• Proposed a deep learning U-Net model to reduce random noise in Laplace-domain seismic data.

• U-Net outperforms traditional SVD for noise reduction in synthetic and field seismic data.

• Improves subsurface imaging quality by enhancing Laplacedomain full waveform inversion accuracy.

Seismic exploration is a fundamental tool for subsurface investigations, particularly in hydrocarbon exploration. It involves acquiring, processing, and interpreting seismic data to create detailed images of subsurface structures. However, various types of noise often compromise seismic data quality, significantly reducing the signal-to-noise ratio (SNR) and complicating the interpretation process. Noise in seismic data generally falls into two broad categories: coherent and incoherent noise (Yilmaz, 2001). Coherent noise includes energy that is systematically correlated, such as ground roll and multiples, whereas incoherent noise is random and uncorrelated, often referred to as random noise (Zhang et al., 2020).

To enhance the quality of seismic data, researchers have developed various traditional methods to attenuate noise. These methods typically transform data into different domains where the separation of signal and noise becomes more straightforward. Widely used techniques include Fourier, wavelet, and curvelet transforms. The Fourier transform, fundamental in seismic data processing, filters out noise by transforming data into the frequency domain. However, its effectiveness is limited when dealing with non-stationary signals due to the lack of time-frequency localization. This limitation often necessitates combining Fourier transforms with other methods to improve performance in complex noise environments (Xue et al., 2019). Wavelet transforms address some limitations of the Fourier transform by offering both time and frequency localization, making them more effective for non-stationary seismic signals. Techniques such as the continuous wavelet transform (CWT) and synchrosqueezed wavelet transform have been successfully used to attenuate ground roll and random noise while preserving the signal (Li et al., 2017, Xue et al., 2019). Curvelet transforms have emerged as powerful tools for seismic noise attenuation, particularly for coherent noise like ground roll. They offer a multiscale, multidirectional representation of seismic data, allowing better differentiation between signal and noise, leading to superior noise suppression while preserving important signal features (Liu et al., 2018, Zhang et al., 2018).

Despite their effectiveness, these traditional methods often struggle to balance noise attenuation and signal preservation, particularly in complex geological settings. To overcome these challenges, researchers have employed advanced techniques like singular value decomposition (SVD) and empirical mode decomposition (EMD) to enhance denoising, especially in environments where traditional approaches may falter (Kopsinis and McLaughlin, 2009, Chen and Ma, 2014, Gan et al., 2015). SVD-based methods, such as structure-oriented singular value decomposition, have shown promise in attenuating random noise while preserving structural features of seismic data (Gan et al., 2015). EMD-based methods provide adaptive decomposition of signals, making them suitable for processing nonstationary and nonlinear data (Kopsinis and McLaughlin, 2009, Chen and Ma, 2014).

Dictionary learning-based methods have also gained traction, providing a more adaptive approach to noise suppression by learning sparse representations of seismic data (Zhu et al., 2015, Almadani et al., 2021). These methods represent a significant advancement over traditional techniques, particularly in environments with low SNR or complex noise characteristics.

In recent years, deep learning has emerged as a powerful tool for noise attenuation in seismic data, offering several advantages over traditional methods. Deep learning-based methods learn to distinguish between signal and noise through training on large datasets, enabling models such as convolutional neural networks (CNNs) to effectively attenuate noise while preserving important seismic signal features (Zhu et al., 2019, Saad and Chen, 2020). Moreover, recent developments have introduced self-supervised and unsupervised learning techniques that are particularly effective in seismic noise attenuation. These methods allow for noise suppression without the need for clean reference datasets, making them highly adaptable to various seismic data scenarios (Yang et al., 2021, Liu et al., 2022, Meng et al., 2022). Additionally, U-Net architectures enhanced with residual blocks have been employed for seismic random noise suppression, enabling noise reduction while preserving signal fidelity (Zhong et al., 2022). These advancements highlight the potential of deep learning in overcoming the limitations of traditional denoising methods.

