Let us assume that I am trying to represent a complex-valued variable, such as electric permittivity in a lossy medium (ε_c = ε′ + iε′′), as a random variable. For example, this lossy medium is a rectangular-shaped tunnel. I want to know what steps or considerations are needed when:

(1) Representing the complex-valued random variable using the Karhunen–Loève expansion, as opposed to a real-valued random variable. What is done differently for the complex case?

(2) Applying this approach specifically to a tunnel environment, such as a rectangular tunnel, for modeling spatially varying random permittivity. Are there special considerations or modifications needed for tunnel geometries?

Please provide any mathematical, computational, or practical details that researchers may need to answer these aspects.