Variational Mode Decomposition (VMD) was proposed by Dragomiretskiy and Zosso (2013). It is an adaptive signal processing method. Different from traditional signal processing methods based on Fourier transform or wavelet transform, VMD decomposes complex multi-component signals into multiple Intrinsic Mode Functions (IMFs) with different central frequencies. These IMFs have specific physical meanings and excellent properties, enabling the effective extraction of features from different frequency components in the signal. As a result, VMD has been widely applied in the field of signal processing. The VMD method seeks a set of mode functions through variational optimization. Its objective function aims to minimize the sum of the bandwidths of all IMFs, with the constraint that the sum of all IMFs is equal to the original signal. The formula is shown in Equations 1, 2: f min = min u k , ω k ∑ k ∂ t δ t + j π t * u k t e − j ω k t 2 2 (1) s . t   ∑ k = 1 K u k = f t (2)

Where δ t is the Dirac function, ∗ denotes the convolution operation, ∂ t denotes the derivative with respect to time, j is the imaginary unit, and f t is the original signal, decomposed into k modal functions u k t , each of which corresponding to a center frequency ω k .

In this study, the VMD method is used to decompose the logging curve data into sub-series with different frequencies, which respond to the overall trend, and local change characteristics of the logging curve. The high complexity and strong nonlinear characteristics of this time series can be solved by this processing.

Convolutional Neural Networks (CNNs) represent a class of deep neural networks characterized by their convolutional operations, originally developed for and predominantly applied in computer vision tasks. The core idea of CNN is to utilize convolutional operation to achieve the extraction of local features (e.g., edges, textures). Through the stacking of multiple convolutional layers, CNN can gradually abstract from low-level features to high-level semantic features, realizing the hierarchical expression of data features. Compared with the traditional fully connected neural network, CNN significantly reduces the number of model parameters through the weight sharing strategy and effectively avoids the overfitting problem (Wu et al., 2024). In this study, the well logging curves are presented in a 1D sequence format, where each data point corresponds to a specific depth. The local correlation between adjacent depth points (such as the sudden signal change at the interface between sandstone and mudstone) directly reflects the subtle geological changes. The 1D-CNN is designed to scan the 1D well logging sequence using a 1D convolution kernel, thereby capturing local spatial features. This method is directly similar to the way two-dimensional convolutional neural networks capture textures and edges in images.

The core component of CNN is the convolutional layer, which generates feature maps through convolution operations (Figure 1). The input data passes through the convolution kernel, and the mathematical expression is shown in Equation 3: Y i , j = ∑ m = 0 M − 1 ∑ n = 0 N − 1 X i + m , j + n · W m , n + b (3)

Where M × N denotes the convolution kernel size, W is the weight, and b is the bias term.

A conventional CNN framework typically comprises three fundamental components: (1) an input layer for raw image data reception, (2) a convolutional module containing sequentially arranged convolutional operations, nonlinear activation units, and downsampling layers for hierarchical feature extraction and dimensional reduction, and (3) a fully-connected classifier for final feature-to-label transformation.

Long Short-Term Memory network (LSTM) is a special kind of Recurrent Neural Network (RNN) proposed by Hochreiter and Schmidhuber (1997), the core of which is to solve the problem of gradient vanishing and explosion encountered by RNN when dealing with long sequence data. Long Short-Term Memory (LSTM) networks are suitable for processing temporal data and are widely used in speech processing and natural language processing. LSTM controls the flow of information through three gate structures: forgetting gate, memorizing gate, and output gate (Figure 2).

Oblivion Gate: selectively forgets historical information about the input. This component receives the information h t − 1 of the previous time state and the current input x t , expressed as the formula is shown in Equation 4: f t = σ W f · h t − 1 , x t + b f (4)

Where f t is the output information of the forgetting gate, σ is the activation function, W f is the weight, and b f is the bias.

Input Gate: Controls the addition of current input information to the memory cell. The update of the information in the memory cell and the generation of a new memory value is controlled by an activation function, expressed by the formula is shown in Equations 5, 6: i t = σ W i · h t − 1 , x t + b i (5) C t ∼ = tanh W C · h t − 1 , x t + b C (6)

Where i t is the output of the input gate, C t ∼ is the candidate memory value at the current moment, and tanh is the activation function.

Output gate: the output gate selectively filters the composite information derived from both the memory cell state and current input features, thereby determining the final output representation and generating an updated hidden state through a gating mechanism, which is expressed by Equations 7, 8: o t = σ W o · h t − 1 , x t + b o (7) h t = o t · ⁡ tanh C t (8)

Where o t is the output of the output gate and h t is the hidden state at the current moment.

Memory updating: the memory cell state undergoes dynamic updating through the coordinated operation of both input and forget gates, which collectively regulate the incorporation of new information and the retention of historical context, which is expressed by Equation 9: C t = f t · C t − 1 + i t · C t ∼ (9)

In this study, the LSTM’s ability to preserve depth-dependent dependencies allows it to model geological transitions that are not apparent from shallow features alone.

