Depth Discrimination for Low-Frequency Sources Using a Horizontal Line Array of Acoustic Vector Sensors Based on Mode Extraction

Transcription

1 Article Deth Dicriination for Low-Frequency Source Uing a Horizontal Line Array of Acoutic Vector Senor Baed on ode Extraction Guolong Liang 1,,3, Yifeng Zhang 1,,3, Guangu Zhang 1,,3, *, Jia Feng 1,,3 and Ce Zheng 1,,3 1 Acoutic Science and Technology Laboratory, Harbin Engineering Univerity, Harbin 151, China; liangguolong@hrbeu.edu.cn (G.L.); zhangyifeng1@hrbeu.edu.cn (Y.Z.); fengjia@hrbeu.edu.cn (J.F.); zhengce_uae@hrbeu.edu.cn (C.Z.) Key Laboratory of arine Inforation Acquiition and Security (Harbin Engineering Univerity), initry of Indutry and Inforation Technology, Harbin 151, China 3 College of Underwater Acoutic Engineering, Harbin Engineering Univerity, Harbin 151, China; * Correondence: zhangguangu@hrbeu.edu.cn; Tel.: Received: 18 Augut 18; Acceted: 9 October 18; Publihed: 3 October 18 Abtract: Deth dicriination i a key rocedure in acoutic detection or target claification for low-frequency underwater ource. Conventional deth-dicriination ethod ue a vertical line array, which ha diadvantage of oor obility due to the ize of the enor array. In thi aer, we rooe a deth-dicriination ethod for low-frequency ource uing a horizontal line array (HLA) of acoutic vector enor baed on ode extraction. Firt, we etablih linear equation related to the odal alitude baed on odal beaforing in the vector ode ace. Second, we olve the linear equation by introducing the total leat quare algorith and etiate odal alitude. Third, we elect the ower ercentage of the low-order ode a the deciion etric and contruct teting hyothee baed on the odal alitude etiation. Coared with a calar enor, a vector enor irove the deth dicriination, becaue the ode weight are ore aroriate for doing o. The reented linear equation and the olution algorith allow the ethod to aintain good erforance even uing a relatively hort HLA. The contructed teting hyothee are highly robut againt iatched environent. Note that the ethod i not aroriate for the winter tyical ound eed waveguide, becaue the characteritic of the ode differ fro thoe in downward-refracting ound eed waveguide. Robutne analyi and iulation reult validate the effectivene of the rooed ethod. Keyword: deth dicriination; horizontal line array; acoutic vector enor; ode extraction 1. Introduction In underwater acoutic, how to dicriinate between hallow ource and dee one continue to be an active reearch area and i known a the roble of deth dicriination or claification. Source deth dicriination ha any alication, including ubarine, arine biology, and unanned underwater vehicle [1]. There are two aroache to olving the roble of ource deth dicriination. The firt aroach i baed on the deth etiation. The traditional way to etiate ource deth i atched-field roceing, which require iultaneou earche for the ource range and deth uing the range-deth abiguity function. However, becaue of (i) the deendence of deth etiation on range etiation and (ii) the well-known environental iatch roble, etiating ource deth i generally not robut [ 4]. To addre thi iue, Yang [4] rooed atched-ode roceing (P) to etiate ource deth indeendently of ource range. Note Senor 18, 18, 396; doi:1.339/

2 Senor 18, 18, 396 of that oe ode are le enitive to the iatch environent, and the ource deth can be etiated by atching thoe ode with correonding relica ode. Therefore, P olve the environental iatch roble to a certain extent. To do o ore effectively, the data-baed atched-ode ource localization ethod wa rooed [5]. The ain advantage of that ethod i that it i otentially free of the environental iatch roble becaue the wavenuber and deth function are etiated fro data. However, the frequency of the oving ource ut be known a riori, which i difficult with aive onar yte. Next, unlike the firt aroach baed on deth etiation, the econd aroach conider deth dicriination a a binary hyothei tet; exale include the atched-ubace ethod [6,7], the cintillation index (SI) [8], and the traed energy ratio [9] to irove the robutne of deth dicriination. ot of the aforeentioned reearch wa baed on a vertical line array (VLA). However, we are intereted herein in a horizontal line array (HLA) of acoutic vector enor. Coared with a VLA, an HLA i ore convenient to be laced and i ore adatable to it environent. The ability to reolve ode i deterined by the effective horizontal aerture, which i the available horizontal aerture rojected along the ource aziuth. However, the required HLA i quite unrealitic in ot ractical cae, which ha led to earch for a new way to extract ode. Becaue the aerture of an HLA i liited, vector enor are conidered in ource deth dicriination. A vector enor eaure iortant vector coonent of the acoutic field, uch a the article velocity along three orthogonal axe (the x, y, and z axe), which cannot be eaured by a ingle calar reure enor [1,11]; the radial article velocity ued herein i a cobination of the x and y article velocitie. The vector inforation eaured by a vector enor can lead to better ource deth dicriination reult. Becaue it i difficult to etiate ource deth effectively with a relatively hort HLA, we intead conider the deth dicriination roble a a binary hyothei tet. Becaue it i difficult to odel the ound roagation accurately, the ethod ued hould be able to dicriinate ource deth even if the HLA i relatively hort. Therefore, we rooe herein a ethod for ource deth dicriination that ue an HLA of vector enor and i baed on ode extraction for ource with low-frequency line ectra. We begin by uing a noral ode odel to decribe the acoutic vector field. Next, we rooe a ode-extraction ethod baed on a TLS algorith. Finally, according to odal alitude etiation, we rooe deth-dicriination ethod baed on the ower ercentage (PP) of low order ode. The originality of the reent aer i reflected in two aect, naely (i) the dicriinator baed on vector inforation and (ii) the rooed ode-extraction ethod. Becaue the energy leakage of other ode i conidered beide that of the ode elected for extraction, the rooed ode-extraction ethod can reduce the required ize of array aerture. Therefore, the rooed ethod can dicriinate ource deth robutly with a relatively hort HLA. Thi aer i organized a follow. In Section, we uarize the related literature on deth dicriination, which i conidered a a binary hyothei tet. In Section 3, we rooe a ethod for extracting odal alitude that can be ued to dicriinate ource deth. In Section 4, we rooe a ethod for ource deth dicriination baed on the PP of low-order ode. In Section 5, we evaluate the robutne of the ethod in different cae. We reent a contrat exerient in Section 6, and we conclude the aer in Section 7.. Related Work Variou ethod have been rooed recently for ource deth dicriination roble, which i conidered a a binary hyothei tet, and we uarize that literature briefly in thi ection. The ethod can be divided into three grou baed on the tye of enor array, naely ource deth dicriination uing (i) a VLA, (ii) an HLA, and (iii) vector enor. In the firt grou, Preu et al. rooed a atched ubace ethod to deth dicriination [6,7]. They ued the difference in energy rojected in the ode ubace between the hallow ource and the dee one. Reult of ource deth dicriination with a dicrete VLA of liited aerture are deontrated in the SwellEX-96 exerient [7]. The SI can reflect the fluctuation of odal function, which i cloely related to ource deth. Preu et al. dicriinated ource deth baed

