Information geometry analysis example for absolute and relative transmission loss in a shallow ocean

A major goal of underwater acoustical modeling is to gain insight into how acoustical measurements, such as transmission loss (TL), relate to environmental parameters and, thus, estimate what environmental information is or is not encoded in acoustical data. Intuitively, if physically distinct seabeds lead to nearly identical acoustical data, those data cannot be used to distinguish the seabeds. Model parameters associated with these indistinguishable seabed properties are said to be unidentifiable. Parameter identifiability can inform both experimental design and model selection. This paper demonstrates an information geometry approach to choosing experimental design such that information about environmental parameters is maximized.

Information geometry^1–3 is a branch of mathematics that combines statistics and information theory with differential geometry. A multi-parameter model can be interpreted geometrically as a Riemannian manifold, known as the model manifold. The model manifold exists as a curved, high-dimensional hyper-surface in an ambient “data space,” where each point on the manifold corresponds to a specific model prediction. The dimensionality of the model manifold is the number of parameters to be inferred, where the parameters act as coordinates on the manifold. The dimensionality of the ambient data space corresponds to the number of model predictions. In this paper, we use a two-parameter TL model with a vertical line array (VLA) of 15 receivers, resulting in a two-dimensional (2D) model manifold embedded in a 15-dimensional (15D) ambient data space.

The geometric structure of the model manifold connects the information content of data to model parameters. Distance on the model manifold quantifies the statistical distinguishability of different model predictions. The Fisher information matrix is a distance metric on the model manifold, providing a connection to other information-theoretic tools for parameter identifiability analysis such as Cramér-Rao bounds. Importantly, however, the model manifold connects this local measure of identifiability with nonlocal and global properties of the parameter space. The distance between different parameter values in the ambient data space incorporates all nonlinearities of the model into a single measure of statistical distinguishability.

Additionally, the model manifold encodes global parameter sensitivities within the structure of its finite boundaries, a nonlinear effect. Model manifold boundaries correspond to physically interpretable simplified models, such as completely removing sediment layers from a more complex geoacoustic profile. Algorithms such as the manifold boundary approximation method (MBAM)^4,5 can be utilized to find these manifold boundaries, identifying reduced-order models that can retain model accuracy while being physically interpretable. The visualizations in this paper motivate starting cases for testing MBAM on ocean sound propagation models. The effectiveness of model reduction methods such as MBAM rely on the fact that model manifolds in general contain a hierarchy of widths, giving model manifolds a “ribbon-like” structure. This structure is equivalent to saying that multi-parameter models, in general, manifest parameter sensitivities spanning many orders of magnitude, a phenomenon known as sloppiness,^6–8 which is characterized by roughly log-linear spaced eigenvalues of the Fisher information matrix. Wide dimensions of the model manifold correspond to stiff parameter combinations to which the model is most sensitive, while thin dimensions correspond to sloppy parameter combinations.

In summary, the model manifold directly connects the information content of data to model parameters in both local and global ways. Thus, the model manifold approach extends traditional, local sensitivity analyses in ways that have potential applications for model selection and experimental design. For experimental design, the model manifold depends on independent variables such as frequency and source-receiver geometry, in addition to the model parameters and, thus, quantifies the potential information content in data for different experimental setups. A more “spread out” model manifold corresponding to some experimental design choice indicates that choice of acoustical data contains more information about parameters of interest, making parameter inference more accurate. Information geometry is, therefore, a natural framework to explore questions of experimental design.

This paper demonstrates the utility of the model manifold in guiding experimental design by quantifying the impact that using relative TL has on the information content of data for the example of a 15 element VLA in a shallow ocean environment. The TL model manifold is generated by a range-independent normal-mode model, and model manifold visualizations are shown of 2D slices of the data space at different receiver depths to illustrate the impact of experimental design. Principal Component Analysis (PCA) is employed to obtain representative 2D visualizations of the principal variations across a 15D model manifold. Additionally, Euclidean distances on the manifold between seabed types are calculated in the 15D data space. Visualizations of the model manifold for both absolute and relative TL for different receiver depths show that, in this example, relative TL model manifolds are more compact, with seabeds types that are less statistically distinguishable and do not follow expected similarity trends.

This work uses a relatively simple ocean environment with an isovelocity water sound speed of 1500 m/s, water depth of 75 m, and a seabed consisting of a 35 m sediment layer over a half space. The sediment layer is parameterized using the compressional sound speed $c p$⁠, density $ρ$⁠, and attenuation $α p$⁠, which is held fixed at $α p = 0.63$ dB/m/kHz in this paper. This value corresponds to the estimated value of attenuation for silt given in Jensen et al.:¹⁰ for $c p = 1575$ m/s, $α p = 1.0$ dB/ $λ = 0.63$ dB/m/kHz, which is larger than for most sediments. The experimental design parameters include $z s = 6$ m, $r = 3$ km, 15 receiver depths z evenly spaced between 5 and 75 m, and a frequency of 100 Hz. A table of the ORCA parameters used, including half space parameters, is included in the supplementary material.

