Singular Vector Analysis Research Articles

Predictability in process-based ensemble forecast of influenza.

Process-based models have been used to simulate and forecast a number of nonlinear dynamical systems, including influenza and other infectious diseases. In this work, we evaluate the effects of model initial condition error and stochastic fluctuation on forecast accuracy in a compartmental model of influenza transmission. These two types of errors are found to have qualitatively similar growth patterns during model integration, indicating that dynamic error growth, regardless of source, is a dominant component of forecast inaccuracy. We therefore examine the nonlinear growth of model initial error and compute the fastest growing directions using singular vector analysis. Using this information, we generate perturbations in an ensemble forecast system of influenza to obtain more optimal ensemble spread. In retrospective forecasts of historical outbreaks for 95 US cities from 2003 to 2014, this approach improves short-term forecast of incidence over the next one to four weeks.

Singular vector-based targeted observations of chemical constituents: description and first application of the EURAD-IM-SVA v1.0

Abstract. Measurements of the large-dimensional chemical state of the atmosphere provide only sparse snapshots of the state of the system due to their typically insufficient temporal and spatial density. In order to optimize the measurement configurations despite those limitations, the present work describes the identification of sensitive states of the chemical system as optimal target areas for adaptive observations. For this purpose, the technique of singular vector analysis (SVA), which has proven effective for targeted observations in numerical weather prediction, is implemented in the EURAD-IM (EURopean Air pollution and Dispersion – Inverse Model) chemical transport model, yielding the EURAD-IM-SVA v1.0. Besides initial values, emissions are investigated as critical simulation controlling targeting variables. For both variants, singular vectors are applied to determine the optimal placement for observations and moreover to quantify which chemical compounds have to be observed with preference. Based on measurements of the airship based ZEPTER-2 campaign, the EURAD-IM-SVA v1.0 has been evaluated by conducting a comprehensive set of model runs involving different initial states and simulation lengths. For the sake of brevity, we concentrate our attention on the following chemical compounds, O3, NO, NO2, HCHO, CO, HONO, and OH, and focus on their influence on selected O3 profiles. Our analysis shows that the optimal placement for observations of chemical species is not entirely determined by mere transport and mixing processes. Rather, a combination of initial chemical concentrations, chemical conversions, and meteorological processes determines the influence of chemical compounds and regions. We furthermore demonstrate that the optimal placement of observations of emission strengths is highly dependent on the location of emission sources and that the benefit of including emissions as target variables outperforms the value of initial value optimization with growing simulation length. The obtained results confirm the benefit of considering both initial values and emission strengths as target variables and of applying the EURAD-IM-SVA v1.0 for measurement decision guidance with respect to chemical compounds.

On the construction of a direct numerical simulation of a breaking inertia-gravity wave in the upper mesosphere

A systematic approach to the direct numerical simulation (DNS) of breaking upper mesospheric inertia-gravity waves of amplitude close to or above the threshold for static instability is presented. Normal mode or singular vector analysis applied in a frame of reference moving with the phase velocity of the wave (in which the wave is a steady solution) is used to determine the most likely scale and structure of the primary instability and to initialize nonlinear “2.5-D” simulations (with three-dimensional velocity and vorticity fields but depending only on two spatial coordinates). Singular vector analysis is then applied to the time-dependent 2.5-D solution to predict the transition of the breaking event to three-dimensional turbulence and to initialize three-dimensional DNS. The careful choice of the computational domain and the relatively low Reynolds numbers, on the order of 25,000, relevant to breaking waves in the upper mesosphere, makes the three-dimensional DNS tractable with present-day computing clusters. Three test cases are presented: a statically unstable low-frequency inertia-gravity wave, a statically and dynamically stable inertia-gravity wave, and a statically unstable high-frequency gravity wave. The three-dimensional DNS are compared to ensembles of 2.5-D simulations. In general, the decay of the wave and generation of turbulence is faster in three dimensions, but the results are otherwise qualitatively and quantitatively similar, suggesting that results of 2.5-D simulations are meaningful if the domain and initial condition are chosen properly.

