Fitting a smooth curve to 2D, a surface to 3D, and a manifold to 4D irregular point cloud data is becoming a common practice in many engineering and science applications. Piecewise-polynomial spline functions provide a powerful tool applicable to interpolation and approximation problems. This study presents the least squares B-spline approximation (LSBSA) theory, which is a generalized version of the spline interpolation and can be applied to any irregularly scattered point cloud data at knots specified by the user. The formulation allows to apply the well-established body of knowledge of least squares theory to the B-spline approximation. This for example has the benefit of embedding quality control measures such as hypothesis testing and proper error propagation to assess the quality of the approximation problem. The method is applicable to many 1D curve, 2D surface and 3D manifold fitting problems of which both simulated and real data are used to illustrate the efficacy of the proposed theory. In particular, its real-world applications to multi-beam echo-sounder bathymetric data, digital terrain modeling and Greenland ice sheet deformation monitoring will be highlighted. The performance of the method for linear, quadratic, cubic and quartic spline functions will be investigated. The primary application of LSBSA lies in its ability to perform 3D manifold fitting for deformation monitoring. This capability provides the possibility of monitoring changes in continuous spatial and temporal domains. The Python and Matlab source codes of LSBSA are freely accessible at https://github.com/tudelft4d/lsbsa.

Permanent GNSS stations continuously monitor Earth's crust movements in horizontal and vertical directions. The recorded data include deterministic variations, including linear trends, periodic signals, and offsets, alongside stochastic variations represented by various noise models. Accurately detecting deterministic behaviors depends on a realistic estimation of the observation noise model. A new multivariate algorithm based on Monte Carlo singular spectrum analysis (MCSSA) is developed to analyze the multiple channels of time-series data simultaneously (e.g., different position components or data from multiple stations), considering noise correlations without being limited to a specific noise model. Testing on simulated GNSS data showed that, by increasing the number of channels, the algorithm could accurately identify dominant annual and semiannual components in the presence of colored noise. The results also indicated that unrealistic assumptions about the GNSS position time-series noise model can be misleading in the MCSSA hypothesis testing. Applying the algorithm to real Greenland GNSS data confirmed the significance of annual and semiannual harmonic patterns when white plus flicker noise (FLWN) combinations were considered as the stochastic behavior of data. In the univariate analysis of the vertical position time series contaminated with random walk noise, none of the annual and semiannual signals were interpreted to be significant. At the same time, the proposed multivariate algorithm successfully identified the annual signal but lacked sufficient channels (stations) to confirm the significance of the semiannual signal in the presence of random walk noise. The multivariate analysis has confirmed the significance of semiannual signal across 21 time series contaminated with FLWN, which univariate analysis missed due to a high level of colored noise.