Research on Evaporation Duct Height Prediction Modeling in the Yellow and Bohai Seas Using BLA-EDH

Abstract

Evaporation Duct Height (EDH) is a crucial parameter in evaporation duct modeling, as it directly influences the strength of the waveguide trapping effect and significantly impacts the over-the-horizon detection performance of maritime radars. To address the limitations of low prediction accuracy and limited interpretability in existing deep learning models under complex marine meteorological conditions, this study proposes a surrogate model, BLA-EDH, designed to emulate the output of the Naval Postgraduate School (NPS) model for real-time EDH estimation. Experimental results demonstrate that BLA-EDH can effectively replace the traditional NPS model for real-time EDH prediction, achieving higher accuracy than Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) models. Random Forest analysis identifies relative humidity (0.2966), wind speed (0.2786), and 2-m air temperature (0.2409) as the most influential environmental variables, with importance scores exceeding those of other factors. Validation using the parabolic equation shows that BLA-EDH attains excellent fitting performance, with coefficients of determination reaching 0.9999 and 0.9997 in the vertical and horizontal dimensions, respectively. This research provides a robust foundation for modeling radio wave propagation in the Yellow Sea and Bohai Sea regions and offers valuable insights for the development of marine communication and radar detection systems.

1. Introduction

An atmospheric duct is a special refractive index profile formed within the atmospheric boundary layer, where the refractive index gradually decreases with height [1,2]. This structure bends the propagation path of electromagnetic waves downward toward the Earth’s surface, with a curvature exceeding that of the Earth itself. Atmospheric ducts can be categorized into evaporation ducts, surface ducts, and elevated ducts [3]. In marine environments, seawater evaporation often causes a rapid decrease in humidity with height, which is the primary factor in the formation of evaporation ducts [4]. As a result, the occurrence of evaporation ducts is relatively frequent over the ocean [5]. Evaporation ducts can trap electromagnetic waves within a specific atmospheric layer, enabling over-the-horizon propagation [6]. This phenomenon significantly impacts radio communications, radar detection, and navigation systems by extending propagation distances, reducing signal attenuation, and introducing signal strength variations under different environmental conditions [7,8]. The EDH is the key parameter that determines whether electromagnetic waves can be effectively trapped within the duct layer, directly influencing the propagation range and signal quality. Therefore, accurate EDH prediction is essential for optimizing maritime communication systems, enhancing radar performance, and improving the reliability of navigation technologies.

Traditional methods for EDH prediction are primarily based on physical theoretical models. These include the structural model derived from Monin–Obukhov similarity theory [9], the Paulus–Jeske (P–J) model [10], and the Musson–Gauthier–Bruth (MGB) model developed by Météo France, which relies on a mesoscale forecasting system and employs analytical techniques [11]. Other notable approaches include the Refractivity from Clutter (RFC) atmospheric duct inversion method [12] and the NPS model developed by the U.S. Naval Postgraduate School [13]. These models estimate EDH by establishing relationships between meteorological parameters—such as temperature, humidity, and wind speed—and the atmospheric refractive index.

However, the predictive accuracy of these physical models is often limited in complex marine environments, where it becomes challenging to capture the nonlinear vertical variations of meteorological variables. In response, recent research has explored direct generation of refractive index profiles or digitized spatial fields of waveguides from meteorological observations. These efforts aim to construct high-resolution refractivity and duct height maps that can visually support over-the-horizon propagation analysis [14]. With the advancement of deep learning, data-driven approaches have shown increasing promise in EDH prediction. For instance, Zhu et al. [15] employed a Multilayer Perceptron (MLP) model to predict EDH from meteorological inputs, achieving satisfactory results under simple conditions. Zhao et al. [16] proposed a Long Short-Term Memory (LSTM)-based model that effectively captured temporal dependencies in the input data and improved predictive performance. In another study, Zhao et al. [17] utilized a Backpropagation Neural Network (BPNN) for EDH estimation, which delivered reasonable accuracy but lacked robustness in dynamic environments. More recently, Yang and Shi [18] developed a multidimensional deep learning framework that enhanced generalization across both spatial and temporal domains. Nevertheless, many existing models fail to fully leverage attention mechanisms to adaptively focus on the most relevant meteorological features, which compromises both prediction accuracy and generalization capability.

To improve the accuracy of EDH prediction and support real-time forecasting, this study introduces a deep learning model that integrates Bidirectional Long Short-Term Memory (Bi-LSTM) networks with an attention mechanism, referred to as BLA-EDH. Traditional LSTM models, constrained by fixed time windows, often struggle to differentiate the contributions of significant events from stable periods in long temporal sequences [19]. In contrast, Bi-LSTM networks can capture both forward and backward dependencies in time series data, thereby enabling more comprehensive modeling of complex meteorological dynamics. The attention mechanism enhances this capability by applying adaptive weighting to each time step, allowing the model to focus more effectively on features most relevant to EDH prediction and thus improving forecasting accuracy [20]. Additionally, Random Forest is employed for attribution analysis due to its ability to rank feature importance, its robustness in handling high-dimensional and nonlinear relationships, and its low sensitivity to multicollinearity [21]. Compared with other post-hoc interpretation methods, such as SHAP and LIME, Random Forest’s out-of-bag error evaluation mechanism avoids biases introduced by additional assumptions, offering a more objective means of identifying the dominant environmental variables associated with EDH [22].

