Tangent Linear Approximation Research Articles

A tangent linear approximation of the ignition delay time. II: Sensitivity to thermochemical parameters

The tangent linear approximation (TLA) developed in Almohammadi et al. (Combust. Flame 230, 111426) is extended to estimate the sensitivity of the ignition delay time with respect to species enthalpies and entropies. The proposed method relies on integrating the linearized system of equations governing the evolution of the state vector’s partial derivatives with respect to variations in thermodynamic parameters. The sensitivity of the ignition delay time is estimated through a linearized approximation of a temperature functional. The TLA approach is applied to three gas mixtures, H2, n-butanol, and iso-octane, reacting in air under adiabatic, constant-volume conditions. The numerical experiments indicate that the linearized approximation of the ignition delay time’s sensitivity is in excellent agreement with the finite-difference estimates. This is also the case for sensitivity estimates obtained using the TLA approach. Further, significant computational speed-ups are achieved with the TLA approach, and the method scales well with the number of perturbed parameters. In the case of the H2 mechanism, TLA is about ten times faster than finite differences, and this enhancement becomes even more substantial when more complex mechanisms are considered.

A tangent linear approximation of the ignition delay time. I: Sensitivity to rate parameters

A tangent linear approximation is developed to estimate the sensitivity of the ignition delay time with respect to individual rate parameters in a detailed chemical mechanism. Attention is focused on a gas mixture reacting under adiabatic, constant-volume conditions. The uncertainty in the rates of elementary reactions is described in terms of uncertainty factors, and are parameterized using independent canonical random variables. The approach is based on integrating the linearized system of equations governing the evolution of the partial derivatives of the state vector with respect to individual random variables, and a linearized approximation is developed to relate the ignition delay sensitivity to the scaled partial derivatives of temperature. The efficiency of the approach is demonstrated through applications to chemical mechanisms of different sizes. In particular, the computations indicate that for detailed reaction mechanisms the TLA leads to robust local sensitivity predictions at a computational cost that is order-of-magnitude smaller than that incurred by finite-difference approaches based on one-at-a-time rate perturbations.

Adjoint-Based Observation Impact Estimation with Direct Verification Using Forward Calculation

Abstract The adjoint-based observation impact estimation method has been providing essential information to improve data assimilation systems (DASs) in numerical weather prediction (NWP). This paper has two purposes. The first is to verify the four approximations used in the method: iterative construction of the Kalman gain adjoint operator, the tangent linear (TL) approximation of a forecast model, ignoring cross terms of observation impacts, and approximations for incremental DASs. The second is to add new information to our knowledge of observation impacts. For the verification of the adjoint-based method, we use the TL-based observation impact estimation method that can calculate the same quantity as the adjoint-based method without adjoint calculations. Results of these verifications and observation impact estimations performed on the global NWP system of the Japan Meteorological Agency are as follows. First, observation impacts calculated using the adjoint-based method agree well with those from the TL-based method, with the correlation coefficient between them exceeding 0.97. Second, estimated observation impacts are consistent with previous studies in many aspects. There are also system-dependent properties, such as relatively small impacts from GPS radio occultation data above 13 km. Furthermore, new aspects of observation impacts are found: 1) the probability density function of the observation impact agrees well with the scalar theory when giving the experimental value of the observation and the forecast error standard deviations; 2) later observations in the data assimilation window have larger positive impacts; and 3) impacts of AMSU-A in the Southern Hemisphere are more than double those in the Northern Hemisphere.

Strongly Coupled Data Assimilation Experiments with Linearized Ocean–Atmosphere Balance Relationships

Coupled data assimilation is emerging as a target approach for Earth system prediction and reanalysis systems. Coupled data assimilation may be indeed able to minimize unbalanced air–sea initialization and maximize the intermedium propagation of observations. Here, we use a simplified framework where a global ocean general circulation model (NEMO) is coupled to an atmospheric boundary layer model [Cheap Atmospheric Mixed Layer (CheapAML)], which includes prognostic prediction of near-surface air temperature and moisture and allows for thermodynamic but not dynamic air–sea coupling. The control vector of an ocean variational data assimilation system is augmented to include 2-m atmospheric parameters. Cross-medium balances are formulated either through statistical cross covariances from monthly anomalies or through the application of linearized air–sea flux relationships derived from the tangent linear approximation of bulk formulas, which represents a novel solution to the coupled assimilation problem. As a proof of concept, the methodology is first applied to study the impact of in situ ocean observing networks on the near-surface atmospheric analyses and later to the complementary study of the impact of 2-m air observations on sea surface parameters, to assess benefits of strongly versus weakly coupled data assimilation. Several forecast experiments have been conducted for the period from June to December 2011. We find that especially after day 2 of the forecasts, strongly coupled data assimilation provides a beneficial impact, particularly in the tropical oceans. In most areas, the use of linearized air–sea balances outperforms the statistical relationships used, providing a motivation for implementing coupled tangent linear trajectories in four-dimensional variational data assimilation systems. Further impacts of strongly coupled data assimilation might be found by retuning the background error covariances.

