Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticity

TL;DR

The work advances the inverse problem for elastic media by extending Dix’s sphere-data reconstruction from Riemannian to Finsler geometry. It develops a constructive ODE framework along a reference geodesic to recover the curvature operator $R(t)$ and Jacobi fields from forward-sphere data, even in the absence of isotropy. When the data on $U$ agrees under an isometry, the paper proves that the curvature and Jacobi evolution agree along backward geodesics and that surface-normal coordinates yield a determined local metric, enabling a local reconstruction of the Finsler structure. This furnishes a practical path to inferring anisotropic elastic parameters from wave-front data and highlights the role of directional geometry in seismology and elasticity applications, where qP-waves propagate along Finsler geodesics. The results thus provide a rigorous, locally constructive bridge between measured sphere data and the local geometric and material parameters of anisotropic media.

Abstract

Dix formulated the inverse problem of recovering an elastic body from the measurements of wave fronts of point sources. We geometrize this problem in the context of seismology, leading to the geometrical inverse problem of recovering a Finsler manifold from certain sphere data in a given open subset of the manifold. We solve this problem locally along any geodesic through the measurement set.

Figures (3)

• Figure 1: The setting of theorem \ref{['thm:RU']}: We start from a point $x\in U$ and a unit vector $v\in T_xM$ at time $t=0$ and follow the geodesic backwards into the unknown $M\setminus U$. At all times $t$ we have the normal plane $N_t$ to $\dot\gamma(t)$ as a subset of $T_{\gamma(t)}M$. We can identify all these planes canonically with $N_0$ through parallel transport. This identification makes the operators act on just $N_0$ instead of a bundle of planes along $\gamma$. The data we use is the visible smooth surfaces normal to the geodesic near the initial point $x$, pictured in light gray.
• Figure 2: The setting of theorem \ref{['thm:Riemann']}: The surface normal exponential map of a visible smooth sphere $\Sigma$ follows geodesics from $\Sigma$ backwards into the manifold. If the geodesics are lifted to the tangent bundle (indicated with gray arrows), the "lifted normal exponential map" $\overline{\varphi_\alpha}\colon\Omega\times\mathbb{R}\to TM$ is always an immersion. The so defined (local) submanifold of $TM$ is always smooth but the usual normal exponential map $\varphi_\alpha$ fails to be a local diffeomorphism at the center point of the sphere (on the right) and points conjugate to it (between the center and $\Sigma$). These points are focal to $\Sigma$. At these conjugate points the submanifold of $TM$ has a small projection to the base but has many points in the same fiber, illustrated by several arrows at the intersection points. The gray arrows define the vector field $G$, and it is a local vector field on $M$ when the surface normal coordinates actually give local coordinates.
• Figure 3: Setting up the initial conditions begins by setting the matrix $\mathbf{j}$ to be the identity on the line $L_0$. The derivative $\mathbf{y}$ is determined by the shape operator included in the data. We then avoid the conjugate locus as follows. Instead of using the whole $t$-axis which contains conjugate points at $t=t_i$, we only use the segments $L_i$ on that axis. To work around the conjugate point at $(0,0)$, we take the values of $\mathbf{j}$ and $\mathbf{y}$ on the lower part of $L_0$ and solve the Jacobi equation to propagate the data to the upper part of the $L_1'$, which is depicted as light green with an arrow. We extend the values of $\mathbf{j}$ smoothly to the whole of the shifted line $L_1'$ and choose $\mathbf{y}$ as required by the data. Once we are past the conjugate point, we repeat the same process to get back to the $t$-axis. This is repeated at all conjugate points. Solving the Jacobi equation behaves well across conjugate points, whereas smooth extension does not. This way we got to prescribe the initial data smoothly without worrying about consistent singular behaviour at conjugate points.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 5
• Theorem 6
• Remark 7
• Remark 8
• Lemma 9
• proof