This article presents a spatial environmental inversion scheme using broadband impulse signals with deep learning (DL) to model a single spatially-varying sediment layer over a fixed basement. The method is applied to data from the Seabed Characterization Experiment 2022 (SBCEX22) in the New England Mud-Patch (NEMP). Signal Underwater Sound (SUS) explosive charges generated impulsive signals recorded by a distributed array of bottom-moored hydrophones. The inversion scheme is first validated on a range-dependent synthetic test set simulating SBCEX22 conditions, then applied to experimental data to predict the lateral spatial structure of sediment sound speed and its ratio with the interfacial water sound speed. Traditional geoacoustic inversion requires significant computational resources. Here, a neural network enables rapid single-signal inversion, allowing the processing of 1836 signals along 722 tracks. The method is applied to both synthetic and experimental data. Results from experimental data suggest an increase in both absolute compressional sound speed and sound speed ratio from southwest to northeast in the NEMP, consistent with published coring surveys and geoacoustic inversion results. This approach demonstrates the potential of DL for efficient spatial geoacoustic inversion in shallow water environments.

Accurate prediction of ocean acoustic wave propagation requires detailed knowledge of the underwater environment. More specifically, in shallow-water environments, such as coastal waters, wave propagation is affected by bottom interaction with the seabed. Thus, knowledge of the seafloor composition and layering is crucial. The description of the seafloor is typically simplified using a geoacoustic model composed of geoacoustic parameters. Finding the optimal parameters that best approximate a geoacoustic model to the real environment in question is called geoacoustic inversion. Geoacoustic inversion problems are solved using a multitude of traditional inversion techniques (Chapman and Shang, 2021), and due to the complexity of the problem, these methods usually require significant computational power and some amount of manual analysis.

The advent of deep learning (DL) during the last decades has proven to be one of the biggest advancements in recent history. The backbone that makes DL possible is the use of artificial neural networks (NNs) which are trained to perform specific tasks. Apart from being used in everyday life in the form of Large-Language-models, DL has been used to answer scientific questions as well by adapting NNs and fine-tuning them to perform the needed tasks. Moreover, since most of the computational power and time is invested in training the NN, the inference speed needed to generate results from a trained NN is extremely fast.

DL methods have demonstrated the ability to analyze complex acoustic data with great success (Bianco , 2019). In ocean acoustics, DL is now used to solve the direct problem and predict the acoustic field, both for a ray tracing context (Li and Chitre, 2023; Mallik , 2022) or a modal propagation context (Li and Chitre, 2022; Varon , 2023). It is also used to solve ocean acoustic inverse problems. Advances in source localization using DL have been made (Chen and Schmidt, 2021; Niu , 2017), with practical applications for localizing explosives (Van Komen , 2022) or baleen whales (Goldwater , 2023). DL has also been used for environmental characterization. Early pioneering works in the late 1990s proposed the use of NNs for tasks such as seabed inversion (Benson , 2000; Caiti and Jesus, 1996) and water column tomography (Stephan , 1998). The recent advances in DL have revitalized interest in NNs for ocean acoustic environmental characterization. Notable examples include the ability to create a set of seabed types for seabed classification (Neilsen , 2021), reconstructing water sound speed profiles using data-driven tomography (Jin , 2022), or improving geoacoustic inversion results in shallow water from data recorded on vertical arrays (Liu , 2022). Physics-informed NNs are also used to predict parameters of the underlying physical equations that model underwater acoustic propagation, showcasing the ability to predict pressure fields from sparse pressure measurements (Yoon , 2024).

This paper is a continuation of previous published work that uses a NN to perform geoacoustic inversion on data recorded by a single hydrophone in shallow water (Vardi and Bonnel, 2024). Here, we make use of the significantly increased inference speed to perform geoacoustic inversion on hundreds of acoustic signals, originating from different combinations of spatially distributed sources and receivers. Using these hundreds of spatially distributed inversions, the spatial dependence of a geoacoustic parameter, such as the sound speed of a sediment layer, can be derived.

