Semi-Supervised Training for (Pre-Stack) Seismic Data Analysis

Abstract

A lack of labeled examples is a problem in different domains, such as text and image processing, medicine, and static reservoir characterization, because supervised learning relies on vast volumes of these data to perform successfully, but this is quite expensive. However, large amounts of unlabeled data exist in these domains. The deep semi-supervised learning (DSSL) approach leverages unlabeled data to improve supervised learning performance using deep neural networks. This approach has succeeded in image recognition, text classification, and speech recognition. Nevertheless, there have been few works on pre-stack seismic reservoir characterization, in which knowledge of rock and fluid properties is fundamental for oil exploration. This paper proposes a methodology to estimate acoustic impedance using pre-stack seismic data and DSSL with a recurrent neural network. The few labeled datasets for training were pre-processed from raw seismic and acoustic impedance data from five borehole logs. The results showed that the acoustic impedance estimation at the well location and outside it was better predicted by the DSSL compared to the supervised version of the same neural network. Therefore, employing a large amount of unlabeled data can be helpful in the development of seismic data interpretation systems.

1. Introduction

Labeled data are essential for improving the performance and generalization of supervised deep learning algorithms in real-world problems. Different domains, such as healthcare [1,2], natural language processing (NLP) [3,4], and reservoir characterization [5,6], are significantly affected, due to privacy concerns and the need for expert annotations; labeling is also time-consuming and rather costly. However, there is a large amount of unlabeled data available in these domains: medical records, images, and genetic information in healthcare [7,8], and 3D seismic data in reservoir characterization [9,10]. Nevertheless, unsupervised learning techniques have shown the ability to learn hidden features from vast amounts of unlabeled data, and they can be used to enhance supervised learning.

Seismic data and well-log data analysis are complex because of the heterogeneous structure of the Earth’s subsurface [11]. To model this kind of data requires an adequate capability to discover nonlinear relationships between the input and output of an artificial neural network (ANN). Deep neural networks (DNN) are suitable tools for these tasks. These models are widely utilized in geophysics for seismic data interpretation, such as data inversion [12,13,14], data quality improvement [15,16,17,18,19], geological feature classification, and petrophysical property prediction. Various studies have employed geological feature classification; for example, Cunha [20], Islam [21], and Yang [22] used convolutional neural networks (CNNs) for seismic fault detection, salt body identification, and seismic horizon tracking, respectively. Moreover, multi-layer perceptrons (MLP) have been applied to detect gas chimneys [23] and for lithology and fluid classification [24]. On the other hand, for petrophysical property prediction, such as permeability, mineralogy, and porosity, CNNs [25,26,27], Boltzmann machine [28], DNNs [29], gated recurrent neural networks (RNN) [30], and bi-directional long short-term memory (BiLSTM) [27] have been utilized.

Semi-supervised learning (SSL) methods can improve the performance of machine learning models by utilizing unlabeled data, particularly when supervised data are scarce. This has been shown since the early wrapper SSL methods, passing through pre-training methods and later generative methods [31]. Self-training and co-training are wrapper methods that are still used for object detection [32], image classification [33], segmentation tasks for medical images [34], and image annotation [35]. These pre-training methods, which use deep neural networks, have been applied for brain tumor classification [36] using autoencoders, epilepsy detection [37] with CNN, brain functional network classification [38], and image classification on small datasets [39], with the latter using transformer networks. Generative adversarial networks (GANs), which involve a natural unsupervised generative method but extended to semi-supervised methods, have applications in facial image recognition [40] and hate speech detection [41].

The SSL approach is termed deep semi-supervised learning (DSSL) when it involves deep neural network architectures. Recently, there has been active research in training DSSL models for reservoir characterization. For example, Pratama [6] detected the presence of channels or non-channels with a CNN using 3D post-stack seismic data. Seismic facies were classified from elastic features by Asghar [9] using a DNN and with deep autoencoders by Liu [42]. In addition, seismic facies were detected from post-stack seismic data by Su-Mei [43] using a CNN. From pre-stack seismic data, Song [10] identified gas-bearing employing k-nearest neighbors (kNN) and a CNN. In Liu [42], the authors used some unlabeled elastic features for pre-training. Dou [44] detected faults with CNN using synthetic and field seismic data, while Liu [45] classified facies from post-stack seismic data with a GAN.

