06-Target localization using information fusion in WSNs-based Marine search and rescue

<a href="https://doi.org/10.1016/j.aej.2023.01.028">Alexandria Engineering Journal (2023) 68, 227–238</a><img src="/media/202408//1724838576.4254231.png" /><table><tr><td><img src="/media/202408//1724838576.690357.png" /></td></tr></table>Alexandria UniversityAlexandria Engineering Journalwww.elsevier.com/locate/aej <a href="http://www.sciencedirect.com/science/journal/11100168">www.sciencedirect.com</a><img src="/media/202408//1724838576.804097.jpeg" /><img src="/media/202408//1724838576.857533.png" />HOSTED BY<img src="/media/202408//1724838576.8625731.jpeg" />ORIGINAL ARTICLETarget localization using information fusion in WSNs-based Marine search and rescue<img src="/media/202408//1724838576.8692422.png" /><img src="/media/202408//1724838576.888427.png" /><img src="/media/202408//1724838576.895883.png" /><a href="http://crossmark.crossref.org/dialog/?doi=10.1016/j.aej.2023.01.028&amp;domain=pdf">check for updates</a>Xiaojun Mei <a href="#bookmark1">a</a>,b,c, Dezhi Han <a href="#bookmark1">a</a>,b,c, Yanzhen Chen <a href="#bookmark1">d</a>, Huafeng Wu <a href="#bookmark1">e</a>,<a href="#bookmark1">*</a>, Teng Ma <a href="#bookmark1">f</a>a College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China b Shanghai Ship and Shipping Research Institute Co., Ltd., Shanghai 200135, Chinac National Engineering Research Center of Ship &amp; Shipping Control System, Shanghai 200135, China dMarine Design &amp; Research Institute of China, Shanghai 200011, Chinae Merchant Marine College, Shanghai Maritime University, Shanghai 201306, Chinaf Science and Technology on Underwater Vehicle Laboratory, Harbin Engineering University, Harbin 150001, ChinaReceived 9 February 2022; revised 27 October 2022; accepted 13 January 2023 Available online 20 January 2023<table><tr><td></td><td><img src="/media/202408//1724838576.919612.jpeg" /></td></tr><tr><td>KEYWORDSTarget localization;Marine search and rescue (MSR);Wireless sensor networks (WSNs);Information fusion;Received signal strength (RSS);Time of arrival (TOA)</td><td>Abstract Marine search and rescue (MSR) is considered the last line of defense for human life at sea. Recently, a prospective MSR strategy based on wireless sensor networks (WSNs) has been developed, and distress-stricken individuals can be located utilizing various localization methods. Nevertheless, the accuracy cannot satisfy the requirement of related departments, especially when employing a single measurement localization technique, such as received signal strength (RSS)- based technology,in a dynamic and complicated ocean environment. To this end,a scheme inspired by information fusion is developed, which incorporates RSS and time of arrival (TOA) information. The maximum likelihood (ML)-based localization problem is then converted into a hybrid measure- ment alternative nonnegative constrained least squares (HM-ANCLS) framework. Moreover, the paper develops a two-step linearization localization approach (TLLA) to determine the target loca- tion. The ﬁrst step proposes a slight computation method (SCM) that relies on an active set approach to address the framework. In the second step, the paper presents an error correction approach based on the ﬁrst-order Taylor series expansion to reﬁne the solution. In addition, the paper conducts the Cramr-Rao low bound (CRLB) and the computational complexity of the hybrid scheme. Simulations reveal that TLLA outperforms other state-of-the-art approaches in var- ious situations.® 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (<a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">http://creativecommons.org/</a> <a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">licenses/by-nc-nd/4.0/</a>).</td></tr><tr><td></td><td><img src="/media/202408//1724838576.928869.png" /></td></tr></table><img src="/media/202408//1724838576.933283.jpeg" />* Corresponding authors.E-mail address: <a href="mailto:hfwu@shmtu.edu.cn">hfwu@shmtu.edu.cn</a> (H. Wu).Peer review under responsibility of Faculty of Engineering, Alexandria University.1. IntroductionMarine search and rescue (MSR), the last line of defense for human life at sea, plays an indispensable role in maritime transportation <a href="#bookmark1">[1]</a>. An MSR organization would rescue people in distress at sea and, if necessary, by cooperation with<a href="https://doi.org/10.1016/j.aej.2023.01.028">https://doi.org/10.1016/j.aej.2023.01.028</a>1110-0168 ® 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (<a href="http://creativecommons.org/licenses/by-nc-nd/4.0/">http://creativecommons.org/licenses/by-nc-nd/4.0/</a>).neighboring MSR organizations via a variety of means, includ- ing dead reckoning <a href="#bookmark1">[2]</a>, remote sensing <a href="#bookmark1">[3]</a>, and wireless sensors networks (WSNs)-based technologies <a href="#bookmark1">[4]</a>. Among the measures for MSR, technology based on WSNs enables distressed indi- viduals to actively disclose their location, which boosts the rate of rescue in comparison to other techniques <a href="#bookmark1">[5]</a>. Overboard individuals carrying out the survival equipment with sensors embedded, known as targets, can be ﬁgured out via localiza- tion technique in WSNs-based MSR scheme. The system struc- ture for the scheme is depicted in <a href="#bookmark1">Fig. 1</a>. With its cost- effectiveness, ﬂexibility, and scalability, the WSNs-based MSR is seen as a potential method for maritime rescue <a href="#bookmark1">[6]</a>.Notwithstanding, localization in WSNs-based MSR is not an easy task. The difﬁculty arises in implementing the mission in such a highly dynamic ocean environment, along with the uncertain noise. The references therein <a href="#bookmark1">[7–10]</a> have illustrated a fast-paced introduction to this challenge. The localization accuracy might deteriorate if straightly applying the approaches in the terrestrial WSNs <a href="#bookmark1">[11–13]</a>. Therefore, some methods have been investigated to improve localization accuracy in the ocean environment. However, it has been well veriﬁed in literature since at