3 Senor 18, 18, of on the value of SI [8]. However, SI can be ued to dicriinate ource deth only if thoe range are ufficiently cloe, ince it i influenced by the ource-receiver range beide ource deth. Conidering the ill-oed roble of ode filtering when the water colun i not well-aled by the VLA, Conan et al. [9] rooed a robut ethod for dicriinating the ource deth uorted by reviou work introduced in [6 8]. When the VLA an only 5% of the water colun, that ethod outerfor thoe rooed in [6,7], a i evaluated by the receiver oerating characteritic curve. The ethod entioned above in thi aragrah are good at deth dicriination, but they require a VLA, oething that i difficult to ue with a obile latfor. In addition, it i difficult to to a VLA fro tilting, oething that degrade the deth dicriination. Coared with a VLA, an HLA i ore conveniently laced and ore adatable to it environent. However, the fact that an HLA cannot ale in the deth direction ake deth dicriination with an HLA a colex tak. Therefore, deth-dicriination ethod baed on an HLA uually undererfor thoe baed on a VLA. Yang obtained the wavenuber ectru fro the ynthetic aerture beaforing and the dicriinated deth fro the tructure of the wavenuber ectru [1]. However, that ethod can be ued only for cooerating ource, becaue it require (i) the frequency of the original ignal to be known a riori and (ii) a contant ource eed. Baed on the tructure of the wavenuber ectru, Reeder rooed an efficient aroach for clutter-deth dicriination [1], which i ued ainly for active onar. Siulation howed that thi technique could dicriinate between a clutter ource in the water colun and one on the eabed. Preu et al. achieved ource dicriination by uing inforation about the ource range, but it i difficult to obtain uch inforation accurately with aive onar [13]. The data reult of the ethod deontrated that the tet tatitic in [13] could dicriinate between (i) a ource towed at a deth of 1 and (ii) the naturally radiated ignature of two urface hi in the ae environent. Li earated the ode by uing the beaforing in ode ace [14], whereuon atched-ode roceing wa ued to etiate the ource deth. In iulation, that ethod erfored well when the HLA wa relatively long and the ignal-to-noie ratio (SNR) wa relatively high, but the erforance decreaed draatically with a horter HLA becaue of the low reciion of ode extraction; ultiately, that ethod failed to dicriinate the ource deth effectively. In thi aer, we dicu the cae of a aive ource and a relatively hort HLA, a cae to which none of the above ethod can be alied directly. In recent year, attention ha been aid to vector enor, which can obtain vector and calar inforation ynchronouly and can be ued for deth dicriination becaue of their acoutic intenity tructure and caacity to ure the iotroic noie. To dicriinate ource deth, Du ued the local angle of the interference triation directly fro the LOFAR diagra with the hel of two-dienional dicrete Fourier tranfor to dicriinate ource deth [15]. Baed on [15], Yang ued double vertical aive vector enor to irove the dicriination erforance [16], uing the active coonent of cro-ectru of reure and radial article velocity to dicriinate ource deth. However, although the ethod of both Du and Yang are caable of deth dicriination, they are baed on the Pekeri waveguide, the ue of which i iractical in an actual environent, eecially for tyical uer hallow water waveguide. 3. The ode Extraction ethod In thi ection, we focu on a ethod of ode extraction for an HLA of vector enor. In Section 3.1, we decribe the acoutic vector field uing the noral ode odel. In Section 3., we illutrate the odal beaforing in the vector ace. Baed on the odal beaforing, we etablih a relationhi between the odal alitude and the odal beaforing outut in the for of linear equation. In Section 3.3, we introduce the TLS algorith to olve thoe linear equation and thu obtain the odal alitude etiation. The odal alitude etiation will be ued to deciion etric, which are the core for deth dicriination.

4 Senor 18, 18, of 3.1. Acoutic Vector Field According to noral ode odel, the acoutic vector field can be exreed a a finite u of roagative ode. In the hallow waveguide, the ignal on vector enor n can be exreed a [17] j 4 rn jkrn e e ( r, z )= X ( z ) ( z ) n ( r, z ) (1) n r n r 8 1 krn j 4 rn jkrn e e v ( r, z ) X ( z ) ( z ) k n ( r, z ) () r n r r n r 8 1 krn where ( rn, z ) i the ound reure and vr ( rn, z ) i the radial article velocity. n ( rn, z r ) i the noie in the reure channel, and nr ( rn, z r ) i the noie in the radial-article-velocity channel. We aue that the noie ter in Equation (1) and () are indeendent and identically ditributed zero-ean colex circular Gauian rocee and are channel-indeendent. We alo aue that the noie i herically iotroic, oething that reviou aer generally aue [1,18]. z i the ource deth, z r i the deth of enor array, r n i the range between the ource and enor n, and i the water denity. i the angular frequency of ource, X i the ource colex alitude at, i the nuber of the roagative ode, i the attenuation coefficient, k i the horizontal wavenuber, and 3.. odal Beaforing in Vector Sace with HLA i the odal function of ode. A hown in Figure 1, the HLA corie of N enor; the ditance between two adjacent enor i d, and the aziuth of the ource i. Taking the enor cloet to the ource (Senor 1 in Figure 1) a the reference enor (with ditance r r 1 to the ource), the acoutic vector field i ale at range r r ( n 1) d in. (3) n k d in ri Figure 1. Phae difference noral ode i; i the aziuth of the ource. oreover, with conidering the far-field condition, the attention for geoetric and vicou can be neglected along the HLA. Therefore, we can obtain the aroxiation for Equation (1) and ():

