## 1 Introduction  
In the exploration and development of tight carbonate reservoirs, fracture development plays a crucial role in determining reservoir permeability and the formation of reservoir space. The presence of fractures induces anisotropy in the subsurface medium, which manifests as variations in elastic properties depending on the direction. To quantitatively characterize this anisotropy, specific anisotropy parameters for different types of media are proposed ([PERSON], 1986; [PERSON], 1996; [PERSON], 1997) [1, 2, 3]. The petrophysical characterization of fractured media is a critical aspect of understanding the behavior of reservoirs, particularly in terms of permeability and anisotropy. Several models have been proposed to represent the effects of fractures on the elastic properties of subsurface materials. These include [PERSON]'s micro-fracture model ([PERSON], 1980, 1981) [4, 5], [PERSON]'s linear slip model ([PERSON], 1980) [6], and the moving mesh method ([PERSON] and [PERSON], 2022) [7]. Among the most widely used are [PERSON]'s micro-fracture model and [PERSON]'s linear slip model, both of which provide essential frameworks for representing fractured rock as equivalent anisotropic media. [PERSON] further introduced the concepts of normal weakness \(\delta_{N}\) and tangential weakness \(\delta_{T}\) to quantitatively assess the anisotropy induced by fractures, offering a means to characterize fracture-related anisotropy in terms of material behavior. Building upon this, [PERSON] (2000) developed a relationship between the fracture weakness parameter [8], the Thomsen anisotropy parameters, and fracture density, providing a unified theoretical foundation for describing the anisotropic effects of fractures in a comprehensive manner. These advancements have significantly improved our ability to model and interpret the complex anisotropic behavior of fractured reservoirs, offering valuable insights into reservoir exploration and development (Figure 1) [9]. The azimuthal AVO (Amplitude Versus Offset) inversion has emerged as a powerful and reliable technique for predicting fracture parameters, playing a pivotal role in characterizing subsurface anisotropy and heterogeneity ([PERSON], 2015; [PERSON], 2022; [PERSON], 2024) [10; 11; 12]. Central to the success of azimuthal AVO inversion is the accurate estimation of reflection coefficients, which are highly sensitive to fracture characteristics, such as orientation and density. The response of fractured media is strongly dependent on the orientation of the fractures relative to the seismic wave propagation direction. When fractures are vertically oriented, the medium can be effectively modeled as a horizontally layered transversely isotropic (HTI) medium. Numerous studies have focused on deriving and refining reflection coefficient models for HTI media to better understand the seismic response of such systems ([PERSON], 1997; [PERSON] et al., 2017) [13; 14]. In contrast, when fractures are horizontally oriented, the medium can be treated as a vertically layered transversely isotropic (VTI) medium, and extensive research has been conducted on the reflection coefficients for VTI media, providing insights into the seismic behavior of such systems ([PERSON] and [PERSON], 2004; [PERSON] et al., 2019) [15; 16]. When fractures are inclined, the resulting anisotropy becomes more intricate, and the medium is typically described using the tilted transversely isotropic (TTI) model. This model captures the more complex interactions between fracture geometry and seismic waves, and recent advancements have enhanced the understanding of fracture tilt effects on the elastic properties of the medium ([PERSON] et al., 2021; [PERSON] et al., 2024) [17; 18]. In this study, we derive the reflection coefficients for TTI media, considering the influence of a set of inclined fractures with rotational invariance. Our approach is grounded in the assumption of weak anisotropy and the theory of seismic scattering.