The role of eddy feedback in the excitation of the NAO

In this paper, the role of eddy feedback in the formation and excitation of the NAO (North Atlantic Oscillation) is examined in a linear barotropic model with an empirical closure for synoptic eddy and low-frequency flow (SELF) feedback. The EOF-space based model captures the linear dynamics of the barotropic model linearized with respect to climatological mean flow very accurately with relative low-order truncations. The parameterized empirical linear low-dimension SELF-feedback operator also depicts the relation between eddy-induced streamfunction tendencies and the large-scale circulation to a reasonable degree. Singular vector analysis (SVD) is performed on the linear low-dimension dynamic operator with SELF-feedback to exam the leading low-frequency mode of the system. It is shown that the leading SVD mode (the left vector) of the linear low-dimension dynamic operator with smallest SVD value resembles the observed NAO pattern. The role of eddy feedback in the excitation and maintenance of NAO is then further examined by relating the effective external forcing patterns to the right vector of the leading SVD mode of the linear dynamic operator. It is shown that local dipolar external forcing at middle latitudes over North Atlantic is effective in exciting the NAO-like anomaly pattern, whereas the effectiveness of the external forcing over different areas sensitively depends on the eddy feedback.

Singular vector decomposition for sensitivity analyses of tropospheric chemical scenarios

Abstract. Observations of the chemical state of the atmosphere typically provide only sparse snapshots of the state of the system due to their insufficient temporal and spatial density. One possibility for optimisation of the state estimate is to target the observation of those parameters that have the largest potential of resulting in forecast improvements. In the present work, the technique of singular vector analysis is applied to atmospheric chemical modelling in order to identify the most sensitive chemical compounds during a given time period and prioritise them for measurement. Novel to the current work is the fact that, in the application of singular vector analysis, not only the initial values but also the emissions are considered as target variables for adaptive observation strategies. This specific application of singular vector analysis is studied in the context of a chemistry box model allowing for validation of its new features for two chemical regimes. The time and regime dependence of the ozone (O3) and peroxyacetyl nitrate (PAN) formation potential of individual volatile organic compounds (VOCs) is investigated. Results show that the combined sensitivity of O3 and PAN to individual VOCs is strongly dependent on the photochemical scenario and simulation interval used. Particularly the alkanes show increasing sensitivities with increasing simulation length. Classifying the VOCs as being of high, medium, little or negligible importance for the formation of O3 and PAN allows for the identification of those VOCs that may be omitted from measurement. We find that it is possible to omit 6 out of 18 VOCs considered for initial value measurement and 4 out of 12 VOCs considered for emission measurement. The omission of these VOCs is independent of photochemical regime and simulation length. The VOCs selected for measuring account for more than 96% and 90% of the O3 and PAN sensitivity to VOCs, respectively.

Fast Higher-Order MR Image Reconstruction Using Singular-Vector Separation

Medical resonance imaging (MRI) conventionally relies on spatially linear gradient fields for image encoding. However, in practice various sources of nonlinear fields can perturb the encoding process and give rise to artifacts unless they are suitably addressed at the reconstruction level. Accounting for field perturbations that are neither linear in space nor constant over time, i.e., dynamic higher-order fields, is particularly challenging. It was previously shown to be feasible with conjugate-gradient iteration. However, so far this approach has been relatively slow due to the need to carry out explicit matrix-vector multiplications in each cycle. In this work, it is proposed to accelerate higher-order reconstruction by expanding the encoding matrix such that fast Fourier transform can be employed for more efficient matrix-vector computation. The underlying principle is to represent the perturbing terms as sums of separable functions of space and time. Compact representations with this property are found by singular-vector analysis of the perturbing matrix. Guidelines for balancing the accuracy and speed of the resulting algorithm are derived by error propagation analysis. The proposed technique is demonstrated for the case of higher-order field perturbations due to eddy currents caused by diffusion weighting. In this example, image reconstruction was accelerated by two orders of magnitude.