5 Senor 18, 18, of r n jrn j r r in the denoinator and e e in the nuerator. The ound reure and radial article velocity aled by the HLA can be exreed a [11,14] T 1 where ( r, z ) (, ) N r z n (4) v n (5) r r r T v n n1( r, zr ) n N ( r, zr ) T r vr1( r, z) vr N ( r, z), n T r nr1( r, zr ) nr N ( r, zr ) n. ( r, z ),, and (, ) rn r z can be exreed a 1( r, z ) N ( r, z), T r r1 r, ( r, z ) (, ) N r z T, j 4 e e ( r, z ) X ( z ) ( z ) n r d ( z ) ( z ) e r jk ( n1) d in jk [ r( n1) d in ] r k r (6) j 4 e ( r, z ) X ( z ) ( z ) k rn r d ( z ) ( z ) e v r jk ( n1) d in e jk [ r( n1) d in ] r k r (7) where d and d v are the odal alitude of reure and radial article velocity, reectively, and can be exreed a j 4 r jkr e e d ( r, z) X z (8) 8 k r j 4 v 8 kr r jkr e e d ( r, z ) X k z. (9) Here, we review briefly the ethod rooed in [14], which extract the odal alitude by odal beaforing. Figure 1 how that the odal hae difference between different enor varie with the wavenuber, where i odal hae difference i. Baed on conventional beaforing, hae coenation i ued to achieve in-hae tacking of odal alitude of ecified order, while other coonent are offet ore or le [14]. When the hae baed on wavenuber i i coenated, only the order i ter i in-hae tacking of odal alitude, while the other are offet ore or le. Therefore, the odal alitude can be etiated by 1 d r z e N jki ( n1) d in i n(, ). (1) Ni ( zr ) n1 Note that the odal alitude can be etiated by odal beaforing with a relatively long HLA effectively [14]. How to etiate odal alitude with a relatively hort HLA will be illutrated a follow. We deduce the relationhi between odal alitude and beaforing

6 Senor 18, 18, of outut and extend the odal beaforing to the vector ace. Beaforing outut of ode i can be exreed a N 1 i b ( i) n( z) e N n1 jk ( n1) d in N j 4 jk in [ ( ) in ] r c k 1 ( ( ) in ) ( r ) 1 ki d N e j k ki d z z e X e nb ( i) k N r in c[ ( k ki ) d in ] (11) where 1 and n ( ) b d ( r, z ) f ( k ) n ( i) i b N in c[ ( k k ) d in ] 1 f k z e N 1 in c[ ( k ki ) d in ] i N 1 j ( k ki ) d in ( i ) ( r ), (1) i i the noie of odal-reure beaforing outut i. When the HLA i relatively long, Equation (1) can be aroxiated a f i ( zr ) i ( ki ). (13) i In thi cae, Equation (11) reduce to Equation (1), becaue excet the elected ode (ode i), the other coonent can be ignored. When the HLA i relatively hort, the aroxiation in Equation (13) no longer alie and therefore Equation (1) i a ecial iue of Equation (11). Siilarly, we have 1 b ( i) v ( z ) e d ( r, z ) f ( k ) n ( i) N jki ( n1) d in v r rn r v i bv N n1 1 (14) where n ( ) bv i i the noie in odal beaforing outut i of the radial article velocity. Here, we rewrite Equation (11) and (14) in atrix for a b d n F (15) b where d n b b (1)... b ( ) d ( r, z )... d ( r, z ) 1 n (1) n ( ) b b b T T b d n T F (16) vr vr bv b b (1)... b ( ) vr vr vr, T, v d r v1( r, z)... dv ( r, z) f1 k1 f k1, T n n (1) n ( ), and bv bv bv d, F f1 k f k T Fro Equation (15) and (16), if atrix F i reverible, the odal alitude can be etiated 1 1 directly a d F b and d F b. However, a reverible F deand a very long HLA, vr vr.,

7 Senor 18, 18, of and in ractice the HLA cannot be a long a we require. Therefore, a novel TLS-baed algorith will be introduced to olve d and d for a ingular or ill-conditioned F in Section Etiation of odal Alitude Baed on TLS v r According to Equation (15) and (16), ode extraction can be viewed a the roble of olving the linear equation, where F i the coefficient atrix, b and b are contant vector, and v r d and d are the unknown vector. Uually, a leat-quare algorith i ued to olve the roble v r when the contant atrix contain an error b. Such an algorith i effective when F i a robut atrix (with lower conditioning nuber). However, F ay be ill-conditioned (with higher conditioning nuber) in the cae under conideration herein, and the error F of the coefficient atrix ay degrade the olution reciion draatically. Therefore, conidering both b and F, we introduce a TLS algorith to olve the equation. The baic rincile of the reent TLS algorith i to olve the otiization roble a [] F, b, d in F, b F b. t. F F d b b for which we ue the aroach given by [] to calculate the etiation of d. The detailed te are a follow. B b, F. Uing ingular value decooition B i Firt, we decooe the atrix T B UΣV. decooed a Second, the nuber q of the ain ingular value i deterined. Setting a threhold, the ain ingular value i elected when the noralized ingular value i larger than, a i i, where i the axiu ain ingular value. Third, we calculate the etiation of d. Let V 1 corie colun q 1 to 1 of V a V1 vq, q,, +1 v + v +1, where v q+1 i colun q + 1 of V, v n i row n of V 1, and let V be the atrix, which i obtained by reoving v 1 fro V 1. The etiation of d i then calculated by V v H ˆ 1 H v1v1 d, where ˆd denote the etiation of d. 4. Source Deth Dicriination In thi ection, by analyzing the characteritic of odal deth function, we dicriinate the ource deth baed on odal alitude etiation. In Section 4.1, we briefly introduce the fraework of deth-dicriination ethod. In Section 4., we dicu the characteritic of the odal deth function in a tyical uer waveguide. Baed on the characteritic of odal deth function, we rooe PP of low-order ode, which can be ued to dicriination roble. In Section 4.3, we build the deciion etric and uarize the rocedure for the ource deth dicriination Fraework of Deth-Dicriination ethod Here, we briefly introduce two fraework of deth-dicriination aroache. In Figure, we divide deth-dicriination ethod into two fraework, naely one baed on the ource deth etiation (red dahed box on the left) and the other baed on binary teting hyothee (blue dahed box on the right). (17)

