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## Abstract

The Liouville equation provides the framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system originating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics. Its solution describes the temporal development of the probability density function of the state vector of a given dynamical model. Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need to generate a large number of realizations within ensemble forecasting. These benefits, however, are obtained only at the expense of considering high-dimensional problems.

The purpose of this work, presented in two pans, is to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. After a review of the basic form of the Liouville equation, for the case that the dynamical system considered is represented by a set of coupled ordinary nonlinear first-order (nonstochastic) differential equations that are generic for meteorologically relevant situations, the general *analytical* solution of the Liouville equation is presented in this first part. This explicit solution allows one, at least in principle, to express in analytical terms the time evolution of the probability density function of the state vector of a given meteorological model.

Several properties of the general solution are discussed. As an illustration, the general solution is used to solve the Liouville equation relevant for a one-dimensional nonlinear dynamical system. The fundamental role of the Liouville equation in the context of predicting forecast skill is emphasized.

## Abstract

The Liouville equation provides the framework for the consistent and comprehensive treatment of the uncertainty inherent in meteorological forecasts. This equation expresses the conservation of the phase-space integral of the number density of realizations of a dynamical system originating at the same time instant from different initial conditions, in a way completely analogous to the continuity equation for mass in fluid mechanics. Its solution describes the temporal development of the probability density function of the state vector of a given dynamical model. Consideration of the Liouville equation ostensibly avoids in a natural way the problems inherent to more standard methodology for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need to generate a large number of realizations within ensemble forecasting. These benefits, however, are obtained only at the expense of considering high-dimensional problems.

The purpose of this work, presented in two pans, is to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. After a review of the basic form of the Liouville equation, for the case that the dynamical system considered is represented by a set of coupled ordinary nonlinear first-order (nonstochastic) differential equations that are generic for meteorologically relevant situations, the general *analytical* solution of the Liouville equation is presented in this first part. This explicit solution allows one, at least in principle, to express in analytical terms the time evolution of the probability density function of the state vector of a given meteorological model.

Several properties of the general solution are discussed. As an illustration, the general solution is used to solve the Liouville equation relevant for a one-dimensional nonlinear dynamical system. The fundamental role of the Liouville equation in the context of predicting forecast skill is emphasized.

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## Abstract

The Liouville equation represents the consistent and comprehensive framework for the treatment of the uncertainty inherent in meteorological forecasts. By its very nature, consideration of the Liouville equation avoids problems that are inherent to commonly used methods for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need for the generation of large ensemble sizes within ensemble forecasting.

The general analytical solution of the Liouville equation presented in the first part of this work is used here to find the solution of the Liouville equation relevant for two low-dimensional nonlinear dynamical systems and to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. The analytical solution of the Liouville equation in these examples, namely, the time-dependent probability density function, is discussed and compared with corresponding results obtained by stochastic-dynamic prediction and ensemble forecasting. The negative effect of the unavoidable higher-moment discard within stochastic-dynamic prediction is quantitatively demonstrated. It is also shown that a large number of ensemble members is necessary to obtain accurate estimates of statistics, such as means and (co)variances, even in the low-dimensional situations considered.

It is concluded that, due to its fundamental role in dealing with initial state uncertainty in dynamical models, the Liouville equation must be considered as a highly valuable and useful guideline during the process of developing a coherent methodology for forecasting forecast skill that minimizes deficiencies of currently used methods.

## Abstract

The Liouville equation represents the consistent and comprehensive framework for the treatment of the uncertainty inherent in meteorological forecasts. By its very nature, consideration of the Liouville equation avoids problems that are inherent to commonly used methods for predicting forecast skill, such as the need for higher-moment closure within stochastic-dynamic prediction, or the need for the generation of large ensemble sizes within ensemble forecasting.

