Estimation of Acoustic Channel Impulse Response at Low Frequencies Using Sparse Bayesian Learning for Nonuniform Noise Power

Sparse Bayesian learning (SBL) has been extended to estimate acoustic channel impulse responses (CIRs) at low frequencies, where matched filter (MF)-based CIR estimation suffers from low resolution due to a limited frequency band. In this study, the extended SBL was developed to account for nonuniform noise power in a signal model for the CIR via a formulation that considers inconsistent noise and multiple measurements, which cannot be handled in the conventional SBL. The extended SBL is applied to simulated and measured acoustic data and then compared with the MF and existing SBLs. With the advancement in the schemes, the time resolution and denoising are enhanced; especially, the results of the extended SBL on the simulated data show that it clearly distinguishes the two adjacent arrivals with moderate errors in the estimated time delays. Additionally, as a result of applying it to the measured array data, the extended SBL can achieve a high resolution in the CIR estimation, which retains assured arrivals with a single measurement, while some arrivals are weakened by the insufficient measurement for the extended SBL.

The CS has been previously used to estimate acoustic CIRs 100 by utilizing the sparse arrivals in acoustic channels [6], [12], 101 [16], [17], [19]. Meanwhile, the CS-based DOA estimations 102 were recently advanced by the SBL, which can find the 103 DOAs into sensor array without additional post-processing 104 [21], [22], [23], [24], [25], [26], [27]. To the best of our 105 knowledge, the SBL has not been used in estimating the CIRs. 106 Thus, in this study, to extract the CIRs from a measurement 107 (a signal emitted by a transducer, transmitted through the 108 ocean waveguide, and received by a hydrophone), the SBL 109 is adopted due to its properties that are exhibited in the DOA 110 estimation as well as free manual hyperparameters and the 111 properties of SBL are demonstrated in the CIR estimation. 112 A linear system for the CIR estimation is established by 113 expressing the frequency domain, which is the TF and the 114 candidate arrivals have distinct time delays. Note that noise 115 in the linear system is scaled by the source spectrum, which 116 results in nonuniform noise powers over the measurements 117 that are relative to the frequency response of the received sig-118 nal. The conventional SBL should be modified to accommo-119 date the nonuniform noise, which was attempted by several 120 researchers [26], [27], [28]. In the present study, a rigorous 121 mathematical expression that accounts for the nonuniform 122 noise is derived, and the associated updating rule is obtained, 123 which is different from the previous studies by using the 124 heuristic approach [26], [27] or considering only the real val-125 ues and a single measurement [28]. Particularly, in [26] and 126 [27], the previously established stochastic likelihood function 127 was exploited, which provided an asymptotically efficient 128 estimate of noise. To the best of our knowledge, no studies 129 have been conducted on the SBL where the basic expres-130 sions for noise are rigorously extended for nonuniform noise 131 to estimate the amplitudes and the time delays of arrivals 132 in the CIRs. In Section II, the linear system for the CIRs 133 is derived in the frequency domain using the definition of 134 the TF. The conventional SBL is extended to account for 135 the nonuniform noise powers in the linear system, which is 136 discussed in Section III. The novel SBL is examined in terms 137 of the time resolution enhancement and the noise reduction 138 with simulated and measured acoustic data, which are shown 139 in Sections IV and V, and the results are compared with the 140 results from the MF and the existing SBLs. To investigate the 141 performances of the conventional and the extended SBLs for 142 uniform and nonuniform noise cases, we performed numer-143 ical experiments and discuss the validation regions of the 144 SBLs. Section VI summarizes the present study. When underwater sound that is emitted from a source is 150 transmitted through the ocean, the signal that arrives at the 151 receiver contains multiple signals that pass through differ-152 ent paths that arise from the waveguide from the interfaces 153 99012 VOLUME 10,2022 (sea bottom and surface) and the scatterers (submerged 154 objects and fish schools