8 Senor 18, 18, of z ( ) R z z z z R( z ) R( z ) R( z) Figure. Procedural diagra for ource deth dicriination. Left (red dahed box): deth dicriination baed on deth etiation. Right (blue dahed box): deth dicriination baed on binary teting hyothee. For the fraework baed on the ource deth etiation, the deth dicriination can be detailed a follow. Uually thi fraework etiate ource deth firt, then erfor deth dicriination. After etiating the ource deth z, a deth threhold z (uually 5 1 ) i et to deterine whether a ource i hallow or dee. The ource dicriination roble can be addreed a [9] H z z : H z z. 1 : When H i true, the ource deth i aller than z and the ource i ditinguihed a a hallow one. By contrat, H 1 correond to a dee one. Equation (18) i the hyical bai for dicriinating ource deth baed on etiated ource deth. For the other baed on binary teting hyothee, the deth dicriination ethod are robut than the one baed on deth etiation. The baic idea of thi fraework i to find another hyical quantity related to z. In thi way, ource deth can be dicriinated without etiating ource deth z. We aue that the hyical quantity can be exreed a R( z ). Therefore, the ource-dicriination odel i written a [9] H : R( z ) H1 : R( z ) where i the deciion threhold. Herein, we rooe the deth-dicriination ethod baed on the econd fraework. In Equation (19), we ee that the core of the ethod for dicriinating ource deth i electing the deciion etric. In the next ub-ection, we chooe aroriate deciion etric by analyzing the characteritic of odal deth function and rooe PP of low-order ode to dicriinate ource deth. (18) (19)

9 Senor 18, 18, of 4.. Power Percentage of Low-Order ode We conider an exale of a tyical hallow water waveguide with a downward-refracting ound eed to analyze the characteritic of odal deth function. The water deth i 88. Figure 3a how the ound eed rofile (SSP) of a tyical hallow water waveguide, and the correonding odal deth function for ode 1 4 and are hown in Figure 3b. The low-order odal deth function uch a ode 1 4 reent nearly inuoidal function below the tranition oint and decay evanecently in alitude toward the urface above the tranition oint [6,7]. The tranition oint fro inuoidal to evanecent decay becoe hallower with the ode index [6,7]. For the high-order ode uch a ode hown in Figure 3b, the tranition oint i at the urface, and only function that are inuoidal in alitude are reent [6,7]. Thi henoenon of the traing of low order ode i fundaental to the rooed ethod. (a) (b) (c) (d) Figure 3. Characteritic of hallow water waveguide. (a) Sound eed rofile (SSP) of tyical uer hallow water waveguide. (b) Deth deendence of ode Function 1 4 and for tyical uer hallow water waveguide. (c) Deth deendence of ode Function 1 4 and for tyical winter hallow waveguide. (d) Noralized ode ower for tyical uer hallow water waveguide. Note that the henoenon decribed above ertain to the tyical uer waveguide but not to a tyical iovelocity winter waveguide. In the cae of the Pekeri odel, the ode function can be exreed a [17]

10 Senor 18, 18, of where the vertical wavenuber by c ( z) ( z) in( z) () can be obtained fro the dierion equation, which i given k ; the tudier have been dicu how k i obtained [1]. Fro Equation (), we ee that the ode function i the inuoidal in the water waveguide regardle of whether the ode i of low order or high order. Therefore, the henoenon decribed for a tyical uer waveguide doe not ertain to a tyical iovelocity winter waveguide, a hown in Figure 3c. The rooed ethod i ainly uitable for a tyical uer waveguide, rather than tyical winter waveguide becaue we ue the characteritic of the ode function. Figure 3d how the noralized ode ower for a tyical uer hallow water waveguide, which indicate the value of where P ax i the axiu value of ( z ( ) ). The noralized ode ower i defined a 1log z 1 Pax ( z ) with all oible value of and z. For hallow ource, the noralized ower of the low-order ode i relatively low and that of highorder ode i relatively high. For the dee ource, the noralized ower of the low-order ode and high-order ode ha an aroxiate agnitude. That i to ay, for hallow ource, the low-order ode contribute only a all ercentage of the ower aong all the ode, wherea for dee ource, the low-order ode contribute a oderate ercentage of ower of all ode. Therefore, we conider dicriinating the ource by uing the PP of the low-order ode. Given that the odal function cannot be obtained without knowing the deth, we ue the correlation between the odal alitude of reure and radial article velocity to calculate the ode ower, and the PP of low-order ode i given by where R( z ) * r d( r, z) dv ( r, z) e z 1 1 * r d( r, z) dv ( r, z) e z 1 1 R( z ) i defined a the PP of low-order ode. Fro Equation (1), we ee (1) R( z ) alo deend on the ource range r. The curve in Figure 4 decribe the characteritic of the PP. The black line rereent the PP of the low-order ode without attenuation, which i indeendent of the ource range, and the other three line rereent the PP of low-order ode with different ource range. For each line, the ower ercent increae draatically with ource deth below and then fluctuate lightly above. The deth threhold z i uually elected fro the range 5 15, and there i a one-to-one correondence between PP and ource deth in that range, allowing the correonding PP to be deterined directly. Herein, we take z = 1 (a hown in Figure 4). Becaue the ource range influence the PP lightly, we can deterine the PP threhold with auing ource range.,