This paper presumably reports the first use of machine learning (ML) to automatically process a large ocean acoustic dataset, provides hundreds of seabed inversion results, and combines them to build a map of sediment sound speed. However, estimating the spatial variability of the geoacoustic properties of the seabed is not new. Potty and Miller (2003) were among the first to perform spatial inversions for seabed acoustics, using a genetic algorithm with neighborhood approximation to invert compressional wave speeds from variations in travel-time dispersion during the 1996–1997 Shelf Break Primer Experiment. Spatial inversion schemes were also applied to data obtained during the Shallow Water Experiment (SW06) in the New Jersey Shelf where a three-dimensional (3D) sediment model was obtained by combining seismic reflection measurements with a perturbative inversion process based on modal wavenumbers (Ballard , 2010). Similar perturbative inversions were applied on modal time-frequency dispersion data obtained during the Modal Mapping Experiment (MOMAX) using a set of drifting buoys, enabling an estimate of a 3D sound speed profile in the water column (Ballard , 2014), as well as a joint 3D estimation of water and seabed geoacoustic properties (Rajan and Frisk, 2020). However, due to the high computational cost needed for multiple geoacoustic inversions, the methods mentioned previously all rely on perturbative (i.e., linearized) inversion algorithms, and as such, require a strong prior knowledge of the environment. More recently, a maximum-entropy method was applied with full-waveform inversions to estimate the marginal probability distributions for the top of the sediment sound speed and sediment sound speed gradient from signals originating at different range radii from the receiver (Knobles , 2020). However, the same need for prior knowledge of the environment and high computational cost schemes were needed to achieve a spatially meaningful inversion result. Here, the use of DL alleviates the computational cost issue. This enables one to cast the problem as a multidimensional parameter search, mimicking what is now routinely done in non-linear geoacoustic inversion (Bonnel and Lavery, 2024; Dosso, 2002).

To demonstrate the feasibility of our method, we applied it to experimental data collected during the Seabed Characterization Experiment (SBCEX) in the New England Mud-Patch (NEMP), specifically in the Central Mud-Patch (CMP). SBCEX is a multi-year (from 2015 to 2022) inter-institutional effort to understand the effects of mud sediments on shallow water acoustic propagation. This paper focuses on data collected in 2022. Although very few results are available from the 2022 experiment (notable exceptions are Bonnel , 2024; Dahl and Dall'Osto, 2022; Dahl , 2023), an overview of the results obtained with data from 2017 can be found in Wilson (2020) and Wilson (2022). Multiple CMP inversion results and analyses were already published using the 2017 data, which will help in establishing a strong repertoire of knowledge for assessing and comparing results in the NEMP. Importantly, ex-situ core logger data (Chaytor , 2022) together with in situ sound speed measurements (Yang and Jackson, 2020) are available from the 2017 experiment. More recent in situ sound speed measurements are also available from the 2022 experiment (Garcia , 2024) as well as a set of local inversion results derived from reflection coefficient data (Belcourt , 2020; Jiang , 2023). Those will enable some comparison with the seabed spatial variability estimated in this paper using ML.

The data used here consists of acoustic recordings of Signal Underwater Sound (SUS) explosives (Dall'Osto , 2023; Wilson , 2020) captured by a distributed set of receivers called TOSSITs, all moored to the seafloor (Zitterbart , 2022). The reliability of the proposed spatial inversion scheme is assessed on simulated data, informed by prior information on the experimental area. Then, it is applied to the experimental data obtained during SBCEX22 to invert the spatial structure of the mud sound speed and the sound speed ratio (SSR) in the NEMP. The inversion results suggest that both the sediment sound speed and the SSR between the sediment and the interfacial water sound speed increase from the southwest to the northeast. The experimental results are consistent with the published results obtained in the NEMP at a discrete set of locations, both from coring surveys and geoacoustic inversion results.

The SBCEX22 experiment was performed in the New England Mud Patch, about 100 km south of Martha's Vineyard in Massachusetts, USA. Here, we focus on a subset of the experiment, performed at the CMP between May 17 and May 19, 2022, where 17 single hydrophones were deployed and distributed over an area of approximately 250 km2 (see Fig. 1). These single hydrophones are deployed on TOSSITs (Zitterbart , 2022): simple devices that are moored to the seafloor. The embedded hydrophone on each moored TOSSIT is located 1 m above the seafloor.

FIG. 1.

(Color online) SUS-TOSSIT tracks on top of the mud layer thickness and water depth in the Central Mud-Patch. Water depth and layer thickness data courtesy of Goff (2019).

FIG. 1.

(Color online) SUS-TOSSIT tracks on top of the mud layer thickness and water depth in the Central Mud-Patch. Water depth and layer thickness data courtesy of Goff (2019).

Close modal

During the experiment, SUS explosive charges were detonated at 71 different locations in the CMP, (see Fig. 1). At each location, up to six SUS charges were detonated. The nominal detonation depth of each SUS charge is 18.3 m. These charges act as a strong impulsive acoustic source, loud enough to be recorded by the TOSSITs. Only signals that propagated between 0.5 and 10 km are considered.

The CMP was heavily surveyed in the past. We make use of previous sonar surveys of the area to determine the seafloor depth variation and the mud-thickness from two-way travel time data (Goff , 2019). Both seafloor depth and mud thickness were derived from two-way travel time data using a water sound speed of 1490 m/s, and the spatial map is presented in Fig. 1.

Multiple conductivity, temperature and depth (CTD) measurements were taken between May 17 and May 19, 2022, and they are presented in Fig. 2(a). Not all CTDs are taken in the same location and at the same time, therefore, it can be concluded that the sound speed structure in the water column varies in both space and time, enhancing the complexity of the inversion task (Dosso and Bonnel, 2023).

