15.1 Minimum variance reference system

The minimum variance reference system, i.e., a reference system with one of its axes aligned with a direction along whit the field has the smallest fluctuations (Sonnerup and Cahill, 1967 ). This method provides information on the spatial distribution of the fluctuations of a given vector.

Given a generic field B(x,y,z), the variance of its components is

2 2 2 2 2 2 < B x > - < Bx > ;< B y > - < By > ;< B z > - < Bz > .

Similarly, the variance of B along the direction $S$ would be given by

VS = < B2S > - < BS >2 .

Let us assume, for sake of simplicity, that all the three components of $B$ fluctuate around zero, then

< Bx >= < By >= < Bz >= 0 ===> < BS >= x < Bx > +y < By > +z < Bz >= 0.

Then, the variance $VS$ can be written as

VS = < B2 >= x2 < B2 > +y2 < B2 > +z2 < B2 > +2xy < BxBy > + S x y z

+2xz < BxBz > +2yz < ByBz >,

which can be written (omitting the sign of average $<>$ ) as

VS = x(xB2 + yBxBy + zBxBz) + y(yB2 + xBxBy + zByBz) + z(zB2 + xBxBz + yByBz). x y z

This expression can be interpreted as a scalar product between a vector $S(x, y,z)$ and another vector whose components are the terms in parentheses. Moreover, these last ones can be expressed as a product between a matrix $M$ built with the terms $2 Bx$ , $2 B y$ , $2 B z$ , $BxBy$ , $BxBz$ , $ByBz,$ and a vector $S(x, y,z)$ . Thus,

V = (S,M S), S

where

( ) x S =_ y z

and

( ) Bx2 BxBy BxBz M =_ BxBy By2 ByBz . 2 BxBz ByBz Bz

At this point, $M$ is a symmetric matrix and is the matrix of the quadratic form $VS$ which, in turn, is defined positive since it represents a variance. It is possible to determine a new reference system $[x,y,z]$ such that the quadratic form $VS$ does not contain mix terms, i.e.,

'2 '2 '2 '2 '2 '2 VS = x Bx + y By + z B z .

Thus, the problem reduces to compute the eigenvalues $c i$ and eigenvectors $V~ i$ of the matrix $M$ . The eigenvectors represent the axes of the new reference system, the eigenvalues indicate the variance along these axes as shown in Figure 102 .

Figure 102:

Original reference system $[x,y,z]$ and minimum variance reference system whose axes are $V1$ , $V2$ , and $V3$ and represent the eigenvectors of $M$ . Moreover, $c1$ , $c2$ , and $c3$ are the eigenvalues of $M$ .

At this point, since we know the components of unit vectors of the new reference system referred to the old reference system, we can easily rotate any vector, defined in the old reference system, into the new one.