In reduced order modeling, the basis is generally built from Eulerian data, e.g., velocity, vorticity, or streamfunction. Thus, ROMs generally are Eulerian ROMs. Of course, these Eulerian ROMs can efficiently approximate the corresponding Eulerian fields. However, Lagrangian fields (e.g., the FTLE field) might be better approximated by Lagrangian ROMs, i.e., ROMs whose bases are built from both Eulerian and Lagrangian data. The new α-ROM and λ-ROM are Lagrangian ROMs, since the FTLE weighted inner products (23) and (25) use both Eulerian **data** (i.e., the vorticity ω) and Lagrangian **data** (i.e., the FTLE λ). The input Eulerian **data** (i.e., the vorticity field) help the resulting FTLE-ROMs yield an accurate approximation of the output Eulerian **data** (i.e., the vorticity, streamfunction, and velocity fields computed by the FTLE-ROM). On the other hand, the input Lagrangian **data** (i.e., the FTLE field) "nudge" the FTLE-ROMs toward an accurate approximation of the output Lagrangian **data** (i.e., the FTLE field obtained from the FTLE-ROM velocity field). Thus, we expect the new α-ROM and λ-ROM to yield more accurate FTLE approximations than the standard G-ROM (17). In Section 5, we investigate the new α-ROM and λ-ROM, as well as the standard G-ROM, in the FTLE computation of a test problem that uses the QGE as mathematical model.