Table 1. State-of-the-art methods compared to our method.

Methods	Seismic Data	ANN	Learning	Task
Geological feature detection [6]	Post-stack	CNN	DSSL	Binary classification
Facies classification [9]	Post-stack	DNN	DSSL	Muticlass classification
Facies classification [42]	Pre-stack (indirect use)	Deep autoencoder	DSSL	Muticlass classification
Facies identification [43]	Post-stack	CNN	DSSL	Muticlass classification
Gas-bearing prediction [10]	Pre-stack	CNN	DSSL	Binary classification
Fault detection [20]	Post-stack	CNN	Supervised	Binary classification
Salt bodies classification [21]	Post-stack	CNN	Supervised	Binary classification
Gas chimney detection [23]	Pre-stack	MPL	Supervised	Binary classification
Porosity prediction [27]	Pre-stack	CNN	Supervised	Estimation
Multiple physical parameters prediction [30]	Post-stack (indirect use)	Gated RNN	Supervised	Estimation
Mineralogy prediction [29]	Well-log data was used	DNN	Supervised	Estimation
Acoustic impedance prediction (Our method)	Pre-stack	LSTM	DSSL	Estimation

summarizes recent DSSL and supervised methods closely related to our proposed method.

In this work, we present a new method for training a DSSL model for unlabeled 3D pre-stack seismic data, which are seismic traces organized into gathers, also known as common depth points (CDP). CDP gathers are constructed from a systematic arrangement of controlled seismic events. In contrast to the stacked traces in post-stack seismic data, these gathers encapsulate a richer seismic dataset. This kind of information allows us to significantly enhance the performance of neural networks in seismic data analysis [46,47]. Research on using pre-stack seismic data for reservoir characterization in DSSL is still ongoing, as well as semi-supervised learning employing shallower models [48,49,50,51,52,53,54].

Well-log data provide a direct and comprehensive depiction of subsurface conditions, offering detailed insights into rock and fluid properties for the depth of a borehole [55]. These properties can serve as a valuable resource for labeling pre-stack seismic data and facilitating the creation of labeled datasets, but only for seismic data near the wells. However, the availability of such datasets remains constrained due to factors like the high costs associated with well drilling and data ownership by companies, which affect subsurface modeling using deep neural networks. A valuable petroacoustic property derived from well-log data is the absolute acoustic impedance (AI), which plays a pivotal role in characterizing reservoirs for lithology, hydrocarbon potential, fluid content, and more [56]. Given the importance and challenges associated with acquiring AI, the intersection of technology and data, specifically DNN and pre-stack seismic data, holds the potential to enhance the estimation and generation of valuable data for oil exploration, which is the primary focus of our work.

This study contributes to addressing the problem of training a DSSL model to estimate the absolute AI from CDP seismic data using limited labeled information derived from well-log data. This is a challenge for both computer science and geophysics. While previous related works successfully classified or estimated various geological features or petrophysical properties, predicting well-log absolute AI from pre-stack seismic data has yet to be fully explored. Extracting underlying features directly from unlabeled data is also an aspect that has not been thoroughly investigated. A substantial amount of unlabeled pre-stack seismic data are generated for the pre-training stage of the DSSL method, which enhances the training convergence speed, especially with limited labeled instances. Additionally, we have included a data augmentation step to expand the availability of labeled data by leveraging the nearest gathers to the well-log data. The nearest gathers exhibit similar patterns in their seismic traces, allowing us to generate additional labeled instances from the same well-log data.

The DSSL approach employed in this study utilizes a multi-layer long short-term memory (LSTM) architecture, treating seismic data as sequential or time-series information. To circumvent random initialization, a greedy layer-wise pre-training strategy was employed, to facilitate the gradual learning of data representations in an unsupervised manner, layer by layer. Additionally, the model was fine-tuned using a limited number of labeled instances obtained from well-log data. In addition to configuring the optimizer, loss function, and neural network size, introducing dropout hyperparameters at each layer was crucial in enhancing the model’s learning capability concerning the data. Furthermore, adopting a linear-epoch gradual-warm-up scheme proved instrumental in accelerating the training process when handling vast volumes of unlabeled data. To assess the effectiveness of the DSSL method, we also implemented a supervised learning approach for comparative analysis of the results and predictions.