In this study, we utilize ERA5 data from the entire year of 2023 to develop a deep learning-based surrogate model that emulates the NPS model output and enables real-time estimation of EDH-like behavior, with a focus on interpretability and operational usability. Section 2 and Section 3 outline the original ERA5 dataset and the methods used for data processing. The primary model is detailed in Section 4. The conclusions and future outlook of this study are presented in Section 5.

2. Data and Methodology

The ECMWF ERA5 dataset, produced by the European Centre for Medium-Range Weather Forecasts through its Integrated Forecasting System, is the fifth-generation global climate reanalysis dataset. It offers a spatial resolution of 31 km and a temporal resolution of one hour, enabling comprehensive capture of the spatiotemporal variations of meteorological variables [23]. The temporal coverage extends from 1950 to the present, facilitating long-term climate trend analysis and extreme weather event research, making it a vital tool for global scientific and operational applications.

ERA5 data is extensively used in meteorological environmental analysis and communications engineering. Jiang et al. [24] used ERA5 data to analyze the accuracy and reliability of precipitation data over mainland China, Shen et al. [25] evaluated the performance of ERA5 data in extreme precipitation events in the Yangtze River Delta region, and Wang et al. [26] utilized ERA5 reanalysis data to study the spatiotemporal characteristics of lower atmospheric ducts in the Arctic region.

In this study, we use the full-year 2023 data from the Yellow Bohai Sea area (116° E–126° E, 32° N–42° N) to train the model. Compared with sparse in situ observations, the ERA5 dataset provides a continuous and uniform global meteorological field, which is essential for constructing a high-precision spatiotemporal refractive index profile of the Yellow Bohai Sea area. This dataset offers stable and reliable inputs for the deep learning model, ensuring its prediction accuracy under complex oceanic meteorological conditions. The selected core meteorological parameters include air temperature, wind speed, and pressure, as detailed in

Table 1. Meteorological variables in ERA5 reanalysis data.

Reanalysis Variables	Level	Horizontal Resolution
Air temperature (K)	2 m	$0.{25}^{\circ }\times 0.{25}^{\circ }$
Relative humidity (% *)	2 m (derived)
Sea-surface temperature (K)	Sea surface
Sea-level pressure (Pa)	Sea surface
Dewpoint temperature (K)	2 m
U component of wind (m/s)	10 m
V component of wind (m/s)	10 m

* Relative humidity (RH) is derived from 2 m air temperature and dewpoint temperature as detailed in the methodology section.

.

Relative humidity (RH) is derived from 2-m air temperature (t2m) and dewpoint temperature, both converted to degrees Celsius, using the following empirical formulas:

$$e_s=6.11\times {10}^{\left({\displaystyle \frac{7.5\times T}{T+237.3}}\right)}$$

(1)

$$e=6.11\times {10}^{\left({\displaystyle \frac{7.5\times T_d}{T_d+237.3}}\right)}$$

(2)

$$RH={\displaystyle \frac{e}{e_s}}\times 100\%$$

(3)

The raw ERA5 temperature and dewpoint values were originally in Kelvin and were converted to degrees Celsius before applying the above formulas. In the above equations, both T and $T_d$ represent near-surface air temperature and dewpoint temperature, respectively, converted from Kelvin to degrees Celsius (°C). The variable $e_s$ denotes the saturation vapor pressure (hPa), and e is the actual vapor pressure (hPa). $RH$ stands for relative humidity, expressed as a percentage.

3. Evaporation Duct Diagnostic Model

In this study, to ensure the accuracy and credibility of the deep learning model’s prediction results, the forecasted values from the NPS model are used as targets for emulating and training the BLA-EDH surrogate model. The goal is to approximate the output behavior of the NPS model rather than to predict ground-truth EDH values observed in situ. The NPS model integrates several key meteorological data sources and constructs a parameterization scheme for the turbulent boundary layer based on Monin–Obukhov similarity theory, significantly improving the prediction accuracy of the evaporative waveguide height. Moreover, the COARE 3.5 algorithm embedded within the NPS model further enhances its accuracy under stable meteorological conditions, effectively correcting the tendency of traditional models to overestimate EDH values, thereby ensuring reliability across different meteorological scenarios. The NPS model has demonstrated high accuracy and stability in various applications related to atmospheric waveguides and over-the-horizon communication. Comparative analyses of experimental and evaluation results indicate that the NPS model consistently achieves reliable performance under diverse environmental conditions, making it a valuable benchmarking tool [27].

