On passive recovery of structured elastic density and initial states
Abstract
We study simultaneous recovery of the (variable) mass density, initial displacement, and initial velocity for the three-dimensional isotropic elastic wave equation with known constant Lamé parameters. The data are the complete displacement trace on an enclosing boundary. We first assume that the density-weighted initial displacement and velocity have fixed known profiles in one spatial direction. The s0 and s1 coefficients of the zero-frequency Laplace expansion identify these weighted states. The s2 and s3 coefficients then give static Lamé orthogonality identities for the density difference. The exact difference expansion through order s3 has an O(s4) remainder, uniformly on bounded spatial sets and bounded density-contrast and weighted-state classes. Under alignment of the two initial states and a nonzero moment of the density-weighted initial velocity, the two density identities reduce to a constant-vector static transform. Two opposite elastic null phases give uniqueness for one or two fixed vertical density profiles when the profile family is two-sided Laplace nondegenerate. When λ+μ≠0, each nonzero normal root has partial multiplicities $2$ and $1$ and admits a length-two Jordan chain. The associated polynomial--exponential Lamé mode produces derivatives of the bilateral profile transforms. The resulting Hermite--Laplace system gives uniqueness for aligned classes with up to four fixed vertical profiles and independent horizontal coefficients, provided that the profile system is Hermite--Laplace nondegenerate. Distinct translations of one compactly supported profile provide an explicit four-profile class. We also prove rigidity of the alignment reduction and exhibit an infinite-dimensional kernel for the reduced transform on unrestricted densities.
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BibTeX
@misc{gao2026passive,
title = {{On passive recovery of structured elastic density and initial states}},
author = {Yixian Gao and Hongyu Liu and Yang Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14215},
eprint = {2609.14215},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14215},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On passive recovery of structured elastic density and initial states
AU - Yixian Gao
AU - Hongyu Liu
AU - Yang Liu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14215
UR - https://arxiv.org/abs/2609.14215
AB - We study simultaneous recovery of the (variable) mass density, initial displacement, and initial velocity for the three-dimensional isotropic elastic wave equation with known constant Lamé parameters. The data are the complete displacement trace on an enclosing boundary. We first assume that the density-weighted initial displacement and velocity have fixed known profiles in one spatial direction. The s^0 and s^1 coefficients of the zero-frequency Laplace expansion identify these weighted states. The s^2 and s^3 coefficients then give static Lamé orthogonality identities for the density difference. The exact difference expansion through order s^3 has an O(s^4) remainder, uniformly on bounded spatial sets and bounded density-contrast and weighted-state classes. Under alignment of the two initial states and a nonzero moment of the density-weighted initial velocity, the two density identities reduce to a constant-vector static transform. Two opposite elastic null phases give uniqueness for one or two fixed vertical density profiles when the profile family is two-sided Laplace nondegenerate. When λ+μ≠0, each nonzero normal root has partial multiplicities $2$ and $1$ and admits a length-two Jordan chain. The associated polynomial--exponential Lamé mode produces derivatives of the bilateral profile transforms. The resulting Hermite--Laplace system gives uniqueness for aligned classes with up to four fixed vertical profiles and independent horizontal coefficients, provided that the profile system is Hermite--Laplace nondegenerate. Distinct translations of one compactly supported profile provide an explicit four-profile class. We also prove rigidity of the alignment reduction and exhibit an infinite-dimensional kernel for the reduced transform on unrestricted densities.
ER -