Laplace-domain full waveform inversion (FWI) is a powerful technique for recovering large-scale subsurface velocity models and is known for its robustness to initial models (Shin and Cha, 2008, Shin and Ha, 2008). However, the Laplace-domain wavefield is highly sensitive to noise, especially near the first-arrival signal, due to the exponential damping in the Laplace transform (Shin and Cha, 2008). This damping amplifies small-amplitude noise, making noise attenuation in the Laplace domain crucial for successful inversion.

Traditional noise attenuation methods, such as SVD in the Laplace domain, have been applied to mitigate this issue (Ha and Shin, 2021a). However, these methods may not fully exploit the potential of learning-based approaches in capturing complex patterns in the data. Supervised deep learning methods, like the U-Net architecture, can leverage labeled data to learn intricate relationships between noisy and clean data, potentially outperforming unsupervised techniques like SVD in denoising tasks.

This study aims to advance the current understanding and application of noise attenuation techniques by developing a novel deep learning-based method specifically for denoising seismic data in the Laplace domain. By leveraging the strengths of deep learning in noise suppression and applying it within the Laplace domain, we seek to achieve more effective noise attenuation than traditional SVD methods, ultimately enhancing the accuracy of seismic imaging and inversion processes.

We employed a modified U-Net architecture (Ronneberger et al., 2015) for denoising Laplace-domain seismic wavefields (Fig. 1). The network follows the standard structure of contracting and expanding paths. In the contracting path, each block consists of two successive 3×3 convolutional layers with ReLU activation, followed by a max-pooling layer that halves the input size. The number of kernels doubles at each block, enabling the model to learn hierarchical features from the data. At the bottom of the U-Net, we added dropout layers (rate 0.5) to two 3×3 convolutional layers for regularization, forming the bottleneck connecting the contracting and expanding paths. The expanding path is symmetrical but uses upsampling convolutions to restore the spatial dimensions. We included concatenation layers to merge learned feature maps from the contracting path with their corresponding blocks in the expanding path, facilitating the preservation of spatial information.

Figure 1. Architecture of a denoising U-Net.

The network input comprises noisy logarithmic Laplacedomain wavefields with dimensions corresponding to 100 shots and 128 receivers for a single damping constant, represented in one channel. The labels are the corresponding noise-free logarithmic wavefields.

2.1. Data Generation

We generated training data using 2D acoustic Laplacedomain wave propagation modeling for computational efficiency (Shin and Cha, 2008). The Laplace-domain wavefield modeling directly computes the Laplacetransformed wavefields without the need for long timedomain simulations followed by Laplace transformation, which can be computationally intensive due to the necessity of long recording times for stability (Ha and Shin, 2013, Ha and Shin, 2021a).

To simulate diverse subsurface conditions, we created 36,096 P-wave velocity models representing the Gulf of Mexico environment. Fig. 2 displays four sample velocity models. The models consisted of 360 × 100 grids with grid sizes varying randomly from 10 m to 125 m. Varying the grid size enhances the network's robustness by exposing it to data with different spatial resolutions and sampling rates, improving its generalization ability across different acquisition scenarios.

Figure 2. Sample velocity models used to generate training data.

We simulated a towed-streamer acquisition with 100 shots and 128 receivers per shot. The first source was located at the 138th grid point and the last at the 336th, with 2-grid intervals between sources and 1-grid intervals between receivers. The minimum offset between a source and the nearest receiver was 2 grids. We used one of six damping constants ranging from 2 to 12 s⁻¹ at intervals of 2 s⁻¹ for each training sample. The Delta function was used as the Laplace-domain source. Since source wavelet convolution in the time domain corresponds to a constant multiplication in the Laplace domain for a single damping constant, the source wavelet does not affect denoising in the Laplace domain.