The NPS evaporation waveguide model developed by the U.S. Naval Institute employs a computational method based on the vertical distribution characteristics of boundary layer meteorological elements. The model constructs a turbulent boundary layer parameterization scheme by integrating parameters such as wind speed, air pressure, air temperature, humidity, and sea surface temperature at different observation levels near sea level. This is combined with the Monin–Obukhov similarity theory, ultimately enabling the quantitative assessment of the evaporative waveguide height [13]. This model uses boundary layer similarity theory to construct vertical profiles of temperature, humidity, and pressure. It derives the vertical distribution of the refractive index based on physical relationships between atmospheric refractivity and temperature, humidity, and pressure. By locating and correcting the minima in the refractive index profile, it determines the duct top height, significantly enhancing the physical rigor of diagnosing atmospheric super-refraction phenomena. In the model, the vertical profiles of near-surface temperature T. and specific humidity q are expressed using the following specific functional forms:

$$T\left(z\right)=T_0+{\displaystyle \frac{{\theta }_{*}}{\kappa }}\left[ln\left({\displaystyle \frac{z}{z_{0}t}}\right)-{\psi }_h\left({\displaystyle \frac{z}{L}}\right)\right]-{\eta }_{d}z,$$

(4)

$$q\left(z\right)=q_0+{\displaystyle \frac{q_{*}}{\kappa }}\left[ln\left({\displaystyle \frac{z}{z_{0}t}}\right)-{\psi }_h\left({\displaystyle \frac{z}{L}}\right)\right],$$

(5)

In the equation, $T\left(z\right)$ and $q\left(z\right)$ represent the air temperature and specific humidity at height z, respectively. When accounting for the effects of seawater salinity and relative humidity, $T_0$ and $q_0$ represent the sea surface temperature and specific humidity, respectively. ${\theta }_{*}$ and $q_{*}$ are the characteristic scales for potential temperature $\theta$ and specific humidity q. $\kappa$ is the von Kármán constant. $z_{0}t$ is the temperature roughness height. ${\psi }_h$ is the universal temperature function. ${\eta }_d$ is the dry adiabatic lapse rate, approximately equal to $0.00976\mathrm{K}/\mathrm{m}$. L represents the Obukhov length. The relationship between water vapor pressure e and specific humidity q defines the vapor pressure profile, as given by Equation (3).

$$e={\displaystyle \frac{qP}{\epsilon +\left(1-\epsilon \right)q}},$$

(6)

In the equation, $\epsilon$ is a constant (usually 0.622). By substituting the derived water vapor pressure e into Equation (4), the relationship between the refractive index, temperature, pressure, and water vapor pressure is expressed as

$$N={\displaystyle \frac{77.6}{T}}\left(P+4810{\displaystyle \frac{e}{T}}\right),$$

(7)

In the investigation of evaporation duct height, taking into account the curvature effect of the Earth’s surface on the electromagnetic wave propagation path, the modified refractive index M is employed to represent the modified refractive index profile of the evaporation duct:

$$M=N+{\displaystyle \frac{h}{R_e}}\times {10}^6=N+0.157h,$$

(8)

In the equation, M represents the modified refractive index, measured in M units, which refers to the modified refractive index profile; N is measured in N units, referring to the refractive index itself. $R_e$ is the Earth’s average radius (commonly set to 6371 km), and h is the altitude (in meters). The model identifies the evaporation duct height (EDH) by adjusting the location of the minimum value in the refractive index profile of M.

3.1. Data Standardization

To eliminate the influence of dimensional differences among meteorological parameters during model training, and to prevent features with larger numerical ranges from disproportionately affecting the model—thus improving overall performance and stability—this study adopts the Standard Scaler to standardize both the feature matrix and target variable. The data mean is adjusted to 0, and the standard deviation to 1. The feature variable X includes all input features, y represents the predicted output. These variables are treated as estimators. The feature variable is standardized to have a mean of 0 and a standard deviation of 1. The standardization formula is as follows:

$$Z_i={\displaystyle \frac{X_i-{\mu }_X}{{\sigma }_X}},$$

(9)

In this equation, $Z_i$ represents the standardized feature value, $X_i$ is the original feature value, ${\mu }_X$ the sample mean estimator of the feature variable X, and ${\sigma }_X$ is the sample standard deviation estimator of the feature variable X. The formulas for calculating the mean and standard deviation are as follows:

$${\mu }_X={\displaystyle \frac{1}{n}}\sum _{i=1}^n{X}_i,$$

(10)

$${\sigma }_X=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(X_i-{\mu }_X)}^2},$$

(11)

Here, n represents the sample size.

The target variable y is similarly standardized, adjusting its mean to 0 and standard deviation to 1. The standardization formula is as follows:

$$Z_j={\displaystyle \frac{y_j-{\mu }_y}{{\sigma }_y}},$$

(12)

In the equation, $Z_j$ is the standardized target value, $y_j$ is the original target value, ${\mu }_y$ is the mean of the target variable y, and ${\sigma }_y$ is the standard deviation of the target variable y.

