Supervised-Learning-Based Method for Restoring Subsurface Shallow-Layer Q Factor Distribution

Abstract

The distribution of shallow subsurface quality factors (Q) is a crucial indicator for assessing the integrity of subsurface structures and serves as a primary parameter for evaluating the attenuation characteristics of seismic waves propagating through subsurface media. As the complexity of underground spaces increases, regions expand, and testing environments diversify, the survivability of test nodes is compromised, resulting in sparse effective seismic data with a low signal-to-noise ratio (SNR). Within the confined area defined by the source and sensor placement, only the Q factor along the wave propagation path can be estimated with relative accuracy. Estimating the Q factor in other parts of the area is challenging. Additionally, in recent years, deep neural networks have been employed to address the issue of missing values in seismic data; however, these methods typically require large datasets to train networks that can effectively fit the data, making them less applicable to our specific problem. In response to this challenge, we have developed a supervised learning method for the restoration of shallow subsurface Q factor distributions. The process begins with the construction of an incomplete labeled data volume, followed by the application of a block-based data augmentation technique to enrich the training samples and train the network. The uniformly partitioned initial data are then fed into the trained network to obtain output data, which are subsequently combined to form a complete Q factor data volume. We have validated this training approach using various networks, all yielding favorable results. Additionally, we compared our method with a data augmentation approach that involves creating random masks, demonstrating that our method reduces the mean absolute percentage error (MAPE) by 5%.

1. Introduction

Subsurface media exhibit an absorption and attenuation effect on seismic waves which primarily consists of geometric attenuation and physical attenuation. Physical attenuation is the main factor leading to the reduction in seismic data resolution, and this absorption and attenuation effect is typically characterized by the quality factor Q. Therefore, accurate estimation of the Q factor is of paramount importance. Numerous studies have been conducted by scholars on the estimation methods for estimating the Q factor. Futterman established the well-known Futterman model, positing that the Q factor is a formation medium parameter used to characterize the absorption and attenuation of wave propagation [1]. Stacey and Gladwin proposed a time-domain method for estimating Q values by studying the phenomenon of pulse variation during seismic wave propagation [2]. Subsequent improvements to their method involved estimating the Q factor by examining the width of the pulse [3]. Janssen later investigated the relationship between travel time differences and dispersion, creating a wavelet simulation method [4]. Engelhard introduced the analytic signal method [5]. Subsequently, Tonn concluded, through algorithm consolidation, that the quality performance of actual seismic data influences the accuracy and reliability of Q factor estimation [6].

The absence of Q factor data in subsurface vibration fields can lead to a reduction in seismic data resolution and imaging accuracy [7]. Consequently, an effective solution is to perform data imputation, replacing missing values with estimated ones [8]. In recent years, with the continuous development of deep learning, deep neural networks have been applied across various domains [9,10]. Seismic researchers have begun to explore the direct application of image processing methods to seismic data reconstruction. For instance, dictionary learning was initially used for denoising and super-resolution reconstruction in related fields. Qin et al. proposed deep internal learning for interpolating regularly sampled aliased seismic data, generating training samples solely from the remaining undersampled seismic data [11]. Chai et al. utilized deep learning to extend 2D data reconstruction to 3D data, circumventing many assumptions inherent in traditional non-intelligent reconstruction methods (e.g., linearity, sparsity, and low-rankness) [12]. Mu et al. employed a combination of self-attention mechanisms and generative adversarial networks for seismic signal reconstruction and denoising [13]. Chang et al. proposed a dual-domain conditional generative adversarial network for interpolation, utilizing seismic datasets in the frequency domain and discrete Fourier transform datasets as network input vectors [14]. Gao et al. employed a Swin Transformer convolutional residual network for simultaneous denoising and interpolation [15]. The effectiveness of these networks, ranging from basic U-net and GANs to subsequent Transformers, in interpolation hinges on the availability of large datasets as larger datasets yield better deep learning models [16,17]. However, for the specific task of repairing the Q factor in this study, which is targeted at a single region, data augmentation is necessary to create more samples for training the network. Among existing single-sample data augmentations, geometric and color transformations are not applicable to sparse subsurface Q factors. For multi-sample augmentation, the experimental data lack sufficient samples to employ interpolation-based methods such as SMOTE [18] and MIXUP [19].