least <a href="#bookmark1">[14]</a> that only one measurement used in localizationcan easily cause signiﬁcant performance deteriorations or evenfailures on a large scale <a href="#bookmark1">[15]</a>. Although some methods inspired by information fusion have been proposed in terrestrial WSNs, the localization accuracy cannot satisfy the requirement of related departments in the WSNs-based MSR scheme due to the dynamic and complicated ocean environment.To this end, we propose a two-step linearization localiza- tion approach (TLLA) for WSNs-based MSR by fusing the hybrid measurement. With a linearization operation exploited, the original localization problem is reshaped into a hybrid measurement alternative nonnegative constrained least square (HM-ANCLS) framework. Subsequently, the TLLA addressesthe framework by exploiting the two-round procedure. In the former round, a slight computation method (SCM) that relies on an active set approach is developed, whereas an error cor- rection method (ECM) based on the ﬁrst-order Taylor series expansion is presented in the later round. Besides, we conduct the Cramr-Rao low bound (CRLB) of the hybrid scheme to evaluate the proposed method.The main contributions of the paper can be concluded as follows:1) The original localization problem in terms of the hybrid scheme with highly convex is converted into an HM-ANCLS framework via a linearization operation.2) The TLLA that integrates the SCM with ECM is pre- sented for localization in WSNs-based MSR, in which the accuracy is improved compared with some state-of-the-art methods in a simulated dynamic environment.3) The CRLB of the information fusion scheme is con- ducted to evaluate the performance of the proposed method.The rest of the paper is organized as follows. The related works are illustrated in Section 2,whereas Section 3 formulates the localization problem using the hybrid measurement. In Section 4, we introduce the proposed method, i.e., TLLA. Moreover, the CRLB of the hybrid measurement scheme and the computational complexity are discussed in Section 5. Comprehensive simulation results in different scenarios com- pared with some state-of-the-art methods are demonstrated in Section 6. In Section 7, we conclude the paper.2. Related worksLocalization techniques generally contain two types,i.e., range- based and range-free. It has been acknowledged in the research since at least <a href="#bookmark1">[16,17]</a> that the localization accuracy of range- based techniques is better than that of range-free techniques.<img src="/media/202408//1724838576.951272.png" />Fig. 1 WSNs-based MSR system structure.Target localization using information fusion in WSNs-based Marine search and rescue 229Sensors with some equipment facilitated could be aware of the distance towards the target, and the position would be acquired bysomemeans,forinstance,timeofarrival(TOA),timeofﬂight (TOF), time difference of arrival (TDOA), angle of arrival (AOA), received signal strength (RSS)-based approaches. Due to its outperformance, range-based techniques are widely used for localization in the ocean environment <a href="#bookmark1">[18]</a>.To mention a few, R. Diamant et al. <a href="#bookmark1">[19]</a> have presented a TOF-based graph localization approach to assist a diver in dis- tress by formalizing the task as a non-convex, multi-objective constraint optimization problem. C. Zhao et al. <a href="#bookmark1">[20]</a> proposed a TOA-based three-stage localization method in which the localization error is compensated via correction parameters. To further improve the localization accuracy, C. Zhao et al. <a href="#bookmark1">[21]</a> developed a second-order TDOA (STDOA) and a general- ized STDOA (GSTDOA) to tackle the problem of signal per- iod decrease in the ocean environment. In contrast to TDOA, TOA, and AOA, with no synchronization or array require- ment in localization <a href="#bookmark1">[22]</a>, RSS-based techniques were widely used, especially for WSNs-based MSR. H. Wu et al. <a href="#bookmark1">[5]</a> con- verted the localization problem into a status estimation issue based on the RSS measurements, wherein an enhanced particle ﬁlter method (EPFM) for localization in MSR was presented. But the computational time of EPFM is signiﬁcant. In this case, X. Mei et al. <a href="#bookmark1">[7]</a> then present a robust localization method with a relatively low computational complexity. The localiza- tion problem is formed into a generalized trust regional sub- problem (GTRS), and a bisection method is proposed.Nevertheless, the localization accuracy of the RSS-based technique may degrade or even fail in the ocean environment due to its latent defects <a href="#bookmark1">[23,24]</a>. Inspired by the information fusion fundamental, research using hybrid measurements for localization, for instance, hybrid RSS and TOA information, has attracted scholars to explore <a href="#bookmark1">[25–30]</a>. For instance, M. Hosseini <a href="#bookmark1">[25]</a> proposed a multilateration technique to locate a target using hybrid measurement. R. Diamant et al. <a href="#bookmark1">[26]</a> pre- sented a hybrid scheme along with the non-line-of-sight (NLOS) link detection technique. Also, S. Tomic et al. <a href="#bookmark1">[27]</a> addressed localization problems in the NLOS environment. They have proposed a novel alternating algorithm by applying a squared range (SR) and weighted least squares (WLS) crite- rion, the so-called SRWLS.However, the localization accuracy of the existing hybrid methods is relatively low and cannot meet the MSR mission. Moreover, the computational complexity of the existing meth- ods is signiﬁcant, which may increase the computing time for location determination. As we know, the target location may be changeable over the winds or currents. If the localization procedure consumes too much time, the location we eventually get may not be the true one at the current time slot. In this case, computational efﬁciency is another vital factor for WSNs-based MSR. The existing hybrid methods cannot well balance the trade-off between accuracy and efﬁciency. More- over, to the best of our