161
where a k and τ k are the amplitude and time delay of arrival, 162 respectively, and K is the number of (dominant) arrivals. 163 The received signal, y (t), is represented by the convolution 164 between the source waveform s (t) and the acoustic CIR h (t) 165 with ubiquitous noise n(t) as follows: 166 a k s (t − τ k ) + n(t).
(2) 167 A known source waveform is exploited to estimate the CIR 168 when operating an active sonar system, which uses a trans-169 ducer and a receiver that have almost flat frequency responses 170 in the source bandwidth. By using the Fourier transform (FT),

171
(2) is denoted in the frequency domain as follows: Equation (5) is rearranged by adopting the following vector 206 and matrix notation: y = Ax + n, where y, x, n, and A are the 207 measurement, unknown, noise, and transformation matrix, 208 respectively. It is worth noting that the signal model cannot 209 be in the form of a linear system without dividing S (ω) 210 in (3), because the corresponding unknown that has S (ω) x n 211 as an element varies according to the equality conditions. 212 The measurement elementŶ m and the noise element N m are 213 Y (ω) and N (ω) at the angular frequency ω m , respectively. 214 ω 1 and ω M correspond to the lowest and highest frequencies, 215 respectively, which are used for the SBL-based CIR estima-216 tion and belong to the source frequency band. The number of 217 equations, M, is smaller than the number of unknowns, N. The 218 SBL is used to determine a solution of the underdetermined 219 linear system (i.e., the amplitudes and the time delays of 220 arrivals that constitute of the CIR).

221
In contrast, a linear system can be constructed in the time 222 domain, where the transformation matrix comprises of the 223 time-shifted source waveforms as its columns and M is equal 224 to N. In this study, the solution of the frequency domain linear 225 system is preferred because less equality conditions cause a 226 more refined CIR with a sparser solution by recovering less 227 meaningful arrivals than the given equality conditions.

228
Note that the noise powers in (5) (i.e., variances of N m ) 229 are inconsistent due to the fluctuating ocean noise spectrum 230 being scaled by the source spectrum [29], which violates the 231 assumption of constant noise power over the measurements in 232 the conventional SBL [20]. Therefore, the conventional SBL 233 should be modified to treat the nonuniform noise powers, 234 which is described in the following section. Hereafter, the 235 conventional and expanded SBL are referred to as uniform 236 noise SBL (UN-SBL) and nonuniform noise SBL (NN-SBL), 237 respectively.

239
An SBL was first introduced by Tipping [20] for classifica-240 tion and regression in machine learning. It was later adopted 241 for high-resolution beamforming in underwater acoustics, 242 where noise powers over measurements (frequency domain 243 signals at sensors in array) are assumed to be constant, as in 244 the original paper, except for a few studies [26], [27], where 245 the formulations in the SBL were modified approximately to 246 treat the nonconstant noise. However, in this study, the SBL 247 is rigorously expanded by deriving mathematical expressions 248 for the probabilities in the SBL for inconsistent noise powers 249 VOLUME 10, 2022 regarding complex value and multiple measurements. After-250 wards, the results for the SBL using different approaches for 251 the nonconstant noise are compared with each other.

252
In the SBL framework, the unknown (x) and noise (n) 253 are treated as random vectors, which consist of independent 254 random variables, and they are assumed to follow a zero- The NN-SBL starts by deriving an analytic expression for the 279 posterior probability p x | y, γ s , γ n , which is denoted below 280 using Bayes' theorem:

282
where p y | x, γ n is the likelihood function, and p x | γ s is 283 the prior function. Furthermore, p y | γ s , γ n is the evidence 284 (marginal likelihood) that is used to evaluate the hidden vari-285 ables of the variances.

286
The noise is an independent random variable that follows a n is a diagonal matrix that results from the independent 298 noise making off-diagonal components zeros, and its diag-299 onal components are γ n (i.e., n = diag γ n ).

300
The prior corresponds to the distribution of the unknown 301 comprising independent random variables as follows: where s is a diagonal matrix whose diagonal components 305 equal γ s (i.e., s = diag γ s ). The components of x are acti-306 vated when the corresponding components of γ s have non-307 zero values. During the application of the SBL for solving the 308 linear system, γ s appears sparsely, which is advantageous to 309 enhance the resolution and suppressing noise [21], [22], [23], 310 [26], [27].