The paper is organized as follows: Section 2 describes the methodology for data pre-processing, the neural network setup, and critical concepts. Section 3 briefly presents the experimental setup. Section 4 compares the DSSL and supervised learning and discusses the results. Finally, the conclusions are given in Section 5.

2. Methodology

In this research, we introduce a methodology that utilizes extensive unlabeled data, particularly pre-stack seismic data, where limited available labels were obtained from well-log measurements. The aim is to estimate the absolute AI from seismic pre-stack time migration (PSTM) cubes using a deep semi-supervised learning framework founded upon LSTM networks, as illustrated in Figure 1.

The DSSL network depicted in Figure 2 shows the flow of data throughout the pre-trained and output stages of the neural network. Initially, pre-stack unlabeled seismic data are received to learn hidden features during the pre-training phase (Section 2.6). This phase involves the three LSTM layers and the next two dense layers. Subsequently, the entire output stage and pre-trained stage are fine-tuned with pre-stack labeled seismic data. The training dataset used in our study was meticulously constructed, following a series of thorough preparation (Section 2.1) and pre-processing (Section 2.2) procedures.

2.1. Dataset Preparation: Integration and Alignment

Data preparation involved integrating and aligning seismic (measured in time) and well-log data (measured in meters, referred to as depth). Seismic data were organized as a volumetric cube, whereas well-log information consisted of rows with borehole petrophysical measurements. The integration process entailed combining the seismic and well-log data in the same domain through well-log-related details, such as the well-track and a time-depth transformation [57]. It is important to note that the time-depth transformation had been obtained in a previous analysis. The alignment phase focused on optimizing the correlation between the same events identified in the seismograms (seismic data) and the petrophysical data (synthetic seismogram). Then, once the closest seismic trace was aligned with the well-log data, the AI was used as the target dataset for the seismic data in the well’s vicinity. Figure 3 illustrates the alignment performed for the first well-log information labeled Well-1.

2.2. Seismic Data Pre-Processing

The pre-processing phase entailed preparing the datasets for training the neural network. This involved the seismic cube as a source of features and the AI data as the target variable. A set of traces per CDP, or gather, from the seismic cube were selected, as depicted in Figure 4. A gather from the land seismic acquisition comprised 40 traces, and experimentation revealed that the neural network exhibited improved performance when trained on 18 traces per gather, following comparisons with configurations employing 6, 12, and 24 traces. Subsequently, these selected traces and the target were subjected to rescaling, resulting in values within ranges of $[-1,1]$ and $[0,1]$, respectively. In addition, seismic traces had a lower sample rate than the AI data. Consequently, an interpolation process was performed, ensuring both had the same number of samples.

To provide a seismic context for each target sample, the dataset used for training was constructed by relating a window of seismic samples to each target sample, as illustrated in Figure 5. From the initial seismic point to the final one, a moving window slides across a predefined number of samples (in this case, 53) along each gather’s traces. After each windowing operation, the window slides one sample. In this way, each target sample is related to the midpoint of the window length. This approach yielded a large number of examples from each CDP. Compiling these examples spanning across multiple CDPs created the dataset.

In Figure 6, each cell represents a CDP. Test examples were obtained by selecting the gather corresponding to the location of the borehole, while training examples were extracted from adjacent CDPs. Both test and training examples were consistently labeled using the well-log AI data. This approach resulted in a larger number of labeled examples compared to a single gather-based strategy using the well-log AI, which would have yielded fewer labeled examples. In turn, the labeled dataset was employed for supervised training purposes.

The dataset is aptly termed “labeled” because, from a geoscientist’s perspective, AI is a petroacoustic property exhibiting distinct stratigraphic units for various subsurface materials. For instance, the impedance values for hydrocarbon-bearing sands typically fall within the range of 17,500 to 21,500 (ft/s)*(g/cc), while water-bearing sands are found in the range of 22,000 to 24,500 (ft/s)*(g/cc). Shale encompasses values spanning from 24,500 to 27,500 (ft/s)*(g/cc) [56,58]. However, we would like to point out that this categorization is beyond the primary scope of this study, where we treated AI as individual data samples.

