Implicit structural modeling using sparse and unevenly distributed data is essential for various scientific and societal purposes, ranging from natural source exploration to geological hazard forecasts. Most advanced implicit approaches formulate structural modeling as least squares minimization or spatial interpolation, using various mathematical methods to solve for a scalar field that optimally fits all the inputs under an assumption of smooth regularization. However, these approaches may not reasonably represent complex geometries and relationships of structures and may fail to fit a global structural trend when the known data are too sparse or unevenly distributed. Additionally, solving a large system of mathematical equations with iterative optimization solvers could be computationally expensive in 3-D. To deal with these issues, we propose an efficient deep learning method using a convolution neural network to create a full structural model from the sparse interpretations of stratigraphic interfaces and faults. The network is beneficial for the flexible incorporation of geological empirical knowledge when trained by numerous synthetic models with realistic structures that are automatically generated from a data simulation workflow. It also presents an impressive characteristic of integrating various types of geological constraints by optimally minimizing a hybrid loss function in training, thus opening new opportunities for further improving the structural modeling performance. Moreover, the deep neural network, after training, is highly efficient for the generation of structural models in many geological applications. The capacity of our approach for modeling complexly deformed structures is demonstrated by using both synthetic and field datasets in which the produced models can be geologically reasonable and structurally consistent with the inputs.

A geological model structurally consistent with the subsurface is essential for understanding the subsurface spatial organization and quantitatively simulating geological processes for a wide variety of Earth science applications well

Recently, more and more implicit structural modeling methods have been proposed for constructing geological models because of their efficient, updatable, and reproducible characteristics

The discrete smooth interpolation (DSI) is one class of implicit methods that computes structural models by discretizing the scalar function on a volumetric mesh

The existing approaches exhibit many promising characteristics; however, reproducing the structures of highly deformed regions remains a challenging task regarding geological consistency because the modeling reliability depends on the availability and quality of the observed data. Structural interpolation fully guided by mathematical equations might not always produce a geologically valid model given sparse or unevenly sampled data (sparse or clustered) in some complex geological circumstances. Corresponding structural models often have erroneous geometries that are incompatible with geological knowledge and have spatial relationships with relevant structures. This problem is mainly attributed to the limited constraints that are permitted in structural interpolants in which all the data and knowledge are mathematically represented as a form of linear constraints to compute a continuous scalar function as smoothly as possible. Although this assumption is helpful to derive a unique model, imposing such a smoothness criterion might compromise the influence of local structural variations and negatively impact the modeling accuracy of highly variant structures

Our implicit modeling method produces a volumetric scalar function as an implicit representation of all the geological structures from input structural data by using a convolutional neural network (CNN). Trained with numerous synthetic data, the network can be applied to field structural data to efficiently predict a geologically reasonable model that matches the inputs well.

Our network has a U-shaped architecture that consists of encoder and decoder branches shown in panel

The normalized hidden feature representations computed in each spatial scale of the 3-D structural modeling networks.

In this study, we present a deep learning method using a convolutional neural network (CNN) as an alternative to conventional implicit structural modeling. Deep learning is a type of data-driven and statistical approach that estimates an implicit function that maps inputs to outputs from past experiences or example data by minimizing given quality criteria

As is shown in Fig.

We organize this paper as follows. In Sect. 2, we describe the CNN architecture designed for implicit modeling and its associated loss function definition. In Sect. 3, we introduce the methodology used to automatically generate training samples and simulate the partially missing horizons. Sections 4 and 5 include both synthetic and real-world case studies to verify the performance of our network in representing complex geological structures. Section 6 presents the promising characteristics of our CNN approach and its current limitations and possible improvements that we will focus on in future research. Finally, we summarize the work in Sect. 7.

In this section, we describe the CNN architecture and its associated loss function used in training the network to generate implicit structural models.

Our developed CNN architecture uses a U-shaped framework modified, from UNet and its associated variants

We show the proposed 2-D and 3-D CNNs with the same architecture in Fig.

