## 4 Methods and the Representation of Results  
To provide better control over the hammer impact and to prevent the hopping effect, a test automation device (TAD) was designed and 3D printed. The designed device is depicted in Figure 1, in the configuration for in situ measurements. The device consists of a main structure, a sleeve for attaching the hammer, a support structure, a counter-reaction spring, and a force regulator. The hammer is attached to the structure by a sleeve, which is pivoted to the main structure so that it can rotate around a single axis. The main structure integrates a mechanism for impact force regulation, which essentially consists of an arc with an adjustable circumference that has the function of a hard stop, that is, a limit for the rotation of the hammer and sleeve. Its effect is to limit the maximum height at which the operator can lift the hammer and, thus, the resulting impact force. Indeed, the measurement is made by manually lifting the hammer to the angle allowed by the limit and letting it fall back on the impedance head. The counter-reaction spring is properly attached to the hammer and to the structure to provide the appropriate counter-reaction and prevent hopping after the initial impact. As can be seen in Figure 2, there are actually two impact angles, the angle between the axis of the hammer handle and the ground plane and the angle between the axis of the hammer head and the vertical axis. The control of those angles is achieved by adjusting the height of the main structure along the bracket and adjusting the rotation of the hammer in the sleeve seat. A typical DS spectrum obtained with the hammer in combination with the TAD is reported in faded gray in Figure 3. The figure presents the three spectra corresponding to three independent in situ measurement sessions on the same road surface point. In light  
Figure 1: Example of the experimental setup in the case of in situ measurement. The impedance head is attached to the ground via a threaded mounting stud in this case. The impact hammer, the impedance head, and TAD are depicted. For illustrative purposes, the optional weight was included for the hammer. blue, the medium 1/3 octave band spectrum of the three results is reported, while in orange, the sigmoidal model proposed by [36] is shown. Below the 1 kHz band, there is a complete overlapping between the 1/3 spectrum, the model, and the three fine band spectra. For higher frequencies, the three fine band spectra are subjected to variation from the model, but they overlap themselves. Starting from 2500 Hz and above, the three spectra begin to be subjected to random oscillations which are not overlapping. Otherwise, the medium 1/3 octave band spectrum mitigates this behavior, exhibiting very good agreement with the model. Based on that evidence, the authors propose representing the DS spectrum results in 1/3 octave bands to enhance the readability of the results and to facilitate fitting with the sigmoidal model from the literature (Equation (8)). The DS third-octave spectrum is obtained by averaging the spectra of three independent measurements performed on the same measurement point. The spectrum of each measurement is calculated from the narrow-band spectrum obtained from ten consecutive impacts following the ISO 7626-5:2015 [34]. The error bars associated with the band's value of the medium spectrum represent the standard deviation between the three original spectra. Figure 3: Fit of the trend in 1/3 octave bands with the sigmoid model. Shaded in gray are the frequency spectra of the stiffnesses measured in the three individual measurements. At a low frequency, the unmediated trend is in agreement with the model. At a high frequency, the trends of the three measurements are no longer superimposable, but averaging over the band still allows for good agreement with the model. Figure 2: Depiction of the angles of impact.