311
The evidence can be derived using γ s and γ n with a linear 312 system. The Gaussian-distributed unknown and noise in the 313 linear system render the evidence normally distributed as 314 follows: The mean of y is equal to a zero vector because each mean of 317 the unknown and noise is the zero vector, y = A x + n = 318 0. y is calculated with the definition of the covariance matrix 319 as follows: Note that the nonuniform noise power leads to different forms 322 of the likelihood function and the evidence from them in the 323 UN-SBL, which subsequently result in a distinct posterior 324 probability.

325
By substituting (8)-(10) in (7), the posterior probability 326 that considers the inconsistent noise variances is derived 327 using the Woodbury matrix identity as follows: 332 As expected, the posterior probability follows a Gaussian 333 distribution, and its mean and covariance matrix are calcu-334 lated using the estimated γ s and γ n with measurement y. 335 x est is equivalent to µ x , where the posterior probability is the 336 maximum.

337
B. ESTIMATE OF THE VARIANCES γ s AND γ n 338 As previously described, γ s and γ n must be estimated to 339 obtain the solution via the SBL. The variances that best 340 describe the given measurement are used for the solution, and 341 they are calculated as follows: p γ s , γ n | y equals p y | γ s , γ n p γ s p γ n . When there 344 are no preferences over the signal and noise, which is the 345 situation in this study (i.e., both p γ s and p γ n follow a 346 uniform distribution) [20], (13) becomes the same problem 347 of finding γ s and γ n to maximize p y | γ s , γ n as follows: 349 Herein, the expectation-maximization (EM) algorithm is  The expectation of lnp y|x, γ n from the perspective of the 371 posterior probability is denoted as follows, which the terms 372 relevant to γ n are retained: The extremum of the function Q n γ n is obtained as 377 follows: value as in the UN-SBL, the constant noise powerγ n is 386 expressed as follows: v 2 is the Euclidean norm of the vector v. This is the 389 average of the noise powers over the measurement (i.e., 390 γ n = M m=1 γ n m /M ) and it equals the previous result 391 ((22) in Ref. [21] for a single measurement). The noise 392 power was previously estimated with a representative value as 393 the average, whereas, the power was expressed individually 394 according to the measurement component in this study. When 395 the variation of the noise powers is noticeable, (19) should 396 accommodate the probabilities of y, which was considered in 397 this study. This is achieved at the cost of the increased number 398 of variables (the noise variances), and it is proportional to the 399 equality conditions in (5).

400
To estimate γ s and γ n , the posterior probability 401 p x | y, γ s , γ n (or µ x and x ) is required, and it is expressed 402 using γ s and γ n with the measurement, which they are 403 entangled to each other. Thus, the first step in the EM 404 algorithm is to generate p x | y, γ s , γ n with initial small 405 values γ s and γ n (initial guesses for γ s and γ n ), which 406 allows x to be non-singular. In the next step, using µ x 407 and x in the posterior probability,γ s andγ n are computed 408 using (17) and (19), respectively, which are used to update 409 the posterior probability. The expectation and maximization 410 steps are iterated until convergence, and they correspond to 411 the first and second terms in the second line of (6). The final 412 γ s andγ n are used to evaluate x est , which is equivalent to the 413 final µ x .

416
Unlike the previous studies [20], [21], [22], [23], the noise 417 power is estimated using the corresponding element in the 418 measurement vector, which reduces the performance of the 419 NN-SBL at low SNRs. To alleviate this problem, multiple 420 measurements can be exploited when the underwater sounds 421 are repeatedly emitted from a transducer for a relatively short 422 duration or in a stationary ocean environment.