For semi-supervised learning, the data pre-processing was similar to that of the supervised learning. However, there were slight differences in the approach. It was similar because it involved taking the same number of traces per CDP, as well as rescaling, interpolation, and windowing operations. However, while the labeled data remained consistent, the unlabeled data underwent pre-processing without including AI information, comprising features exclusively. The extent of the CDPs utilized for unlabeled examples is visually represented in Figure 7, which shows that $65\%$ of the data were allocated for training and the remaining $35\%$ were reserved for testing during the pre-training phase. Following that, the labeled dataset was employed for fine-tuning in the subsequent stages of the analysis.

The process described above was initially carried out for a single well and was later expanded to include five drilled wells. The temporal range from which the examples were extracted for each well is shown in

Table 2. Time interval from which the examples were extracted.

Time (ms)	Well-1	Well-2	Well-3	Well-4	Well-5
Start	2375	2408	2402	2582	2640
End	2584	2648	2691	2831	3013

. After this stage, all the generated data were consolidated into two principal subsets: the labeled and unlabeled datasets, each comprising training and test examples. The comprehensive dataset composition is presented in

Table 3. Pre-stack seismic dataset.

Dataset	Labeled	Unlabeled
Pre-stack Seismic (164,100 Full)	10,800 training	98,280 training
	2100 test (well)	52,920 test

for reference.

2.3. Neural Network Setup

2.3.1. Supervised Learning Case

The tuning of neural network hyperparameters for pre-stack seismic data was started from scratch, due to the absence of a predefined set of parameters regarding the number of features, sequence length, or the volume of examples designated for training and validation. This tuning process led to the adoption of the following architecture for supervised learning: three layers of LSTM units, each comprising 1024 units, with a dropout rate of 0.75. This architecture was further complemented by a dense layer with 1024 units, followed by an additional layer with a single unit. The model was compiled using the loss function of the mean absolute error and the Adam optimizer, an efficient stochastic optimization method for deep learning problems.

Additionally, the hyperparameters were determined by the linear-epoch gradual-warm-up (LEGW) method (Section 2.8). In the case of supervised learning, it was determined that employing a batch size of 32, 1 warm-up epoch, and a learning rate of 0.001, all scaled by a factor of 2 except for the warm-up epoch, produced the most favorable loss values.

2.3.2. Deep Semi-Supervised Learning Case

The architecture of DSSL was similar to the one described earlier, but with a few differences. In the pre-training phase, the network architecture comprised three layers of LSTM units, analogous to the supervised learning setup. However, the output stage was different and included a dense layer with 1024 units and an additional layer with 18 units, aligning with the number of features. The model was created using the Adam optimizer and the mean squared error loss function. The validation loss was used as the monitor metric during compilation. Later on, a fine-tuning architecture was established by appending a dense layer with a single unit to the pre-training network. The remaining hyperparameters for fine-tuning mirrored those employed in supervised learning.

The LEGW method was used to determine hyperparameters throughout the semi-supervised learning phases. In the pre-training phase, a batch size of 2048, 1 warm-up epoch, and a learning rate 0.001 were designated as baseline values. It is worth noting that the substantial unlabeled dataset availability supported a large batch size. Although these hyperparameters were not scaled, the pivotal aspect of LEGW was utilizing a warm-up epoch, enabling effective learning despite the large batch size. Furthermore, the LEGW method was applied consistently with the supervised learning during the fine-tuning phase.

The extensive unlabeled data drove the acceleration of neural network training by implementing the TensorFlow multi-worker strategy, which employed 2 GPUs from separate servers. Additionally, the implementation of the LEGW method played a pivotal role in optimizing GPU memory utilization.

2.4. Inline Pre-Processing and Prediction

After completing the training phase, our objective was to predict the acoustic impedance from the seismic inline that corresponded to the well’s location. We achieved this by pre-processing the seismic data along the inline as unlabeled data. We followed the methodology outlined in Section 2.2, treating each CDP individually. Within a designated time range, seven gathers were systematically pre-processed along the inline, with the well situated in the central gather. We repeated this pre-processing procedure for five distinct inlines, each associated with a specific borehole location.

The pre-processed seismic data from the inline served as the input for the neural network, which had previously been trained using the DSSL approach. The trained neural network predicted AI samples from each input example derived from the CDP data. This prediction process was carried out individually for each CDP within the inline.

