# Identification of Topographic Seismic Site Periods in Sloping Terrains
## 1 Introduction  
For the seismic design of buildings, defining the maximum demand for acceleration and displacement and identifying the periods during which these maximums occur play an important role in the development of the design response spectra for a given condition. The fundamental period of a terrain, \(T_{0}\), is one of the parameters used to define the period of vibration in which the largest peak soil amplification can be expected. Therefore, for terrains with a horizontal surface, different seismic design codes use \(T_{0}\)[1, 2], either directly or indirectly, as one of the parameters to define the design response spectra according to the seismic site classification. Among other factors, it has been established that steep or complex terrain may be subject to potential seismic response amplifications, which would represent an additional demand on the structure compared to that which occurs in a low-slope condition. Recent studies have indicated that seismic amplification is significantly influenced by the input signal frequency and that current standards, such as Eurocode, might underestimate this effect. It suggests that a greater emphasis on topographic characteristics, along with soil properties, should be considered to accurately assess potential earthquake damage [3]. Since the 1960s, seminal works by [PERSON] and [PERSON] [4] and [PERSON] [5] established the importance of understanding topographic effects on seismic site response, particularly in sloping terrains, which has led to both numerical and experimental studies on the topographic effects on seismic site response [6, 7, 8, 9, 10, 11]. Among other factors, it has beenestablished that steep or complex terrain may be subject to potential seismic response amplifications, which would represent an additional demand on the structure compared to that which occurs in a low-slope condition [8; 11; 12; 13; 14]. However, few studies have focused on identifying whether the topography of a site induces an amplification peak associated with a characteristic period on the sloping terrain, which is not observed in the free field on horizontal terrain. In this regard, [PERSON] and [PERSON] [4] suggest that the natural period of the soil column behind the crest of a slope can condition the amplification of motion, even more than the terrain geometry can. On the other hand, several studies agree with the formulation proposed by [PERSON] [5] and [PERSON] and [PERSON] [15], who indicate that the periods of amplification depend on the wavelength-to-ridge dimension ratio. Thus, if the wavelength of the seismic waves is significantly greater than the geometric dimensions of the terrain, there would be no topographic effects associated with it. To identify a characteristic period, \(T_{0}\), in an earth dam, [PERSON] and [PERSON] [16] developed an analytical solution based on a two-dimensional \"shear-beam analysis\". For the case of a dam with a triangular cross-section with a height \(H\) and shear wave velocity \(Vs\), they found that \(T_{0}\sim 2.61\frac{H}{Vs}\). Similarly, based on the study by [PERSON] et al. [14], [PERSON] [17] used the Rayleigh method to propose an analysis for the case of a symmetric triangular terrain with a height of \(H\) and an inclination angle of \(\beta\), where he proposed that \(T_{0}\sim\frac{(Human\beta)}{(0.7 Vs)}\). For the case of a sloping terrain, which is the focus of the present study, [PERSON] et al. [18] used the term \"topographic frequency, \(f_{i}\)\" to describe the frequency of ground motion at which the maximum topographic amplification occurs, which is equal to \(f_{t}=\frac{Vs}{5H}\). They observed this occurs when the \(H/\Lambda\) ratio is 0.2, where \(H\) represents the slope height, and \(\lambda\) denotes the wavelength of the motion.