Michel Y (2013a) Estimating deformations of random processes for correlation modelling in a limited area model. Quarterly Journal of the Royal Meteorological Society 139(671):534-547

Michel Y (2013b) Estimating deformations of random processes for correlation modelling: methodology and the one-dimensional case. Quarterly Journal of the Royal Meteorological Society 139(672):771-783

Mikulevicius R, Rozovskii B (2004) Stochastic Navier-Stokes equations for turbulent flows. SIAM Journal on Mathematical Analysis 35(5):1250-1310

Mitchell L, Gottwald G (2012) Data assimilation in slow-fast systems using homogenized climate models. Journal of the atmospheric sciences 69(4):1359-1377

Neves W, Olivera C (2015) Wellposedness for stochastic continuity equations with ladyzhenskaya-prodi-serrin condition. Nonlinear Differential Equations and Applications NoDEA 22(5):1247-1258

Ng GH, McLaughlin D, Entekhabi D, Ahanin A (2011) The role of model dynamics in ensemble Kalman filter performance for chaotic systems. Tellus A 63(5):958-977

Orszag S (1970) Analytical theories of turbulence. Journal of Fluid Mechanics 41(02):363-386

Pannekoucke O, Massart S (2008) Estimation of the local diffusion tensor and normalization for heterogeneous correlation modelling using a diffusion equation. Quarterly Journal of the Royal Meteorological Society 134(635):1425-1438

Pannekoucke O, Emili E, Thual O (2014) Modelling of local length-scale dynamics and isotropizing deformations. Quarterly Journal of the Royal Meteorological Society 140(681):1387-1398

Papadakis N, Mémin E, Cuzol A, Gengembre N (2010) Data assimilation with the weighted ensemble Kalman filter. Tellus A 62(5):673-697

Papanicolaou G, Kohler W (1974) Asymptotic theory of mixing stochastic ordinary differential equations. Communications on Pure and Applied Mathematics 27(5):641-668

Parzen E (1962) On estimation of a probability density function and mode. The annals of mathematical statistics 33(3):1065-1076

Peavoy D, Franzke C, Roberts G (2015) Systematic physics constrained parameter estimation of stochastic differential equations. Computational Statistics & Data Analysis 83:182-199

Pegion P, Bates GM G, Hamill T, Kolczynski W, Whitaker J, Zhu Y (2016) Stochastic parametrization development in the noaa/ncep global forecast system. In: ECMWF/WWRP Workshop: Model Uncertainty, ECMWF, Reading

Penland C (2003a) Noise out of chaos and why it won't go away. Bulletin of the American Meteorological Society 84(7):921

Penland C (2003b) A stochastic approach to nonlinear dynamics: A review (extended version of the article-" noise out of chaos and why it won't go away"). Bulletin of the American Meteorological Society 84(7):925-929

Penland C, Matrosova L (1994) A balance condition for stochastic numerical models with application to the El Nino-southern oscillation. Journal of climate 7(9):1352-1372

Penland C, Sardeshmukh P (1995) The optimal growth of tropical sea surface temperature anomalies. Journal of climate 8(8):1999-2024

Pinier B, Mémin E, Laizet S, Lewandowski R (2019) Stochastic flow approach to model the mean velocity profile of wall-bounded flows, DOI 10.1103/PhysRevE.99.063101, URL https://link.aps.org/doi/10.1103/PhysRevE.99.063101

Plant R, Craig GC (2008) A stochastic parameterization for deep convection based on equilibrium statistics. Journal of the Atmospheric Sciences 65(1):87-105

Pope S (1983) A lagrangian two-time probability density function equation for inhomogeneous turbulent flows. The Physics of fluids 26(12):3448-3450

Pope S (1994) Lagrangian PDF methods for turbulent flows. Annu Rev Fluid Mech 26:23-63

Pope S, Chen Y (1990) The velocity-dissipation probability density function model for turbulent flows. Physics of Fluids A: Fluid Dynamics 2(8):1437-1449

Prévôt C, Röckner M (2007) A concise course on stochastic partial differential equations, vol 1905. Springer