The model manifold is an N-dimensional surface consisting of all possible model predictions for N variable parameters, embedded in an M-dimensional data space with perpendicular axes, in this case, corresponding to TL at M choices of receiver depth. For this example, two environmental parameters, sediment sound speed $c p$ and density $ρ$⁠, are varied (N = 2), with sampled points shown in the parameter space in Fig. 1(a). Model manifolds are constructed by calculating the TL for each of these parameter choices at two different receiver depths (M = 2). The model manifolds corresponding to TL at receiver depths of 5 and 65 m, and 25 and 30 m, are shown in Figs. 1(b) and 1(c), respectively. These model manifolds demonstrate the characteristic ribbon-like structure of a sloppy model due to the nonlinear model transformation. These model manifolds are qualitatively different, signaling differences in information content between channels; this observation holds for selection of any subset of receiver depths.

The model manifolds in Figs. 1(b) and 1(c) retain the same adjacency relations of the colored edges in Fig. 1(a), other than the apparent self-intersection of the manifold, which in general disappears in higher dimensional data spaces. (Note that any true self-intersection of the model manifold does not correspond to any continuous shorter path between parameter choices in parameter space; practically this just introduces additional ambiguities in projecting measured data onto the model manifold.)

The location for five seabed sediment types (mud, clay, silt, sand, gravel) are also indicated on the parameter space and the TL manifolds in Fig. 1. (The parameter values are listed in the supplementary material.) Distance between two points on the model manifold in data space quantifies the statistical distinguishability between those parameter values. The Euclidean distance between model outputs for two seabed environments in data space is referred to as “information distance” in this paper. This is not to be confused with geodesic distance between seabeds along the model manifold. Seabed types with a small information distance, such as mud and clay in Fig. 1(b), exhibit a degree of similarity that can pose a challenge in geoacoustic inversion. In contrast, mud and clay have larger information distances from the other three seabeds—silt, sand, and gravel—making them more distinguishable from those seabed types.

Also, notice that the model manifolds in Figs. 1(b) and 1(c) have a long dimension, corresponding to the more identifiable sound speed parameter, while the short dimension corresponds to the sloppier density parameter. This distinction is especially true in Fig. 1(c) for predictions made at nearby receiver depths of 25 and 30 m; the very narrow portion of the manifold indicates that, for TL > 72 dB, these receiver depths contain no information about the density.

Due to measurement uncertainty and ambient noise in measured TL, data typically do not lie on the model manifold. Black dots in Figs. 1(b) and 1(c) represent hypothetical noisy TL data, and the black arrows represent possible projections of this “data” onto the model manifold. Geoacoustic inversion projects noisy data onto the model manifold by acting as a non-unique pseudo-inverse, corresponding to a specific choice of loss function in the optimization process, often the sum of squares error that minimizes the Euclidean distance between observed data and model predictions.

To reduce the noise in the data that is common to all measurements, relative TL is often used in ocean acoustic applications instead of absolute TL. One way to calculate relative TL is to subtract the TL for one channel from the other channels, reducing the effective data space dimension from M to $M − 1$⁠. Using relative TL should in practice bring noisy data closer to the model manifold. The remaining distance between the data point and the manifold could be caused by model mismatch or sources of uncertainty that vary between measurements, such as array tilt for VLAs, individual sensor noise, and sound speed variability. Thus, using relative TL for the axes of data space corresponds to a different noise model where the noise is limited to those sources that are different between measurements.

As an example, Fig. 1(d) shows a relative TL model manifold. The relative TL manifold is the absolute TL manifold in Fig. 1(b) with axes of relative TL at depths of 65 m and 5 m, with the TL at a depth of 30 m [y axis of Fig. 1(c)] subtracted from each channel. Notice that the relative TL manifold appears to be more compressed than the original manifold, with smaller information distances than for the absolute TL manifolds. This compression of the relative TL model manifold is explored more quantitatively in Sec. 3. While absolute TL is often considered an exclusively positive quantity, relative TL can be positive or negative because of how it is defined here. In this application, the total dB distance in data space is most relevant, not whether the values are positive or negative.

