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( Judd et al. 2004b ) to the Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Rosmond 1991 ). In section 2 we first provide a brief overview of some important properties of nonlinear dynamical systems from a geometric point of view. It is noted that model states can be expected to evolve toward an attracting manifold, which is of a lower dimension than the entire state space. We argue that forecast errors can be resolved into two distinct components: one due to initial
( Judd et al. 2004b ) to the Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Rosmond 1991 ). In section 2 we first provide a brief overview of some important properties of nonlinear dynamical systems from a geometric point of view. It is noted that model states can be expected to evolve toward an attracting manifold, which is of a lower dimension than the entire state space. We argue that forecast errors can be resolved into two distinct components: one due to initial
1. Motivation Numerical weather forecasting errors grow with time as a result of two contributing factors. First, atmospheric instabilities amplify uncertainties in the initial conditions, causing indistinguishable states of the atmosphere to diverge rapidly on small scales. This phenomenon is known as internal error growth. Second, model deficiencies introduce errors during the model integration leading to external error growth. These deficiencies include inaccurate forcings and
1. Motivation Numerical weather forecasting errors grow with time as a result of two contributing factors. First, atmospheric instabilities amplify uncertainties in the initial conditions, causing indistinguishable states of the atmosphere to diverge rapidly on small scales. This phenomenon is known as internal error growth. Second, model deficiencies introduce errors during the model integration leading to external error growth. These deficiencies include inaccurate forcings and
the observations are assumed random. In numerical weather prediction (NWP) models, the nudging coefficients are typically tuned manually. Those coefficients are analogous to the matrix function multiplying the white noise in the stochastic differential equation; in the optimal solution it depends on the instantaneous uncertainty in the dynamical system, which includes model error. Because nudging does not consider that uncertainty, it can be tuned to be resistant to model error. In practice
the observations are assumed random. In numerical weather prediction (NWP) models, the nudging coefficients are typically tuned manually. Those coefficients are analogous to the matrix function multiplying the white noise in the stochastic differential equation; in the optimal solution it depends on the instantaneous uncertainty in the dynamical system, which includes model error. Because nudging does not consider that uncertainty, it can be tuned to be resistant to model error. In practice
1. Introduction The Kalman filter provides a coherent closed description for the evolution of errors in a data-assimilation cycle. Estimation errors evolve and grow with the dynamics of the forecast model and are also due to differences between that model and the atmosphere. The errors are periodically reduced because of the assimilation of observations. The Kalman filter is a theoretically attractive algorithm for the optimal estimation of atmospheric states ( Ghil et al. 1981 ). For an
1. Introduction The Kalman filter provides a coherent closed description for the evolution of errors in a data-assimilation cycle. Estimation errors evolve and grow with the dynamics of the forecast model and are also due to differences between that model and the atmosphere. The errors are periodically reduced because of the assimilation of observations. The Kalman filter is a theoretically attractive algorithm for the optimal estimation of atmospheric states ( Ghil et al. 1981 ). For an
research forecasts across a range of scales from short-range to seasonal and annual predictions ( Berner et al. 2008 , 2009 , 2011 ; Charron et al. 2010 ; Doblas-Reyes et al. 2009 ; Tennant et al. 2011 ; Weisheimer et al. 2011 ). It is well known that certain aspects of systematic model error seem to be persistent not only in different model versions of the same model, but also across models developed at different climate centers. For example, many climate models underpredict the observed
research forecasts across a range of scales from short-range to seasonal and annual predictions ( Berner et al. 2008 , 2009 , 2011 ; Charron et al. 2010 ; Doblas-Reyes et al. 2009 ; Tennant et al. 2011 ; Weisheimer et al. 2011 ). It is well known that certain aspects of systematic model error seem to be persistent not only in different model versions of the same model, but also across models developed at different climate centers. For example, many climate models underpredict the observed
