TITLE:
Construction of Equivalent Functions in Anisotropic Radon Tomography
AUTHORS:
William Menke
KEYWORDS:
Geotomography, Radon Transform, Acoustic Anisotropy, Non-Uniqueness, Null Space
JOURNAL NAME:
Applied Mathematics,
Vol.10 No.1,
January
25,
2019
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 (Bc,Bs) and/or an arbitrary (Cc,Cs) ; 2) a (Bc,Bs) can always be found with the
same d(u, θ) as an arbitrary A, and
furthermore, infinite families of them exist; 3) a (Cc,Cs) can always be found
with the same d(u, θ) as an arbitrary A, and
furthermore, infinite families of them exist; 4) a (Bc,Bs) can always be found with the
same d(u, θ) as an arbitrary (Cc,Cs) , and vice versa; and
furthermore, infinite families of them exist; and 5) given an arbitrary
isotropic reference slowness function m0(x,y), “null
coefficients” (Bc,Bs) can be constructed for which d(u, θ) is identically zero (and similarly for Cc,Cs ). We provide explicit methods
of constructing each of these “equivalent functions”.