High-Resolution Imaging: An Approach by Compensating Absorption and Dispersion in Prestack Time Migration With Effective Q Estimation and Fresnel Zone Identification Based on Deep Learning

Abstract

We have developed a migration scheme that can compensate absorption and dispersion with effective estimation and Fresnel zone identification based on deep learning. We use the U-Net neural network technology in deep learning to automatically identify Fresnel zones from compensated migrated dip-angle gathers and obtain the optimal aperture for migration, avoiding the tedious task of manually modifying the boundaries of Fresnel zones. Instead of the interval factor, we used an effective parameter to compensate absorption and dispersion. The effective is estimated using VSP well data and surface seismic velocity data. The proposed scheme can be incorporated into conventional seismic data processing workflow. A field data set was employed to validate the proposed scheme. Higher resolution imaging results with low noise levels are obtained.

1 Introduction

The dissipation of seismic energy is caused by the anelasticity of the subsurface medium, which will decrease the amplitude and modify the phase. In this dissipative medium, as the propagation distance of the seismic wave increases, the attenuation of the seismic wave becomes more serious. Therefore, seismic waves in deep and ultra-deep stratums face the problem of lower resolution due to dissipation. It is crucial to find an appropriate method to eliminate the absorption and dispersion effects of seismic waves for higher resolution. We commonly use the quality factor -related methods to compensate absorption and dispersion in seismic data processing, and most of them can be divided into two categories: one is the inverse filtering (Hargreaves and Calvert, 1991; Wang, 2002; Ferber, 2005; Cavalca et al., 2011; Chen et al., 2014; Zhang et al., 2014; Dai et al., 2018; Shi et al., 2019; Sangwan and Kumar, 2021), and the other is the anelastic prestack migration based on the viscoacoustic wave equation (Zhang and Wapenaar, 2002; Xie et al., 2009; Zhang et al., 2013; Guo et al., 2016; Wang et al., 2018; Zhang et al., 2021). In the first category, inverse filtering is based on the theory of 1-D wave backpropagation and cannot calculate the seismic wave propagation path accurately. In the second category, anelastic prestack depth migration (PSDM) utilizes the viscoacoustic wave equation to simulate wave propagation with dissipation in the wavefield extrapolation, which is a more accurate and consistent way; however, the calculation load is huge, and the interval model is difficult to obtain. Because of the effectiveness of prestack time migration (PSTM) in imaging complex structures without strong velocity variations, various -compensated methods based on the PSTM structure have been developed (Zhang et al., 2013; Zhang et al., 2016; Wu et,al., 2019). These methods employ effective parameters, rather than the interval model used in the depth migration approach, and the estimation of an effective model is easier to achieve than that of an interval model.

An optimal migration aperture can improve the signal-to-noise ratio (S/N) of imaging results. Schleicher et al. (1997) pointed out that the Fresnel zone is an optimal migration aperture. The signal outside the Fresnel zone does not contribute to imaging but brings noise and artifacts, which reduces the quality of the imaging results (Chen 2004; Marfurt 2006; Klokov and Fomel 2012a; Yu et al., 2013). However, the low S/N of field data and underground complex structures make an accurate Fresnel zone estimation challenging. In recent years, some articles have realized the estimation of Fresnel zones in a simple domain, by constructing a migrated dip-angle gather in the time or depth domains (Zhang et al., 2016; Li et al., 2018; Cheng et al., 2020). Zhang et al. (2016) has applied conventional PSTM to generate migrated dip-angle gathers for Fresnel zone estimation during deabsorption of the PSTM process. Cheng et al. (2020) used a modified VGGNet (A convolutional neural network was developed by the University of Oxford’s Visual Geometry Group and Google DeepMind in 2014) to extract Fresnel zones from migrated dip-angle gathers, which is a useful attempt at deep learning for Fresnel zone estimation. However, these Fresnel zone estimation methods are all suitable for dip-angle gathers generated by conventional migration methods, and little research has been carried out on that using compensated dip-angle gathers with a high resolution generated by compensated migration methods.