423
The NN-SBL that uses a single measurement has been 424 expanded to exploit the common time delays of arrivals over 425 the multiple measurements [22], [23]. The first step for the 426 expansion is to modify the posterior probability to accommo-427 date the multiple measurements as follows: other, and the likelihood function, which is the prior proba-436 bility, and evidence for multiple measurements can then be 437 stated as follows:  (21), the posterior probability for the multiple 450 measurements is derived as follows:

464
As shown in (26) and (27), the means of the signal and the 465 noise powers over multiple measurements are used for the 466 update, which benefits the estimation at low SNRs. In con-467 trast, the average of the noise variances over frequency com-468 ponents and multiple measurements can be used to derive the 469 updating rule for the uniform noise power case as follows:

495
A single arrival is presumably in a received signal for 496 the convenient evaluation of the errors from the SBLs. The 497 longest observation time was 200 ms, and the signal with a 498 frequency band between 250 Hz and 750 Hz, which corre-499 spond to the lowest and highest frequencies in (5), respec-500 tively, randomly arrives between 20 ms and 180 ms in the 501 simulation. The ideal TF without noise is produced with the 502 transformation matrix multiplied by x, which comprises of 503 a single non-zero element (value of one) that corresponds to 504 the random arrival time. The transformation matrix is formed 505 with the discretized time delays (along columns) and the 506 angular frequencies (along rows) using a sampling frequency 507 of 3,000 Hz (four times the maximum frequency). There are 508 601 unknowns (N ) and 100 equality conditions (M ).

509
The following noise is added to the noise-free ideal TF to 510 examine the robustness of the SBL-based estimator. where U (a, b) is the uniform distribution between a and b. 514 The subscripts m and l denote the lth measurement at ω m . 515 N P (ω m ) determines the SNR of the measurement at ω m . The 516 noise variances are constant over multiple measurements due 517 to the stationary ocean environment, so N P (ω m ) is indepen-518 dent of the measurement time (or measurement number l). 519 For a simulation with uniform noise power, N P (ω m ) becomes 520  of N P to examine the robustness of the SBLs (Fig. 1(a)).

545
For a fixed N P , the trial is repeated 50 times; the means of 546 SNR a and estimation error are values on the x and y axes of 547 Fig. 1(a), respectively. While the noise that follows uniform 548 distribution in the simulation deviates from the assumption of 549 SBL (Gaussian distribution), the SBLs show superior perfor-550 mances as in [26], [27], and [28]. Some estimation results at 551 SNRs less than −11 dB for both SBLs have the largest peaks 552 at time delays that are different from the true values, which 553 increase the estimation error. As expected, the error becomes 554 insignificant with an increase in the SNR (or a decrease 555 in N P ). At the SNRs above −11 dB, the SBLs accurately 556 estimate the time delays, and the estimation error becomes 557 almost zero, which the order of error is 10 −14 ms. Fig. 1(b) 558 shows the CIR from the SBLs for a specific trial with an 559 SNR a of approximately −14 dB. While minor peaks appear 560 and the maximum amplitude is considerably less than the true 561 value of 1 in the CIR, the arrival can be clearly distinguished 562 from the noise, and the time delay that corresponds to the 563 maximum amplitude matched the true value (vertical dash-564 dotted line in Fig. 1(b)).
565 Fig. 1(c) includes the corresponding noise variances of 566 the CIRs in Fig. 1(b). They are calculated using (27) and 567 (28) for the NN-SBL and the UN-SBL, respectively. They 568 are then compared with the measured noise variances based 569 on (30). N P is independent of the frequency, so the measured 570 noise variances fluctuate slightly around −14 dB, which is 571 in good agreement with the results from the NN-SBL. The 572 constant noise variances from the UN-SBL are located at 573 the center of the fluctuation. While the NN-SBL requires 574 more noise variances to be estimated using the EM algorithm 575 with the same number of equality conditions, it delivers a 576 performance that is similar to that the performance of the 577 UN-SBL in the environment of uniform noise power due to 578 the sufficient measurements, as shown in Fig. 1. The two 579 SBLs show different trends in regards to estimating the CIRs 580 under insufficient measurements. The NN-SBL underesti-581 mates the arrival amplitudes, which the opposite happens in 582 the UN-SBL. Furthermore, the UN-SBL inclines to include 583 the fake arrivals by noise due to the simplified uniform noise 584 assumption. Each scheme has pros and cons when using 585 insufficient measurements, which are shown in Fig. 6. 586 As described in (4), the noise with the TF is scaled by 587 the FT of the source waveform, which incurs nonuniform 588 noise variances over the measurement along the frequency. 589 VOLUME 10, 2022  its performance is similar to the performance that is illustrated 609 in Fig. 1(a).

