Fitting Laplace Process Parameters for Non-equidistant Road Roughness Data

Road roughness is the most important source of vertical loads for road vehicles. To capture this during durability engineering, various mathematical models for describing road profiles have been developed. The Laplace process has turned out to be a suitable model, which can describe road profiles in a more flexible way than e.g., Gaussian processes. The Laplace model essentially contains two parameters called C and ν (to be explained below), which need to be adapted to represent a road with certain roughness properties. Usually, local road authorities provide such properties along a road on sections of constant length, say, 100 m. Often the ISO 8608 roughness coefficient or the IRI (International Roughness Index) are used. In such cases, there are well known explicit formulas for finding the parameters C and ν of the Laplace process, which best fits the road under certain assumptions. Besides local road authorities there are also other sources of roughness data, for instance commercial providers of digital maps. Sometimes, the information is given for road sections of varying length such that these formulas do not apply anymore. Therefore, this paper suggests a method to fit the Laplace parameters in the more general setting of non-equidistant roughness data. Neither the roughness indicators nor the Laplace model are modified, just the process of parameter fitting is adapted. The task is reduced to solving a maximum-likelihood problem. In addition, an explicit formula approximating the maximum-likelihood solution for the Laplace parameters C and ν is derived. This extends the explicit formulas known for data on an equidistant grid. The methods and results are validated based on roughness data for roads of a total length of 1270 km. Road surface properties are also highly relevant for vehicle vibration, ride comfort, or handling. However, this is not considered further in this paper.