# A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections
## 1 Introduction  
Seismic wave propagation in thin-bed models has been a hot topic of interest to crustal, exploration and engineering seismologists for many years [1; 2; 3]. In geodynamics study, stable deposited basins of the upper crust are thin-bed models under low-frequency seismic detection [1]. In fossil energy exploration, large or relatively simple structural reservoirs have been widely explored and become depleted and exhausted at present [4]. Locating and detecting the existence of stratigraphic lithologic reservoirs are gaining more and more attention. For some sedimentary basins, such as Williston Basin in America [5; 6; 7], Alberta Basin in Canada [8; 9], SongLiao and Tarim Basins in China [10; 11], lithologic reservoirs are composed of thin beds with the thicknesses below the vertical resolution limits [12]. Meanwhile, thin-bed problems are also faced in engineering exploration, such as monitoring CO\({}_{2}\) storage at Sleipner in the Norwegian North Sea [2; 13; 14; 15]. Therefore, seismic responses of a thin bed, which carry information of stratigraphic structure, lithology and pore fluid, are significantly important to crustal, exploration and engineering research. Differed significantly from single-interface cases, seismic responses of a thin bed are composed of reflections from the top and bottom interfaces, including the interbed multiples and converted reflections [16; 17]. Therefore, current AVO analysis techniques based on the Zoeppritz equations and their corresponding approximations for a single interface are generally inefficient for the thin-bed reflections [18]. The reflectivity theory and seismic characteristics of thin-bed models have been discussed for several decades [12; 19; 20; 21; 22; 23; 24; 25]. Normal seismic responses of a thin bed are widely discussed and mainly focused on the relationship of reflected amplitude and thin-bed thickness [12; 24; 25; 26; 27]. For arbitrary incidence, [PERSON] and [PERSON] derived an analytical approximation of a thin bed in acoustic media without considering the interbed multiples [20]. [PERSON] and [PERSON] extended thin-bed AVO analysis from acoustic media to elastic media based on [PERSON] and [PERSON]'s work [21]. [PERSON] et al. gave approximate formulas of thin-bed reflections in elastic media under the assumptions of small incidence angles and weak impedance contrasts [22]. These approximations show as relatively compact forms, but all ignore the contributions of interbed multiples and converted waves. [PERSON] and [PERSON] ever pointed out when the contrast of elastic properties between the thin layer and its surrounding rock increases, it is necessary to consider the contributions of the interbed multiples and converted waves [17]. The contributions of interbed waves on thin-bed reflections have been considered by many researchers. [PERSON] [28] and [PERSON] [29] proposed a matrix method that transfers displacements and stresses through successive layers in multi-layered media. [PERSON] established the propagator matrices of displacements and stresses in multi-layered media as the seismic waves defined by displacement potential functions [19]. [PERSON] et al. presented a closed-form Green's function in the frequency wavenumber domain for a layered, elastic half-space model for SH wave propagation [30]. [PERSON] et al. derived displacement reflection/transmission (R/T) coefficient equations of a thin bed and simplified them into pseudo-Zoeppritz equations under thin-bed assumption [23]. [PERSON] et al. presented reflection coefficients due to incident plane SH-wave at an anisotropic magnetoelastic layer sandwiched between two inhomogeneous viscoelastic half-spaces [31]. [PERSON] et al. gave reflection and transmission of plane waves through isotropic medium sandwiched between two highly anisotropic half-spaces [32]. [PERSON] et al. studied the reflection and transmission of a plane wave through a fluid layer of finite width sandwiched between two dissimilar monoclinic elastic half-spaces [33]. [PERSON] et al.