3.2. Evaluation Metrics

The Root Mean Square Error (RMSE) measures the square root of the average of the squared differences between the predicted values and the actual values. The formula is as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2},$$

(13)

The Mean Absolute Error (MAE) measures the average absolute difference between the predicted values and the actual values. The lower the MAE, the smaller the gap between the model’s predictions and the actual values, indicating better model performance. The formula is as follows:

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|,$$

(14)

The coefficient of determination $R^2$ evaluates the proportion of variance explained by the model. The value of $R^2$ ranges from 0 to 1, with values closer to 1 indicating better model fit. The formula is as follows:

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}},$$

(15)

In the three formulas, n represents the sample size, $y_i$ is the true value of the i-th sample, and ${\widehat{y}}_i$ is the predicted value of the i-th sample. $\overline{y}$ is the average of all true values.

4. Main Model

4.1. BLA-EDH Model

In the Yellow Sea and Bohai Sea region, atmospheric duct meteorological data exhibit significant spatiotemporal inhomogeneity and temporal dependency. Seasonal variations and meteorological fluctuations lead to complex temporal dependencies in duct height changes, which are not random but instead follow spatial patterns and regional differences. We propose an emulation framework for the BLA-EDH model, which learns to replicate the EDH output from the NPS model using meteorological inputs that capture the complex characteristics of atmospheric duct conditions in the Yellow Sea and Bohai Sea. The bidirectional LSTM considers both forward and backward temporal dependencies, allowing it to capture long-term and short-term trends more comprehensively, thereby enhancing the model’s adaptability to dynamic meteorological features. The attention mechanism enables the model to automatically focus on key time steps, improving its ability to identify important meteorological patterns. With this structure, the model effectively handles the temporal and spatial non-homogeneity of Yellow Sea and Bohai Sea data, improving the accuracy and robustness of evaporation duct height predictions. The model architecture is illustrated in Figure 1.

The BLA-EDH model is constructed using the Keras framework with a TensorFlow backend. It includes two stacked Bidirectional LSTM layers with 128 and 64 hidden units, respectively. Each Bi-LSTM layer is followed by batch normalization and a dropout layer (rate = 0.3) to prevent overfitting. A custom attention mechanism is applied to the output sequence, followed by a dense layer with 64 ReLU units and a final linear output node.

The model is trained using the Adam optimizer (learning rate = 0.0001), with mean squared error as the loss function. Training is conducted for up to 200 epochs, using early stopping (patience = 15) and learning rate reduction on plateau (patience = 5, factor = 0.5). The batch size is 32, and the minimum learning rate is set to $1\times {10}^{-6}$.

The core components of the BLA-EDH architecture are designed to address the complex temporal and nonlinear characteristics of meteorological data: (1) The Bi-LSTM layer captures the bidirectional dynamic features of the time series, enabling the model to effectively represent the complex meteorological evolution process. (2) The attention mechanism uses Softmax to convert any real-valued vector into a probability distribution, ensuring that the sum of all attention weights is 1. It computes the weight coefficients for each time step, enabling the model to focus adaptively on time segments strongly associated with EDH mutation, weighting the features of each time step to emphasize the influence of critical time steps. (3) The dense layer further fits the nonlinear relationships by extracting the weighted features from the attention mechanism layer. (4) The output layer provides the final prediction of EDH and returns the attention weights to help analyze the importance of features.

During the model training process, a two-step stratified sampling method is used for data preprocessing to avoid overfitting, reduce data bias, and ensure the model’s generalization ability. The environmental parameters and corresponding EDH dataset (2023 data) computed by the NPS model are divided into training and temporary sets with a 60:40 ratio, and the temporary set is further split into validation and test sets with a 50:50 ratio. This division ensures that the test set remains fully independent of the training process.

The model inputs the preprocessed standardized data $X=\left[x_1,x_2,\cdots ,x_t\right]$, where $x_t$ represents the input environmental variables at time step t, and the output hidden state $h_t$ is expressed by Equation (13). The model predicts the EDH value based on the output hidden state $h_t$.

$$h_t=\left[\overrightarrow{h_t};\overleftarrow{h_t}\right],$$

(16)

${\overrightarrow{h}}_t$ represents the forward LSTM hidden state, which contains the sequence information from time step 1 to time step t. ${\overleftarrow{h}}_t$ is the output of the backward LSTM, with hidden and cell states similar to the forward LSTM. The data is processed from right to left, and the two vectors are concatenated at each time step to form the final hidden state $h_t$. The formulas for ${\overrightarrow{h}}_t$ and ${\overleftarrow{h}}_t$ are shown below:

$${\overrightarrow{h}}_t={\overrightarrow{o}}_t\ast \tanh h\left({\overrightarrow{c}}_t\right),$$

(17)

$${\overleftarrow{h}}_t={\overleftarrow{o}}_t\ast \tanh \left({\overleftarrow{c}}_t\right),$$

(18)

The cell state $c_t$ is represented as

$$c_t=f_t\ast c_{t-1}+i_t\ast {\tilde{c}}_t,$$

(19)

