Construction of Equivalent Functions in Anisotropic Radon Tomography

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”.


Introduction
This paper addresses the non-uniqueness of the two-dimensional inverse Radon transform, when the real-valued function ( ) , , m x y θ being transformed is presumed to be anisotropic; that is, varying with polar angle θ as well as with posi- As we will discuss further below, the presence of weak anisotropy results in slowness with three modes of angular variation: isotropic, 2θ and 4θ (with polar angle θ), which are described by a total of five spatially-varying "component functions" [6] [7] [8]. Previously, a modal analysis was used to prove that insufficient information is contained in travel time measurements to uniquely determine all five component functions [9]. Also previously, analytic formula were derived for the spatially-distributed 2θ components equivalent to (in the sense of having the same travel time as) an impulsive isotropic component, and vice versa [10]. While these results indicate that every isotropic mode has a 2θ equivalent, and vice versa, it does not provide a simple method for constructing equivalent modes, and it leaves unresolved the issues regarding 4θ non-uniqueness.
We address these issues here.

Isotropic Travel Time Tomography with the Radon Transform
The line integral is taken over the straight line L with arc length  , parameterized by its perpendicular distance u to the origin and the counter-clockwise angle θ that the perpendicular makes with the x-axis. We limit our discussions here to two-dimensional functions drawn from Schwatz space [11] in which the Fourier transform is a linear isomorphism [12] [13]. This restriction is usually acceptable in geotomography, where slowness functions can be assumed to rapidly decrease towards zero outside of a restricted area of interest. As usual, we will refer to the travel time ( ) 0 , d u θ with 0 θ held constant as a "projection".
The Projection Slice Theorem [14] shows that the one-dimensional Fourier transform u  of a projection, which takes u into u k , is the two-dimensional Fourier Transform of the slowness function, evaluated on a line of angle 0 θ in the wavenumber plane: A fan filter satisfying ( ) 1 F θ ≤ does not increase the overall energy of the Fourier transform, so by Plancherel's theorem [15] the energy in the ( ) The fan filter must also obey the symmetry condition

Anisotropic Travel Time Tomography with the Radon Transform
The slowness function associated with weak anisotropy [ contains three "modes" of angular variation specified by a total of five spatially-varying "coefficient functions". The coefficient function A specifies the iso- The trigonometric functions can be moved outside the Radon transforms, since they do not vary along the transform's straight line integration path: Note that the term ( )

Isotropic Mode Equivalent to an Anisotropic Mode
Consider a case where only the isotropic component function ( ) , A x y is nonzero, and another case where only the 2θ anisotropic component functions, Since B d is just the sum of fan-filtered versions of functions whose Radon transforms are presumed to exist, we are assured that its inverse Radon transform exists, too. Hence, an A can always be found that is equivalent to an arbitrary ( ) Hence, an A can always be found that mimics a set of ( ) , C S C C . We demonstrate Equation (7)

Isotropic Mode Equivalent to an Anisotropic Mode
First, note the identity: Now consider one case where only ( ) , A x y is nonzero: and Note that this approach relies upon the zeros in the would follow. Evidentially, many such pairs of function exist, including: Similarly, the ( ) can always be found that is equivalent to an arbitrary A, and furthermore, that infinite families of such pairs exist. By replacing 2θ with 4θ in the above argument, we conclude that a ( ) , C S C C can always be found that is equivalent to an arbitrary A, and furthermore, that infinite families of such pairs exist. We demonstrate Equations (14) and (16)   to Equation (14) and (16) with n = 2. The equivalence of the travel times has been verified by numerical calculation (not shown).

Two-Theta Mode Equivalent to a Four-Theta Mode, and Vice Versa
We can find the ( ) As an example, we find a ( )

Anisotropic Null Slowness Functions
We have demonstrated in Section 5 that two slowness functions, say This argument can be extended to the 4θ mode simply by replacing 2θ with 4θ.
Hence, given an isotropic reference slowness function

Conclusions
A weakly-anisotropic acoustic slowness function on a plane has three modes of angular variability: isotropic, 2θ with 4θ (where θ is polar angle) that are de-  In geotomography, non-uniqueness is usually handled through the addition of prior information that regularizes the problem by singling out one "best" slowness function from among all slowness functions consistent with the travel time data [18] [19] [20]. The choice of regularizing constraint is informed both by knowledge of the physical system that is being imaged and nature of the Applied Mathematics regularization is preferred over second-derivative regularization, because the latter has an oscillatory response [21]. Smallness regularization [22] of slowness functions believed to contain strong two-theta and four-theta modes should be avoided, because it will tend to roughen them by suppressing the contribution of null components that while not observable, contribute to the smoothness of the slowness function that is being imaged.