## 1 Introduction  
Piles are widely used as the foundations of infrastructures. The seismic response of piles plays an important role in the aseismic design of superstructures [1; 2; 3; 4; 5; 6; 7; 8]. The investigation of the pile-soil dynamic interaction in the absence of a superstructure under a seismic wave incidence, known as a kinematic interaction, has been the focus of numerous studies [9; 10; 11; 12]. Various theoretical approaches have been developed to investigate the lateral kinematic interaction of the pile-soil system. However, less attention has been paid to the vertical kinematic interaction. The vertical seismic motion at the pile head is often assumed as the free-field soil motion in the practical design of the pile-soil system. In fact, the scattering of seismic waves induced by the pile foundation will significantly modify the seismic response, which allows the pile motion to vary from the free-field motion. Moreover, from the observation evidence from the Northridge and Kobe earthquakes, [PERSON] and [PERSON] [13] found that many failure modes are mainly caused by vertical earthquake motion. Hence, investigations of the vertical kinematic response of the pile, which is typically induced by the vertical P-wave, are an important research subject. Over the past few decades, methods have been developed to investigate the vertical kinematic response of the pile. [PERSON] and [PERSON] [14] investigated the seismic response using the boundary element method. [PERSON] and [PERSON] [15] developed a boundary integral equation method to investigate the dynamic response of a thin-walled pile embedded in isotropic soil subject to a vertical incidence of the P-wave. Later, this work was extended to the study of a pile embedded in a transversely isotropic half-space with the aid of ring-loadsGreen's functions [5; 16]. Pioneered by the work of [PERSON] and [PERSON] [17], [PERSON] and [PERSON] [18] developed a rod-on-dynamic Winkler foundation model to investigate the kinematic pile response under a vertical incidence of the P-wave. Based on the pioneering work by [PERSON] and [PERSON] [19], [PERSON] et al. [20] proposed a continuum elastodynamic solution to investigate the vertical kinematic pile-soil interaction. Then, this method was extended to investigate the kinematic response of piles considering three-dimensional wave scattering [9; 21]. Later, the solution of floating piles was proposed [22]. Based on a foundation model proposed by [PERSON] and [PERSON] [23], [PERSON] et al. [24] and [PERSON] et al. [25] developed a new continuum elastodynamic solution to investigate the vertical kinematic pile-soil interaction. It is worth noting that the surrounding soil in most of the previous studies was assumed to be homogeneous or layered homogeneous. However, due to the construction disturbance effect, the soils around the pile are squeezed and pushed away, and the soil properties vary gradually within the region adjacent to the pile in the radial direction. The effects of pile driving on soil properties have been investigated for the past 50 years [26; 27; 28; 29; 30]. Many of the previous studies (including field measurements of shear-wave velocity, field vane shear tests, and unit weight comparisons) have addressed soil properties obtained before and after pile installation [31; 32]. All studies have shown that the soil properties are changed in the radial direction during pile-sinking [27]. Over the last few decades, the construction effects have been considered in the study of the inertial response of piles. [PERSON] and [PERSON] [33] proposed an analytical model to study the vertical and torsional vibration of piles, which allows the soil properties to change continuously within the disturbed zone in the radial direction. However, these methods can only simulate the case that the soil properties change in a particular form.