## 1 Introduction  
Seismic inversion refers to the process of inferring various physical property parameters of rocks using seismic and well log data [1]. Specifically, seismic elastic parameter inversion aims to deduce elastic parameters, such as P-wave velocity (Vp) and density, which are two of the most important subsurface properties. Seismic inversion plays a vital role in various applications, including oil and gas exploration, mineral resource prospecting, and groundwater resource management [2, 3, 4, 5]. In recent years, deep learning (DL) methods, known for their efficiency, flexibility, accuracy, and adaptability, have become integral parts of geophysical exploration workflows [6]. These data-driven methods have been successfully applied in various tasks, such as seismic resolution enhancement [7], seismic lithology prediction [8], reservoir characterization [9], seismic fault detection [10, 11], seismic data reconstruction [12, 13], denoising [14, 15], and seismic inversion [16, 17, 18]. In practice, the inversion problem is characterized by nonlinearity and multiple solutions [19]. Traditional inversion methods, often known as model-driven methods, heavily rely on prior knowledge and physical models that link seismic data to model parameters [20, 21, 22]. Furthermore, traditional approaches still require the appropriate initial models to avoid being trapped in local optima. In contrast, DL-based methods surpass traditional approaches by exploiting their capacity to learn the intricate nonlinear relationship between seismic data and elastic parameters. This results in improved performance and accuracy [23]. The learning ability of artificial neural networks (ANNs) is limited due to the shallow network structure. Deep neural networks (DNNs) with more complex network structures can learn complex mapping relationships better and overcome the deficiency of ANNs. Convolutional neural networks (CNNs) have emerged as powerful tools for seismic inversion due to their exceptional capability to extract features from data. [PERSON] [16] developed a semi-supervised inversion model utilizing a two-layer CNN and a forward model to invert post-stack seismic data. Similarly, [PERSON] [17] applied fully convolutional networks (FCNs) with two convolutional layers to invert impedance with satisfactory results. Building on the concept of FCNs, [PERSON] [24] proposed the fully convolutional residual network (FCRN), which combines the flexibility and simplicity of FCNs with the advantages of residual blocks. Experimental results strongly support the effectiveness of FCRN in achieving high inversion accuracy and noise immunity. A time series is a sequence of values observed over time and arranged in chronological order [25]. A seismic trace is in a natural time series. Based on this inception, recurrent neural networks (RNNs) [26] are used for seismic inversion to model time-related information among seismic traces and elastic parameters. RNNs have a feedback loop that allows retaining memory and incorporating temporal information. For RNN models, there can potentially be gradient disappearance or gradient explosion. On this point, long short-term memory (LSTM) [27] and gated recurrent unit networks (GRUs) [28] are proposed to mitigate the drawbacks of RNNs. They have been successfully implemented for seismic inversion [29; 30]. [PERSON] [31] utilized a single-layer LSTM and one fully connected layer to model the correlation between seismic data and the large-scale density parameter. [PERSON] [32] introduced a bidirectional LSTM (BiLSTM) layer into CNN for P-impedance inversion. On the other hand, GRU demonstrated superiority over LSTM due to fewer parameters and faster convergence. [PERSON] [33] employed a stack of three layers of bidirectional GRU (BiGRU) in the timing module to extract low-frequency information from seismic data and ensure smooth network output. [PERSON] [34] proposed a deep convolutional GRU (DCGRU) architecture that combines CNN and GRU.