Seismic impedance inversion based on cycle-consistent generative adversarial network

Yu-Qing Wang ,Qi Wang ,Wn-Kai Lu ,*,Qiang G ,Xin-Fi Yan

a The Institute for Artificial Intelligence,Tsinghua University (THUAI),Beijing,100084,China

b State Key Laboratory of Intelligent Technology and Systems,Tsinghua University,Beijing,100084,China

c Beijing National Research Center for Information Science and Technology (BNRist),Tsinghua University,Beijing,100084,China

d The Department of Automation,Tsinghua University,Beijing,100084,China

e The Research Institute of Petroleum Exploration and Development,China National Petroleum Corporation (CNPC),Beijing,100083,China

Keywords:Seismic inversion Cycle GAN Deep learning Semi-supervised learning Neural network visualization

ABSTRACT Deep learning has achieved great success in a variety of research fields and industrial applications.However,when applied to seismic inversion,the shortage of labeled data severely influences the performance of deep learning-based methods.In order to tackle this problem,we propose a novel seismic impedance inversion method based on a cycle-consistent generative adversarial network (Cycle-GAN).The proposed Cycle-GAN model includes two generative subnets and two discriminative subnets.Three kinds of loss,including cycle-consistent loss,adversarial loss,and estimation loss,are adopted to guide the training process.Benefit from the proposed structure,the information contained in unlabeled data can be extracted,and adversarial learning further guarantees that the prediction results share similar distributions with the real data.Moreover,a neural network visualization method is adopted to show that the proposed CNN model can learn more distinguishable features than the conventional CNN model.The robustness experiments on synthetic data sets show that the proposed method can achieve better performances than other methods in most cases.And the blind-well experiments on real seismic profiles show that the predicted impedance curve of the proposed method maintains a better correlation with the true impedance curve.

1.Introduction

Seismic impedance inversion is an effective method used for stratigraphic interpretation and reservoir prediction,which can help engineers analyze the spatial structure and physical properties of underground strata.Over the last 50 years,a variety of researches have been conducted to solve the seismic inversion problem.We can divide them into three kinds of methods:model-based methods,data-based methods,and learning-based methods.Model-based methods assume that the relationship between seismic data and inversion targets obeys some deterministic model such as the Robinson seismic convolution model (RSCM).It is the earliest way to solve the inversion problem and is now widely adopted in the seismic exploration industry.Recursive inversion(Lindseth,1979) and trace integration (Ferguson and Margrave,1996) compute impedance directly from the predicted reflection coefficients.Generalized linear inversion (GLI) (Cooke and Schneider,1983) and constrained sparse spike inversion (CSSI)(Sacchi,1997)iteratively improve the inversion model according to the reconstruction error between the calculated seismic data and the recorded seismic data.However,the recorded seismic data have limited frequency bandwidth and the inversion target has broad bandwidth,leading to the inversion problem underdetermined.To avoid the influence of inaccurate initial models,some data-based methods are proposed,such as well-logging constrained methods(Carron,1989),sparsity constrained method(Yuan et al.,2015),and geostatistical inversion methods (Gouveia and Scales,1998).Another famous data-based model is the artificial neural network(Lu et al.,1996) based method.By supposing that there exists a nonlinear mapping between seismic data and the inversion target,ANN is proposed to model the inversion process without defining a specific model which may introduce modeling errors.Recently,Deep learning (LeCun et al.,2015) has led to groundbreaking advancements in many fields such as image classification,image segmentation,and machine translation.Compared to other methods,the convolutional neural network(CNN)contains deeper layers and more sophisticated structures,which leads to stronger abilities of feature extraction and structure representation.The development of deep learning algorithms helps to improve the performance of data-driven seismic inversion methods by adopting convolutional neural networks.Das et al.(2018,2019)propose a 1D CNN to predict high-resolution impedance data from seismic data and conduct robustness experiments on synthetic data sets.Wu et al.(2018) and Huang et al.(2018) propose to improve FWI based on CNN.Some more complex network structures such as Generative adversarial network (GAN) (Mosser et al.,2018;Li and Luo,2019),recurrent neural network (RNN) (Richardson,2018),Unet(a special fully convolutional neural network)(Xu et al.,2019;Wang et al.,2020)and long short-term memory(LSTM)(Guo et al.,2019;Sun et al.,2019;Alfarraj et al.,2019) are also adopted for seismic inversion.Besides,due to the sophisticated structure of CNN,the trained CNN models are always hard to interpret and it causes difficulty in measuring the performance of CNN on new data sets.To address this problem,Bayesian deep learning-based inversion methods (Choi et al.,2020;Luo et al.,2020) are proposed to estimate the uncertainty of inversion results and evaluate the effectiveness of CNN models on new data sets.

