ .

As shown in Figure 4, a counting rule within a scanning window is adopted in this article, using the total number of “1”s wherein as a statistic. Denote the discrete width and height of the window by $\mathcal{S}$ and K, where $1\le \mathcal{S}\le M$ . The window will scan $M-\mathcal{S}+1$ times. The statistics within the window at the nth time is calculated as

$$D_n=\sum _{m=n}^{n+\mathcal{S}-1}\sum _{k=1}^K{x}_{\mathrm{mk}}$$

(9)

$$=\sum _{m=n}^{n+\mathcal{S}-1}{X}_m$$

(10)

where

$$X_m\stackrel{\Delta }{=}\sum _{k=1}^K{x}_{\mathrm{mk}}.$$

(11)

The maximal scan statistic $I(\mathcal{S})$ is defined as

$$I(\mathcal{S})=D_{n^{*}}\stackrel{\Delta }{=}\max \{D_n|1\le n\le M-\mathcal{S}+1\}$$

(12)

where $D_{n^{*}}=\sum _{m=n^{*}}^{n^{*}+\mathcal{S}-1}{X}_m$ .

Figure 4.

Data table at the fusion center. x(m, k) or x_mk denotes the decision made by the kth sensor of sensor array m, that is, sensor (m, k). The solid circles refer that the corresponding decisions are “1”s.

We call the particular window with the maximal statistic $I(\mathcal{S})$ the maximal window.

It is obvious that each sensor may have different false alarm probability and detection probability. In this case, $\mathrm{Pr}\{D_n=i|0\le i\le \mathcal{S}·K\}$ is intricate. For ease of calculation, we assume that $\forall m,k$ , $p_{f_{\mathrm{mk}}}=p_f$ , and $p_{d_{\mathrm{mk}}}=p_d$ .

When $\mathcal{S}$ is given, rewrite the observations $\{x_{\mathrm{mk}}\}$ within the maximal window to a vector x. The optimal decision rule is given by a likelihood ratio test as

$$\frac{P(\mathbf{x}|H_1)}{P(\mathbf{x}|H_0)}\underset{H_0}{\overset{H_1}{\gtrless }}\frac{P(H_0)(C_{10}-C_{00})}{P(H_1)(C_{01}-C_{11})}$$

(13)

where $C_{\mathrm{ij}}$ represents the cost of choosing $H_i$ when $H_j$ is true. The quantity on the left side of the equation is the likelihood ratio, and the right side is the Bayes optimum threshold. Under the minimal error probability criterion ( $C_{00}=C_{11}=0$ and $C_{10}=C_{01}=1$ ), the corresponding log-likelihood ratio test can be expressed as ²⁵

$$\log \frac{P(H_1)}{P(H_0)}+\sum _{{\mathcal{K}}_1}\log \frac{p_d}{p_f}+\sum _{{\mathcal{K}}_0}\log \frac{1-p_d}{1-p_f}\underset{H_0}{\overset{H_1}{\gtrless }}0$$

(14)

where ${\mathcal{K}}_i\stackrel{\Delta }{=}\{(m,k):x_{\mathrm{mk}}=i\}$ .

Note that $|{\mathcal{K}}_1|=I_M(\mathcal{S})$ , and $|{\mathcal{K}}_0|=\mathcal{S}·K-I_M(\mathcal{S})$ , where $|·|$ denotes the cardinality of the set. When $p_d>p_f$ (generally, it holds in reality), by substituting and simplifying, we obtain the optimal decision rule described by

$$I(\mathcal{S})\underset{H_0}{\overset{H_1}{\lessgtr }}\mathrm{Th}$$

(15)

where

$$\mathrm{Th}=\frac{\log \frac{P(H_0)}{P(H_1)}+\mathcal{S}·K·\log \frac{1-p_f}{1-p_d}}{\log \frac{p_d(1-p_f)}{p_f(1-p_d)}}.$$

(16)

We assume that $P(H_0)/P(H_1)=1$ , thus the first item of the numerator of equation (16) is zero.

If the fusion center declares ${\mathcal{H}}_1$

$$m^{*}\stackrel{\Delta }{=}*{\arg }_{m}max\{X_m|n^{*}\le m\le n^{*}+\mathcal{S}-1\}$$

(17)

and it indicates that the event happens nearby the geometric plane passing the line which goes through the m^*th sensor array and the corresponding acoustic source.