Raynaud L, Boisserie M, Bouttier DL F, Ladabie C, Michel Y (2016) Model error representation in meteo france ensemble nwp system. In: ECMWF/WWRP Workshop: Model
Michel Y (2013a) Estimating deformations of random processes for correlation modelling in a limited area model. Quarterly Journal of the Royal Meteorological Society 139(671):534–547

Michel Y (2013b) Estimating deformations of random processes for correlation modelling: methodology and the one-dimensional case. Quarterly Journal of the Royal Meteorological Society 139(672):771–783

Mikulevicius R, Rozovskii B (2004) Stochastic Navier–Stokes equations for turbulent flows. SIAM Journal on Mathematical Analysis 35(5):1250–1310

Mitchell L, Gottwald G (2012) Data assimilation in slow-fast systems using homogenized climate models. Journal of the atmospheric sciences 69(4):1359–1377

Neves W, Olivera C (2015) Wellposedness for stochastic continuity equations with ladyzhenskaya–prodi–serrin condition. Nonlinear Differential Equations and Applications NoDEA 22(5):1247–1258

Ng GH, McLaughlin D, Entekhabi D, Ahanin A (2011) The role of model dynamics in ensemble Kalman filter performance for chaotic systems. Tellus A 63(5):958–977

Orszag S (1970) Analytical theories of turbulence. Journal of Fluid Mechanics 41(02):363–386

Pannekoucke O, Massart S (2008) Estimation of the local diffusion tensor and normalization for heterogeneous correlation modelling using a diffusion equation. Quarterly Journal of the Royal Meteorological Society 134(635):1425–1438

Pannekoucke O, Emili E, Thual O (2014) Modelling of local length-scale dynamics and isotropizing deformations. Quarterly Journal of the Royal Meteorological Society 140(681):1387–1398

Papadakis N, Mémin E, Cuzol A, Gengembre N (2010) Data assimilation with the weighted ensemble Kalman filter. Tellus A 62(5):673–697

Papanicolaou G, Kohler W (1974) Asymptotic theory of mixing stochastic ordinary differential equations. Communications on Pure and Applied Mathematics 27(5):641–668

Parzen E (1962) On estimation of a probability density function and mode. The annals of mathematical statistics 33(3):1065–1076

Peavoy D, Franzke C, Roberts G (2015) Systematic physics constrained parameter estimation of stochastic differential equations. Computational Statistics & Data Analysis 83:182–199

Pegion P, Bates GM G, Hamill T, Kolczynski W, Whitaker J, Zhu Y (2016) Stochastic parametrization development in the noaa/ncep global forecast system. In: ECMWF/WWRP Workshop: Model Uncertainty, ECMWF, Reading

Penland C (2003a) Noise out of chaos and why it won't go away. Bulletin of the American Meteorological Society 84(7):921

Penland C (2003b) A stochastic approach to nonlinear dynamics: A review (extended version of the article-" noise out of chaos and why it won't go away"). Bulletin of the American Meteorological Society 84(7):925–929

Penland C, Matrosova L (1994) A balance condition for stochastic numerical models with application to the El Nino-southern oscillation. Journal of climate 7(9):1352–1372

Penland C, Sardeshmukh P (1995) The optimal growth of tropical sea surface temperature anomalies. Journal of climate 8(8):1999–2024

Pinier B, Mémin E, Laizet S, Lewandowski R (2019) Stochastic flow approach to model the mean velocity profile of wall-bounded flows, DOI 10.1103/PhysRevE.99.063101, URL https://link.aps.org/doi/10.1103/PhysRevE.99.063101

Plant R, Craig GC (2008) A stochastic parameterization for deep convection based on equilibrium statistics. Journal of the Atmospheric Sciences 65(1):87–105

Pope S (1983) A lagrangian two-time probability density function equation for inhomogeneous turbulent flows. The Physics of fluids 26(12):3448–3450

Pope S (1994) Lagrangian PDF methods for turbulent flows. Annu Rev Fluid Mech 26:23–63

Pope S, Chen Y (1990) The velocity-dissipation probability density function model for turbulent flows. Physics of Fluids A: Fluid Dynamics 2(8):1437–1449

Prévôt C, Röckner M (2007) A concise course on stochastic partial differential equations, vol 1905. Springer