As a data-driven algorithm,the practical application of CNN is limited by the number of labeled samples.In order to enhance the performance of CNN when the number of tag datasets is limited,data augmentation (Ding et al.,2017;Jia et al.,2017) and semisupervised learning (Chen et al.,2018;Chen et al.,2019;Xu et al.,2019) have also been proposed.In the actual seismic exploration process,the stratigraphic physical parameters are usually obtained by drilling.Due to the high cost of drilling and hard acquisition of well-logging parameters,the number of labeled samples that can be used for inversion is small.The application of CNN in seismic inversion is limited by the insufficient number of well-logging samples.Therefore,it is necessary to improve the generalization ability of CNN when only a few numbers of labeled training data are available.

This paper is an extension and improvement of Wang et al.(2019b).In this paper,we address the problem of seismic impedance inversion when the number of labeled samples is small and propose a novel inversion method based on a 1D cycle-consistent generative adversarial network (Cycle-GAN).The proposed 1D Cycle-GAN contains two 1D generative subnets and two 1D discriminative subnets.The generative subnets are used to model the seismic forward and inversion process and they are reversed for each other.The discriminative subnets can extract the distribution of the true data sets and constrain that the distributions of the generated data are identical to the distributions of the true data.And the experimental results on synthetic data show that the Cycle-GAN based inversion algorithm obtains better performance than other CNN based algorithms.The experiments on real seismic data also obtain promising inversion results.Besides,we adopt a neural network visualization method (Erhan et al.,2009;Mordvintsev et al.,2015;Wang et al.,2019a) to interpret the learned features in the trained CNN model and illustrate that the proposed Cycle-GAN model can learn more distinguishable features than conventional CNN model.The main contribution of this paper can be summarized as follows:

(1) We propose a novel seismic impedance inversion method based on a 1D cycle-consistent generative adversarial network.

(2) The robustness of the proposed method under different Signal-to-Noise Ratios (SNRs),numbers of labeled data,and cut-off frequencies of the estimated low-frequency impedances are studied.And different types of inversion methods are compared.

(3) We adopt a neural network visualization method to visualize the learned features in the trained CNN model and compare the visualized features of Cycle-GAN with the conventional open-loop CNN model.

The rest of the paper is organized as follows:First,we review the recent research works related to our paper.Second,we describe the proposed Cycle-GAN-based inversion method,the training procedure,and the adopted neural network visualization method in detail.Third,the experimental setups and results on both synthetic and real data sets are demonstrated and other deep learning-based inversion methods are compared.Last,we draw conclusions based on the conducted experiments.

2.Related works

2.1.Convolutional neural network

Due to the powerful feature extraction capability of Convolutional Neural Networks(CNNs),it has been widely applied in many research fields and achieved superior performances (LeCun et al.,2015).CNN is usually composed of a feature extraction stage and a task-specific stage.The feature extraction stage is designed to extract features from the input data,and it usually contains stacked convolutional layers.The convolutional layers convolve the local regions of the input images with filter kernels and then are followed by non-linear activation layers to generate the output features.The adopted activation functions are essential for acquiring nonlinear expressions of the input signals and enhancing the representation ability.The task-specific stage differs with different types of tasks.For the classification tasks,the task-specific stage is usually composed of several fully-connected layers with Sigmoid or Softmax functions.And for the regression tasks,it usually contains convolutional layers with Tanh activation functions.The parameters of CNNs are optimized based on the defined loss functions and the backpropagation algorithms.More details of CNNs can be found in LeCun's work (LeCun et al.,2015).

2.2.CNN-based seismic inversion

CNNs have been widely adopted in seismic inversion.Das et al.(2018,2019)propose an effective synthetic data generating method based on geostatistical simulation and train a simple two-layer CNN model on the generated data sets which share statistically similar facies proportions,rock-physics relations,and source wavelet properties with the true well-logging data.Benefit from the geostatistical simulation method,enough labeled data can be generated to train the CNN model and it partially resolves the small sample size problem.Alfarraj et al.(2019) and Wang et al.(2020)both adopt a semi-supervised learning framework for seismic impedance inversion.They propose to simultaneously model the seismic forward process and the inversion process based on CNN,thus the forward CNN model can serve as a geophysical constrain to the inversion CNN model and the solution space can be reduced.Similarly,a recent work by Sun et al.(2020)trains a hybrid network,which contains a theory-guided wave propagation network to regularize the inversion CNN model,for pre-stack seismic inversion and can effectively reduce the degree of freedom within the neural network.In this paper,we propose a Cycle-GAN-based inversion method.Compared to the previous works (Alfarraj et al.,2019;Wang et al.,2020),we further adopt two discriminators to constrain the distribution consistency between the generated data and the true data.