FIG. 2.

(Color online) (a) Experimental sound speed profiles of the water column measured between May 17 and May 19, 2022, in the Central Mud Patch. (b) Simulated 3D water sound speed field c0(x,y,z).

FIG. 2.

(Color online) (a) Experimental sound speed profiles of the water column measured between May 17 and May 19, 2022, in the Central Mud Patch. (b) Simulated 3D water sound speed field c0(x,y,z).

Close modal

The spatial inversion scheme proposed in this paper involves a multi-stage process. First, a NN is trained to invert geoacoustic parameters along each track. The predictions made using the trained NN are then used to invert the spatial variability of the geoacoustic parameters of interest. The parameters of interest in this work are the mud sound speed, the interfacial water sound speed, and the ratio between them. It should be noted that, even though the NN is used as part of the spatial inversion scheme, it is trained on simulated acoustic signals that propagate in a spatially invariant environment, as was done by Vardi and Bonnel (2024).

To demonstrate the feasibility of the proposed approach, the work follows the SBCEX22 study of the NEMP. Every step in the inversion pipeline is informed by the knowledge gathered on the NEMP, but the method proposed here is not exclusive to it.

To train the inversion NN, a training dataset was synthesized using computer simulations. The dataset is composed of simulated pressure time-series acoustic signals originating from a SUS charge and traveling to receivers 1 m from the seafloor (just like the TOSSITs used in SBCEX22). KRAKEN was used to simulate these signals; a normal-modes-based acoustic propagation simulation program (Porter, 1992), suited for use in shallow underwater environments (Jensen, 2011). These simulations were run using a variety of parameterized underwater environments, which were all modeled using a geoacoustic model composed of a single sediment layer over a bottom half-space.

Both the sediment layer of thickness h1 and the bottom half-space are defined by their constant sound speed c, bulk density ρ, and compressional wave attenuation α. The values of interest, sought for in the inversion scheme, form the parameter grid which is used to generate a variety of different environments through which an acoustic signal would theoretically propagate. The geoacoustic parameters of interest are the interfacial water sound speed cw=c0(z=h0), sediment sound speed c1, layer density and layer thickness of the sediment layer, ρ1 and h1 respectively, and the bottom half-space sound speed and density cb and ρb, respectively. A total of 25 000 environments were generated using the parameter grid defined with the values presented in Table I.

TABLE I.

Geoacoustic parameter table used to generate underwater environments for training dataset simulations.

Parameter Unit Bounds/value Step
Water column 
c0(z=5m)  m/s  [1482, 1494]  Random 
c0(z=20m)=c0(z=40m)  m/s  [1482, 1494]  Random 
c0(z=h0)  m/s  [1482, 1494]  Random 
Seafloor depth h0  [60, 80]  Random 
Water density ρ0  kg/m3  1000  Fixed 
Mud layer 
Thickness h1  [5, 15] 
Sound speed c1  m/s  [1450, 1510] 
Density ρ1  kg/m3  [1500, 1700]  0.1 
Attenuation α1  dB/λ  [0.01, 0.07]  Random 
Bottom half-space 
Sound speed cB  m/s  [1700, 1900]  20 
Density ρB  kg/m3  [1900, 2100]  0.1 
Attenuation αB  dB/λ  0.25  fixed 
Sound receiver 
Source depths zs  [10, 30] 
Source range r  km  [0.5, 10]  0.25 
Receiver depth zr  1 m from bottom   
Parameter Unit Bounds/value Step
Water column 
c0(z=5m)  m/s  [1482, 1494]  Random 
c0(z=20m)=c0(z=40m)  m/s  [1482, 1494]  Random 
c0(z=h0)  m/s  [1482, 1494]  Random 
Seafloor depth h0  [60, 80]  Random 
Water density ρ0  kg/m3  1000  Fixed 
Mud layer 
Thickness h1  [5, 15] 
Sound speed c1  m/s  [1450, 1510] 
Density ρ1  kg/m3  [1500, 1700]  0.1 
Attenuation α1  dB/λ  [0.01, 0.07]  Random 
Bottom half-space 
Sound speed cB  m/s  [1700, 1900]  20 
Density ρB  kg/m3  [1900, 2100]  0.1 
Attenuation αB  dB/λ  0.25  fixed 
Sound receiver 
Source depths zs  [10, 30] 
Source range r  km  [0.5, 10]  0.25 
Receiver depth zr  1 m from bottom   

Additionally, to increase the ability of the NN to generalize to dynamic ocean conditions, the water sound speed c0(z) and the compressional attenuation α1 were randomly varied. The water sound speed was defined at four discrete depth points: at 0, 20, 40 m, and at the seafloor h0 with values randomly picked between 1482 and 1494 m/s. These values were informed by the CTDs taken during the experiment [see Fig. 2(a)]. Additionally, we set c0(20)=c0(40)c0(0) in order to better represent the water column behavior seen in Fig. 2(a).

