Mathematical physics presents us with many partial differential equations, the heat flow equation, the acoustic wave equation, seismic, electromagnetic, and others, many whose essential expression is found in the viscous scalar wave equation. Concepts of imaging tell us to identify a linear operator, essentially a matrix, and then use its transpose (adjoint) to make an image.
This paper shows how to build the adjoint of the viscous wave equation from its finite-difference representation. Inversion techniques use the operator and its adjoint to solve a wide variety of problems. Here we cover basic principles only, up to and including the stage of coding the operator and its adjoint and proving the adjoint's precision to be ``full word''.