In general, from a list of experimental wavelengths there are two main ways in which to then produce a list of energy levels.
1. Fit a Hamiltonian (model with molecular constants) for each electronic state and then from this model create a list of energies (term values which you may have seen). Problem with this method is that the model created for each individual electronic state does not take into account perturbations due to the other electronic states-i.e. "its uncoupled".
2. Essentially MARVEL works as creating a network of energy levels from measured frequencies (an energy of 0.000 is applied for a "base") thus cutting out the "middle process" of fitting a Hamiltonian. The idea is that the energy levels produced are "self-consistent": each energy level is linked to others by 1 or more measurements of frequencies to other states. Thus instead of producing " ctitious" energy levels from a Hamiltonian which has been fitted to an electronic state, the list of energies produced by MARVEL can be regarded as pure experimentally measured levels.
The MARVEL approach \citep{jt412,07CsCzFu.marvel,12FuCsi.method} is a sophisticated methodology that allows extraction of experimental energy levels, and associated uncertainties, from a (usually large) set of experimental transition frequencies.
The methodology is similar to traditional approaches based on the Ritz principle, such as `combination differences', but is a more sophisticated, computational, near-black-box approach. The MARVEL program takes as input formatted assigned transitions. The program then constructs the experimental spectroscopic networks (SNs) \citep{11CsFuxx.marvel,12FuCsxx.marvel,14FuArMe.marvel,16ArPeFu.marvel,16CzFuAr.marvel} which contains all inter-connected transitions. For each SN, the assigned transition data is then inverted to find the energy levels. The uncertainties of the transition frequencies weight this inversion process using a robust reweighting procedure advocated by \citet{Watson03} allowing MARVEL to yield the uncertainty of each extracted energy level.
The MARVEL software takes as input assigned, measured transitions, with estimated uncertainties, and outputs assigned energy levels together with recommended uncertainties. However, often there is no consistent set of energy levels that produce the input transitions within the estimated uncertainties. This can occur due to typographic or digitisation errors, mis-assignments and under-estimated uncertainties for the transitions. For this reason, the master list of \Marvel\ input transitions should be gradually increased with issues resolved as new transitions are added to the master file. \Marvel\ produces new recommended uncertainties. If these are less than twice the original uncertainties, we generally adopt these recommended uncertainties. If there is a very large difference in the recommended uncertainty, we look for typographic and digitisation errors; if none are found, we then assume mis-assignment and put a negative in front of the transition frequency, thus retaining the data but not utilising it in the MARVEL algorithm for future runs. Transitions initially discarded in this way can be reconsidered later in the process.
It is important throughout and particularly at the final stage that the trends and patterns in the energy levels are validated using available means. In previous studies this has often been against energies calculated theoretically; here we are more reliant on trends such as reasonably systematic quadratic increase in energy with J, approximately linear increase with vibrational quantum number and so forth.