VLIDORT User's Guide

Here, E is the 4 x 4 identity matrix, and E ~ the 4N x 4N identity matrix. The (⊥) superscriptindicates the conjugate transpose. The link between the eigenvector X ~α and the solution vectorsin Eq. (2.35) is through the auxiliary equations:~ ± ~ −1α = MW12⎡~1⎢E±⎣ kα~S+⎤ ~⎥X⎦α. (2.44)Eigenvalues occur in pairs { ± kα}. As noted by Siewert [Siewert, 2000b], both complex variableand real-variable eigensolutions may be present. Left and right eigenvectors share the samespectrum of eigenvalues. Solutions may be determined with the complex-variable eigensolverDGEEV from the LAPACK suite [Anderson, et al., 1995]. DGEEV returns eigenvalues plus leftandright-eigenvectors with unit modulus.In the scalar case, the formulation of the eigenproblem is simpler (see [Spurr, 2002] forexample). The eigenmatrix is symmetric and all eigensolutions are real-valued. In this case, theeigensolver module ASYMTX [Stamnes et al., 1988] is used. ASYMTX is a modification of theLAPACK routine for real roots; it delivers only the right eigenvectors. For the vector case, thereare circumstances (pure Rayleigh scattering for example) where complex eigensolutions areabsent, and one may then use the faster ASYMTX routine. We return to this point in section3.4.3.The complete homogeneous solution in one layer is a linear combination of all positive andnegative eigensolutions:~ ~I ; (2.45)~4N~ +~ −∑{ LαWαexp[ −kαx]+ MαWαexp[ −kα( Δ x)]}++( x)= D−α = 1~4N~ −~ +∑{ LαWαexp[ −kαx]+ MαWαexp[ −kα( Δ − x)]}−I−( x)= D. (2.46)α = 1~ −~ + ~Here, D = diag{D,D,...,D}and D = E. The use of optical thickness Δ − x in the secondexponential ensures that solutions remain bounded [Stamnes and Conklin, 1984]. The quantities{ Lα, M α} are the constants of integration, and must be determined by the boundary conditions.In equations (2.45) and (2.46), some eigensolutions will be complex, some real. It is understoodthat when we use these expressions in the boundary value problem (section 2.3.1), we computethe real parts of any contributions to the Stokes vectors resulting from complex eigensolutions.~Thus if { k +α, W α} is a complex solution with (complex) integration constant Lα , we require:~ − −k~~α x− −kαx− −kαxRe[ LαWαe ] = Re[ Lα]Re[ Wαe ] − Im[ Lα]Im[ Wαe ]. (2.47)From a bookkeeping standpoint, one must keep count of the number of real and complexsolutions, and treat them separately in the numerical implementation. In the interests of clarity,we have not made an explicit separation of complex variables, and it will be clear from thecontext whether real or complex variables are under consideration.2.2.2. Linearization of the eigenproblemWe require derivatives of the above eigenvectors and separation constants with respect to someatmospheric variable ξ in layer n. From (2.38) and (2.39), the eigenmatrix Γ ~ is a linear function21