The general analytical solution of the Liouville equation presented in the first part of this work is used here to find the solution of the Liouville equation relevant for two low-dimensional nonlinear dynamical systems and to investigate the potential usefulness of the Liouville equation in the context of predicting forecast skill. The analytical solution of the Liouville equation in these examples, namely, the time-dependent probability density function, is discussed and compared with corresponding results obtained by stochastic-dynamic prediction and ensemble forecasting. The negative effect of the unavoidable higher-moment discard within stochastic-dynamic prediction is quantitatively demonstrated. It is also shown that a large number of ensemble members is necessary to obtain accurate estimates of statistics, such as means and (co)variances, even in the low-dimensional situations considered.

It is concluded that, due to its fundamental role in dealing with initial state uncertainty in dynamical models, the Liouville equation must be considered as a highly valuable and useful guideline during the process of developing a coherent methodology for forecasting forecast skill that minimizes deficiencies of currently used methods.

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## Abstract

Total energy *E* as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for *E* is derived for the QG model formulation of Marshall and Molteni. While *E* is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total energy norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total energy or just the kinetic energy norm are briefly illustrated and contrasted. It is argued that *E* might preferably be used (rather than kinetic energy) in predictability and data assimilation studies that are based on the QG model considered here.

## Abstract

Total energy *E* as the sum of kinetic and available potential energies is considered here for quasigeostrophic (QG) dynamics. The discrete expression for *E* is derived for the QG model formulation of Marshall and Molteni. While *E* is conserved by the nonlinear unforced model equations, an analogous expression in terms of perturbed fields is, in general, not conserved for tangent-linearized versions of the model, thereby allowing for growth (or decay) in this total energy norm. Examples for structures linearly growing optimally (i.e., the so-called singular vectors) in terms of either the total energy or just the kinetic energy norm are briefly illustrated and contrasted. It is argued that *E* might preferably be used (rather than kinetic energy) in predictability and data assimilation studies that are based on the QG model considered here.

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## Abstract

The semigeostrophic potential vorticity equation is derived in a vector-based notation for the shallow-water and primitive equations, as models for atmospheric flow. The derivation proceeds from knowledge of the functional form of potential vorticity and starts directly from a vector form of the governing equations. The method makes use of highly involved vector identities and provides for a clearer picture of the nature of the final result, as compared to a component-based derivation. It is, however, limited by the need to know a priori the appropriate functional vector form of the dynamically relevant quantity, such as potential vorticity.

## Abstract

The semigeostrophic potential vorticity equation is derived in a vector-based notation for the shallow-water and primitive equations, as models for atmospheric flow. The derivation proceeds from knowledge of the functional form of potential vorticity and starts directly from a vector form of the governing equations. The method makes use of highly involved vector identities and provides for a clearer picture of the nature of the final result, as compared to a component-based derivation. It is, however, limited by the need to know a priori the appropriate functional vector form of the dynamically relevant quantity, such as potential vorticity.

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## Abstract

Variational data assimilation systems require the specification of the covariances of background and observation errors. Although the specification of the background-error covariances has been the subject of intense research, current operational data assimilation systems still rely on essentially static and thus flow-independent background-error covariances. At least theoretically, it is possible to use flow-dependent background-error covariances in four-dimensional variational data assimilation (4DVAR) through exploiting the connection between variational data assimilation and estimation theory.

This paper reports on investigations concerning the impact of flow-dependent background-error covariances in an idealized 4DVAR system that, based on quasigeostrophic dynamics, assimilates artificial observations. The main emphasis is placed on quantifying the improvement in analysis quality that is achievable in 4DVAR through the use of flow-dependent background-error covariances. Flow dependence is achieved through dynamical error-covariance evolution based on singular vectors in a reduced-rank approach, referred to as reduced-rank Kalman filter (RRKF). The RRKF yields partly dynamic background-error covariances through blending static and dynamic information, where the dynamic information is obtained from error evolution in a subspace of dimension *k* (defined here through the singular vectors) that may be small compared to the dimension of the model’s phase space *n*, which is equal to 1449 in the system investigated here.