Secondary Instabilities in Breaking Inertia–Gravity Waves

Abstract The three-dimensionalization of turbulence in the breaking of nearly vertically propagating inertia–gravity waves is investigated numerically using singular vector analysis applied to the Boussinesq equations linearized about three two-dimensional time-dependent basic states obtained from nonlinear simulations of breaking waves: a statically unstable wave perturbed by its leading transverse normal mode, the same wave perturbed by its leading parallel normal mode, and a statically stable wave perturbed by a leading transverse singular vector. The secondary instabilities grow through interaction with the buoyancy gradient and velocity shear in the basic state. Which growth mechanism predominates depends on the time-dependent structure of the basic state and the wavelength of the secondary perturbation. The singular vectors are compared to integrations of the linear model using random initial conditions, and the leading few singular vectors are found to be representative of the structures that emerge in the randomly initialized integrations. A main result is that the length scales of the leading secondary instabilities are an order of magnitude smaller than the wavelength of the initial wave, suggesting that the essential dynamics of the breaking might be captured by tractable nonlinear three-dimensional simulations in a relatively small triply periodic domain.

Further analysis of singular vector and ENSO predictability in the Lamont model—Part II: singular value and predictability

This is the second part of the 148 years (1856–2003) singular vector analysis, as an extension of part I (Cheng et al. 2009 Clim Dyn, doi: 10.1007/s00382-009-0595-7 ), in which a fully physically based tangent linear model has been constructed for the Zebiak-Cane model LDEO5 version. In the present study, relationships between the singular values and prediction skill measures are investigated for the 148 years. Results show that at decadal/interdecadal time scales, an inverse relationship exists between the singular value (S1) and correlation-based skill measures whereas an in-phase relationship exists between the S1 and MSE-based skill measures. However, the S1 is not a good measure or predictor of prediction skill at shorter time scales such as the interannual time scale and for individual prediction. To explain these findings, S1 was decomposed into linear perturbation growth rate (L1) and linearized nonlinear perturbation growth rate (N1), which are controlled by the opposite underlying model dynamical processes (the linear warming and the nonlinear cooling). An offsetting effect was found between L1 and N1, which have opposite contributions to the S1 (i.e., S1 ≈ L1 − N1). The “negative” perturbation growth rate −N1 (denoted as NN1) probably is the consequence of the unrealistic nonlinear cooling in the LDEO5 model. Although the correlations of the actual prediction skill to both the L1 and the NN1 are good, their opposite signs lead to a weak relationship between S1 and actual prediction skill. Therefore, either L1 or N1/NN1 is better than S1 in measuring actual prediction skill for the LDEO5 model.

A novel technique for nonlinear sensitivity analysis: application to moist predictability

AbstractA new nonlinear technique is described to compute the sensitivity of synoptic perturbation growth to environmental moisture. The perturbation growth is defined using a nonlinear generalization of singular vectors called Nonlinear Singular Vectors (NLSV). For a given atmospheric state evolving in time, the nonlinear sensitivity method consists in maximizing the growth rate of perturbations by seeking both the optimal NLSV perturbations and the most favorable spatial distribution of the moisture field. This results in a new atmospheric state that differs initially only by the water vapour field. The NLSV computed along this new state has the largest possible growth rate for all possible water vapour fields. We apply this method to a simulation of the moist primitive equations. For the particular case we study, we obtain a moistening of the lower troposphere and an amplification of energy of optimal perturbations three times larger than the amplification of optimal perturbations computed without altering the water vapour field. The optimal perturbations are similar to the perturbations of the unmodified water vapour case. A noteworthy property is that the complete saturation of the atmosphere leads to a smaller increase in amplification rate, which means that the water vapour field is strongly tied to the NLSV structure. Mechanisms explaining these results are discussed. This technique overcomes the limitations (in particular the linearity assumption) of moist singular vector analysis using moist norms or adjoint sensitivity analysis. It can be applied to diagnose sensitivity to other fields as well. Copyright © 2009 Royal Meteorological Society

Further analysis of singular vector and ENSO predictability in the Lamont model—Part I: singular vector and the control factors