Where $f_t$ is the forget gate, which determines how much information from the previous memory cell at time step $t-1$ needs to be forgotten, represented by

$$f_t=\sigma \left(W_f·\left[h_{t-1},x_t\right]+b_f\right),$$

(20)

$i_t$ is the input gate, denoted by

$$i_t=\sigma \left(W_i·\left[h_{t-1},x_t\right]+b_i\right),$$

(21)

${\tilde{c}}_t$ is the candidate memory, represented by

$${\tilde{c}}_t=tanh\left(W_c·\left[h_{t-1},x_t\right]+b_c\right),$$

(22)

$o_t$ is the output gate, determining the influence of the current cell state on the final hidden state,

$$o_t=\sigma \left(W_o·\left[h_{t-1},x_t\right]+b_o\right),$$

(23)

In the formulas, $\sigma$ is the Sigmoid activation function, which maps the input between 0 and 1, indicating how much information is retained. The obtained hidden state is used as the input for the attention mechanism, which computes the attention score $e_t$, represented as

$$e_t=score\left(h_t\right)=tanh\left(W·h_t+b\right),$$

(24)

In this formula, W is the learned weight matrix, and b is the bias term used to adjust the output. The score $e_t$ is normalized using the softmax function to obtain the attention weight ${\alpha }_t$ for each time step t, where ${\alpha }_t$ represents the influence of that time step on the output:

$${\alpha }_t={\displaystyle \frac{\exp \left(e_t\right)}{{\sum }_{t=1}^T\exp \left(e_t\right)}},$$

(25)

${\alpha }_t$ performs a weighted sum of the hidden states $h_t$ over all time steps to obtain the final weighted hidden state $\widehat{h}$, represented as

$$\widehat{h}=\sum _{t=1}^T{\alpha }_t·h_t,$$

(26)

$\widehat{h}$ integrates information from all time steps in the sequence, focusing on important time steps based on the attention weights. Finally, $\widehat{h}$ is used for prediction, outputting the final predicted value $\widehat{y}$, as represented by the formula:

$$\widehat{y}=W_{dense}·\widehat{h}+b_{dense},$$

(27)

In the formula, $W_{\mathrm{dense}}$ is the weight matrix of the fully connected layer, determining the importance of each feature in the input vector $\widehat{h}$ for the output. $\widehat{h}$ is the output of the attention layer, and $b_{\mathrm{dense}}$ is the bias of the fully connected layer.

4.2. Random Forest Model

To verify the validity of the attention mechanism in feature selection and quantify the contribution of meteorological parameters to EDH prediction—thereby enhancing the interpretability of the BLA-EDH model—this paper employs the Random Forest algorithm to analyze the importance of meteorological parameters. The model is built using the Scikit-learn framework, with 100 decision trees and a fixed random seed of 42 to ensure reproducibility. The model structure is shown in Figure 2.

In the data preprocessing phase, meteorological parameters including relative humidity, wind speed, air temperature, sea surface temperature, and sea-level pressure were used as input data. The corresponding EDH values calculated by the NPS model were also included. All variables were standardized using the same Standard Scaler method described in Section 3.1, ensuring consistency with the deep learning models. The resulting dataset was then randomly divided into training, validation, and test sets in a 6:2:2 ratio. A Random Forest regression model was trained using this standardized dataset. During training, each decision tree performed split operations based on a specified impurity reduction criterion. The prediction formula for each tree is as follows:

$${E\hat{D}H}_i=f_{tree}\left(X_{norm}\right),$$

(28)

${E\hat{D}H}_i$ is the predicted value of the i-th decision tree for EDH, $f_{tree}\left(X_{norm}\right)$ is the prediction function for each tree, $X_{norm}$ is the input feature vector. In the prediction process, every tree relies on input environmental variables like temperature, humidity, wind speed, and others to evaluate at various split nodes, ultimately yielding a local prediction ${E\hat{D}H}_{leaf}$ for EDH at the terminal node, followed by the formula:

$${E\hat{D}H}_{\mathrm{leaf}}={\displaystyle \frac{1}{N_{\mathrm{leaf}}}}\sum _{i=1}^{N_{\mathrm{leaf}}}ED{H}_i,$$

(29)

In this formula, $N_{\mathrm{leaf}}$ is the number of samples in the leaf node, and $ED{H}_i$ is the EDH value of the i-th sample in that leaf node. Each tree provides a prediction result, and the final prediction is obtained by averaging the results from all decision trees. The ensemble formula for regression problems is as follows:

$${E\hat{D}H}_{\mathrm{RF}}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{E\hat{D}H}_i,$$

(30)

In this formula, ${E\hat{D}H}_{\mathrm{RF}}$ is the final regression prediction of the random forest, ${E\hat{D}H}_i$ is the prediction result from the i-th decision tree, and N is the total number of decision trees.