In light of this, to better reconstruct the values of the Q factor, this paper utilizes the Futterman absorption and attenuation model as a foundation, employing the spectral ratio method [20] and the joint algebraic reconstruction algorithm [21] to calculate the values at the points along the seismic wave propagation path. Then, leveraging the characteristics of deep learning, the paper aims to repair the Q factor values in areas not traversed by the wave during propagation. To address the issue of limited data, this study adopts a block-based selection method, fixing the size of the selection, and using this method to extract samples from the initial Q factor data volume obtained through empirical formulas and the partially true data volume after solving, to construct input/label pairs for training the network. After training, the data volume obtained through empirical formulas is evenly divided into blocks, which are individually input into the network. The outputs are then concatenated to obtain the fully repaired Q factor field.

2. Data Restoration Based on the Principle of Convolutional Neural Networks

From reference [22], we learn that the objective of traditional seismic data restoration is to recover complete data $\mathrm{s}$ from incomplete seismic data $\mathrm{d}$, which can be represented as follows:

$$\mathbf{d}=\mathrm{P}\circ \mathbf{s}$$

(1)

Among them, $\mathrm{P}$ represents the observation matrix, with only 0 and 1 internal elements, representing the actual positions of elements that can be observed on the original complete data $\mathbf{s}$; $\circ$ represents the multiplication of corresponding positional elements.

In supervised seismic data restoration methods, seismic data (typically referring to synthetic seismic records) are often represented as images. Therefore, image processing techniques are commonly employed in engineering applications. Convolutional neural networks (CNNs) have been found to outperform other basic networks in image processing. Moreover, CNNs are considered as mapping functions that map from missing data to complete data, which necessitates a substantial number of input/label pairs for training. We will represent the missing data as ${\mathbf{d}}^{\left(\mathrm{k}\right)}$ and represent the complete data labels as ${\mathbf{s}}^{\left(\mathrm{k}\right)}$ (k = 1, 2, 3, …, N, indicates the number of samples). The loss function for optimizing network parameters can be expressed as follows:

$$\mathbf{L}\mathbf{o}\mathbf{s}\mathbf{s}=\frac{1}{\mathrm{N}}\sum _{\mathrm{k}=1}^{\mathrm{N}}\frac{1}{\mathrm{M}\times \mathrm{N}}{‖{\mathbf{o}}^{\left(\mathrm{k}\right)}-{\mathbf{s}}^{\left(\mathrm{k}\right)}‖}_{\mathrm{F}}^2$$

(2)

where ${\mathbf{o}}^{\left(\mathrm{k}\right)}$ represents the output of the CNN:

$${\mathbf{o}}^{\left(\mathrm{k}\right)}=\mathbf{C}\mathbf{N}\mathbf{N}\left({\mathbf{d}}^{\left(\mathrm{k}\right)},\mathsf{\theta }\right)$$

(3)

In the above two equations, $\mathrm{M}\times \mathrm{N}$ is the number of elements in all samples; ${‖·‖}_{\mathrm{F}}^2$ represents the square of the F-norm; ${‖{\mathbf{o}}^{\left(\mathrm{k}\right)}-{\mathbf{s}}^{\left(\mathrm{k}\right)}‖}_{\mathrm{F}}^2$ can represent the distance between ${\mathbf{o}}^{\left(\mathrm{k}\right)}$ and ${\mathbf{s}}^{\left(\mathrm{k}\right)}$; and $\mathsf{\theta }$ represents the parameters of a CNN.

For the mapping relationship between input and output elements in each sample, ${\mathbf{o}}^{\left(\mathrm{k}\right)},{\mathbf{s}}^{\left(\mathrm{k}\right)}$ in the loss function above can be represented as ${\mathbf{o}}_i^{\left(\mathrm{k}\right)},{\mathbf{s}}_i^{\left(\mathrm{k}\right)}$ and represent (2) as:

$$\mathbf{L}\mathbf{o}\mathbf{s}\mathbf{s}=\frac{1}{\mathrm{N}}\sum _{k=1}^{\mathrm{N}}\frac{1}{\mathrm{M}\times \mathrm{N}}\sum _{i=1}^{\mathrm{M}\times \mathrm{N}}{\left({\mathbf{o}}_i^{\left(\mathrm{k}\right)}-{\mathbf{s}}_i^{\left(\mathrm{k}\right)}\right)}^2$$

(4)

Thus, $\left\{{\mathbf{d}}_i^{\left(\mathrm{k}\right)},{\mathbf{s}}_i^{\left(\mathrm{k}\right)}\right\}$ can be seen as a training pair in a supervised CNN, with the loss function being the average squared distance between the corresponding position elements. By using this loss function for back-propagation, a network which can repair missing data can be trained. For the Q factor values that need to be fixed in this article, there are no complete data. At the same time, combined with the loss function calculation formula mentioned above, it can be inferred that the iterative loss function created by the distance between the elements in some labels and the elements corresponding to the output position can also reconstruct the complete data.