The CNN-BiLSTM model combines the advantages of convolutional neural network (CNN) and bi-directional long short-term memory network (BiLSTM) (Redwan et al., 2025). The hybrid architecture leverages complementary strengths: CNN demonstrates superior capability in extracting localized temporal patterns, while the bidirectional LSTM (BiLSTM) effectively captures long-range dependencies through its dual-directional processing, enabling comprehensive utilization of inter-strata contextual information.

In this paper, a VMD-CNN-BiLSTM cascade model is constructed by combining VMD, CNN, and BiLSTM, and the model structure is shown in Figure 3. Each VMD component is independently fed into a CNN-BiLSTM model, whose output is a subcomponent of the final DT curve; these are then summed to reconstruct the complete acoustic profile. Specifically, the VMD method is used to decompose different logging curves into multiple secondary curves with different frequencies, and construct the prediction model of logging curves with different frequencies, for each model, firstly, the secondary logging curves of the frequency are used as the input of CNN, and after feature extraction of the local features by CNN, BiLSTM receives the sequences processed by CNN, and extracts the before-and-after correlation information of the logging data, and then, the BiLSTM processed features are input to a fully connected layer to map to the secondary acoustic curve of the same frequency, and finally, the secondary acoustic curves obtained from each model are superimposed to obtain the final acoustic reconstruction curve.

In this study, a composite loss function combining mean square error (MSE) and soft dynamic time regularization loss (Soft-DTW) (Cuturi and Blondel, 2017; Jiang et al., 2022) is used to balance the optimization objective of local point-by-point error and global morphological similarity in time series forecasting. The traditional MSE loss is sensitive to point-by-point errors, but it is difficult to capture the overall trends and interrelationships of the sequences; Soft-DTW introduces a differentiable approximation to the classical dynamic time warping algorithm through continuous relaxation of alignment paths, thereby enabling direct integration with neural network training frameworks that traditional DTW cannot support due to its non-differentiable nature.

The mean square error (MSE) represents a fundamental loss metric in regression analysis, quantifying the expected squared deviation between predicted and observed values across temporal data points. The formula is shown in Equation 10: L M S E = 1 N ∑ i = 1 N y i ^ − y i 2 (10)

Where y i ^ is the predicted value of the ith data point, y i is the corresponding true value, and N is the total number of data points.

While MSE offers computational advantages including straightforward gradient computation and stable optimization convergence, its underlying assumption of temporal independence limits its ability to capture sequential dependencies and global morphological patterns in time-series data.

Soft Dynamic Time Regularization Loss (Soft-DTW), on the other hand, is an improvement of traditional DTW, aiming at solving its problem of non-trivial discrete path search. The conventional DTW algorithm employs dynamic programming to identify the optimal nonlinear alignment path between temporal sequences, providing an effective measure of global morphological similarity. However, its discrete optimization nature precludes direct integration with neural network architectures for end-to-end learning. Soft-DTW transforms the discrete path selection into continuous probability distributions based on the softmax function through the introduction of the smoothing relaxation technique, making the loss function microscopic. Specifically, a similarity matrix M is constructed between the predicted sequence and the actual sequence, as shown in Equation 11: M i , j = Y i ^ − Y j | 2 2 (11)

Where M i , j denotes the squared Euclidean distance for the ith and jth data points.

The accumulation matrix A is then computed by dynamic programming, as shown in Equation 12: A i , j = M i , j + log exp A i − 1 , j + exp A i , j − 1 + exp A i − 1 , j − 1 (12)

Where each element A i , j represents a smooth approximation of the minimum cumulative distance from the starting point (1,1) to (i,j).

The final Soft-DTW loss is the normalized value of the element A T , T in the lower right corner of the accumulation matrix, where T is the length of the sequence. This loss function is insensitive to the time axis offset and can effectively capture the global characteristics of the sequence such as trend and period.

The MSE is finally combined with Soft-DTW to construct the composite loss function, as shown in Equation 13: L t o t a l = λ L M S E + 1 − λ · L S o f t ‐ D T W (13)

Where λ is used to regulate the weights of the two losses as hyperparameters of the model.

The data of this experiment comes from the open-source logging dataset of Xu et al. (2024), which includes four wells, GY1, C21, SYY1, and YX58, in GL block, and the purpose layer of this experiment is the QSK layer, and the total depth of the four wells is about 1,542 m. The logging curves include the natural gamma (GR), the deep lateral resistivity (RD), the shallow lateral resistivity (RS), the density (DEN), the neutron (CNL), as input features, and acoustic time difference curve (DT) as prediction curve, with a longitudinal sampling rate of 0.125 m.