The results show that the use of flow-dependent background-error covariances based on the RRKF leads to improved analyses compared to a system using static background-error statistics. That latter system uses static background-error covariances that are carefully tuned given the model dynamics and the observational information available. It is also shown that the performance of the RRKF approaches the performance of the extended Kalman filter, as *k* approaches *n*. Results therefore support the hypothesis that significant analysis improvement is possible through the use of flow-dependent background-error covariances given that a sufficiently large number (here on the order of *n*/10) of singular vectors is used.

## Abstract

Variational data assimilation systems require the specification of the covariances of background and observation errors. Although the specification of the background-error covariances has been the subject of intense research, current operational data assimilation systems still rely on essentially static and thus flow-independent background-error covariances. At least theoretically, it is possible to use flow-dependent background-error covariances in four-dimensional variational data assimilation (4DVAR) through exploiting the connection between variational data assimilation and estimation theory.

This paper reports on investigations concerning the impact of flow-dependent background-error covariances in an idealized 4DVAR system that, based on quasigeostrophic dynamics, assimilates artificial observations. The main emphasis is placed on quantifying the improvement in analysis quality that is achievable in 4DVAR through the use of flow-dependent background-error covariances. Flow dependence is achieved through dynamical error-covariance evolution based on singular vectors in a reduced-rank approach, referred to as reduced-rank Kalman filter (RRKF). The RRKF yields partly dynamic background-error covariances through blending static and dynamic information, where the dynamic information is obtained from error evolution in a subspace of dimension *k* (defined here through the singular vectors) that may be small compared to the dimension of the model’s phase space *n*, which is equal to 1449 in the system investigated here.

The results show that the use of flow-dependent background-error covariances based on the RRKF leads to improved analyses compared to a system using static background-error statistics. That latter system uses static background-error covariances that are carefully tuned given the model dynamics and the observational information available. It is also shown that the performance of the RRKF approaches the performance of the extended Kalman filter, as *k* approaches *n*. Results therefore support the hypothesis that significant analysis improvement is possible through the use of flow-dependent background-error covariances given that a sufficiently large number (here on the order of *n*/10) of singular vectors is used.

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## Abstract

The Eady model with rigid upper lid is considered. The resonant, linearly amplifying solution that exists in the situation of neutral normal modes when an infinitely thin potential vorticity (PV) perturbation is located precisely at the steering level of a zero-PV neutral mode is explicitly derived. This resonant solution is discussed by partitioning the solution into nonzero-PV and zero-PV contributions. It possesses key observed properties of a growing baroclinic structure. The partitioning of the resonant solution clearly demonstrates the existence of modal growth in a short-wave Eady setting. In contrast to the semi-infinite model, a nonamplifying zero-PV contribution is necessary, in addition to the resonant part, to ensure vanishing vertical velocity at both the upper and the lower lid.

## Abstract

The Eady model with rigid upper lid is considered. The resonant, linearly amplifying solution that exists in the situation of neutral normal modes when an infinitely thin potential vorticity (PV) perturbation is located precisely at the steering level of a zero-PV neutral mode is explicitly derived. This resonant solution is discussed by partitioning the solution into nonzero-PV and zero-PV contributions. It possesses key observed properties of a growing baroclinic structure. The partitioning of the resonant solution clearly demonstrates the existence of modal growth in a short-wave Eady setting. In contrast to the semi-infinite model, a nonamplifying zero-PV contribution is necessary, in addition to the resonant part, to ensure vanishing vertical velocity at both the upper and the lower lid.

