Appendix B L2 Inverse Theory With Complex Quantities

APPENDIXB L2 INVERSE THEORY WITH

COMPLEX QUANTITIES

Some inverse problems (especially those that deal with data that has been operated on by Fourier or other transforms) involve complex quantities. Many of these problems can readily be solved with a simple modification of the theory (the exception is the part involving inequality constraints, which we shall not discuss). The definition of the L 2 norm must be changed to accommodate the complex nature of the quantities. The appropriate change is to define the squared length of a vector v to be "v,,~ = vHv, where vH is the Hermetian transpose, the transpose of the complex conjugate of the vector v. This choice ensures that the norm is a nonnegative real number. When the results of L 2 inverse theory are rederived for complex quantities, the results are very similar to those derived previously; the only difference is that all the ordinary transposes are replaced by Hermetian transposes. For instance, the least squares solution is (B.I) 275

Appendix B

276

Note that all the square symmetric matrices of real inverse theory now become square Hermetian matrices: [cov x]i)

=

J [Xi -

(Xi) ]H[Xi - (Xi) ]P(x) dx, [GHG], [GGH], etc.

(B.2)

These matrices have real eigenvalues, so no special problems arise when deriving eigenvalues or singular-value decompositions. The modification made to Householder transformations is also very simple. The requirement that the Hermetian length of a vector be invariant under transformation implies that a unitary transformation must satisfy THT = I. The most general unitary transformation is therefore T = I - 2vvHjvHv, where v is any complex vector. The Householder transformation that annihilates the jth column of G below its main diagonal then becomes

° 'r,

=

°

{I - 10'1(10'1 ~ 1Gj)

where 10'1 =

* means

~

Gj,j -

0' [0, ... ,0, GZ - 0'*,

G!+I.j • . . .

,Gtl}

Gj + 1 ,J

±

'-j

1Gil

(B.3)

"complex conjugate," and the phase of a is chosen to be 1! away from the phase of Gj j • This choice guarantees that the transformation is in fact a unitary transformation and that the denominator is not zero.