This study employs a comprehensive set of quantitative metrics to assess the model’s performance in curve reconstruction tasks. Four key evaluation indicators are utilized: Mean Squared Error (MSE), Mean Absolute Error (MAE), Coefficient of Determination (R²), and Mean Absolute Percentage Error (MAPE). Among these metrics, MSE and MAE primarily measure the magnitude of absolute deviations between the reconstructed curves and their corresponding ground truth values. The former emphasizes the overall degree of dispersion, while the latter highlights the magnitude of the average deviation. To evaluate the model’s curve-fitting capability from the perspective of variance interpretation, the R² quantifies how well the reconstructed curves explain the variance in the original data. MAPE is a relative error measure that eliminates the impact of dimensions. The formula for MSE is shown in Equation 10 and the calculations for the other indicators are shown in Equations 15–17: MAE = 1 m ∑ i = 1 m y i ^ − y i (15) R 2 = 1 − ∑ i = 1 m y i ^ − y i 2 ∑ i = 1 m y ¯ − y i 2 (16) MAPE = 1 m ∑ i = 1 m y i − y i ^ y i × 100 % (17)

Where m is the number of samples, y i denotes the true value of the ith sample, and y i ^ is the corresponding model prediction.

Lower values of MSE, MAE, and MAPE indicate higher reconstruction accuracy, while an R² value closer to 1 reflects better curve-fitting performance.

The workflow for reconstructing the DT curve based on the model proposed in this paper is shown in the Figure 12. First, preprocessing is performed on the collected curves, including outlier removal and normalization. Then, the optimized CNL, RS, and GR curves are decomposed using VMD. Additionally, the DT curve to be predicted also undergoes decomposition. On this basis, prediction models are established for each component IMF. Finally, all predicted values are superimposed to obtain the final reconstructed curve.

The experiment was run on Window11 operating system, using Python programming language and Tensorflow deep learning framework, using a computer configuration of NVIDIA RTX 3090 with 16G video memory. The model uses the above combination function as the loss function and the Adam optimizer, and the individual hyperparameters of the experiment are set as follows, with the Batchsize set to 32, the initial learning rate set to 0.01, λ set to 0.7, and 64 training iterations.

Figure 13 shows the trend of the loss values of different component prediction models after VMD decomposition with the number of iterations, and it can be seen that with the increase of the number of training times, the loss values of the model are decreasing, and gradually converge to a stable state, and finally reach the convergence state.

The trained model is applied to two test wells, thus demonstrating the practical application effect of the model.

Figures 14, 15 respectively present the scatter plots comparing the predicted and measured values of the acoustic time difference (DT) for the SYY-1 well and the YX58 well. The scattered points are mainly distributed near the 45° diagonal line, indicating a good linear consistency between the two. Specifically, for Figure 14 (SYY-1 well), the coefficient of determination R² is 0.8783, and the slope of the regression equation is 0.8231; for Figure 15 (YX58 well), the coefficient of determination R² is 0.7142, and the slope of the regression equation is 0.6354. There is a certain difference in the prediction results of the two wells. This is because the similarity of the logging curves between the YX58 well and the training well is lower than that of the SYY1 well, and at the same time, the formation thickness of the YX58 well is much smaller than that of the training well (Wu et al., 2024), reflecting the differences in sedimentary characteristics. Therefore, the difference in the prediction results of the two wells is acceptable. Furthermore, single-well logging curves are plotted. In the DT curve track of Figure 16, the blue and yellow curves denote the true and predicted values, respectively. The results demonstrate that the proposed method achieves accurate predictions for the acoustic curves in both wells. Specifically, the predicted curve closely follows the overall trend of the true curve, effectively capturing its amplitude variations. Furthermore, the model reliably reproduces key features, including peak and valley positions, indicating strong agreement between predictions and ground truth. This indicates that the model can effectively uncover the correlations among different logging curves, and learn the local undulation characteristics, long-distance variation trends, and sequential correlations of the curves.

To validate the effectiveness of the proposed method, we conducted comparative experiments against several conventional machine learning approaches, including Support Vector Regression (SVR), Random Forest, and standard LSTM models. The performance metrics of these models on the validation set are presented in Table 1, enabling a comprehensive evaluation of their relative strengths and limitations.

From the data in the table, it can be seen that the VMD-CNN-BiLSTM model shows the optimal performance with MSE of 30.30, MAE of 4.20, MAPE of 0.041, and the fitting effect of R² reaches 0.67. Compared with the other models, it has reduced the error metrics, significantly improved the prediction accuracy, and the best reconstruction of acoustic wave curves.

The performance of the models can be further visualized from the corresponding test well prediction curve plots (Figure 17), where the green curve represents the predicted value and the blue curve represents the true labeled value. The SVR model exhibits significant deviations from the true curves in both morphological structure and absolute values, particularly in high-variability segments where prediction errors are most pronounced. While the Random Forest model demonstrates measurable improvement over SVR, it still shows limitations in capturing fine-scale variations. The LSTM model achieves superior performance overall, faithfully reconstructing the general trend of true curves across most depth intervals. However, its predictive accuracy diminishes in zones of rapid curve fluctuation, notably around the 2000 m depth region where complex patterns emerge. The VMD-CNN-BiLSTM model is the closest to the real labeled curves, and it can reconstruct the overall trend of the curves and the local peaks and valleys in the whole depth range. The results show that the VMD-CNN-BiLSTM model has better application in logging curve reconstruction.