Tracking Time-Varying Multipath Channels forActive Sonar Applications

T rac king Time-V arying Multipath Channels for A ctiv e Sonar Applications Ash w ani K oul † , Gustaf Hendeby † , Isaac Skog ‡ † Division of A utomatic Con trol, Link öping Univ ersity , Link öping, Sw eden {ash w ani.k oul, gustaf.hendeb y}@liu.se ‡ Division of Comm unication Systems, KTH Ro y al Institute of T ec hnology , Stockholm, Sw eden, and F OI-Sw edish Defence Researc h Agency , Stockholm, Sw eden sk og@kth.se Abstract—Reliable detection and tracking in active sonar require accurate and ecient learning of the acoustic m ul- tipath background environmen t. Conv entionally , background learning is p erformed after transforming measuremen ts in to the range–Doppler domain, a step that is computationally exp ensiv e and can obscure phase-coheren t structure useful for monitoring and tracking. This pap er prop oses a framew ork for learning and trac king the multipath background directly in the raw measurement domain. Starting from a wideband Doppler linearization of the impulse resp onse of a time-v arying m ultipath channel, a state-space mo del with a heteroscedastic measuremen t equation is derived. This mo del enables channel trac king using an extended Kalman lter (EKF), and unknown mo del parameters are learned from the marginalized lik eliho o d. The statistical adequacy of the prop osed mo dels is assessed via a p -v alue signicance test. Finally , this pap er in tegrates the learned c hannel model into a sequential likelihoo d-ratio test for target detection. BELLHOP-based simulations show that the prop osed mo del better captures channel dynamics induced b y sea-surface uctuations and transmitter and receiver drift, yielding more reliable detection in time-v arying shallow-w ater en vironments. Index T erms—extended Kalman lter (EKF), heteroscedas- tic noise, signicance test, sequential lik eliho o d ratio test (SLR T), underwater acoustics. I. Introduction A ccurate mo deling of the m ultipath background is essen tial for reliable active-sonar detection and track- ing. In shallo w w aters, rep eated surface and b ottom in teractions pro duce strong rev erb eration and coherent m ultipath propagation whose statistics v ary ov er time due to surface dynamics and platform drift. This nonstation- arit y makes estimating multipath bac kground dicult, increasing false-alarm rates, and masking weak target re- turns. This, in turn, makes reliable detection and trac king dicult in lo w signal-to-noise ratio (SNR) regimes [1], [2]. Classical sonar pip elines map the received time series to range-Doppler via matc hed ltering (often with Doppler lter banks) and apply constant false alarm rate (CF AR)- t yp e thresholding [3]–[5]. Their performance relies on the in terference statistics b eing approximately stationary o v er the CF AR training windo w, an assumption that can This w ork was partially supp orted by the W allen berg AI, A u- tonomous Systems and Softw are Program (W ASP) funded by the Knut and Alice W allen berg F oundation. T ransmitter Node Receiver Node T arget Fig. 1: Illustration of the considered bistatic sonar scenario with a time-v arying multipath channel. break down under time-v arying, rev erb eration-dominated m ultipath, leading to threshold mismatc h and degraded detection. F or bistatic sonar, as illustrated in Fig. 1, the structured m ultipath background can evolv e due to sensor-no de drift and surface motion [2], [6]. Under suc h conditions, con v en tional range-Doppler pro cessing requires frequent matc hed