2.2. Noise Generation

We added random noise to the clean Laplace-domain wavefields. To simulate realistic seismic ambient noise, we generated Gaussian random noise in the time domain for each trace over 10 s with a 1 ms sampling rate. We calculated the first-arrival traveltime for each shot of each velocity model (Shin et al., 2003), muted noise before the first arrival and transformed the noise-only wavefields to the Laplace domain. The muting process is essential because noise before the first arrival can dominate the Laplacedomain wavefields due to exponential damping, masking the useful signals (Shin and Cha, 2008). We varied the noise level scale for each model to expose the network to a range of SNR conditions. The data generation process for each trace is expressed as:

$$x_{ij}\left(s\right)=y_{ij}\left(s\right)+\lambda n_{ij}\left(s\right)=y_{ij}\left(s\right)+\lambda {\int }_0^T{n}_{ij}\left(t\right)e^{-st}dt$$(1)

where x is the noisy wavefield, y is the clean wavefield, n is random noise, s is the damping constant, T is maximum recording time, λ is the scaling factor controlling the noise level, and i and j are the source and receiver indices, respectively.

2.3. Data Preprocessing

We took the logarithm of the absolute value of the Laplace-domain noisy wavefields to obtain the logarithmic wavefield, which served as the network input. The logarithm reduces the large scale differences between near- and faroffset traces resulting from exponential damping in the Laplace-domain, facilitating network training and allowing the use of single-precision floating-point numbers (Ha and Shin, 2021b). The noisy logarithmic Laplace-domain wavefields served as network input, with clean logarithmic wavefields as labels:

$$\hat{y}=N(\tilde{x},\theta )$$(2)

where $\hat{y}$ is the predicted wavefield, N is the denoising network, $\tilde{x}$ represents the noisy logarithmic wavefields, and θ denotes network parameters. Fig. 3 shows two input and label samples from the validation data.

Figure 3. Inputs (top), labels (middle), and their profiles (bottom) from two validation samples. The profiles are extracted from the 51st shots indicated by dashed lines. The damping constants used are 10 s⁻¹ and 2 s⁻¹, and the grid sizes are 60.6 m and 70.6 m, respectively.

2.4. Network Training

We used 32,000 samples for training and 4,096 for validation. The network was implemented using TensorFlow (Abadi et al., 2016) and trained on four Nvidia GeForce RTX 3090 GPUs. The batch size per GPU was 32, with an initial learning rate of 1.0 × 10⁻³, decreasing by 1% after each epoch to ensure convergence. We used the Adam optimizer (Kingma and Ba, 2015) with AMSGrad (Reddi et al., 2018) to minimize the mean squared error (MSE) loss between predicted data and labels. Training for 100 epochs took 1 h 14 m 47 s. Fig. 4 displays denoised validation data after 100 epochs, demonstrating the network’s ability to attenuate most random noise in the wavefields, although small discrepancies remain at far offset due to lower SNR in those regions.

Figure 4. Predicted results (top) and profiles of inputs (bottom) shown in Fig. 3.

3.1. Denoising Noisy Laplace-Domain Wavefields

We tested the denoising U-Net using the Pluto velocity model (Stoughton et al., 2001) (Fig. 5). We generated time-domain shot gathers using 100 sources with a towedstreamer acquisition geometry. The Ricker source wavelet had a maximum frequency of 20 Hz, and we used a grid size of 50 ft (15.24 m). We employed an acoustic finite difference modeling scheme, second-order in time and eighth-order in space. The intervals between sources were 600 ft (182.88 m), with a minimum offset of 200 ft (60.96 m). Each shot gather contained 128 receivers at the 200 ft intervals, resulting in a maximum offset of 25,600 ft (7.8 km).

Figure 5. The Pluto velocity model (Stoughton et al., 2001).

We generated random noise for each shot in the time domain and muted noise before the first arrival to prevent early-time noise from dominating the Laplace-domain wavefields (Shin and Cha, 2008). Fig. 6 shows noisy, clean, and denoised wavefields using the SVD (Ha and Shin, 2021a) and U-Net methods for damping constants of 4, 6, 8, and 10 s⁻¹. For SVD denoising, we retained the three largest singular values. We evaluated the denoising performance by calculating the MSE loss between the denoised data and the clean data (Table 1). The MSE of the noisy data was 0.5870, the MSE of SVD-denoised data was 0.0935, and the MSE of U-Net-denoised data was 0.0372. These results confirm that the deep learning-based denoising more closely approximates the clean data compared to SVD denoising.