After training is completed, the Random Forest evaluates the importance of each feature by measuring its contribution to error reduction. The importance $I_j$ of meteorological parameter $X_j$ is computed based on the impurity reduction at split nodes. A commonly used method for calculating feature importance is given by

$$I_j={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\Delta }_{\mathrm{imp}}\left(X_j\right),$$

(31)

In this formula, ${\Delta }_{\mathrm{imp}}\left(X_j\right)$ represents the contribution of feature $X_j$ to impurity reduction in the i-th tree, and N is the number of trees.

The ranking of feature importance provides a data-driven physical explanation for the atmospheric duct formation mechanism and reveals the model’s sensitivity to different meteorological parameters. This not only improves the model’s computational efficiency and generalization ability but also enhances the interpretability of the BLA-EDH model.

5. Results and Discussion

5.1. Analysis of BLA-EDH Model Results

The 2023 ERA5 data were processed through the NPS model to compute the corresponding meteorological variables and EDH values, which were then used as inputs for the prediction models. The EDH prediction results generated by the MLP, LSTM, and BLA-EDH models were compared, as shown in Figure 3.

As shown in Figure 3, the BLA-EDH model exhibits the closest alignment with the NPS results, indicating its superior ability to capture the fluctuation trends of the EDH data generated by the NPS model. The LSTM model captures the overall trend but shows larger deviations at local peaks and troughs, suggesting a reduced sensitivity to rapid EDH fluctuations. Due to its lack of temporal modeling capabilities, the MLP model produces smoother predictions with limited variation, making it difficult to reflect subtle changes in the actual data. The BLA-EDH model maintains better visual consistency with the NPS reference curve when dealing with rapid changes in meteorological conditions and complex temporal dependencies. This improved alignment is attributed to its bidirectional architecture, which captures both forward and backward dependencies in the time series, thereby enhancing the model’s contextual awareness. The performance advantage of the BLA-EDH model stems from its architectural design. The Bi-LSTM component enables the model to learn dependencies in both temporal directions, improving its understanding of evolving meteorological patterns. Meanwhile, the attention mechanism applies dynamic weighting to time steps, allowing the model to focus on key moments that contribute most significantly to EDH changes. This synergy enables the model to effectively capture the time-varying nature of complex and volatile atmospheric conditions.

To analyze the impact of different components on model performance, ablation experiments and model comparisons were conducted. The following configurations were tested: MLP, LSTM, Bi-LSTM, MLP with Attention, LSTM with Attention, and the complete BLA-EDH model. All models were trained under identical settings, and their performance was evaluated using RMSE, MAE, and $R^2$. These experimental results provide an important basis for model optimization and practical application. The specific performance metrics are presented in

Table 2. Parameters assessed by the ablation experiment model.

Model	RMSE	MAE	R2
MLP	0.6823	0.2520	0.9692
LSTM	0.7592	0.5180	0.9619
Bi-LSTM	0.7013	0.1974	0.9675
MLP+A	0.7214	0.2590	0.9656
LSTM+A	0.6214	0.1938	0.9745
BLA-EDH	0.4492	0.1442	0.9867

.

As shown in Table 2, the proposed BLA-EDH model significantly outperformed all baseline models in EDH prediction, achieving the lowest RMSE (0.4492), lowest MAE (0.1442), and the highest $R^2$ score (0.9867). The model was trained using a full year of ERA5 reanalysis data from the Yellow and Bohai Seas (2023), covering a wide range of seasonal and synoptic meteorological conditions. This broad temporal coverage supports the model’s generalization capability across diverse atmospheric environments. The ablation results demonstrate the effectiveness of each module. Introducing Bi-LSTM reduced MAE by 21.7% compared to MLP and 61.9% compared to LSTM, while attention improved the model’s focus on key features, further enhancing prediction accuracy. The combination of Bi-LSTM and attention yielded the best performance, confirming their synergistic effect. Adding attention to LSTM reduced RMSE by 18.2% and MAE by 62.6%, and to Bi-LSTM reduced RMSE by 35.9% and MAE by 27.0%. These results indicate that attention helps the model focus on key features and dynamic changes, contributing to more accurate predictions. Compared to models reported in the literature, such as the LSTM-based model by Zhao et al. [16] (RMSE = 0.7592, MAE = 0.5180) and the MLP model by Zhu et al. [15] (RMSE = 0.6823, MAE = 0.2520), the BLA-EDH model outperforms them in both RMSE (0.4492) and MAE (0.1442), with the highest $R^2$ (0.9867), demonstrating superior accuracy and robustness.

Furthermore, in Section 5.4, we validated the physical reliability of the predicted EDH values by applying them to parabolic equation-based radio wave transmission loss simulations. The results confirm that the predicted values produce realistic and consistent wave propagation patterns over varying ranges and altitudes. This demonstrates that the BLA-EDH model not only performs well statistically but also provides physically meaningful predictions suitable for practical applications.

5.2. Comparative Analysis of Model Efficiency

To validate the applicability of the model in real-time prediction scenarios, an efficiency test was conducted. The experimental platform was equipped with an Intel^® Core™ i5-8250U processor and the PyCharm 2023 runtime environment. The efficiency comparison experiment executed prediction tasks for the BLA-EDH model and the traditional NPS algorithm using meteorological parameter datasets generated by the NPS model based on ERA5 data. The experimental results are shown in

Table 3. Comparison of model efficiency.