Raynaud L, Boisserie M, Bouttier DL F, Ladabie C, Michel Y (2016) Model error representation in meteo france ensemble nwp system. In: ECMWF/WWRP Workshop: Model
Uncertainty, ECMWF, Reading

Resseguier V (2017) Mixing and fluid dynamics under location uncertainty. PhD thesis,
Université Rennes 1

Resseguier V, Mémin E, Chapron B (2015) Reduced flow models from a stochastic Navier-Stokes representation. Annales de l'ISUP

Resseguier V, Mémin E, Chapron B (2017a) Geophysical flows under location uncertainty, part I: Random transport and general models. Geophysical & Astrophysical Fluid Dynamics 111(3):149-176

Resseguier V, Mémin E, Chapron B (2017b) Geophysical flows under location uncertainty, part II: Quasi-geostrophic models and efficient ensemble spreading. Geophysical & Astrophysical Fluid Dynamics 111(3):177-208

Resseguier V, Mémin E, Heitz D, Chapron B (2017c) Stochastic modelling and diffusion modes for POD models and small-scale flow analysis. J of Fluid Mech In press

Resseguier V, Pan W, Fox-Kemper B (2019) Data-driven versus self-similar parameterizations for stochastic lie transport and location uncertainty Submitted, to be published in arXiv soon

Reynolds C, Leutbecher M, Batt L, Chen S, Christensen H, Klasa C, Pegion P, Plant B, Raynaud L, Roberts N, Sandu I, Singleton A, Sommer M, Swinbank R, Tennant W, Theis S (2016) Reports from working group 3 : What are the pros/cons of existing model uncertainty schemes and how should these be measured? In: ECMWF/WWRP Workshop: Model Uncertainty, ECMWF, Reading

Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics 27(3):832-837

Sakov P, Haussaire JM, Bocquet M (2017) An iterative ensemble kalman filter in the presence of additive model error. Quarterly Journal of the Royal Meteorological Society

Sapsis T (2013) Attractor local dimensionality, nonlinear energy transfers and finite-time instabilities in unstable dynamical systems with applications to two-dimensional fluid flows. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 469(2153), DOI 10.1098/rspa.2012.0550

Sapsis T, Lermusiaux P (2012) Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty. Physica D: Nonlinear Phenomena 241(1):60-76

Sapsis T, Majda A (2013a) Blending modified Gaussian closure and non-Gaussian reduced subspace methods for turbulent dynamical systems. Journal of Nonlinear Science 23(6):1039-1071

Sapsis T, Majda A (2013b) Statistically accurate low-order models for uncertainty quantification in turbulent dynamical systems. Proceedings of the National Academy of Sciences 110(34):13705-13710

Sapsis T, Majda A (2013c) A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems. Physica D: Nonlinear Phenomena 252:34-45

Sardeshmukh PD, Sura P (2009) Reconciling non-gaussian climate statistics with linear dynamics. Journal of Climate 22(5):1193-1207

Sardeshmukh PD, Compo GP, Penland C (2015) Need for caution in interpreting extreme weather statistics. Journal of Climate 28(23):9166-9187

Sawford B (1991) Reynolds number effects in lagrangian stochastic models of turbulent dispersion. Physics of Fluids A: Fluid Dynamics 3(6):1577-1586

Scheuerer M, Hamill TM (2015) Variogram-based proper scoring rules for probabilistic forecasts of multivariate quantities. Monthly Weather Review 143(4):1321-1334

Schumann U (1995) Stochastic backscatter of turbulence energy and scalar variance by random subgrid-scale fluxes. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, vol 451, pp 293-318

Separovic L, Charron M, Erfani A, Gagnon N, Zadra A, Vaillancourt P (2016) Model error representation in the canada ensemble prediction systems. In: ECMWF/WWRP Workshop: Model Uncertainty, ECMWF, Reading

Shutts G (2005) A kinetic energy backscatter algorithm for use in ensemble prediction systems. Quarterly Journal of the Royal Meteorological Society 612:3079-3012

Shutts G (2015) A stochastic convective backscatter scheme for use in ensemble prediction systems. Quarterly Journal of the Royal Meteorological Society 141(692):2602-2616
Uncertainty, ECMWF, Reading

Resseguier V (2017) Mixing and fluid dynamics under location uncertainty. PhD thesis,
Université Rennes 1