For each environment, different source depths zs and distance from receiver r were used (defined in Table I) to simulate different source-receiver geometries. The receiver depth zr was known to be 1 m from the seafloor, and as such, it was fixed to be zr=h01. The total number of different geometries used was 440 (see Table I). In total, using all possible combinations of parameters, 11 000 000 acoustic signals were generated.

The generated acoustic signals are 2 s-long, sampled at 1000 Hz. Thus, the broadband simulations were performed at frequencies from 0.5 to 500 Hz with 0.5 Hz steps, resulting in 1000 simulation runs using KRAKEN for each signal. A total of 41 orthogonal modal depth functions Ψm(z,fn) and their associated wave numbers krm(fn) were calculated for each frequency bin fn. The modal depth functions Ψ adhere to the normalization condition such that
(1)
where in our specific case, water density is assumed to be constant, thus, ρ(zs)=ρ0.
Using normal-mode theory, range-independent frequency-domain acoustic pressure signals were calculated with (see Jensen, 2011, Chap. 5)
(2)
where S(zs,f) is the depth-dependent SUS source signature synthesized using instructions detailed in Wilson (2020).

An inverted Fourier transform was applied to each signal pf to transfer it to the time domain pt(r,zr,zs;t). The time domain signal was then normalized by its maximum absolute value. Last, the signal was time-shifted to have |pt(t0)|=0.1, where t0 is the first time that the absolute value of the signal is above an amplitude of 0.1.

The NN is used to locate the source and invert geoacoustic parameters using a recorded acoustic signal that propagated along a source-receiver track. The inversion NN architecture and training procedure are very similar to that described by Vardi and Bonnel (2024), but with some notable differences, such as the addition of a pooling layer, some changes in the properties of the layer, and a different data augmentation procedure. All of these will be described here.

To train a NN, Flux.jl was used; a Julia Programming Language package for DL (Innes , 2018). The NN architecture is composed of a chain of one-dimensional- (1D) convolutional layers, a max-pooling layer, and two dense layers. The activation function applied to the output of each layer is the ReLU (Ramp) function. Between each layer, a batch normalization layer is placed after the output of a layer, but before the activation function. The specifics of the NN architecture are given in Fig. 3, including the size and number of kernels for the convolutional layers and the number of weights for the dense layers. More information on NN architectures and layer types can be found in Aggarwal (2023).

FIG. 3.

(Color online) Layers and their parameters used as part of the inversion NN architecture.

FIG. 3.

(Color online) Layers and their parameters used as part of the inversion NN architecture.

Close modal
The training dataset is made up of labeled acoustic signals pti that were synthesized using the instructions presented in Sec. III A. Each signal pti is labeled using a vector of inversion parameters, which were used to generate the signal,
(3)
Some parameters needed to generate the signals, e.g., water Sound Speed Profile (SSP), are not included in ui; they are considered nuisance parameters.
The role of the NN is to find an approximate inverse operator such that
(4)
where F is the forward operator F(ui)=pti and F̂1 is an approximation to the non-linear inverse operator.

To achieve this, the NN is optimized through a training process, utilizing the synthesized training dataset to estimate the inverse operation. Before training the NN, each signal goes through an augmentation process. First, Gaussian noise is added to the signal to simulate a realistic recorded signal. Noise is added using a randomly chosen signal-to-noise ratio (SNR) with a value randomly chosen between 10 and 25 dB. The SNR is calculated over the entire signal and is defined in dB, 10log10(Psignal/Pnoise), where P is the variance of the signal. The signal is then time-shifted with a randomly chosen value between −50 and 300 ms, to enable a shift-invariant NN behavior (negative values shift the signal left and positive values shift the signal right). Shift-invariant NNs are robust to time shifts of the input signal, a desirable property, as the exact beginning of a signal in the time window is not precisely known. Moreover, it forces the NN to focus on the time dispersion of the signal, a behavior attributed to normal-mode propagation in shallow water.

To train the NN, mini-batches of 512 training samples were fed to the NN, and the distance between the true and predicted labels was computed using a mean square error loss function. After each batch, the ADAM optimizer was used to minimize the loss function by tweaking the NN's weights and kernels with backpropagation. An epoch had lapsed once all the training dataset was fed entirely through the NN in mini-batches. The NN was trained for a total of 50 epochs. The ADAM optimizer was initially used with a learning rate of 103 but was changed every epoch using a Cosine-Annealing parameter scheduler with a period of ten epochs and a final value of 105.

To make sure that the NN was not overfitting on the training data, the performance of the NN was evaluated on a validation dataset every 1000 minibatches. The validation dataset was not used for training and was created by extracting 0.01% of the signals included in the training dataset prior to starting the learning process. The NN was said to generalize well if the loss function on the validation data set was minimized during the training process.