knowledge, localization for MSR using information fusion in such a highly dynamic ocean environ- ment has not been fully addressed.In this context, the paper develops TLLA that can have a relatively good localization accuracy on the premise of compu- tational efﬁciency for MSR. The original highly convex maximum-likelihood (ML)-based localization problem is reshaped into the HM-ANCLS framework with a linearization operation. To ﬁgure out the solution of the framework, thepaper further divided TLLA into two parts. The ﬁrst part is the coarse estimation based on SCM, whereas the second is the reﬁned estimation based on ECM. It is worth noting that SCM is avariant of least squares (LS). The solution may drop to the local minimum. In this case, ECM based on the ﬁrst- order Taylor series expansion is developed to reﬁne the solu- tion. With the two-step estimation, localization accuracy can be guaranteed. Also, as mentioned above, SCM comes from LS, whereas ECM is from the Taylor series expansion. Since the computational complexity of LS and Taylor series expan- sion is considered linear to the dimension <a href="#bookmark1">[31]</a>, the proposed method’s computational efﬁciency can also be guaranteed. The simulation results in section 6 also reveal the outperfor- mance of the TLLA compared with other methods.3. Problem formulationThe localization scenario can be referred to as <a href="#bookmark1">Fig. 1</a>, where the overboard people carrying out the survival equipment with the sensors are recognized as the targets. The rescue helicopter deploys the sensors with global positioning systems (GPS) in the area of interest, known as anchors. Furthermore, all these sensors, including anchors and targets, connect via a particular protocol <a href="#bookmark1">[32]</a>. Then, the mission for searching the targets is then transformed into the localization problem in the network <a href="#bookmark1">[5]</a>.Assume there are N sensors with GPS information (an- chors) and a target that needs to be located in the area of inter- est. The positions of the ith sensors and the target at time t area = [a1 ; a2]T andxt = [x ; x]T, respectively, where T repre-sents the transpose operation andi = 1; ... ; N. Without loss of generality, anchors could receive the radio signal with the hybrid information of RSS and TOA from the target at each time slot, which can be modeled as <a href="#bookmark1">[27]</a>Pi = <img src="/media/202408//1724838576.982709.png" />ﬄﬄﬄﬄ<img src="/media/202408//1724838576.988128.png" /><img src="/media/202408//1724838576.992296.png" /><img src="/media/202408//1724838577.022156.png" />}) —10atlog10 <img src="/media/202408//1724838577.043052.png" /> 十 y ; (1)d = Ⅱ xt — a Ⅱ十 η ; (2)where Pi denotes the received signal power of the ith sensorfrom the target at timet, d0 is a reference distance(Ⅱ xt — a Ⅱ ≥ d0), P is the transmit power of the target attimet, PL(d0 ) indicates the loss in signal strength when the ref- erenced distanced0 = 1 m, at represents the path loss exponentat timet, Ⅱ . Ⅱ is the l2 norm, y and η are the measurementnoises of RSS and TOA, respe~ctively, where the~y aremodeledas Gaussian distribution yN (0; σ) and ηN (0; δ) if weassume the corresponding variances of each at time t areasymptotically equal, i.e., σ ≈ σi ≈ σ andδ ≈ δi ≈ δ .Suppose the RSS and TOA measurement information are independently distributed. Given the observation vectorPt = [Pi]T anddt = [d]T, the joint probability density func-tion (PDF) is given asp(Pt ; dt |xt ) = p(Pt |xt )p(dt |xt ) N 1<img src="/media/202408//1724838577.154381.png" />= Yi=1 √<img src="/media/202408//1724838577.201258.png" />&gt;ﬃﬃ<img src="/media/202408//1724838577.237657.png" /><img src="/media/202408//1724838577.263561.png" /><img src="/media/202408//1724838577.2995992.png" /><img src="/media/202408//1724838577.443987.png" /><img src="/media/202408//1724838577.5039601.png" />ﬃt — Pt 十 10atlog Ⅱxt —a Ⅱ)2 δ2 十 (dt — Ⅱ xt — at Ⅱ )2 σ2 9&gt;. exp &lt;&gt; — <img src="/media/202408//1724838577.537652.png" /> &gt;=.: ; (3)With the maximizing operation of the joint PDF exploited, the maximum likelihood (ML) estimator of xt is derived asF(t ) = argxmt in Σi<img src="/media/202408//1724838577.578466.png" />1(Pi — P 十 10atlog10 <img src="/media/202408//1724838577.610063.png" />2 δ2 十 (d — Ⅱ xt — a Ⅱ )2 σ2× <s> </s>2σ2 δ2 .(4)Nevertheless, it is still challenging to ﬁgure out the solution in <a href="#bookmark1">(4)</a> with a substantial computational complexity due to its high non-convexity. In this context, we develop an alternative scheme to solve the highly non-convex problem under the HM- ANCLS framework.4. Proposed two-linearization localization approachIn this section, the proposed TLLA is described in three parts: in the ﬁrst part, we illustrate the linearization procedure for constructing the HM-ANCLS framework from the original localization problem, followed by the SCM on solving the framework, while the last part proposes the ECM based on the ﬁrst-order Taylor series expansion (considered as the sec- ond linearization operation) for modifying the solution obtained by SCM.4.1. HM-ANCLS frameworkWe ﬁrstly make a transformation by applying the simple manipulations from <a href="#bookmark1">(1)</a> such thatμ .Ⅱ xt — a Ⅱ = λ . 10<img src="/media/202408//1724838577.6565099.png" /> ; (5)where μ = 10<img src="/media/202408//1724838577.692299.png" />t = d0 10<img src="/media/202408//1724838577.777546.png" /> .<a id="bookmark1"></a>When the noise is relatively small, and the right side of <a href="#bookmark1">(5)</a> can be approximated using the ﬁrst-order Taylor series expan- sion <a href="#bookmark1">[31]</a> asμ .Ⅱ xt — a Ⅱ = λt . (1 十 <img src="/media/202408//1724838577.9522681.png" /> σ); (6)Given the assumption that σi = σ andδi = δ, the problem in<a href="#bookmark1">(4)</a> could be converted intoargxmt in Σi<img src="/media/202408//1724838577.973799.png" />1 (λt — μ .Ⅱ xt — a Ⅱ )2 十 Σi<img src="/media/202408//1724838577.9781172.png" />1 (d — Ⅱ xt — a Ⅱ )2 .