Distance difference representations of Riemannian manifolds

Let $M$ be a complete Riemannian manifold and $F\subset M$ a set with a nonempty interior. For every $x\in M$, let $D_x$ denote the function on $F\times F$ defined by $D_x(y,z)=d(x,y)-d(x,z)$ where $d$ is the geodesic distance in $M$. The map $x\mapsto D_x$ from $M$ to the space of continuous functions on $F\times F$, denoted by $\mathcal D_F$, is called a distance difference representation of $M$. This representation, introduced recently by M. Lassas and T. Saksala, is motivated by geophysical imaging among other things. We prove that the distance difference representation $\mathcal D_F$ is a locally bi-Lipschitz homeomorphism onto its image $\mathcal D_F(M)$ and that for every open set $U\subset M$ the set $\mathcal D_F(U)$ uniquely determines the Riemannian metric on $U$. Furthermore the determination of $M$ from $\mathcal D_F(M)$ is stable if $M$ has a priori bounds on its diameter, curvature, and injectivity radius. This extends and strengthens earlier results by M. Lassas and T. Saksala.