ltering, whic h causes a computational burden. In addition, Doppler mismatc h and windowing can spread coheren t multipath energy across range-Doppler bins, making background evolution harder to mo del. These eects motiv ate op erating directly on raw measuremen ts and explicitly tracking the time-v arying m ultipath back- ground. Instead of op erating in the range-Doppler domain, sev eral approaches work directly with raw sensor mea- suremen ts, enabling lik eliho o d-based formulations such as track-before-detect, and b earing-only tracking using the maximum a posteriori (MAP) method. Op erating directly with raw measuremen ts has sho wn improv ed p erformance in lo w-SNR regimes [7], [8]. Moreov er, raw- domain pro cessing retains the phase-coherent structure of the receiv ed eld. In [9], w eak targets are detected by tracking the time- v arying channel impulse response (CIR) using a blo ck- up dated sparse estimator. Ev en though suc h mo dels ac hiev e go o d performance, in high-rev erb eran t environ- men ts with no structure and sparsity , mo del mismatch will limit p erformance. It can fav or faster, non-structured adaptiv e up dates, resulting in fewer target detections. In this pap er, a dierent approac h for target detection is prop osed. Instead of estimating the CIR and then com- puting the residuals for target detection, the time-v arying m ultipath channel is trac k ed using an extended Kalman Filter (EKF). Using a wideband Doppler linearization appro ximation, the m ultipath background is mo delled directly using the ra w measuremen ts. With this, a detector is designed for weak target detection based on the changes in the marginal likelihoo d of the measuremen ts [2], [10]. T racking the multipath channel is done by mo deling the time-v arying components using a heteroscedastic measure- men t model for the sensor measurements. Dierent models are prop osed, and the signicance of eac h of the mo dels is ascertained b y a p -v alue signicance test. Hence, the resulting framew ork enables sim ultaneous trac king of the time-v arying m ultipath bac kground and detection of a target. T o that end, the main contributions of this paper are: • A ra w-measuremen t-domain bac kground mo del for activ e sonar in which Doppler eects are captured through a physically in terpretable, state-dep endent co v ariance structure. • A EKF-based framework for trac king the m ultipath c hannel, and learning h yperparameters in the pro- p osed channel mo del. • A sim ulation-based ev aluation demonstrating the sta- tistical adequacy of the prop osed background mo del via p -v alue signicance testing, and its impact on detection p erformance in time-v arying environmen ts. Repro ducible research: The co de and datasets used to generate the results in this pap er are av ailable at https://github.com/ASHKoul/Bcg_SLRT. I I. Signal Mo del Consider the bistatic setup in Fig. 1. The receiv ed data consist of a time-v arying multipath bac kground and a p ossible target return. T o enable likelihoo d-based target detection and trac king, the mo deling of the multipath bac kground, its statistical prop erties, and its time evo- lution has to b e p erformed. A. General Contin uous Time Channel Mo del Let s ( t ) denote the transmitted wa veform. A standard con tin uous-time m ultipath model for the received signal, in a noise-free en vironmen t, is x ( t ) = N p  i =1 a i ( t ) s  t − τ i ( t )  , t ∈ [0 , t f ] , (1) where τ i ( t ) and a i ( t ) are the time-v arying dela y and amplitude of the i th arriv al, N p is the n um b er of