3. Design of the Q Factor Distribution Restoration Method

We first utilize empirical formulas to obtain an initial shallow subsurface Q factor field model and establishe input values through a block-based approach. Subsequently, a fixed-input deep learning network model is constructed, with the Q factor values at discrete grid points along ray paths of seismic waves propagating from shallow underground explosions serving as labels. After training, the network can repair the Q factor values at all grid points within the region.

By calculating the Q factor values along the propagation path, which conform to the Futterman absorption and attenuation characteristics, we consider the Q factor at the grid points traversed by the ray as true values. To compute the complete Q factor field, we employ a deep learning approach to construct a restoration network framework for the Q factor field. However, for the original data, the sample size is too sparse. Randomly removing data pixels to augment the sample size still results in a high missing rate for individual samples, and the reconstruction accuracy cannot be guaranteed. In practical applications, it is impossible to determine whether there is a correlation between each pixel for a specific physical quantity, and it is not feasible to reconstruct the missing parts by setting a correlation function. To address these issues, we propose a solution:

Utilizing a block-based technique, we fix the network input size, allowing for data augmentation by partitioning the original data. We also use the Q factor values obtained from existing empirical formulas as network inputs, which accelerates convergence and reduces error. The partitioned data serve as samples for training the network model. To obtain output data consistent with the input size, the network model should be designed as an encoder–decoder network. The encoder extracts deep-level features from the samples, which are then expanded into output with the same shape as the original data by the decoder. The overall dataset construction and network training pseudo-code are as follows (Algorithm 1):

Algorithm 1. Dataset acquisition and network training

Input: seismic amplitude spectrum: $A_i(i=1,\cdots ,M)$, the reference signal amplitude spectrum: $A_0$ Output: the complete quality factor field: $Q$       Initialize: $Q_{input}$       the spectral ratio method to obtain the tomography equation set:       For $i=1,2,\cdots ,M$ do                                          $A_i=A_0\underset{j=1}{\overset{N}{\mathsf{\Pi }}}exp(\frac{-\pi f{t}_{ij}}{Q_j})(j=1,2,\cdots ,M)$       the fitting of the slope of the logarithmic spectral ratio versus frequency:                                         ${k^{\prime }}_i=ar{g}_{k}min{\left(ln\right(\frac{A_i}{A_0})-kf-b)}^2$       end for       the acquisition of the tomography equation:                                                 $\mathbf{L}\mathbf{q}={\mathbf{K}}^{\prime }$       the solution of the tomography equation: $Q_{true}$       While $(K^k-K^{\prime }{)<=10}^{-5}$ AND $times=2$ do                                         $Q_j^{k+1}=SART(q^k,K^k)$       end while       the construction of samples based on a block-based approach:                       $<Q_{input}^l,X_{label}^l,mas{k}_{label}^l>=\mathit{s}\mathit{e}\mathit{g}\mathit{m}\mathit{e}\mathit{n}\mathit{t}(Q_{input},Q_{ture};size,step)(l=1,2,\cdots ,N_{data})$       For $l=1,2,\cdots ,N_{epochs}$ do                                                           $r^{\left(l\right)}=\mathbf{C}\mathbf{N}\mathbf{N}(Q_{input},\theta )$                                         $o^{\left(l\right)}=mas{k}^{\left(l\right)}·r^{\left(l\right)}$               $s^{\left(l\right)}=mas{k}^{\left(l\right)}·Q_{ture}^{\left(l\right)}$                                         $loss(\theta )=\frac{1}{N}{\displaystyle \sum _{k=1}^N\frac{1}{\left|{\omega }_k\right|}{\displaystyle \sum _{j\in {\omega }_k}{(o^{\left(l\right)}-s^{\left(l\right)})}^2}}$                                                           $\theta \leftarrow \theta -\eta \nabla loss$ end for

3.1. Framework for Q Factor Restoration Based on Prior Information

In the task of seismic data interpolation, the CNN can be regarded as a mapping function that maps from input to output, utilizing the mapping relationship between input values and true values to reconstruct the physical parameters of the target region. Therefore, for the restoration of Q factors in the shallow underground vibration field, we decide to use empirical formulas of Q factors as inputs and incomplete true Q factor data as labels. The iterative loss function adopts the mean squared error (MSE) between the output and the missing data labels containing true values, which can be achieved through masking.

