Quantitative analysis and prediction of the sound field convergence zone in mesoscale eddy environment based on data mining methods
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Abstract: The mesoscale eddy (ME) has a significant influence on the convergence effect in deep-sea acoustic propagation. This paper use statistical approaches to express quantitative relationships between the ME conditions and convergence zone (CZ) characteristics. Based on the Gaussian vortex model, we construct various sound propagation scenarios under different eddy conditions, and carry out sound propagation experiments to obtain simulation samples. With a large number of samples, we first adopt the unified regression to set up analytic relationships between eddy conditions and CZ parameters. The sensitivity of eddy indicators to the CZ is quantitatively analyzed. Then, we adopt the machine learning (ML) algorithms to establish prediction models of CZ parameters by exploring the nonlinear relationships between multiple ME indicators and CZ parameters. Through the research, we can express the influence of ME on the CZ quantitatively, and achieve the rapid prediction of CZ parameters in ocean eddies. The prediction accuracy (R) of the CZ distance (mean R: 0.9815) is obviously better than that of the CZ width (mean R: 0.8728). Among the three ML algorithms, Gradient Boosting Decision Tree has the best prediction ability (root mean square error (RMSE): 0.136), followed by Random Forest (RMSE: 0.441) and Extreme Learning Machine (RMSE: 0.518).
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Key words:
- convergence zone /
- mesoscale eddy /
- statistic analysis /
- quantitative prediction /
- machine learning
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Figure 5. Characteristics of sound propagation with a warm eddy: the sound propagation loss (a) and sound ray (b) distribution. In b, the red line represents the refracted sound, and the green line represents the reflected sound.
Figure 6. Characteristics of sound propagation with no eddy: the sound propagation loss (a) and sound ray (b) distribution. In b, the red line represents the refracted sound, and the green line represents the reflected sound.
Table 1. The description of eddy parameters and sound source position
Parameter Definition Setting Eddy intensity/(m·s−1) it is characterized by the sound velocity difference between the eddy center and the edge value range is –50 m/s to +50 m/s, gap is +5 m/s;
+ indicates warm eddy and – indicates cold eddyEddy radius/km it is characterized by the horizontal distance between the eddy center and the edge value range is 20 km to 300 km, gap is 20 km Depth of sound source/m literal meaning value range is 100 m to 400 m, gap is 20 m Distance of sound source/km it is characterized by the horizontal distance of the sound source from the eddy center (vertical central axis) value range is 0 km to 150 km, gap is 15 km 
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Table 2. Regression analysis between CZ distance and eddy intensity
Model Depth/m Regression equation R2 F Sig. RMSE Linear regression model 100 y = 0.07615x + 53.15 0.992 14598.965 0 0.226 200 y = 0.08775x + 49.94 0.998 19823.564 0 0.307 300 y = 0.09979x + 47.14 0.988 10479.997 0 0.181 Quadratic regression model 100 y = 0.00024x2 + 0.07615x + 52.92 0.999 13874.476 0 0.079 200 y = 0.00032x2 + 0.08775x + 49.64 0.998 14462.807 0 0.121 300 y = 0.00017x2 + 0.04979x + 46.98 0.996 9134.502 0 0.096
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Table 3. Regression analysis between CZ distance and eddy radius
Model Depth/m Regression equation R2 F Sig. RMSE Linear regression model 100 y = 0.01253x + 54.59 0.752 1124.873 0 0.625 200 y = 0.01651x + 51.37 0.839 1952.309 0 0.626 300 y = 0.00886x + 48.02 0.793 1267.598 0 0.392 Quadratic regression model 100 y = −0.00011x2 + 0.04102x + 52.93 0.963 11269.375 0 0.244 200 y = −0.00012x2 + 0.04471x + 49.73 0.973 14633.754 0 0.261 300 y = −0.00011x2 + 0.02719x + 46.96 0.979 15124.561 0 0.127
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Table 4. Regression analysis between the CZ width and eddy intensity
Model Depth/m Regression equation R2 F Sig. RMSE Linear regression model 100 y = −0.03299x + 5.979 0.973 12531.632 0 0.264 200 y = −0.04228x + 9.829 0.978 13105.415 0 0.256 300 y = −0.06683x + 14.51 0.184 / / 4.549 Quadratic regression model 100 y = 0.00022x2 − 0.03299x + 5.77 0.942 9975.143 0 0.183 200 y = 0.00017x2 − 0.04228x + 9.67 0.966 10132.472 0 0.211 300 y = 0.00292x2 − 0.06683x + 11.68 0.454 / / 3.787 Note: / indicates invalid value.
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Table 5. Regression analysis between the CZ width and eddy radius
Model Depth/m Regression equation R2 F RMSE Linear regression model 100 y = −0.00197x + 5.022 0.411 / 0.254 200 y = −0.00475x + 8.996 0.721 1024.143 0.256 300 y = 0.00516x + 12.81 0.789 1322.219 0.231 Quadratic regression model 100 y = 0.000032x2 − 0.01243x + 5.63 0.788 1387.458 0.133 200 y = 0.000019x2 − 0.01103x + 9.36 0.789 1269.655 0.226 300 y = −0.0000015x2 − 0.0054x + 12.82 0.787 1463.914 0.236 Note: / indicates invalid value.
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Table 6. Parameters set of the three ML algorithms
ML algorithm Parameter setting RF the number of decision trees, the maximum depth of each tree, and the number of node samples are set as 100, 30, and 10; the remaining parameters are left with the default values GBDT the maximum depth of the tree is 5; maximum iteration is 5; learning rate is 0.1; the remaining parameters are left with the default values SaD-ELM the number of hidden layers is 3; the number of neurons in each hidden layer is 10; the excitation function is “sig.”; the population number, variation probability and crossover probability are set as 30, 0.5, and 0.4; the remaining parameters
are left with the default values
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Table 7. Error measures of the predicted CZ parameters
CZ parameter Assessing indicator RF GBDT SaD-ELM CZ distance RMSE 0.4412 0.1353 0.5182 CC 0.9771 0.9982 0.9694 NSE 0.9149 0.9829 0.9134 CZ width RMSE 1.3345 1.3093 1.3642 CC 0.8733 0.8782 0.8671 NSE 0.6047 0.6231 0.5023
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Table 8. CZ parameters calculated by ML and simulation
Eddy Distance of the first CZ/km Width of the first CZ/m GBDT BELLHOP GBDT BELLHOP Eddy A 56.3 63.9 9.2 4.8 Eddy B 60.5 62.3 11.3 8.3 Eddy C 57.1 59.3 9.5 5.2 Eddy D 52.8 61.7 10.9 6.4 Eddy E 55.6 63.8 12.1 7.2
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