To allow for more accurate discussion of information distances in Sec. 3, an alternate method for obtaining 2D visualizations of the model manifold is introduced. The 2D visualizations in Figs. 1(b) and 1(c) each correspond to observations from two receivers. Inclusions of data from additional channels (e.g., 15 receiver depths) are more informative, but difficult to visualize. Low-dimensional visualizations of high-dimensional spaces are obtained via PCA. PCA finds a new basis for data space aligned with the directions in which the model output varies most. Points sampled from the manifold in 15D are first translated to be centered at the origin by subtracting the mean TL of all the data points for each channel. The shifted points are collected into the columns of a matrix D, and a singular value decomposition is performed: $D = U Σ V T$⁠, where U and V are unitary matrices. The columns of V define the new basis for data space and are known as the principal components. $Σ$ is a diagonal matrix of the singular values of D, which are the standard deviations of the sampled points projected onto each principal component. The first two principal components typically describe most of the variability in D, so the projection of the model manifold onto the first two principal components yields a low-dimensional visualization of the main features of the model manifold.

For the case of 15 receiver depths, a 2D PCA visualization of the model manifold is displayed in Fig. 2(a). The axes of Fig. 2(a) represent linear combinations of TL at all 15 depths projected onto the first two principal component directions, which captures 91.2% of the variation in the original 15D data. For Fig. 2(a) and the other PCA manifold visualizations in the paper, the exact values on the axes are not important, only the total distance in dB between points; the units are still dB, allowing for comparison to the widths of the manifolds shown in Fig. 1. The model manifold in Fig. 2(a) is wider and appears less folded over itself than the manifolds in Figs. 1(b) and 1(c), indicating an increase in the quantity of information relevant to distinguishing seabed parameters. The information distances between the five marked seabeds in the 15D data space are given in Fig. 2(b). These information distances differ slightly from the apparent distances in Fig. 2(a) based on the first 2D PCA model manifold because the information distances in Fig. 2(b) are calculated in the 15D data space.

The impact of using relative TL instead of absolute TL on the informativity of data for a VLA is explored in this section. Relative TL model manifolds in Figs. 3(a)–3(c) are created by subtracting the TL at reference depths of 20, 30, and 65 m, respectively, from all 15 channels. The relative TL manifold in Fig. 3(d) is calculated differently by subtracting from each channel the average model predictions across all 15 VLA elements, which can be thought of as subtracting the mean TL. All four relative TL manifolds are then projected into 2D using PCA. Information distances are calculated on the relative TL manifolds in 14D (or 15D) data space, and shown in Figs. 3(e)–3(h). Additionally, the supplementary material document contains model manifolds and information distance matrices relative to all 15 receiver depths on the VLA, as well as a table with average and median information distances, a plot presenting the PCA singular values, and a table indicating the percentage of variance explained by the 2D PCA visualizations.

Several notable changes in model geometry and parameter identifiability occur when using relative TL on the VLA. First, as seen in the relative TL model manifold in Fig. 1(d), the relative TL model manifolds in Figs. 3(a)–3(d) are more folded and compact than the absolute TL manifold in Fig. 2(a). For example, the model predictions in the first principal component in Figs. 3(b)–3(d) vary by two-thirds the dB range as Fig. 2(a). This compression of the relative TL model manifold is due both to additional folding and the overall reduced scale of the model manifold. Comparison of information distances in the full data space can provide a quantitative measure of the compression of the model manifold.

Relative TL model manifolds information distances, shown in Figs. 3(e)–3(h), are smaller than the absolute TL information distances shown in Fig. 2(b). First, the maximum relative TL information distances [darkest blue elements of Fig. 3(e)–3(h)] are 30% to 70% smaller than the absolute TL maximum information distance in Fig. 2(b). Additionally, the average information distances for relative TL are less than for absolute TL. The average information distance in Fig. 2(b) (excluding self-correlation) is 56.6 dB. In contrast, the average information distances in Figs. 3(e)–3(h) for relative TL obtained by subtracting TL at reference depths 20, 30, and 65 m, and by subtracting the mean TL, are 31.6, 20.1, 22.5, and 18.9 dB, respectively. Smaller information distances indicate that, in this example, data associated with different seabed parameters are less distinguishable than when using absolute TL.

Most importantly, the respective ordering of the information distances has changed. For absolute TL, the information distances follow the overall trend of increased reflectivity of the seabed, where gravel is furthest from, and, therefore, most distinguishable from, mud and clay, with information distances of $∼$ 90 dB (Fig. 2). However, for the relative TL manifolds in Figs. 3(a)–3(d), mud and clay are much closer to gravel in data space, and closer than gravel is to sand and silt, due to folding of the model manifold. This trend becomes clear upon inspection of the information distance matrices in Figs. 3(e)–3(h): mud and clay are anywhere between 5 and 15 dB closer to gravel than gravel is to sand and silt. Thus, the overall identifiability of the seabed parameters is reduced when using relative TL, due to the subtracting out of common features, which increases the folding of the model manifold. In context of Eq. (1), it is not clear exactly what structure in the data is lost when using relative TL; however, qualitatively it appears that some overall bottom loss due to the sediment has been removed. The practical implication of this seabed reordering is that geoacoustic inversions to determine seabed parameters may be more challenging for relative than absolute TL, depending on how much noise reduction is obtained by use of relative TL.

