Improving the Seismic Impedance Inversion by Fully Convolutional Neural Network

Abstract

Applying deep neural networks (DNNs) to broadband seismic wave impedance inversion is challenging, especially in generalizing from synthetic to field data, which limits the exploitation of their nonlinear mapping capabilities. While many research studies are about advanced and enhanced architectures of DNNs, this article explores how variations in input data affect DNNs and consequently enhance their generalizability and inversion performance. This study introduces a novel data pre-processing strategy based on histogram equalization and an iterative testing strategy. By employing a U-Net architecture within a fully convolutional neural network (FCN) exclusively trained on synthetic and monochrome data, including post-stack profile, and 1D linear background impedance profiles, we successfully achieve broadband impedance inversion for both new synthetic data and marine seismic data by integrating imaging profiles with background impedance profiles. Notably, the proposed method is applied to reverse time migration (RTM) data from the Ceduna sub-basin, located in offshore southern Australia, significantly expanding the wavenumber bandwidth of the available data. This demonstrates its generalizability and improved inversion performance. Our findings offer new insights into the challenges of seismic data fusion and promote the utilization of deep neural networks for practical seismic inversion and outcomes improvement.

1. Introduction

Deep learning (DL) is widely utilized in seismic data processing and interpretation due to its robust nonlinear transformation capabilities [1]. Currently, supervised learning remains the predominant approach in DL [2]. However, obtaining labels for real-world data within the framework of supervised learning can be a challenging task. Consequently, the training dataset may not be exhaustive and fail to encompass all possible scenarios encountered in real-world situations [3]. The generalization from synthetic and labeled data to observed and unlabeled data presents challenges and difficulties in supervised learning [4,5,6]. Nevertheless, it is crucial for a method to be suitable for all cases encountered. Researchers strive to train deep neural networks (DNNs) that remain effective on novel datasets acquired from different survey areas, various acquisition systems, and processed using alternative methodologies.

Seismic wave impedance is determined by the density and velocity of geological media, and is fundamental in oil and gas reservoir exploration. The difference in impedance between adjacent layers influences the reflected response at interfaces. Both qualitative and quantitative techniques for impedance inversion can accurately describe structures using physical parameters of geological media, interpret subsurface seismic profiles, and predict oil and gas reservoirs [7,8,9]. These inversion techniques are based on forward modeling approaches such as wave equations or convolutional modeling techniques [10,11]. Impedance inversion seeks to compute an inverse operator within each forward modeling framework to obtain accurate impedance values from raw seismic records. This process involves solving a typical inverse problem characterized by strong nonlinearity, non-uniqueness, and instability. Full-waveform inversion (FWI) is a widely used technique with great potential for obtaining velocity profiles from raw seismic records that capture comprehensive information from the wavefield. To achieve improved inversion through FWI, it is necessary to reduce the nonlinearity of the forward modeling operator [12,13], enhance regularization of well logging [14,15], and obtain a better initial impedance model [16,17,18]. Obtaining high-precision inversion directly from raw records poses challenges. Numerous studies have explored DL methods for seismic data processing and impedance inversion [19,20,21,22,23,24]. To ensure reliable results, it is valuable to incorporate physics-driven approaches [25,26,27,28], augmentation data [29], enhance loss functions [30,31,32], and optimize neural network structures [33,34].

The results of imaging and inversion can be obtained through various processing methods applied to raw data. For example, migration and imaging techniques can offer valuable insights into the location of reflected interfaces and scatter points in the subsurface [35,36]. Tomography and well logging can be utilized to invert coarse physical parameters. These imaging and inversion results are relatively easy to obtain and capture diverse characteristics of subsurface structures across different spatial wavenumber bands. Considering the ill-conditioned nature of FWI, effectively improving inversion performance can be achieved by integrating imagings and inversions from different wavenumber bands into seismic impedance profiles [37,38].