The quality factor is closely related to the rock properties of the formation, water saturation, seismic wave amplitude and frequency, and other factors; therefore, calculating the value accurately is very difficult. To meet the demand for in seismic data processing, many methods have been developed to estimate . The estimation method was initially proposed using a vertical seismic profile (VSP) (Tonn, 1991) and crosswell data (Neep et al., 1996). These methods can obtain a small amount of values because VSP and cross-well data are not always available in the field, and we prefer to estimate the value from surface reflection seismic data. A variety of methods have been proposed to estimate the value from surface seismic data, and most of them can be divided into two categories: one is the wavelet information–based method (Quan and Harris, 1997; Dasgupta and Clark, 1998; Zhang et al., 2013; Bettinelli Pet al., 2014), which is employed in the time or frequency domain (e.g., the frequency shift method and spectral-ratio method) and demonstrates good performance for estimation of the value, whereas often suffers from noise and wavelet interferences, and the other one is the tomography inversion–based method (Brzostowski and McMechan, 1992; Shen et al., 2018). In the first category, Zhang et al. (2013) estimated the value using surface seismic data by constant migration scanning; however, the implementation complexity of this method limits its broader application. In the second category, the widely used ray-based tomography can estimate the Q value for the dominant frequency with expensive calculation cost and local instability (Cavalca et al., 2011; Shen and Zhu, 2015; Dutta and Schuster, 2016). Full-waveform inversion (FWI) is another popular inversion approach using waveform rather than travel-time, but it requires an accurate initial model and burdens a huge computational expense (Kamei and Pratt, 2008).

This article takes the estimations of the optimal aperture and effective model as the research focus in the compensated PSTM, which is arranged as follows: first, we introduce a modified PSTM scheme with compensation based on the effective ; second, we propose a Fresnel zone identification scheme based on compensated migrated dip-angle gathers using deep learning; third, we present an estimation approach of the effective model for the compensated PSTM. Finally, we demonstrate our scheme with a field data set.

2 PSTM With Compensation Based on Effective Q

By following Zhang et al. (2013), a modified PSTM with compensation based on the effective Q model is expressed aswhere is the Fourier transform of the p^th prestack trace, and represent the travel times from the shot and receiver to the imaging point, respectively, is the two-way vertical travel time, represents the whole migration aperture, represents the starting travel time of the migration aperture, and is the effective parameter. Eq. 1 denotes a compensated migration impulse response of a seismic trace. Summation of the impulse responses of all seismic traces yields a compensated migration result. The two -related terms in Eq. 1 are the frequency-dependent dispersion and amplitude attenuation correction terms, respectively, which are different from the conventional PSTM. In Eq. 1, the size of the migration aperture has an important influence on the signal-to-noise ratio of the imaging result, and the accuracy of effective determines the quality of the compensation result. In view of these two aspects, this article proposes a method of using deep learning to pick up the optimal aperture and a method of quickly obtaining the effective model using VSP data and seismic velocity data. These two methods, together with the modified PSTM with compensation, form a seismic data imaging workflow that is specifically used for high-resolution imaging of prestack seismic data.

3 Identification of Fresnel Zones Using Deep Learning

Three separate sections are considered to introduce the theory of deep learning–based automated Fresnel zone extraction. The first section makes a review of migrated dip-angle gathers.

The second section introduces the architecture of the deep neural network adopted, including the design of U-Net input and output patterns and different types of layers in the network. The final section gives the loss function and training details in seeking optimal weights and biases of the network.

3.1 Review of Migrated Dip-Angle Gathers

Summing the migrated traces within Fresnel zones can produce a high S/N imaging profile. The migrated dip-angle gather supplies a simple domain that makes a visual pickup of Fresnel zones possible, which are constructed by sorting and summing the migrated results in the time or depth domains according to the dip angle (Zhang et al., 2016; Cheng et al., 2020). Different from the conventional migrated dip-angle gather, the compensated migrated dip-angle gather has the characteristics of high resolution and thin events. Therefore, the label data and training parameters of the trained network for conventional migrated dip-angle gathers must be relabeled and trained respectively when the neural network is applied to identify Fresnel zones using compensated migrated dip-angle gathers. In the next section, we will discuss how to use compensated dip-angle gathers to determine Fresnel zones of 2D seismic data. Figure 1 shows the geometrical relationship about the dip angle (Zhang et al., 2016). The angle can be expressed as follows:where denotes the dip-angle related to travel time at the imaging point I; is the root mean-square velocity at the imaging point; and represent the travel times from the shot () and receiver () to the imaging point I, respectively; and T represents the one-way vertical travel time. We obtained a 1D dip-angle gather by summing the migrated traces with dip angles () over the 2D imaging result. This process can be expressed aswhere n denotes the number of seismic traces, denotes a half-derivative of the i^th prestack seismic trace, and is the corresponding compensation factor.

