The problem of forecasting the behavior of a complex dynamical system through analysis of observational time-series data becomes difficult when the system expresses chaotic behavior and the measurements are sparse, in both space and/or time. Despite the fact that this situation is quite typical across many fields, including numerical weather prediction, the issue of whether the available observations are 'sufficient' for generating successful forecasts is still not well understood. The goal of this study is to develop the assimilation schemes using the time series observation. Several approaches to Lorenz 96 chaotic dynamics and geophysical fluid dynamics have been proposed. Chapter 1 and 2 contain the problem statement of data assimilation and description of two nonlinear models, the Lorenz 96 model, a toy model commonly used in data assimilation, and the shallow water model, a fast but relatively realistic model in geophysical fluid dynamics. The details of the simulations are provided. In Chapter 3, we will systematically compare the data assimilation methodology in literature. We will focus on the advantages and disadvantages of each method. In Chapter 4, the time delayed nudging method with the time delayed observation data is presented. We show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. In particular, we find that this estimate of 70% can be reduced to about 33% using time delays, and even further if Lagrangian drifter locations are also used as measurements. Chapter 5, we apply the sequential adversarial method on Lorenz 96 model. We find that the adversarial method can highly reduce the task burden of the parameter tuning in the data assimilation algorithm. We obtain a substantial improvement in the traditional sequential data assimilation scheme. Chapter 6 will be the conclusion and discussion on future work.