The MaxAbsScaler function from the sci-kit-learn library was used to revert the predicted inline data to their original representation. This scaling-back procedure restored the AI values to their original form, undoing the normalization applied during the earlier stages of data processing.

2.5. Deep Semi-Supervised Learning

Semi-supervised learning is a valuable approach applied in scenarios where annotated datasets are scarce, but where a substantial volume of unlabeled data are available. The main objective was to use this unlabeled data to improve the performance of a model that had been trained using a limited number of annotated examples [59]. DNNs have demonstrated significant success in various domains such as image classification [60], NLP [61], and speech recognition [62], primarily when well-prepared, extensive, and labeled datasets are accessible. However, real-world applications often need more labeled examples, as in reservoir characterization, where labeled data are derived from a limited number of drilled wells.

Semi-supervised learning methods are crucial in the successful implementation of DNNs. These methods do not only rely on annotated datasets but also leverage a substantial volume of unlabeled data, which leads to a subfield called deep semi-supervised learning [63]. This approach effectively addresses the constraints imposed by limited labeled data and enhances the applicability of DNNs in real-world scenarios.

2.6. The Greedy Layer-Wise Pre-Training

Greedy layer-wise pre-training is an innovative approach in the realm of DSSL. It was initially introduced by Bengio [64] for deep belief networks and then adapted for LSTM networks by Xu [65]. When applied to LSTMs, this pre-training technique effectively uses unlabeled data to initialize weights, bringing them closer to favorable local minima within multi-layer LSTM networks. This initialization process significantly improves the generalization capability of DNNs [66].

The complete procedure consists of two stages: pre-training and fine-tuning.

• Train the first LSTM layer as an LSTM autoencoder.
• Utilize the output of the last layer as the input for the subsequent LSTM autoencoder layer.
• Iterate through step 2 until the desired number of initialized layers is achieved.
• Channel the output of the final LSTM layer into a newly introduced supervised layer.
• In a supervised learning context, fine-tune all parameters within this deep network.

This methodology progressively builds a deep network through layer-wise pre-training, with each layer learning representations from the preceding layer, ultimately culminating in a fine-tuned network for the specific task.

2.7. Long Short-Term Memory

The Long Short-Term Memory (LSTM) architecture, which is a recurrent neural network model tailored for sequential data, relies on a crucial internal component called the cell state $c^t$, acting as a memory. This architecture features three sigmoidal gates—the input gate $i^t$, the output gate $o^t$, and the forget gate $f^t$—which control the reading or modification of the cell state. At each timestep t, the sigmoidal gates, cell state, and output $h^t$ undergo updates when receiving input $x^t$ and the output $h^{t-1}$, as follows:

$$i^t=\sigma (W^{xi}{x}^t+W^{hi}{h}^{t-1}+W^{ci}{c}^{t-1}+b^i),$$

(1)

$$f^t=\sigma (W^{xf}{x}^t+W^{hf}{h}^{t-1}+W^{cf}{c}^{t-1}+b^f),$$

(2)

$$c^t=f^t{c}^{t-1}+i^{t}tanh(W^{xc}{x}^t+W^{hc}{h}^{t-1}),$$

(3)

$$o^t=\sigma (W^{xo}{x}^t+W^{ho}{h}^{t-1}+W^{co}{c}^t+b^o),$$

(4)

$$h^t=o^{t}tanh\left(c^t\right),$$

(5)

$h^{t-1}$ and $c^{t-1}$ represent the output and cell state from the previous timestep. $Ws$ and $bs$ denote the weights and biases, respectively [65,67,68]. The output $h^t$ can be fed into an output layer or dense layer (feedforward network) to calculate the network’s final output

$$y^t=\varphi (W{h}^t+b),$$

(6)

in this context, W and b represent the weights and bias of the dense layer, respectively. The activation function $\varphi$ computes the output y at time step t. The LSTM architecture in this study was trained using both supervised and unsupervised learning approaches.

2.8. Linear-Epoch Gradual-Warm-up

The linear-epoch gradual-warm-up (LEGW) method is designed to facilitate data parallelism and allows LSTM architectures and CNNs to harness the advantages of large-batch training. This approach leverages distributed processing to effectively accelerate the training process.

