  We exploit local quantum estimation theory to investigate the measurement of
linear and quadratic coupling strengths in a driven-dissipative optomechanical
system. For experimentally realistic values of the model parameters, we find
that the linear coupling strength is considerably easier to estimate than the
quadratic one. Our analysis also reveals that the majority of information about
these parameters is encoded in the reduced state of the mechanical element, and
that the best estimation strategy for both coupling parameters is well
approximated by a direct measurement of the mechanical position quadrature.
Interestingly, we also show that temperature does not always have a detrimental
effect on the estimation precision, and that the effects of temperature are
more pronounced in the case of the quadratic coupling parameter.
