Geological structures are spatially correlated. The closer the distance, the stronger the correlation, and conversely, the weaker the correlation. The correlation of seismic data is reflected in the temporal correlation in the vertical direction of seismic traces and the spatial correlation in the horizontal direction between the central trace and the adjacent traces. Based on the spatiotemporal characteristics of the seismic data, the inversion framework in Figure 1 is constructed using a supervised learning approach. The inversion framework shown in Figure 1 consists of three main modules: the feature extraction module, the regression module, and the forward module.

In the training phase, the input of the inversion network is the seismic data of the well-side trace and the 2 k nearby seismic data centered on it. The feature extraction module extracts the temporal and spatial features of the seismic data of the well-side trace and the 2 k nearby seismic traces by Attention U-Net. The regression module is used to map the data from the feature domain (spatiotemporal feature series) to the target domain (predicted acoustic impedance), while the forward module is used to map the data from the feature domain to the target domain (forward 2 k + 1 traces seismic data). Referring to the structure of the multi-task learning of Mustafa et al. (2021), the inverse network learns two tasks simultaneously: the predicted acoustic impedance data and forward seismic data. By sharing representations between the two tasks, especially if they are related to each other, we bias the network to learn more generalizable features.

The loss of the entire inversion network is the mean square error between the predicted acoustic impedance data and the actual acoustic impedance ( l 1 ), and the mean square error between the 2 k seismic data in and around the well-side traces and the predicted 2 k + 1 seismic data ( l 2 ), with the total loss shown in Eq. 3: l = α l 1 + β l 2 (3) where α and β are weighting factors that control the effects of acoustic impedance losses and seismic losses, respectively.

The inversion results are evaluated quantitatively by calculating the mean square error (MSE) and the coefficient of determination (R ²) of the actual and predicted acoustic impedance.

Mean Squared Error (MSE): MSE is the average of the squared sum of the errors of the corresponding points of the predicted data and the real data, and the smaller the value indicates that the predicted data fits better with the original data, which is defined as: M S E = 1 N ∑ i = 1 N y i − y ^ i 2 2 (4) where y i , y ^ i denote the actual acoustic impedance and predicted impedance, respectively, and N is the number of data.

Determination Coefficient (R ²): R ² is a measure of the goodness of fit between variables that takes into account the mean square error between predicted and actual data. Its range of values is [0, 1], and the larger the value, the better the fit between the variables, the more the independent variable explains the dependent variable, and the more the independent variable contributes to the overall variation. It is defined as: R 2 y , y ^ = 1 − ∑ i = 1 N y i − y ^ i 2 ∑ i = 1 N y i − μ y 2 (5) where y i , y ^ i , and μ y represent the actual acoustic impedance, predicted acoustic impedance, and the average of the actual acoustic impedance, respectively. When R ² is closer to 1, the stronger the correlation between the predicted and actual acoustic impedance is.

The Marmousi2 model is an extension of the original Marmousi model for amplitude variation with offset (AVO) analysis (Martin et al., 2002). The original Marmousi model has been widely used to validate inversion and imaging algorithms. The researchers added more complex structures representing hydrocarbon regions to the model and increased the number of strata, resulting in a new model, the Marmousi2 model, which has a width of 17 km and a depth of 3.5 km. The model is accompanied by synthetic seismic data, which are obtained by convolutional forward simulations of the model’s reflection coefficients using seismic wavelets.

The acoustic impedance model was obtained by multiplying the density and p-velocity models of the Marmousi2 data. The seismic data and acoustic impedance profiles are shown in Figures 5A, B, with 2,721 traces and 688 sampling points per trace in the seismic profile and 2,721 traces and 688 sampling points per trace in the acoustic impedance profile. The colors in Figure 5A represent seismic amplitude values, and the colors in Figure 5B represent the acoustic impedance values. Twenty traces of acoustic impedance are uniformly extracted as pseudo-well data, and for each pseudo-logging curve, 2 k + 1 seismic traces centered on the well-side trace and with k as the radius will be obtained. This paper sets k to 3, and each pseudo-logging curve corresponds to 7 seismic traces with a depth of 688 sampling points. The inverse network is trained using seismic data and pseudo-well data, the training epoch is set to 700, and the batch size is 20 for each iteration. In each training iteration, the weight coefficients in the loss function α and β are set to 1. The total training loss of the previously described inverse network is calculated and back-propagated through the network.