Once the baseline hyperparameters have been identified that produce the optimal loss, the LEGW method enables the LSTM network to operate with a larger batch size, while maintaining the loss without further tuning. The method achieves this by scaling three key hyperparameters by a factor of k: the batch size (Bs), the warmup epochs (We), and the learning rate (Lr). Specifically, the learning rate is increased by a factor of $\sqrt{k}$, while a factor of k increases the batch size and warm-up epochs [69]. Significantly, the scaling of these hyperparameters may vary slightly between supervised learning and DSSL methods within the context of this work.

3. Experimental Setup

3.1. Real Seismic Data

The proposed methodology was rigorously tested using real seismic data from Mexico, specifically, a pre-stack seismic time migration cube. The survey area encompassed 141.47 km² and featured 491 inlines and 461 crosslines. The data spanned a time range of 1251 ms, with a sampling rate of 4 ms and a bin size of 25 m. Each CDP consisted of approximately 40 traces. Furthermore, the dataset was enriched by incorporating well-log information obtained from five drilled wells.

A pre-processing stage, described in Section 2, was meticulously applied to prepare the raw data for training, resulting in two subsets: one labeled and one unlabeled, as seen in Table 3.

3.2. Experiments

Our experimental setup involved using two different approaches: DSSL and supervised learning. The primary objective was to assess the performance of DSSL, considering its utilization of both the substantial volume of unlabeled data and the limited labeled data from Table 3. As a reference point, supervised learning was used, which relied only on the available labeled data.

Neural network configurations for the DSSL and supervised learning approaches were consistent with the specifications described in Section 2.3. These configurations were deliberately chosen to guarantee consistency and allow for a meaningful comparison between the two methodologies.

4. Results and Discussion

4.1. Neural Networks Learning Curves

Learning curves of the DSSL and the supervised method highlighted loss behavior of the training and validation (or test) datasets over 225 epochs, as shown in Figure 8. A notable observation was that the DSSL method’s performance was more stable than supervised learning. Specifically, the loss values achieved were 0.0187 for DSSL and 0.0247 for supervised learning. This improvement in the DSSL performance was due to the use of unlabeled data during the pre-training phase, which effectively stabilized the network weights [65], thus enhancing the fine-tuning phase.

4.2. Predictions at Well Locations

The improvement achieved with the DSSL method is even more evident when comparing the absolute AI predictions for five different wells with those obtained using the supervised learning approach. Figure 9 provides a visual representation of these comparisons.

For instance, when considering the prediction for Well-1, the DSSL method exhibited a slight deviation in accuracy within the range of 2500 ms to 2550 ms (as shown in Figure 9a). In contrast, the supervised prediction displayed inaccuracies both before and after the 2500 ms mark and beyond the 2550 ms point (as shown in Figure 9b). This advantage of DSSL over supervised learning is similar to that of Well-3.

Regarding Well-2, while the DSSL method appeared to provide a more accurate prediction within the range of 2500 ms to 2600 ms, the differences between the two methods’ predictions were relatively subtle. This pattern also holds for Well-4 and Well-5, where the DSSL method consistently demonstrated a competitive performance compared to supervised learning.

4.3. Evaluating Predictions at Well Location

A comprehensive set of evaluation metrics and visualization tools were employed to visualize and quantify the difference between the absolute AI derived from well data and the predictions generated by the neural network. These included crossplots, the Pearson correlation (PC), and the mean square error (MSE).

Figure 10 shows the crossplots, highlighting the positive correlation between the well-derived absolute AI and the predictions generated by the neural network for both the supervised and DSSL methods evaluated for each well. Overall, the crossplots associated with the supervised approach exhibited a higher dispersion, indicating a lower precision for its predictions.

This observed pattern was particularly evident in the case of Well-1 (Figure 10b); however, it extended to all wells. Notably, in some areas of the crossplots, the points appear even more scattered in the supervised method. This consistent trend reinforces the conclusion that the DSSL method outperformed (e.g., Well-1 Figure 10a) its supervised counterpart. This advantage is evident in the more concentrated clustering of points within the crossplots, signifying an enhanced predictive accuracy.

The predictions generated by the DSSL method exhibit a stronger correlation with the well-log absolute AI when compared to the predictions produced by the supervised approach for the PC analysis. Additionally, the MSE evaluations conducted on the normalized values of the well AI and predictions confirm these findings. The MSE values indicate that the prediction errors were smaller for the DSSL method, as confirmed by the data presented in

Table 4. PC and MSE measurements of DSSL and supervised predictions. For all PCs, the p-value < 0.05.