signican t arriv als. In shallow w ater, these arriv als include the direct path and strong surface and bottom interactions, yielding a reverberation-dominated multipath background in addition to the target returns. Assuming the channel v aries slowly ov er the time duration t f , the multipath amplitude and delay can b e appro ximated as a i ( t ) ≈ a i , τ i ( t ) ≈ τ io + η i t, t ∈ [0 , t f ] , (2) where τ io is the initial dela y and η i is the dela y-rate. Then t − τ i ( t ) ≈ (1 − η i ) t − τ io = β i ( t − ¯ τ i ) , (3) with β i ≜ 1 − η i and ¯ τ i ≜ τ io β i . (4) Substituting (4) in to (1), x ( t ) can b e approximated as x ( t ) ≈ N p  i =1 a i s  β i ( t − ¯ τ i )  . (5) T o separate the p otential target components from the m ultipath bac kground, (5) can b e split as x ( t ) = x b ( t ) + x o ( t ) , (6a) where x b ( t ) ≜ N b  i =1 a i s  β i ( t − ¯ τ i )  (6b) denotes the m ultipath bac kground, and x o ( t ) ≜ N p  i = N b +1 a i s ( β i ( t − ¯ τ i )) (6c) is the multipath structure formed due to the target- reected signal. Here, for notational conv enience, the arriv als are ordered such that N b and N o arriv als are attributed to bac kground and target-induced comp onen ts, resp ectiv ely , with N p = N b + N o . Since the target amplitudes are muc h smaller relative to the multipath bac kground, the target multipath is appro ximated by its dominan t target arriv al as x o ( t ; ¯ τ o , β o ) ≈ a o s ( β o ( t − ¯ τ o )) . (7) B. Wideband Linearized Mo del The m ultipath bac kground can be further approximated using a rst-order wideband Doppler linearization (see App endix A), x b ( t ) ≈ N b  i =1  a i s ( t − ¯ τ i ) + a i r i ( t − ¯ τ i ) ˙ s ( t − ¯ τ i )  , (8a) = N b  i =1  a i s ( t − ¯ τ i ) + a i r i u ( t − ¯ τ i )  , (8b) where r i ≜ ln β i and u ( t ) ≜ t ˙ s ( t ) . This approximation is accurate when β ≈ 1 . Equation (8a) is bi-linear in the unkno wn parameters a i and r i and can b e interpreted as a linear conv olutional mo del driven by the tw o known w a v eforms s ( t ) and u ( t ) . This simplies estimation and trac king of the time-v arying multipath background. C. Discrete-Time Linearized Channel Mo del Dening the sampled transmitted w av eform and the cor- resp onding time deriv ative comp onent with the sampling p eriod ∆ t as s [ n ] ≜ s ( n ∆ t ) and u [ n ] ≜ n ∆ t ˙ s ( n ∆ t ) . (9) With this, the zero-padded T oeplitz matrices S and U of size N × N l can b e dened as [ S ] n,l ≜ s [ n − l ] and [ U ] n,l ≜ u [ n − l ] , (10) where n = 0 , · · · , N − 1 , l = 0 , · · · , N l − 1 , N ≜ ⌈ t f / ∆ t ⌉ , and N l is the length of the sampled de- la y grid. Giv en these quantities and dening  x b ≜  x b (0) · · · x b (( N − 1)∆ t )  ⊤ , a discrete time approxi- mation in v ector form of the time contin uous mo del (8a) is giv en b y  x b ≈ S a + U Λ a r , (11) where a =  a 1 · · · a N l  ⊤ and r =  r 1 · · · r N l  ⊤ are N l ( N l ≫ N p ) length vectors collecting the multipath amplitudes and log time-scaling parameters, resp ectively . F urther, Λ a ≜ diag( a ) . D. Sto chastic Multipath Background Mo del The environmen tal uctuations captured by the time scaling parameter r can b e decomp osed into r ≜ c + d, (12) where c and d mo del the Doppler eect unique to each signal path and the common Doppler eects, resp ectiv ely . Assume that c and d are zero-mean random v ariables with the distribution c ∼ N (0 , σ 2 c I ) , and d ∼ N (0 , σ 2 d 11 ⊤ ) , (13) where 1 = [1 · · · 1] ⊤ . Then the measurements, condi- tioned on the amplitude v ector a , can b e mo deled as y b = S a + U Λ a r + e, e ∼ N (0 , σ 2 e I ) , (14) where σ 2 e denotes the pow er of the ambien t noise, y b is the collection of discrete-time measurements when the target is absen t, and y b | a ∼ N ( S a, R ( a )) , R ( a ) ≜ σ 2 e I + Σ b ( a ) , (15) with Σ b ( a ) = Σ c ( a ) + Σ d ( a ) . (16) Here, Σ c ( a ) ≜ σ 2 c U Λ a Λ ⊤ a U ⊤ , and Σ d ( a ) ≜ σ 2 d U aa ⊤ U ⊤ . (17) are the heteroscedastic cov ariance matrices, and the mea- suremen t mo del b ecomes heteroscedastic because R ( a ) dep ends on the unknown background amplitudes a . E. State-Space Mo del Due to surface dynamics and sensor no des’ drift, the m ultipath amplitudes v ary from ping to ping. Moreo ver, in shallow water, the multipath background is typically smo oth or diused in delay , and estimating a full N l - tap vector a b ecomes computationally exp ensiv e when N l ≫ N p . Therefore, a low-dimensional parameterization is imp osed by dening a ≜ B θ , (18) where θ is a length- M vector of unkno wn basis w eigh ts and B is a xed N l × M basis dictionary . The dictionary B is constructed using Gaussian basis functions, [ B ] l,m ≜ exp  − ( l ∆ t − µ m ) 2 2 σ 2 m  , (19) where µ m and σ m is the center and the length scale of the m th basis, resp ectively . The cen ters µ m are placed uniformly ov er the delay grid, whereas the length scale is set σ m ≈ 0 . 42 /B W , with B W as the bandwidth of the transmitted signal. Substituting (18) in to (15) giv es the measuremen t mo del y b | θ ∼ N  H θ , R ( θ )  , H ≜ S B , (20) where R ( θ ) can b e computed b y substituting a = B θ in (15). With this, let θ k and y b,k denote the basis co ecien t and the measuremen t vector at ping k , resp ectiv ely . The ev olution of the basis co ecien ts across pings is mo deled as a random-w alk process. The resulting state-space mo del under the bac kground-only h yp othesis is given as θ k +1 = θ k + w k , w k ∼ N (0 , σ 2 q I ) , (21a) y b,k = H θ k + v k , v k ∼ N (0 , R ( θ k )) , (21b) where w k and v k are mutually indep endent, and v k cap- tures b oth am bien t noise and state-dep enden t bac kground uctuations with R ( θ k ) = σ 2 e I + Σ b ( θ k ) . I II. Parameter Estimation The state-space mo del in (21) enables sequential esti- mation of the latent m ultipath bac kground state θ k , whic h describ es the multipath channel. A. Amplitude T rac king The time-v arying multipath channel amplitudes a are estimated by tracking the lo w-dimensional co ecient v ector θ k . The tracking of θ k is performed using an EKF, as outlined in Alg. 1. The measuremen t cov ariance R ( θ k ) is state-dependent due to the heteroscedastic noise mo del. T o obtain a tractable up date, a locally constan t appro ximation of the cov ariance is used by ev aluating the co v ariance at the predicted latent state ˆ θ k | k − 1 , i.e., R k ≜ R ( ˆ θ k | k − 1 ) . (22) Algorithm 1: EKF for estimating the amplitude θ and calculating the marginal lik eliho od. Input : Measuremen ts { y k } N k k =1 , measuremen t matrix H , h yp erparameter set Θ , mo del M . Initialize: ( ˆ θ 0 | 0 , P 0 | 0 ) // F or mo del M 1 for k = 1 , . . . , N k do // Time up date 2 ˆ θ k | k − 1 ← ˆ θ k − 1 | k − 1 3 P k | k − 1 ← P k − 1 | k − 1 + σ 2 q I // State-dep enden t cov ariance 4 R k ← R M  ˆ θ k | k − 1  // Measuremen t up date 5 ν k ← y k − H ˆ θ k | k − 1 6 Σ k ← H P k | k − 1 H ⊤ + R k 7 K k ← P k | k − 1 H ⊤ Σ − 1 k 8 ˆ θ k | k ← ˆ θ k | k − 1 + K k ν k 9 P k | k ← ( I − K k H ) P k | k − 1 // Log-lik eliho od increment 10 ℓ k ( y k ; Θ) ≜ log p ( y k | y 1: k − 1 ; Θ) 11 ≈ − 1 2  ν ⊤ k Σ − 1 k