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## Abstract

This paper explores the relationship between the quality and value of imperfect forecasts. It is assumed that these forecasts are produced by a primitive probabilistic forecasting system and that the decision-making problem of concern is the cost-loss ratio situation. In this context, two parameters describing basic characteristics of the forecasts must be specified in order to determine forecast quality uniquely. As a result, a scalar measure of accuracy such as the Brier score cannot completely and unambiguously describe the quality of the imperfect forecasts. The relationship between forecast accuracy and forecast value is represented by a multivalued function—an accuracy/value envelope. Existence of this envelope implies that the Brier score is an imprecise measure of value and that forecast value can even decrease as forecast accuracy increases (and vice versa). The generality of these results and their implications for verification procedures and practices are discussed.

## Abstract

This paper explores the relationship between the quality and value of imperfect forecasts. It is assumed that these forecasts are produced by a primitive probabilistic forecasting system and that the decision-making problem of concern is the cost-loss ratio situation. In this context, two parameters describing basic characteristics of the forecasts must be specified in order to determine forecast quality uniquely. As a result, a scalar measure of accuracy such as the Brier score cannot completely and unambiguously describe the quality of the imperfect forecasts. The relationship between forecast accuracy and forecast value is represented by a multivalued function—an accuracy/value envelope. Existence of this envelope implies that the Brier score is an imprecise measure of value and that forecast value can even decrease as forecast accuracy increases (and vice versa). The generality of these results and their implications for verification procedures and practices are discussed.

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## Abstract

Optimal perturbations, also referred to as singular vectors (SVs), currently constitute an important guideline for the generation of initial ensembles to be used for ensemble prediction. The optimality of these perturbations refers to their property of maximizing prespecified quadratic measures of error growth, given that tangent-linear error evolution is assumed. The goal of ensemble prediction is the accurate prediction of the uncertainty of forecasts made with dynamical numerical weather prediction models.

In the present paper the theoretical justification for the use of SVs in ensemble prediction systems is investigated. It is shown that, in a tangent-linear framework, SVs—constructed using covariance information valid at the initial time—evolve into the eigenvectors of the forecast error covariance matrix valid for the end of the optimization interval. As such, SVs represent the most efficient means for predicting the forecast error covariance matrix, given a prespecified number of allowable (tangent-linear) model integrations. Such optimal prediction is of particular importance in light of the fact that the forecast error covariance matrix is summarizing important information about the probability density function of the model state at a given future time.

Based on the above result, optimal covariance prediction through appropriately determined SVs is demonstrated here for a three-dimensional Lorenz model, as well as for a barotropic model of intermediate dimensionality, both within a perfect-model framework. In the case of the barotropic model it is found that less than 15% of the SVs suffice to account for more than 95% of the total final error variance. Viewed differently, at least 80% of the final error variance is accounted for by retaining those SVs that are amplifying in terms of an enstrophy norm. In addition, variances and covariances predicted through SVs agree closely with independently obtained Monte Carlo estimates, as long as the tangent-linear approximation is sufficiently accurate.

Further, the problem of approximating the forecast error covariance matrix in the presence of a state-independent model-error representation is briefly considered. The paper is concluded with a summary of the results and a discussion of their possible implications on data assimilation procedures and on the further development of ensemble prediction systems.

## Abstract

Optimal perturbations, also referred to as singular vectors (SVs), currently constitute an important guideline for the generation of initial ensembles to be used for ensemble prediction. The optimality of these perturbations refers to their property of maximizing prespecified quadratic measures of error growth, given that tangent-linear error evolution is assumed. The goal of ensemble prediction is the accurate prediction of the uncertainty of forecasts made with dynamical numerical weather prediction models.

In the present paper the theoretical justification for the use of SVs in ensemble prediction systems is investigated. It is shown that, in a tangent-linear framework, SVs—constructed using covariance information valid at the initial time—evolve into the eigenvectors of the forecast error covariance matrix valid for the end of the optimization interval. As such, SVs represent the most efficient means for predicting the forecast error covariance matrix, given a prespecified number of allowable (tangent-linear) model integrations. Such optimal prediction is of particular importance in light of the fact that the forecast error covariance matrix is summarizing important information about the probability density function of the model state at a given future time.

