Examples of nonuniqueness for an inverse problem of geophysics

Abstract

Two different velocity profiles and a source term are constructed such that the surface data are the same for all times. The governing equation is c −2( x) u tt − Δu = f( x, t) in D×[0, ∞), u = 0 for t < 0, u N = 0 on ∂D, D ⊂ R + n ≔ {x : x n > 0} . The data are the values u( x, t), ∀ x ϵ S, ∀ t > 0. Here S is the part of ∂D which lies on the plane x n = 0, D = { x : a j ≤ x j ≤ b j , 1 ≤ j ≤ n, a n = 0}.