ν k + log | Σ k |  12 L ( Y N k ; Θ) =  N k k =1 ℓ k ( y k ; Θ) Output : p osterior estimates { ˆ θ k | k , P k | k } N k k =1 , and log marginal lik eliho od L ( Y N k ; Θ) . B. Hyp erparameter Learning Not all the parameters, such as { σ q , σ c , σ d } , in the mo del (21) ma y b e known before hand. These parameters, here denoted by Θ , can b e learned from the data by maximizing the log marginal lik eliho od [11]. That is ˆ Θ = arg max Θ L ( Y b,N k ; Θ) , (23) where Y b,N k = { y b,k } N k k =1 . Here, the approximation of L ( Y b,N k ; Θ) can b e calculated via the EKF; see Alg. 1. IV. Ev aluation Metho dology A cen tral question is whether explicitly mo deling time- v arying background uctuations yields a statistically sig- nican t improv emen t ov er an ambien t noise-only baseline, i.e., σ c = σ d = 0 , and whether this translates in to more reliable target detection. T o that end, mo del signicance and target detection capabilit y are ev aluated using the follo wing four co v ariance structures M 0 : R M 0 ( θ k ) = σ 2 e I , (24a) M c : R M c ( θ k ) = σ 2 e I + Σ c ( θ k ) , (24b) M d : R M d ( θ k ) = σ 2 e I + Σ d ( θ k ) , (24c) M cd : R M cd ( θ k ) = σ 2 e I + Σ c ( θ k ) + Σ d ( θ k ) , (24d) A. P -v alue Signicance T est Let M ∈ {M 0 , M c , M d , M cd } index the candidate co v ariance mo del for R ( θ k ) in (21b), and y b,k = H θ k + v M ,k , v M ,k ∼ N (0 , R M ( θ k )) . (25) Then the log likelihoo d-ratio statistic for eac h extended mo del M j , j ∈ { c, d, cd } relative to M 0 is T j ( Y b,N k ) ≜ log max Θ j p ( Y b,N k ; Θ j , M j ) max Θ 0 p ( Y b,N k ; Θ 0 , M 0 ) (26a) = L ( Y b,N k ; ˆ Θ j , M j ) − L ( Y b,N k ; ˆ Θ 0 , M 0 ) . (26b) The h yp erparameter set for each mo del is Θ cd =  σ 2 q σ 2 c σ 2 d  ⊤ , Θ c =  σ 2 q σ 2 c  ⊤ , Θ d =  σ 2 q σ 2 d  ⊤ , Θ 0 = σ 2 q , where σ 2 q is presen t under all mo dels and σ 2 c or σ 2 d augmen t the cov ariance structure relative to M 0 . The noise v ariance σ 2 e is assumed known from data segments with no transmitted pulse. Using Wilks’ theorem, the distribution of the test statistics under the n ull h yp othesis is approximated as 2 T j ( Y b,N k ) | M 0 a ∼ χ 2 ζ j , ζ j = dim(Θ j ) − dim(Θ 0 ) . (27) Hence, the p -v alue is computed as p j = Pr  χ 2 ζ j ≥ 2 T j  (28) with M 0 b eing preferred to M j unless p j ≤ α . B. T arget Detection P erformance Let  x o,k represen t a collection of discrete-time samples for the target reection during the k th ping, and is dened as  x o,k =  x o,k [0] · · · x o,k [ N − 1]  ⊤ , (29) where x o,k [ n ] =  0 , k < ¯ k a o,k s ( β o,k ( n ∆ t − ¯ τ o,k )) , k ≥ ¯ k (30) and { a o,k , ¯ τ o,k , β o,k } are the target parameters during the k th ping. F urthermore, ¯ k represents the unknown onset ping index where the target is assumed to appear. With this, the discrete-time measuremen t v ector is giv en as y k = H θ k +  x o,k + v k . (31) Our goal is to quantify the b enet of explicitly modeling and tracking the multipath background b efore p erforming target detection. Since the target onset time is unkno wn, a sequential likelihoo d ratio test (SLR T) can be emplo y ed delay [s] 1.34 1.36 1.38 1.4 1.42 time [s] 0 2 4 6 8 10 12 14 16 18 -50 -40 -30 -20 -10 0 20 log 10 |CIR| (a) Scenario 1. delay [s] 1.34 1.36 1.38 1.4 1.42 time [s] 0 2 4 6 8 10 12 14 -50 -40 -30 -20 -10 0 20 log 10 |CIR| (b) Scenario 2. delay [s] 1.34 1.36 1.38 1.4 1.42 time [s] 0 5 10 15 -50 -40 -30 -20 -10 0 20 log 10 |CIR| (c) Scenario 3. Fig. 2: Sim ulated CIRs for three scenarios. The trajectories for the stationary and moving target