	PC		MSE
Well	DSSL	Supervised	DSSL	Supervised
1	0.9938	0.9837	$1.15\times {10}^{-4}$	$3.05\times {10}^{-4}$
2	0.9937	0.9851	$2.05\times {10}^{-4}$	$4.83\times {10}^{-4}$
3	0.9882	0.9721	$2.46\times {10}^{-4}$	$5.92\times {10}^{-4}$
4	0.9807	0.9800	$3.33\times {10}^{-4}$	$3.52\times {10}^{-4}$
5	0.9892	0.9791	$4.46\times {10}^{-4}$	$8.67\times {10}^{-4}$

. These quantitative metrics further support the superior performance of the DSSL method in producing predictions that align more closely with the actual well AI values.

4.4. Predictions beyond Well Location

We carried out a thorough comparative analysis of the outcomes of the two methods, extending our evaluation beyond the absolute AI derived from the well. We leveraged the well’s AI as a reference point to facilitate this analysis. Specifically, our approach involved the neural network to predict the AI of the seismic inline in the vicinity of the well.

For Well-1, Figure 11a displays the seismic inline with both synthetic and absolute AI data from Well-1. It is important to point out that this AI dataset served as the training target for the neural network. Subsequently, Figure 11b,c show the predictive outcomes for the seismic inline obtained through the DSSL and supervised methods, respectively. These figures align with the AI values at the well location along the seismic inline. Figure 11b, which shows the output of the DSSL model, displays a noteworthy capability to predict AI values, even at distances extending further from the well’s designated location. The gray arrows indicate an increased predictive range, distinguishing it from the supervised model shown in Figure 11c.

Conversely, regarding Well-2, both Figure 12b,c show a similar pattern in the predictions generated by the two methods when viewed from this perspective. The projections appeared to align closely in this scenario. In contrast, the improved prediction results of the DSSL are evident for Well-3 when comparing Figure 13b with Figure 13c.

Regarding Well-4, as shown in Figure 14, the predictions rendered by both methods displayed a high degree of similarity, particularly near the well’s AI values. However, as we extended our analysis further from the well location, as seen in Figure 14b, the DSSL method demonstrated superior predictive capabilities. This enhancement in predictive performance with the DSSL model is also noticeable in the inline prediction results for Well-5, as seen in Figure 15b.

4.5. Analytical AI

In addition to the AI predictions generated by the neural network, a comparative analysis was conducted between two types of AI computations: absolute AI and analytical AI. The latter was computed using a low-frequency model derived from well-log data along with band-limited seismic data. The open-source library Pylops facilitated the computational process for this analytical AI [70].

Figure 16 shows this comparison across Wells 1 to 5. It shows that the analytical AI behavior was similar to that of the other AI datasets. However, it does not align as closely with the well-derived AI values as observed in the predictions generated by the neural network.

4.6. Discussion

Obtaining acoustic impedance data from pre-stack seismic data using a DNN like our DSSL model is challenging, due to the limited availability of labeled data. However, the use of unlabeled data showed a significant improvement in the overall performance of the neural network, as seen in the learning curves in Figure 8. Furthermore, this improvement extended to the accurate prediction of absolute AI values at well locations, as confirmed by the results of the crossplots, PC, and MSE.

Moreover, a comparison at the inline level for each well provided strong evidence that the DSSL method outperformed the other methods in predicting absolute AI values when working with pre-stack seismic data located at a considerable distance from the well’s position. This was found to be the case for most of the wells analyzed. This suggests that the pre-training phase effectively extracted underlying features from unlabeled pre-stack seismic data, which in turn helped in fine-tuning the model, even when only a limited quantity of pre-stack labeled seismic data were available. However, it is worth mentioning that, in some instances, such as that shown in Figure 12, the predictions generated by both methods exhibited substantial similarity, which suggests that the neural network may not have needed to obtain additional hidden features from unlabeled data to model the intricate relationship between the AI and seismic data for this particular well and the selected CDPs.

