Joint geoacoustic inversion based on Pearson correlation coefficient constraints Open Access

Ocean acoustic modal group velocity dispersion data and recorded signal waveform data are widely used in the field of geoacoustic inversion,^1,2 source localization,³ and environmental inversion.⁴ However, geoacoustic inversion based on modal dispersion involves travel-time information but not information on seabed attenuation needed for amplitude information.⁵ Many characteristics of the acoustic field can be extracted from the broadband signals and utilized in geoacoustic inversions. Furthermore, different characteristics exhibit varying sensitivities to geoacoustic parameters. Previous joint inversion studies employed the multi-stage inversion approach⁶ to address the issue of specific sensitivity between various parametrizations and characteristics. However, errors in parameters at different stages are also propagated, making it difficult to adequately assess the error of inversion results.⁷ Waveform inversions utilize relative time or time correction among multimodal arrivals,^8–10 the inversion accuracy of which decreases with the increase in uncertainty error, i.e., the periodic jump phenomenon.

As the criterion principle of modal dispersion inversion is based on phase comparison, it is unable to provide an estimation of attenuation.⁵ In a single receiver modal context, attenuation must be estimated using mode amplitude considerations, such as modal amplitude ratios inversion⁶ or broadband waveform inversion. When one data feature cannot simultaneously invert all model parameters, joint inversion methods are generally used. Therefore, this Letter carries out a joint geoacoustic inversion of modal group velocity dispersion and the full waveform envelope obtained via the Hilbert transform (FWH), combining the two by incorporating a constraint based on the Pearson correlation coefficient (PCC)¹¹ and Wasserstein metric.⁸ However, it should be emphasized that joint inversion of different quantities derived from the same measurements may not always be ideal unless these quantities exhibit suitable independence.

The remainder of this Letter is organized as follows: Section 2 provides a short description of previous work related to modal dispersion geoacoustic inversion and waveform inversion. We describe our proposed joint inversion approach, including the formulation of the objective function and the optimization strategy. The performance and analysis of joint geoacoustic inversion are presented in Sec. 3. Finally, conclusions are given in Sec. 4.

In the traditional waveform inversion approach based on the Bartlett metric or $L 2$ norm misfit function, the periodic jump phenomenon of waveform signal is inevitable. More specifically, when the temporal shift between the synthesized waveform and the observed waveform exceeds half a period, the objective function is prone to falling into multiple local minima. This will affect inversion performance in terms of convergence and precision. Possible solutions to this challenge could include the implementation of the Wasserstein metric as the objective function. Compared to the point-by-point comparison of signals in the $L 2$ norm, the Wasserstein metric^19,20 used as the objective function allows for global signal quality comparisons, thus helping to alleviate the problem of local minima.

Figure 1(a) illustrates the overview of the joint inversion framework and outlines the relationships involved. This study implements warping and Hilbert transformations on T-F signals to extract dispersion curves and waveform envelope, respectively, for use as inputs in the joint inversion. The process involves dividing the full parameter space into two separate subspaces that are optimized and searched for using distinct genetic algorithms (GA). In essence, the dispersion inversion and waveform inversion are carried out independently, but are combined through a joint constraint, which takes account of the PCC between the parameter populations.

For each iteration, when an algebraic update is applied to a population in the dispersion inversion dataset, the PCC joint constraint is also updated and replaces the joint constraint in the objective function of the FWH inversion dataset, and vice versa.

Therefore, Eqs. (9) and (10) establish the objective function of the joint inversion under L₂ norm (L₂-J), while Eqs. (9) and (11) establish it under Wasserstein metric.

The simulation is range-independent, and the environmental characteristics are based on the general features observed in the Shallow Water 2006 (SW06) experiment.²⁵ The simulated environmental parameters are given in Fig. 1(b) and corresponding parameter settings are shown in Table 1. We now demonstrate the numerical experiment by presenting actual pulse simulations obtained by a source pulse with the frequency band 50–200 Hz, emitted at depth 22 m, and recorded 7 km away at depth 20 m, with a signal-to-noise ratio (SNR) of 6 dB. $SNR = 10 log ( σ s 2 / σ b 2 )$ can be expressed as a logarithmic ratio of the signal power $σ s 2$ to the Gaussian white noise power $σ b 2$ for the same bandwidth. The GA method with a population of 100 and a crossover rate of 0.5 is used to optimize $J D , FWH ( m )$⁠.

