irregular grid
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Spatial imaging of ground penetrating radar (GPR) measurement data is a difficult computational problem that is time consuming and requires substantial memory resources. The complexity of the problem increases when the measurements are performed on an irregular grid. Such grid irregularities are typical for handheld or flying GPR systems. In this paper, a fast and efficient method of GPR data imaging based on radial basis functions is described. A compactly supported modified Gaussian radial basis function (RBF) and a hierarchical approximation method were used for computation. The approximation was performed in multiple layers with decreasing approximation radius, where in successive layers, increasingly finer details of the imaging were exposed. The proposed method provides high flexibility and accuracy of approximation with a computational cost of N·log (N) for model building and N·M for function evaluation, where N is the number of measurement points and M is the number of approximation centres. The method also allows for the control smoothing of measurement noise. The computation of one high-quality imaging using 5000 measurement points utilises about 5 s on an Intel Core i5-7200U CPU 2.5 GHz, 8 GB RAM computer. Such short time enables real-time image processing during field measurements.

The adaptive approximations for some characteristic of random functions defined on arbitrary irregular grids are discussed in this paper. The mentioned functions can be examined as flows of random real values associated with an irregular grid. This paper considers the question of choosing an adaptive enlargement of the initial grid. The mentioned enlargement essentially depends on the formulation of the criterion in relation to which adaptability is considered. Several criteria are considered here, among which there are several criteria applicable to the processing of random flows. In particular, the criteria corresponding to the mathematical expectation, dispersion, as well as autocorrelation and cross-correlation of two random flows are considered. It is possible to consider criteria corresponding to various combinations of the mentioned characteristics. The number of knots of the initial (generally speaking, irregular) grid can be arbitrary, and the main grid can be any subset of the initial one. Decomposition algorithms are proposed, taking into account the nature of the changes in the initial flow. The number of arithmetic operations in the proposed algorithms is proportional to the length of the initial flow. Sequential processing of the initial flow is possible in real time.

We have developed a general method to obtain the equiripple and the least-squares finite-difference (FD) operator weights to compute arbitrary-order derivatives from arbitrary sample locations. The method is based on the complex-valued Remez exchange algorithm applied to three cost functions: the total error, the relative error, and the group-velocity error. We evaluate the method on three acoustic FD modeling examples. In the first example, we assess the accuracy obtained with the optimal coefficients when propagating acoustic waves through a medium. In the second example, we propagate a wave through an irregular grid. In the final example, we position a source and receiver at arbitrary locations in-between the modeling grid points. In the examples using regular grids, the equiripple solution to the relative cost function performs best. It obtains marginally (4%–10%) better results compared to the second-best option, the least-squares solution for the relative cost function. The least-squares solution for the relative error produced the only stable and accurate results also in the example of modeling on an irregular grid.

<p>The spatialization of meteorological variables when the ground network is scattered and the relief is disturbed is a major issue for watershed hydrology or for the characterization of agricultural water consumption. The aim of this study is to set up the SAFRAN re-analysis system on the Tensift catchment area in Morocco. To this end, all the meteorological measurements acquired on the site between 2004 and 2014 by several organisations were gathered in a single database and quality control was carried out.  SAFRAN was then assessed according to a leave-one-out approach, which consists of removing a station from the database and comparing the re-analysis with the data from this station. It was also compared to another technic for meteorological variables spatialization named MICROMET (Liston et al., 2006). Particular attention was paid on the mountainous areas. In order to reproduce the high climate variability in this area, SAFRAN is also set up with an irregular grid up to 1 km resolution and compared to the regular version (8 km grid point). The results show that the re-analysis on the irregular grid is much better than on the regular grid, especially in the mountains. For example, the validation at the Aremd mountain station (2058 m) shows that the bias and RMSE on the surface temperature decreased from -4.8°C and 6.2°C for the regular grid to 0.6°C and 3.6°C for the irregular grid. Likewise, for precipitation, the correlation coefficient is improved by more than 23% for the regular grid. Concerning the visible radiation, MICROMET is strongly biased compared to the measurements carried out at the Aremd station (86 W/m²) whereas for SAFRAN, the bias is only 48W/m². Our current work concerns the mapping of vertical soil-vegetation-atmosphere exchanges over the catchment area using SAFRAN forcing on the irregular grid. The challenge is notably to represent irrigation, which strongly modifies the surface water states.</p>