Construction of Equivalent Functions in Anisotropic Radon Tomography

Abstract

We consider a real-valued function on a plane of the form m(x,y,θ)=A(x,y)+Bc(x,y)cos(2θ)+Bs(x,y)sin(2θ)+Cc(x,y)cos(4θ)Cs(x,y)sin(4θ) that models anisotropic acoustic slowness (reciprocal velocity) perturbations. This “slowness function” depends on Cartesian coordinates and polar angle θ. The five anisotropic “component functions” A (x,y), Bc(x,y), Bs(x,y), Cc(x,y) and Cs(x,y) are assumed to be real-valued Schwartz functions. The “travel time” function d(u, θ ) models the travel time perturbations on an indefinitely long straight-line observation path, where the line is parameterized by perpendicular distance u from the origin and polar angle θ; it is the Radon transform of m ( x, y, θ ) . We show that: 1) an A can always be found with the same d(u, θ ) as an arbitrary (B c ,B s ) and/or an arbitrary (C c ,C s ) ; 2) a (B c ,B s ) can always be found with the same d(u, θ ) as an arbitrary A, and furthermore, infinite families of them exist; 3) a (C c ,C s ) can always be found with the same d(u, θ ) as an arbitrary A, and furthermore, infinite families of them exist; 4) a (B c ,B s ) can always be found with the same d(u, θ ) as an arbitrary (C c ,C s ) , and vice versa; and furthermore, infinite families of them exist; and 5) given an arbitrary isotropic reference slowness function m 0 (x,y) , “null coefficients” (B c ,B s ) can be constructed for which d(u, θ ) is identically zero (and similarly for C c ,C s ). We provide explicit methods of constructing each of these “equivalent functions”.