To address these issues, this study focuses on using a fully convolutional neural network (FCN) with a novel data fusion and generalization approach to enhance the precision of impedance profile estimation, which is distinct from previous methods. The U-Net is a kind of FCN with a two-channel input that allows for flexible input data size [39]. It aims to enhance the precision of impedance profile estimation by fusing imaging profiles with background impedance profiles. Additionally, we improve the generalization performance through a pre-processing strategy based on histogram equalization. Moreover, incorporating smooth input and implementing iterative techniques during the testing phase prove advantageous in enhancing the inversion process. By optimizing our experimental approach, we successfully demonstrate that the trained network can effectively generalize from synthetic data to field data. Also, integrating various seismic data processing techniques, we ensure optimal utilization of unique insights offered by each dataset.

This paper begins by training a U-Net model on pre-processed synthetic datasets, which consist of post-stacked imaging profiles and 1D linear background impedance profiles. Then, without any fine-tuning, the trained U-Net is directly applied to pre-processed test datasets containing various types of imaging profiles (e.g., RTM profile) and background profiles (e.g., 1D logging, 2D Gaussian-blurred profiles), resulting in successful inversions. Furthermore, incorporating iterative processing during the testing stage proves effective in enhancing inversion performance. The marine seismic data, which were released by NOPIMS [40], are utilized to validate the generalization and inversion performance of the proposed method. Finally, based on these numerical experiments, this research provides valuable recommendations for applying neural network methods to similar tasks.

2. Methods

2.1. Fully Convolutional Neural Network (FCN)

The FCN, a variant of convolutional neural networks (CNNs), was initially designed for the purpose of image semantic segmentation [41]. Distinguishing itself from traditional CNNs, the FCN excludes fully connected layers and instead constructs its framework solely using convolutional and transposed convolutional layers (also known as deconvolutional layers). This architectural innovation offers several advantages: primarily, it utilizes convolutional layers to extract data features and subsequently employs transposed convolutional layers to restore feature maps to match the input data size. FCN’s ability to handle variable-sized inputs and generate consistent outputs is particularly beneficial for seismic data, which often exhibit complex spatial characteristics and require seamless processing across different scales in impedance inversion tasks. This characteristic renders FCN exceptionally adaptable and efficient in scenarios requiring seamless integration from input to output across varying scales, thereby expanding their applicability in diverse computational and analytical tasks [14,42,43,44].

Building upon the author’s previous research [20], this study employs the U-Net architecture, a member of the Fully Convolutional Networks (FCNs). Given the utilization of supervised learning in this investigation, maintaining strict consistency in test data size relative to training data would introduce significant complexities in practical applications. Such rigidity could necessitate procedures like patch-based prediction followed by stitching, potentially resulting in visible seams in the output. However, provided that GPU memory constraints are accommodated, U-Net can directly process input data of varying sizes without prior resizing. This flexibility enables direct computation and subsequent output of inversion results regardless of input dimensions, thereby circumventing issues associated with standardizing data size and enhancing the model’s practical utility and efficiency in real-world applications.

As depicted in

Table 1. Architecture of U-Net.

Layer	Input (BS, W, H, C)	Type	Output (BS, W, H, C)
1	(1, 200, 200, 2)	Input	(1, 200, 200, 64)
2	(1, 200, 200, 64)	Forward convolution	(1, 100, 100, 128)
3	(1, 100, 100, 128)	Forward convolution	(1, 50, 50, 256)
4	(1, 50, 50, 256)	Forward convolution	(1, 25, 25, 512)
5	(1, 25, 25, 512)	Forward convolution	(1, 12, 12, 1024)
6	(1, 12, 12, 1024)	Self attention	(1, 12, 12, 1024)
7	(1, 12, 12, 1024)	Transposed convolution	(1, 25, 25, 512)
8	(1, 25, 25, 512)	Transposed convolution	(1, 50, 50, 256)
9	(1, 50, 50, 256)	Transposed convolution	(1, 100, 100, 128)
10	(1, 100, 100, 128)	Transposed convolution	(1, 200, 200, 64)
11	(1, 200, 200, 64)	Output	(1, 200, 200, 1)

, the U-Net architecture comprises 11 components, including an input layer, output layer, forward convolution layer, transposed convolution layer, and self-attention block. Additionally, Table 1 and

Table 2. Details of layers.

