Observability of Initial States in One-Dimensional Inviscid Flows With Shocks

We investigate the observability of initial states in one-dimensional, compressible, inviscid flows with shocks using two-dimensional variational (2DVar) data assimilation and sparse pressure measurements. The Sod shock tube problem is used as a test case, sustaining a canonical wave structure with a shock wave, contact discontinuity, and rarefaction fan. To characterize sensitivity pathways from observations of pressure to the initial state, we analyze adjoint dynamics along acoustic and entropic characteristics in the shock tube and compute initial-time adjoint fields. These fields are assembled to form an observation Jacobian at optimality, which enables a decomposition of state space into linearly observable and unobservable modes. Full-state reconstruction experiments demonstrate that 2DVar reliably reduces errors in the observable directions, while components in the null space persist, reflecting a multiplicity of initial states that are mutually consistent with sparse pressure measurements. An alternative, low-dimensional parameterization of the initial state, based on uniform driver and driven gas states, yields robust reconstructions, and the posterior covariance of the estimated parameters is accurately predicted, up to a scale factor, by the Cramér–Rao bound associated with the reduced-order observation Jacobian. These results may inform the design of control vectors and measurement arrays for data assimilation in shock-dominated flows.