To estimate the uncertainties of the NN predictions, eight different NNs were trained using the procedure described previously. Each NN was initialized with different weights and a random number seed generator. The final output from the resulting model was defined as the mean of the outputs from all individual NNs. The uncertainty was estimated by calculating the standard deviation of each output label from the eight NNs, resulting in a final output u¯i±σi where σ is a vector of standard deviations for each of the labels in the output vector u.

Once individual inversion results were obtained using the trained NNs, they were used to invert the spatial dependence of the sound speed. In this paper, we are most interested in inverting for the sediment sound speed and SSR. Individual inversion results are assumed to be average values along each track. Spatial inversions for the sediment sound speed c1(x,y) or the water sound speed at the interface cw(x,y) were performed separately. Thus, the following description of the inversion scheme is presented for a general compressional sound speed c, but applies to both cases.

The spatial inversion scheme is inspired by the linear inverse theory for tomography problems (Aster , 2019). As such, the inverse solution is the solution to the linear equation
(5)
where dRntracks×1 is the vector of travel times along each track, mRncells×1 is the vector of average slowness 1/c¯, and GRntracks×ncells is composed of ray path lengths within corresponding cell blocks.

To work with tracks that were used during SBCEX22, the experimental area is converted to a Cartesian coordinate system. The conversion is approximated using the Universal Transverse Mercator (UTM) coordinate system. The new coordinate system is normalized by both the maximum values of the converted x and y track coordinates to have the southwest corner of the area originate at (0, 0).

A two-dimensional (2D) Cartesian grid is then discretized using discrete steps of 0.7 km in both x and y directions, creating a grid of ncells cells. The matrix G is then constructed by calculating the path lengths in each cell for each track. Track density is the number of tracks crossing a single cell, and can be visualized spatially in Fig. 4 for the SBCEX22 tracks.

FIG. 4.

(Color online) Track density representing the number of track crossings in the discretized experimental area in Cartesian coordinates.

FIG. 4.

(Color online) Track density representing the number of track crossings in the discretized experimental area in Cartesian coordinates.

Close modal
The vector d represents the average travel time across each track, which might be more intuitive to describe using the calculated sum,
(6)
where Gij is the path length of track i in cell j and mj is the average slowness 1/c¯j in cell j.
The solution to the linear problem is the regularized least squares solution,
(7)
where Cd is the data covariance matrix, Hx and Hy are regularization terms, and λ is a regularization constant. The choice of regularization is the second-order finite difference matrices for both x and y directions, resulting in a second-order Tikhonov regularization scheme. The second-order difference matrix in the x direction is defined as Hx, and in the y direction as Hy.

The data covariance matrix Cd is constructed using the uncertainty estimates σ from the output of the trained model. For each track i, the predictions for the sediment and interfacial water sound speeds are c¯1i±σc1i and c¯wi±σcwi, respectively. The mean of the standard deviations for all tracks σ¯c=(1/ntracks)i=1ntracksσci is then used to construct the data covariance matrix Cd=I/σ¯c2. This particular approach further regularizes the problem by increasing the smoothness of the inverted plane of varying sound speed.

The value of the optimal regularization parameter λ̂ is found using an iterative method. To prevent over or underfitting the noisy data d, λ is optimized over χ2 such that χ2ndata,
(8)
where
(9)
Typically, for geoacoustic inversion in mud, it is useful to present the results in terms of the SSR. To recover the spatially varying SSR, which is defined as ϒ, one has to invert for both parameters and use the ratio between them as
(10)

To gain confidence in the spatial inversion results from the experimental data, the spatial inversion scheme, described in Sec. III C, was first applied in a simulated case that mimicked acoustic propagation in an environment similar to CMP.

Acoustic signals propagating through a range-dependent environment along SBCEX22 tracks were synthesized. The range-dependent environment was simulated using multiple range-dependent geoacoustic parameters, informed by the information collected on the NEMP. The environment was modeled using the same 1-layer over bottom half-space geoacoustic model used previously. The spatially varying geoacoustic parameters were the depth of the water h0(x,y), the thickness of the mud, h1(x,y), the speed of the sound of the water, c0(x,y,z), and the speed of the sound of the sediment, c1(x,y).

Both the water depth h0 and the mud thickness h1 follow the two-way travel time data presented in Fig. 1. The water sound speed field c0 was simulated using a smoothly varying function derived from real-world CTDs taken during the experiment. Two CTDs that were taken on 17 May and 19 May 2022, were interpolated in depth and space, where one CTD was located at the origin (0, 0) and the other CTD at the northeast extrema (xmax,ymax), creating a 3D interpolation, and the resulting function c0(x,y,z) is visualized in Fig. 2(b). The sediment sound speed c1 was simulated using a synthetically generated 2D polynomial that is inspired by the results obtained on the experimental data, which will be presented later in Sec. V.