al. 1998 ), the presumed benefit of the EnKF method is its ability to provide flow-dependent estimates of the background-error covariances through ensemble covariances so that the observations and the background are more appropriately weighted during the assimilation. Encouraging results have been reported in both observing system simulation experiments (OSSEs) and experiments with real numerical weather prediction (NWP) models and observations; in the numerical predictions from global to
al. 1998 ), the presumed benefit of the EnKF method is its ability to provide flow-dependent estimates of the background-error covariances through ensemble covariances so that the observations and the background are more appropriately weighted during the assimilation. Encouraging results have been reported in both observing system simulation experiments (OSSEs) and experiments with real numerical weather prediction (NWP) models and observations; in the numerical predictions from global to
1. Introduction Despite the significant efforts made in climate modeling since phase 3 of World Climate Research Programme (WCRP) Coupled Model Intercomparison Project (CMIP3), large systematic errors in precipitation, clouds, and water vapor remain in the simulated climate mean state for majority of the climate models in the recent CMIP5 ( Jiang et al. 2012 ; Li et al. 2012 ; Klein et al. 2013 ). Understanding the origin of these systematic errors is challenging because nonlinear feedback
1. Introduction Despite the significant efforts made in climate modeling since phase 3 of World Climate Research Programme (WCRP) Coupled Model Intercomparison Project (CMIP3), large systematic errors in precipitation, clouds, and water vapor remain in the simulated climate mean state for majority of the climate models in the recent CMIP5 ( Jiang et al. 2012 ; Li et al. 2012 ; Klein et al. 2013 ). Understanding the origin of these systematic errors is challenging because nonlinear feedback
no perfect model in reality. Several studies have indicated that considering model errors is important for improving the skill of numerical weather forecasting and climate prediction ( Buizza et al. 1999 ; Palmer 2000 ; Orrell et al. 2001 ; Orrell 2005 ; Palmer et al. 2009 ; Duan et al. 2013 ; Vannitsem 2014 ). In view of this, new ensemble forecasting methods have been designed to address forecast uncertainties caused by model errors. For example, the ECMWF proposed a stochastically
no perfect model in reality. Several studies have indicated that considering model errors is important for improving the skill of numerical weather forecasting and climate prediction ( Buizza et al. 1999 ; Palmer 2000 ; Orrell et al. 2001 ; Orrell 2005 ; Palmer et al. 2009 ; Duan et al. 2013 ; Vannitsem 2014 ). In view of this, new ensemble forecasting methods have been designed to address forecast uncertainties caused by model errors. For example, the ECMWF proposed a stochastically
1. Introduction The growth of systematic errors in general circulation models (GCMs) remains one of the central problems in producing accurate predictions of climate change for the next 50 to 100 years ( Randall et al. 2007 ). Although great advances in global climate modeling have been made in recent decades ( Solomon et al. 2007 ), there are still large uncertainties in many processes such as clouds, convection, and coupling to the oceans and the land surface (e.g., Cubasch et al. 2001
1. Introduction The growth of systematic errors in general circulation models (GCMs) remains one of the central problems in producing accurate predictions of climate change for the next 50 to 100 years ( Randall et al. 2007 ). Although great advances in global climate modeling have been made in recent decades ( Solomon et al. 2007 ), there are still large uncertainties in many processes such as clouds, convection, and coupling to the oceans and the land surface (e.g., Cubasch et al. 2001
1. Limitations of performance metrics One of the primary objectives of measurement or model evaluation is to quantify the uncertainty in the data. This is because the uncertainty directly determines the information content of the data (e.g., Jaynes 2003 ), and dictates our rational use of the information, be it for data assimilation, hypothesis testing, or decision-making. Further, by appropriately quantifying the uncertainty, one gains insight into the error characteristics of the measurement
1. Limitations of performance metrics One of the primary objectives of measurement or model evaluation is to quantify the uncertainty in the data. This is because the uncertainty directly determines the information content of the data (e.g., Jaynes 2003 ), and dictates our rational use of the information, be it for data assimilation, hypothesis testing, or decision-making. Further, by appropriately quantifying the uncertainty, one gains insight into the error characteristics of the measurement
