I have an equation of the form t=X^T D^-1 X (1), where X is a 3D vector X=[X Y Z] and D^-1 is the inverse of a 3x3 diffusion tensor D=[Dxx Dxy Dxz; Dxy Dyy Dyz; Dxz Dyz Dzz]. X^T means that the vector is transposed.
The 3D vector of X is the 3D distance for a set of points from the origin and t is the time for a pulse to get there.
Assuming that one of the eigenvectors is vertical (Dxz and Dyz are zero), equation (1) becomes
t=Dxx^-1 X^2+Dyy^-1 Y^2+Dzz^-1 Z^2+2 Dxy^-1 X Y (2)
The problem is to find the eigenvectors of D-1.
The principal horizontal axes of D will be those where the non-diagonal term Dxy approaches zero. The principal horizontal axes (x, y) are derived from the geographical system (xg, yg) by a rotation of θ about the z axis (θ is expressed counterclockwise in degrees between x axis and xg axis).
Knowing t and X for a set of points, a linear regression can be applied to equation (2).
My problem is that I assume that a multiple linear regression will be efficient, but I don't know how the angle θ is expressed in equation (2) so as to perform the multiple linear regression for each angle θ and see where Dxy approaches zero.
I would be grateful if anyone can help me with this.