(7)Further squaring the range of each term, the problem is then derived asargxmt in Σi<img src="/media/202408//1724838577.983011.png" />1 ({λt }2 —{μ }2 xt 十 2{μ }2 {a }Txt —{μ }2 Ⅱ a Ⅱ2 )2 十 Σi<img src="/media/202408//1724838577.988223.png" />1 ( {d }2 — xt 十 2{a }Txt — Ⅱ a Ⅱ2 ); (8)where xt = Ⅱ xt Ⅱ2 .Let θt = [x ; x ; xt]T be the estimated variables. With theconstraint ofθt ≥ 0, the original problem in <a href="#bookmark1">(4)</a> is then trans- formed into the HM-ANCLS framework with a single right- hand side (RHS) vector,<table><tr><td colspan="2">argminⅡ Atθt — Bt θt ≥ 0</td><td rowspan="2">Ⅱ2 ;— {μ }2...— {μ}2—1...—1</td><td rowspan="2">l 「 I I I I I I I I I I I I I I 	I ; Bt = I I I I I I I I I I I I I I I 」 l</td><td rowspan="2">{μ }2 Ⅱ a Ⅱ2 — {λt }2...{μ}2 Ⅱ a Ⅱ2 — {λt }2 Ⅱ a Ⅱ2 —{d }2...Ⅱ a Ⅱ2 —{d}2</td><td rowspan="2">(9)l IIIIIIIII . II I I I I I」(10)</td></tr><tr><td>where「 I I I I I I IAt = I I I I I I I I l</td><td>2 {μ }2 {a }T...2 {μ}2 {a}T 2 {a }T...2 {a}T</td></tr></table>4.2. Slight computation method (SCM)To ﬁgure out the solution in <a href="#bookmark1">(9)</a>, the SCM that relies on an active set approach is introduced. A solid solution with rela- tively high accuracy can be obtained in a ﬁnite number of iter- ations compared with others <a href="#bookmark1">[33]</a>.In SCM, we deﬁne the setΩ = {1; 2; 3}, in which each value indexes the columns of At and the rows ofθt . In addition, we divide the set Ω into two subsets named the active set Ψ and the passive set Y such thatΨ U Y = Ω .Theorem. <a href="#bookmark1">[34]</a>: Assume a vector ζt ∈ R3×1 is a solution of the problem in <a href="#bookmark1">(9)</a> deﬁned as.argmin Ⅱ Atζt — Bt Ⅱ2 ; s.t. ζt ≥ 0 (11)if and only if there exists a vector rt ∈ R3×1 and a partition of the integers 1 through 3 into subsets Ψ and Y in case that with rt = (At )T (Atζt — Bt )ζ = 0 for j ∈ Ψ ; ζ &gt; 0 for j ∈ Y; ( <a href="#bookmark2">12</a>)r ≥ 0 for j ∈ Ψ ; r = 0 for j ∈ Y; (13)where j indicates the index value of the setY.Then, the vector ζ that satisﬁesζ &gt; 0 for j ∈ Y and ζ = 0 for j ∈ Ψ ; (14)is the solution of the problemargminⅡ Aζ — Bt Ⅱ2 ; (15)θtwhere A is a 2N × 3 matrix and deﬁned ascolumn j of A = { column; <img src="/media/202408//1724838578.008169.png" /> ;j ∈ Y ; (16)And the dual vector xt = —rt = (At )T (Bt — Atζ) wouldsatisfyx = 0; j ∈ Y and x ≤ 0; j ∈ Ψ . (17)The problem results in <a href="#bookmark1">(9)</a> would be the ﬁnal solution if and only if the conditions in Theorem are satisﬁed.Speciﬁcally, SCM consists of an outer loop and an inner loop. In the outer loop step, we should ﬁgure out the solution of <a href="#bookmark1">(15)</a> and search an index q ∈ Ψ subjectTarget localization using information fusion in WSNs-based Marine search and rescue 231tow = max {w : j ∈ Ψ}. Subsequently, the index q would bemoved from the set Ψ to the setY.In the inner loop, a new index k ∈ Y that we need to ﬁgure out such thatθ<img src="/media/202408//1724838578.057874.png" />/ (θ<img src="/media/202408//1724838578.066408.png" />—ζ<img src="/media/202408//1724838578.069325.png" />) = min {θ/ (θ—ζ) : ζ ≤ 0,j ∈ Y},βt = θ<img src="/media/202408//1724838578.077049.png" />/ (θ<img src="/media/202408//1724838578.097224.png" />—ζ<img src="/media/202408//1724838578.099026.png" />),(18)ζt = θt 十 βt (ζt — θt ).The entire procedure of SCM could be concluded in Algo- rithm SCM.<table><tr><td>Algorithm SCM:</td></tr><tr><td>1. Initiation:θt = 03×1 ,Ψ = {1, 2, 3},Y =,xt = (At )T (Bt — Atθt )</td></tr><tr><td>2. Do while (Ψ≠ and 彐 j ∈ Ψ with x &gt; 0)</td></tr><tr><td>3. Find an index q ∈ Ψ such that w = max {w : j ∈ Ψ}</td></tr><tr><td>4. Pass the index q from the set Ψ to the set Y</td></tr><tr><td>5. Solve the problem of <a href="#bookmark1">(15)</a> and deﬁne ζ = 0 for j ∈ Ψ</td></tr><tr><td>6. Do while (ζ ≤ 0 for any j ∈ Y)</td></tr><tr><td>7. Find an index k ∈ Y with <a href="#bookmark1">(18)</a></td></tr><tr><td>8. Pass the set Y to the set Ψ for all indices j ∈ Y suchthatθ = 0.</td></tr><tr><td>9. Solve the problem of <a href="#bookmark1">(15)</a></td></tr><tr><td>10. End While (ζ &gt; 0 for all j ∈ Y)</td></tr><tr><td>11. θt = ζt and xt = (At )T (Bt — Atθt )</td></tr><tr><td>12.End While (Ψ is empty or w ≤ 0 for all j ∈ Ψ)</td></tr></table>4.3. Error correction method (ECM)Theoretically, SCM can obtain a suboptimal solution t byexchanging the index of potential from the passive set and the active set. However, the potential solution is generally acquired by the variant of LS, which may drop to the local minimum. Therefore, the ECM is presented to boost the per- formance of SCM. We ﬁrst reconstruct the optimized function asΦt = 夕t θt 十 Δt , (19)whereΦt = [ (ζ)2 , (ζ)2 , ζ]T,θt = [ (x)2 , (x)2]T,夕t = [I2, 12×1]T,and「 (ζ)2 —(x)2 l 「 (ζ 十 x)(ζ — x) l 「2x (ζ — x) lΔt = I (ζ)2 —(x)2 I = I (ζ 十 x)(ζ — x) I ≈ I 2x (ζ — x) I .l ζ — xt 」 l ζ — xt 」 l ζ — xt 」(20)Then, the corresponding estimate θt could be obtained by θt = {(夕t )T夕t }—1 (夕t )TΦt . (21)Since <a href="#bookmark1">(21)</a> are the estimates of (x)2 and (x)2 , we need toemploy the signum function to modify the sign of the corre-sponding solution, i.e.,t = [sign (ζ) √<img src="/media/202408//1724838578.1125991.png" /><img src="/media/202408//1724838578.120516.png" />, sign (ζ) √<img src="/media/202408//1724838578.141694.png" /><img src="/media/202408//1724838578.153682.png" />]T. (22)However, it should be noted that one more square root operation is implemented in <a href="#bookmark1">(22)</a> to ﬁgure out the ﬁnal solu- tion, which might exacerbate the estimation error inζt .In this context, we develop a squaring root-free method that is a linearization approach based on the ﬁrst-order Taylor series expansion.Recalling from the problem in <a href="#bookmark1">(9)</a>, we assume the estimate ^obtained by SCM isθCM . The corresponding cost function canbe rewritten asJ = (Atθt — Bt )TAtθt — Bt= (θt — CM )T (At )TAt (θt — CM ). (23)where θt = [ , , t]T.After using the ﬁrst-order Taylor series expansion ofθt — CM for θ:2 approachingt , we can acquireθt — CM = θt'' θ:2 = t — CM十 <img src="/media/202408//1724838578.1659782.png" /> '' θ:2 = t (θ:2 — t ) = Θ 十 I (θ:2 — t ), (24)whereΘ = θt'' θ:2 = t — CM =「I0 l 0 I ,Ⅱ2 — t 」[ 2(<img src="/media/202408//1724838578.183815.png" />)T ].l Ⅱ t(25)I = <img