Comment on ‘The magnetotelluric phase tensor’ by T. Grant Caldwell, Hugh M. Bibby and Colin Brown Free

Given the magnetotelluric transfer function Z = X+iY the phase tensor is given by

This real second rank tensor can also be expressed in term of three rotational invariants and one rotation angle. In the appendix of Caldwell et al. (2004) the formulae to calculate the invariants Φ_max, Φ_min and β are given as an equivalent definition for Φ_max and Φ_min is given by Bibby (1986, 2005), In certain cases the determinant of the phase-tensor can become negative. The recommendation by Caldwell et al. is to use the absolute value in these cases instead to avoid problems while calculating the intermediate quantity Φ₂ which should be real. Through this manipulation the expressions for Φ_max and Φ_min become incorrect for . Short algebraic manipulation of eq. (5) shows that in this case and similarly for Φ_min. There are two simple ways to avoid problems with negative determinants: Either use the formulae (10) and (11) given by Bibby (2005) that avoid the calculation of the determinant altogether, or circumvent the calculation of Φ₂ by inserting (3) into (5) and (6). The resulting expressions are and are correct in all cases.