GPS and Inertial Navigation Systems are used for positioning and attitude determination in a wide range of applications. Over the last few years, a number of low cost inertial sensors have become available. Although they exhibit large errors, GPS measurements can be used correct the INS and sensor errors to provide high accuracy real-time navigation. The integration of GPS and INS measurements is usually achieved using a Kalman filter. The measurement and process noise matrices used in the Kalman filter represent the stochastic properties of the GPS and INS systems respectively. Traditionally they are defined a priori and remain constant throughout a processing run. In reality, the stochastic properties of the system vary depending on factors such as vehicle dynamics and environmental conditions. This is particularly an issue for low cost inertial sensors where the initial sensor errors can be large, and experience significant temporal variation. This paper investigates three adaptive Kalman filtering algorithms that can be used to improve the estimation of the stochastic properties of a low cost INS. The algorithms are tested using a low cost Crossbow MEMS IMU integrated with carrier phase GPS for a marine application. The adaptive Kalman filtering algorithms are shown to reduce the dependence on the a priori information used in the filter. This results in a reduction in the time required to initialise the sensor errors and align the INS, and results in an improvement in navigation performance.