# P-Wave Reflection Approximation of a Thin Bed and Its Application
## 1 Introduction  
A large percentage of oil and gas reservoirs exist as thin beds, as their thicknesses are below seismic resolution [1]. Studies show that thin-bed amplitude versus offset/amplitude versus angle (AVO/AVA) responses depend on the thin-bed thickness and may differ significantly from that of a single interface [2, 3]. Therefore, traditional AVO/AVA analysis and inversion methods, which are based on Zoeppritz equations for a single interface, are ineffective for thin-bed seismic simulation and inversion. The main problem associated with thin beds is the interference effects of all the reflecting waveforms, including converted waves and multiples.  
Thin-bed reflection/transmission (R/T) coefficients in isotropic media have been studied for several decades, with discussions mainly focused on the relationship between thin-bed reflections and thickness [4, 5, 6, 7, 8, 9, 10]. For the cases of arbitrary incidence, [PERSON] [11], [PERSON] [12] and [PERSON] [13] studied reflections and transmissions of layered elastic media and provided solutions with matrix propagation. However, the complexity of their propagator matrix limited its application. [PERSON] and [PERSON] [14] and [PERSON] and [PERSON] [3] presented a time-delayed R/T method and deduced that thin-bed reflections were the total stack responses of primary reflections at the top interface and the intra-bed multiples time-delayed to the primary reflection. [PERSON] et al. [15] gave three types of approximate analytical formulas for thin-bed PP-wave reflectivity based on the work of [PERSON] and [PERSON] [3] by ignoring converted waves and intra-bed multiples. [PERSON] et al. [16] and [PERSON] [17] built up the reflection and transmission matrices recursively by starting at the top interface and workingdown through the successive interfaces iteratively. [PERSON] and [PERSON] [10] gave an analytical formula of thin-bed seismic responses in acoustic media by ignoring multiple waves. [PERSON] and [PERSON] [18] extended the work of [PERSON] and [PERSON] [10] into elastic media and used amplitude spectra to achieve thin-bed pre-stack spectral inversion. [PERSON] and [PERSON] [19] utilized [PERSON]'s layered elastic medium theory to develop a three-layer case and pointed out that thin-bed effects are significantly different in elastic media than in acoustic media, especially for small incidence angles.  
The authors mentioned above gave a clear description of thin-bed responses. However, their thin-bed R/T coefficient formulas for arbitrary incidence are generally too complex for applications or have insufficient accuracy in that they disregarded the effects of thin-bed intra-bed multiples and converted waves. [PERSON] et al. [20] gave thin-bed true R/T coefficients defined by displacement and approximated them to be in a quasi-Zoeppritz matrix form under the thin-bed assumption. After that, they [21] gave second-order series approximation for PP-wave reflection of a thin bed and estimated thin-bed properties, including a P-wave impedance ratio and thickness by statistical data. Considering that analytic approximations of reflection coefficients are important in AVO analysis and inversion for a single interface [22; 23], we further simplify second-order series approximation under a thin-bed assumption for thin-bed AVO/AVA analysis and inversion in this paper.  
First, we give a P-wave reflected approximation of a thin bed under a thin-bed assumption based on second-order series approximation. Then, we test the accuracy of the linear analytic approximation in this paper derived through numerical simulations of four representative thin-bed models. Finally, we exhibit the applicability of our approximate formula in thin-bed inversion of synthetic seismic data.