The self-supervised training framework based on observed signals, as depicted in Figure 1, can be divided into two stages: training and testing. In the training phase, the parameters of the CNN are trained. After training, the data are re-partitioned and input into the trained network to construct a complete Q factor field, which is also used to test the effectiveness of the restoration.

The original data (depicted in Figure 1 as ‘input value of empirical formula’) are $\mathbf{b}$ in Formula (5). The data are then subjected to chunking procedures, which involve dividing them into manageable segments for further processing. Simultaneously, the data with true values (depicted in Figure 1 as ‘labels with true value’) are partitioned in the same manner. Each block is assigned a serial number ${\mathbf{b}}^{\left(\mathrm{k}\right)},{\mathbf{q}}^{\left(\mathrm{k}\right)}$, and then the corresponding sequences are combined into input/target pairs. The labels in the pairs are filtered, denoted as ${\left\{{\mathbf{b}}^{\left(\mathrm{k}\right)};{\mathbf{q}}^{\left(\mathrm{k}\right)};{\mathbf{m}}^{\left(\mathrm{k}\right)}\right\}}_{\mathrm{k}=1}^{\mathrm{N}}$, where $\mathrm{N}$ represents the total number of samples after filtering, and ${\mathbf{m}}^{\left(\mathrm{k}\right)}$ represents the mask matrix for the k-th sample. This mask matrix is a binary matrix containing only 0 and 1 values.

We input ${\mathbf{b}}^{\left(\mathrm{k}\right)}$ into the network to obtain the predicted complete data, denoted as ${\mathbf{o}}^{\left(\mathrm{k}\right)}$, according to the following formula:

$${\mathbf{r}}^{\left(\mathrm{k}\right)}=\mathbf{C}\mathbf{N}\mathbf{N}\left({\mathbf{b}}^{\left(\mathrm{k}\right)},\mathsf{\theta }\right)$$

(5)

where $\mathsf{\theta }$ represents the parameters to be trained in the CNN. After obtaining the predicted complete data, we compare the parts with existing true values to the corresponding parts predicted by the network. In a more detailed perspective, and combining with the content of the previous section, for the prediction of signal points, we have the following formula:

$${\mathbf{r}}_i^{\left(\mathrm{k}\right)}=\mathbf{C}\mathbf{N}\mathbf{N}\left({\mathbf{b}}_i^{\left(\mathrm{k}\right)},\mathsf{\theta }\right)$$

(6)

where ${\mathbf{r}}_i^{\left(\mathrm{k}\right)}$ represents the value of the i-th element generated by the k-th sample. For the predicted values generated at the corresponding positions of the ${\mathsf{\omega }}_{\mathrm{k}}$ true values of the k-th sample, they can be represented as follows:

$${\mathbf{o}}^{\left(\mathrm{k}\right)}={\mathbf{m}}^{\left(\mathrm{k}\right)}\circ {\mathbf{r}}^{\left(\mathrm{k}\right)}$$

(7)

where $‘\circ ’$ denotes the Hadamard product, which signifies the element-wise multiplication of corresponding elements. Therefore, the true part of ${\mathbf{q}}^{\left(\mathrm{k}\right)}$ can still be calculated through masking:

$${\mathbf{s}}^{\left(\mathrm{k}\right)}={\mathbf{m}}^{\left(\mathrm{k}\right)}\circ {\mathbf{q}}^{\left(\mathrm{k}\right)}$$

(8)

Finally, the network parameters are optimized by minimizing the average distance between predicted points and true values:

$$\widehat{\mathsf{\theta }}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{\mathsf{\theta }}{\min }\frac{1}{\mathrm{N}}\sum _{k=1}^{\mathrm{N}}\frac{1}{\left|{\mathsf{\omega }}_k\right|}\sum _{j\in {\mathsf{\omega }}_k}{\left({\mathbf{o}}_j^{\left(\mathrm{k}\right)}-{\mathbf{s}}_j^{\left(\mathrm{k}\right)}\right)}^2$$

(9)

Following training, the CNN is capable of directly reconstructing the missing portions as shown in Figure 2; the initial data are re-divided into several equal blocks. These blocks are sequentially fed into the trained CNN, yielding an equal number of reconstructed values. These reconstructed blocks are then assembled into a matrix o, which aligns with the original data in shape and size. Ultimately, the true values replace the predicted values at corresponding positions to yield the final interpolation outcome (where $\mathbf{M}$ denotes the mask for the true value segments in $\mathbf{q}$).