The decoder branch includes the five spatial scales (from

Figure

The network provides an attractive characteristic to integrate various structural constraints by minimizing the corresponding errors between the predicted and reference models. To make geologically valid predictions, we combine element-wise accuracy with multi-scale structural similarity to define a hybrid loss function. We introduce the notations and formal definitions used in this loss function. Let

A total of four pairs of 3-D training data samples are shown. The first row shows 3-D synthetic implicit structural models used as labels in training our 3-D network. The second and third rows, respectively, display the fault and sparse horizon points extracted from the models (first row), which are used together as inputs to the CNN.

A total of four pairs of 2-D training data samples are shown. The first row displays 2-D synthetic implicit structural models used as labels in training our 2-D CNN. The second and third rows, respectively, show the fault images and sparse horizon points extracted from the label models (first row), which are used together as inputs to the CNN. It is worth noting that the points denoted by the same color in each image in the third row correspond to the same horizon.

In many geologically related regression problems, mean square error (MSE) and mean absolute error (MAE) are commonly used to measure, element-wise, the accuracy of the solutions

Although MAE outperforms MSE in geological modeling scenarios, the results are still not optimal. CNN trained by using MAE alone might not correctly capture the geometrical features that are represented by the distribution of the neighboring points, while blurring high-frequency and sharp structural discontinuities. Thus, the two models with similar MAE might appear to have significantly distinct structures, which negatively impacts the optimization of the CNN's parameters. To alleviate such smooth effectiveness, we use a hybrid loss function by combining MAE with structural similarity

The standard deviation

MS-SSIM highlights structural variations, focusing on a neighborhood of point

Our CNN architecture is beneficial for the flexible incorporation of empirical geological knowledge in a supervised learning framework with numerous structural models that are all automatically generated from an automatic data simulation workflow. We randomly delete some segments from the models to obtain the partially missing horizons similar to the modeling objects collected from field observations. In training our network, the incomplete horizons, together with the faults, are used as inputs to predict a structural field under the supervision of the full model.

Training (cyan) and validation (orange) curves of using our developed hybrid loss

We apply the trained CNN to the five geological models

A challenge of applying the supervised learning method is the preparation of a great deal of example data and especially the corresponding geological labels for training the network. In the structural modeling, the training dataset should incorporate structurally varying geological models as much as possible to enable the CNN to learn representative knowledge for achieving its reliable generalization in real-world applications. However, it is hardly possible to manually label all geological structures in a field survey because the ground truth of the subsurface is inaccessible. To solve this problem, we use an automatic workflow to simulate geological structural models with some typical folding and faulting features that are controlled by a set of random parameters

The input structural data of our network consist of scattered points that are gridded into a volumetric mesh with valid annotations on structures and zeros elsewhere. As displayed in Figs.

As the geological interfaces are implicitly embedded in the scalar field with the iso-values and can be obtained by iso-surface extraction methods, we adopt jittered sampling

In this section, we present the geological structural models derived from our CNN for both synthetic and field data applications to demonstrate its modeling performance.

A quantitative comparison between our network and the widely used powerful networks using various quality metrics. For each of the quality metrics, the best performance is highlighted in bold. The proposed network (DeepISMNet) is marked with an asterisk to distinguish it from the others. Note: GFLOPs is the giga floating point operations per second.

A quantitative analysis of our network trained with the distinct loss functions using multiple modeling quality metrics. For each of the quality metrics, the best modeling result is highlighted in bold.

Considering the that coordinate ranges of the field geological datasets can be much different from each other, we rescale every structural model to obtain the normalized one that ranges from zero to one. This normalization is implemented by first subtracting the minimum and then dividing its maximum and thus would not change its geological structures. When normalizing the structural data, we assign the scattered points on the same geological interface to the corresponding iso-values of the normalized model. In training the network, we formulate these normalized training samples in batches and set the batch size to

Furthermore, we evaluate the modeling stability of our network in terms of the perturbations of the input structures created from the same geological model. In this experiment, we randomly choose

When the CNN is well trained, the modeling experiences and knowledge learned from the synthetic dataset are implicitly embedded in the network parameters. To verify its modeling performance, we apply the trained CNN to the five synthetic structural models not included in the training dataset.
As is shown in Fig.