Resseguier V, Mémin E, Chapron B (2015) Reduced flow models from a stochastic Navier-Stokes representation. Annales de l'ISUP

Resseguier V, Mémin E, Chapron B (2017a) Geophysical flows under location uncertainty, part I: Random transport and general models. Geophysical & Astrophysical Fluid Dynamics 111(3):149-176

Resseguier V, Mémin E, Chapron B (2017b) Geophysical flows under location uncertainty, part II: Quasi-geostrophic models and efficient ensemble spreading. Geophysical & Astrophysical Fluid Dynamics 111(3):177-208

Resseguier V, Mémin E, Heitz D, Chapron B (2017c) Stochastic modelling and diffusion modes for POD models and small-scale flow analysis. J of Fluid Mech In press

Resseguier V, Pan W, Fox-Kemper B (2019) Data-driven versus self-similar parameterizations for stochastic lie transport and location uncertainty Submitted, to be published in arXiv soon

Reynolds C, Leutbecher M, Batt L, Chen S, Christensen H, Klasa C, Pegion P, Plant B, Raynaud L, Roberts N, Sandu I, Singleton A, Sommer M, Swinbank R, Tennant W, Theis S (2016) Reports from working group 3 : What are the pros/cons of existing model uncertainty schemes and how should these be measured? In: ECMWF/WWRP Workshop: Model Uncertainty, ECMWF, Reading

Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics 27(3):832-837

Sakov P, Haussaire JM, Bocquet M (2017) An iterative ensemble kalman filter in the presence of additive model error. Quarterly Journal of the Royal Meteorological Society

Sapsis T (2013) Attractor local dimensionality, nonlinear energy transfers and finite-time instabilities in unstable dynamical systems with applications to two-dimensional fluid flows. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 469(2153), DOI 10.1098/rspa.2012.0550

Sapsis T, Lermusiaux P (2012) Dynamical criteria for the evolution of the stochastic dimensionality in flows with uncertainty. Physica D: Nonlinear Phenomena 241(1):60-76

Sapsis T, Majda A (2013a) Blending modified Gaussian closure and non-Gaussian reduced subspace methods for turbulent dynamical systems. Journal of Nonlinear Science 23(6):1039-1071

Sapsis T, Majda A (2013b) Statistically accurate low-order models for uncertainty quantification in turbulent dynamical systems. Proceedings of the National Academy of Sciences 110(34):13705-13710

Sapsis T, Majda A (2013c) A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems. Physica D: Nonlinear Phenomena 252:34-45

Sardeshmukh PD, Sura P (2009) Reconciling non-gaussian climate statistics with linear dynamics. Journal of Climate 22(5):1193-1207

Sardeshmukh PD, Compo GP, Penland C (2015) Need for caution in interpreting extreme weather statistics. Journal of Climate 28(23):9166-9187

Sawford B (1991) Reynolds number effects in lagrangian stochastic models of turbulent dispersion. Physics of Fluids A: Fluid Dynamics 3(6):1577-1586

Scheuerer M, Hamill TM (2015) Variogram-based proper scoring rules for probabilistic forecasts of multivariate quantities. Monthly Weather Review 143(4):1321-1334

Schumann U (1995) Stochastic backscatter of turbulence energy and scalar variance by random subgrid-scale fluxes. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, vol 451, pp 293-318

Separovic L, Charron M, Erfani A, Gagnon N, Zadra A, Vaillancourt P (2016) Model error representation in the canadian ensemble prediction systems. In: ECMWF/WWRP Workshop: Model Uncertainty, ECMWF, Reading

Shutts G (2005) A kinetic energy backscatter algorithm for use in ensemble prediction systems. Quarterly Journal of the Royal Meteorological Society 612:3079-3012

Shutts G (2015) A stochastic convective backscatter scheme for use in ensemble prediction systems. Quarterly Journal of the Royal Meteorological Society 141(692):2602-2616
Smagorinsky J (1963) General circulation experiments with the primitive equation: I. the basic experiment. Monthly Weather Review 91:99-165

Snyder C, Bengtsson T, Morzfeld M (2015) Performance bounds for particle filters using the optimal proposal. Monthly Weather Review 143:4750-4761

