In the 30 years of its availability, satellite altimetry has established itself as an important tool for understanding the Earth system. Originally developed for oceanography and geodesy, it has also proven valuable for monitoring water level variation of lakes and rivers. However, when using altimetry for inland waters, there is always a critical issue: retracking, i.e., the procedure in which the range from the satellite to the water surface is (re)estimated. The current retracking methods heavily rely on single waveforms, which results in a high sensitivity to every individual peak in the waveform and in a strong dependency on the waveform’s shape. Here, we propose the bin-space-time (BiST) retracking method that moves beyond finding a single point in a 1-D waveform and instead seeks a retracking line within a 2-D radargram, for which the temporal information over different cycles is also considered. The retracking line divides the radargram into two segments: the left (Front) and the right-hand side (Back) of the retracking line. Such a segmentation approach can be interpreted as a binary image segmentation problem, for which spatiotemporal information can be incorporated. We follow a Bayesian approach, exploiting a probabilistic graphical model known as a Markov random field (MRF). There, the problem is arranged as a maximum a posteriori estimation of an MRF (MAP-MRF), which means finding a retracking line that maximizes a posterior probability density or minimizes a posterior energy function. Our posterior energy function is obtained by a prior energy function and a likelihood energy function, both of them depending on signal intensity and bin: 1) the prior: the bin-space energy function defined between first-order neighboring pixels of a radargram modeling the spatial dependency between their labels for given intensities and bins and 2) the likelihood: the temporal energy function of a pixel for labeling Front or Back given its overall temporal evolution. The realization of the field with the minimum sum of the bin-space and the temporal energy functions is then found through the maxflow algorithm. Consequently, the retracking line, which defines the boundary between the Back and Front region, is obtained. We apply our method to both pulse-limited and synthetic aperture radar (SAR) altimetry data over nine lakes and reservoirs in the USA with different sizes and different altimetry characteristics. The resulting water level time series are validated against in situ data. Across the selected case studies, on average, the BiST retracker improves the root-mean-square error (RMSE) by approximately 0.5 m compared to the best existing retracker. The main benefit of the proposed retracker, which operates in bin, space, and time domains, is its robustness against unexpected waveform variations, making it suitable for diverse inland water surfaces.

Normalized Cut is a well-established divisive image segmentation method, which we adapt in this paper for the segmentation of laser point clouds in urban areas. Our focus is on polyhedral objects with planar surfaces. Due to its target function, Normalized Cut favours cuts with "short cut lines" or "small cut surfaces", which is a drawback for our application. We therefore modify the target function, weighting the similarity measures with distance-dependent weights. We call the induced minimization problem "Distance-weighted Cut" (DWCut). The new target function leads to a generalized eigenvalue problem, which is slightly more complicated than the corresponding problem for the Normalized Cut; on the other hand, the new target function is easier to interpret and avoids some drawbacks of the Normalized Cut. We point out an efficient method for the numerical solution of the eigenvalue problem which is based on a Krylov subspace method. DWCut can be beneficially combined with an aggregation in order to reduce the computational effort and to avoid shortcomings due to insufficient plane parameters. We present examples for the successful application of the Distance-weighted Cut principle and evaluate its results by comparison with the results of corresponding manual segmentations.