Type	Input shape	Operation	Activation	Channel	Filter	Strid	Padding	Output
Input	(BS, NVX, NVZ, 2)	Convolution	ReLU	64	3 × 3	1	Same	(BS, NVX, NVZ, 64)
Forward convolution	(BS, W, H, C)	Convolution	ReLU	C	3 × 3	1	Same	(BS, W, H, C)
	(BS, W, H, C)	Max pool	-	-	2 × 2	2	-	(BS, W/2, H/2, C)
	(BS, W/2, H/2, C)	Convolution	ReLU	2C	3 × 3	1	Same	(BS, W/2, H/2, 2C)
Self attention	(BS, W, H, C)	Convolution	Softmax	1	1 × 1	1	Same	(BS, W, H, C)
Transposed convolution	(BS, W/2, H/2, 2C)	Convolution	ReLU	C	2 × 2	2	Same	(BS, W, H, C)
	(BS, W, H, C)	Channel-wise concatenation	-	C+C	-	-	Same	(BS, W, H, 2C)
	(BS, W, H, 2C)	Convolution	ReLU	C	3 × 3	1	Same	(BS, W, H, C)
	(BS, W, H, C)	Convolution	ReLU	C	3 × 3	1	Same	(BS, W, H, C)
Output	(BS, NVX, NVZ, 64)	Convolution	ReLU	1	3 × 3	1	Same	(BS, NVX, NVZ, 1)

provide details on the input and output size, operation type, activation function, and filter size for each layer. While BS represents batch size, W, H, and C denote the width, height, and channels of the data, respectively. Figure 1 illustrates the proposed U-Net architecture.

2.2. Loss Function

In the domains of machine learning and statistics, Mean Square Error (MSE) serves as a widely employed metric for evaluating the performance of regression models. It quantifies the average square discrepancy between predicted values and actual values. The formula defining MSE is as follows:

Given a set of predicted values ${\widehat{y}}_i$ and corresponding true values $y_i$ for $i=1,2,\dots ,n$, the MAE is calculated as

$$\begin{array}{c}\hfill MSE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2\end{array}$$

(1)

where n denotes the total number of observations. MSE offers a straightforward interpretation of the square error magnitude in predictions, so it is more sensitive to the misfit than MAE, rendering it an invaluable tool for assessing model accuracy and performance.

2.3. Pre-Processing Strategy

The histogram equalization [45] is predominantly referenced in the proposed pre-processing strategy. The workflow is listed in Figure 2, and the specific algorithm is presented in Algorithm 1.

Algorithm 1 Algorithm of pre-process.

