The effectiveness of the breeding method in determining growth and decay characteristics of certain solutions to the Kortweg-de Vries (KdV) equation and the Nonlinear Schrödinger equation is investigated. Bred vectors are a finite amplitude, finite time generalization of Leading Lyapunov Vectors (LLV), and breeding has been used to predict large impending fluctuations in many systems, including chaotic systems. Here, the focus is on predicting fluctuations associated with extreme waves. The bred vector analysis is applied to the KdV equation with two types of initial conditions: soliton collisions, and a Gaussian distribution which decays into a group of solitons. The soliton solutions are stable, and the breeding analysis enables tracking of the growth and decay during the interactions. Furthermore, this study with a known stable system helps validate the use of breeding method for waves. This analysis is also applied to characterize rogue wave type solutions of the NLSE, which have been used to describe extreme ocean waves. In the results obtained, the growth rate maxima and the peaks of the bred vector always precede the rogue wave peaks. This suggests that the growth rate and bred vectors may serve as precursors for predicting energy localization due to rogue waves. Finally, the results reveal that the breeding method can be used to identify numerical instabilities.

   Effective simulation of diurnal variability is an important aspect of many geophysical data assimilation systems. For the Martian atmosphere, thermal tides are particularly prominent and contribute much to the Martian atmospheric circulation, dynamics and dust transport. To study the Mars diurnal variability (or thermal tides), the GFDL Mars Global Climate Model (MGCM) with the 4D-Local Ensemble Transform Kalman Filter (4D-LETKF) is used to perform a reanalysis of spacecraft temperature retrievals. We find that the use of a "traditional" 6-hr assimilation cycle induces spurious forcing of a resonantly-enhanced semi-diurnal Kelvin waves represented in both surface pressure and mid-level temperature by forming a wave 4 pattern in the diurnal averaged analysis increment that acts as a "topographic" stationary forcing. Different assimilation window lengths in the 4D-LETKF are introduced to remove the artificially induced resonance. It is found that short assimilation window lengths not only remove the spurious resonance, but also push the migrating semi-diurnal temperature variation at 50 Pa closer to the estimated "true" tides even in the absence of a radiatively active water ice cloud parameterization. In order to compare the performance of different assimilation window lengths, short-term to long-term forecasts based on the hour 00 and 12 assimilation are evaluated and compared. Results show that during NH summer, it is not the assimilation window length, but the radiatively active water ice cloud that influences the model prediction. A "diurnal bias correction" that includes bias correction fields dependent on the local time is shown to effectively reduce the forecast root mean square differences (RMSD) between forecasts and observations, compensate for the absence of water ice cloud parameterization, and enhance Martian atmosphere prediction. The implications of these results for data assimilation in the Earth's atmosphere are also discussed.