FIGURE 1

The dip-angle gather shows a curved reflected event (Klokov and Fomel 2012b), and its vertex is the stationary-phase point (Cheng et al., 2020). The Fresnel zone is within half a wavelength near the stationary-phase point. Since the Fresnel zone is easy to identify in the dip-angle gather, we can pick it up through the dip-angle gather and obtain a high S/N migrated result by summing the Fresnel zones, but in practice, estimating Fresnel zones through dip-angle gathers will become challenging because dip-angle gathers will become correspondingly more complicated due to the low S/N of field data and underground complex structures, especially for imaging results with compensating absorption and dispersion since their dip-angle gathers differ in S/N and resolution from those generated by conventional migration, which add additional complexity. Many manual modifications to the Fresnel zone boundaries are required, which is a time-consuming and difficult task.

3.2 U-Net Architecture

Deep learning can think and process data just like the human brain, showing its superior capability in many fields in recent years (LeCun et al., 2015). It has multi-layer nonlinear activation function, which can discover hidden features in complex high-dimensional data by simulating signal transformation. Convolutional neural networks (CNNs) are currently the most successful and extensive application in deep learning, which connect input and output through multi-layer convolution. U-Net, a special type of CNN, was originally an auto-encoder–decoder network designed for medical image segmentation (Ronneberger, et al., 2015; A. Sevastopolsky, 2017; Wu et al., 2019; Zhang et al., 2021). We use U-Net to identify the left and right boundaries in the dip-angle gathers as the boundaries of Fresnel zones because one important reason is that U-Net can deliver a satisfactory performance even if the size of the training set is not very large. As shown in Figure 2, the main structure of the network includes two parts, down (encoder) and up (decoder), presenting a symmetrical form. Different levels of networks have different functions. The shallow layer is employed to solve the pixel positioning problem, while the deep layer is used to classify pixels. In the contraction path on the left, each step consists of two 3 × 3 convolution layers, followed by a rectified linear unit (ReLU) (Nair and Hinton, 2010; Krizhevsky et al., 2012) and a 2 × 2 max-pooling operation with stride 2 for downsampling. Symmetrically, each step on the right expansive path consists of a 2 × 2 upsampling operation with the same stride and two convolutional layers to halve feature channels. The sigmoid activation function is applied to the last channel feature vectors to produce a probability map of the output with the same size as the input. The skip connection is used in each upsampling operation, instead of directly monitoring and loss back-transmission on high-level semantic features, to integrate more low-level features into the finally recovered feature map. After building the network, we feed small volumes of seismic images generated by PSTM with compensation, together with corresponding labels. Each data volume contains 128 2D images with a size of 128 × 128. In order to avoid the odd-sized feature map encountered by the pooling layer, the same-padding convolution process was adopted in each step of the network.

FIGURE 2

3.3 Loss Function and Training

Network training uses a loss function to represent the difference between the true Fresnel zones and the predictions. The update of the parameters in the network is realized via the loss backpropagation (Rumelhart et al., 1986; Hecht-Nielsen, 1989), which is commonly used in the gradient descent optimization algorithm to iteratively adjust weights and biases of the neurons by calculating the gradient of the loss function. We consider the Fresnel zone identification problem as a binary segmentation problem; in other words, the output of the network is a probability distribution of 0–1, and the binary cross-entropy loss function is generally adopted:where n is the number of pixels, denotes the true binary labels (0 or 1), and is the prediction probabilities (0 < < 1) computed from the sigmoid activation in the last convolutional layer. The boundary occupies a relatively small proportion of the entire imaging region, resulting in a high imbalance between zero (no boundary) and one (boundary). To overcome this issue, we apply a class-balanced binary cross-entropy loss function (Xie and Tu, 2015; Wu et al., 2021) to adjust the imbalance so that the network is not trained or converged to predicted only zeros.where and , and represent the number of pixels of boundaries and non-boundaries in the label data sets, respectively. denotes the total number of pixels in the label data sets. The class-balanced binary cross-entropy loss can help the network converge in the correct direction by introducing the class-balancing weight on a per-pixel term basis.