2.3.Generative adversarial network

Generative adversarial network(GAN)(Goodfellow et al.,2014)includes a generative model and a discriminative model.By imposing an adversarial loss function,the generative model can learn to generate fake data as similar as the real data while the discriminative model tries to distinguish between the fake data and the real data.The loss function of GAN adopts a two-player minimax game strategy.It has been proven that the minimax game reaches its optimal when the discriminator is unable to distinguish between fake data and real data,and the generator can generate fake data sharing the same distribution as the real data(Goodfellow et al.,2014).Since the proposal of GAN,a series of variants has been proposed to adapt different applications,such as cGAN(Mirza et al.,2014),DCGAN (Radford et al.,2015),InfoGAN (Chen et al.,2016),WGAN (Arjovsky et al.,2017) and Cycle-GAN (Zhu et al.,2017;Yi et al.,2017;Luo et al.,2017).Among them,Cycle-GAN has done a great job in unpaired image translation tasks.It contains two generative neural networks and two discriminative networks.The former is used to translate images between domainXto domainYand the latter is used to constrain distribution consistency in the two domains.Compared with GAN,Cycle-GAN includes a new kind of loss function named cycle-consistency loss to ensure that the input image can be reconstructed after processed by the two generative networks.For the seismic inversion problem,the welllogging data can be obtained as labeled data.Therefore,we modify the loss function of Cycle-GAN to make it suitable for semisupervised learning.

3.Methodology

In this section,we first introduce the proposed Cycle-GAN structure for seismic impedance inversion,the adopted generative CNN models,and the discriminative CNN models.Then the designed loss functions are described in detail.Later,we summarize the training procedure.And finally,the adopted neural network visualization method is introduced.

3.1.Network structures

The adopted Cycle-GAN structure for seismic impedance inversion is depicted in Fig.1.Similar to the cycle-GAN structure in image translation (Zhu et al.,2017;Yi et al.,2017;Luo et al.,2017),the adopted Cycle-GAN model contains two signal domains:seismic domain and impedance domain.The seismic domain includes the seismic data and the impedance domain includes the acoustic impedance data.The main target of Cycle-GAN is to implement the translation between the seismic domain and the impedance domain.Two generative subnets and two discriminative subnets are used in the Cycle-GAN model.As shown in Fig.1,the generative subnets are denoted as forward-CNN and backward-CNN,the discriminative subnets are denoted as D1 and D2.From the perspective of geophysics,the forward-CNN models the seismic forward process,which computes seismic data from impedance data:z→d,and the backward-CNN models the seismic inversion process,which maps seismic data to the high-frequency part of impedance data:d→zH.Notice that in order to ease the inversion task and better control the inversion results,the low-frequency part of impedance datazLis provided by the traditional interpolation method (Wang et al.,2020).Specifically,the low-frequency part of near-well traces are computed from well-logging data,and the low-frequency data between near-well traces are interpolated along seismic events.The forward process and inversion process are reversed for each other,thus during the training,the forward-CNN serves as a good regularization term for backward-CNN.According to the theory of GAN (Goodfellow et al.,2014),the discriminative network D1 is adopted to constrain the distribution consistency between the predicted impedance data and the true impedance data,the discriminator D2 is used to constrain the consistency between the predicted seismic data and the true seismic data.

Fig.1.The architecture of Cycle-GAN.

Fig.2.The neural network structure of generative subnets.

Fig.3.The neural network structure of discriminative subnets.

The adopted neural network structures of generative subnets and discriminative subnets are illustrated in Fig.2 and Fig.3.As it shows in Fig.2,a 1D U-net structure (Ronneberger et al.,2015;Wang et al.,2020)is adopted for both the forward-CNN model and backward-CNN model.The input size and output size are both set to ben×1.The 1D U-net structure includes an encoding process and a decoding process.The encoding layers are denoted asei，i=1，2，···，L,whereLrepresents the number of encoding layers.The white arrow in Fig.2 represents the convolution operation:

wheredenotes the convolution kernel of thejthfeature map at layere1,cdenotes the number of feature maps.The size of the convolution kernel is set to 3 and the stride of convolution is set to 2.Therefore,the output size of layere1is×c.The red arrow in Fig.2 represents the “ReLU+conv+BN” operation:

whererepresents the convolution kernel of thejthfeature map at layerei,the number of feature maps at layereiis set to 2i-1c,BN（·）represents the batch normalization operation.Correspondingly,the decoding layers are denoted asdi，i=1，2，···，L.The yellow arrow in Fig.2 represents the “ReLU+resize+conv+BN”operation:

wheredenotes the convolution kernel of thejthfeature map at layerdi,the number of feature maps at layerdiis set to 2L-i-1c,resize（·）is the up-sampling operation and concat（·）implements the skip connections to merge multi-scale information.The output of the generative subnets is obtained as follows:

wherekodenotes the convolution kernel of the output layer.For the forward-CNN model,the input is acoustic impedancezand the output target is seismic datad.For the backward-CNN model,the input is seismic datadand the output is the high-frequency part of impedancezH.By adding the predictedzHwithzL,the absolute impedance can be obtained.