The other geoacoustic parameters that define the environment are constant and are presented in Table II.

TABLE II.

Geoacoustic parameters used for simulating the range-dependent synthetic test dataset.

Label Unit Value
Source depth  18.3 
Receiver depth  h0(x,y) − 1 
ρ1  kg/m3  1.6 
α1  dB/λ  0.05 
ρb  kg/m3 
αb  dB/λ  0.25 
Label Unit Value
Source depth  18.3 
Receiver depth  h0(x,y) − 1 
ρ1  kg/m3  1.6 
α1  dB/λ  0.05 
ρb  kg/m3 
αb  dB/λ  0.25 
For range-dependent simulations, acoustic signals were synthesized using the Adiabatic approximation of the normal-mode theory (see Jensen, 2011, Chap. 5),
(11)
where R=||r0,r1|| with r0=[x0,y0] and r1=[x1,y1] being the source and receiver positions, respectively. The modal depth functions Ψ(r,z) and associated modal wave-numbers kr(r) are now range-dependent. To solve Eq. (11), both Ψ and kr were calculated using KRAKEN on a discreet set of uniformly spaced coordinates along a track i. At each coordinate, the geoacoustic parameters between the source position r0 and receiver position r1 vary according to the simulated range-dependent environment. For each mode m, the wave number krm was interpolated in range, to be used in the line integral in Eq. (11). Following a trial-and-error procedure, the discretized tracks were spaced with uniform steps of 1 km (except the last section of the track), a choice made after experimenting with more finely spaced intervals that did not produce more accurate range-dependent simulations.
Given that the simulated fields for c1 and cw were spatially dependent, it is assumed here that the predicted sound speeds obtained using trained NN are the average slowness 1/c¯. Therefore, for assessing the inversion performance on the test dataset, the average simulated slowness had to be calculated for each track i using the line integral,
(12)

The individual inversion results for each track i are presented in Fig. 5 for the sediment sound speed c¯̂1, the interfacial water sound speed c¯̂w, and the SSR ϒ¯̂=c¯̂1/c¯̂w, together with the corresponding individual bias errors and normalized bias errors (NBEs) between predicted and simulated values. Individual predictions can be observed to succumb to inversion errors (Fig. 5). First, there is a mean bias error for c¯̂1 and more so for c¯̂w. The normalized mean bias error (NMBE) and the mean bias error are −0.34% and −5.0 m/s for c¯̂1, respectively, and −0.27% and −4.1 m/s for c¯̂w, respectively. Using the ratio between them reduces the mean bias error, with the SSR having a mean bias error of −0.0006 and NMBE of −0.062%, an order of magnitude lower.

FIG. 5.

(Color online) Individual predictions against the real simulated values for the sediment sound speed c¯1, interfacial water sound speed c¯w, and the ratio between them ϒ¯̂ for each track. A histogram of the errors between predicted and simulated average values is presented for each of the three quantities. Red dashed lines are the mean error and the blue dashed lines are the ±1 standard deviations. The NBE is also presented as a histogram for each of the three quantities.

FIG. 5.

(Color online) Individual predictions against the real simulated values for the sediment sound speed c¯1, interfacial water sound speed c¯w, and the ratio between them ϒ¯̂ for each track. A histogram of the errors between predicted and simulated average values is presented for each of the three quantities. Red dashed lines are the mean error and the blue dashed lines are the ±1 standard deviations. The NBE is also presented as a histogram for each of the three quantities.

Close modal

The intermediate inversion results were fed through the spatial inversion scheme to generate the spatial inversion predictions for ĉ1(x,y) and ĉw(x,y). The spatial inversion for the SSR (ϒ) is the ratio between the two spatial inversions ϒ̂=ĉ1(x,y)/ĉw(x,y). All spatial inversion results are presented in Fig. 6. The bias and absolute errors between the simulated and predicted fields are presented in Fig. 7.

FIG. 6.

(Color online) Simulated and predicted fields for the sediment sound speed c1(x,y), interfacial water sound speed cw(x,y), and the SSR ϒ(x,y) using signals from the synthetic test simulations.

FIG. 6.

(Color online) Simulated and predicted fields for the sediment sound speed c1(x,y), interfacial water sound speed cw(x,y), and the SSR ϒ(x,y) using signals from the synthetic test simulations.

Close modal
FIG. 7.

(Color online) The bias and absolute errors between the simulated and predicted fields for the sediment sound speed c1(x,y), interfacial water sound speed cw(x,y), and the SSR ϒ(x,y) using signals from the synthetic test simulations.

FIG. 7.

(Color online) The bias and absolute errors between the simulated and predicted fields for the sediment sound speed c1(x,y), interfacial water sound speed cw(x,y), and the SSR ϒ(x,y) using signals from the synthetic test simulations.