Based on the above result, optimal covariance prediction through appropriately determined SVs is demonstrated here for a three-dimensional Lorenz model, as well as for a barotropic model of intermediate dimensionality, both within a perfect-model framework. In the case of the barotropic model it is found that less than 15% of the SVs suffice to account for more than 95% of the total final error variance. Viewed differently, at least 80% of the final error variance is accounted for by retaining those SVs that are amplifying in terms of an enstrophy norm. In addition, variances and covariances predicted through SVs agree closely with independently obtained Monte Carlo estimates, as long as the tangent-linear approximation is sufficiently accurate.

Further, the problem of approximating the forecast error covariance matrix in the presence of a state-independent model-error representation is briefly considered. The paper is concluded with a summary of the results and a discussion of their possible implications on data assimilation procedures and on the further development of ensemble prediction systems.

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## Abstract

The spectrum of finite-time most unstable structures, also referred to as singular vectors (SVs), is computed for a regional, mesoscale primitive-equation model. The number of growing SVs present in this spectrum is of interest for investigating mesoscale predictability since it provides an estimate of the dimension of the unstable subspace of the model phase space. This dimension is used to critically assess the contrasting conclusions that have been reached by different authors in mesoscale predictability studies.

Computations are carried out for two different synoptic cases (explosive cyclogenesis over the North Atlantic and Alpine lee cyclogenesis) using two different norms. The first is loosely related to total perturbation energy and the second measures the energy of rotational normal modes only. The latter is designed to reduce the influence of geostrophic adjustment on the measure of growth. The models used are the tangent-linear and adjoint components of the dry-adiabatic version of the primitive-equation regional model denoted as the National Center for Atmospheric Research Mesoscale Adjoint Modeling System version 1. Spectra are obtained for a 24-hour optimization time interval by partially solving the relevant eigenproblems in an iterative fashion using a Lanczos algorithm. The spectra relevant for the first norm are found to possess a very large number of growing SVs, a considerable part of which, however, is growing purely due to adjustment processes. The number of growing structures relevant for the second norm is between 150 and 200, which is approximately 3% of the total degrees of freedom allowed for the perturbations in this situation. This percentage becomes as small as 0.25% if all degrees of freedom are taken into account.

The first few SVs amplify by factors between 5 and 10 and are strongly related to the synoptic situation under consideration, being quite localized and possessing baroclinic structure. Also, similar characteristics are found for the first few SVs, independent of which norm is being used.

Based on the small number of growing perturbations, it is concluded that it is quite likely that a randomly chosen perturbation will decay because its projection on the growing part of the spectrum is small. Nevertheless, this does not necessarily imply that mesoscale circulations are more, predictable than synoptic-scale circulations due to neglected larger growing scales, the neglect of physical processes like convection, and the short timescale considered. Implications of the results regarding the role of SVs for data assimilation and ensemble prediction are briefly discussed.

## Abstract

The spectrum of finite-time most unstable structures, also referred to as singular vectors (SVs), is computed for a regional, mesoscale primitive-equation model. The number of growing SVs present in this spectrum is of interest for investigating mesoscale predictability since it provides an estimate of the dimension of the unstable subspace of the model phase space. This dimension is used to critically assess the contrasting conclusions that have been reached by different authors in mesoscale predictability studies.

Computations are carried out for two different synoptic cases (explosive cyclogenesis over the North Atlantic and Alpine lee cyclogenesis) using two different norms. The first is loosely related to total perturbation energy and the second measures the energy of rotational normal modes only. The latter is designed to reduce the influence of geostrophic adjustment on the measure of growth. The models used are the tangent-linear and adjoint components of the dry-adiabatic version of the primitive-equation regional model denoted as the National Center for Atmospheric Research Mesoscale Adjoint Modeling System version 1. Spectra are obtained for a 24-hour optimization time interval by partially solving the relevant eigenproblems in an iterative fashion using a Lanczos algorithm. The spectra relevant for the first norm are found to possess a very large number of growing SVs, a considerable part of which, however, is growing purely due to adjustment processes. The number of growing structures relevant for the second norm is between 150 and 200, which is approximately 3% of the total degrees of freedom allowed for the perturbations in this situation. This percentage becomes as small as 0.25% if all degrees of freedom are taken into account.