src="/media/202408//1724838578.223474.png" /> '' θ:2 = t (θ:2 — t ) =Replacing the corresponding term in <a href="#bookmark1">(23)</a> with <a href="#bookmark1">(24)</a>, the function can be reshaped intoJ = {Θ 十 I (θ:2 — t )}T (At )TAt {Θ 十 I (θ:2 — t )}. (26) Taking the derivative of <a href="#bookmark1">(26)</a> in terms ofθ:2, then we have<img src="/media/202408//1724838578.2605078.png" /> = 2IT (At )TAt I (θ:2 — t ) 十 2IT (At )TAt Θ . (27)By forcing <a href="#bookmark1">(27)</a> to 0, the corrected solution can be obtained asorrected = t — {IT (At )TAt I }—1IT (At )TAt Θ . (28)5. Evaluation of the hybrid scheme for localizationIn this section, we conduct the CRLB for the hybrid measure- ment scheme to calibrate the proposed method. Moreover, the computational complexity of TLLA is discussed and compared with other state-of-the-art approaches.5.1. Cramr-Rao low bound (CRLB)Theoretically, CRLB is considered the benchmark of anyunbi- ased estimators <a href="#bookmark1">[35]</a>. Thus, we conduct the CRLB of the hybrid RSS and TOA scheme in this part.In what concerns RSS and TOA, the corresponding mea- surements at each time slot can be expressed as<table><tr><td>「 Pt = I l</td><td>Ptr1...PtrN</td><td>l 「 d lI ; dt = I ... I .」 ld 」</td><td>(29)</td></tr></table>Because the noises follow the Gaussian distribution, the PDF of each term can be stated aspRSS (Pi |xt ) = (2兀<img src="/media/202408//1724838578.3112488.png" />1/2σ exp { — <img src="/media/202408//1724838578.354135.png" />; (30)where Λ = Pi — P 十 10atlog10 <img src="/media/202408//1724838578.554726.png" /> .pTOA (d |xt ) = (2兀<img src="/media/202408//1724838578.595049.png" />1/2δ exp { — <img src="/media/202408//1724838578.602016.png" />; (31)where H = d — Ⅱ xt — a Ⅱ.To take the logarithm of each term, and then we havelnpRSS (Pi |xt ) = — <img src="/media/202408//1724838578.610603.png" />ln2兀 — Nlnσ — <img src="/media/202408//1724838578.613989.png" /> (Λ)2 . (32)lnpTOA (d|xt ) = — <img src="/media/202408//1724838578.6186528.png" />ln2兀 — Nlnδ — <img src="/media/202408//1724838578.627799.png" /> (H)2 . (33)The partial derivative of <a href="#bookmark1">(32) and (33)</a> with subject to xt when d0 = 1 m are∂ lnpRSS (Pi |xt ) = 10at x —a1 ∂x σ2 ln10 Ⅱxt —a Ⅱ2 ;∂ lnpRSS (Pi |xt ) = 10at x —a2 ∂x σ2 ln10 Ⅱxt —a Ⅱ2 ;(34)∂ lnpTOA (d | xt ) x —a1 ∂x = δ2 Ⅱxt —a Ⅱ2 ;∂ lnpTOA (d | xt ) x —a1 ∂x = δ2 Ⅱxt —a Ⅱ2 .As for N anchors, then we have「		10at x —at12 σ2 ln10 Ⅱxt —a Ⅱ2		10at x —a σ2 ln10 Ⅱxt —a Ⅱ2l III ; I」∂ lnpRSS (Pt|xt) = II...		10at x —a2 σ2 ln10 Ⅱxt —aⅡ2...		10at x —a1 σ2 ln10 Ⅱxt —aⅡ2∂xt I I l		x —at12 δ2 Ⅱxt —a Ⅱ2(35)		x —a δ2 Ⅱxt —a Ⅱ2「I= I I I ll III . I」∂ lnpTOA (dt|xt) ∂xt......		x —a1 δ2 Ⅱxt —a Ⅱ2		x —a2 δ2 Ⅱxt —a Ⅱ2The Fisher information matrix (FIM) can be obtained asFIMRSS = <img src="/media/202408//1724838578.647566.png" />T <img src="/media/202408//1724838578.6569622.png" /> ; FIMTOA = <img src="/media/202408//1724838578.661674.png" />T <img src="/media/202408//1724838578.664175.png" /> .(36)CRLB is the trace of the inverse of the FIM <a href="#bookmark1">[35]</a>. In the hybrid measurement scheme, RSS and TOA are assumed to be statistically independent. In this case, the CRLB of the hybrid scheme can be added up by each FIM according to <a href="#bookmark1">[36]</a>. Then, the CRLB of the hybrid RSS and TOA can be expressed asCRLB = Tr (FIM—1) = Tr [(FIMRSS 十 FIMTOA )—1]. (37)5.2. Computational complexitySeveral state-of-the-art methods are discussed for the compar- ison in terms of computational complexity, including WLS in <a href="#bookmark1">[37]</a>, SRWLS in <a href="#bookmark1">[27]</a>, linear least squares (LLS-I) in <a href="#bookmark1">[38]</a>, Semi-Deﬁnite Relaxation (SDP) in <a href="#bookmark1">[28]</a>, the proposed SCM, and TLLA. It is noteworthy that both WLS and LLS-I are based on LS, wherein the computational complexity isO(N). Regarding SRWLS, it needs to ﬁgure out the inverse of the diagonal matrix in each bisection procedure. In this case, the computational complexity is relatively more signiﬁcant than that of WLS and LLS-I. Suppose τ is the maximum number of iterations, then the computational complexity of SRWLS is O(τN) if the worst condition happens. When it comes to SDP, the solution of which is obtained by the standard interior-point methods. Assuming the worst case occurs, the computational complexity of SDP should beO {√-log <img src="/media/202408//1724838578.675025.png" />N(N 十 3)2 十 N2 (N 十 3)2 十 N3 )}; (38)where ε denotes the solution precision. On average, the worst- case complexity of SDP could beO(N4.5 ).In SCM, the corresponding solution is also solved by LS with two rounds. Therefore, the computational complexity of SCM isO(N2 ). While more extra operation needs to be han- dled in terms of <a href="#bookmark1">(28)</a>, the computational complexity of ECM would beO(N). In this case, the computational complexity of the two-step linearization method, i.e., TLLA, should beO(N(1 十 N)). <a href="#bookmark1">Table 1</a> summarizes the considered methods. Although the computational complexity of the proposed two-step linearization method is higher than that of some methods, the localization accuracy and computational time are satisﬁed with the MSR mission from the simulation results demonstrated in next section.6. Simulation results and discussion6.1. Simulation environment and the calibrationTo evaluate the performance of the proposed method, we carry out the simulations in different scenarios in Matlab R2021b compared with WLS, SRWLS, LLS-I, SDP, and the CRLB in <a href="#bookmark1">(37)</a>. The corresponding simulation environment is illus- trated in <a href="#bookmark1">Table 2</a>.<table><tr><td>Table 1 Summary complexity.