The structure of discriminative subnets is shown in Fig.3.We adopt a 1D neural network structure similar to AlexNet(Krizhevsky et al.,2012).It contains an encoding process similar to the generative subnets,whose encoding process is implemented by equation(1) and (2),and followed by fully connected layers.The computation of the two fully connected layers can be written as follows:

wheredenotes the weights of fully connected layeri,denotes the bias,Flatten（·）is the function that reshapes the input data into a vector,Sigmoid（·）is the non-linear activation function.The output size is set to 1.For discriminator D1,the input is the predicted impedance or the real impedance.For discriminator D2,the input is the predicted seismic data or the real seismic data.The two generative subnets and two discriminative subnets are constructed according to the structure in Fig.1.

3.2.Loss functions

The originally proposed Cycle-GAN (Zhu et al.,2017;Yi et al.,2017;Luo et al.,2017) structure is used to tackle unpaired image translation problems and it mainly includes two kinds of losses:cycle-consistent loss and adversarial loss.For the seismic inversion problem,the well-logging data can provide labeled data for training.Therefore,the proposed Cycle-GAN-based impedance inversion method is actually under the semi-supervised learning framework.A new kind of loss,named estimation loss,is adopted to constrain the prediction consistency on the near-well traces.

In this paper,we denote the labeled seismic data set asDand its corresponding target impedance asZ,whose number is small.The unlabeled seismic data set is denoted asD*.As displayed in Fig.1,the cycle-consistency loss is used to constrain that the forward-CNN and backward-CNN are reversed for each other.It can be calculated both on labeled data and unlabeled data:

where（·）denotes the computation of backward-CNN with weightsWB,（·）denotes the computation of forward-CNN with weightsWF,d∈Drepresents the labeled seismic data,d*∈D*represents the unlabeled seismic data,zH∈ZHrepresents the highfrequency part of the impedance data,zL∈ZLrepresents the lowfrequency part of the impedance data,∈represents the estimated low-frequency part for the unlabeled seismic data.From the above equation,we can see that the information contained in unlabeled data can be extracted during the training process.

The estimation loss is calculated based on the predicted results and the labeled targets and it can be written as:

The estimation loss is used to constrain the convergence on labeled data sets.

The adversarial loss encourages the generative subnets to generate fake data similar to real data and encourages the discriminative subnets to distinguish between the generated fake data and real data.The adversarial game obtains its optimum solution when the distribution of the generated data is identical to the distribution of the true data and the discriminators cannot distinguish between the generated data and the real data (Goodfellow et al.,2014).The adversarial loss can be written as:

wherefWD1（·）denotes the computation of discriminative subnet D1 with weightsWD1,fWD2（·）denotes the computation of discriminative subnet D2 with weightsWD2.The adversarial lossLD1can guarantee that the inversion results of the unlabeled seismic data share similar distributions with the true impedance data.

In summary,the overall loss functions for the two generative subnets can be written as:

where λ1，λ2＞0 are constant parameters to balance different loss terms.And the overall loss functions for discriminators are shown in equation (11) and (12).

3.3.Training procedure

The training procedure for Cycle-GAN is illustrated in Table 1.Similar to the training process of GAN(Goodfellow et al.,2014),we iteratively update the generative subnets and discriminative subnets.The iteration number of generative subnets and discriminative subnets are set to,respectively.The overall number of training iterations is.Adam(Kingma and Ba,2014) with initial learning rate α is adopted to update the weights of subnets.After optimization,the trained Cycle-GAN is used to obtain the inversion results.

Table 1 Cycle-GAN training procedure.

Table 2 Training parameter settings.

Table 3 Parameter settings of generative subnets and discriminative subnets.