Performance analysis

According to the local detection model described in the previous section, we can get the knowledge of each sensor’s false alarm probability $p_f$ and the detection probability $p_d$ .

Under $H_0$ , local decisions made by sensors are independent and identically distributed (i.i.d.). Therefore, we have

$$\mathrm{Pr}\{X_m=i|{\mathcal{H}}_0\}=\left(\begin{array}{c}K\hfill \\ \multicolumn{1}{c}{i}\end{array}\right)p_f^i(1-p_f{)}^{K-i}$$

(18)

where $0\le i\le K$ . The system-level false alarm probability is calculated as

$$P_f=\mathrm{Pr}\{I(\mathcal{S})\ge \mathrm{Th}|{\mathcal{H}}_0\}.$$

(19)

Likewise, the system-level detection probability is calculated as

$$P_d=\mathrm{Pr}\{I(\mathcal{S})\ge \mathrm{Th}|{\mathcal{H}}_1\}.$$

(20)

Because the local decisions made by the sensors under $H_1$ are not i.i.d. but target dependent, we analyze $P_d$ via simulations.

The equations for calculating $P_f$ are similar to those by Song et al., ¹³ thus we could use the recursive methods presented by Glaz ²⁶ and Karwe and Naus ²⁷ to estimate $P_f$ as Song et al. ¹³ did. However, the recursive methods are not efficient and usually take much time for calculations, especially when the parameters become larger. Therefore, we make a trivial change here. Once D_n is calculated, a decision denoted by ${\mathfrak{D}}_n$ is made based on equation (21)

$${\mathfrak{D}}_n=\{\begin{array}{cc}1\hfill & \mathrm{if}{D}_n\ge \mathrm{Th}\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}.\hfill \end{array}$$

(21)

Accordingly, equation (19) is rewritten as

$$\begin{array}{cc}\multicolumn{1}{c}{P_f}& =\mathrm{Pr}\{I(\mathcal{S})\ge \mathrm{Th}|{\mathcal{H}}_0\}\hfill \\ \multicolumn{1}{c}{}& =P(\sum _{n=1}^{M-\mathcal{S}+1}{\mathfrak{D}}_n\ge 1|{\mathcal{H}}_0).\hfill \end{array}$$

(22)

Based on Katenka et al., ¹² we arrive at

$$P_f\approx 1-\left(\frac{1-\sum _{n=1}^{M-\mathcal{S}+1}{\mu }_n}{\sqrt{\sum _{n=1}^{M-\mathcal{S}+1}{\delta }_n^2+\sum _{i\ne j}\mathrm{Cov}({\mathfrak{D}}_{\mathfrak{i}},{\mathfrak{D}}_{\mathfrak{j}})}}\right)$$

(23)

where $\mathrm{Cov}({\mathfrak{D}}_{\mathfrak{i}},{\mathfrak{D}}_{\mathfrak{j}})$ is the covariance between ${\mathfrak{D}}_{\mathfrak{i}}$ and ${\mathfrak{D}}_{\mathfrak{j}}$ , and

$$\{\begin{array}{c}{\mu }_n=E({\mathfrak{D}}_n|{\mathcal{H}}_0)=P({\mathfrak{D}}_n=1|H_0)\hfill \\ {\delta }_n^2=P({\mathfrak{D}}_n=1|{\mathcal{H}}_0)(1-P({\mathfrak{D}}_n=1|{\mathcal{H}}_0))\hfill \end{array}$$

(24)

where $P({\mathfrak{D}}_n=1|{\mathcal{H}}_0)=\sum _{j=\mathrm{Th}}^{\mathcal{S}·K}\left(\begin{array}{c}\mathcal{S}·K\hfill \\ \multicolumn{1}{c}{j}\end{array}\right)p_f^j{(1-p_f)}^{\mathcal{S}·K-j}$ .

