@article{AsgarimehrWickertReich2018,
  author    = {Asgarimehr, Milad and Wickert, Jens and Reich, Sebastian},
  title     = {TDS-1 GNSS Reflectometry},
  series = {IEEE journal of selected topics in applied earth observations and remote sensing},
  volume    = {11},
  journal   = {IEEE journal of selected topics in applied earth observations and remote sensing},
  number    = {11},
  publisher = {Inst. of Electr. and Electronics Engineers},
  address   = {Piscataway},
  issn      = {1939-1404},
  doi       = {10.1109/JSTARS.2018.2873241},
  pages     = {4534 -- 4541},
  year      = {2018},
  abstract  = {This study presents the development and a systematic evaluation study of GNSS reflectometry wind speeds. After establishing a wind speed retrieval algorithm, UK TechDemoSat-1 (TDS-1) derived winds, from May 2015 to July 2017, are compared to the Advanced Scatterometer (ASCAT). ERA-Interim wind fields of the European Centre for Medium-range Weather Forecasts (ECMWF) and in situ observation from Tropical Atmosphere Ocean buoy array in the Pacific are taken as reference. One-year averaged TDS-1 global winds demonstrate small differences with ECMWF in a majority of areas as well as discuss under- and overestimations. The pioneering TDS-1 winds demonstrate a root-mean-squared error (RMSE) and bias of 2.77 and -0.33 m/s, which are comparable to the RMSE and bias derived by ASCAT winds, as large as 2.31 and 0.25 m/s, respectively. Using buoys measurements as reference, RMSE and bias of 2.23 and -0.03 m/s for TDS-1 as well as 1.40 and -0.68 m/s for ASCAT are obtained. Utilizing rain microwave-infrared estimates of the Tropical Rainfall Measuring Mission, rain-affected observation of both ASCAT and TDS-1 are collected and evaluated. Although ASCAT winds show a significant performance degradation resulting in an RMSE and bias of 3.16 and 1.03 m/s, respectively, during rain condition, TDS-1 shows a more reliable performance with an RMSE and bias of 2.94 and -0.21 m/s, respectively, which indicates the promising capability of GNSS forward scattering for wind retrievals during rain. A decrease in TDS-1-derived bistatic radar cross sections during rain events, at weak winds, is also demonstrated.},
  language  = {en}
}
@article{AsgarimehrZavorotnyWickertetal.2018,
  author    = {Asgarimehr, Milad and Zavorotny, Valery and Wickert, Jens and Reich, Sebastian},
  title     = {Can GNSS Reflectometry Detect Precipitation Over Oceans?},
  series = {Geophysical research letters},
  volume    = {45},
  journal   = {Geophysical research letters},
  number    = {22},
  publisher = {American Geophysical Union},
  address   = {Washington},
  issn      = {0094-8276},
  doi       = {10.1029/2018GL079708},
  pages     = {12585 -- 12592},
  year      = {2018},
  abstract  = {For the first time, a rain signature in Global Navigation Satellite System Reflectometry (GNSS-R) observations is demonstrated. Based on the argument that the forward quasi-specular scattering relies upon surface gravity waves with lengths larger than several wavelengths of the reflected signal, a commonly made conclusion is that the scatterometric GNSS-R measurements are not sensitive to the surface small-scale roughness generated by raindrops impinging on the ocean surface. On the contrary, this study presents an evidence that the bistatic radar cross section sigma(0) derived from TechDemoSat-1 data is reduced due to rain at weak winds, lower than approximate to 6 m/s. The decrease is as large as approximate to 0.7 dB at the wind speed of 3 m/s due to a precipitation of 0-2 mm/hr. The simulations based on the recently published scattering theory provide a plausible explanation for this phenomenon which potentially enables the GNSS-R technique to detect precipitation over oceans at low winds.},
  language  = {en}
}
@article{LeungLeutbecherReichetal.2019,
  author    = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.},
  title     = {Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier},
  series = {Journal of the atmospheric sciences},
  volume    = {76},
  journal   = {Journal of the atmospheric sciences},
  number    = {12},
  publisher = {American Meteorological Soc.},
  address   = {Boston},
  issn      = {0022-4928},
  doi       = {10.1175/JAS-D-19-0057.1},
  pages     = {3883 -- 3892},
  year      = {2019},
