For the analysis of me series data from hydrology, we used a recently developed technique that is by now widely known as the Hilbert-Huang transform (HHT). Specifi cally, it is designed for nonlinear and nonsta onary data. In contrast to data analysis techniques using the short-me, windowed Fourier transform or the con nuous wavelet transform, the new technique is empirically adapted to the data in the following sense. First, an addive decomposi on, called empirical mode decomposi on (EMD), of the data into certain mul scale components is computed. Second, to each of these components, the Hilbert transform is applied. The resul ng Hilbert spectrum of the modes provides a localized me-frequency spectrum and instantaneous ( me-dependent) frequencies. In this study, we applied the HHT to hydrological me series data from the Upper Rur Catchment Area, mostly German territory, taken during a period of 20 yr. Our fi rst observa on was that a coarse approxima on of the data can be derived by trunca ng the EMD representaon. This can be used to be er model pa erns like seasonal structures. Moreover, the corresponding me-frequency energy spectrum applied to the complete EMD revealed seasonal events in a par cular apparent way together with their energy. We compared the Hilbert spectra with Fourier spectrograms and wavelet spectra to demonstrate a be er localiza on of the energy components, which also exhibit strong seasonal components. The Hilbert energy spectrum of the three measurement sta ons appear to be very similar, indica ng li le local variability in drainage.Abbrevia ons: EMD, empirical mode decomposi on; HHT, Hilbert-Huang transform; IMF, intrinsic mode func on.Given empirical data, the detection and parameterization of multiscale patterns and shapes in the measurements is an important task. Specifi cally, to study the eff ect of patterns on water and solute fl uxes, temporal and spatial data have to be analyzed at various stages so that their parameterization can eventually be used in simulation fl ux models.Th is study was part of a SFB/TR32 project (Transregional Collaborative Research Centre 32, www.tr32.de; verifi ed 3 June 2010) for which the overall objective is to improve our knowledge about the mechanisms leading to spatial and temporal patterns in energy and matter fl uxes of the soil-vegetation-atmosphere system. Part of the objectives is the determination, description, and analysis of patterns derived from diff erent sources. For instance, the large spatial and temporal variability of soil moisture patterns is determined by factors like atmospheric forcing, topography, soil properties, and vegetation, which interact in a complex, nonlinear way (see, e.g., Grayson and Blöschl, 2001;Western et al., 2004). Th us, a very large number of continuous soil moisture measurements are necessary to adequately capture this variability. In the framework of the TR32, a dense soil moisture sensor network for monitoring soil water content changes at high spatial and temporal scales has been set up (see Bogena et al., ...

The Hilbert–Huang-Transform (HHT) has proven to be an appropriate multiscale analysis technique specifically for nonlinear and nonstationary time series on non-equidistant grids. It is empirically adapted to the data: first, an additive decomposition of the data (empirical mode decomposition, EMD) into certain multiscale components is computed, denoted as intrinsic mode functions. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time-frequency spectrum and instantaneous (time-dependent) frequencies. For the first step, the empirical decomposition of the data, a different method based on local means has been developed by Chen et al. (2006). In this paper, we extend their method to multivariate data sets in arbitrary space dimensions. We place special emphasis on deriving a method which is numerically fast also in higher dimensions. Our method works in a coarse-to-fine fashion and is based on adaptive (tensor-product) spline-wavelets. We provide some numerical comparisons to a method based on linear finite elements and one based on thin-plate-splines to demonstrate the performance of our method, both with respect to the quality of the approximation as well as the numerical efficiency. Second, for a generalization of the Hilbert transform to the multivariate case, we consider the Riesz transformation and an embedding into Clifford-algebra valued functions, from which instantaneous amplitudes, phases and orientations can be derived. We conclude with some numerical examples.

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We investigate on the use of the Domain Embedding Method (DEM) for the forward modelling in EIT. This approach is suitably configured to overcome the model meshing bottleneck since it does not require that the mesh on the domain is adapted to the boundary surface. This is of crucial importance for, e.g., clinical applications of EIT, as it avoids tedious and time-consuming (re-)meshing procedures. The suggested DEM approach can accommodate arbitrary yet Lipschitz smooth boundary surfaces and is not limited to polygonal domains. For the discretisation purposes, we employ B-splines as they allow for arbitrary accuracy by raising the polynomial degree and are easy to implement due to their inherent piecewise polynomial structure. Numerical experiments confirm that a B-spline discretization yields, similarly to conventional Finite Difference discretizations, increasing condition numbers of the system matrix with respect to the discretisation levels. Fortunately, multiresolution ideas based on B-splines allow for optimal wavelet preconditioning.

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