$$\mathbf{y}=\left(1-\mathbf{M}\right)\circ \mathbf{o}+\mathbf{M}\circ \mathbf{q}$$

(10)

3.2. Construction of Data Body

According to the Futterman absorption attenuation model [23], viscoelastic media exhibit an absorption and attenuation effect on the propagation of seismic waves. Assuming the amplitude of the vibration wave at a reference point is $A_0$, and, after propagating $x$ meters due to absorption attenuation, the amplitude decreases to $A$, then it can be expressed as:

$$A=A_0{e}^{-\alpha x}$$

(11)

where $\alpha$ represents the attenuation coefficient. Assuming a frequency $f$, velocity v, and quality factor Q, the attenuation coefficient can be computed using Equation (12).

$$\alpha =\frac{\pi f}{Qv}$$

(12)

After discretizing the testing area into grids, sensors are deployed on the surface. With a total of M sensors and N grids, if the i-th ray traverses j-th grids, then combining Equations (11) and (12), the amplitude spectrum of the signal received by the i-th sensor can be derived as Equation (13), one of these paths is shown in Figure 3:

$$A_i(f)=A_0(f){\displaystyle \prod _{j=1}^N\exp \left(\frac{-\pi f{t}_{ij}}{Q_j}\right)}$$

(13)

In the equation, $t_{ij}$ represents the travel time of the i-th ray in the j-th network. Taking the logarithm of both sides of the equation, and employing the spectral ratio method [20], a system of equations is established by fitting the linear relationship between the logarithmic spectral ratio and frequency within the effective frequency range:

$$k_i^{\prime }=\underset{k}{\arg \min }{(L_i(f)-kf-b)}^2$$

(14)

$$L_i(f)=\ln \left(\frac{A_i}{A_0}\right)$$

(15)

Here, k and b represent the true slope and intercept of the line to be fitted, and ${\mathrm{k}}_i^{\prime }$ is the slope of the fitted line within the effective range of the logarithmic spectral ratio between the i-th sensor and the reference signal. Thus, the attenuation equation set is given by:

$$k_i^{\prime }={\displaystyle \sum _{j=1}^N{q}_j{l}_{ij}}$$

(16)

$$q_j=\frac{1}{Q_j}$$

(17)

where ${\mathrm{q}}_{\mathrm{j}}$ represents the reciprocal of the Q factor of the j-th grid. Further expressing Equation (17) in matrix form, we obtain the attenuation tomography model:

$$Lq=K^{\prime }$$

(18)

Here, $\mathrm{L}={\left({\mathrm{l}}_{ij}\right)}_{\left(\mathrm{M}-1\right)\times \mathrm{N}}$ is called the attenuation tomography kernel matrix, $\mathrm{q}={\left({\mathrm{q}}_j\right)}_{\mathrm{N}\times 1}$ is the attenuation feature vector, ${\mathrm{K}}^{\prime }={\left({\mathrm{k}}_i\right)}_{\left(\mathrm{M}-1\right)\times 1}$ is the slope of the logarithmic spectral ratio, and the elements of the $\mathrm{L}$ matrix are related to the time at which the propagation lines pass through the grid between the source and sensors. Not all grids are intersected by rays, so the $\mathrm{L}$ matrix is not full rank, making this equation underdetermined. Using standard algorithms, we can only obtain the Q factor values at the ray intersection points.

We employ the SART (Simultaneous Algebraic Reconstruction Technique) [21], which is an iterative reconstruction algorithm that integrates the advantages of both the ART (Algebraic Reconstruction Technique) [24] and the SIRT (Simultaneous Iterative Reconstruction Technique) [25], to calculate the Q factor values along the propagation paths. The core idea of the SART algorithm is to update each pixel of the image after computing the projection errors for all paths at each projection angle, thereby reducing the noise introduced by the ART algorithm. The iterative formula for the SART algorithm is as follows:

$$q_j^{k+1}=q_j^k+\frac{\lambda }{{\displaystyle \sum _{i\in I_{\varphi }}{l}_{ij}}}·{\displaystyle \sum _{i\in I_{\varphi }}\frac{l_{ij}·\left(k_i^{\prime }-{\displaystyle \sum _{j=1}^J{l}_{ij}{q}_j^k}\right)}{{\displaystyle \sum _{j=1}^J{l}_{ij}}}}$$