are also shown in (a). under dierent mo dels ( M 0 , M c , M d , M cd ) and ev aluate the resulting detection p erformance. T o that end, let H 0 and H 1 denote the background-only and target-presen t h yp otheses for a given mo del M : H 0 : y k = H θ k + v M ,k , (32a) H 1 : y k = H θ k +  x o,k + v M ,k , (32b) Here, the fo cus is on quan tifying the detection b enet for dierent background mo dels, and therefore the target w a v eform parameters ( ¯ τ o,k , β o,k ) and amplitude a o,k are assumed kno wn; only the onset ping index is unkno wn. No w, the sequen tial log-likelihoo d ratio is computed as G k +1 = G k + γ k , ∀ k = k o , k o + 1 , . . . (33a) with γ k ≜ log p ( y k | y k o : k − 1 ; ˆ Θ , H 1 ) p ( y k | y k o : k − 1 ; ˆ Θ , H 0 ) , (33b) where G k o = 0 and k o denotes the ping at whic h the test is (re)started, and the marginal likelihoo ds are calculated via the EKF. Here, the notation M is suppressed as the mo del remains the same in both hypotheses. The stopping rule is giv en as      G k ≤ h 0 : restart the test at k o ← k + 1 h 0 < G k < h 1 : monitor G k ≥ h 1 : declare detection . (33c) The thresholds h 0 and h 1 are selected to meet the desired false-alarm rate and detection-delay tradeo. F or the rst time, the SLR T is started with k o = 1 . With h 0 = 0 , the SLR T b ecomes equiv alent to the Page’s test [2] for detecting the signals with unkno wn start times. C. Simulation Setup T o assess the statistical adequacy of the prop osed bac kground mo dels and their impact on target detection, sev eral time-v arying underwater c hannel scenarios are sim ulated using BELLHOP [12], [13]. Time v ariabilit y is induced by (i) a rough sea surface syn thesized from a wind-wa v e sp ectrum [14] and (ii) random transmitter T ABLE I: System and en vironmen t conguration used to generate the receiv ed signal. Parameter V alue Signal type Linear frequency-mo dulated (LFM) chirp Number of pings 100 Pulse duration 25 ms Pulse rep etition interv al (PRI) 0.12 s Carrier frequency 3 kHz Sampling frequency 15 kHz Bandwidth (BW) 4 kHz Ocean depth 50 m Field generation mo de Time varying Sound sp eed prole Iso-velocity (1500 m/s) Bottom type A coustic half-space Bottom density 1.7 g/cm 3 Bottom attenuation 0.5 dB/ λ Bottom roughness 0.05 Sound sp eed in b ottom 1650 m/s Distance b etw een sensor nodes 2 km Sensor no de depth 2 m (from ocean surface) Sea-surface sp ectrum JONSW AP Signicant wa ve height H s 1 m Peak perio d T p 4 s Peak enhancement factor γ 3.3 Node drift per axis σ pos 0.20 m and receiver drift ab out their nominal p ositions, with indep enden t horizontal-plane motion. T able I summarizes the acoustic, en vironmen tal, and geometric parameters used to generate the receiv ed signal. With this, three signicant o cean scenarios are simu- lated • Scenario 1: a time-inv arian t baseline with a at sea surface and xed transmitter and receiv er p ositions. • Scenario 2: a surface-w a v e case with a moving surface and xed transmitter and receiv er p ositions. • Scenario 3: a fully time-v arying case with a mo ving surface and drifting transmitter and receiver p osi- tions. Figs. 2(a)–(c) show the corresp onding c hannel impulse resp onses (CIRs) for these o ceanic scenarios. T o simulate a target return under these o cean scenarios, t w o target-motion conditions are considered: • A stationary target with xed target delay . • A mo ving target that crosses the bistatic baseline (transmitter-receiv er line), pro ducing a time-v arying target dela y . The stationary target case is ev aluated b ecause b oth