11 Senor 18, 18, of Figure 4. Evolution of ower ercentage (PP) of low order ode with ource deth in hallow water waveguide for different ource range. The vertical line indicate the dicriination deth z choen in Section 5. In the rooed ethod, the radial article velocity i ued to cancel the ode weighting by the horizontal wavenuber. If we ue the reure inforation only, then only the odal alitude of reure can be obtained. The deciion etric in that cae i given by R r e * z d( r, z) d ( r, z) 1 1 k z * r d ( r, z) d ( r, z) e z 1 1 k. () Note that the atternuation ter increae with ode order, wherea k decreae. Thi ean that the low-order ode have all weight, which i not the cae in ractice. In addition, becaue the noie of the radial article velocity i indeendent of that of reure, the rooed dicriinator i better in ureing noie than the dicriinator in Equation (). Therefore, uing inforation about the radial article velocity in the dicriinator irove the ability to dicriinate ource deth. Herein, we focu on the unifor HLA of vector enor; if the array i ore colex, there ight be a better way to take advantage of the article velocity, an iue that we will be urued in future work Source Dicriation Baed on ode Extraction The reviou analyi lead u to etiate the PP of low-order ode a a deciion etric for ource deth dicriination. Uing the etiation of odal alitude dˆ ˆ ˆ [ d1,, d ] T and dˆ [ dˆ,, dˆ ] T by the ode-extraction ethod, v v1 v

12 Senor 18, 18, of ˆ ˆ* d r z dv r z 1 ˆ( ) ˆ ˆ* d r z dv r z 1 R z (, ) (, ) (, ) (, ) (3) where i the nuber of low-order ode. The low-order ode are defined a thoe with hae eed lower than the axiu ound eed in the water colun [9], and i deterined by c ( ) c, c ( 1) cax (4) ax where c ( ) i the hae eed of the ode of order, and ax c i the axiu ound eed. In thi cae, ode 1 9 are the low-order ode and the ret are the high-order ode. The dicriination i erfored by coaring the deciion etric to a deciion threhold, and the deciion threhold i et to R( z ). H : Rˆ ( z ) R( z ) H : Rˆ ( z ) R( z ) 1 where z i et 1. H denote the hallow ource, and H 1 denote the dee one. To facilitate undertanding of the rooed ethod, we uarize it diagraatically in Figure 5 and with the following decrition. The rooed ource-dicriination ethod baed on ode extraction corie the following te. Ste 1. We ileent beaforing in the vector ode ace and obtain the outut of ound reure b and radial article velocity b in Equation (11) and (14). Ste. The coefficient atrix F i contructed. Ste 3. We etiate the odal alitude ˆd and v r (5) ˆd v r of ound reure and vibration velocity in Equation (15) and (16) utilizing the TLS algorith. Ste 4. We deterine the threhold of low-order ode baed on Equation (4) and calculate the deciion etric according to Equation (1). Ste 5. We deterine the deth threhold with auing ource-receiver range. z and calculate the deciion threhold Ste 6. We erfor ource deth dicriination according to Equation (5). When R ˆ z the ource i a urface ource; otherwie, it i a dee one. R z,

13 Senor 18, 18, of Ste 1 Beaforing in vector ode ace i erfored b b vr The coefficient atrix i deterined The relationhi between odal beaforing outut and the odal alitude i contructed a follow b F d n b b F d n vr vr bv F Ste Ste 3 The odal alitude are etiated by TLS algorith d The deciion etric i calculated by ˆ ˆ * d ( r, z ) d v ( r, z ) ˆ 1 R z ˆ ˆ * d ( r, z ) d ( r, z ) No Shallow ource dv v 1 ˆR z R z ˆR z Ye Dee ource R z Ste 6 Ste 4 The nuber of low order ode i deterined The deciion threhold i deterined by electing the threhold of deth Ste 5 5. Robutne Analyi Figure 5. Diagra of ource deth dicriination baed on ode extraction. In thi ection, we evaluate the robutne of the ethod. The iulation environent i etablihed a follow. The SSP i hown in Figure 3a. We conider the botto with a ound eed of 165 /, a denity of 1.76 g/c 3, and an attenuation of.8 db /, where i the wavelength. The deth of the waveguide i 88, the HLA contain 1 enor, and the ditance between two adjacent enor atifie d. The ource i 5.1 k fro the HLA with an aziuth of the ource 6. The deth of the HLA i 5, wherea that of the ource i elected randoly in each iulation with the variation range 88. The ource frequency i 35 Hz and the SNR i db. Fro Equation (4), the SNR i defined a [9] SNR 1 log 1 E n. (6) The noie field can be exreed a a ueroition of roagating lane wave fro all oible direction [11]. To cobine the reure and radial article velocity, both hould be in the

