SUMMARYIn the field of state space estimation and data assimilation, the Kalman filter (KF) and the extended Kalman filter (EKF) are among the most reliable methods used. However, KF and EKF require the storage of, and operations with, matrices of size n ×n, where n is the size of the state space. Furthermore, both methods include inversion operations for m ×m matrices, where m is the size of the observation space. Thus, KF methods become impractical as the dimension of the system increases. In this paper, we introduce a variational Kalman filter (VKF) method to provide a low storage, and computationally efficient, approximation of the KF and EKF methods. Furthermore, we introduce a variational Kalman smoother (VKS) method to approximate the fixed-lag Kalman smoother (FLKS) method. Instead of using the KF formulae, we solve the underlying maximum a posteriori optimization problem using the limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method. Moreover, the LBFGS optimization method is used to obtain a low storage approximation of state estimate covariances and prediction error covariances. A detailed description of the VKF and VKS methods with LBFGS is given. The methodology is tested on linear and nonlinear test examples. The simulated results of the VKF method are presented and compared with KF and EKF, respectively. The convergence of BFGS/LBFGS methods is tested and demonstrated numerically.

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Extracting digital elevation models (DTMs) from LiDAR data under forest canopy is a challenging task. This is because the forest canopy tends to block a portion of the LiDAR pulses from reaching the ground, hence introducing gaps in the data. This paper presents an algorithm for DTM extraction from LiDAR data under forest canopy. The algorithm copes with the challenge of low data density by generating a series of coarse DTMs by using the few ground points available and using trend surfaces to interpolate missing elevation values in the vicinity of the available points. This process generates a cloud of ground points from which the final DTM is generated. The algorithm has been compared to two other algorithms proposed in the literature in three different test sites with varying degrees of difficulty. Results show that the algorithm presented in this paper is more tolerant to low data density compared to the other two algorithms. The results further show that with decreasing point density, the differences between the three algorithms dramatically increased from about 0.5 m to over 10 m.