Inversion of Higher Dimensional Radon Transforms of Seismic-Type
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Inversion of Higher Dimensional Radon Transforms of Seismic-Type Hiroyuki Chihara1 Received: 14 December 2019 / Accepted: 7 July 2020 / © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of the independent variables, and are imposed some symmetry with respect to the axes. These transforms are higher dimensional version of generalization of the parabolic Radon transform and the hyperbolic Radon transform arising from seismology. We prove the inversion formulas for these transforms under some vanishing and symmetry conditions of functions. Keywords Radon transform · Inversion formula · Seismology Mathematics Subject Classification (2010) Primary 44A12 · Secondary 65R10 · 86A15 · 86A22
1 Introduction Let n be a positive integer. Fix arbitrary c = (c1 , . . . , cn ) ∈ Rn and α1 , . . . , αn , β > 1. Set α = (α1 , . . . , αn ) for short. Let (x, y) = (x1 , . . . , xn , y), (s, u) = (s1 , . . . , sn , u) ∈ Rn+1 = Rn × R be independent variables of functions. We study the inversions of the integral transforms Pα f (s, u), Qα f (s, u) and Rα,β f (s, u) of a function f (x, y). These are the integrations of f (x, y) on some special families of hypersurfaces. We do not deal with the general hypersurfaces. The precise definitions of our transforms are the following. Firstly, Pα f (s, u) is defined by n αi Pα f (s, u) = f x, si |xi − ci | + u dx Rn
i=1
Hiroyuki Chihara
[email protected] 1
College of Education, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan
H. Chihara
which is
Pα f (s, u) =
Rn
x + c,
f
n
si |xi | + u dx. αi
i=1
Pα f (s, u) is the integration of f on a hypersurface P (α; s, u) = (x, y) ∈ R
n+1
: y=
n
si |xi − ci | + u , αi
i=1
and dx is not the standard volume of P (α; s, u) induced by the Euclidean metric of Rn+1 . P (2, . . . , 2; s, u) is a paraboloid if si = 0 for all i = 1, . . . , n. In particular, when n = 1, P2 f (s, u) is an integration over a parabola in R2 , and is called the parabolic Radon transform of f in seismology. We assume that f (x1 , . . . , xi−1 , −xi +ci , xi+1 , . . . , xn , y) = f (x1 , . . . , xi−1 , xi +ci , xi+1 , . . . , xn , y) (1) for i = 1, . . . , n, that is, f (−x + c, y) = f (x + c, y). If we split f (x) into the even and odd parts in some xi with respect to the hyperplane xi = ci in Rn , then the contribution of the odd part to Pαf becomes 0, and the injectivity of Pα fails to hold. Secondly, Qα f (s, u) is defined by n Qα f (s, u) = f x, si (xi − ci )|xi − ci |αi −1 + u dx Rn
which is
i=1
Qα f (s, u) =
Rn
f
x + c,
n
si xi |xi |
αi −1
+ u dx.
i=1
Qα f (s, u) is the integration of f on a hypersurface n n+1 αi −1 : y= si (xi − ci )|xi − ci | +u Q (α; s, u) = (x, y) ∈ R i=1
for f and the measure i
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