3.4.Neural network visualization

In order to understand what kind of features the neural networks have learned by stacked convolutional layers,we adopt a neural network visualization method to visualize the trained inversion model(Erhan et al.,2009;Mordvintsev et al.,2015;Wang et al.,2019a).One of the important features in CNNs is that a unit in the feature maps is only related to a region of the input,and this region is named as the“receptive field”of the unit(Luo et al.,2016).Take the CNN model in Fig.2 for example,when the size of the convolution kernel is set to 3 and the stride is set to 2,the receptive field size of each layer can be computed as follows:

whererirepresents the receptive field size of layerei.It can be seen that as the layer goes deeper,the receptive field size grows.The goal of neural network visualization is to find the special patterns in the receptive field that trigger the unit on the feature maps of each layer.Therefore,the target function of neural network visualization can be designed as follows:

wheredenotes the optimized parameters of backward-CNN,Riis defined as the receptive field of a specific unit on layereiandcomputes the activation value of the unit.We adopt a gradient ascent algorithm to solve the above maximum optimization problem:

where η represents the learning rate.However,the visualization results of deeper layers are severely affected by checkerboard artifacts (Odena et al.,2016).In order to mitigate these artifacts,we initialize the gradient ascent method by selecting samples from the training data setsDandD*:

where φ（·）represents the cropping operation.To deal with the above problem,we repeatedly select samples from data sets and choose the kind of patterns that maximize the target function.After the optimization of (18) and (16),the obtained receptive field patterncan be used to interpret the unit on the feature map of layerei.

4.Results

In this section,we describe the parameter settings of our method,demonstrate the inversion results,and compare them with other methods on both synthetic data sets and real data sets.Moreover,we conduct robustness experiments on the influences of noise,the numbers of labeled samples,and the cut-off frequencies ofzL.Also,the neural network visualization results are displayed and compared.

Fig.4.The synthetic profile.(a) The synthetic impedance profile.(b) The synthetic seismic profile.(c) The low frequency impedance profile.

4.1.Synthetic data examples

A 2D synthetic profile,shown in Fig.4,is adopted to examine our method.Fig.4a displays the synthetic impedance profile,the traces located at CDP 75,225,375,525 are regarded as labeled tracesZand the rest traces are unlabeled traces.Fig.4b displays the synthetic seismic profile,which is generated from Fig.4a by convolving with a 20 Hz Ricker wavelet,the 4 traces at CDP 75,225,375,525 are labeled tracesDand the rest traces are unlabeled tracesD*.Fig.4c displays the low-frequency impedance profile whose cut-off frequency is 5.3 Hz,the 4 traces at CDP 75,225,375,525 are labeled tracesZLand the rest traces are unlabeled traces.

Table 2 illustrates the training parameter settings,including the setting of total training epochs,the training batch sizeNbatch,the iteration number of generators,the iteration number of discriminators,the initial learning rate α and the weighting parameters λ1,λ2.We give two parameter setting strategies in Table 2,where“Cycle-GAN-0”represents the parameter settings of the originally proposed Cycle-GAN(Zhu et al.,2017;Yi et al.,2017;Luo et al.,2017)and“Cycle-GAN”represents the parameter settings of our proposed Cycle-GAN.Furthermore,we refer to “Cycle-GANRefine” as firstly adopting “Cycle-GAN-0” to obtain an initial inversion model and then fine-tuning it with “Cycle-GAN”.In the experiments,we find that Cycle-GAN-Refine tends to get better inversion results than other methods.

The parameter settings of generative subnets and discriminative subnets are shown in Table 3,whereLdenotes the number of encoding layers,ndenotes the input and output size,cdenotes the number of feature maps at layere1.Notice that parameterLof backward-CNN is set to 5 whileLof forward-CNN is set to 3,it is because that the seismic forward process is based on a simple convolution model and is much easier to model than the seismic inversion process.The loss curves of Cycle-GAN during training are displayed in Fig.5a,from which we can see that the loss values decrease as the epoch increases,and after 500 epochs,the decrease of loss values slows down.

Table 4 Comparison of different methods based on the average MSE of 5 repeated experiments.The optimal result is highlighted with bold red font and the suboptimal result is highlighted with bold font.

Fig.5.The loss and log MSE curve during the training.(a)The loss curve of Cycle-GAN.(b)The log MSE curves of the predicted impedance on unlabeled data sets during training(Red curve:Cycle-GAN,blue curve:Open-loop CNN).It can be seen that the testing error of Cycle-GAN decreases faster than those of Open-loop CNN.

Fig.6.Box plot of the MSE value in the 5 repeated experiments shown in Table 4.

