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This paper presents a nonlinear adaptive observer to estimate the annular pressure profile throughout the well bore continuously while drilling. A simple mechanistic model is presented that captures the dominant phenomena of the drilling system and forms the basis for model-based observer and control design. A new nonlinear adaptive observer is developed for state estimation and parameter estimation. Simulation results obtained on a high fidelity drilling simulator based on the real data from an off-shore drilling operation of North Sea well are presented to demonstrate the effectiveness of the proposed estimation schemes.