  Observability quantification is a key problem in dynamic network sciences.
While it has been thoroughly studied for linear systems, observability
quantification for nonlinear networks is less intuitive and more cumbersome.
One common approach to quantify observability for nonlinear systems is via the
Empirical Gramian (Empr-Gram) -- a generalized form of the Gramian of linear
systems. In this technical note, we produce three new results. First, we
establish that a variational form of nonlinear systems (computed via perturbing
initial conditions) yields a so-called Variational Gramian (Var-Gram) that is
equivalent to the classic Empr-Gram; the former being easier to compute than
the latter. Via Lyapunov exponents derived from Lyapunov's direct method, the
technical note's second result derives connections between existing
observability measures and Var-Gram. The third result demonstrates the
applicability of these new notions for sensor selection/placement in nonlinear
systems. Numerical case studies demonstrate these three developments and their
merits.
