  The goal of quantitative elastography is to identify biomechanical parameters
from interior displacement data, which are provided by other modalities, such
as ultrasound or magnetic resonance imaging. In this paper, we analyze the
stability of several linearized problems in quantitative elastography. Our
method is based on the theory of redundant systems of linear partial
differential equations. We analyze the ellipticity properties of the
corresponding PDE systems augmented with the interior displacement data; we
explicitly characterize the kernel of the forward operators and show
injectivity for particular linearizations. Stability criteria can then be
deduced. Our results show stability of shear modulus, pressure and density;
they indicate that singular strain fields should be avoided, and show how
additional measurements can help in ensuring stability.