</td><td>of</td><td>the</td><td>computational</td></tr><tr><td>Method</td><td></td><td></td><td>Complexity</td></tr><tr><td>WLS</td><td></td><td></td><td>O(N)</td></tr><tr><td>SRWLS</td><td></td><td></td><td>O(τN)</td></tr><tr><td>LLS-I</td><td></td><td></td><td>O(N)</td></tr><tr><td>SDP</td><td></td><td></td><td>O (N4.5 )</td></tr><tr><td>SCM</td><td></td><td></td><td>O (N2 )</td></tr><tr><td>TLLA</td><td></td><td></td><td>O(N(1 十 N))</td></tr></table>of interest is large, or the noise is relatively high, the localiza- tion accuracy may deteriorate dramatically. In this case, it can be seen from <a href="#bookmark1">Fig. 2</a> that the performance of TLLA with only RSS measurement is the worst.It is worth noting that the positions of the target and anchors are changeable due to the currents and winds on the sea surface. In this context, we randomly deploy the anchors and the target initiatively and use the random walk model <a href="#bookmark1">[39]</a> to mimic the dynamics of all nodes, which, in other words, means the position of all nodes is not ﬁxed at each time slot. The rest of the ﬁxed parameters are set asτ = 1000,d0 = 1 m.As the calibration for the performance, the root means square error (RMSE) is conducted asHowever the drawback of the RSS-based technique can be eliminated by information fusion. As <a href="#bookmark1">Fig. 2</a> shows, the perfor- mance of the hybrid measurement scheme is better than that of only the RSS-based approach. Interestingly, the proposed method with TOA measurement only seems to outperform most hybrid schemes, which proves the effectiveness of the proposed two-step linearization approach. Although SCM can perform relatively satisfactorily, the solution could be modiﬁed with ECM. The localization accuracy is further improved by the two-step linearization-based method, i.e., TLLA, and close to CRLB. The outperformance of the pro- posed method, i.e., TLLA, in terms of variable anchors can be illustrated further with cumulative distribution function (CDF) in <a href="#bookmark1">Fig. 3</a>. The proposed two-step linearization method--------------------------------------RMSE = <img src="/media/202408//1724838578.690853.png" />Σ<img src="/media/202408//1724838578.697097.png" />a Ⅱ t — xt Ⅱ. (39)where t and xt denote the actual position and the estimate ateach time slot, respectively, and tmax is the maximum time that we set tmax = 1000 s in the simulations.6.2. Senario with variable anchorsThe RMSE versus variable anchors is depicted in <a href="#bookmark1">Fig. 2</a> with the parameters in <a href="#bookmark1">Table 3</a>. Theoretically, more available mea- surement information can be acquired with the increase of the anchors. Therefore, the performance of the methods is improved in the simulation as expected. It should be empha- sized that the RSS model ﬁgures out the measurement based on the loss of signal intensity recalling from <a href="#bookmark1">(1)</a>. Once the areawith information fusion can achieveⅡ t — xt Ⅱ≤ 5.03 m,Ⅱ t — xt Ⅱ ≤ 3.95 m,Ⅱ t — xt Ⅱ ≤ 3.53 m andⅡ t — xt Ⅱ ≤ 3. 17 m at almost 95 % inN = 6,N = 8,N = 10,andN = 12, respectively. In contrast, other state-of-the-art methods can reach the same probability at a relatively high localization error.<table><tr><td colspan="2">Table 3 Parameters in the scenario with variable anchors.</td></tr><tr><td>Parameter</td><td>Value</td></tr><tr><td>σ</td><td>5 dB</td></tr><tr><td>δ</td><td>5 m</td></tr><tr><td>at</td><td>3.5</td></tr><tr><td>P</td><td>—55 dBm</td></tr><tr><td>Sidelength</td><td>100 m</td></tr></table><table><tr><td colspan="3">Table 2 Simulation environment.</td></tr><tr><td rowspan="2">Hardware</td><td>Memory</td><td>CPU</td></tr><tr><td>8 GB</td><td>Intel (R) Core (TM) i7-8550U, CPU1.8 GHz</td></tr><tr><td>Software</td><td>Platform MatlabR2021b</td><td>Operation System Windows 10</td></tr></table><img src="/media/202408//1724838578.7425878.png" /><img src="/media/202408//1724838578.7453492.png" /><img src="/media/202408//1724838578.750998.png" /><img src="/media/202408//1724838578.754326.png" /><img src="/media/202408//1724838578.7565749.png" /><img src="/media/202408//1724838578.757975.png" /><img src="/media/202408//1724838578.762484.png" /><img src="/media/202408//1724838578.781645.png" /><img src="/media/202408//1724838578.788722.png" /><img src="/media/202408//1724838578.8099892.png" /><img src="/media/202408//1724838578.849795.png" />Fig. 2 RMSE versus variable anchors.Fig. 3 CDF of variable anchors.6.3. Scenario with variable noises<table><tr><td colspan="2">Table 4 Parameters in the scenario with variable noises.</td></tr><tr><td>Parameter</td><td>Value</td></tr><tr><td>N</td><td>8</td></tr><tr><td>at</td><td>3.5</td></tr><tr><td>P</td><td>-55 dBm</td></tr><tr><td>Sidelength</td><td>100 m</td></tr></table>The RMSE versus variable noises of RSS and TOA is shown in <a href="#bookmark1">Fig. 4</a> with parameters in <a href="#bookmark1">Table 4</a>. Moreover, it is worth noting that we assume the values of the variances of RSS and TOA are the same for convenience. With variances rising, the local- ization accuracy of the methods degrades, as shown in <a href="#bookmark1">Fig. 4</a>. The rate of deterioration of TLLARSS is relatively more promi- nent than that of the others due to the absence of information fusion technology and the inherent defects of the RSS-based technique.<img src="/media/202408//1724838578.899221.png" /><img src="/media/202408//1724838578.916967.png" /><img src="/media/202408//1724838578.9396532.png" /><img src="/media/202408//1724838578.9892778.png" /><img src="/media/202408//1724838579.12315.png" /><img src="/media/202408//1724838579.171369.png" /><img src="/media/202408//1724838579.1985972.png" /><img src="/media/202408//1724838579.236211.png" /><img src="/media/202408//1724838579.260841.png" /><img src="/media/202408//1724838579.301389.png" /><img src="/media/202408//1724838579.4812758.png" /><img src="/media/202408//1724838579.5652611.png" /><img src="/media/202408//1724838579.652372.png" /><img src="/media/202408//1724838579.69171.png" />Among the methods, WLS and SRWLS seem to have good robustness over the change in the noise. However, the localiza- tion accuracy of WLS is not satisfactory compared with SRWLS. Although information fusion technology is used in SCM, the localization accuracy of SCM is similar to that of TLLATOA. From the comparison of SCM, TLLATOA, and TLLARSS, we can see that the localization accuracy of the TOA-based technique is higher than that of the RSS-based technique. Also, the second step of correction, i.e., ECM, seems to decrease the localization error. In this case, the localization accuracy of TLLA with the information fusion technology beats others and is close to CRLB. The outperformance of TLLA can be clearly seen in <a href="#bookmark1">Fig. 5</a>, wherein<img src="/media/202408//1724838579.763315.png" /><img src="/media/202408//1724838579.821776.png" />TLLA can reach the error thatk t - xt k ≤ 1.89 m, k t - xt k≤ 3.23 mk t - xt k ≤ 4. 