The reference hydrophone used to calculate relative TL changes the model manifolds, effectively increasing or decreasing the information contained in the relative TL data. For example, contrast using TL relative to 30 m [Figs. 3(b) and 3(f)] to using TL relative to 20 m [Figs. 3(a) and 3(e)]. The average information distance in Fig. 3(f) is 20.1 dB, while the average information distance in Fig. 3(e) is 31.6 dB. Therefore, parameter inference using TL relative to 30 m may be more challenging, with larger uncertainty in parameter estimates, than if a reference depth of 20 m was used. Specifically, the information distance between mud and clay for absolute TL is 7.3 dB, as shown in Fig. 2(b). The information distances between mud and clay for reference depths of 30 and 20 m, respectively, are 4.6 and 7.1 dB [Figs. 3(f) and 3(e)]. Thus, using a reference depth of 30 m yields a model manifold in which mud and clay are less distinguishable than when using absolute TL, while using a reference depth of 20 m yields a distinguishability of mud and clay nearly identical to the absolute TL case. Information distances for averaged relative TL in Fig. 3(h) show similar compression as Fig. 3(f), with the distance between mud and clay being 4.5 dB, and an even smaller average information distance of 18.9 dB. Thus, if relative TL is used, distances between seabed locations on the model manifolds can be examined to indicate which method and which reference depth maximizes the information with respect to the parameter of interest.

While more studies need to be done looking at relative TL for changing different experimental design parameters, this work provides insights into the loss of information content that may occur when using relative TL for a VLA. In this example, because of decreased and mixed up information distances from using relative TL, care should be taken when using parameter values inferred from relative TL. The increased uncertainties from using relative TL may propagate as these values are subsequently used in, for example, source ranging. Thus, the advantages of relative TL in reducing uncertainty due to noise in the data have to be balanced with the potential decrease in information content about the seabed parameters.

This work provides an example of how model manifolds can be used for optimal experimental design. The model manifold, an information geometry tool for parameter identifiability analysis, has been constructed for TL from the ORCA normal-mode sound propagation model for a shallow water case with a 15 element VLA at a single frequency, source depth and range. The wide dimension of the “ribbon-like” model manifolds correspond to the identifiable sound speed parameter $c p$⁠, while the thin dimension of the model manifolds corresponds to density $ρ$⁠, in some situations, indicating a sloppy parameter. The PCA method provides a way to obtain a 2D model manifold visualization projected from a 15D data space, while retaining maximal variation of the model output.

Comparisons of the model manifolds for absolute and relative TL reveals significant insights into information loss and changes in the model (manifold) structure that come from using relative TL, in the case of receivers on a VLA relative to one receiver depth, and for the case of subtracting the mean TL across VLA elements. These relative TL model manifolds are more compact than absolute TL manifolds and have smaller information distances in data space. Smaller information distances indicate that relative TL contains less information to distinguish certain seabed environments, with implications for geoacoustic inversion for seabed parameters. The compression of the relative TL model manifold can be seen as a symptom of having some overall influence of the seabed environment removed from the acoustical data. Substantially, when using relative TL, the ordering of information distances also changes such that information distance does not follow the overall trend of seabed reflectivity observed when using absolute TL. With relative TL, mud and clay appear closer to gravel than gravel is to sand and silt, implying that mud and clay are statistically more similar to gravel. With an understanding of the limitations of relative TL, the model manifold approach allows for selection of the reference depth that maximizes the information content in the relative TL data. Future work is required to explore the implications of relative TL for different experimental design choices, for example, considering TL relative to different source depths, frequencies, and source ranges (such as on a horizontal line array).

Other future work includes considering higher dimensional models (and therefore model manifolds), such as including more sediment layers and varying sediment attenuation in addition to sound speed and density. These higher dimensional models will not be amenable to direct visualization, but similar analyses can be performed by visualizing 2D slices of the model manifold. Additionally, future work will utilize differential geometry tools such as geodesics to obtain reduced order models in these more complex cases.

See supplementary material for seabed parameters, relative TL model manifolds and information distance matrices for all 15 reference depths, a table of mean and median information distances for different reference depths, and PCA singular values and explained variance percentages for all relative TL model manifolds.

The authors acknowledge the U.S. Office of Naval Research for their support of this work under Grant No. N00014-21-S-B001. We also thank the reviewers and the Associate Editor for their helpful suggestions.

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.