The first few SVs amplify by factors between 5 and 10 and are strongly related to the synoptic situation under consideration, being quite localized and possessing baroclinic structure. Also, similar characteristics are found for the first few SVs, independent of which norm is being used.

Based on the small number of growing perturbations, it is concluded that it is quite likely that a randomly chosen perturbation will decay because its projection on the growing part of the spectrum is small. Nevertheless, this does not necessarily imply that mesoscale circulations are more, predictable than synoptic-scale circulations due to neglected larger growing scales, the neglect of physical processes like convection, and the short timescale considered. Implications of the results regarding the role of SVs for data assimilation and ensemble prediction are briefly discussed.

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## Abstract

The sufficiency relation, originally developed in the context of the comparison of statistical experiments, provides a sound basis for the comparative evaluation of forecasting systems. The importance of this relation resides in the fact that if forecasting system *A* can be shown to be sufficient for forecasting system *B*, then all users will find *A*'s forecasts of greater value than *B*'s forecasts regardless of their individual payoff structures.

In this paper the sufficiency relation is applied to the problem of comparative evaluation of prototypical climate forecasting systems. The primary objectives here are to assess the basic applicability of the sufficiency relation in this context and to investigate the implications of this approach for the relationships among the performance characteristics of such forecasting systems.

The results confirm that forecasting system *A* is sufficient for forecasting system *B* when the former uses more extreme probabilities more frequently than the latter. Further, in terms of the relatively simple forecasting systems considered here, it is found that system *A* may be sufficient for system *B* even if the former uses extreme forecasts less frequently, provided that *A*'s forecasts are—to a certain degree—more extreme than *B*'s forecasts. Conversely, system *A* cannot be shown to be sufficient for system *B* if the former users less extreme forecasts more frequently than the latter. The advantages of the sufficiency relation over traditional performance measures in this context are also demonstrated.

Several issues related to the general applicability of the sufficiency relation to the comparative evaluation of climate forecasts are discussed. Possible extensions of this work, as well as some implications of the results for verification procedures and practices in this context, am briefly described.

## Abstract

The sufficiency relation, originally developed in the context of the comparison of statistical experiments, provides a sound basis for the comparative evaluation of forecasting systems. The importance of this relation resides in the fact that if forecasting system *A* can be shown to be sufficient for forecasting system *B*, then all users will find *A*'s forecasts of greater value than *B*'s forecasts regardless of their individual payoff structures.

In this paper the sufficiency relation is applied to the problem of comparative evaluation of prototypical climate forecasting systems. The primary objectives here are to assess the basic applicability of the sufficiency relation in this context and to investigate the implications of this approach for the relationships among the performance characteristics of such forecasting systems.

The results confirm that forecasting system *A* is sufficient for forecasting system *B* when the former uses more extreme probabilities more frequently than the latter. Further, in terms of the relatively simple forecasting systems considered here, it is found that system *A* may be sufficient for system *B* even if the former uses extreme forecasts less frequently, provided that *A*'s forecasts are—to a certain degree—more extreme than *B*'s forecasts. Conversely, system *A* cannot be shown to be sufficient for system *B* if the former users less extreme forecasts more frequently than the latter. The advantages of the sufficiency relation over traditional performance measures in this context are also demonstrated.

Several issues related to the general applicability of the sufficiency relation to the comparative evaluation of climate forecasts are discussed. Possible extensions of this work, as well as some implications of the results for verification procedures and practices in this context, am briefly described.