We compare our method with model-based inversion (Russell et al.,1991) and other deep learning-based inversion methods,including Semi-CRNN(Alfarraj et al.,2019),LSTM,Open-loop CNN,and Closed-loop CNN (Wang et al.,2020),where LSTM and Openloop CNN adopts the conventional CNN training strategy that only use one single deep learning model to learn the inversion process based on the loss function in(10),the inversion CNN model of Open-loop CNN is the same as the one used in our Cycle-GAN model.We repeat the experiments 5 times under different initialization weight parameters.The average Mean square error(MSE)of the inversion results are illustrated in Table 4 and the box plots of MSE in the 5 repeated experiments are displayed in Fig.6.It can be seen that Cycle-GAN-Refine obtains the lowest average MSE and Cycle-GAN obtains the sub-lowest average MSE.The deep learningbased inversion methods achieve better inversion results than the traditional model-based inversion method.Cycle-GAN-0 is actually trained on unpaired data sets,thus the inversion error of Cycle-GAN-0 is higher than other methods.From Fig.6,we can see that the inversion errors of Closed-loop CNN and Cycle-GAN are quite close,but Cycle-GAN-Refine always tends to get better inversion results than other methods.Furthermore,we compare the testing errors of Open-loop CNN and Cycle-GAN during the training process.As it is shown in Fig.5b,the testing error of Cycle-GAN decreases faster than Open-loop CNN,it can be inferred that benefit from the semi-supervised learning scheme and the constraints of discriminators,Cycle-GAN can learn reasonable inversion models faster than Open-loop CNN.

Table 5 Comparison under different numbers of labeled traces.The optimal results are highlighted with bold red font and the suboptimal results are highlighted with bold font.

Fig.7.The 2D inversion results of different methods.The top row displays the inversion results of model-based inversion,Semi-CRNN,LSTM,and Open-loop CNN.The bottom row displays the inversion results of Closed-loop CNN,Cycle-GAN-0,Cycle-GAN and Cycle-GAN-Refine.

Fig.8.The inversion error maps of different methods.The top row displays the error maps of model-based inversion,Semi-CRNN,LSTM,and Open-loop CNN.The bottom row displays the error maps of Closed-loop CNN,Cycle-GAN-0,Cycle-GAN and Cycle-GAN-Refine.

Fig.9.The inversion results of different methods at CDP 75,225,375,525,50,150,300,and 400.The red lines represent the true impedance,the magenta,blue,green,and black lines represent the inversion results of model-based inversion,Semi-CRNN,Cycle-GAN-0 and Cycle-GAN-Refine.

Figs.7-9 display inversion results of the above methods.The top rows in Figs.7 and 8 show the inversion results of model-based inversion,Semi-CRNN,LSTM,and Open-loop CNN.And the bottom rows in Figs.7 and 8 show the inversion results of Closed-loop CNN,Cycle-GAN-0,Cycle-GAN,and Cycle-GAN-Refine.As marked by the black circles in Fig.7,there are some discontinuities along seismic events in the inversion results of Semi-CRNN,LSTM,and Open-loop CNN,and there are some blurs in the inversion results of modelbased inversion,Closed-loop CNN,and Cycle-GAN.The inversion profile of Cycle-GAN-Refine is clearer and more continuous along seismic events.Besides,it can be seen from the error maps in Fig.8 that Cycle-GAN-Refine obtains lower inversion errors than other methods.Fig.9 shows the predicted impedance sequences of different methods at 4 labeled traces(CDP 75,225,375,525)and 4 unlabeled traces (CDP 50,150,300,400).For labeled traces,the inversion results of Semi-CRNN and Cycle-GAN-Refine correlate well with the true impedance,while the predicted impedance sequences of model-based inversion have lower resolution,and those of Cycle-GAN-0 show some misfits with the true impedance sequence.For unlabeled traces,the misfitting between the inversion results and the true impedance increases.However,the predicted sequences of Cycle-GAN-Refine correlate better with the true impedance than other methods.

Moreover,we further reconstruct the seismic profile based on the predicted impedance profile.If the inverted impedance profile is correct and the forward model is reliable,the reconstructed seismic profile should consist well with the synthetic seismic profile.We compare the reconstruction results of Semi-CRNN,Closed-loop CNN,Cycle-GAN-0,Cycle-GAN,and Cycle-GANRefine,the comparison results are illustrated in Table 4 and Fig.10.From Table 4,we can see that Cycle-GAN-Refine achieves the lowest reconstruction errors.And similar conclusions can be drawn from Fig.10.The lower reconstruction errors of our methods indicate that the predicted impedance profile of our method is more reliable than other methods.

Fig.10.The reconstructed seismic data and corresponding error maps of different methods.The top row displays the reconstructed seismic data of Semi-CRNN,Closed-loop CNN,Cycle-GAN-0,Cycle-GAN and Cycle-GAN-Refine.The bottom row displays the corresponding error maps.

Table 6 Comparison under different SNR values of the input seismic data.The optimal results are highlighted with bold red font and the suboptimal results are highlighted with bold font.