16 m and k t - xt k ≤ 5.06 m at<img src="/media/202408//1724838579.83323.png" /><img src="/media/202408//1724838579.8372831.png" /><img src="/media/202408//1724838579.8393679.png" /><img src="/media/202408//1724838579.8574688.png" />95 % over the change of variance value from 1 to 7. On the contrary, the others could approach the same probability with a relatively high error. For instance, SCM achieves the sameprobability with the error thatk t - xt k ≤ 2.28 m,k t - xt k ≤ 4.35 m,k t - xt k ≤ 5.74 m and k t - xt k ≤Fig. 5 CDF of variable noises.6. 17 m.6.4. Scenario with variable side lengths (SL) of the areaIt should be noted that anchors and the target would drift on the sea surface due to the impact of the currents and winds. In this case, the area for locating might be changeable. Assumethe communication radius of the sensors is above 500 m. We conduct the simulation of variable side lengths of the area, as shown in <a href="#bookmark1">Fig. 6</a>, for MSR with parameters in <a href="#bookmark1">Table 5</a>. The more signiﬁcant the area is, the more severe the path loss effect would have in the RSS-based technique due to its attri- bution. In this case, the performance of TLLARSS seems to be the worst among the methods.Interestingly, the localization accuracy of WLS, SRWLS, and SDP degrades over the rise in side length, whereas LLS- I, SCM, TLLATOA, and TLLA are more robust to the change of the side length. The localization accuracy of TLLATOA is similar to that of LLS-I and SCM. The proposed method with the information fusion technology, i.e., TLLA, seems to be better than the others, which enables securing the error less than 5 m at 95 % on average, as shown in <a href="#bookmark1">Fig. 7</a>, compared with that of, for instance, 7 m for LLS-I, SCM,and TLLATOA.<img src="/media/202408//1724838579.87651.png" />6.5. Scenario with variable path loss exponents (PLE)M.R. Gholami et al. <a href="#bookmark1">[40]</a> have studied that the PLE may change over the temperature, pressure, or humidity ranging from 2 to 6. In this context, simulations in variable PLE are carried out in this part, where parameters are shown in <a href="#bookmark1">Table 6</a>. It is worth noting that there is no PLE involved in the TOA- based technique, which, in other words, means the change of PLE will not have any positive or negative effects on localiza- tion accuracy subject to the TOA-based method. Thus, weFig. 4 RMSE versus variable noises of RSS and TOA.Target localization using information fusion in WSNs-based Marine search and rescue 235<img src="/media/202408//1724838579.8832672.png" /><img src="/media/202408//1724838579.892256.png" />remove the corresponding results of TLLATOA that only use the TOA-based method to ﬁgure out the measurements in this part.Moreover, the RMSE versus variable PLE and the corre- sponding CDF are depicted in <a href="#bookmark1">Fig. 8</a> and <a href="#bookmark1">Fig. 9</a>, respectively. Theoretically, the larger the PLE is, the more similar the indoor channel condition that the environment approaches <a href="#bookmark1">[41]</a>. In this case, the localization accuracy is improved signif- icantly over the rise in the PLE in TLLARSS. With the TOA- based measurements fused in the localization, the localization accuracy seems robust to the PLE for most methods except for SDP. Among these methods, the performance of TLLA appears to be better than the others and can secure the error within 4.5 m on average in variable PLE, as shown in <a href="#bookmark1">Fig. 9</a>.<img src="/media/202408//1724838579.903748.png" />6.6. Scenario with variable transmit power (TP)To further demonstrate the effectiveness of the proposed method, we carry out the simulations in different TP with parameters in <a href="#bookmark1">Table 7</a>, as shown in <a href="#bookmark1">Fig. 10</a>. There is no TP involved in the calculation of the TOA-based model <a href="#bookmark1">(2)</a>. Thus, the results of TLLATOA are not illustrated in this part. It can be seen from <a href="#bookmark1">Fig. 10</a> that all methods seem to be robust to theFig. 6 RMSE versus variable SL of the area.<table><tr><td colspan="2">Table 5 Parameters in the scenario with variable sl.</td></tr><tr><td>Parameter</td><td>Value</td></tr><tr><td>σ</td><td>5 dB</td></tr><tr><td>δ</td><td>5 m</td></tr><tr><td>at</td><td>3.5</td></tr><tr><td>P</td><td>-55 dBm</td></tr><tr><td>N</td><td>8</td></tr></table><table><tr><td colspan="2">Table 6 Parameters in the scenarios with variable ple.</td></tr><tr><td>Parameter</td><td>Value</td></tr><tr><td>σ</td><td>5 dB</td></tr><tr><td>δ</td><td>5 m</td></tr><tr><td>Sidelength</td><td>100 m</td></tr><tr><td>P</td><td>-55 dBm</td></tr><tr><td>N</td><td>8</td></tr></table><img src="/media/202408//1724838579.914529.png" /><img src="/media/202408//1724838579.921601.png" />Fig. 7 CDF of variable SL of the area.Fig. 8 RMSE versus variable PLE.<img src="/media/202408//1724838579.938037.png" /><img src="/media/202408//1724838579.951723.png" /><img src="/media/202408//1724838579.95331.png" /><img src="/media/202408//1724838579.964919.png" /><img src="/media/202408//1724838579.96892.png" /><img src="/media/202408//1724838579.976009.png" /><img src="/media/202408//1724838579.98306.png" /><img src="/media/202408//1724838579.997855.png" /><table><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr></table><img src="/media/202408//1724838580.0131772.png" />Fig. 9 CDF of variable PLE.change of TP, albeit some ﬂuctuations occur in terms of SDP. The average localization error of TLLA is below 2.5 m com- pared with TLLARSS for 7 m, WLS for 6.5 m, SDP for 5 m, SRWLS for 4 m, SCM for 3 m, and LLS-I for 3.5 m. Besides, the outperformance of the proposed method can also be seen in <a href="#bookmark1">Fig. 11</a>, where TLLA can secure the localization error below 4.3 m at 95 % compared with that of 5.8 m for LLS-I, 5.5 m<table><tr><td colspan="2">Table 7 Parameters in the scenario with variable tp.