(19)

The Q factor values obtained through the aforementioned method conform to Futterman’s absorption attenuation characteristics. We consider the Q factor values at the grid points traversed by the propagation path to be accurate. In the entire discrete grid, the calculated values are assigned to their respective positions, with the rest set to zero, serving as the labeled data body. Simultaneously, a mask matrix is constructed. Then, the empirical initial values combined with this form a complete input/label set for block operation, serving as data for network training. We choose Li’s empirical formula [26], which is the fitting relation function obtained by Li after synthesizing a lot of data, where $V_p$ represents the P wave velocity in $\mathrm{m}/\mathrm{s}$.

$$Q=14{\left(V_p/1000\right)}^{2.2}$$

(20)

3.3. Dataset Expansion

Upon obtaining the complete dataset, only the grids traversed by the propagation path optimize their quality factor values. To interpolate the quality factors for grids not covered by the propagation path using deep learning, we face challenges due to limited samples, which hinder effective neural network training. Therefore, we propose a block-based data augmentation method.

For an incomplete three-dimensional dataset with a shape of $\mathrm{N}\times \mathrm{N}\times \mathrm{N}$ (with missing parts represented as 0), each divided block has a shape of $\mathrm{n}\times \mathrm{n}\times \mathrm{n}$. Then, blocks with missing rates lower than a specified threshold are selected. Each block is then converted into a one-dimensional vector and used as input or output for the network. The displacement between each block is set to 1, providing flexibility for using the same network to reconstruct three-dimensional data. Subsequently, the parameters of the deep learning network are trained, and uniformly divided blocks are input into the network to predict outputs. Finally, the evenly divided blocks are concatenated to obtain the reconstructed data.

Assuming that complete three-dimensional data are divided into blocks and the sliding step size is $\mathrm{t}$, the following formula can be obtained:

$$\mathrm{L}=\frac{\mathrm{N}-\mathrm{n}}{\mathrm{t}}+1$$

(21)

In the formula, $\mathrm{L}$ represents the number of divisions that can be made by an edge. The above complete three-dimensional data can be divided into a total of blocks, and each filtered block is used as a sample to train the network.

For our experiment, as illustrated in Figure 4, the initial values obtained from the empirical formula and the partial true data body solved in Section 3.2 are subjected to block operations. At this stage, the input and labels correspond one to one. When blocks that meet the criteria are selected from the label blocks, the corresponding input blocks are also chosen, together serving as the input/label pairs for network training.

3.4. Network Structure

The primary network architecture we employ is the renowned U-Net. The specific network structure is depicted in Figure 5. Both the input and output of the entire network are one-dimensional tensors; therefore, prior to the commencement of training, unfolding processing is flattened for each sample tensor. The red rectangles represent the feature extraction module with residual connections, consisting of two blocks. Each block follows a “convolution (conv), rectified linear unit (ReLU), convolution (conv), rectified linear unit (ReLU)” structure, with residual connections added. The first block of each layer requires a convolution operator of size 1 at the connection point to unify the channel numbers, represented by gray cubes. The orange rectangles represent multi-scale skip connections, enabling the model to better utilize feature information from both high and low levels. The green rectangles represent the decoding module, which follows the same “convolution (conv), rectified linear unit (ReLU), convolution, rectified linear unit” connection method as the block modules above. Purple triangles indicate downsampling, implemented using 1D convolutions, while orange triangles represent upsampling operators, achieved using 1D deconvolutions. Finally, a convolution operator of size 1 (1D; thus, size = 1) is added as the output layer. The output layer inherits channel dimension information, yielding the reconstruction result.

4. Experiment and Result Analysis

The primary objective of this experiment is to solve for the quality factor values in specific areas not traversed by the propagation path and assess their proximity to the true values. Since the quality factors for locations not hit by rays cannot be inferred using the aforementioned method, and there is no existing dataset of Q factors or data for other regions, we decide to deploy several sensors and uniformly select 20% of them as test data for the trained network. Specifically, we utilize the Q factor values at the grid points traversed by these 20% of the propagation paths as the portions that require repair. Post-network training, the repaired values are obtained and compared with the true values at the same locations that were excluded. A smaller error indicates that the Q factor values at other grid locations are closer to the true values. In this experiment, the mean absolute percentage error (MAPE) and Root-Mean-Square Error (RMSE) are employed as evaluation metrics. Smaller MAPE and RMSE values signify better performance.

