Seismic poststack impedance inversion using geophysics-informed deep-learning neural network Open Access

Poststack seismic impedance inversion is a process of transforming recorded seismograms into acoustic impedance (AI), which is the product of rock density and P-wave velocity. The traditional poststack seismic impedance inversion methods can be classified into two categories: deterministic and stochastic inversion methods. The deterministic inversion method includes model-based inversion (Cooke and Schneider, 1983; Gelfand and Larner, 1984). The stochastic inversion methods include band-limited inversion (Lindseth, 1979; Oldenburg et al., 1983), colored inversion (Lancaster and Whitcombe, 2000), and sparse spike inversion (SSI) (Kormylo and Mendel, 1983; Chi et al., 1984; Russell, 1988). Model-based inversion starts with an initial model and outputs one of many different acceptable solutions (Russell, 1988). Nonuniqueness is the main challenge for model-based inversion (Cooke and Cant, 2010). Band-limited inversion regards the deconvolved seismic data as the reflectivity and obtains the impedance by reclusively integrating the reflectivity (Oldenburg et al., 1983). Building a proper low-frequency model is one of the challenges for band-limited inversion (Ferguson and Margrave, 1996). Colored inversion is designed to match the spectrum of seismic with acoustic spectra of well data (Lancaster and Whitcombe, 2000). The assumptions of colored inversion are that the wavelet of seismic data is zero phase and the reflectivity series of well data is the same as that of seismic data of all the region. The advantages of colored inversion include low computation cost, easy to be implemented, and robust to noise. SSI (Kormylo and Mendel, 1983; Chi et al., 1984) assumes a layer-cake model for the subsurface and the reflectivity of subsurface can be represented by small number of “pulse” and the objective of SSI is obtain high-resolution impedance. Researchers have developed methods to address the uniqueness (e.g., Chopra, 2001) and lateral continuity (e.g., Pereg et al., 2017) of impedance estimated using SSI.

Recently, machine-learning (ML) and deep-learning (DL) algorithms have been used to compute seismic impedance from seismic data or seismic attributes. The ML/DL methods can be classified into supervised neural network (SNN) and unsupervised (physics-informed) neural network methods. The inputs of SNN algorithms include the impedance logs of sparse wells (and simulated synthetic AI logs) and the seismograms at wellbore locations (and synthetic seismograms computed from simulated synthetic AI logs and a user-defined wavelet). The impedance logs of sparse wells (and simulated synthetic impedance logs) are treated as labels during the training processing and the objective of training process is minimizing the difference between the labels and outputs from the ML/DL algorithms. Considering that impedance well logs are usually sparse within a seismic survey, Das et al. (2019) first simulate various synthetic impedance logs by applying sequential Gaussian simulation to sparse impedance logs. Then, Das et al. (2019) obtain a DL model that is trained using the simulated and sparse impedance logs (labels) and corresponding synthetic and real seismograms. Finally, Das et al. (2019) predict the seismic impedance by applying the trained DL modes to seismic volumes. Wang et al. (2020) use closed-loop convolutional neural network (CNN) to compute seismic impedance. The closed-loop CNN consists of a forward CNN, which predicts impedance from seismic data, and a backward CNN, which maps the predicted impedance to seismic data. Wang et al. (2021) improve the performance of impedance inversion by combining the transferring learning with closed-loop fully convolutional residual network. Zeng et al. (2021) point out that including seismic attributes as additional labels in the loss function helps to improve the accuracy of seismic impedance predicted using ML algorithm. Fernandes et al. (2024) conclude that seismic data preconditioning is important for DL-based inversion. To improve the sensitivity of DL algorithms to the boundary of different lithologies, Cao et al. (2022) modify the CNN of Das et al. (2019) to include reflectivity in the loss function and obtain better performance in predicting seismic impedance. To reduce the effect of sparsity of wells on the prediction of seismic impedance, Biswas et al. (2019) propose the physics-guided network for elastic parameters estimation by modifying the CNN of Das et al. (2019). The inputs of the physics-guided network are seismic data, a low-frequency model, and a wavelet. The outputs of the physics-guided network are elastic parameters. The training process of the CNN is minimizing the difference between seismograms and synthetic seismograms that are computed based on the output elastic parameters from the CNN. The neural network of Das et al. (2019) only contains two layers of convolution and one fully connect layer and is applied to seismic traces independently. Thus, additional processing might be needed to improve the lateral continuity of estimated elastic parameters.