14 Senor 18, 18, of ae unit [1]. The agnitude of reure and article velocity are related by / c v under the lane-wave aroxiation, where c i the ound eed [1]. Therefore, caling the radial article velocity by c allow u to define it in reure unit, which i the o-called reure-equivalent radial article velocity. Herein, we conider only the reure and radial article velocity. The ower of the noie in channel of reure and reure-equivalent radial article velocity hould be identical [11]. The robability of the correct dicriination (PCD) i et to K c be the criteria. The PCD i defined a PCD = 1% K, where K c i the nuber of correct dicriination and K i the total nuber of exerient. We ued 1 onte Carlo iulation (for each cae) to evaluate the robutne of the ethod [19] Influence of Nuber of the Senor and the SNR on Perforance The nuber of enor and the SNR are the two ain influence on the PCD of the rooed ethod. Theoretically having a large nuber of enor decreae the ingularity of atrix F and increae the etiation reciion of the odal alitude, while a high SNR enure that the received ignal i of high quality and irove the PCD. Figure 6a how the conditioning nuber of atrix F varie with the nuber of enor. Figure 6b how how the PCD varie with the nuber of enor for different SNR. Figure 7 how the odal alitude etiation reult. When the SNR exceed 1 db (include 1 db), the ethod can dicriinate the ource deth with ufficient nuber of enor. When the nuber of enor i aller than 1, the high conditioning nuber bring high error of odal alitude a hown in Figure 7a. The conditioning nuber decreae with the nuber of enor a hown in Figure 6a. Then, the odal alitude etiation reciion increae with the nuber of the enor. When the nuber of enor i larger than 1, the odal alitude etiation error i relatively all a hown in Figure 7b. It can be een that when the nuber of enor i 15, the odal alitude etiated are alot the ae a the real one. Increaing the nuber of enor ha little benefit to etiate odal alitude. Therefore, with fewer than 1 enor, the PCD rie draatically a the nuber of enor i increaed, tabilizing when there are ore than 1 enor. When the SNR i db, the ethod fail to dicriinate ource deth. The noie level i relatively high coared with the ower of low-order ode. In fact, the TLS-baed ethod olve the equation by iniizing the nor of atrix error and noie. The exceive noie level ake the otiization roble no longer valid. The odal alitude etiation ethod fail to work. Therefore, the PCD i around 5% in the cae of the SNR i db. (a) (b)

15 Senor 18, 18, of Figure 6. Conditioning nuber and erforance evolution with the nuber of enor for different SNR. The teted nuber of enor range fro 1 to, and SNR i, 1,, and 3 db. (a) Conditioning nuber of atrix F veru nuber of enor. (b) PCD veru nuber of enor. (a) (b) Figure 7. odal alitude etiated by TLS algorith with (a) 5 enor and (b) 15 enor. Deth of ource and HLA are 5. SNR i db. 5.. Influence of HLA Deth on Perforance According to Equation (1), f i related to the odal function z r. Therefore, the deth of the HLA can affect the ingularity of atrix F and influence the PCD. For a relatively long HLA, Equation (1) can be aroxiated by Equation (13), allowing atrix F to be exreed a F diag{ ( z ), ( z ) ( z )}. (7) 1 r r r Fro Equation (15), the error of the reure odal alitude i the etiator bia where nn B( ) { ˆ } d E d d. The covariance atrix of dˆ d F b d and 1 ˆd i K E{ dˆ dˆ } E{ dˆ } E{ dˆ } = F K ( F ) (8) H H 1 1 H nn K i the atial covariance of the noie vector and n n n 1 zr zr zr K I, nn n n i the ower of the noie []. According to Equation (7) and (8), the covariance atrix i K diag{, }. When the HLA i located near the water urface, the ower ( ) ( ) ( ) of the low-order ode i quite all and thu give a very large error covariance. Thi doe uch to exlain why the erforance of the algorith i not good when the HLA i near the water urface. For a hort HLA, we ue the conditioning nuber of F to analyze the erforance. Figure 8a how how the conditioning nuber of F varie with HLA deth. The PCD i lotted againt deth in Figure 8b. The conditioning nuber ha an effect on the odal alitude etiation and PCD. When the HLA i located near the urface, the conditioning nuber i high and the etiation of low-order odal alitude i alot zero, a hown in Figure 9a. Thi ean that the etiated value of PP i aller than the real one. Therefore, a large nuber of dee ource are dicriinated a hallow one. In that cae, the PCD i low. Figure 8a how the conditioning nuber decreae draatically with HLA deth when the deth i le than 1 and tabilized once the deth exceed 1. The accuracy of the odal alitude etiation increae with the decreaing conditioning nuber. When the HLA deth exceed 1, the odal alitude etiation ha a

16 Senor 18, 18, of higher accuracy a hown in Figure 9b and the etiated value of PP i cloe to the real one. Therefore, the correonding PCD increae with the HLA deth when the deth i le than 1. Deite ocillating, the PCD retain relatively high if the HLA i dee enough. For the iulation environent dicued in thi ection, an HLA deth of the 1 enure a high PCD (~9%). (a) (b) Figure 8. Conditioning nuber and PCD influenced by the deth of HLA. (a) Conditioning nuber of atrix F veru HLA deth. (b) PCD veru HLA deth. (a) (b) Figure 9. odal alitude etiated by the TLS algorith, when HLA deth are (a) 5 and (b) 5. SNR i db Influence of Source Aziuth and the Aziuth Etiation Error Perforance The ource aziuth and HLA aerture deterine the effective array aerture. When the ource i located at the end-fire direction of HLA, the effective array aerture achieve it axiu value, thereby benefiting odal beaforing. When the ource i located at the abea direction of the HLA, the bet ource aziuth etiation i obtained. Fro Equation (1), the ode earation i related to ( k k ) d in. When the ource i located at the end-fire direction of HLA, ( k k ) d in achieve it axiu value without conidering the error of the ource aziuth etiation. In that cae, the conditioning nuber of atrix F i low and the PCD i high. Otherwie, when the ource i located at the abea direction of HLA, ( k k ) d in i zero. atrix F i coletely noninvertible, and the ethod fail. i i i

17 Senor 18, 18, of Figure 1a how the conditioning nuber of atrix F for ource with different aziuth, and Figure 1b how the PCD for different aziuth. The conditioning nuber decline draatically for aziuth below 3 and then tabilize, o the PCD rie draatically for aziuth below 3 and then tabilize. In fact, we can exlain the roble fro the oint of the effective array aerture. When the ditance between two adjacent enor i ket, the ource aziuth and the nuber of enor have a iilar effect on the effective aerture. Therefore, the effect of the two on the erforance of the rooed ethod i iilar. (a) (b) Figure 1. The conditioning nuber and the erforance evolution with the ource aziuth. (a) The conditioning nuber of atrix F veru the ource aziuth. (b) PCD veru the ource aziuth. For a ource aziuth of 6 degree, we conider ource aziuth etiation error of 3 degree. Figure 11 how how the PCD varie with the ource aziuth etiation error in that range, fro which the PCD clearly decline with the ource aziuth etiation error. When the etiation error exceed 1, the PCD i below 9%. Therefore, the ethod require the aziuth to be etiated with high reciion, which can be done by increaing the nuber of enor. Figure 11. Perforance evolution with the aziuth etiation error. The ource aziuth etiation range fro to 3 degree.

