Spatial analysis of modern soil compaction roller measurement values

Modern compaction rollers monitor soil properties by observing vibrational characteristics of the soil. A vibrating drum traverses the compaction site measuring soil stiffness and collecting GPS coordinates that are together termed roller measurement values (RMVs). These RMVs can be modeled as a random spatial field and additively decomposed into any sensible combination of mean terms, spatial terms, spline terms, and ridge regression terms. The goal of this modeling is to implement intelligent compaction for quality control and quality assurance purposes. Proper modeling of such data (stationarity, anisotropy,.) is then of paramount concern. Each layer of the compaction site can be modeled by the n-vector y = Xβ+α+γ+ ɛ, where Xβ is a low-order (linear) polynomial trend, α is a mean term estimated using ridge regression or splines modeling the large-scale variation, γ is a (zero-mean Gaussian) spatial process modeling the small-scale variation, and ɛ is the noise. Here, X is the (n × p) design matrix with rank p. There are many general approaches to working with such an additive mixed model, including a backfitting procedure for maximum-likelihood estimation and generalized cross-validation. Due to computational complexity of maximum-likelihood estimation a backfitting procedure, Furrer and Sain (2009) [1], was extended to the more general models used here and employed in the estimation. The extended backfitting procedure has been shown to converge and the iterative least-squares estimates have been shown to converge to the generalized least-squares estimate. A simulation study has been conducted to analyze estimates of this general model using a penalized likelihood and generalized cross-validation (GCV) approach as well. Results of the cross-validation study using a spline structure indicate there are some random fields that can be generated that do not have a minimum GCV.