610
While the UN-SBL has a maximum peak, which corre-611 sponds to the true arrival time discriminated from minor 612 peaks, it underestimates the arrival amplitude at a low SNR of 613 approximately −14 dB (Fig. 2(b)). This problem is amelio-614 rated by the NN-SBL exploiting the high SNR measurement 615 components, which is shown in Fig. 2(c), where the noise ing outliers, and their amplitudes follow uniform distribution 627 between five and ten times the largest value of the transmitted 628 signal. The ideal TF with a flat response along the frequency 629 is substituted with a TF with an uneven response that reflects 630 the frequency-dependent attenuation [30]. The amplitudes of 631 the TF in the frequency band that are used for the estimation 632 are random numbers that are uniformly selected between 633 0.9 and 1.1. The FT of the received signal divided by that of 634 the chirp signal is the measurement, and 20 measurements 635 are applied for the estimation via the SBLs. As shown in 636 Fig. 3(c), considerable peaks in the MF result arising from 637 the noise with the outliers are significantly diminished by 638 the NN-SBL, whereas the sparse solution from the UN-SBL 639 also provides a clearer CIR compared to the MF result, which 640 the noticeable peaks remain, and they are detrimental to the 641 accurate evaluation of the true arrival. The inset displays the 642 difference between the NN-SBL and the UN-SBL results near 643 the true arrival.

644
As expected, the source spectrum causes the AWGN to 645 have different SNRs along the frequency (Fig. 3(d)). A source 646 frequency component with a larger value tends to yield a 647 higher SNR m and vice versa, which results in SNR m having 648 a similar shape to that of the source spectrum. When a sound wave propagates in the ocean, it arrives at a 652 receiver with a time delay in the continuous domain. Fur-653 thermore, the predefined on-grid bases (or columns) of the 654 transformation matrix induce inevitable errors in the linear 655 system. To mitigate the problem due to the basis match, a finer 656 grid with a higher sampling frequency can be adopted, but it 657 increases the computational burden as well as the similarity of 658 the adjacent bases in the transformation matrix, which leads 659 better for an SBL-based CIR estimation, which is similar to 685 an MF.

686
The similarity decreases with an increase in n, which is 687 shown in Fig. 4(a), and it includes similarities for all pairs 688 of two columns in the transformation matrix, which is used 689 for Figs. 1-3. Thus, the arrival with an off-grid time delay is 690 represented by the bases near the arrival.

