Correction of Geometric Anisotropy (in Two Dimensions)

Consider a geometric anisotropy in two dimensions (e.g., the horizontal space). The directional graph of the ranges $a_/alpha$ is elliptical. Let $(x_u, x_v)$ be the initial rectangular coordinates of a point, $/varphi$ the angle made by the major axis of the ellipse with the coordinate axis $Ox_u$, and $/lambda > 1$ the ratio of anisotropy of the ellipse. The three following steps will transform this ellipse into a circle and, thus, reduce the anisotropy to isotropy.

(i)

The first step is to rotate the coordinate axes by the angle $/varphi$ so that they become parallel to the main axes of the ellipse. The new coordinates $(y_1, y_2)$ resulting from the rotation can be written in matrix notation as

where $R_/varphi$ is the matrix of rotation through the angle $/varphi$.

(ii)

The second step is to transform the ellipse into a circle with a radius equal to the major range of the ellipse. This is achieved by multiplying the coordinate $y_2$ by the ratio of anisotropy $/lambda > 1$. The new coordinates $(y_1', y_2')$ are then deduced from the coordinates $(y_1, y_2)$ by the matrix expression

(iii)

The initial orientation of the coordinate system is then restored by rotation through the angle $-/varphi$, and the final transformed coordinates $(x_u', x_v')$ are given by

The final transformed coordinates $(x_u', x_v')$ can then be derived from the initial coordinates $(x_u, x_v)$ by the transformation matrix $A$ which is the product of the three matrices $R_{-/varphi}/lambda R_{/varphi}$, i.e.,

with

and

If ${/boma h}$ is any vector in the two-dimensional space with initial coordinates $(h_u, h_v)$, then, to obtain the value of the anisotropic semi-variogram $/gamma({/boma h}) = /gamma(h_u, h_v)$, we first calculate the transformed coordinates $(h_u', h_v')$ from

and then we substitute these coordinates in the isotropic model $/gamma'(/vert{/boma h}/vert)$, which has a range equal to the major range of the directional ellipse, i.e.,