Table 1 . MSE losses of the noisy and denoised Pluto data calculated with the clean data.

Data	MSE loss
Noisy data	0.5870
Denoised (SVD)	0.0935
Denoised (U-Net)	0.0372

Figure 6. Noisy, clean, and denoised wavefields (a) using the SVD and (b) U-Net for damping constant of 4, 6, 8, and 10 s⁻¹.

3.2. Denoising Field Data

We also tested denoising on a Gulf of Mexico field dataset. Fig. 7 shows a shot gather from 399 total gathers. Each shot gather contains 408 receivers at 25 m intervals, with 50 m intervals between sources. The minimum offset is 137 m, and the maximum offset is 10.3 km. We extracted 100 distributed shot gathers from 399 total gathers and interpolated their Laplace-transformed wavefields to an 80 m grid. We interpolated dead traces using simple linear interpolation with adjacent traces. The resulting interpolated data comprise 100 shots and 128 receivers. We used four damping coefficients from 4 to 10 s⁻¹ at intervals of 2 s⁻¹. Fig. 8 shows the original and denoised data using SVD and U-Net methods. Both denoised datasets exhibit smooth wavefield variations.

Figure 7. A shot gather from the Gulf of Mexico dataset.

Figure 8. The original and denoised data using SVD and U-Net for damping constants of 4, 6, 8, and 10 s⁻¹. The number of traces is 128, and the grid size is 80 m.

3.3. Full Waveform Inversion Using Denoised Data

We performed Laplace-domain full waveform inversion using the denoised data to assess the effectiveness of the denoising methods in improving subsurface imaging. We first inverted the synthetic Pluto data using noisy data, SVD-denoised data, and U-Net-denoised data. The inversion minimizes the logarithmic objective function for each damping constant:

$$E\left(s\right)=\frac{1}{2}{\sum _{i,j}^{N_{src,}{N}_{rcv}}\left(\log \frac{u_{ij}\left(s\right)}{d_{ij}\left(s\right)}\right)}^2,$$(3)

where u is the modeled data, d is the observed data, N_src is the number of sources and N_rcv is the number of receivers. We used four damping constants from 4 to 10 s⁻¹ in the inversion, with an inversion grid size of 200 ft (60.96 m). We employed the Nesterov accelerated gradient method using sloth parameterization (Park et al., 2020).

Fig. 9 shows the initial velocity model and an inversion result using clean data. Small salt bodies on the left and right are not recovered due to limited offset. Fig. 10 displays inversion results using the noisy data, SVDdenoised data, and U-Net-denoised data. The inversion using noisy data failed to recover the salt shapes correctly (Fig. 10a). The inversion result from SVD-denoised data (Fig. 10b) was similar to that from noisy data, as SVD tends to smooth the noisy wavefields but retains the general trends, including residual noise. In contrast, the inversion result from U-Net-denoised data (Fig. 10c) showed improved recovery of the salt shapes, closer to the inversion result using clean data (Fig. 9b). We compared the error histories during inversion (Fig. 11). The inversion using U-Net-denoised data achieved a lower final error compared to the inversion using SVD-denoised data and noisy data, indicating a better fit to the observed data.

Figure 9. Laplace-domain waveform inversion for the Pluto velocity model. (a) The initial velocity model and (b) the inversion result using clean data.

Figure 10. Inversion results using (a) the noisy data, (b) SVD-denoised data, and (c) U-Net-denoised data.

Figure 11. Error histories of Laplace-domain inversions.

We also inverted the Gulf of Mexico field data using noisy data, SVD-denoised data, and U-Net-denoised data. We used four damping constants from 4 to 10 s⁻¹ with an 80 m grid size. We used only 100 shots for inversion among 399 shots for comparison. Fig. 12a shows the initial velocity model and Fig. 12b, 12c, 12d show the inversion results of noisy data, SVD-denoised data, and U-Net-denoised data. We interpolated the velocity models to a 25 m grid for subsequent analysis. The results from noisy data and SVD-denoised data are similar (Ha and Shin, 2021a), but the salt diapir shape differs slightly in the velocity model from U-Net-denoised data, showing a slenderer salt body.