Using unlabeled data to improve the performance of models has been proven effective in various applications of DSSL, including lithofacies classification [9,42]. Nevertheless, a regression model becomes imperative in scenarios where the labeling is not categorical, such as in the case of petrophysical property estimation, specifically absolute AI, as explored in this study. Furthermore, it is important to note that the prediction of other reservoir properties, such as permeability, mineralogy, and porosity, was previously undertaken by Tembely [25], Kim [29], and Yang [27], respectively. These predictions were accomplished within the framework of supervised learning. In our work, we utilized DSSL to predict absolute AI, a petrophysical property that offers crucial insights into subsurface properties and contributes to reservoir characterization [71,72]. No other methods were found that estimated acoustic impedance using DSSL. While all DSSL methods are typically associated with classification tasks, there are no supervised methods that estimate acoustic impedance. Therefore, we built a supervised learning version of our DSSL to have a comparison baseline for the results.

In the literature focused on DSSL for lithofacies classification, it is common to use CNN architectures, along with other DNNs. However, our study took a different approach and used a LSTM recurrent neural network. When working with seismic data, we encountered significant challenges, particularly in determining the optimal balance between sequence length and the number of features per example derived from seismic gathers. The right sequence length is essential to enable the neural network to discern meaningful patterns and, concurrently, optimize training efficiency. The latter is particularly significant when working with extensive volumes of unlabeled data.

We noticed a significant trend during our analysis of the appropriate number of features or traces per CDP. Specifically, we found that data scarcity tends to occur primarily at the initial offsets, rather than within the selected range. To address this issue we adopted a strategic approach. We deliberately retained offsets within the range, even in cases where data were absent. This decision may seem contradictory, but it was based on the notion that these data-deficient offsets might potentially contribute to the model as noise [73,74], and that this, paradoxically, may enhance the overall performance of the neural network.

Developing this study presented several challenges. First, integrating seismic and well-log data separately was time-consuming and required detailed adjustments before pre-processing. This integration had to be repeated for five well-log datasets. Additionally, the neural network had to be designed from scratch and tuned for large amounts of unlabeled data. Furthermore, both pre-stack and well-log data are complex datasets that present challenges in establishing the relationships between them. However, our method, which includes both non-supervised and supervised stages, effectively addressed this complexity compared to using only supervised methods.

The effectiveness of the DSSL method becomes apparent when making predictions within seismic regions that share similar patterns. However, it is worth noting that the accuracy of forecasts may decrease as the seismic data extend farther away from this reference area. Furthermore, allocating a portion of the well-log data as reference or test data is important to accurately evaluate a neural network’s prediction performance.

The proposed methodology has important applications in accessing valuable data that are necessary for reservoir characterization. These valuable data are often limited by various factors and include a wide range of petrophysical properties such as permeability, mineralogy, and porosity, as well as important geological features like gas chimneys, lithology, and fluid-bearing zones.

A potential future approach would be to decrease the amount of unlabeled data by selecting crucial samples from the entire dataset. This would allow absolute AI estimation across the entire volume. It is important to note that a significant portion of unlabeled data were extracted from small cubes. In addition, it is worth highlighting that implementing the down-sampling method helped to reduce the total number of examples.

5. Conclusions

A lack of annotated instances is a significant challenge in many fields, including geophysics for seismic interpretation, such as reservoir characterization. Building an effective modeling capability to characterize the subsurface requires a large amount of labeled data. Unfortunately, obtaining such data is often too expensive.

To address this issue, we took the following steps. First, we used deep semi-supervised learning to make the most of the large amount of unlabeled data. This improved the overall performance of the model within the constraints of a limited supply of labeled data. Second, we expanded our labeled dataset by establishing connections between well-log data and the specific gather from the same location, as well as neighboring gathers in the vicinity. This helped us to train the model more effectively. Third, we developed a method to obtain unlabeled data from pre-stack seismic time migration data. This method helped the neural network learn more efficiently, even when there was a limited amount of labeled data.

We presented a methodology that uses the DSSL approach to estimate the absolute AI from pre-stack seismic data. It is worth noting that absolute AI is mainly determined using a limited set of well-log data, whereas seismic data resources are considerably more abundant.

After conducting our experiments, we found that the prediction outcomes were consistent with the known information obtained from well-logs and other analytical methods. This observation led us to conclude that our methodology holds the potential to be a valuable tool for experts seeking access to this type of critical information.