691
To demonstrate the orthogonal-like property of the bases 692 in the CIRs, the estimation errors are computed using two 693 different sampling frequencies of 3,000 Hz and 6,000 Hz, 694 as shown in Fig. 4(b), where the chirp signal, as shown in 695 Fig. 3, is used for the source waveform, and AWGN is added 696 to the received signal that is computed from the convolution 697 of the source waveform with the CIRs that have a time delay 698 in the continuous domain as well as an uneven frequency 699 response, as shown in Fig. 3. The overall estimation errors 700 increase due to the basis mismatch as opposed to Fig. 2(a), 701 where an ideal TF with an on-grid time delay is used for the 702 simulation. However, the largest estimation error of 0.16 ms 703 occurs at an SNR a of −16 dB with a low sampling frequency. 704 It is approximately half of the time difference between the 705 two consecutive samples (sampling interval t = 1/f s ) 706 discretized with a sampling frequency of 3,000 Hz. This 707 indicates that the major arrivals that were estimated from 708 the SBL are located near the true arrival even at the lowest 709 SNR of the simulation. As the SNR increases, both estimation 710 errors for the different sampling frequencies decrease and 711 converge to 0.08 ms and 0.04 ms, which are equal to a 712 quarter of the sampling intervals that corresponds to the sam-  Fig. 4(b). It also suffers 729 less from the basis mismatch, which is unlike the sparse signal [22], [23], [24], [25], [26], [27].   Figs. 1-3. The similarity decreases with increase in distance between the columns, which makes the Gram matrix resemble the identity matrix. (b) The mean estimation error according to the SNR for simulations using different sampling frequencies of 3,000 Hz and 6,000 Hz. Off-grid arrival times in the simulation lead to a basis mismatch resulting in performance degradation of the NN-SBL. However, the orthogonal-like columns enable the NN-SBL to have the estimated arrivals near the true ones even at the lower sampling frequency with less time resolution.
for a convenient comparison, where both SNR a high and 768 low frequencies are approximately −4.5 dB. In particular, 769 the existing SBLs [26], [27], which can treat heteroscedastic 770 noise, are applied to the same data for a comparison, which 771 are hereinafter referred to as heteroscedastic-noise SBL 772 (HN-SBL). The vertical dash-dotted lines indicate the true 773 arrival time delays that correspond to the direct (the first) and 774 the scattered (the second) signals.

775
At high frequencies (sampling frequency of 80 kHz), the 776 MF can separate two arrivals due to the broadband signal 777 whereas the overall offsets are induced by the noise. The 778 noise also weakened the second arrival from the UN-SBL, 779 The direct arrival is divided into two arrivals in HN-SBL. Furthermore, as shown in the inset, the noise weakens the scattered signals in the SBL results. In particular, the scattered signals from the HN-SBL are distributed over the time delay, and it results in the weakest scattered signal. On the other hand, the NN-SBL recovers the arrivals clearly from the contaminated synthetic data, whereas the amplitude of the scattered signal is slightly reduced. (b) The mean estimation results from the MF, UN-SBL, NN-SBL, and HN-SBL at 1 kHz. At a low frequency, the estimation results are different according to the trials, and their mean is spread over the time delay, so the UN-SBL results are more distributed, which makes the mean blunt. Furthermore, the first arrival from the HN-SBL is considerably deviated from the true one, which is displayed in the inset, and the fictitious peak emerges ahead of scattered signal. As in the high frequency case, the NN-SBL demonstrates the best performance among the SBLs.
whereas that the second arrival from the NN-SBL remained  It is worth noting that the conventional SBL is corrected to 815 treat the heteroscedastic noise heuristically in Refs. [26] and 816 [27], and the performance degradation might be attributed to 817 the heuristic derivation. In this study, the rigorous expansion 818 of the conventional SBL is derived to consider the nonuni-819 form noise exactly, which leads to the clearest CIR estimation 820 among the SBLs. The shallow-water acoustic variability experiment 2015 825 (SAVEX15) was conducted in the northeastern East China 826 Sea [32], and the sound waves that were traveling within a 827 shallow acoustic waveguide were measured at high and low 828 frequencies. In this study, the low-frequency signals, which 829 were recorded with a sampling frequency of 100 kHz, were 830 used to examine the NN-SBL by comparing its results with 831 the results from the MF and the existing SBLs, such as the 832 UN-SBL and the HN-SBL. A single acoustic measurement is used for the estimation. The schemes are more sophisticated, so the acoustic structures (e.g., 'x' shapes that are formed by the upward and downward incoming arrivals to the VLA) are observed more clearly. The noise in the MF result is noticeably removed in the SBL results. In particular, the most possible arrivals remain in the NN-SBL result at the cost of their weakening due to the insufficient measurement. Meanwhile, the HN-SBL estimates the CIRs too sparsely and filters out almost all the true arrivals except for intensive arrivals in the middle of the water column. Afterwards, the CIRs around the dotted lines are used for a performance comparison in terms of resolution and reestablishment of the CIR from the measurement.
was separated from the transducer at a distance of approx- which is sufficient to prevent aliasing at a low frequency.