the m ultipath and the target exhibit similar Doppler v alues. Sp ecically , in the case of Scenario 1, the delay resp onse of the target matc hes with the multipath bac kground as sho wn in Fig. 2(a). V. Results and Discussion With the simulated datasets, the statistical adequacy and target detection p erformance are ev aluated across in terference-to-noise ratio (INR) and signal-to-noise ratio (SNR), resp ectively , using N MC Mon te Carlo realizations. F or eac h candidate bac kground mo del in (24), a p -v alue test in (28) is ev aluated. Signicance is declared when p ≤ 0 . 05 . Here, the INR and SNR are dened as INR = 10 log 10  ∥  x b ∥ 2 2 N σ 2 e  , (34a) SNR = 10 log 10  ∥  x o ∥ 2 2 N σ 2 e  . (34b) The SLR T is applied to detect the target onset, which o ccurs after an initial target-free perio d. Specically , the rst 40 pings (background-only) are used to learn the mo del hyperparameters Θ , after which the sequential test is p erformed with the target in tro duced from ping 41 . The target detection p erformance is ev aluated b y the proba- bilit y of detection P d as a function of SNR, mean time- to-detection (MTD) vs. SNR, computed only for target detections o v er N M C trials. Also, an empirical cumulativ e distribution function (CDF) is computed, which sho ws at whic h p ost-onset time delay index the target has been detected mostly . The SLR T threshold h 1 is calibrated empirically under H 0 , for all models and SNR v alues, to ac hiev e a designed false-alarm probabilit y P fa = 0 . 05 , and all rep orted P d , MTD, and CDF results are estimated using N MC = 200 trials. Firstly , the signicance of the prop osed mo dels is ev aluated for INR from 0 to 30 [dB]. It is observed that in Scenario 1, M c and M cd are rejected for all tested INR. In contrast, in Scenarios 2 and 3, the extensions M c , M d , and M cd yield p -v alues b elow 10 − 3 for essentially all INR v alues. This indicates substantial structure b eyond M 0 in the time-v arying scenarios. Ev en though the models in ( 24) are low-dimensional appro ximations rather than exact mo dels, the observed small p -v alues in Scenarios 2 and 3 suggest that the prop osed cov ariance comp onents capture structure beyond M 0 in the sim ulated data. The detection p erformance for the dierent background mo dels is shown in Fig. 3, Fig. 4, and Fig. 5. The ambien t noise-only baseline M 0 yields near constant MTD (with a small decrease) and low er P d across the SNR sweep. It suggests that a homoscedastic measurement co v ariance is insucient even under near-static conditions. With INR = 30 dB xed, p erformance gains with increasing SNR primarily reect improv ed separabilit y betw een the target return and the learned bac kground structure. The b enet of explicitly modeling the multipath back- ground for the target detection is pronounced in all sce- narios. Mo dels that include the common-mo de comp onent, i.e., M d and M cd , ac hiev e consisten tly shorter MTD and higher P d for b oth stationary and moving targets. The CDF s corrob orate this by showing earlier rises (faster detection) and higher plateaus (few er misses) for the M d and M cd mo dels. Also, the analysis for Scenario 2 is not pro vided as there w ere marginal dierences in p erformance for dierent mo dels, and provided no extra insight, which is not captured b y Scenarios 1 and 3. Ov erall, the BELLHOP-generated results fav or model- ing the common-mo de term parameterized b y σ d . Ev en when the mo del M c pro vides only incremental improv e- men t, accounting for common-mo de uctuations