@article{PanetKuroishiHolschneider2011,
  author    = {Panet, Isabelle and Kuroishi, Yuki and Holschneider, Matthias},
  title     = {Wavelet modelling of the gravity field by domain decomposition methods: an example over Japan},
  series = {Geophysical journal international},
  volume    = {184},
  journal   = {Geophysical journal international},
  number    = {1},
  publisher = {Oxford Univ. Press},
  address   = {Oxford},
  issn      = {0956-540X},
  doi       = {10.1111/j.1365-246X.2010.04840.x},
  pages     = {203 -- 219},
  year      = {2011},
  abstract  = {With the advent of satellite gravity, large gravity data sets of unprecedented quality at low and medium resolution become available. For local, high resolution field modelling, they need to be combined with the surface gravity data. Such models are then used for various applications, from the study of the Earth interior to the determination of oceanic currents. Here we show how to realize such a combination in a flexible way using spherical wavelets and applying a domain decomposition approach. This iterative method, based on the Schwarz algorithms, allows to split a large problem into smaller ones, and avoids the calculation of the entire normal system, which may be huge if high resolution is sought over wide areas. A subdomain is defined as the harmonic space spanned by a subset of the wavelet family. Based on the localization properties of the wavelets in space and frequency, we define hierarchical subdomains of wavelets at different scales. On each scale, blocks of subdomains are defined by using a tailored spatial splitting of the area. The data weighting and regularization are iteratively adjusted for the subdomains, which allows to handle heterogeneity in the data quality or the gravity variations. Different levels of approximations of the subdomains normals are also introduced, corresponding to building local averages of the data at different resolution levels. We first provide the theoretical background on domain decomposition methods. Then, we validate the method with synthetic data, considering two kinds of noise: white noise and coloured noise. We then apply the method to data over Japan, where we combine a satellite-based geopotential model, EIGEN-GL04S, and a local gravity model from a combination of land and marine gravity data and an altimetry-derived marine gravity model. A hybrid spherical harmonics/wavelet model of the geoid is obtained at about 15 km resolution and a corrector grid for the surface model is derived.},
  language  = {en}
}
@article{GaciZaourarBriqueuetal.2011,
  author    = {Gaci, Said and Zaourar, Naima and Briqueu, Louis and Holschneider, Matthias},
  title     = {Regularity analysis applied to sonic logs data a case study from KTB borehole site},
  series = {Arabian journal of geosciences},
  volume    = {4},
  journal   = {Arabian journal of geosciences},
  number    = {1-2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1866-7511},
  doi       = {10.1007/s12517-010-0129-y},
  pages     = {221 -- 227},
  year      = {2011},
  abstract  = {Borehole logs provide in situ information about the fluctuations of petrophysical properties with depth and thus allow the characterization of the crustal heterogeneities. A detailed investigation of these measurements may lead to extract features of the geological media. In this study, we suggest a regularity analysis based on the continuous wavelet transform to examine sonic logs data. The description of the local behavior of the logs at each depth is carried out using the local Hurst exponent estimated by two (02) approaches: the local wavelet approach and the average-local wavelet approach. Firstly, a synthetic log, generated using the random midpoints displacement algorithm, is processed by the regularity analysis. The obtained Hurst curves allowed the discernment of the different layers composing the simulated geological model. Next, this analysis is extended to real sonic logs data recorded at the Kontinentales Tiefbohrprogramm (KTB) pilot borehole (Continental Deep Drilling Program, Germany). The results show a significant correlation between the estimated Hurst exponents and the lithological discontinuities crossed by the well. Hence, the Hurst exponent can be used as a tool to characterize underground heterogeneities.},
  language  = {en}
}