855
The CIRs along the hydrophone depths produced acoustic 856 structures by the waveguide, which included the 'x' shapes 857 that were formed by the incoming arrivals to the VLA from 858 the upward and downward directions. They appeared more 859 clearly with the improvement of the scheme (Fig. 6). The SBLs inherently derive a high-resolution CIR with less noise 867 by solving the linear system of (5). For the visualization in 868 this study, the SBL results were purposely convolved with 869 an autocorrelation of the LFM signal, and the corresponding 870 envelopes were used, which is illustrated in Fig. 6. The direct 871 results for the SBLs were subsequently compared with the 872 results from the MF in Fig. 7. 873 While the arrivals around 80 ms and 105 ms have high 874 intensities due to the refractive sound speed profile in the 875 SAVEX15, the arrivals that form the 'x' shapes are more 876 focused on the performance comparison. The major arrivals 877 were found to use the MF, but they entailed noise overall. 878 The noise around the 'x' shapes caused the arrivals to appear 879 stretched over the relative time delay. The noise near the lower 880 left part of the second 'x' shape masked the arrivals. These 881 are detrimental to the clear detection of the arrival directions 882 to the VLA when time-domain beamforming is performed. 883 The SBLs tend to select bases of the transformation matrix 884 that correspond to the actual arrivals, so the noise in the 885 measurement is suppressed during the iterative estimation of 886 the signal powers in the SBLs. Thus, in the UN-SBL result, 887 the noise is remarkably removed, and the masked arrivals 888 around the second 'x' shape are more clearly observed. The 889 performance in terms of the noise reduction is improved 890 by the extended SBL, which is due to the more complex 891 approach to noise in the measurement. Most of the noise in 892 the MF result is diminished with the more advanced scheme, 893 as shown in Figs. 6(a) and 6(c). This is achieved at the cost 894 of the weakened arrivals. The NN-SBL estimates multiple 895   The SBL, which was originally developed for classification 933 and regression, was extended to estimate a high-resolution 934 CIR with a limited frequency band.

935
In this study, the SBL was used to solve a linear system of 936 CIRs, which were established by the physics of underwater 937 sound propagation. The omnipresent noise was scaled using 938 the source spectrum, which resulted in nonuniform noise 939 powers in the measurement violating a basic assumption in 940 the conventional SBL. To treat the inconsistent noise, the SBL 941 was expanded mathematically to accommodate the noise, 942 which was examined using simulated data that included a 943 single arrival. The expanded SBL (referred to as the NN-SBL) 944 delivered a superior performance with a more apparent arrival 945 estimation than the existing SBLs (referred to as the UN-SBL 946 and the HN-SBL) at a low SNR.

947
An inherent problem of the SBL is the basis mismatch 948 that was caused by the discretized columns corresponding 949 to the on-grid arrival times that did not precisely match the 950 true off-grid arrival times. The orthogonal-like bases of the 951 transformation matrix in the linear system fortunately ensure 952 that the estimated arrival is close to the true value. The syn-953 thetic data, which was sampled with two different sampling 954 frequencies, was used to support the properties of the bases. 955 The NN-SBL has an arrival around the true one even at a low 956 sampling rate with less time resolution.

957
A numerical experiment was conducted to demonstrate the 958 high-resolution CIR via the SBLs, where a scattered signal 959 from a finite submerged target was simulated with the direct 960 arrival. When a low-frequency source was used with a fre-961 quency band of 1 kHz, the MF could not distinguish between 962 the two arrivals with a travel distance difference of 2 m, which 963 was due to its low time resolution being proportional to the 964 source frequency band. On the contrary, the SBLs separated 965 the scattered signal from the direct signal via the sparse 966 estimation of the CIR linear system. The NN-SBL exhibited 967 a better noise-reduction performance when the multiple mea-968 surements that were obtained over time were available.

969
The extended SBL was applied to the SAVEX15 data 970 that was measured using a VLA at low frequencies 971 (0.5-2 kHz) with the other schemes (MF, UN-SBL, and 972 VOLUME 10, 2022