地震波反演是利用地震波的到达时间、振幅和波形等信息获取地下介质构造、岩性和物性特征的有效手段。基于波动方程的地震反演方法利用正演模拟技术不断迭代更新模型参数，这通常需要大量的数值模拟和优化计算，耗费大量的计算资源和时间。近年来，以傅里叶神经算子(Fourier neural operator，FNO)为代表的神经算子学习引起了广泛关注。然而，在复杂介质地震波反演中，原始FNO结构无法有效学习地质结构变化剧烈的波场信息，导致其反演结果准确性不高。

为了提升FNO在复杂地质模型下学习地震波场信息的准确性和泛化性能，提出了一种新颖的声波速度反演方法−卷积自编码傅里叶神经算子(CAE-FNO)。CAE-FNO利用编码器进行特征提取，并基于FNO进行高效训练，以更好地捕捉波场的细微特征并提高预测精度。CAE-FNO在网络训练过程中逐层减小傅立叶模的规模，从而有效减少网络参数的数量，同时增强网络的泛化能力。

通过对均匀、非均匀、层状和Marmousi2等模型进行数值实验验证，结果表明：CAE-FNO的反演精度优于FNO及其变体UFNO和UNO。在均匀介质模型中，CAE-FNO的速度反演结果相对误差为1.3%，而UFNO与UNO的反演结果相对误差分别为1.7%、2.3%，FNO的误差高达10.1%。在非均匀模型中，CAE-FNO准确反演地质结构和速度变化位置，而UFNO和UNO在速度变化剧烈区域的误差相对较大。层状模型中，CAE-FNO能够清晰区分不同层间的微小速度变化，而FNO无法明显区分。在Marmousi2模型的平滑区域和突变区域，CAE-FNO较UFNO和UNO更能准确捕捉不规则的速度变化界面，FNO则无法有效处理这些区域的速度突变与细节变化。CAE-FNO通过更低的损失函数值和更高的反演精度，展示了其在复杂介质反演中的优势，为地震反演技术提供新的研究思路。

Seismic wave inversion serves as an effective method for obtaining the characteristics of structures, lithology, and physical properties of subsurface media using the arrival times, amplitude, and waveforms of seismic waves. Seismic inversion methods based on wave equations iteratively update model parameters using forward modeling. This generally involves extensive numerical simulations and optimization calculations, requiring large quantities of computational resources and time. In recent years, neural operators for deep learning, represented by the Fourier neural operator (FNO), have gained widespread attention. However, the original FNO structure fails to effectively learn the wavefield information with sharp changes in geological structures in the seismic wave inversion of complex media, leading to low accuracy of inversion results.

To enhance the accuracy and generalization performance of FNO in learning seismic wavefield information under complex geological models, this study developed a novel acoustic velocity inversion method—Convolutional autoencoder embedded with Fourier neural operator (CAE-FNO), which utilized an encoder for feature extraction and performed efficient training based on FNO to effectively capture the fine features of the seismic wavefield and improved prediction accuracy. During the network training, the CAE-FNO method progressively reduced the size of the Fourier mode, thus effectively reducing the number of network parameters while enhancing the generalization capability of the network.

The numerical experiments on homogeneous, heterogeneous, layered, and Marmousi2 models demonstrate that the CAE-FNO method exhibited significantly higher inversion accuracy than FNO and its variants UFNO and UNO. The experiments on the homogeneous model revealed that the velocity inversion results of the CAE-FNO method had a relative error of 1.3% and those of UFNO, UNO, and FNO exhibited relative errors of 1.7%, 2.3%, and up to 10.1%, respectively. In the experiments on the heterogeneous model, CAE-FNO yielded accurate inversion results of geological structures and velocity change positions, whereas UFNO and UNO exhibited higher errors for zones with sharp velocity fluctuations. During the experiments on the layered model, CAE-FNO clearly distinguished minor velocity changes between layers, while FNO failed. For both smooth zones and zones with abrupt changes in the Marmousi2 model, CAE-FNO exhibited higher accuracy in capturing irregular interfaces with velocity changes than UFNO and UNO, while FNO failed to effectively handle the abrupt changes in velocity and detail changes in these zones. Therefore, the CAE-FNO method, demonstrating small loss functions and high accuracy, enjoys advantages in the inversion of complex media, providing a novel research philosophy for seismic inversion.