Examples of nonuniqueness for an inverse problem of geophysics

AbstractTwo 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)utt−Δu = f(x, t) in D×[0, ∞), u = 0 for t < 0, uN = 0 on ∂D, D ⊂ R+n ≔ {x : xn > 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 xn = 0, D = {x : aj ≤ xj ≤ bj, 1 ≤ j ≤ n, an = 0}