Considering that the deterministic and stochastic inversions are well documented, a full reviewing of the traditional seismic impedance inversion methods is beyond the scope of our paper. Instead, we choose SSI and model-based inversion as the representative methods and compare them with DL neural networks. Our paper begins with introducing a typical workflow to estimate seismic impedance using supervised DL neural network. Next, we proposed the geophysics-informed neural network (GINN) to estimate the seismic impedance from seismic data. To match the magnitude between the output impedance directly from neural network (value ranges from 0 to 1) with the impedance logs, the proposed method consists of a linear operation to output impedance prior combining it with low-frequency model. The proposed neural network is trained using the whole seismic volume. As a result, our method successfully preserves the lateral continuity of elastic parameters without additional processing. We apply those four methods to synthetic and real seismic data. We conclude with the advantages and limitations of each method and areas for future algorithm development.

Figure 1 shows a typical workflow of seismic impedance inversion using SNN (e.g., Wu et al., 2021). It consists of two phases: training and inference phases. The training process aims to obtain a well-trained neural network using seismic data and corresponding labels (impedance logs). The prediction process is estimating the impedance by applying the well-trained neural network to seismic data.

In this paper, the loss function of the neural network is shown in equation 2 and the architecture of the neural network is shown in Figure 2. The encoder and decoder paths of U-Net (Ronneberger et al., 2015) consist of four layers. Each layer of encoder consists of two successive convolutional operators, a rectified linear unit (ReLU) activate function (Nair and Hinton, 2010), and a max-pooling operator. Each layer of decoder consists of a deconvolution operator, two convolutional operators, and an ReLU function. The convolution kernels of encoder and decoder are set as $5×1$ and the convolution kernels in the last hidden layer are set as $1×1$⁠, respectively. The stride of encoder and decoder is set as one. The number of kernels of the first, second, third, and fourth layers of encoder path are set as 16, 32, 64, and 128, respectively. The number of kernels of the first, second, third, and fourth layers of decoder path are set as 128, 64, 32, and 16, respectively.

Figure 3 shows the proposed GINN, which is used to estimate the seismic impedance from seismic data. The inputs of GINN include seismic data, wavelet, and a low-frequency model of impedance. The output of the GINN is impedance. Instead of updating the weights of neural network by comparing the “correct” impedance and output impedance, we compare input seismic data with synthetic seismic data that are computed from output impedance and input wavelet. U-Net is used as the neural network to estimate the seismic impedance from seismic data (Ronneberger et al., 2015). The encoder and decoder paths consist of seven layers. Each layer in the encoder path consists of convolution, a batch normalization regularizer, an ReLU, and a max-pooling operation. The filter numbers of layers 1–7 (from shallow to deep) in the encoder path are 32, 64, 128, 256, 512, 512, and 1024, respectively. Each layer in the decoder path consists of up-convolutions and concatenations with high-resolution features from the decoder path. The filter numbers of layers in the decoder path from deep to shallow are 1024, 512, 512, 256, 128, 64, and 32, respectively. We adopt the Sigmoid activation function in the final output layer.

To evaluate the performance of GINN, the proposed method is applied to synthetic data and real seismic data. The batch size in the training procedure is set as 16 and the epochs are set as 50 in both experiments. The convolution filter size in Biswas et al. (2019) is set as [$nt_w×1$] for the neural network, where $nt_w$ is the time sample number of wavelet. In our experiments, we find that the size of the filters does not affect the accuracy of estimated impedance, and we set the convolution filter size as [$3×1$] for both experiments after testing.

The impedance model used for synthetic testing is modified from the impedance model that was generated by Zeng et al. (1998). The Zeng et al. (1998) impedance model was designed to represent a lower Miocene progradational microtidal shore-zone system within the Powderhorn field in Calhoun County, Texas. The AI model is composed of interbedded 14 sandstone units (low AI) and 15 shale units (high AI) (Figure 4a). We choose an Ormsby wavelet (5–10–40–45 Hz) as the source wavelet (Figure 4b) to produce synthetic seismograms (Figure 4c). The seismic section shown in Figure 4c consists of 512 traces and each trace contains 192 samples with a 4 ms sampling rate. The yellow vertical line in Figure 4c indicates the location of blind well that is used to evaluate the inverted results in the following experiments. In synthetic experiments, we use correlation coefficient between the true impedance and inverted impedance to quantify the accuracy of inverted impedance.