Denote the collection of the sensors related to D_n by L_n, that is, $L_n=\{(m,k)|n\le m\le n+\mathcal{S}-1,1\le k\le K\}$ . $\mathrm{Cov}({\mathfrak{D}}_{\mathfrak{i}},{\mathfrak{D}}_{\mathfrak{j}})$ is the dependence between ${\mathfrak{D}}_i$ and ${\mathfrak{D}}_j(j\ne i)$ , which results from the intersection between L_i and L_j. Let n_ij denote the number of the sensors within the intersection, then

$$n_{\mathrm{ij}}=\{\begin{array}{cc}(S-|i-j|)·K\hfill & \mathrm{if}1\le |\mathrm{i}-\mathrm{j}|\le \mathcal{S}-1\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}\hfill \end{array}.$$

(25)

Denote the number of positive decisions made in the intersection by A, the number of positive decisions made in L_i but not in L_j by B and the number of positive decisions made in L_j but not in L_i by C. Note that A, B, and C are independent. Therefore

$$\begin{array}{c}\multicolumn{1}{c}{E({\mathfrak{D}}_i{\mathfrak{D}}_j)=P({\mathfrak{D}}_i={\mathfrak{D}}_j=1)}\\ \multicolumn{1}{c}{=\sum _{k=0}^{n_{\mathrm{ij}}}[P(A=k)P(B\ge \mathrm{Th}-k)P(C\ge \mathrm{Th}-k)]}\end{array}$$

(26)

where

$$P(A=k)=\left(\begin{array}{c}{n}_{\mathrm{ij}}\hfill \\ \multicolumn{1}{c}{k}\end{array}\right)p_f^k(1-p_f{)}^{n_{\mathrm{ij}}-k}$$

and

$$\begin{array}{c}P(C\ge \mathrm{Th}-k)=P(B\ge \mathrm{Th}-k)\hfill \\ =\sum _{q=\mathrm{Th}-k}^{\mathcal{S}·K-n_{\mathrm{ij}}}\left(\begin{array}{c}\mathcal{S}·K-n_{\mathrm{ij}}\hfill \\ \multicolumn{1}{c}{q}\end{array}\right)p_f^q(1-p_f{)}^{\mathcal{S}·K-n_{\mathrm{ij}}-q}\hfill \end{array}.$$

Then the covariance is given by

$$\mathrm{Cov}({\mathfrak{D}}_i,{\mathfrak{D}}_j)=\{\begin{array}{cc}E({\mathfrak{D}}_i{\mathfrak{D}}_j)-{\mu }_i{\mu }_j\hfill & \mathrm{if}{n}_{\mathrm{ij}}>0\hfill \\ \multicolumn{1}{c}{0}& \mathrm{otherwise}.\hfill \end{array}$$

(27)

By plugging equation (27) into equation (23), the estimate of $P_f$ is obtained.

Experimental results

All the simulations are done by MATLAB. The parameters associated with the real environment (e.g. the mean radius of the solid particulate a_s) are decided according to the practical situations of Xiamen Port.

Local detection

Simulation parameters and performance metrics

The usable frequency range of the transducer is from 20 to 25 kHz, and the code duration is 10 ms. Other simulation parameters are listed in Table 2. The model to simulate the falling process of the dredged sediments is given in another paper, ²⁸ which is needed to decide the value of $\Delta x_{\ell }$ described in Figure 1.

Table 2.

Simulation parameters for local detection scheme.

Parametric description	Symbol	Value
Sea surface temperature	T	21.5°C
Density of pure seawater	${\rho }_{\omega }$	1.25 kg/m3
Mean radius of solid particulate	as	3.9 × 10−6 m
Density of solid particulate	${\rho }_s$	2650 kg/m3
Depth of the transducer	d	10 m

The depth of the navigation channel is set to 16 m.

As it is a binary detection problem (binary hypothesis testing), the metrics should be the detection probability (i.e. $p_d$ , the higher the better) and the false alarm probability (i.e. $p_f$ , the lower the better).

Numerical results

To start with, we discuss the effectiveness and reliability of the feature selection and classification processes. Figures 5–7 have shown that the three features described by equation (8) can be used to discern the scenarios of illegal dumping sediments (yellow box-plots shown in these figures) and the contrary (green box-plots shown in these figures). The values of each box-plot are obtained from the results over 10³ simulation runs. As shown in Figure 5, the main value range of $E_p$ (from the lower quartile $Q1$ to the upper quartile $Q3$ ) under the scenarios of illegal dumping sediments is quite distinct from that under the contrary situations, and the ranges are little affected by the dumping volumes and the dumping locations. Similar results can be found in Figures 6 and 7.