As tabulated in Table

We use a synthetic model

It might not be surprising that the CNN trained with a synthetic dataset works well to produce a geologically valid and consistent model by using the structural data generated from the same workflow for creating the training dataset. In this section, we further present modeling results of our trained network applied on real-world data that are acquired at different geological surveys to demonstrate our proof of concept. The modeling objects collected from field observations or seismic data are required to convert into the uniformly sampling grids to obtain the input structural data of our CNN.

In most cases, the structural data collected from field surveys are discrete and not necessarily located on the sampling grid of the model, such that we need a preprocessing step that scatters the structural data into a volumetric mesh with annotations. To scatter the structural data, we simply shift the horizon and fault interpretations to their nearest sampling grids in the model and obtain the associated scattered points. In both the synthetic and the field data applications, the annotation of fault scattered points is straightforward, by simply assigning ones near the faults and zeros elsewhere. However, although the points on horizons can be assigned to the corresponding iso-values of the model in the synthetic data experiments, this might not be feasible when modeling real-world geology from structural interpretations. As the ground truth of geological structures is typically inaccessible before modeling, how to properly annotate the horizon data remains a problem.

We implement a numerical experiment using the horizon data labeled with different iso-values in a synthetic structural model (Fig.

We apply the trained CNN to 2-D field data to verify its modeling performance using the structural data with geometrical patterns distinct from the training data. The input structural data are manually interpreted from the seismic images that are acquired from the WESTCAM dataset. This dataset is acquired in regions with closely spaced and complexly crossing faults with large slips in which the seismic images are of low resolution due to insufficient coverage and data stacking. The ambiguous seismic reflections shown in Fig.

Application in a 2-D seismic field dataset. We display seismic images

As shown in Fig.

Application in an outcrop field dataset. We display two outcrop images

The second 2-D field data experiment uses the dataset acquired from a geological survey and mineral exploration of the Araripe Basin in the region of the Borborema Province in northeastern Brazil

The modeling results presented in Fig.

Using the automatically simulated dataset, we train a 3-D modeling network with the same architecture as the 2-D CNN above to correctly capture the geometrical characteristics of 3-D geology. To validate its modeling ability, we apply the trained CNN to 3-D field data and construct a full structural model from unevenly sampled scattered points obtained from seismic interpretation. The first seismic data sampled in regions with complexly deformed structures have relatively low resolution and signal-to-noise ratio. As shown in Fig.

The first real-world data application. We display seismic volumes

The second real-world data application. We display seismic volumes

The modeling results shown in Fig.

The second 3-D real-world case study is of a conformably folded and layered model with numerous faults that are curved and complexly intersected with each other. As shown in Fig.

In this section, we discuss the modeling characteristic of our method and its abilities for structural uncertainty analysis, along with the current limitations. We also demonstrate a potential improvement that we will focus on in future research to incorporate extra structural constraints in the CNN-based structural modeling.

When modeling complex geological structures, the reliability of the implicit methods is heavily dependent on the quality and availability of the input structural data. However, the heterogeneously distributed structural data pose an ill-posed problem in that there exist multiple plausible structural models which fit the inputs equally. Therefore, a data uncertainty analysis is necessarily critical when looking for an optimal solution, especially for the noisy and hard-to-reconcile structural observations

Geological uncertainty analysis. We display multiple sets of modeling elements interpreted from the vertical boreholes and the outcrop observations

We adopt various combinations of modeling objects with the horizons and faults interpreted from the boreholes and the outcrop observations shown in Fig.

Our CNN architecture permits the flexible incorporation of varying types of geological information by defining an appropriate loss function to measure the modeling error for every structural constraint. The input data of our method are not limited to horizons and faults and can include the structural angular observations in modeling process. In this section, we use the structural angular information that represents local orientations of geological layers to permit geometrical relationships in the gradient of the scalar function to be considered. The loss function of the orientation constraint aims to measure the angle errors between the directional derivatives of the predicted model and the orientation observations using cosine similarity. We adopt the second-order accurate central differences method

The faults and incomplete horizons, together with the structural orientations, are used as inputs