The first experiment of synthetic testing is designed to evaluate the effect of estimated wavelet on the accuracy of estimated impedance. It is well known that it is impossible to obtain an accurate source wavelet in real practice. Instead, researchers usually adopt a statistical wavelet (estimated from seismic data) or deterministic wavelet (estimated from seismic and well data). We use a statistical wavelet that is computed from the seismic data shown in Figure 4c. The black and red curves in Figure 5 are the accurate wavelet shown in Figure 4b and the estimated statistical wavelet from seismic data, respectively. Note that there is a slight difference between those two wavelets. The seismic data in our first experiment are noise-free data that is shown in Figure 4c. We use three “wells” to build the low-frequency model of impedance for the GINN. The three wells are evenly located within the seismic section (located at the trace numbers 128, 256, and 384). Figure 6a shows the inverted seismic impedance using the extracted statistical wavelet. Note that the negligible difference between the inverted impedance shown in Figure 6a and the true impedance shown in Figure 4a. Figure 6b shows the inverted impedance at the location of blind well (indicated by the yellow line in Figure 4c). The black and red curves in Figure 6b are the true impedance log and the estimated impedance at the location of blind well, respectively. Note that the nearly perfect match between the black and red curves in Figure 6b. The average correlation coefficient between the estimated (Figure 6a) and true impedance (Figure 4a) is 0.997. This high coefficient indicates that we obtain a nearly perfect impedance estimation if the estimated wavelet in the inversion is slightly different from the true wavelet.

The second experiment of synthetic testing is designed to evaluate the phase of the estimated wavelet on the accuracy of estimated impedance. In this testing, we assume that the seismic wavelet has a constant phase (45°) when we estimate wavelet (Figure 7a). The input seismic data and low-frequency model are the same as the first experiment. Figure 7b shows the estimated results by applying GINN to the noise-free seismic data shown in Figure 4 and the wavelet shown in Figure 7a. Note that the inverted impedance fails to characterize the subtle geologic features such as the thin bed indicated by the white arrows in Figure 7b. The average correlation coefficient between the estimated (Figure 7b) and true impedance (Figure 4a) declines to 0.796. The first and second experiments together demonstrate that the accuracy of the phase of estimated wavelet is important for proposed GINN to estimate the impedance from seismic data.

The third experiment of synthetic testing is designed to evaluate the effect of the white noise contained in the seismic data on the accuracy of estimated impedance. We design five scenarios with different signal-to-noise ratios (S/Ns) for the seismic data (20, 10 5, 2, and 1 dB) (Figure 8). The number and location of wells are the same as the first experiments (three wells at the location of trace numbers 128, 256, and 384). In this experiment, we compare the inverted results using the proposed GINN, SNN, and SSI. Figures 9, 10, and 11 show the impedance inverted using GINN, SNN, and SSI, respectively. Figure 12 shows the comparison of inverted impedance at the blind well location. The black, red, green, and blue curves in Figure 12 represent the true impedance, impedance inverted using GINN, impedance inverted using SNN, and impedance inverted using SSI, respectively. Table 1 shows the average correlation coefficients between the inverted impedance and the true impedance. We obtain the following four conclusions by comparing the results shown in Figures 9–12 and Table 1: (1) GINN has best estimated impedance if the S/N of seismic data is greater than 2 dB, (2) the results of SNN are not as good as that of GINN and S/N due to limited numbers of wells (only three wells used to simulate synthetic wells), (3) SSI is the most robust method to the white noise, and (4) the lateral continuity of SNN and SSI is better than GINN. Figure 12 demonstrates that the white noise severely distorts the impedance inverted using GINN and SNN at the locations (indicated by the black arrows) where the true impedance has subtle or no changes.

The fourth experiment of synthetic testing is designed to evaluate the effect of number of wells used in the inversion on the accuracy of estimated impedance. It is well known that the accuracy of low-frequency model increases with number of wells in real practice. As a result, the accuracy of inverted impedance also increases for traditional inversion methods. In this experiment, we test five scenarios with total number of wells as 1, 3, 5, 10, and 20, respectively. The wells are evenly distributed within the seismic section (Table 2). To exclude the effect of other factors on the inverted impedance, the inversion is applied to the noise-free seismic data. Figures 13, 14, and 15 show the inverted impedance using GINN, SNN, and SSI, respectively. Figure 16 shows the inverted impedance at the blind well location (indicated by the yellow vertical line in Figure 4c) using GINN, SNN, and SSI, respectively. Table 3 shows the average correlation coefficients between the inverted impedance and the true impedance. We obtain the following two conclusions by comparing the results shown in Figures 13–16 and Table 3: (1) the number of wells, which is used to build the low-frequency model, has limited effects on GINN if the S/N of seismic data is free of noise and (2) the number of wells has the most effects on SNN.

