ABSTRACT
In this sense, the symbols x and y may represent various physical quantities, not their values. We say equivalently that we invert a set of equations.
We need to distinguish the true value of x from the retrieved value, which I denote
by . Throughout this book, I represent a retrieved or estimated quantity with a superior hat,3 and a measurement that includes noise, with a prime. The recipe for calculating from y' is an algorithm: a prescribed sequence of arithmetic operations that will, after a finite number of operations, produce an answer. Equation (1) is linear; the inversion algorithm will also be linear. Even within this limited framework, there is a lot to be said about different retrieval methods, or different inversion algorithms. If we replace equation (1) with a nonlinear vector equation, we may want to find a linear inversion algorithm, or we may want to find a nonlinear one.