Input: Imaging profiles $IMG=\{{\mathit{x}}_{\mathbf{11}},{\mathit{x}}_{\mathbf{21}},{\mathit{x}}_{\mathbf{31}},...,{\mathit{x}}_{\mathit{m}\mathbf{1}}\}$       Background impedance profiles $BG=\{{\mathit{x}}_{\mathbf{12}},{\mathit{x}}_{\mathbf{22}},{\mathit{x}}_{\mathbf{32}},...,{\mathit{x}}_{\mathit{m}\mathbf{2}}\}$       where $x_{ij}$ represents the data in $j^{th}$ channel of $i^{th}$ input. Output: $IM{G}_{norm}$, $B{G}_{norm}$   1: for  $i\in [1,m]$  do   2:     ${\mathit{x}}_{\mathit{i}\mathbf{1}}=((({\mathit{x}}_{\mathit{i}\mathbf{1}}-\min \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right))/(\max \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right)-\min \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right))-0.5)\times 2$   3:     $m_1,m_2,m_3=\min \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right),\max \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right),\mathrm{std}\left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right)$   4:     while $True$ do   5:         if $\mathrm{std}\left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right)\ge 0.15$ and $\mathrm{abs}\left(\mathrm{mean}\left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right)\right)\le 0.01$ then   6:            break   7:         else   8:            $cli{p}_{min}=-0.8+m_3,cli{p}_{max}=0.8+m_3$, Hyper-parameter $0.8$ is determined by dataset.   9:            $lo{c}_{min},lo{c}_{max}=\mathrm{where}\left({\mathit{x}}_{\mathit{i}\mathbf{1}}>cli{p}_{min}\right),\mathrm{where}({\mathit{x}}_{\mathit{i}\mathbf{1}}<cli{p}_{max})$ 10:          ${\mathit{x}}_{\mathit{i}\mathbf{1}}\left[lo{c}_{min}\right],{\mathit{x}}_{\mathit{i}\mathbf{1}}\left[lo{c}_{max}\right]=cli{p}_{min},cli{p}_{max}$ 11:       end if 12:     end while 13:     $m_1,m_2,m_3=\min \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right),\max \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right),\mathrm{std}\left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right)$ 14:     ${\mathit{x}}_{\mathit{i}\mathbf{1}}=((({\mathit{x}}_{\mathit{i}\mathbf{1}}-\min \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right))/(\max \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right)-\min \left({\mathit{x}}_{\mathit{i}\mathbf{1}}\right))-0.5)\times 2$ 15:     ${\mathit{x}}_{\mathit{i}\mathbf{2}}={\mathit{x}}_{\mathit{i}\mathbf{2}}/5000{\rho }_c$, ${\rho }_c$ is constant density. 16:     ${\mathit{x}}_{\mathit{i}\mathbf{2}}=GaussianFilter({\mathit{x}}_{\mathit{i}\mathbf{2}},40)$ 17: end for 18: return $IM{G}_{norm}$, $B{G}_{norm}$

2.4. Test Strategy

During the training stage, U-Net is trained using augmented 200 × 200 forward modeling data incorporating data augmentation techniques for both label generation and imaging profile forward modeling consistent with prior research by authors [20,21]. In this study, we extract both maximum and minimum values from impedance labels to form a 1D linear profile that represents background impedance. This methodology facilitates the development of a simplified yet representative baseline model essential for understanding underlying geological structures and guiding controlled inversion procedures. The utilization of extremum values derived from impedance labels effectively highlights substantial contrasts within subsurface regions, thereby improving interpretability and efficacy of subsequent inversions. Figure 3 shows a decreasing loss curve during validation and illustrates that the network becomes convergent.

Throughout our experimentation, we made a concerted effort to incorporate a diverse range of geological models, such as the SEG/EAGE Salt model, Marmousi, and Ceduna sub-basin. We also utilized various imaging profiles and different types of background impedance profiles. The SEG/EAGE Salt model and Marmousi are widely recognized for their standard seismic exploration numerical model capabilities; hence they are employed to generate synthetic imaging profiles and background impedance profiles. The Ceduna sub-basin provides publicly available imaging profiles and migration velocity [46]. These datasets served as the foundation for conducting our experiments, with detailed information presented in

Table 3. Notes of testing data.

Model	Type of Imaging Profiles	Type of Background Impedance Profiles
SEG/EAGE Salt		1D linear
and Marmousi	Synthetic	1D logging
		2D Gaussian-blurred
Ceduna sub-basin	RTM	Migration velocity with constant density

. Our primary objective was to rigorously evaluate the generalizability of the proposed method by subjecting it to a wide array of conditions and scenarios. By doing so, we aimed to ascertain its adaptability and effectiveness across varying geological contexts, pre-processing strategies, and imaging methodologies. This comprehensive assessment not only validates the robustness of our approach but also offers valuable insights into its versatile applicability in real-world geophysical investigations.

The sequence of 1D linear, 1D logging, and 2D Gaussian-blurred profiles is chosen during the testing stage. The reason is that, for 1D, linear data are consistent with the training dataset, while the high wavenumber information increases in 1D logging and 2D Gaussian-blurred data, leading to progressively stronger constraints. Consequently, this testing strategy aims to evaluate the performance of the generalization.