Table 4 summarizes the advantages and limitations of GINN, SSN, and SSI according to the synthetic examples. The S/N is the dominant factor that affects the accuracy of predicted impedance. The accuracy of wavelet phase is more important for GINN than the accuracy of the wavelet “shape” in the time domain. The accuracy of impedance predicted using SNN noticeably increases with the number of wells used in the training process. SSI demonstrates the most robust capability to noise.

To evaluate the performance of our method on real seismic data, we apply it to the seismic data acquired over the North Sea. The seismic data are depth-migrated seismic data. Each seismic trace has 501 samples with an 8 ft sampling interval. Figure 17 shows one vertical seismic section across five wellbore locations. The vertical yellow lines in Figure 17 indicate the locations of wells that are used to build the low-frequency model of impedance. The BRH-1 well (indicated by red vertical line) is selected as the blind well to evaluate the estimated impedance.

We have three estimated impedances in this real example application: (1) the impedance estimated using SSI, (2) the impedance estimated using SNN proposed by Cao et al. (2022), and (3) the impedance inverted using GINN. Figure 18a–18c shows the estimated impedance using SSI, SNN, and GINN, respectively. Note that the impedance estimated using DL algorithms has higher vertical resolution than the impedance inverted using SSI. The white arrows in Figure 18 indicate representative “artifact” of impedance that is introduced by the multiple noise of seismic data. Figure 19 compares the inverted AI using different methods at the locations of four wells (UHZ-1, ZPR-1, SWO-1, and AMR-1). Figure 20 shows the inverted AI at the location of the blind well. The black, green, blue, and red curves in Figures 19 and 20 represent the true AI, estimated AI using SSI, estimated AI using SNN, and estimated AI using GINN, respectively. The red arrow in Figure 19 indicates that the high-impedance layer estimated by both DL algorithms is thinner than the impedance well log. However, the high-impedance layer estimated by SSI is thicker than the true scenario (indicated by the red arrow in Figure 19). The yellow arrows in Figures 19 and 20 indicate the “inaccurate impedance” corresponding to the waveform of multiples. We can obtain the following observations by comparing the inverted impedance at blind well location (Figure 20). (1) The impedance computed using SSI has the best correlation with impedance log at frequency band. (2) The impedance computed using SNN is generally higher than the impedance log, whereas the impedance computed using SSI is lower than impedance log (indicated by the purple arrow in Figure 20). (3) The impedance computed using GINN and SSI has a higher correlation with the impedance log when compared with SNN at the target reservoir zone (indicated by the purple color in Figure 20). Figure 21 shows the synthetic seismogram at the wellbore locations. The black, green, blue, and red curves are the real seismic trace at the well bore locations, synthetic seismogram of SSI, synthetic seismogram of SNN, and synthetic seismogram of GINN, respectively. We computed the synthetic seismograms shown in Figures 21 and 22 using the estimated impedance and the wavelet that is used to estimate the impedance from seismic data. Figures 21 and 22 indicate that the synthetic seismogram of GINN almost has a perfect match with real seismic traces. Note that real seismic data always have noise. However, it is beyond the capability of GINN to differentiate noise from signal for real seismic data and GINN would produce artifact impedance that corresponds to the noise contained in seismic data.

In this paper, we proposed the GINN to estimate the seismic impedance from seismograms. The synthetic experiments demonstrate that statistical wavelet is sufficiently good to produce a good impedance estimation if the S/N of seismic data is greater than 2 dB. However, the accuracy of the phase of estimated wavelet has an obvious effect on the accuracy of inverted impedance. Thus, we suggest that researchers pay special attention to determining the phase of seismic wavelet. The noise-free synthetic experiment demonstrates that GINN is less sensitive to the accuracy of a low-frequency model. The noisy synthetic experiment demonstrates that SSI is the most robust method to the white noise if the S/N of seismic data is low (smaller than 2 dB). The number of wells experiment demonstrates that SNN needs relatively large numbers of wells to produce high accurate impedance. The real seismic data in this paper have a relatively high S/N and the impedance estimated using GINN is better than that of SSI and SNN. The synthetic and real seismic applications together demonstrate that GINN is very promising in seismic impedance inversion if the seismic data have high S/N. However, GINN would “overfit” the noise contained in the seismic data if the energy of noise is stronger than the seismic signal.

This research was partially supported by the National Natural Science Foundation of China (grant no. 42325403).

Data associated with this research are available and can be obtained by contacting the corresponding author.

Biographies and photographs of the authors are not available.