In this study, singular vector analysis was performed for the period from 1856 to 2003 using the latest Zebiak-Cane model version LDEO5. The singular vector, representing the optimal growth pattern of initial pertur- bations/errors, was obtained by perturbing the constructed tangent linear model of the Zebiak-Cane model. Variations in the singular vector and singular value, as a function of initial time, season, ENSO states, and optimal period, were investigated. Emphasis was placed on exploring relative roles of linear and nonlinear processes in the optimal per- turbation growth of ENSO, and deriving statistically robust conclusions using long-term singular vector analysis. It was found that the first singular vector is dominated by a west-east dipole spanning most of the equatorial Pacific, with one center located in the east and the other in the central Pacific. Singular vectors are less sensitive to initial conditions, i.e., independence of seasons and decades; while singular values exhibit a strong sensitivity to initial conditions. The dynamical diagnosis shows that the total linear and nonlinear heating terms play opposite roles in controlling the optimal perturbation growth, and that the linear optimal perturbation is more than twice as large as the nonlinear one. The total linear heating causes a warming effect and controls two positive perturbation growth regions: one in the central Pacific and the other in the eastern Pacific; whereas the total linearized nonlinear advection brings a cooling effect controlling the negative perturbation growth in the central Pacific.

Improved ENSO Prediction by Singular Vector Analysis in a Hybrid Coupled Model

Abstract In this study, a method based on singular vector analysis is proposed to improve El Niño–Southern Oscillation (ENSO) predictions. Its essential idea is that the initial errors are projected onto their optimal growth patterns, which are propagated by the tangent linear model (TLM) of the original prediction model. The forecast errors at a given lead time of predictions are obtained, and then removed from the raw predictions. This method is applied to a realistic ENSO prediction model for improving prediction skill for the period from 1980 to 1999. This correction method considerably improves the ENSO prediction skill, compared with the original predictions without the correction.

Application of singular vector analysis to the Kuroshio large meander

Singular vector analysis is applied to the formation of the Kuroshio large meander south of Japan. We use a simulation run of an ocean general circulation model (OGCM), in which a large meander occurs, as the background state. Singular vectors are calculated by the Lanczos method in which the tangent linear and adjoint models of the OGCM are used. We apply an incremental correction approach for setting up the initial perturbation and averaging for the final norm in order to get modes reflecting nonlinear physics by filtering out high‐frequency variations. The perturbation extracted as the leading right singular vector affects the formation of the large meander most if it is set to the background state two months before. When the perturbation is added, it enhances an anticyclonic eddy southeast of Kyushu in the upper layer, which induces downwelling north of it and develops an anticyclonic eddy in the deep layer. The anticyclonic eddy produces a cold anomaly at the downstream edge, and produces baroclinic instability there, resulting in further development of the large meander. In contrast, the anticyclonic eddy in the deep layer is weakened when the perturbation is subtracted, the development of the cold anomaly in the deep layer is delayed, and consequently the meander is not so well developed. Thus singular vector analysis reveals the role of the anticyclonic eddy approaching southeast of Kyushu in the upper layer, and indicates the importance of the anticyclonic eddy in the deep layer.

The impact of atmospheric nonlinearities on the fastest growth of ENSO prediction error

In this paper we explore the impact of atmospheric nonlinearities on the optimal growth of initial condition error of El Nino and the Southern Oscillation (ENSO) prediction using singular vector (SV) analysis. This is performed by comparing and analyzing SVs of two hybrid coupled models (HCMs), one composed of an intermediate complexity dynamical ocean model coupled with a linear statistical atmospheric model, and the other one with the same ocean model coupled with a nonlinear statistical atmosphere. Tangent linear and adjoint models for both HCMs are developed. SVs are computed under the initial conditions of seasonal background and actual ENSO cycle simulated by the ocean model forced with the real wind data of 1980–1999. The optimization periods of 3, 6 and 9 months are individually considered. The results show that the first SVs in both HCMs are very similar to each other, characterized by a central east-west dipole pattern spanning over the entire tropical Pacific. The spatial patterns of the leading SV in both HCMs are not sensitive to optimization periods and initial time. However, the first singular value, indicating the optimal growth rate of prediction error, displays considerable differences between the two HCMs, indicating a significant impact of atmospheric nonlinearities on the optimal growth of ENSO prediction error. These differences are greater with increasing optimization time, suggesting that the impact of atmospheric nonlinearities on the optimal growth of prediction error becomes larger for a longer period of prediction.