The Marmousi2 model has a complex stratigraphic structure and contains many different subsurface layered media models. The mean square error function is chosen as the loss function to measure the mean square error of the predicted and real acoustic impedance. ADAM is chosen as the optimizer, and ADAM adaptively sets the learning rate during training, with the initial learning rate set to 0.001. A weight decay of 0.0001 is chosen to limit the L₂ norm of the weights from becoming too large, reducing the risk of overfitting the network. The network’s training is implemented in the PyTorch framework, and GPUs are applied to accelerate the computation. Finally, the trained inverse network is used for acoustic impedance inversion.

In order to prove the effectiveness of this paper’s method, the inversion results of this paper’s inversion method are compared with the inversion results of the commonly used deep learning inversion methods, including the inversion method based on CNN (Das et al., 2019), the inversion method based on 1D TCN (Mustafa et al., 2019), and the inversion method based on 1D U-Net. This 1D U-net model is constructed into the same network structure as the U-net proposed in this paper, but it lacks an attention mechanism. These inverse networks are set up with the same training conditions, training data, and hyperparameters. The inversion result of the method based on CNN is shown in Figure 6A, the inversion result of the method based on 1D TCN is shown in Figure 6C, the inversion result of the method based on 1D U-Net is shown in Figure 6E, and the inversion result of the method proposed in this paper is shown in Figure 6G. Figures 6B, D, F, H correspond to the residual difference between each network’s inverse acoustic impedance and the real acoustic impedance.

As shown in Figure 6, the inversion results shown in Figures 6E, G have a higher similarity to the real model than the inversion results in Figures 6A, C. Moreover, Figure 6G has stronger horizontal continuity and weaker visible jitter in both horizontal and vertical directions for the inverse acoustic impedance profile than Figure 6E, the water layer at the top of the figure also clearly shows a relatively better inversion of Figure 6G. The partition interface and fault location in different strata are the main locations where the inversion results show errors, according to the residual profiles. In comparison to other figures in Figure 6, the inversion method in this paper can also invert the convolution structure in the model well, and the inversion results are more continuous and closer to the actual acoustic impedance, as well as more accurate in predicting the location of the faults. In most locations, the error is lower than that of other inversion methods. This is due to the effective use of the inversion network proposed in this paper for the spatial correlation of seismic data’s horizontal direction.

In order to compare the details of the inversion results of different methods from the microscopic level, the representative Trace No. 570 (corresponding to the position around x = 3,565 m) and Trace No. 1400 (corresponding to the position around x = 8,747 m) are selected for inversion.

At these two locations, the acoustic impedance values obtained by four inversion methods were compared. Figures 7A–D shows the inversion results of Trace No. 570 using the conventional CNN inversion method, the 1D TCN inversion method, the 1D U-Net inversion method, and the method proposed in this paper, with the red and black lines representing the true impedance and acoustic impedance inversion results, respectively. Similar to the inversion results of the four networks mentioned above for all seismic traces, the inversion result of the method in this paper is relatively better. The inversion result in Figure 7A has a large inversion error at a large depth, the inversion result in Figure 7B is very different from the true value, and the inversion result in Figure 7C changes too drastically, whereas the difference between the inversion result and the actual acoustic impedance in Figure 7D is very small, with the two curves almost overlapping.

Figures 8A–D corresponds to the inversion results of the above four methods for Trace No. 1400 seismic trace, respectively, and the conclusions are consistent with Figure 7. The inversion results of Figures 8A, B in the figure deviate more from the true values. The inversion results of Figures 8C, D are in better agreement with the actual curves, but between sampling points 0 and 100, the inversion of Figure 8D is better, while the curve change of the inversion result of Figure 8C is too drastic. This further validates the performance of the inversion network proposed in this paper.

In order to objectively and quantitatively evaluate the reliability of the inversion results of the four methods, the coefficients R ² and MSE are used as evaluation criteria. Table 1 shows the MSE and R ² between the acoustic impedance inversion results of different methods in Figure 6 and the actual acoustic impedance.