Model	Single File Processing(s)	Total File Processing(s)	Efficiency GAP
BLA-EDH	2.89–4.45	12.29	-
NPS	1701.39–5044.59	10,536.26	3 orders of magnitude

, where the BLA-EDH model has a stable single-file processing time ranging from 2.89 to 4.45 s, with a total running time of 12.29 s. In contrast, the traditional NPS algorithm requires 1701.39 to 5044.59 s to process a single file under the same hardware environment, demonstrating a significant efficiency gap and a substantial improvement in computational efficiency. The results confirm that the BLA-EDH model is capable of performing high-frequency data real-time predictions within a short time frame, providing a practical technical solution for rapid decision-making under extreme weather conditions.

5.3. Meteorological Parameter Sensitivity

To quantify the sensitivity of the BLA-EDH model to environmental parameters and identify which meteorological features have the most significant impact on the prediction results, we compared the mean attention weights of time steps in the BLA-EDH model (representing the global importance of different features over time) with the feature importance scores calculated by the Random Forest regression algorithm. The results are shown in Figure 4.

The Random Forest analysis indicates that relative humidity is the dominant predictor of EDH, followed by wind speed and the 2 m dew point temperature. Relative humidity strongly influences the atmospheric refractive index, and high-humidity conditions favor the formation of super-refractive layers essential for evaporation ducts [28]. Wind speed regulates boundary-layer turbulence, thereby shaping the temperature and humidity profiles that control duct height. Temperature further modifies the refractive index gradient and thus duct characteristics [29]. Other factors, including sea surface temperature and air pressure, play smaller but non-negligible roles: air pressure modulates surface-layer stability, while sea surface temperature, closely linked to evaporation, alters atmospheric water vapor distribution [30].

Overall, relative humidity, temperature, and wind speed are the dominant meteorological factors influencing EDH prediction. In the Random Forest model, relative humidity ranks first in importance, whereas it ranks second in the attention weights. This discrepancy arises because the attention mechanism captures temporal dynamics and distributes importance across local time steps, while the global evaluation in Random Forest reduces such temporal dependence. The attention-based feature analysis confirms that relative humidity, temperature, and wind speed are the key predictors of EDH, consistent with the Random Forest results. The high degree of agreement between the two methods strengthens the interpretability of the proposed BLA-EDH model, which not only delivers accurate EDH predictions but also integrates an internal feature selection mechanism aligned with the physical principles of atmospheric duct formation.

5.4. Model Analysis and Validation

To assess the effect of the EDH predicted by the BLA-EDH model on transmission loss calculations, we input the predicted EDH values into the parabolic equation model. In oceanic environments with atmospheric ducts, accurate calculation of transmission loss is essential for signal quality. The parabolic equation is designed based on the study by [31]. This model uses the known EDH values to evaluate transmission loss and simulate signal propagation characteristics under complex atmospheric conditions [32]. The fundamental parameters in the parabolic equation were: A frequency of 15 GHz, typically used in communication systems due to its favorable propagation characteristics, transmitter antenna height of 5 m, beamwidth of 0.7°, and antenna elevation angle of 0°. The EDH predicted by the model was 13.46 m, while the EDH calculated by the NPS model was 13.6 m. We assessed the engineering applicability of the predicted EDH by comparing the transmission loss profiles (lateral distance-loss, vertical height-loss) and global statistical metrics between the BLA-EDH and NPS models along the same propagation path. The results are presented in Figure 5.

As shown in Figure 5, the path loss distribution of the NPS and BLA-EDH models indicates that in the near-surface duct region and at longer propagation distances, the spatial distribution of path loss from the BLA-EDH model was highly consistent with that of the NPS model, demonstrating its excellent spatiotemporal adaptability and prediction stability. The experimental results show that within a propagation distance of 0 to 200 km, the BLA-EDH model can accurately reproduce the path loss characteristics of the NPS model, further validating its effectiveness in radio wave propagation prediction.

To further compare the spatial differences between the two models, a systematic evaluation of the error distribution in complex atmospheric duct scenarios was conducted. Using the parabolic equation method, the path loss differences between the two models were obtained, as shown in Figure 6.

Figure 6 shows the heatmap of the path loss differences between the NPS and BLA-EDH models, with the differences calculated as the absolute difference in path loss between the NPS and BLA-EDH models. In Figure 6a, within a propagation distance of 0 to 200 km and a height range of 0 to 300 m, the path loss differences were mostly below 5 dB (with over 80% of the area dominated by blue), indicating high similarity in signal attenuation between the two models in that region. Here, the path loss is expressed in decibels (dB), a relative logarithmic unit commonly used in radio wave propagation to represent signal attenuation. It is calculated as $L=10{\log }_{10}(P_t/P_r)$, where $P_t$ and $P_r$ denote the transmitted and received power, respectively. The path loss differences slightly increased to values between 5 and 10 dB for distances above 150 km and heights above 300 m, mainly at the edges of the duct propagation. These differences were localized and likely resulted from the models’ differing treatment of refractive index gradients or boundary conditions.

