With the restrictions on online downhole measuring during petroleum drilling, mathematical reconstruction of the bottomhole pressure (BHP) is desired to improve control of the pressure profile throughout the well. With sufficient pressure management, increasingly tight pressure margins, that are imposed by the worsening accessibility of the remaining oil reservoirs, might be met. A regularized and adaptive Moving Horizon Estimation (MHE) approach is proposed to estimate the BHP. The observer comes with abilities to adapt to the degree of excitation via thresholded singular value decomposition, and means for filtering using an a priori estimate in the cost function. The employed observer model is nonlinear, as required by the process, and chosen to be third order to mitigate computational load. The algebraic BHP equation includes two unknown parameters and an unmeasured state. Accuracy of the model is enhanced by estimation of selected parameters. Testing on a data set from a North Sea well, conducted to investigate the suitability of the approach, has delivered promising results.