Given one thousand images of dip-angle gathers for training and the corresponding true segmentations as labels, training a given model and optimizing the parameters is the goal of training. The labels here are established by manual interpretation and labeling, with labeling ones on true boundaries and zeros elsewhere. Figure 3 shows three randomly seismic images of different dip angle gathers with their corresponding labels. We prepared another 400 dip-angle gathers for validation and testing, of which 60% are used for validation and 40% for testing. In general, a validation set is used to evaluate the model during the training process, fine-tune hyperparameters, and perform model selection, while the testing set is used to evaluate the model. The network takes in the images and outputs 2D boundary distribution probability maps. Cheng et al. (2020) used a modified VGGNet to identify the Fresnel boundary, and the output of his network is a one-dimensional probability distribution map. In our research, we employ the U-net to identify the Fresnel boundary, and its output is a two-dimensional probability distribution map. The U-Net is essentially a fully convolutional network, and its output is different from the VGGNet’s (Wu et al., 2021). Although Cheng’s method is suitable for dip-angle gathers generated by conventional migration methods and the U-net proposed is carried out on compensated dip-angle gathers with high resolution, the steps of the two methods in learning and training are roughly the same, and both need to pre-process the data, and both use training to optimize network parameters. In order to improve the convergence of U-Net training and balance the numerical difference between training data and prediction data, the input image needs to be normalized. Adam method (Kingma and Ba, 2014) was adopted to optimize network parameters, and the default learning rate was set to 0.001. The Adam method is designed to combine the advantages of two methods: AdaGrad (Duchi et al., 2011), which works well with sparse gradients, and RMSProp (Tieleman and Hinton, 2012), which works well in online and non-stationary settings. We can also pick up a proper learning rate manually (Smith L, 2017). We used 60 epochs to train the network, and each epoch processed 1,000 training images. As shown in Figure 4, after 60 training epochs of approximately 22 h, the accuracy of training and validation gradually increases to 95%, while the training and validation loss converges to 0.01. It shows that our network has been trained.

FIGURE 3

FIGURE 4

4 Estimation of Effective

In step 1, we use the centroid frequency shift method (Quan and Harris, 1997) to estimate that readswhere σ² is the variance of the source wavelet; τ is the travel time in the layers; and f_shot and f_geo are the centroid frequencies of the shot point and detection point, respectively. Since this method is sensitive to layering effects and background noise, it is necessary to preprocess the VSP data such as denoising. In addition, try to avoid thin layers, and select some large layers for calculation.

In step 2, we use a Rick wavelet to generate a synthetic trace without attenuation at the location of this VSP well, first. Because the attenuation of shallow seismic data is weak, its dominant frequency can be used as the dominant frequency of the Rick wavelet. Next, we get different migrated imaging traces corresponding to the synthetic trace using PSTM with compensation under a set of regular variable . Based on and multiplied by different weight coefficients, the variable was obtained as 0.5, 0.6, 0.7, 0.8, 0.9,1.0, 1.1, 1.2, 1.3, 1.4, and 1.5 times of . The optimal is selected according to the similarity between the synthetic trace and its corresponding seismic imaging traces.

In step 3, the unit of the parameter in Lee’s empirical formula is km/s, and it is a root mean square velocity. Lee’s formula uses velocity information to estimate the value, which has the characteristics of high efficiency and easy realization in practical application. However, its estimation accuracy is low, and it needs to be corrected by well information.

In the last step, we use all to calibrate and get a model of the whole work area. is derived from VSP data, and its accuracy is higher than that of calculated from Lee’s formula. Using all to calibrate can improve the overall accuracy of the model.

5 Result

5.1 Field Data Example

In this section, we directly use a 2D field data line to analyze and discuss the effective estimation, the identification of Fresnel zones using deep learning, and the imaging with compensation. This line consists of 1,000 CDPs (common depth point) with a CDP spacing of 12.5 m. The data are sampled at 1 ms with a length of 2.5 s.