18 Senor 18, 18, of 5.4. Influence of the Frequency of Source Signal on Perforance The frequency of the ource ignal influence the erforance of the ethod for ource deth dicriination. The nuber of low-order ode and all roagative ode are given in Figure 1a. (a) (b) Figure 1. Nuber of ode and erforance evolution with the ource frequency. (a) Nuber of all roagative ode (blue) and low-order ode (red) veru frequency. (b) PCD veru the frequency of the ource ignal. The effect of ource frequency on PCD i illutrated in Figure 1b. Thi how the PCD deend on the ource frequency. A exected, the ethod i the ot efficient at higher frequencie. Note that, at low frequency, the ethod ha a relatively low PCD. Indeed, the all nuber of involved ode caue the ethod to lack effective inforation. For exale, for a frequency of 5 Hz, the nuber of low-order ode and all roagative ode are one and three, reectively. The rooed ethod ue the PP of the ode. Becaue there i only one low-order ode, the PP i highly influenced by the erforance of etiating thi ode, thereby decreaing the robutne of the rooed ethod. Therefore, a hown in Figure 1b, the PCD i relatively low for frequencie below 1 Hz Perforance in a iatched Environent In an actual ituation, error exit in araeter uch a the SSP and ound eed, denity, and attenuation of the botto becaue of inaccurate eaureent. Therefore, it i neceary to analyze the erforance of the ethod in a iatched environent. In thi ub-ection, the reviou environental odel entioned before i ued to iulate data. Suoing a ractical where thi odel i not available, the dicriinator i built with a different environental odel (iatch odel). Therefore, the dicriinator doen t atch with data. We conider two environent tye: with iatched SSP in the water colun and with iatched botto araeter. In the firt one, we focu on the cae of iatched SSP without any iatch in botto araeter. Figure 13 how the SSP of Cae 1 3, where Cae 1 involve the real SSP (no iatch) and Cae and 3 are relaced by the real one adding SSP error [1]. The error of SSP i a linear function related to deth. In Cae, the error in SSP are /, which ean the error in SSP at the urface i 5.8 / and at the botto i 11.6 /. The error in SSP are / in Cae 3. Given that the three cae involve an offet in the SSP only, there i no ajor iact on odal tructure. Adding a rando erturbation to the SSP deth by deth, the additive erturbation i odeled a a zero-ean Gauian roce. The tandard deviation of erturbation in Cae 4 6 are 3, 5, and 1 /, reectively. In the econd one, the real SSP in the water colun i ket, wherea the botto araeter are relaced by the iatched one.

19 Senor 18, 18, of Figure 13. SSP for different iatched environent. Cae 1 involve the actual SSP, Cae and 3 involve iatched SSP. The error in the ound eed are / in Cae, and 11.6 / 3. / in Cae 3. Table 1 lit the effect of environental iatch on the erforance of dicriinator, which include two environent tye. Fro Table 1, we ee that the erforance deteriorate with the SSP error. The PCD decreae to 88.9% when the aued SSP i in Cae, which i till accetable for deth dicriination. However, the PCD decreae further to 58.3% when the aued SSP i in Cae 3, cauing the ethod to loe efficacy. The PCD are 9.8% and 9.7% in Cae 4 and 5, reectively. Becaue of the large tandard deviation of the erturbation, the odal tructure change with the increaing erturbation and the PCD decreae to 84.9%. Beide the iatched SSP, the ethod aintain a high PCD for iatched ound eed, iatched denity, and iatched attenuation of the botto. The iatch of botto araeter ha little effect on the nuber of ode and the odal tructure. On the whole, the rooed ethod can tolerate an SSP error to oe extent and i highly robut againt iatched environent. Table 1. Probability of correct dicriination (PCD) in different iatched environent. SSP Error Error in Sound Seed of Botto ( / ) Error in Denity of Botto (g/c 3 ) Error in Attenuation of Botto ( db/λ ) PCD Cae % Cae 88.9% Cae % Cae 4 9.8% Cae 5 9.7% Cae % Cae % Cae % Cae % Cae % Cae % Cae %