The CNN trained by using the synthetic dataset presents excellent modeling capacities in real-world cause studies to represent complicated geological structures that are distinct from the simulated models. Instead of imposing any explicit mathematical constraints in the conventional implicit methods, our CNN-based structural modeling is implemented by the recursive spatial convolutions with trainable kernel parameters and the loss function related to various geological constraints. The spatial convolutions in the CNN can be viewed as the implicit interpolants used in the traditional approaches, and the only difference is that the parameterization of their kernel functions is optimized through training. As structural modeling is dependent on the analysis of the spatial relations of the observed structures to interpolate new geologically valid structures elsewhere, acquiring representative example data is essential for training the CNN to achieve its reliable generalization performance. Therefore, we adopt an automatic workflow to generate numerous models with realistic structures and simulate partially missing horizons in building the training dataset. It is a significant reason why our network could be applied to the real-world datasets acquired in different geological surveys with distinct structural patterns.

Another improvement of our approach is attributed to the use of a hybrid loss function based on element-wise accuracy and structural similarity when updating the CNN's parameters. To demonstrate the improved modeling performance, we implement a quantitative analysis of our CNN trained with the different loss functions using the multiple quality metrics. The averages of these metrics on the validation dataset are tabulated in Table

Although working well to recover faulted and folded structures, the proposed method might not properly represent other geological structures that are not considered in the training dataset, such as unconformities and igneous intrusions. The trained network also might not correctly construct low dip-angle thrust faults in predicted models because we still do not include this type of fault in the used data generator. Despite the current limitations, the proposed CNN architecture still shows promising potential for computing a geological valid and structurally compatible model honoring the observed structures. Considering that the used training samples are still not sufficiently diverse to support modeling complex and unseen geological settings, future works will focus on expanding the training dataset to a broader range of structural geometries and relationships associated with these settings. For example, we can further augment simulation workflow by adding more complex and various features in the structural models or adopting a recently developed 3-D geological modeling dataset

A CNN-based deep learning method has been used to represent geological structures over the entire volume of interest from typically sparse and hard-to-reconcile structural interpretations. The network is composed of encoder and decoder branches and supplemented with a lightweight depth-wise separable convolution and channel-wise attention to find an optimal tradeoff between modeling accuracy and computational efficiency. The developed CNN architecture leverages the low-ranking nature of the sparse and heterogeneously sampled structural data to adaptively suppress uninformative features by using a linear bottleneck and inverted residual structure in each of the encoded convolutional layers. Our approach is beneficial for the flexible incorporation of empirical geological knowledge constraints in a supervised learning framework using numerous and realistic structural models that are generated from an automatic data simulation workflow. This also provides an impressive characteristic to flexibly integrate multiple types of structural constraints into the modeling by using an appropriate loss function, exhibiting a promising perspective for further improving geological modeling. We verify the effectiveness of the proposed approach by using the case studies acquired in distinct geological surveys, including synthetic examples created by the same workflow for acquiring the training dataset, the randomly created modeling objects without any ground truth of geology, and the structural interpretations obtained from the seismic images. In both synthetic data and real-world data applications, we verify its modeling capacities in representing complex structures with a model geologically reasonable and structurally compatible with the inputs.

To verify the modeling performance of our CNN, we quantitatively measure the differences between the ground truth structural models and predictions by using various regression metrics including SSIM (structural similarity), MSE (mean square error), MAE (mean absolute error), EVS (explained variance score), MSLE (mean square logarithm error), MDAE (median absolute error), and

The synthetic structural models, used for training and validating our network, have been uploaded to Zenodo and are freely available at

XW initiated the idea of this study and advised the research on it. ZB conducted the research and implemented the 2-D and 3-D CNN-based structural modeling algorithms. XW conducted numerical structural simulations to provide synthetic structural models for training. ZB prepared the training datasets from the simulated structural models and carried out the experiments for both synthetic and real-world case studies. ZL, DC, and XY helped design the experiments and advised on result analysis from a geological perspective. ZB and XW prepared the paper, with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been financially supported by the National Science Foundation of China (grant nos. 42050104 and 41974121) and Research Institute of Petroleum Exploration and Development–Northwest (NWGI), PetroChina.

This research has been supported by the National Natural Science Foundation of China (grant nos. 42050104 and 41974121) and the Research Institute of Petroleum Exploration and Development–Northwest (NWGI), PetroChina.

This paper was edited by Thomas Poulet and reviewed by two anonymous referees.