3. Numerical Experiments

3.1. Background Test 1: 1D Linear Profile

The 1D linear profile is determined by the minimum and maximum values of the true model, exhibiting a linear increase in impedance with depth. In Test 1, the imaging profile and background impedance profile remain consistent with those used during the training stage. However, for the proposed trained U-Net model, we introduce a completely new sample from the SEG/EAGE Salt model (refer to Figure 4). Figure 5 illustrates wavenumber spectra for background impedance, inversion, and true models. The U-Net effectively expands the spatial frequency band. As depicted in Figure 6, the Marmousi model is included as part of the training dataset to represent two fundamental geological structures: an anomaly body of high impedance within the Salt model and a layered structure consisting mainly of faults and folds within the Marmousi model. Our proposed method allows for accurate predictions of both location and impedance value of anomaly bodies while also incorporating details such as layer edges, folds, and faults into an impedance model that faithfully represents geological characteristics.

3.2. Background Test 2: 1D Logging Profile

The test is based on 1D logging profiles that are tiled by a single well log trace. There are two scenarios regarding the Salt model: when the well log traverses the anomaly body and when it does not. The corresponding results are presented in Figure 7 and Figure 8, respectively. Figure 7 exhibits a clear boundary but an inaccurate value for the salt dome, while Figure 8 demonstrates a relatively accurate value due to guidance from a high-impedance yet incomplete salt dome. These inversions illustrate the general applicability of our proposed method for 1D logging background impedance profiles. In Figure 9, the initial background velocity of the Marmousi model encapsulates most of the velocity information present in the original model, thereby providing relatively correct constraint information to guide network training. Consequently, this yields better results compared with those obtained with the salt model since an initial background model lacking high-velocity information prevents recovery of high-velocity salt bodies by neural networks. Therefore, for accurate representation of salt domes, having a good initial model becomes crucial.

3.3. Background Test 3: Gaussian-Blurred Profile

The final synthetic test is based on Gaussian-blurred background profiles that closely resemble the characteristics of the true model. As shown in Figure 10 and Figure 11, the inversions exhibit distinct subsurface boundaries and accurate values. The proposed method demonstrates robustness even in cases with Gaussian-blurred backgrounds.

In conjunction with the relative errors depicted in Figure 12 and Figure 13, the proposed method demonstrates its suitability for three distinct background profiles, yielding inversions that closely match the true model in terms of structure and impedance values when compared with those obtained through 1D linear and 2D Gaussian-blurred profiles. The tables of MSE corresponding to Figure 12 and Figure 13 are given in

Table 4. MSE.

First Column	Second Column
1.0946	0.2031
0.8790	0.1561
0.7728	0.1426
0.1901	0.0838

and

Table 5. MSE.

First Column	Second Column
0.2447	0.0892
0.3332	0.1085
0.0295	0.0148

. The values of MSE are rationally decreased.

3.4. Field Test: Ceduna Sub-Basin Marine Seismic Data

The Ceduna sub-basin marine seismic data come from Equinor, who is the sole titleholder of the exploration permit for petroleum 39 (EPP39) [40,46]. The survey area is marked by a red line in Figure 14. It surrounds the Stromlo-1 exploration well, which is located approximately 400 km southwest of Ceduna and 476 km west of Port Lincoln and in a water depth of approximately 2240 m.

The proposed method has been applied to marine seismic data from the Ceduna sub-basin survey. It includes RTM imaging profiles and migration velocity profiles provided by Equinor.

In Figure 15d, the impedance is combined with the location of reflected subsurface to enhance the wavenumber bandwidth of the data. Compared with the previous method shown in Figure 15c, which lacks pre-processing strategies and iterations during testing, our proposed method preserves more high-wavenumber characteristics, as verified by wavenumber spectra (f) and (g).

4. Discussion

4.1. What Does the Width of the Abnormal Impedance Layer Influence?

In this part, we discuss how different logging traces, which show different widths of high-impedance layers, influence inversions by the proposed method; a background profile is utilized to provide a fundamental range of impedance values. Within the SEG/EAGE Salt model, there exists an anomaly body with high impedance. Figure 16 displays results obtained through 1D logging using different widths of high-impedance layers for the background profiles, and

Table 6. MSE.