This numerical example shows the advantage of joint inversion method on reducing geoacoustic inversion uncertainty. To estimate the performance of joint inversion algorithm, we need to consider the calculation time as well as the generation of convergence iteration. For each approach, the GA was run over 100 with different starting values through Monte Carlo sampling. The average running times of the dispersion curve inversion based on L₂ norm (L₂-D), FWH inversion based on L₂ norm (L₂-FWH), L₂-J, and Was-J, are 22.5, 29.3, 45.5, and 37.4 min, respectively. The runtime of joint inversions is significantly higher than that of the individual inversions. According to Fig. 2, the standard deviation of the objective function based on the Was-J is the lowest. The color interval in generations 4–8 is the iteration interval after the implementation of joint constraint (when the value of the objective function mismatch term decreases to one-third of that during the original generation). Based on the inset in Fig. 2, it is observed that the joint term constraint leads to a better value of H in the early stages of the optimization process.

Was-J results can be summarized by parameter distributions derived from 100 optimization runs, as shown in Fig. 3, where the green color indicates the best model. The sediment depth and soundspeed, as well as basement soundspeed, are superimposed on Fig. 3(a) to facilitate comparison. For the soundspeed parameters, the inversion results compared to the true value show a variation range of ± 375 m/s in the basement layer, while in the sedimentary layer, it is only ± 20 m/s. Therefore, the distribution of the basement soundspeed is not included in Fig. 3, due to the lack of representativeness in the results obtained from different inversion methods. When compared to the results of the L₂-J with high uncertainty and standard deviations, the Was-J clearly outperforms. First, it is evident that the mean values of all parameter distributions are in close proximity to the true values. The sediment parameters (⁠ $H$⁠, $c s$⁠, and $α s$⁠) are well estimated. Moreover, the sediment soundspeed $c s$ and attenuation $α s$ exhibit high peakedness and concentration in their unimodal distributions, reflecting the low uncertainty in Was-J inversion. According to Table 1, the overall NRMSE of parameters in Was-J is 4.35 × 10⁻⁵, and deviation range of other parameters, except $α b$⁠, is 0.1%–2.97%, both of which are better than FWH and L₂-J. On the other hand, L₂-FWH method produces the highest NRMSE in most cases and performs poorly in estimating $H$⁠, $ρ s$⁠, and $α b$⁠.

Furthermore, as shown in Fig. 3(c), the uncertainty interval of dispersion curve of mode 4 is large, particularly near the Airy phase. This is attributed to the inaccuracy in extracting the fourth mode during the warping transformation and the parameter sensitivity of this mode. Finally, the parameter distribution of the sediment thickness in Was-J is a unimodal distribution, whereas in the FWH, it is a bimodal distribution (which is not shown limited by the length of the paper). This demonstrates that the joint inversion approach presented in this research successfully combines the advantages of dispersion inversion for sedimentary sound speed with the advantages of FWH for attenuation. Although it is not feasible to directly quantify the uncertainty associated with the joint inversion method involving dependent datasets, conducting a statistical analysis of the estimates obtained from multiple inversions with various initial values, crossovers, and mutations, can provide insights into the overall uncertainty of the inversion results. This analysis reveals that the overall uncertainty, considering the various sources of uncertainty, is relatively minimal for the inversion results.

This joint inversion approach is also applied to the Workshop's 97 geoacoustic inversion benchmark AT cases, with details of the environmental settings and search intervals provided in the reference.²⁶ The results are summarized in Table 2.

This Letter develops a joint inversion approach of modal dispersion and FWH data based on the PCC. Essentially, PCC is a joint constraint on the temporal dispersion characteristics and attenuation dispersion characteristics. By simultaneously coupling these two individual inversions, this joint approach prevents the error propagation associated with two-staged inversion approaches. In addition, this Letter provides further evidence that incorporating the Wasserstein metric to formulate the objective function demonstrates enhanced robustness in joint inversion, resulting in decreased uncertainty of certain inverted parameters, as compared to employing the L₂ norm.

Specifically, we successfully implement the Wasserstein metric in joint geoacoustic inversion, by using the softscaling functions to transform acoustic signals into probability densities. The simulation results demonstrate that the joint inversion approach, especially based on the Wasserstein metric, outperforms these two existing individual inversions in terms of reducing the uncertainty of inversion results. However, this enhanced performance comes at the expense of compromising the accuracy of certain parameters other than sediment attenuation and sediment sound speed, as well as increasing the optimization time. Overall, the results obtained from the benchmark problems are in general agreement with the results of the simulated data experiments, according to Table 2.

The joint inversion method has the potential to improve the reliability and accuracy of the inversion results, but may also increase the uncertainty if the joint inversion involves dependent datasets. Therefore, future work will further seek to analyze the noise sensitivity and compare the performance and quantitative uncertainties under different SNR conditions.

This work was supported by the National Natural Science Foundation of China under Grant No. 41775027. The authors gratefully acknowledge Professor Julien Bonnel for discussion on the geoacoustic inversion using warping operators and Professor Stan Dosso for his invaluable guidance.

The authors have no conflicts to disclose.

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