## 3 Indexes for Gravel Road Quality Assessment  
The fundamental concept of road surface profiles at different mechanical vibration levels is described by the international standard ISO 8608 [69]. The PSD of vertical road profile displacement, \(G_{d}(n)\), or its rate of change by velocity, \(G_{v}(n)\), or acceleration, \(G_{a}(n)\), conforming to spatial frequency, \((n)\), are used for defining the degree of roughness:\[G_{d}(n)=G_{d}(n_{0})\left(\frac{n}{n_{0}}\right)^{-w}, \tag{1}\] \[G_{v}(n)=(2\pi n)^{2}.G_{d}(n),\] (2) \[G_{a}(n)=(2\pi n)^{4}.G_{d}(n), \tag{3}\]  
where \(n\) is the spatial frequency (cycles/m); \(n_{0}\) is the reference spatial frequency (0.1 cycles/m); \(w\) is the pavement waviness indicator, or the exponent of the fitted PSD. For general use, the road roughness indicator usually takes the values of \(1.75<w<2.25\)[70] and for ISO classification assuming constant velocity PSD, \(w=2\). The profile of gravel pavement cannot be approximated with the waviness of \(w=2\) used in ISO 8608, which usually occurs on paved roads [46]. [PERSON] [71] reported that up to 40% of roads had differences in wavebands, and 20% of tested sections had short wave unevenness (\(w>2\)). Another type of specific road usually used for evaluating vehicle dynamic response, known as Belgian paving, also has few brakes in its displacement spectral density characteristic [72]. There, the wavenumber of 6 cycles/meter corresponds with the size of cobbles forming the pavement [73]. Generally, PSD expresses power in a signal per unit frequency [74]. The PSD of road profile displacement can also be expressed by an unevenness index \(C\)[75]:  
\[G_{d}(n)=C.n^{-w}. \tag{4}\]  
Wavelength \(\lambda\) (m) has an inverse value of spatial frequency. In the time domain, frequency is expressed as the ratio between wavelength and vehicle speed \(v\) (m/s), which generally satisfies conditions for light vehicles [76; 77]:  
\[0.5\;\mathrm{Hz}<\frac{v}{\lambda}<15\;\mathrm{Hz} \tag{5}\]  
In the context of vehicle dynamic response and vibration response analysis, it is more convenient to convert excitation from road pavement spatial frequency \(n\) (cycles per meter) to temporal frequency \(f\) (cycles per second or Hz). In this case, the supposed vehicle speed (\(v\)) is involved [78]:  
\[f=n.v. \tag{6}\]  
Then, the PSD of the road pavement, in consonance to temporal frequency \(G_{d}(f)\), is adapted to vehicle speed:  
\[G_{d}(f)=\frac{G_{d}(n)}{v}. \tag{7}\]  
The IRI is the most frequently used single-number indicator usually implemented in road maintenance management plans, ride comfort, or stability assessments. The IRI value is estimated from the reference quarter car model (Figure 1a), also known as the Golden car model [39], driving at 80 km/h. The highest sensitivity for IRI computation was found from 0.065 to 0.42 cycles/m (wavelength from 15 to 2.4 m) (Figure 1b), which also varied from the selected sample interval and integration method [79]. The general IRI computation is based on two components as the velocity of suspension displacement [79; 80]:  
\[\mathrm{IRI}=\frac{1}{L}\int_{0}^{L/v}\abs{\dot{z}_{sm}-\dot{z}_{usm}}dt, \tag{8}\]  
where \(L\) is the length of the evaluated road profile (m), \(v\) is the longitudinal vehicle speed applied for the quarter car model (m/s), \(\dot{z}_{sm}\) is the the velocity of sprung mass (SM) vertical displacement (m/s), \(\dot{z}_{usm}\) is the the velocity of unsprung mass (USM) vertical displacement (m/s), and \(dt\) is the time increment (s). The dynamic load coefficient (DLC) has been used for evaluating vehicle response to ride safety and driving stability.