4.1. Field Experiment

To verify the effectiveness of the proposed algorithm, real data are collected for calculation, and the following experiments are designed for validation. Take any point on the ground surface as the origin, and set the direction perpendicular to the ground as the z-axis direction. Use tools to measure a test area measuring 24 m long and 24 m wide. Then, deploy a seismic sensor array consisting of a square array formed by 144 sensors in total, as shown in Figure 6. Designate the coordinates of the seismic source at position (−0.87, 0.45, −24); additionally, deploy the position at (−0.51, −0.35, −24.31) for measuring vibration signals. Connect an electric spark seismic source device and initiate the explosion. After the explosion, observe the reception of signals to ensure normal signal reception by the sensors. The experimental layout is shown in Figure 6 and Figure 7.

The displacement parameters are obtained after cumulative trapezoidal integration of the measured speed signal. Some sensor signals in the approach area are shown in Figure 8.

According to the spectral ratio feature extraction method proposed above, the reference signal node is selected. The relevant waveform and spectrum of the reference signal are as shown in Figure 9.

In the experiment, according to the requirements of the inversion reconstruction algorithm, the velocity model is determined by measuring the prior information of the shallow subsurface medium and the travel time information. The medium is divided into three layers at depths of 23.91 m, 15.93 m, and 8.10 m, respectively. Subsequently, the grid is discretized with a grid length of 0.8 m. The obtained complete data volume shape is $30\times 30\times 30$. Using source location coordinates, sensor location coordinates, and velocity field model arrival time information, ray paths are searched based on Fermat’s principle, yielding coordinates of ray grid intersections and travel time information of seismic signals collected by each sensor as shown in Figure 10. The effectiveness rate of data in the established grid is 16.18%.

Using the spectral ratio method (Equations (11)–(18)), the signal received by the sensor with the earliest arrival time is used as the reference signal. For each signal received by the sensors, a frequency band ranging from 10 Hz to 80 Hz is selected for logarithmic spectral ratio fitting. From the fitting of lines with different slopes, a model correlating travel time, Q factor, and the corresponding slope of attenuation tomography (Equation (20)) is established. The SART algorithm, presented in Equation (19), is used to solve this model. The relaxation factor is determined by sampling from a Gaussian distribution with a mean of 0.5 and a variance of 1. If the residual difference between each round of calculation and the previous round is less than 2 × 10⁻⁵, it can be considered that the iterative algorithm has reached the convergence condition.

Upon acquiring the quality factor values for grids intersected by rays, by eliminating certain ray paths, we can construct a training dataset using a block-based approach. Using the partitioning method described in Section 3.3, each segment is shaped into a $6\times 6\times 6$, serving as the label for the network, with a displacement size of 1 between each segment. Similarly, the Q factor data bodies derived from empirical formulas are processed in the same manner to form blocks that serve as the network’s input. Ultimately, 8431 input/label pairs are generated as the training set for the network.

After obtaining the input–label pairs, Equation (9) is used as the loss function, and the network parameters are updated using the Adam gradient descent algorithm. The batch size is set to 32 with an initial learning rate of 0.002 for the gradient descent method, which is multiplied by 0.1 every 200 rounds, amounting to a total of 2000 training rounds. An RTX-4070 GPU is used for training, which lasts 1.5 h. During training, the loss gradually decreases and eventually converges. The training with empirically initialized inputs is compared to training with randomly initialized inputs, as illustrated in Figure 11. The blue curve represents the randomly initialized inputs, and the red represents the empirically initialized inputs. Both use the same number of training rounds and hyper-parameter settings. From the graph, it can be observed that using empirical formula initialization inputs leads to faster convergence.

4.2. Experimental Results

After the network training concludes, the original input data of size $30\times 30\times 30$ are evenly divided into 125 blocks of size $6\times 6\times 6$. Each of these blocks represents a three-dimensional array. These three-dimensional arrays are then flattened into one-dimensional arrays using the same flattening process as before. Subsequently, each one-dimensional array is inputted into the network, resulting in 125 outputs of the same size as the input. Next, the output one-dimensional tensors are reshaped back into $6\times 6\times 6$ three-dimensional arrays following the original traversal method. These $6\times 6\times 6$ arrays are then combined to reconstruct the $30\times 30\times 30$ array. Finally, according to Equation (10), the true values are replaced with the corresponding output values at the same positions to obtain the repaired Q factor for the entire region.

