  A non-iterative waveform sensing approach is proposed toward (i) geometric
reconstruction of penetrable fractures, and (ii) quantitative identification of
their heterogeneous contact condition by seismic i.e. elastic waves. To this
end, the fracture support $\Gamma$ (which may be non-planar and unconnected) is
first recovered without prior knowledge of the interfacial condition by way of
the recently established approaches to non-iterative waveform tomography of
heterogeneous fractures, e.g. the methods of generalized linear sampling and
topological sensitivity. Given suitable approximation $\breve\Gamma$ of the
fracture geometry, the jump in the displacement field across $\breve\Gamma$
i.e. the fracture opening displacement (FOD) profile is computed from remote
sensory data via a regularized inversion of the boundary integral
representation mapping the FOD to remote observations of the scattered field.
Thus obtained FOD is then used as input for solving the traction boundary
integral equation on $\breve\Gamma$ for the unknown (linearized) contact
parameters. In this study, linear and possibly dissipative interactions between
the two faces of a fracture are parameterized in terms of a symmetric,
complex-valued matrix $\boldsymbol{K}(\boldsymbol{\xi})$ collecting the normal,
shear, and mixed-mode coefficients of specific stiffness. To facilitate the
high-fidelity inversion for $\boldsymbol{K}(\boldsymbol{\xi})$, a 3-step
regularization algorithm is devised to minimize the errors stemming from the
inexact geometric reconstruction and FOD recovery. The performance of the
inverse solution is illustrated by a set of numerical experiments where a
cylindrical fracture, endowed with two example patterns of specific stiffness
coefficients, is illuminated by plane waves and reconstructed in terms of its
geometry and heterogeneous (dissipative) contact condition.