Figure 5.

The peak energy values between the scenarios of illegal dumping sediments and the contrary (marked as the label “NoDump”). Each box-plot is obtained from the results over 10³ simulation runs: (a) the impact of different dumping volumes. The dumping location is110 m far away from the signal emitting end. (b) The impact of different dumping locations. The volume of the dredged sediments dumped is 50 m ³ . The dumping location is the distance from the signal-emitting end.

Figure 6.

The ratios between 3 dB bandwidth and 50 kHz under the scenarios of illegal dumping sediments and the contrary (marked as the label “NoDump”). Each box-plot is obtained from the results over 10³ simulation runs: (a) the impact of different dumping volumes. The dumping location is110 m far away from the signal emitting end. (b) The impact of different dumping locations. The volume of the dredged sediments dumped is 50 m ³ . The dumping location is the distance from the signal-emitting end.

Figure 7.

The ratio between the energy within the frequency range from 20 to 25 kHz and total energy under the scenarios of illegal dumping sediments and the contrary (marked as the label “NoDump”). Each box-plot is obtained from the results over 10³ simulation runs: (a) the impact of different dumping volumes. The dumping location is110 m far away from the signal emitting end. (b) The impact of different dumping locations. The volume of the dredged sediments dumped is 50 m ³ . The dumping location is the distance from the signal-emitting end.

However, any one of the features would fail to discern the two kinds of scenarios with high accuracy, as their value ranges are not entirely different. Especially when the extreme outliers are generated, miscarriage of justices would be made. Therefore, the following classifiers are tested using all of the features to see which one is preferred: LibSVM, K nearest neighbors (KNN) classifier, Naive Bayes classifier and discriminant analysis classifier.

As shown in Figures 5–7, E_p is much bigger than the other two features, thereby E_p should be processed first. Because the highest value of E_p is not bigger than 20, so we use E_p/20 instead of E_p as a classification feature.

A total of 6900 simulation results under 14 different conditions, that is, no dredged sediments dumped (3000 data) and 50 m ³ dredged sediments dumped at 13 different distances from the emitting end (13 × 300 data), are used for training. Another 23,100 simulation results under 25 different conditions, that is, no dredged sediments dumped (3000 data), 50 m ³ dredged sediments dumped at 13 different distances from the emitting end (13 × 700 data), and 11 different volumes of dredged sediments dumped at the distance 110 m far away from the emitting end (11 × 1000 data), are used for predicting.

Classification results are shown in Figure 8(a). The recognition success rates are judged by a single detecting code. Unfortunately, the false alarm probabilities (i.e. the detection probabilities for the scenario of no dredged sediments dumped in the figure) are too high, which are above 10% on the whole. It is worth noting that, as mentioned before, there are several (here we choose three) evenly spaced detecting codes in a detecting signal. So the judgment should be made by the three detecting codes together. When the minority is subordinate to the majority, the recognition success rates judged by a complete detecting signal are shown in Figure 8(b).

Figure 8.

The recognition success rates of the local sensor detection approach proposed: (a) judged by a single detecting code and (b) judged by a complete detecting signal (three detecting codes).

As depicted in Figure 8(b), false alarm probabilities $(p_f)$ and miss alarm probabilities $(1-p_d)$ are both less than 5%. When we emphasize a low false alarm probability, we can choose Naive Bayes classifier; otherwise, LibSVM is the best choice, as it obtains the highest detection probability $p_d$ .

Global fusion

We hope the system-level detection probability $P_d$ is as high as possible and the system-level false alarm probability $P_f$ is as low as possible. That is, $P_d$ and $P_f$ are the performance metrics of the system proposed. We choose $M=51$ , and $K=3$ . Each $P_d$ value shown in the figures is obtained by 10⁵ Monte Carlo runs. The relationship between $P_f$ $(P_d)$ and the window width $\mathcal{S}$ is shown in Figure 9, where $p_f=0.2$ and $p_d=0.8$ . Note that we use double y axes here. The left y axis belongs to $P_f$ , and the right one belongs to $P_d$ . We have the following observations:

1. $P_d\mathrm{s}$ and $P_f\mathrm{s}$ are decreasing functions of the window width $\mathcal{S}$ . This is not surprising because the larger $\mathcal{S}$ is, the more closely we can approach the limiting case of counting all sensor decisions within the whole network.