Singular Vector Analysis for Atmospheric Chemical Transport Models

Abstract The singular vectors of a chemical transport model are the directions of maximum perturbation growth over a finite time interval. They have proved useful for the estimation of error growth, the initialization of ensemble forecasts, and the optimal placement of adaptive observations. The aim of this paper is to address computational aspects of singular vector analysis for atmospheric chemical transport models. The distinguishing feature of these models is the presence of stiff chemical interactions. A projection approach to preserve the symmetry of the tangent linear–adjoint operator for stiff systems is discussed, and extended to 3D chemical transport simulations. Numerical results are presented for a simulation of atmospheric pollution in East Asia in March 2001. The singular values and the structure of the singular vectors depend on the length of the simulation interval, the meteorological data, the location of the optimization region and the selection of optimization species, the choice of error norms, and the size of the optimization region.

Climatological relevance of leading seasonal singular vectors. Part I: Energy, enstrophy and spatio temporal variability

Fast growing atmospheric Rossby wave trains (RWT modes) are solutions of a Singular Vector (SV) analysis of a damped barotropic vorticity equation for northern winter basic flows. Using 40 DJF basic flows RWT modes are found over four regions of the globe only. Their propagation paths are remarkably constant for development times up to 96 h, but spread for longer times. The RWT mode with the largest 4 day growth rate develops over the North-Pacific (NPAC) region for each of the observed DJF basic flows considered. This mode is referred to as NPAC-mode. The eigenvalues, which are the 4 day kinetic energy growth factors of the NPAC mode, range from 10 to 24 for nearly all of the basic flows. In some exceptional years they are close to 30. The NPAC modes with moderate growth factors could be shown to be approximate solutions of the nonlinear model equation. Hereto a new formulation of the advection term provided by the energy-vortex theory was used. It appears that the constancy of propagation paths and the mode's shapes, the large growth factors and the validity of the linearization assumption up to development times of 96h make the NPAC mode a candidate to explain substantial parts of large scale interannual atmospheric variability in the North Pacific region.

ENSO Predictability of a Fully Coupled GCM Model Using Singular Vector Analysis

Abstract Using a recently developed method of computing climatically relevant singular vectors (SVs), the error growth properties of ENSO in a fully coupled global climate model are investigated. In particular, the authors examine in detail how singular vectors are influenced by the phase of ENSO cycle—the physical variable under consideration as well as the error norm deployed. Previous work using SVs for studying ENSO predictability has been limited to intermediate or hybrid coupled models. The results show that the singular vectors share many of the properties already seen in simpler models. Thus, for example, the singular vector spectrum is dominated by one fastest growing member, regardless of the phase of ENSO cycle and the variable of perturbation or the error norm; in addition the growth rates of the singular vectors are very sensitive to the phase of the ENSO cycle, the variable of perturbation, and the error norm. This particular CGCM also displays some differences from simpler models; thus subsurface temperature optimal patterns are strongly sensitive to the phase of ENSO cycle, and at times an east–west dipole in the eastern tropical Pacific basin is seen. This optimal pattern also appears for SST when the error norm is defined using Niño-4. Simpler models consistently display a single-sign equatorial signature in the subsurface corresponding perhaps to the Wyrtki buildup of heat content before a warm event. Some deficiencies in the CGCM and their possible influences on SV growth are also discussed.

Dynamics of Synoptic Eddy and Low-Frequency Flow Interaction. Part III: Baroclinic Model Results

Abstract In this three-part study, a linear closure has been developed for the synoptic eddy and low-frequency flow (SELF) interaction and demonstrated that internal dynamics plays an important role in generating the leading low-frequency modes in the extratropical circulation anomalies during cold seasons. In Part III, a new linearized primitive equation system is first derived for time-mean flow anomalies. The dynamical operator of the system includes a traditional part depending on the observed climatological mean state and an additional part from the SELF feedback closure utilizing the observed climatological properties of synoptic eddy activity. The latter part relates nonlocally all the anomalous eddy-forcing terms in equations of momentum, temperature, and surface pressure to the time-mean flow anomalies. Using the observational data, the closure was validated with reasonable success, and it was found that terms of the SELF feedback in the momentum and pressure equations tend to reinforce the low-frequency modes, whereas those in the thermodynamic equation tends to damp the temperature anomalies to make the leading modes equivalent barotropic. Through singular vector analysis of the linear dynamical operator, it is highlighted that the leading modes of the system resemble the observed patterns of the Arctic Oscillation, Antarctic Oscillation, and Pacific–North American pattern, in which the SELF feedback plays an essential role, consistent with the finding of the barotropic model study in Part II.