Figure 12. Laplace-domain waveform inversion of Gulf of Mexico field data. (a) The initial velocity model used in Laplace-domain waveform inversion, and inversion results from (b) the original noisy data, (c) SVD-denoised data, and (d) U-Net-denoised data.

Although we cannot definitively determine which velocity structure is more accurate without additional data such as well logs, we can indirectly evaluate the plausibility of the velocity models by comparing error histories and the forward-modeled wavefields with the observed data. Fig. 13 shows that both denoising methods significantly reduced error; however, the inversion result from U-Net-denoised data shows a smaller error than that from SVD-denoised data. Since these three inversion examples used different observed data, a smaller error does not necessarily indicate a better velocity model. Therefore, we compared the inversion results using the same original data. We generated forwardmodeled Laplace-domain wavefields using the inverted velocity models and the original acquisition geometry and compared them with the original observed data.

Figure 13. Error histories of Laplace-domain inversions.

Fig. 14 shows the original data and the modeled data from the inverted velocity models. Table 2 presents the MSE loss between the observed data and the forward-modeled data using all 399 shots, 408 receivers, and four damping constants. Note that the MSE loss is the logarithmic objective function of the FWI multiplied by a constant, as we used logarithmic wavefields to calculate the loss. The wavefields from the inversion result of UNet- denoised data show the smallest error. Although we used denoised data to obtain the velocity models, the forward-modeled data from the inversion results fit the original observed data better than that from the inversion result using the original data. Accordingly, we confirm that denoising helped waveform inversion in the Laplace domain recover an enhanced subsurface velocity model.

Table 2 . MSE losses of the logarithmic forward-modeled data generated from the inversion results calculated with the observed Gulf of Mexico data.

Data of FWI	MSE loss
Original noisy data	2.6636
Denoised (SVD)	2.4736
Denoised (U-Net)	1.9878

Figure 14. The observed data and forward modeled data from the inversion results using the noisy data (Fig. 12b), SVD-denoised data (Fig. 12c), and U-Net-denoised data (Fig. 12d) for damping constants of 4, 6, 8, and 10 s⁻¹. The number of traces is 408 and the grid size is 25 m.

In this study, we explored the application of a deep learning-based U-Net architecture for attenuating random noise in Laplace-domain seismic wavefields, comparing its performance with the traditional singular value decomposition (SVD) method. The results from both synthetic and field data demonstrated that the U-Net method outperforms SVD in noise attenuation, leading to improved inversion results in Laplace-domain full waveform inversion (FWI). We discuss the characteristics of Laplace-domain wavefields, the differences between noise attenuation in the Laplace domain versus the time domain, the advantages of supervised learning methods like U-Net over unsupervised methods such as SVD, and the limitations of the proposed method with suggestions for future research.

4.1. Characteristics of Laplace-Domain Wavefields

Laplace-domain wavefields are obtained by applying the Laplace transform to time-domain seismic data, involving an exponential damping factor. This damping attenuates later arrivals more than earlier ones, resulting in smooth, non-oscillatory wavefields (Ha and Shin, 2013). The smoothness of Laplace-domain wavefields is advantageous for recovering large-scale velocity structures in FWI, as it emphasizes the low-wavenumber components of the model.

However, a significant challenge arises due to the sensitivity of Laplace-domain wavefields to noise, particularly near the first arrival. Even small-amplitude noise before the first arrival can be exponentially amplified, severely contaminating the wavefields (Ha and Shin, 2021a). This amplification occurs because the Laplace transform weighs earlier time samples more heavily, allowing any noise present before the first arrival to dominate the transformed wavefield.