improv es the target detection p erformance despite strong bac k- ground interference. This is consistent with the observed CIR ev olution in Scenarios 2 and 3, where the multipath structure c hanges slowly and coherently o v er time, making a drift-lik e bac kground comp onen t more prominent. VI. Conclusion T o summarize, the presented results indicate that the prop osed wide-band Doppler linearization-based model can reliably describ e Doppler eects due to platform and surface wa v e motion. This enables ecient background learning in the raw acoustic measurement domain to b olster the detection of w eak target(s) em b edded in the m ultipath background. F uture research will fo cus on in te- grating the model into a track-before-detect framew ork for join t background learning, target detection, and tracking. App endix A Wideband linearization of time-v arying multipath signal Dene F ( r, t ) ≜ s ( e r t ) with r = ln β . Then, ∂ F ( r , t ) ∂ r = e r t ˙ s ( e r t ) = t dF ( r, t ) dt With F (0 , t ) = s ( t ) , the solution to the PDE is F ( r, t ) = e r ( t d dt ) s ( t ) , s ( β t ) = e ln β ( t d dt ) s ( t ) . (35) Using the series expansion, e a = 1 + a + a 2 2! + a 3 3! + · · · and taking the rst order approximation, (35) can b e written as s ( β t ) ≈ s ( t ) + ln β t ˙ s ( t ) . (36) References [1] N. J. Willis, Bistatic radar. Raleigh, NC: SciT ech Publishing, Jun. 2005. SNR[dB] 0 5 10 15 20 Meantime(pings)todetection 0 10 20 30 40 50 M 0 M cd M c M d (a) Stationary target, sce- nario 1. SNR[dB] 0 5 10 15 20 Meantime(pings)todetection 0 10 20 30 40 50 M 0 M cd M c M d (b) Mo ving target, scenario 1. SNR[dB] 0 5 10 15 20 Meantime(pings)todetection 0 10 20 30 40 50 60 M 0 M cd M c M d (c) Stationary target, sce- nario 3. SNR[dB] 0 5 10 15 20 Meantime(pings)todetection 0 10 20 30 40 50 M 0 M cd M c M d (d) Mo ving target, scenario 3. Fig. 3: Mean time to detection (MTD) versus SNR at INR = 30 dB. Solid lines show the mean detection delay ov er detected trials, and b ounds indicate the 10–90 p ercentile range of detection delays. SNR [dB] 0 5 10 15 20 P d 0 0.2 0.4 0.6 0.8 1 M 0 M cd M c M d (a) Stationary target, sce- nario 1. SNR [dB] 0 5 10 15 20 P d 0.3 0.4 0.5 0.6 0.7 0.8 M 0 M cd M c M d (b) Mo ving target, scenario 1. SNR [dB] 0 5 10 15 20 P d 0 0.2 0.4 0.6 0.8 1 M 0 M cd M c M d (c) Stationary target, sce- nario 3. SNR [dB] 0 5 10 15 20 P d 0 0.2 0.4 0.6 0.8 1 M 0 M cd M c M d (d) Mo ving target, scenario 3. Fig. 4: Probabilit y of detection P d v ersus SNR at INR = 30 dB. A trial is counted as a missed detection if no alarm is raised b y the nal ping in the test horizon. time 0 10 20 30 40 50 60 CDF 0 0.2 0.4 0.6 0.8 M 0 M cd M c M d (a) Stationary target, sce- nario 1. time 0 10 20 30 40 50 60 CDF 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 M 0 M cd M c M d (b) Mo ving target, scenario 1. time 0 10 20 30 40 50 60 CDF 0 0.2 0.4 0.6 0.8 1 M 0 M cd M c M d (c) Stationary target, sce- nario 3. time 0 10 20 30 40 50 60 CDF 0 0.2 0.4 0.6 0.8 1 M 0 M cd M c M d (d) Mo ving target, scenario 3. Fig. 5: Cumulativ e distribution function (CDF) at an SNR = 10 dB. The plateau level equals P d , and the rise rate reects detection latency . [2] D. Abraham, Underwater Acoustic Signal Pro cessing: Mo deling, Detection, and Estimation. Cham,Switzerland: Springer, F eb. 2019. [3] B. Kalyan and A. Balasuriya, “Sonar based automatic target detection scheme for underwater environmen ts using CF AR techniques: A comparative study ,” in Pro c. of the Int. 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