Figure 5 shows how an optimal value is obtained from the compensated imaging traces. The specific implementation process is as follows: we get the synthetic seismic trace without attenuation, and then use VSP data to estimate the value, which is denoted as . Next, we get different migrated traces corresponding to the synthetic trace by PSTM with compensation using with different weight coefficients. There are eleven weight coefficients used, which are 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5. Multiply by different weight coefficients to obtain the waveforms of compensated imaging traces under different , and then, compare them with their corresponding synthetic seismic trace without attenuation. When the two waveforms are similar, the corresponding value is the optimal . In Figure 5, trace 0 represents a synthetic seismic trace without attenuation, which exists as a reference trace during picking. Trace 1 to trace 11 represent the compensated imaging traces under different . For example, trace 1 is when is multiplied by 1.5, and trace 2 is when is multiplied by 1.4, and so on; trace 11 is when is multiplied by 0.5. On trace 0, six black typical crests are selected, and each crest is surrounded by a set of red dashed lines. The amplitude of the intersection of two red dashed lines with the wave curve is zero. These six crests represent six events. For each wave crest, two red dashed lines that envelop it extend from trace 0 to trace 11. We judge whether is optimal according to whether the two red dashed lines intersect the wave curve at amplitude zero. The principle of picking here is that when is optimal, the waveform of its compensated imaging trace should be closest to that of its corresponding synthetic seismic trace without attenuation. The six red boxes in Figure 5 are the best -labeled. Figure 6 shows the final model of real data, and the value is displayed as the reciprocal of .

FIGURE 5

FIGURE 6

Figure 7 shows part of the prediction results, which are the Fresnel zones predicted from the dip-angle gathers at three CDPs (400, 600, and 800). In order to solve the problem of local unsmoothness of Fresnel zones predicted by deep learning, the prediction results were sparsely processed and only twelve pairs of equally spaced sample points were retained. The Fresnel zones predicted by deep learning was obtained by connecting the sample points with smooth curves. In Figure 7, the red lines are the boundaries manually picked, while the green points on yellow lines are the sample points predicted by U-Net. The predicted boundaries of the Fresnel zones are similar to the manually picked ones, with a smoother curve. After superimposing the Fresnel zones at each CDP within the predicted boundaries, a migrated result with a higher S/N can be obtained. Figure 8 shows the compensated imaging results with different apertures. While Figure 9 shows the detailed comparison of two white boxes in Figure 8. The result with U-Net predicted an optimal aperture has a higher S/N than the result with a constant aperture. The prediction of each dip-angle gather by the trained U-Net requires approximately 0.6 s when using six TITAN Xp GPUs, which is much more efficient than manual picking. Figure 10 shows the comparison between the migration results obtained using conventional PSTM and the PSTM with compensation. The conventional PSTM used a constant aperture, and the compensated PSTM used an optimal aperture predicted by deep learning. Figure 11 shows the detailed comparison of two white boxes in Figure 10. We see the overlay events are well separated by the PSTM with compensation. Figure 12 shows the comparison of dB spectra between the migration sections obtained using conventional PSTM and the PSTM with compensation. The white boxes in Figure 10 are the time windows of the frequency spectrum. Observe that the high frequencies have been recovered well by the PSTM with compensation.

FIGURE 7

FIGURE 8

FIGURE 9

FIGURE 10

FIGURE 11

FIGURE 12

6 Conclusion

We have presented a PSTM scheme that can compensate absorption and dispersion with effective estimation and Fresnel zone identification based on deep learning. Using U-Net to estimate Fresnel zones from compensated migrated dip-angle gathers, we obtain an optimal migration aperture. The predicted boundaries of the Fresnel zones were similar to the manually picked ones, and the migrated result obtained by applying the predicted Fresnel zones exhibited a higher S/N. The effective model is constructed using surface seismic velocity data and VSP well data. The optimal is selected according to the similarity between the synthetic trace and its corresponding seismic imaging traces, which is a quick and effective method. Since the proposed migration scheme can compensate absorption and dispersion, the real data have been imaged with a higher resolution. Here, we discussed how to obtain 1D Fresnel zones from 1D dip-angle gathers for 2D seismic data using deep learning. Because of the high computation cost and memory requirement for 2D dip-angle gathers, it is difficult to directly estimate 2D Fresnel zones from 2D dip-angle gathers for 3D migration. Although 2D Fresnel zones can be represented by incorporating the inline and crossline 1D Fresnel zones from 1D dip-angle gathers obtained from 3D data, this simplified strategy will bring about inaccuracy of migration apertures in other directions except the inline and crossline directions. Therefore, it will be the next research focus to use deep learning to obtain 2D Fresnel zones from 3D data (Aki and Richards, 1980; Bleistein, 1984; Tian, 1990; Xu and Zhang, 2017; Wu and Zuo, 2019).

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