Figure 6b further refines the difference analysis, focusing on the loss consistency within the 100 to 200 km propagation distance and the 100 to 200 m height range. The differences remained under 5 dB in the majority of regions (covering over 90% of the region), with only a few narrow bands reaching up to 15 dB. These regions appeared as regular stripes, suggesting a connection to multipath propagation effects under specific environmental refractive structures.

To verify the model’s stability in different transmission scenarios, the signal strength distributions of the BLA-EDH and NPS models along typical horizontal (5 m height) and vertical (30 km distance) profiles were shown in Figure 7.

As shown in Figure 7a, the loss curves of the BLA-EDH model and the NPS model exhibited a consistent overall trend within the 0 to 200 km transmission distance ($R^2=0.9999$), with an average absolute error of 0.13 dB. The loss curves are nearly overlapped. Figure 7b indicates that within the 100 to 250 m height range in the 30 km vertical profile, the path loss difference between the BLA-EDH and NPS models was only 0.53 dB ($R^2=0.9997$), with highly consistent loss results. This demonstrates the model’s excellent duct effect recognition capability, with slight differences in regions where multipath effects were significant. Nevertheless, the overall fluctuation trend remained consistent, with small differences, indicating that the BLA-EDH model could effectively adapt to complex propagation environments. The experimental results showed that the path loss trends of the BLA-EDH and NPS models were highly consistent under different propagation distances and height conditions. In particular, in low-altitude duct regions and at heights with significant multipath effects, the BLA-EDH model accurately captured the loss variations, demonstrating excellent spatiotemporal consistency. Although minor localized discrepancies existed in some regions, the overall error remained mostly within 5 dB, and no large-scale deviations were visually observed in the heatmaps, indicating that the BLA-EDH model provides stable and reliable predictions in the path loss simulation. In the future, the model’s applicability can be enhanced by introducing environmental correction factors or improving adaptability to terrain-induced refractive variations.

5.5. Temporal Generalization Validation

To further evaluate the temporal generalization capability of the BLA-EDH model, an additional validation was conducted using meteorological data from January to March 2024. Figure 8 presents the EDH predictions generated by the BLA-EDH model alongside the corresponding results from the NPS model for the same period. The model was not retrained but directly applied to the 2024 dataset, simulating a real-world deployment scenario.

The results indicate that the BLA-EDH model maintains high fitting accuracy across different months and environmental conditions. Specifically, the model effectively captures both abrupt changes and stable trends in EDH over time, demonstrating consistent performance even in unseen temporal domains. This confirms the model’s ability to generalize beyond the training year (2023), validating its practical applicability for operational forecasting under temporal variability. These findings highlight the model’s strong temporal robustness. However, since the current training data covers only a single year, the model may still face challenges when applied to extreme or climatologically anomalous years. Future work should consider incorporating multi-year datasets spanning diverse climatic conditions to further improve the model’s adaptability and robustness.

6. Conclusions

This study presents the development of a deep learning-based surrogate model, BLA-EDH, that learns to reproduce the output behavior of the NPS physical model with high fidelity. Rather than predicting real-world EDH directly, BLA-EDH offers a significant improvement in modeling efficiency and operational feasibility for maritime propagation applications. While the traditional NPS model can calculate EDH, it struggles to fully capture nonlinear relationships and complex meteorological variation patterns under challenging marine atmospheric conditions. By integrating Bi-LSTM and attention mechanisms, the BLA-EDH model supplements traditional EDH prediction methods and leverages the strengths of machine learning to handle large-scale, complex meteorological datasets.

Experimental results showed that this model significantly improved the accuracy and computational efficiency of EDH prediction under complex marine meteorological conditions. Compared to the traditional NPS model, BLA-EDH achieved approximately 920 times higher computational efficiency under the same hardware environment. Additionally, the model exhibited significant improvements in accuracy: compared to MLP, RMSE, and MAE were reduced by 34.2% and 42.8%, respectively; compared to LSTM, RMSE, and MAE were reduced by 40.8% and 72.2%, respectively. Random Forest analysis of the sensitivity to different meteorological parameters further enhanced the interpretability of the model. The results indicated that relative humidity, wind speed, and dew point temperature at 2 m were the core meteorological parameters for BLA-EDH in predicting EDH.

This model provides a high-accuracy, low-latency real-time prediction tool for applications such as ocean communications and radar detection, offering potential pathways for future improvements in more complex modeling tasks. This study primarily focused on the Yellow Sea and Bohai Sea; future work will extend to the fusion modeling of multi-source heterogeneous data and explore the model’s generalizability in more complex maritime regions, such as the South China Sea. By further optimizing the model structure and increasing data diversity, its applicability and accuracy in various marine environments are expected to improve further.