Second Column	Third Column
0.8790	0.1561
0.7728	0.1426
0.8258	0.2491
1.0000	0.2057

shows MSE of background models and inversions. The locations of these different wells indicate that the initial model related to the salt dome provides different degrees of prior information. If the logging location does not pass through the salt dome at all, although the outline of the salt dome can be roughly portrayed, the velocity inversion result of the salt dome part is difficult to have relatively accurate results. This part of the experiment shows that the establishment of the initial model requires a macro-priori understanding of the study area.

4.2. What Does the Pre-Process of Imaging Profiles Influence?

The step of pre-processing plays a crucial role in our method, as it enables us to effectively distinguish and analyze the characteristics of images through clear texture obtained from image processing techniques. By applying normalization and equivalization in Figure 17 and

Table 7. MSE.

(a)	(b)	(c)	(d)
0.8782	0.3999	0.8584	0.7821

, we observe significant improvements in the visibility of salt layers and boundaries within RTM imaging profiles. This pre-processing approach ensures a balanced response amplitude between positive and negative values. Consequently, both the training and testing datasets are subjected to the same pre-processing method. Notably, our corresponding test results demonstrate that incorporating pre-processing significantly enhances subsurface resolution and numerical precision compared with using no pre-processing.

4.3. What Does the Smoothing on Background Impedance Profiles Influence?

For the proposed U-Net, sharp input generates more artifacts than smooth ones. In Figure 18 and

Table 8. MSE.

First Column	Second Column
0.8783	0.1309
0.8584	0.1299
0.8266	0.1290
0.7728	0.1291

, the artifacts caused by the high-impedance layer are alleviated with an increased blurred radius. In order to maintain spatial continuity in inversions, it is essential to apply smoothing to the background impedance profile before inputting it into U-Net.

4.4. Does Iteration During the Testing Phase Make Sense?

Iteration during the testing stage entails utilizing the most recent output as the subsequent background input while maintaining a fixed imaging profile. This paper presents a comprehensive approach that encompasses imaging and background profiles in order to achieve an accurate impedance model. It is natural for the output after each iteration to serve as a background profile. Therefore, conducting iterations during the test stage is justified.

The inversions after three iterations, based on background impedance profiles with varying widths of the high impedance layer, are presented in Figure 19. The results from the first iteration exhibit low subsurface resolution and accuracy of impedance values. In iteration 1, although the boundary of the salt model is clearer compared with others, its value is significantly lower than the true value shown in the second row. The fourth row displays numerous artifacts in deep layers and a fractured Salt model. Subsequent iterations provide more detailed information. When comparing different widths of the high-impedance layer in the initial profile, it is observed that a width comparable to that of the true Salt model yields better results. The locations of sublayers, boundaries, and impedance values are much more reliable when compared with the other two scenarios.

5. Conclusions

Focusing on factors that influence inversion quality and the generalizability of deep neural network methods, this study tries to encompass optimizing deep neural network architectures, refining data value ranges and pre-processing techniques, as well as iterative strategies during testing. Through experiments, the proposed method demonstrates robust generalization capabilities on both synthetic and marine seismic data, yielding high-quality inversions across a wide frequency band after training with augmented synthetic datasets. Based on these numerical experiments, we find that U-Net, a type of FCN, exhibits the ability to handle input data during testing that may not strictly adhere to the training data, thereby expanding the applicability of our approach. Taking impedance inversion as an example task, one can enhance the generalizability and performance of U-Net by implementing the following strategies:

1.

Ensure that the input background impedance value range closely approximates the real one;

2.

A smoothly varying input background impedance profile is beneficial for maintaining inversion continuity;

3.

Testing data do not necessarily need to be consistent with the training data (e.g., trained using synthetic post-stack data but tested using RTM data), but the pre-processing techniques should remain consistent;

4.

An iterative approach during testing can effectively enhance inversion precision.

Our study offers valuable insights into addressing challenges posed by unclear background velocity models, thereby enabling the effective application of deep neural networks for practical seismic inversion and improved outcomes.