  abstract  = {The accepted idea that there exists an inherent finite-time barrier in deterministically predicting atmospheric flows originates from Edward N. Lorenz's 1969 work based on two-dimensional (2D) turbulence. Yet, known analytic results on the 2D Navier-Stokes (N-S) equations suggest that one can skillfully predict the 2D N-S system indefinitely far ahead should the initial-condition error become sufficiently small, thereby presenting a potential conflict with Lorenz's theory. Aided by numerical simulations, the present work reexamines Lorenz's model and reviews both sides of the argument, paying particular attention to the roles played by the slope of the kinetic energy spectrum. It is found that when this slope is shallower than -3, the Lipschitz continuity of analytic solutions (with respect to initial conditions) breaks down as the model resolution increases, unless the viscous range of the real system is resolved—which remains practically impossible. This breakdown leads to the inherent finite-time limit. If, on the other hand, the spectral slope is steeper than -3, then the breakdown does not occur. In this way, the apparent contradiction between the analytic results and Lorenz's theory is reconciled.},
  language  = {en}
}
@article{StaniforthWoodReich2006,
  author    = {Staniforth, Andrew and Wood, Nigel and Reich, Sebastian},
  title     = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {132},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {621C},
  publisher = {Wiley},
  address   = {Weinheim},
  issn      = {0035-9009},
  doi       = {10.1256/qj.06.30},
  pages     = {3107 -- 3116},
  year      = {2006},
  abstract  = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes.},
  language  = {en}
}
@article{Reich2006,
  author    = {Reich, Sebastian},
  title     = {Linearly implicit time stepping methods for numerical weather prediction},
  series = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  volume    = {46},
  journal   = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0006-3835},
  doi       = {10.1007/s10543-006-0065-0},
  pages     = {607 -- 616},
  year      = {2006},
  abstract  = {The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.},
  language  = {en}
}
@article{SomogyvariReich2020,
  author    = {Somogyv{\´a}ri, M{\´a}rk and Reich, Sebastian},
  title     = {Convergence tests for transdimensional Markov chains in geoscience imaging},
  series = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences},
  volume    = {52},
  journal   = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences},
  number    = {5},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1874-8961},
  doi       = {10.1007/s11004-019-09811-x},
  pages     = {651 -- 668},
  year      = {2020},
  abstract  = {Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.},
  language  = {en}
}
@article{TaghvaeideWiljesMehtaetal.2017,
  author    = {Taghvaei, Amirhossein and de Wiljes, Jana and Mehta, Prashant G. and Reich, Sebastian},
  title     = {Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem},
  series = {Journal of dynamic systems measurement and control},
  volume    = {140},
  journal   = {Journal of dynamic systems measurement and control},
  number    = {3},
  publisher = {ASME},
  address   = {New York},
  issn      = {0022-0434},
  doi       = {10.1115/1.4037780},
  pages     = {11},
  year      = {2017},
  abstract  = {This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman-Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.},
  language  = {en}
}
@article{deWiljesReichStannat2018,
  author    = {de Wiljes, Jana and Reich, Sebastian and Stannat, Wilhelm},
  title     = {Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise},
  series = {SIAM Journal on Applied Dynamical Systems},
  volume    = {17},
  journal   = {SIAM Journal on Applied Dynamical Systems},
  number    = {2},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1536-0040},
  doi       = {10.1137/17M1119056},
  pages     = {1152 -- 1181},
  year      = {2018},
  abstract  = {The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.},
  language  = {en}
}
@article{AcevedoDeWiljesReich2017,
  author    = {Acevedo, Walter and De Wiljes, Jana and Reich, Sebastian},
  title     = {Second-order accurate ensemble transform particle filters},
  series = {SIAM journal on scientific computing},
  volume    = {39},
  journal   = {SIAM journal on scientific computing},
  number    = {5},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1064-8275},
  doi       = {10.1137/16M1095184},
  pages     = {A1834 -- A1850},
  year      = {2017},
  abstract  = {Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM T. Sci. Comput., 35, (2013), pp. A2013-A2014[ replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.},
  language  = {en}
}
@article{FrankMooreReich2006,
  author    = {Frank, Jason and Moore, Brian E. and Reich, Sebastian},
  title     = {Linear PDEs and numerical methods that preserve a multisymplectic conservation law},
  issn      = {1064-8275},
  doi       = {10.1137/050628271},
  year      = {2006},
  abstract  = {Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]},
  language  = {en}
}