Table 7 Comparison under different cut-off frequencies of ZL.The optimal results are highlighted with bold red font and the suboptimal results are highlighted with bold font.

4.2.Robustness experiments

Fig.11.Comparisons of MSE under different settings.(a) The MSE curves of different methods under different numbers of labeled traces.(b) The MSE curves under different SNR values of the input seismic data.(c) The MSE curve under different cut-off frequencies of ZL.

In order to examine the robustness of our method,we conduct robustness experiments on synthetic data sets and compare our method with Open-loop CNN,Closed-loop CNN,and Cycle-GAN-0.Table 5 and Fig.11a illustrate the comparison results under different numbers of labeled traces.The optimal results are highlighted with the bold red font in Table 5 and the suboptimal results are highlighted with bold font.As the number of labeled traces increases,the inversion errors of Open-loop CNN,Closed-loop CNN,Cycle-GAN,and Cycle-GAN-Refine tend to decrease.Because that Cycle-GAN-0 is trained under the unsupervised scheme,the number of labeled traces has no obvious correlations with the inversion performance of Cycle-GAN-0.It can be seen from Table 5 and Fig.11a that Cycle-GAN-Refine obtains better inversion results than other methods except that when the number of labeled traces equals 50,the inversion error of Cycle-GAN is lower than Cycle-GAN-Refine.And when the number of labeled traces is larger than 2,the performances of Closed-loop CNN and Cycle-GAN are similar.However,the performance of Cycle-GAN gets worse when the number of labeled traces becomes smaller.We can conclude that Cycle-GAN is not robust enough in the case of small numbers of labeled traces and Cycle-GAN-Refine shows better robustness under different numbers of labeled traces.

Table 6 and Fig.11b show the comparison results under different Signal-to-Noise Ratios (SNRs) of the input seismic data.Random Gaussian noise with different variances is added to the input seismic data to obtain different levels of SNRs.From Table 6 and Fig.11b,we can see that when the SNR value is higher than 10 dB,the inversion accuracy of Cycle-GAN-Refine and Cycle-GAN is higher than other methods,which means the trained Cycle-GAN model is robust when the input seismic data contains little noise.But when the SNR value is low,Open-loop CNN obtains better inversion results than other methods,and the performance of Cycle-GAN and Cycle-GAN-Refine drops significantly.Therefore,if Cycle-GAN and Cycle-GAN-Refine are applied to noisy seismic data,noise attenuation methods are first required to enhance the quality of the seismic data.

Table 8 The average correlation coefficients of the visualized features on different encoding layers.

Fig.12.The visualization features of different encoding layers.From left to right are the visualization results of the first 128 features in the layer e1， e3， e4 and e5,respectively.

Fig.13.The correlation matrices of visualized features.(a)and(b)are the correlation matrices of Cycle-GAN and Open-loop CNN at encoding layer e3.(c)and(d)are the correlation matrices of Cycle-GAN and Open-loop CNN at encoding layer e4.

Fig.14.The histograms of correlation matrices.(a)The histogram of the encoding layer e3.(b)The histogram of the encoding layer e4.The blue bars display the histogram of Cycle-GAN and the red bars display the histogram of Open-Loop CNN.

Table 7 and Fig.11c display the comparison results under different cut-off frequencies ofZL.With the increasing of cut-off frequency,the inversion errors of different methods decrease.The inversion accuracy of Cycle-GAN and Closed-loop CNN is quite close under different cut-off frequencies,and the performance of Cycle-GAN-Refine is better than other methods.According to the slop of the MSE curves in Fig.11c,it can be concluded that Cycle-GANRefine shows higher robustness than other methods under different cut-off frequencies ofZL.

4.3.Neural network visualization results

Based on the neural network visualization methods introduced in 3.4,we demonstrate the visualization results of the backward-CNN in Cycle-GAN and compare it with Open-loop CNN in this section.According to equation (17),the receptive field sizes of encoding layerse1，e2，e3，e4，e5can be computed as 3,7,15,31,62.And according to the parameter setting in Table 3,the feature map numbers ofe1，e2，e3，e4，e5are 128,256,512,1024,2048,respectively.Fig.12 displays the first 128 visualization features of layere1，e3，e4ande5,from which we can see that as the layer goes deeper,the learned features become more complicated and structured.