Table 1 shows that this paper employs multichannel inversion, and the method of acoustic impedance inversion by Attention U-Net using spatial correlation performs best in terms of MSE and R ², demonstrating the method’s efficacy.

Gaussian noise of 4%, 8%, and 12% was added to the seismic data to test the adaptability of the method proposed in this paper to noise. Table 2 shows the quantitative evaluation of the inversion results obtained from the different inversion networks in Figure 6 under different noise conditions. As shown in Table 2, the performance of each method’s inversion results decreases as noise increases relative to a noiseless environment, but the performance index of the method proposed in this paper decreases the least. For example, when the noise of the seismic data increases from 4% to 12%, the R ² coefficients of the inversion results of CNN, TCN, U-Net, and the proposed method decreased by 7.43%, 2.85%, 4.83%, and 1.38%, respectively. Observing the changes in MSE data leads to a similar conclusion. It can be seen that the proposed method in this paper has better noise immunity performance compared with other methods.

To further verify the feasibility of the method, this paper conducts experiments with the SEAM model. The SEAM model is open source and also widely used for the validation of deep learning inversion methods (Mustafa et al., 2021). The SEAM model is a 3D seismic survey with very drastic lateral variations in density and longitudinal wave velocity, which is challenging for the inversion algorithm. The SEAM model is constructed based on basic rock properties, such as the volume of shale and sand. It follows the changing trend of shale porosity characteristics in the Gulf of Mexico, which is a better simulation of the actual geological conditions. The density of the SEAM model and the longitudinal wave velocity model are multiplied to obtain the real acoustic impedance model. The seismic data and the real acoustic impedance profiles are shown in Figures 9A, B, respectively, with 501 traces and 688 sampling points per trace in the seismic profile and 501 traces and 688 sampling points per trace in the acoustic impedance profile. 12 traces of acoustic impedance are uniformly extracted from the acoustic impedance model as pseudo-well data, and k is also set to 3, so that each pseudo-logging curve corresponds to 7 seismic traces with a depth of 688 sampling points. The training epoch is set to 400, and the batch size is 12 for each iteration. The network is then trained in the same way as the Marmousi2 model, and the trained network is used to perform acoustic impedance inversion on all seismic traces.

The inversion results are shown in Figure 10. Figures 10A, C, E correspond to the results of the inversion based on the conventional CNN inversion method, the 1D U-Net inversion method, and the inversion of the proposed method in this paper, respectively. Figures 10B, D, F correspond to the residuals between the acoustic impedance and the real acoustic impedance inverted by each method, respectively. As can be seen from the figure, compared with Figures 10C, E has a better effect in displaying the stratigraphic interface in the left half of the depth range of 10,000 m to 14,000 m, and the strata are clearer. Some thin stratigraphic variations can be clearly observed in the upper left part of Figure 10E diagram between 5,000 and 9,000 m depth. For example, at 2,500 m depth in the real model, there is a thin arc-shaped stratigraphy that can be seen more clearly in Figure 10E, whereas it is difficult to see in Figures 10A, C, and Figure 10A does not outline the central uplifted area in the real model better. Although the method in this paper has some errors in the inversion of the SEAM model, the overall effect is better than the other two methods.

Trace No. 179 (corresponding to the vicinity of x = 12,500 m) was selected for the inversion experiment, and the acoustic impedance inversion results of the three inversion methods are shown in Figure 11. Figures 11A–C shows the inversion results of Trace No. 179 using the conventional CNN inversion method, the 1D U-Net inversion method, and the method proposed in this paper, with the red and black lines representing the true impedance and acoustic impedance inversion results, respectively. The proposed method has better inversion results compared with other methods. From Figure 11C, we can see that the inversion result obtained by the proposed method almost completely overlaps with the true impedance, while the inversion result of the CNN deviates from the true value, and the result obtained by the 1D U-Net inversion method also has large deviations, with a large deviation at a small depth.

To quantitatively evaluate the performance of the method proposed in this paper, the MSE and the R ² between the acoustic impedance inversion results and the true acoustic impedance are calculated and presented in Table 3. The data are the MSE and R² between the acoustic impedance and the true acoustic impedance obtained by the inversion of different inversion methods in Figure 10. The data in the table show that the inversion result of the proposed method performs best in terms of MSE and R ², which verifies the effectiveness of the method.