  Georeferenced compositional data are prominent in many scientific fields and
in spatial statistics. This work addresses the problem of proposing models and
methods to analyze and predict, through kriging, this type of data. To this
purpose, a novel class of transformations, named the Isometric
$\alpha$-transformation ($\alpha$-IT), is proposed, which encompasses the
traditional Isometric Log-Ratio (ILR) transformation. It is shown that the ILR
is the limit case of the $\alpha$-IT as $\alpha$ tends to 0 and that $\alpha=1$
corresponds to a linear transformation of the data. Unlike the ILR, the
proposed transformation accepts 0s in the compositions when $\alpha>0$. Maximum
likelihood estimation of the parameter $\alpha$ is established. Prediction
using kriging on $\alpha$-IT transformed data is validated on synthetic spatial
compositional data, using prediction scores computed either in the geometry
induced by the $\alpha$-IT, or in the simplex. Application to land cover data
shows that the relative superiority of the various approaches w.r.t. a
prediction objective depends on whether the compositions contained any zero
component. When all components are positive, the limit cases (ILR or linear
transformations) are optimal for none of the considered metrics. An
intermediate geometry, corresponding to the $\alpha$-IT with maximum likelihood
estimate, better describes the dataset in a geostatistical setting. When the
amount of compositions with 0s is not negligible, some side-effects of the
transformation gets amplified as $\alpha$ decreases, entailing poor kriging
performances both within the $\alpha$-IT geometry and for metrics in the
simplex.
