Joint consistent mapping of high-dimensional geochemical surveys

Interpolated maps of compositional variables (in wt\% or ppm) should guarantee that results give a consistent composition: i.e., at any location, all variables must be positive and sum up to 100\%, and should preserve certain geochemical links. (Co)kriging after a normal score transformation is a common choice, though it does not ensure honoring any positivity, constant sum or geochemical constraints. Moreover, it requires a huge effort of consistent modeling of (cross-)variograms. A more suitable way is the application of log-ratio techniques. A D-component system can be univocally expressed with (D-1) log-ratios (e.g., the additive logratio transformation with common denominator TiO2). Any one-to-one logratio transformation can be applied to sampled data; variograms can be estimated for logratio scores; and cokriging can be applied. Back-transformed interpolated scores map the original system components. Consistently modeled cross-variograms provide the same interpolated compositions whatever logratio transformation is used. However, compositional variograms are best modeled through variation-variograms, the set of direct variograms of all possible logratios of two components. They carry the same information, and can be modeled in the same way, as a full set of direct and cross-variograms of any logratio transformation, but they have several advantages for computation and interpretation. First, being positive functions, logarithmic goodness-of-fit criteria can be used, which give more importance to shorter distances (influencing the interpolation itself) than to the sill, prone to larger fluctuations. Second, they can be estimated with a minimum of complete observations; zeros, missing values, values below detection limit and other irregular data have thus much less influence. Third, using ratios of components, systematic differences might be filtered out. This is important in large geochemical surveys, where all samples may have not been treated in the same labs with the same analytical techniques. Fourth and last, the fitted LMC can be rank-deficient. This allows to obtain all component maps just cokriging a few individual factors and combining them together (compositional factorial cokriging). These procedures are illustrated with the horizon C Kola data set, with 25 components and 605 samples covering most of the Kola peninsula (Finland, Norway, Russia).