To illustrate the drastic reduction in the SNR caused by the Laplace transform, consider a simplified example where both the signal and noise are modeled as impulse functions occurring at different times with different amplitudes. Suppose the signal is an impulse function of amplitude 100 occurring at t_s = 1 s, and the noise is an impulse function of amplitude 1 occurring at t_n = 0.1 s. In the time domain, the SNR is high, calculated as:

$$SN{R}_T=10{\log }_{10}\left(\frac{P_s}{P_n}\right)=10{\log }_{10}\left(\frac{{100}^2}{1^2}\right)=40dB,$$(4)

where P_s and P_n are the power (squared amplitude) of the signal and noise, respectively. After applying the Laplace transform with a damping constant 10 s⁻¹, the transformed amplitudes become:

$$A_s=100e^{-10},A_n=e^{-1}.$$(5)

Calculating the SNR in the Laplace domain yields:

$$SN{R}_L=10{\log }_{10}\left(\frac{P_s}{P_n}\right)=10{\log }_{10}\left(\frac{{\left(100e^{-10}\right)}^2}{{\left(e^{-1}\right)}^2}\right)\approx -38dB.$$(6)

This example demonstrates that, despite a high SNR in the time domain (40 dB), the SNR dramatically decreases to approximately –38 dB in the Laplace domain due to the exponential weighting of earlier time samples. The noise occurring before the first arrival is exponentially amplified, overwhelming the signal and severely degrading the SNR. Therefore, muting the noise before the first arrival is essential to prevent early-time noise from dominating the Laplace-domain wavefields and to preserve the integrity of the seismic signal for inversion.

To address these challenges, we employed several Laplacedomain- specific strategies that differ from time-domain denoising methods. We generated clean training data directly in the Laplace domain using a matrix-based 2D Laplacedomain modeling algorithm for efficiency (Nihei and Li, 2007). Random noise was generated in the time domain to apply muting before the first arrival signal and to ensure the noise had a realistic distribution, as generating noise directly in the Laplace domain leads to a different noise distribution that may not represent actual seismic noise (Ha and Shin, 2021a).

Due to the large amplitude variations between near-offset and far-offset traces resulting from exponential damping, we took the logarithm of the absolute value of the Laplacedomain wavefields. This preprocessing step reduces scale differences and allows the use of single-precision floatingpoint numbers, facilitating network training and reducing computational demands (Ha and Shin, 2021b). Taking the absolute value before applying the logarithm avoids complexvalued data, and Ha and Shin (2021b) demonstrated that this approach does not alter inversion results while accelerating calculations.

Unlike many deep learning-based denoising methods that adopt patch-based schemes, we used whole wavefields as input. Maintaining the global smooth variation trend of Laplace-domain wavefields is crucial for subsequent waveform inversion (Ha and Shin, 2013). This approach ensures that the denoising process preserves the essential characteristics of the wavefields necessary for accurate inversion.

4.2. Noise Attenuation in the Laplace Domain versus Time Domain

Noise attenuation in seismic data is traditionally performed in the time domain, where various methods aim to suppress noise while preserving the true signal. However, denoising in the Laplace domain offers unique advantages and poses specific challenges, particularly for Laplace-domain FWI.

In the Laplace domain, the exponential damping amplifies the impact of early-time noise, which may not be adequately addressed by time-domain denoising methods. One advantage of noise attenuation in the Laplace domain is the targeted reduction of noise that is problematic for Laplace-domain FWI. Since the wavefield's sensitivity to noise is accentuated after the transformation, denoising in this domain can more effectively suppress noise that persists even after timedomain denoising. Additionally, methods designed for the Laplace domain can preserve the smooth, non-oscillatory nature of the wavefield, which is critical for successful inversion.

However, there are disadvantages, including increased complexity due to the need for Laplace transform and specialized processing techniques. Furthermore, muting noise before the first arrival is necessary; without this step, earlytime noise can dominate the Laplace-domain wavefield, rendering denoising ineffective (Ha and Shin, 2021a). These challenges highlight the importance of developing effective denoising methods specifically tailored for the Laplace domain.

4.3. Supervised Learning versus Unsupervised Methods for Denoising

The U-Net denoising method employed in this study is a supervised learning approach that utilizes labeled data— pairs of noisy and clean wavefields—to learn the mapping between them. This supervised training enables the network to capture complex, nonlinear relationships and effectively distinguish between signal and noise. By learning the characteristics of both clean and noisy wavefields, the network can remove noise while preserving essential signal features.

