## 2 Theoretical Background  
### Pile Integrity Test by Impact Echo Method  
The method of pile integrity test (PIT), also known as sonic echo (SE), pulse echo (PE), impulse response (IR), or impact echo (IE) methods, is used for the low-strain integrity testing. The technique is referred to as the low-strain method because the strain induced by the impact is of the order of a few microstrains or less. It is non-destructive testing that is also commonly used for deep foundations, such as bored or drilled shafts, auger cast-in-place piles, driven concrete piles, and concrete-filled pipes [21; 22]. The impact echo method (Figure 1) involves impacting the top of a deep foundation with a hammer to generate a downward traveling compression wave, which reflects back to the surface from changes in stiffness, cross-sectional area, and density. The arrival of the reflected wave energy is sensed by a receiver (accelerometer or vertical geophone). If a defect is present along the shaft, its size and location can be estimated by analyzing the propagation and reflection of the wave signal induced on the foundation by the hammer impact. It is also possible to estimate the depth of the pile toe. Considering the pile to be a one dimensional elastic and isotropic bar, the equation of motion of the stress wave propagating along the axis in a pile-soil system is given by Equations (1) and (2) [23]. \[\frac{\partial p}{\partial t}+\zeta\frac{\partial v}{\partial x}=0 \tag{1}\]  
\[\frac{\partial p}{\partial t}+\zeta\frac{\partial v}{\partial x}=-c\cdot R \tag{2}\]  
where \(p(x,t)\) and \(v(x,t)\) denote the force and particle velocity acting on each pile section, respectively; \(c\) is the wave speed of the pile, which is given by Equation (3); \(\rho\) and \(E\) are the density and Young's modulus of the pile material, respectively; \(\zeta\) is defined as the wave impedance of the pile, \(\zeta=\rho cA\), in which \(A\) is the cross-sectional area; \(R\) is the soil resistance; \(t\) is the time measured from the moment of impact; and \(x\) is the travel time, which is defined by Equation (4). In Equation (4), \(z\) is the depth below the pile top. \[c=\sqrt{\frac{E}{\rho}} \tag{3}\]  
\[x=\int\limits_{0}^{z}\frac{dz}{c} \tag{4}\]  
At low-strain level, the mobilized shaft resistance \(R\) is small and maybe neglected for all practical purposes, in which the low-strain condition is formed as the basis for the integrity testing [1]. With the mobilized shaft resistance being negligible under these conditions, Equation (2) is reduced to Equation (5). \[\frac{\partial\mathrm{p}}{\partial x}+\zeta\frac{\partial v}{\partial t}=0 \tag{5}\]  
Thus, the impedance \(\zeta\) becomes the unique model parameter we seek to recover or interpret, with the wave Equations (1) and (5) now being an inverse medium problem. The change in wave impedance at the defect is mainly due to the change in cross-section area or wave velocity. The changes in impedance can be used to detect potential defects, anomalies, and imperfections, such as major cracks, necking, soil inclusions or voids, low-quality concrete, changes in pile cross-section, and reflections from the pile tip can be used to determine the embedment length. Figure 1: Low-strain pile integrity test by impact echo method. For a perfectly intact and uniform pile with length \(L\) and having no defect or imperfections, the reflected wave from the pile tip propagates a total distance of 2 \(L\), and thus the total travel time \(t\) defined by Equation (4) is given by Equation (6). \[t=\frac{2L}{c} \tag{6}\]  
For the case of cast-in-place concrete drilled piles or bored shafts, the embedment depth or length \(L\) is normally known with sufficient accuracy, such that the propagation time \(t\) in Equation (6) is, therefore, mainly dependent and inversely proportional to the wave velocity of concrete \(c\).