2 Pavement surface

Irregularities in pavement surface are often called rough- ness. All pavement surfaces have some irregularities. The National Cooperative Highway Research Program defines ‘‘roughness’’ as ‘‘the deviations of a pavement surface from a true planar surface with characteristic dimensions that affect vehicle dynamics, ride quality, and dynamic pave- ment loads [21].’’ Factors contributing to roughness, in a general sense, include vertical alignment, cracks, joints, potholes, patches and other surface  distresses.

Newly constructed pavements may be poorly finished or have design features such as construction joints and ther- mal expansion joints, which can be main sources of vehicle vibration [22, 23]. Measurements show that pavement roughness can be modeled as a random field consisting of different wavelength. Considerable effort has been devoted to  describing  pavement  surface  roughness  [24–26].  The peak-to-valley measurements, the average deviation from a straight edge, and the cockpit acceleration are several distinct approaches that have been suggested for pavement surface characterization [27]. Practical limitations of these three approaches could be found in the study of Hsueh and Penzien [28].

Spectrum of a deterministic function referring to its Fourier transform reveals sinusoid components that a deterministic periodic or nonperiodic function is composed of. Spectrum analysis of a deterministic function f ðtÞ can be obtained if and only if this deterministic function sat- isfies Dirichlet condition and absolute integrability condi- tion [29]:

Pavement surface as a random field of elevation does not decay as spatial coordinates extend to infinity; therefore, it does not satisfy inequality (1). For this reason, taking Fourier analysis to a sampled pavement surface does not make too much sense in theory. However, the correlation vector of random field does decay as spatial coordinates extend to infinity. Hence, applying Fourier transform to correlation vector of pavement surface, commonly known as power spectral density, does serve the purpose of spectrum analysis in theory [30]. As such, mathematical description of pavement surface takes place under the theoretical framework of spectral analysis of stochastic process.

The following subsections consist of a general mathe- matical framework for describing pavement surface roughness, a review of statistical description of    pavement surface   roughness,   a  description   of   periodic   joints of According to the Winner–Khintchine theory [31], the following expressions constitute a pair of Fourier transform:

2.1  One-dimensional description

Given the manifest complexity exhibited in pavement surface as a random field, making some assumptions becomes indispensable to simplify mathematical descrip- tion of pavement surface. Commonly used assumptions on surface roughness are that surface roughness is an ergodic and homogeneous random field with elevation obeying Gaussian distribution [31–33]. The assumption of ergo- dicity makes sure that the temporal average of a sample of stochastic process equals to the statistical mean of sto- chastic process,  which enable  one  to obtain the statistical where X represents the distance between any two points along the road. Wavenumber spectrum, SnnðXÞ, is the direct PSD function of wavenumber X , which represents spatial frequency defined by X = 2p/k where k is the wavelength of roughness. Under the assumption of homogeneity, spatial autocorrelation function RnðXÞ is defined by for any x1 and X. Here E½·] is the expectation operator and can be calculated  by characteristics of a stochastic process by measuring only a few  samples.  The  assumption  of  homogeneity    ensures