@article{CotroneiDiSalvoHolschneideretal.2017,
  author    = {Cotronei, Mariantonia and Di Salvo, Rosa and Holschneider, Matthias and Puccio, Luigia},
  title     = {Interpolation in reproducing kernel Hilbert spaces based on random subdivision schemes},
  series = {Journal of computational and applied mathematics},
  volume    = {311},
  journal   = {Journal of computational and applied mathematics},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0377-0427},
  doi       = {10.1016/j.cam.2016.08.002},
  pages     = {342 -- 353},
  year      = {2017},
  abstract  = {In this paper we present a Bayesian framework for interpolating data in a reproducing kernel Hilbert space associated with a random subdivision scheme, where not only approximations of the values of a function at some missing points can be obtained, but also uncertainty estimates for such predicted values. This random scheme generalizes the usual subdivision by taking into account, at each level, some uncertainty given in terms of suitably scaled noise sequences of i.i.d. Gaussian random variables with zero mean and given variance, and generating, in the limit, a Gaussian process whose correlation structure is characterized and used for computing realizations of the conditional posterior distribution. The hierarchical nature of the procedure may be exploited to reduce the computational cost compared to standard techniques in the case where many prediction points need to be considered.},
  language  = {en}
}
@article{AfshariMoeinSomogyvariValleyetal.2018,
  author    = {Afshari Moein, Mohammad J. and Somogyv{\´a}ri, M{\´a}rk and Valley, Beno{\^i}t and Jalali, Mohammadreza and L{\"o}w, Simon and Bayer, Peter},
  title     = {Fracture network characterization using stress-based tomography},
  series = {Journal of geophysical research : JGR},
  volume    = {123},
  journal   = {Journal of geophysical research : JGR},
  number    = {11},
  publisher = {American Geophysical Union},
  address   = {Washington},
  issn      = {2169-9313},
  doi       = {10.1029/2018JB016438},
  pages     = {9324 -- 9340},
  year      = {2018},
  abstract  = {Information on structural features of a fracture network at early stages of Enhanced Geothermal System development is mostly restricted to borehole images and, if available, outcrop data. However, using this information to image discontinuities in deep reservoirs is difficult. Wellbore failure data provides only some information on components of the in situ stress state and its heterogeneity. Our working hypothesis is that slip on natural fractures primarily controls these stress heterogeneities. Based on this, we introduce stress-based tomography in a Bayesian framework to characterize the fracture network and its heterogeneity in potential Enhanced Geothermal System reservoirs. In this procedure, first a random initial discrete fracture network (DFN) realization is generated based on prior information about the network. The observations needed to calibrate the DFN are based on local variations of the orientation and magnitude of at least one principal stress component along boreholes. A Markov Chain Monte Carlo sequence is employed to update the DFN iteratively by a fracture translation within the domain. The Markov sequence compares the simulated stress profile with the observed stress profiles in the borehole, evaluates each iteration with Metropolis-Hastings acceptance criteria, and stores acceptable DFN realizations in an ensemble. Finally, this obtained ensemble is used to visualize the potential occurrence of fractures in a probability map, indicating possible fracture locations and lengths. We test this methodology to reconstruct simple synthetic and more complex outcrop-based fracture networks and successfully image the significant fractures in the domain.},
  language  = {en}
}
@article{RingelSomogyvariJalalietal.2019,
  author    = {Ringel, Lisa Maria and Somogyv{\´a}ri, M{\´a}rk and Jalali, Mohammadreza and Bayer, Peter},
  title     = {Comparison of hydraulic and tracer tomography for discrete fracture network inversion},
  series = {Geosciences},
  volume    = {9},
  journal   = {Geosciences},
  number    = {6},
  publisher = {MDPI},
  address   = {Basel},
  issn      = {2076-3263},
  doi       = {10.3390/geosciences9060274},
  pages     = {17},
  year      = {2019},
  abstract  = {Fractures serve as highly conductive preferential flow paths for fluids in rocks, which are difficult to exactly reconstruct in numerical models. Especially, in low-conductive rocks, fractures are often the only pathways for advection of solutes and heat. The presented study compares the results from hydraulic and tracer tomography applied to invert a theoretical discrete fracture network (DFN) that is based on data from synthetic cross-well testing. For hydraulic tomography, pressure pulses in various injection intervals are induced and the pressure responses in the monitoring intervals of a nearby observation well are recorded. For tracer tomography, a conservative tracer is injected in different well levels and the depth-dependent breakthrough of the tracer is monitored. A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics. The used Metropolis-Hastings-Green algorithm is refined by the simultaneous estimation of the measurement error's variance, that is, the measurement noise. Based on the presented application to invert the two-dimensional cross-section between source and the receiver well, the hydraulic tomography reveals itself to be more suitable for reconstructing the original DFN. This is based on a probabilistic representation of the inverted results by means of fracture probabilities.},
  language  = {en}
}