20 Senor 18, 18, 396 of 6. Contrat Exerient In thi ection, we coare the erforance of the rooed ethod with thoe of two other exiting deth-dicriination ethod, naely the ode ubace rojection (SP) ethod [13] and the odal doain beaforing (DB) ethod [14]. The iulation environent are the ae a that in Section 5. The exerient i ileented with different ize of HLA and in different iatched environent, a decribed in Table. We conider fewer than 1 HLA enor and a hort HLA. Exerient 1 and involve a hort HLA. Exerient 3 and 5 involve iatched environent. Exerient 4 involve a long HLA and no iatched environent. Table. Exeriental environent. Exerient Nuber of Senor SSP Error 1 5 Cae 1 1 Cae Cae 4 5 Cae Cae The exeriental reult are lited in Table 3. For a long HLA and a atched environent (Exerient 4), all three coared ethod achieved high-reciion deth dicriination, but the PCD of the rooed ethod wa lightly lower than that of the SP ethod, becaue the ource range inforation wa ued in the SP ethod. With a horter HLA (correonding to Exerient 1 and ), the erforance deteriorated but the rooed ethod aintained the lowet degree of deterioration aong the ethod. For the HLA with 5 enor, the PCD of the SP ethod decreaed to 8%, the DB ethod failed, and the PCD of the rooed ethod reained above 9%. Becaue the rooed ode extraction ethod i uitable for a hort HLA. With iatched environent (Exerient 3 and 5), the PCD of the SP and DB ethod decreaed by around 11%, wherea that of the rooed ethod decreaed by around only 6%, indicating that our ethod wa one that wa ot robut againt a iatched environent. Overall, the rooed ethod uraed the other two ethod for a hort HLA or a iatched environent. Table 3. Exeriental reult. Exerient PCD of SP ethod PCD of DB ethod PCD of Prooed ethod 1 8.3% 5.% 9.3% 9.7% 73.% 95.3% % 65.5% 88.9% % 9.6% 98.1% % 85.% 97.6% 7. Concluion In thi aer, we addreed the roble of deth dicriination for low-frequency ource uing an HLA of vector enor. Firt, we derived an exreion for beaforing in the vector ode ace and etablihed linear equation related to the odal alitude. We then olved linear equation by introducing a TLS algorith, thereby etiating the odal alitude in high reciion. Finally, we dicriinated the ource deth baed on the extracted odal alitude. To evaluate the erforance of the rooed ethod, we erfored robutne analyi and contrat exerient. The analyi howed that the rooed ethod can achieve a high PCD with ore than 1 enor, an HLA deth in exce of 1, a ource aziuth in exce of 3 degree, a ource aziuth etiation error le than 1 degree, and a frequency above 1 Hz. oreover, it indicated that the rooed ethod i highly robut againt a iatched environent. In the contrat exerient, we coared (i) the rooed ethod, which coute the PP in the vector ode ace and extract the ode by ean of the TLS algorith, with two other ethod, naely (ii) SP, which ue rojection and dicriinate deth in enor ace, and (iii) DB, which

21 Senor 18, 18, of dicriinate ource deth baed on P. The reult of contrat exerient howed that the rooed ethod erfor better for a hort HLA and iatched SSP coared with SP and DB. Thi i becaue (i) working in the ode ace ake the etric le enitive to any iatch and (ii) the ode extraction ethod ued herein i ore uitable to the cae of a hort HLA. In the reent work, the ethod wa baed on a range-indeendent environent. Therefore, the feaibility of the ethod hould be aeed for a range-deendent environent, which i cloer to the actual SSP environent. We alo intend to invetigate how to reduce the deendence on rior knowledge and how to ue the vertical article velocity effectively. Author Contribution: Concetualization, G.L. and Y.Z.; Foral analyi, Y.Z.; Funding acquiition, G.L.; ethodology, Y.Z.; Project adinitration, G.Z.; Reource, J.F.; Suerviion, J.F.; Writing original draft, Y.Z.; Writing review & editing, G.L., Y.Z., G.Z., and C.Z. Funding: Thi work i uorted by the National Key Reearch and Develoent Plan (17YFC369), the National Natural Science Foundation of China (115464), and the Technology of Baic Scientific Reearch Project (JSJL1664B3). Conflict of Interet: The author declare no conflict of interet. Reference 1. Yang, T.C. Source deth etiation baed on ynthetic aerture beafoing for a oving ource. J. Acout. Soc. A. 15, 138, , doi: / Baggeroer, A.B.; Kueran, W.A.; ikhalevky, P.N. An overview of atched field ethod in ocean acoutic. IEEE J. Ocean. Eng. 1993, 18, 41 44, doi:1.119/ Wang, Q.; Wang, Y..; Zhu, G. L. atched field roceing baed on leat quare with a all aerture hydrohone array. Senor 17, 17, 71, doi:1.339/ Yang, T.C. A ethod of range and deth etiation by odal decooition. J. Acout. Soc. A. 1987, 8, , doi:1.111/ Yang, T.C. Data-baed atched-ode ource localization for a oving ource. J. Acout. Soc. A. 14, 135, , doi:1.111/ Preu, V.E.; Backan, D. A atched ubace aroach to deth dicriination in a hallow water waveguide. In Proceeding of the Conference Record of the Forty-Firt Ailoar Conference on Signal, Syte and Couter, Pacfilc Grove, CA, USA, 4 7 Noveber Preu, V.E.; Ward, J.; Richond, C.D. ode filtering aroache to acoutic ource deth dicriination. In Proceeding of the Conference Record of the Thirty-Eighth Ailoar Conference on Signal, Syte and Couter, Pacfilc Grove, CA, USA, 7 1 Noveber Preu, V.E.; odal cintillation index: A hyic-baed tatitic for acoutic ource deth dicriination. J. Acout. Soc. A. 1999, 15, 17 18, doi:1.111/ Conan, E.; Bonnel, J.; Chonavel, T.; Nicola, B. Source deth dicriination with a vertical line array. J. Acout. Soc. A. 16, 14, EL434, doi:1.111/ Paulo, F.; Paulo, S.; Sérgio,.J. Acoutic reure and article velocity for atial filtering of botto arrival. IEEE J. Ocean. Eng. 18, 1 14, doi:.119/joe alcol, H.; Arye, N. Acoutic vector-enor correlation in abient noie. IEEE J. Ocean. Eng. 1, 6, , doi:1.119/ Reeder, D.B. Clutter deth dicriination uing the wavenuber ectru. J. Acout. Soc. A. 14, 135, EL1 EL7, doi:1.111/ Preu, V.E.; Helfrick,.N. Ue of ode ubace rojection for deth dicriination with a horizontal line array: Theory and exeriental reult. J. Acout. Soc. A. 13, 133, , doi: / Li, P.; Zhang, X.H.; Fu, L.F.; Zeng, X.X. A odal doain beaforing aroach for deth etiation by a horizontal array. Acta Phy. Sin. 17, 66, 3 34, doi:1.7498/a Du, J.Y.; Zheng, Y.; Wang, Z.Q.; Cui, H.; Liu, Z.W. Paive acoutic ource deth dicriination with two hydrohone in hallow water. In Proceeding of the OCEANS, Shanghai, China, 1 13 Aril 16.

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