  Adaptive moving spatial meshes are useful for solving physical models given
by time-dependent partial differentialequations. However, special consideration
must be given when combining adaptive meshing procedures with ensemble-based
data assimilation (DA) techniques. In particular, we focus on the case where
each ensemble member evolvesindependently upon its own mesh and is interpolated
to a common mesh for the DA update. This paper outlines aframework to develop
time-dependent reference meshes using locations of observations and the metric
tensors (MTs)or monitor functions that define the spatial meshes of the
ensemble members. We develop a time-dependent spatiallocalization scheme based
on the metric tensor (MT localization). We also explore how adaptive moving
mesh tech-niques can control and inform the placement of mesh points to
concentrate near the location of observations, reducingthe error of observation
interpolation. This is especially beneficial when we have observations in
locations that wouldotherwise have a sparse spatial discretization. We
illustrate the utility of our results using discontinuous Galerkin(DG)
approximations of 1D and 2D inviscid Burgers equations. The numerical results
show that the MT localizationscheme compares favorably with standard
Gaspari-Cohn localization techniques. In problems where the observationsare
sparse, the choice of common mesh has a direct impact on DA performance. The
numerical results also demonstratethe advantage of DG-based interpolation over
linear interpolation for the 2D inviscid Burgers equation.