Close modal

The spatial inversion scheme appears to successfully reconstruct the spatial structure of the simulated field for the sediment sound speed c1(x,y), especially where the track density is highest. This observation can be reinforced by a qualitative comparison of the simulated and predicted fields in Fig. 6 and, more importantly, by looking at the absolute errors in Fig. 7. The absolute errors are significantly lower where the track density is highest on the eastern part, where some of the predicted cells have close to perfect agreement. The errors in the western part are higher, reaching an absolute value of 7.8 m/s, which is expected due to the low track density in that area, where most of the cells do not have track crossings.

The inversion result for the interfacial water sound speed field ĉw(x,y) contains clear spatial bias errors. However, despite the bias errors, the predicted field still trends similarly to the simulated field: an increase in sound speed from west to east, and at a similar rate of increase. Moreover, as previously observed, absolute errors are lower in the eastern part of the area, where the track density is highest.

Most interestingly, the resulting inverted SSR field ϒ̂(x,y) is the most accurate result of the three. The reconstructed field contains the lowest errors, both bias and absolute. In the eastern part, there is good agreement between predicted and simulated fields (Fig. 6). This is reinforced by the low absolute errors seen in Fig. 7, which are also lower in the eastern part.

Another way to visualize the performance of the inversion scheme is by averaging the predicted fields in one of the two dimensions x and y and plotting the resulting curves along the other dimension (Fig. 8). Here, the resulting curves for the simulated and predicted SSRs are in almost perfect agreement. The curves for c1 are also in good agreement, while the curves for cw show a bias error between the simulated and predicted cases.

FIG. 8.

(Color online) Mean simulated (blue) and predicted (orange) sediment sound speed (c1), interfacial water sound speed (cw), and SSR (ϒ) along the x and y axes. The mean is calculated using the fields presented in Fig. 6.

FIG. 8.

(Color online) Mean simulated (blue) and predicted (orange) sediment sound speed (c1), interfacial water sound speed (cw), and SSR (ϒ) along the x and y axes. The mean is calculated using the fields presented in Fig. 6.

Close modal

A case can be made that prediction errors are more pronounced in areas where the track density is lowest (see the track density in Fig. 4). Although quite an intuitive conclusion, to strengthen this claim, we plot the track density as a function of absolute errors, along with a trend line using a linear fit, for all three spatial inversions. These are presented in Fig. 9 

FIG. 9.

(Color online) Absolute errors between simulated and predicted fields in each grid cell for the sediment sound speed c1, interfacial water sound speed cw, and the SSR ϒ, as a function of track density in space (excluding values for track density of 0). A linear trend line is shown with the red line for each of the three quantities. However, note that a linear fit is only statistically significant for the third plot.

FIG. 9.

(Color online) Absolute errors between simulated and predicted fields in each grid cell for the sediment sound speed c1, interfacial water sound speed cw, and the SSR ϒ, as a function of track density in space (excluding values for track density of 0). A linear trend line is shown with the red line for each of the three quantities. However, note that a linear fit is only statistically significant for the third plot.

Close modal

Whereas all three plots show a decreasing trend for absolute errors for increasing track density, a linear fit to these data is statistically significant only for the third case, in the plot for ϒ. Thus, it is advised to restrict results for cells with a track density above a certain threshold, an approach that is taken for the experimental results.

The spatial inversion results for the SBCEX22 dataset were obtained using the same approach proposed in Sec. III C. First, geoacoustic parameters were predicted on individual tracks and presented as histograms (Fig. 10). The individual results were then used to predict the spatial fields for ĉ1,ĉw, and ϒ̂ (Fig. 11). A longitude-latitude coordinate system was used instead of Cartesian, in order for the reader to situate himself more easily in the CMP (Cartesian coordinates were still used for the spatial inversion scheme). The predicted fields show only inversion results in cells where the track density is 3 or higher. The decision is informed by both the results obtained for the simulated case (see Fig. 9) and the fact that the inversion results for cells that do not have crossing tracks can not provide any meaningful information.

FIG. 10.

(Color online) Histograms of predictions for the sediment sound speed c¯̂1, interfacial water sound speed c¯̂w, and the SSR ϒ¯̂ using signals obtained during the SBCEX22 experiment.

FIG. 10.

(Color online) Histograms of predictions for the sediment sound speed c¯̂1, interfacial water sound speed c¯̂w, and the SSR ϒ¯̂ using signals obtained during the SBCEX22 experiment.

Close modal
FIG. 11.

(Color online) Spatial inversion results for the sediment sound speed ĉ1, interfacial water sound speed ĉw, and the SSR ϒ̂ with the mean mean values along the longitude and latitude. Predictions made using signals obtained during the SBCEX22 experiment.

FIG. 11.

(Color online) Spatial inversion results for the sediment sound speed ĉ1, interfacial water sound speed ĉw, and the SSR ϒ̂ with the mean mean values along the longitude and latitude. Predictions made using signals obtained during the SBCEX22 experiment.