</td></tr><tr><td>Parameter</td><td>Value</td></tr><tr><td>σ</td><td>5 dB</td></tr><tr><td>δ</td><td>5 m</td></tr><tr><td>Sidelength</td><td>100 m</td></tr><tr><td>at</td><td>3.5</td></tr><tr><td>N</td><td>8</td></tr></table><img src="/media/202408//1724838580.019815.png" />Fig. 10 RMSE versus variable TP.<img src="/media/202408//1724838580.050978.png" />Fig. 11 CDF of variable TP.for SCM, 14 m for TLLARSS, 9 m for SDP, 10 m for WLS, and 6.5 m for SRWLS.6.7. Computational timeBesides localization accuracy, efﬁciency is another vital factor that needs to be considered in MSR. In this case, we visualize the corresponding computational time in <a href="#bookmark1">Fig. 12</a> (computa- tional time for SDP and SRWLS refer to <a href="#bookmark1">Table 8</a>). It can be seen from <a href="#bookmark1">Fig. 12</a> that the proposed methods, including TLLA, SCM, TLLATOA, and TLLARSS, are worse than that of LLS-I and WLS. However, in terms of computational time, the pro- posed method is much better than SDP and SRWLS, as in <a href="#bookmark1">Table 8</a>. Although the computational time is not the best among the methods, the average computational time is below 2e-4 s, which seems to be tolerant in the actual situation.<img src="/media/202408//1724838580.074709.png" />Fig. 12 RMSE versus variable side lengths of the area.Target localization using information fusion in WSNs-based Marine search and rescue 237<table><tr><td colspan="6">Table 8 summary of the computational time.</td></tr><tr><td rowspan="2">Method</td><td colspan="5">Time for diﬀerent scenarios (s)</td></tr><tr><td>Variable Anchors</td><td>Variable SL</td><td>Variable Noises</td><td>Variable PLE</td><td>Variable TP</td></tr><tr><td>WLS</td><td>3.2678e-5</td><td>3.20856e-5</td><td>3.3250e-5</td><td>3.3757e-5</td><td>3.4213e-5</td></tr><tr><td>SRWLS</td><td>0.0022</td><td>0.0023</td><td>0.0021</td><td>0.0021</td><td>0.0021</td></tr><tr><td>LLS-I</td><td>3.7437e-5</td><td>3.1638e-5</td><td>3.1819e-5</td><td>2.9549e-5</td><td>3.3035e-5</td></tr><tr><td>SDP</td><td>0.4781</td><td>0.4649</td><td>0.4568</td><td>0.6529</td><td>0.4606</td></tr><tr><td>SCM</td><td>1.7817e-4</td><td>1.6831e-4</td><td>1.7161</td><td>1.6150e-4</td><td>1.8801e-4</td></tr><tr><td>TLLARSS</td><td>1.9129e-4</td><td>1.6235e-4</td><td>1.6626e-4</td><td>1.5591e-4</td><td>1.6059e-4</td></tr><tr><td>TLLATOA</td><td>1.6569e-4</td><td>1.3979e-4</td><td>1.4322e-4</td><td>	</td><td>	</td></tr><tr><td>TLLA</td><td>2.0375e-4</td><td>1.7025e-4</td><td>1.7562e-4</td><td>1.8006e-4</td><td>1.8488e-4</td></tr></table>7. ConclusionThis paper proposes a two-step linearization method for tar- get localization in WSNs-based MSR. The hybrid RSS and TOA scheme is presented to eliminate the error with only one measurement exploited on localization. We convert the original localization problem to an HM-ANCLS framework via linearization operation. The SCM is further proposed to ﬁgure out the solution of HM-ANCLS via exchanging the index of the potential from the active set and the passive set. Nevertheless, SCM may drop to the local minimum. In this case, the ECM based on the ﬁrst-order Taylor series expansion is conducted to modify the solution obtained by SCM. To mimic the dynamic situation in the ocean environ- ment, we use the random walk model to make all sensors’ positions changeable at each time slot. Although the pro- posed method’s computational efﬁciency is worse than some methods in some scenarios, the localization accuracy seems to be the most satisfactory among the methods. Moreover, the average computational time of the proposed method for locating a target at each time slot is less than 2e-4 s, which seems to be tolerant in the actual situation. Therefore, combining the results of the localization accuracy and the computational efﬁciency, the proposed two-step linearization method could be the better choice for localization in WSNs- based MSR.However, in the procedure to form the HM-ANCLS frame- work, we have assumed the corresponding variable is non- negative, seeing from equation <a href="#bookmark1">(9)</a>. It means the proposed method works well only when the coordinate reference system is well-deﬁned. After thousands of simulations, we have ﬁg- ured out that the error of the proposed method would increase once the corresponding variables are negative. In future research, we would like to investigate how to eliminate the assumption and maintain accuracy simultaneously. Besides, another assumption has been made in the paper that all infor- mation acquired is complete. Nevertheless, wireless networks are vulnerable to attack, and the transmission links are also unstable in the ocean environment. It is infeasible to obtain complete information for localization. How to locate a target with missing information is promising research for future work as well. Last but not least, the corresponding results are only veriﬁed in the simulation. Another future perspective work that attracts our attention is to verify the proposed method in the natural ocean environment.Declaration of Competing InterestThe authors declare that they have no known competing ﬁnancial interests or personal relationships that could have appeared to inﬂuence the work reported in this paper.AcknowledgementsThis work is supported by the National Natural Science Foun- dation of China (No. 52201401, 52201403, 52102397, 61873160, 61672338, 52071200, and 52001093), the National Key Research and Development Program of China (No. 2021YFC2801002), the Shanghai Committee of Science and Technology, China (No. 23010502000), the Shanghai Post- doctoral Excellence Program, China (No. 2022767), the China Postdoctoral Science Foundation (No. 2022M712027, 2021M700790), and Natural Science Foundation of Shanghai, China ( No. 21ZR1426500).References[1] <a href="http://refhub.elsevier.com/S1110-0168(23)00043-1/h0005">W.-C. Ho, J.-H. Shen, C.-P. Liu, Y.-W. 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