Talagrand O (1999) Evaluation of probabilistic prediction systems. In: Workshop proceedings" Workshop on predictability", 20-22 October 1997, ECMWF, Reading, UK

Tandeo P, Ailliot P, Bocquet M, Carrassi A, Miyoshi T, Pulido M, Zhen Y (2018) Joint estimation of model and observation error covariance matrices in data assimilation: a review. arXiv preprint arXiv:180711221

Thorarinsdottir TL (2017) Verification: assessment of calibration and

Trevisan A, Palatella L (2011) On the Kalman filter error covariance collapse into the
unstable subspace. Nonlinear Processes in Geophysics 18(2):243-250

Trevisan A, Uboldi F (2004) Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle system. Journal of the atmospheric sciences 61(1):103-113

Vallis G (2006) Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation. Cambridge University Press

Veneziani M, Griffa A, Reynolds A, Mariano A (2004) Oceanic turbulence and stochastic
models from subsurface Lagrangian data for the Northwest Atlantic Ocean. Journal of
physical oceanography 34(8):1884–1906

Williams CK, Rasmussen CE (2006) Gaussian processes for machine learning. the MIT Press 2(3):4

Wouters J, Gottwald GA (2018) Stochastic model reduction for slow-fast systems with moderate time-scale separation. arXiv preprint arXiv:180409537

Wouters J, Lucarini V (2012) Disentangling multi-level systems: averaging, correlations and memory. Journal of Statistical Mechanics: Theory and Experiment 2012(03):P03003

Yang Y (2014) Study of variational ensemble methods for image assimilation. PhD thesis,
Rennes 1

Yang Y, Mémin E (2017) High-resolution data assimilation through stochastic subgrid tensor and parameter estimation from 4denvar. Tellus A: Dynamic Meteorology and Oceanography 69(1):1308772

Yang Y, Mémin E (2018) Estimation of physical parameters under location uncertainty using an ensemble 2-expectation-maximization algorithm. Quarterly Journal of the Royal Meteorological Society
Smagorinsky J (1963) General circulation experiments with the primitive equation: I. the basic experiment. Monthly Weather Review 91:99-165

Snyder C, Bengtsson T, Morzfeld M (2015) Performance bounds for particle filters using the optimal proposal. Monthly Weather Review 143:4750-4761

Talagrand O (1999) Evaluation of probabilistic prediction systems. In: Workshop proceedings" Workshop on predictability", 20-22 October 1997, ECMWF, Reading, UK

Tandeo P, Ailliot P, Bocquet M, Carrassi A, Miyoshi T, Pulido M, Zhen Y (2018) Joint estimation of model and observation error covariance matrices in data assimilation: a review. arXiv preprint arXiv:180711221

Thorarinsdottir TL (2017) Verification: assessment of calibration and

Trevisan A, Palatella L (2011) On the Kalman filter error covariance collapse into the unstable subspace. Nonlinear Processes in Geophysics 18(2):243-250

Trevisan A, Uboldi F (2004) Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle system. Journal of the atmospheric sciences 61(1):103-113

Vallis G (2006) Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation. Cambridge University Press

Veneziani M, Griffa A, Reynolds A, Mariano A (2004) Oceanic turbulence and stochastic models from subsurface Lagrangian data for the Northwest Atlantic Ocean. Journal of physical oceanography 34(8):1884-1906

Williams CK, Rasmussen CE (2006) Gaussian processes for machine learning. the MIT Press 2(3):4

Wouters J, Gottwald GA (2018) Stochastic model reduction for slow-fast systems with moderate time-scale separation. arXiv preprint arXiv:180409537

Wouters J, Lucarini V (2012) Disentangling multi-level systems: averaging, correlations and memory. Journal of Statistical Mechanics: Theory and Experiment 2012(03):P03003

Yang Y (2014) Study of variational ensemble methods for image assimilation. PhD thesis, Rennes 1

Yang Y, Mémin E (2017) High-resolution data assimilation through stochastic subgrid tensor and parameter estimation from 4denvar. Tellus A: Dynamic Meteorology and Oceanography 69(1):1308772

Yang Y, Mémin E (2018) Estimation of physical parameters under location uncertainty using an ensemble 2-expectation-maximization algorithm. Quarterly Journal of the Royal Meteorological Society