2. $P_f$ decreases more sharply than $P_d$ . And when $\mathcal{S}>9$ , $P_f=0$ . Therefore, we can accordingly find a topgallant balance value of $\mathcal{S}$ with a proper $P_f$ and an acceptable $P_d$ .

As indicated in Figure 8, local sensors can achieve high $p_d$ and low $p_f$ with whichever classifier used. However, due to the complicate properties of the marine environment, it is unrealistic to reach a comparable level. Therefore, we do not use the $p_d$ and $p_f$ values depicted in Figure 8 for the network simulations. Figure 10 depicts the relationship between the system-level false alarm probability $P_f$ and the local sensor false alarm probability $p_f$ . The red broken curve $P_f=p_f$ is used for comparison. As $p_f$ grows, $P_f$ increases quickly. And when $p_f\ge 0.44$ , $P_f$ is worse than $p_f$ . We also study the influence of $p_d$ over $P_d$ . Let $p_d$ increase from 0.2 to 0.9 with a step length of 0.1, and each corresponding $P_d$ value is obtained by 1 million Monte Carlo runs with $\mathcal{S}=10$ , $p_f=0.2$ , and $K=3$ . The results show that all the system-level detection probabilities are over 0.94.

Figure 9.

The relationship between $P_f$ $(P_d)$ and the window width $\mathcal{S}$ , where $p_f=0.2$ , $p_d=0.8$ , and $K=3$ . Note that the value of $P_f$ is related to the left y axis, and the value of $P_d$ is related to the right y axis.

Figure 10.

The relationship between the system-level false alarm probability $P_f$ and the local sensor false alarm probability $p_f$ , where $p_d=0.8$ , $S=10$ , and $K=3$ .

It is clear from Figure 11 that the performance depends on the threshold Th. Double y axes are also adopted here. We notice that the Bayes threshold is optimal.

Figure 11.

The relationship between $P_f$ $(P_d)$ and the threshold Th, where $p_f=0.2$ , $p_d=0.8$ , $S=10$ , and $K=3$ . Note that the value of $P_f$ is related to the left y axis, and the value of $P_d$ is related to the right y axis. The corresponding Bayes optimum threshold is 16.

Conclusion and future work

This article seeks an adequate solution to the problem of detecting dredgers’ illegal behavior of dumping dredged sediments. The solution is composed of a local sensor detection approach and a fusion strategy. In the local sensor part, an acoustic signal propagation model for sediments dumping channel is built first, and then an effective detection approach is presented. The local false alarm probabilities and miss alarm probabilities are both less than 5% for all candidate classifiers, which could be as low as about 4% and 3%, respectively for LibSVM. When the lowest false alarm probability is preferred, Naive Bayes classifier is a better choice with about 2% false alarm probability.

At the fusion center, a scan statistic is used to fuse the decisions made by the local sensors, by which we can make a tradeoff between the system-level false alarm probability $P_f$ and the system-level detection probability $P_d$ . Results show that we can get an accurate and reliable detection network by choosing a proper window width $\mathcal{S}$ and the corresponding Bayes optimum threshold. When $M=51$ , $K=3$ , $p_f=0.2$ , and $p_d=0.8$ , the optimal scanning window width $\mathcal{S}$ is 10, and the corresponding Bayes optimum threshold Th for the system-level binary hypothesis test is 16. The resultant system-level false alarm probability $P_f$ is 0, and the system-level detection probability $P_d$ is over 0.94.

In the future, to offer a more flexible detection network, the sensors will be deployed on buoys or autonomous underwater vehicles, no longer fixed on the shore. Therefore, it is essential to make the sensors be able to plan their paths, in order to minimize the redundant coverage overlap, that is, maximize the overall detection area. Furthermore, when a dredger is working, the sensors are expected to follow the dredger. Once the algorithms are done, the deployments in the real environment (for example, a harbor) may be performed.