This paper addresses the observability analysis and the optimal design of observation parameters in the presence of noisy measurements and parametric uncertainties. The main underlying frameworks are the nonlinear constrained moving horizon estimator design and the probabilistic certification via randomized optimization. As the perfect observability concept is not relevant under the considered uncertain and noisy context, the notion of almost $\epsilon$-observability is introduced and a systematic procedure to assess its satisfaction for a given system with a priori known measurement noise statistics and parameter discrepancy is sketched. A nice feature in the proposed framework is that the observability is not necessarily defined as the ability to reconstruct the whole state, rather, the more general concept of observation-target quantities is used so that one can analyze the precision with which specific chosen expressions of the state and the parameters can be reconstructed. The overall framework is exposed and validated through an illustrative example.