Close modal

Currently, the literature is sparse with results showing a detailed spatial structure for the sediment sound speed layer or the SSR at the NEMP. However, we will attempt to make a comparison with existing coring survey data and geoacoustic inversion results at discreet locations in the CMP and to compare the spatial trends and individual results as seen in Fig. 10 since they can be considered as non-spatial results with uncertainty and can be compared to more traditional inversion output generated on a single track.

The values of the inverted sediment sound speed field ĉ1(x,y), ranging between 1472 and 1492.0 m/s, increase from the southwest to the northeast. Here, the sediment layer is considered to be the CMP's mud layer. The increase in mud sound speed can be roughly compared to the results obtained by coring data in Chaytor (2022), where the mean grain size of the upper 5 cm of the mud layer that was measured at different sites along the CMP and shows an increase in the mean grain size from west to east. The increase in sediment mean grain size correlates with an increase in compressional sound speed (Bachman, 1985), thus corroborating an increase in mud sound speed from west to east. The inversion results for the sound speed of the mud layer were also obtained by Jiang (2023) for the SC2 site and were spatially compared with the results of Belcourt (2020) for the SWAMI and VC31-2 sites (their locations are shown in Fig. 11). Using results for these three locations, they have suggested that the interfacial sound speed of the mud increases from the northwest to the southeast, nevertheless demonstrating an eastward increase. Interestingly, in our inversion results, a region of local maxima for the sediment sound speed can be seen around the SWAMI site. More data is needed to explore whether this is a real representation of the sediment or an inversion artifact.

Inverted values for the interfacial water sound speed field ĉw(x,y), ranging between 1484.0 and 1500.9 m/s, decrease from southwest to northeast. However, the predicted SSR field ϒ̂ exhibits a spatial structure similar to ĉ1: an increase in values, ranging between 0.981 and 1.005, from the southwest to the northeast, and the 95% credibility interval of the SSR being from 0.9860 to 1.0024 (derived from Fig. 10). Again, a region of local maxima for the SSR can be observed around the SWAMI site, although less noticeable than that for the sediment sound speed. The increase in SSR can be further visualized by averaging along the latitude or longitude (see Fig. 11). These results can be compared with SSR data from both in situ acoustic core head data and ex situ core logger data, provided by Garcia (2024), where they report an increase in SSR from west to east. The same trend is also demonstrated in Jiang (2023), where they report an increase in SSR through the three stations with an eastward trend from northwest to southeast, with 95% credibility intervals for the SSR values being from 0.977 to 0.989 for the VC31-2 station, from 0.982 to 0.990 for the SWAMI station, and from 0.993 to 0.999 for the SC2 station. For individual comparisons, Bonnel (2024) used a trans-D Bayesian inversion method to invert the SSR using impulsive signals generated during SBCEX21 and SBCEX22. The resulting values for the SSR's 95% credibility intervals are reported to be between 0.977 and 1.006 for 2022 and between 0.985 and 1.010 for 2021 for a single track in the CMP, corroborating our own results (see Fig. 10).

Last, in situ sound speed measurements were also reported in Yang and Jackson (2020) following the 2017 experiment, and although water conditions were different then, they also reported an increase in SSR from west to east.

Traditional methods for geoacoustic inversion, while producing reliable inversion results, are hampered by the inference speed. Using DL, we can circumvent this problem by investing computing power in training a NN which can then perform the inversion task at fractions of the cost. The current NN used for this task is not as accurate as what a traditional state-of-the-art inversion scheme would output, but it can still prove very useful when large amounts of data are analyzed. On one hand, inversion on an in situ signal can give useful information on the fly, but also, using sufficiently large datasets, spatial reconstruction of a geoacoustic parameter of choice is proven to be achievable thanks to the newly found inference speed.

In this paper, we introduced a spatial inversion scheme for the sediment sound speed and SSR that utilizes trained NNs. We have applied this method on range-dependent simulated signals with success, and the predicted spatial structures of the sediment sound speed and SSR were in agreement with the simulated environment. The same scheme was then applied to the SBCEX22 data, inverting for the spatial structure of both the sound speed of the mud layer and the SSR. The results have shown an increase in both quantities from southwest to northeast in the CMP. This trend has also been shown in the literature, derived from inversions and coring surveys at discrete locations along the CMP, suggesting an increase in both c1 and SSR from west to east, commensurate with our results.

Our proposed approach relies on low-cost equipment: 17 TOSSITs and a handful of strong impulsive sources (which could be easily replaced by off-the-shelf airguns or other impulsive sources such as rupture discs, see Bonnel , 2023; McNeese , 2020) enabling low-overhead experiments. Rather than relying on experimental complexity, our spatial inversion result is made possible by the use of DL. Similar methods could be massively applied to existing or future data sets, with the hope of enabling new insights into seabed acoustics.

The authors have no conflict of interest to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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