Shear and Static Instability of Inertia–Gravity Wave Packets: Short-Term Modal and Nonmodal Growth

Abstract The problem of nonmodal instabilities of inertia–gravity waves (IGW) in the middle atmosphere is addressed, within the framework of a Boussinesq model with realistic molecular viscosity and thermal diffusion, by singular-vector analysis of horizontally homogeneous vertical profiles of wind and buoyancy obtained from IGW packets at their statically least stable or most unstable horizontal location. Nonmodal growth is always found to be significantly stronger than that of normal modes, most notably at wave amplitudes below the static instability limit where normal-mode instability is very weak, whereas the energy gain between the optimal perturbation and singular vector after one Brunt–Väisälä period can be as large as two orders of magnitude. Among a multitude of rapidly growing singular vectors for this optimization time, small-scale (wavelengths of a few 100 m) perturbations propagating in the horizontal parallel to the IGW are most prominent. These parallel optimal perturbations are amplified by a roll mechanism, while transverse perturbations (with horizontal scales of a few kilometers) are to a large part subject to an Orr mechanism, both controlled by the transverse wind shear in the IGW at its statically least stable altitude, but further enhanced by reduced static stability. The elliptic polarization of the IGW leaves its traces in an additional impact of the roll mechanism via the parallel wind shear on the leading transverse optimal perturbation.

Origin of decadal El Niño–Southern Oscillation–like variability in a coupled general circulation model

The origin of decadal El Niño–Southern Oscillation–like variability (hereinafter referred to as DEV) is analyzed in coupled general circulation model (CGCM) simulations. The CGCM experiments test the null hypothesis for the existence of DEV, that is, that DEV is linear, damped, and stochastically forced by atmospheric internal dynamics. In order to test this hypothesis we employ the interactive ensemble coupling strategy. It is found that the spatially averaged variance of sea surface temperature anomaly (SSTA) and heat content anomaly (HCA) associated with the DEV is linearly related to the amplitude of atmospheric internal dynamics. This result suggests that the DEV is primarily noise forced but that damped coupled feedbacks and/or internal ocean dynamics enhance this variability. The SSTA variability associated with the DEV appears to be of a basin‐type standing mode with a triangular shape. However, the decadal evolution of the HCA associated with the DEV shows a systematic westward propagation from the central North Pacific and the eastern South Pacific. Both a singular vector analysis and noise variance analysis based on the coupled simulation results support the argument that atmospheric noise in the subtropics leads to the excitation of the DEV.

A Hierarchy of Data-Based ENSO Models

AbstractGlobal sea surface temperature (SST) evolution is analyzed by constructing predictive models that best describe the dataset’s statistics. These inverse models assume that the system’s variability is driven by spatially coherent, additive noise that is white in time and are constructed in the phase space of the dataset’s leading empirical orthogonal functions. Multiple linear regression has been widely used to obtain inverse stochastic models; it is generalized here in two ways. First, the dynamics is allowed to be nonlinear by using polynomial regression. Second, a multilevel extension of classic regression allows the additive noise to be correlated in time; to do so, the residual stochastic forcing at a given level is modeled as a function of variables at this level and the preceding ones. The number of variables, as well as the order of nonlinearity, is determined by optimizing model performance.The two-level linear and quadratic models have a better El Niño–Southern Oscillation (ENSO) hindcast skill than their one-level counterparts. Estimates of skewness and kurtosis of the models’ simulated Niño-3 index reveal that the quadratic model reproduces better the observed asymmetry between the positive El Niño and negative La Niña events. The benefits of the quadratic model are less clear in terms of its overall, cross-validated hindcast skill; this model outperforms, however, the linear one in predicting the magnitude of extreme SST anomalies.Seasonal ENSO dependence is captured by incorporating additive, as well as multiplicative forcing with a 12-month period into the first level of each model. The quasi-quadrennial ENSO oscillatory mode is robustly simulated by all models. The “spring barrier” of ENSO forecast skill is explained by Floquet and singular vector analysis, which show that the leading ENSO mode becomes strongly damped in summer, while nonnormal optimum growth has a strong peak in December.