Figure 12 depicts the Q factor distributions along the propagation paths used for testing, as well as the Q factor distributions at the same locations output by the trained network. It can be observed that the overall numerical ranges are similar. Due to the dense arrangement of sensors, multiple paths inevitably intersect at the same grid in the final layer of the medium. To evaluate the discrepancy between the network’s output values and the true values, it is necessary to exclude the grids that are common to both the test and training sets, leaving only the grids intended for testing. Figure 13 illustrates the error between the predicted and true values for the remaining sections after removing the values at the common grids. The final MAPE (mean absolute percentage error) achieved is 0.972%, which indicates that the objective of this experiment is successfully met.

4.3. Comparative Experiment

To validate the effectiveness of the data augmentation method proposed in this paper for the restoration of Q factor values, we also conduct an experiment where we randomly remove true values within the entire region to increase the sample size, using this as input to the network for comparison with our method. The input is set to the overall data with randomly missing true values, with the remaining parts replaced by initial values calculated using empirical formulas. The labels are set to the complete data containing all true values, though not every part of the data has true values. The network structure is consistent with the one employed in this paper, with the sole modification being the replacement of one-dimensional convolution with three-dimensional convolution, while all other operations remain the same. The comparison of MAPE and RMSE evaluation metrics can be seen in

Table 1. MAPE indicators for different approaches.

Method	MAPE	RMSE
Block-Based + Our Model	0.972%	0.753
Random Missingness + Our Model	6.501%	1.701
Block-Based + U-net	0.976%	0.756
Block-Based + CBAM-U-net	0.941%	0.726

, indicating that this method yields higher errors. Upon examining the restored full-region image in Figure 14, it becomes apparent that this method has limited restoration capabilities for Q factor values in areas not traversed by the propagation paths and demonstrates weak edge-mapping abilities.

Additionally, we explore the viability of our training framework using different network structures, implementing alterations specifically within the block modules of the network framework diagram. We first remove the residual connections, resulting in a standard U-net network; another variation involves removing the residual links in the block and appending an attention module at the end of Block2. The same evaluation metrics, MAPE and RMSE, are utilized, and the outcomes are displayed in Table 1. The results indicate that our training method achieves commendable performance across the various generative networks.

4.4. Discussion

In the experiment, a $30\times 30\times 30$ discrete grid is delineated within the known region to perform block-based operations for data augmentation. However, the size of the blocks influences the number of training and testing samples. Our ultimate goal is to determine the accuracy of the output full-region Q factor values, so we only need to assess the magnitude of the evaluation metrics. Based on this concept, we try to discuss the impact of block size on the experimental results. Regardless of the block size, we recommend setting the sliding stride to 1, which allows for the acquisition of as many sample pairs as possible while maintaining a fixed network input size.

Due to the use of a $6\times 6\times 6$ block size for the $30\times 30\times 30$ discrete dataset in the aforementioned experiment, we also attempt varying the block sizes to $4\times 4\times 4$, $5\times 5\times 5$, $8\times 8\times 8$, and $10\times 10\times 10$, comparing the trend of the block size impact on experimental results using MAPE and RMSE as evaluation metrics.

As is evident from

Table 2. The effect of block size on the result.

Block Size	MAPE	RMSE
$4\times 4\times 4$	1.68%	0.972
$5\times 5\times 5$	0.982%	0.771
$6\times 6\times 6$	0.972%	0.753
$8\times 8\times 8$	1.907%	1.373
$10\times 10\times 10$	1.921%	1.468

, a block size of 6 × 6 seems to be the most optimal for our data. Nonetheless, this does not guarantee that this size would be the best for data in other experiments. It is necessary to validate this by experimenting with blocks of different sizes.

5. Conclusions

For the problem of Q factor field restoration under a single data volume, we propose a block-based training framework. It employs existing empirical formulas as network inputs and uses the calculated true values (with missing parts) as network labels. The aim is to learn the mapping relationship between empirical values and true values and to test this mapping under three types of networks. The performance of the network training is assessed by comparing the values at the grid points where the 20% masked propagation paths pass through. This training approach effectively restores the Q factor values in the areas where the seismic wave propagation paths have not traversed. Additionally, this method has several advantages for Q factor restoration: firstly, it can be applied to other seismic data reconstruction problems. For single-sample missing data, it can construct a large number of training pairs, use CNN training, and employ the trained network to repair the original missing or damaged data. Secondly, using prior information as network input values can help shorten training time while achieving relatively accurate results. In the future, further research can be conducted to explore restoration of even sparser data.