In contrast, unsupervised methods like SVD do not require labeled data and rely on assumptions about the data structure, such as the signal being of low rank and the noise being of high rank. SVD decomposes the data matrix into singular values and vectors, and by truncating small singular values, it aims to suppress noise. However, SVD tends to smooth the data and retain the main trends of the wavefield, including residual noise. Consequently, SVDdenoised data may still contain noise patterns that affect the inversion results.

Our findings showed that inversion results using SVDdenoised data were similar to those using noisy data (Figs. 10a and 10b). This similarity arises because SVD maintains the general trends of the wavefield but does not effectively remove all noise components. In contrast, the U-Net denoising method provided cleaner data, leading to improved inversion results (Fig. 10c). The supervised learning approach's ability to learn from labeled data and capture intricate patterns contributes to its superior performance in noise attenuation.

It is important to note that access to ground truth data for training does not necessarily guarantee improved denoising on unseen data. The network's generalization ability depends on the representativeness and diversity of the training dataset. By generating realistic synthetic training data that simulate various subsurface conditions, we enhanced the network's ability to generalize to real-world scenarios, as evidenced by the improved results on field data.

4.4. Network Robustness and Limitations

We enhanced the network's robustness by varying the grid sizes of the training models randomly from 10 m to 125 m. This variation exposed the network to data with different spatial resolutions and sampling rates, improving its ability to generalize across different acquisition geometries and grid settings. In practice, Laplace-domain inversions often use larger grid intervals compared to time-domain inversions because the wavefields and velocities vary smoothly due to the exponential damping (Ha and Shin, 2013).

Our use of whole shot-receiver data as input means the network was trained for a fixed marine streamer sourcereceiver geometry. While this approach preserves the global characteristics of the wavefields, it limits the network's applicability to other acquisition geometries. Applying the method to land data or different acquisition setups may require retraining the network with appropriate geometry. Generalizing the network to different acquisition parameters would enhance its applicability across various seismic exploration scenarios.

A significant limitation of the proposed method is its dependence on muting noise before the first arrival. Without muting, early-time noise dominates the Laplace-domain wavefield due to exponential damping, and the network cannot recover the underlying signals effectively.

4.5. Full Waveform Inversion in the Laplace Domain

The improved noise attenuation achieved by the U-Net denoising method has direct implications for full waveform inversion in the Laplace domain. Cleaner input data lead to more accurate inversion results, as the inversion process relies on minimizing the differences between observed and modeled data. With reduced noise, the inversion algorithm can better focus on recovering the true subsurface velocity structures.

Our results showed that the inversion using U-Net-denoised data produced velocity models that better matched the observed data, as indicated by lower error metrics (Figs. 11 and 13). This improvement was evident in both synthetic and field data examples. The enhanced inversion results can lead to more accurate subsurface imaging, which is crucial for applications such as hydrocarbon exploration.

In the field data inversion, although we cannot definitively determine which velocity structure is more accurate without additional data such as well logs, we observed that the forward-modeled wavefields from the inversion result using U-Net-denoised data exhibited the smallest error when compared to the observed data (Table 2). This suggests that the U-Net denoising method enhances the inversion outcome by providing cleaner data for FWI.

Laplace-domain full waveform inversion is effective for recovering large-scale subsurface velocity models but is sensitive to noise in the data. We compared the denoising methods of SVD and a modified U-Net in the Laplace domain. We trained the network with synthetic noisy and clean logarithmic wavefields generated from velocity models simulating the Gulf of Mexico environment. We then tested the network with both synthetic and field datasets. Comparison of inversion results using noisy data, SVD-denoised data, and U-Net-denoised data as observed data for FWI in the Laplace domain showed that U-Netdenoised data produces superior inversion results. These findings confirm the effectiveness of U-Net denoising in the Laplace domain for improving full waveform inversion outcomes.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A1064432).