Moreover,we compare the visualization results of Cycle-GAN and Open-loop CNN in Fig.13,Fig.14,and Table 8.The correlation coefficients of the visualized features at each layer are computed.A larger correlation coefficient indicates the higher similarity between the two visualized features.Fig.13a and b shows the correlation matrices of the visualized features at layere3.Comparing Fig.13a and b,it can be seen that Open-loop CNN has more large correlation coefficients than Cycle-GAN,and a similar situation can be found in Fig.13c and d.We further draw the histograms of the correlation matrices in Fig.14,from the distribution of correlation coefficients we can see that there are more coefficients close to 1 in the Open-loop histograms than in the Cycle-GAN histograms.Table 8 illustrates the average correlation coefficient values of each encoding layer.The average coefficient values of Cycle-GAN is lower than Open-loop CNN.From the above visualization results,it can be concluded that the learned features of Cycle-GAN are more abundant and diverse than the features of Open-loop CNN.The employment of unlabeled data during the training and the use of discriminators can enrich the features learned by CNN.

4.4.Real data examples

In this section,we adopt a 2D real seismic profile to testify our method.The real seismic profile is shown in Fig.15a.The traces located at CDP 38,314,489,and 603 are near-well traces whose target impedance can be computed from well-logging data.And the rest traces are regarded as unlabeled traces.In order to evaluate our inversion results quantitatively,we further consider the trace located at CDP 489 as blind well.The model parameter settings and the training parameter settings are the same as synthetic data experiments shown in Tables 3 and 4.

Fig.15.The inversion results of the real 2D seismic profile.Top row displays the predicted impedance of model-based inversion,Semi-CRNN and Open-loop CNN.Bottom row displays the predicted impedance of Closed-loop CNN,Cycle-GAN and Cycle-GAN-Refine.

Fig.16.The predicted impedance of different methods at CDP 38,314,489,603.The red lines represent the true impedance,the magenta,blue,green,and black lines represent the inversion results of model-based inversion,Semi-CRNN,Closed-loop CNN and Cycle-GAN-Refine.

We compare our method with model-based inversion (Russell et al.,1991),Semi-CRNN (Alfarraj et al.,2019),Open-loop CNN,and Closed-loop CNN(Wang et al.,2020).The predicted impedance profiles are displayed in Fig.15.Fig.16 compares the predicted impedance of different methods on labeled near-well traces (CDP 38,314,603)and unlabeled blind-well trace(CDP 489).As marked by the black circles in Fig.15,compared with other methods,the inversion results of Cycle-GAN-Refine are more continuous and clearer.We can further see from Fig.16 that,on labeled near-well traces,the inversion results of Semi-CRNN,Closed-loop CNN and Cycle-GAN-Refine are much more accurate than the results of model-based inversion.And on blind well,the predicted impedance curve of Cycle-GAN-Refine correlates better with the true impedance curve,especially between 0.15s and 0.2s.In addition,Table 9 illustrates the comparison of correlation coefficients and MSE on blind well,from which we can see that the predicted impedance of Cycle-GAN-Refine obtains the optimal correlation with the true impedance and Cycle-GAN achieves the suboptimal inversion results.Furthermore,in Fig.17,we investigate the reconstructed seismic profile calculated from the predicted impedance of Cycle-GAN-Refine.It can be seen that the reconstructed seismic profile is well consistent with the real seismicprofile and the reconstructed error shown in Fig.17c is quite low.Therefore,we can conclude that our method can be effectively applied to real seismic profiles.

Table 9 The prediction correlation coefficients and MSE of different methods on blind well.The optimal result is highlighted with bold red font and the suboptimal result is highlighted with bold font.

Fig.17.The reconstructed seismic data of Cycle-GAN-Refine.(a)The real seismic profile.(b)The reconstructed seismic profile of Cycle-GAN-Refine.(c)The corresponding error map of (b).

5.Conclusion

In this article,we propose a seismic impedance inversion method based on a cycle-consistent generative adversarial network(Cycle-GAN).The proposed Cycle-GAN model adopts two generative subnets to model the seismic forward and inversion process and adopts two discriminative subnets to guarantee the distribution consistency between the prediction results and the target data sets.Unlabeled data is used during the training procedure to compensate for the shortage of labeled data.Robustness experiments are conducted on synthetic data sets,including the influence of different SNR values,the number of labeled traces,and the cutoff frequencies of low-frequency impedance.The experimental results show that the proposed method achieves better performance than other deep learning-based methods in most cases.And we adopt a neural network visualization method to interpret the learned features of Cycle-GAN and compare them with the features of Open-loop CNN,the visualization results illustrate that Cycle-GAN can learn more distinguishable and diverse features than Open-loop CNN.Finally,the blind-well experiment on real seismic profile shows that the inversion results obtained by the proposed method are better correlated with the well-logging impedance data than other methods.

Acknowledgements

This research is financially supported by the NSFC (Grant No.41974126 and 41674116),the National Key Research and Development Program of China(Grant No.2018YFA0702501),and the 13th 5-Year Basic Research Program of China National Petroleum Corporation (CNPC) (2018A-3306).