Improving the ensemble transform Kalman filter using a second-order Taylor approximation of the nonlinear observation operator

Abstract. The ensemble transform Kalman filter (ETKF) assimilation scheme has recently seen rapid development and wide application. As a specific implementation of the ensemble Kalman filter (EnKF), the ETKF is computationally more efficient than the conventional EnKF. However, the current implementation of the ETKF still has some limitations when the observation operator is strongly nonlinear. One problem in the minimization of a nonlinear objective function similar to 4D-Var is that the nonlinear operator and its tangent-linear operator have to be calculated iteratively if the Hessian is not preconditioned or if the Hessian has to be calculated several times. This may be computationally expensive. Another problem is that it uses the tangent-linear approximation of the observation operator to estimate the multiplicative inflation factor of the forecast errors, which may not be sufficiently accurate. This study attempts to solve these problems. First, we apply the second-order Taylor approximation to the nonlinear observation operator in which the operator, its tangent-linear operator and Hessian are calculated only once. The related computational cost is also discussed. Second, we propose a scheme to estimate the inflation factor when the observation operator is strongly nonlinear. Experimentation with the Lorenz 96 model shows that using the second-order Taylor approximation of the nonlinear observation operator leads to a reduction in the analysis error compared with the traditional linear approximation method. Furthermore, the proposed inflation scheme leads to a reduction in the analysis error compared with the procedure using the traditional inflation scheme.

The linearisation of maps in data assimilation

For the purpose of linearising maps in data assimilation, the tangent-linear approximation is often used. We compare this with the use of the ‘best linear’ approximation, which is the linear map that minimises the mean square error. As a benchmark, we use minimum variance filters and smoothers which are non-linear generalisations of Kalman filters and smoothers. We show that use of the best linear approximation leads to a filter whose prior has first moment unapproximated compared with the benchmark, and second moment whose departure from the benchmark is bounded independently of the derivative of the map, with similar results for smoothers. This is particularly advantageous where the maps in question are strongly non-linear on the scale of the increments. Furthermore, the best linear approximation works equally well for maps which are non-differentiable. We illustrate the results with examples using low-dimensional chaotic maps.

Robust localisation algorithm for solving neighbour position ambiguity

Localisation error due to sensitivity to position ambiguity, with respect to some neighbouring nodes, is a major problem in wireless localisation. To solve this problem, the tangent linear approximation (TLA) for the nonlinear localisation model is derived, contributing to incorporating the effect of the neighbours' position ambiguity into belief propagation (BP) localisation. Theoretical analysis and numerical results indicate that the TLA-BP algorithm can acquire higher accuracy and convergence probability compared with other typical methods.

A reduced adjoint approach to variational data assimilation

The adjoint method has been used very often for variational data assimilation. The computational cost to run the adjoint model often exceeds several original model runs and the method needs significant programming efforts to implement the adjoint model code. The work proposed here is variational data assimilation based on proper orthogonal decomposition (POD) which avoids the implementation of the adjoint of the tangent linear approximation of the original nonlinear model. An ensemble of the forward model simulations is used to determine the approximation of the covariance matrix and only the dominant eigenvectors of this matrix are used to define a model subspace. The adjoint of the tangent linear model is replaced by the reduced adjoint based on this reduced space. Thus the adjoint model is run in reduced space with negligible computational cost. Once the gradient is obtained in reduced space it is projected back in full space and the minimization process is carried in full space. In the paper the reduced adjoint approach to variational data assimilation is introduced. The characteristics and performance of the method are illustrated with a number of data assimilation experiments in a ground water subsurface contaminant model.

Application of model reduced 4D-Var to a 1D ecosystem model

The model reduced 4D-Var (Vermeulen and Heemink, 2006) is investigated to test its feasibility in ecosystem application. Ecological models are known for their high nonlinearity and issues with non-differentiability. However, since the method is performed in the reduced space, the implementation of the adjoint of the tangent linear approximation of the original model is not required.Twin experiments are conducted in a 1D ecological model. Surface phytoplankton data are used, with 30% log-normally distributed measurement error. Three parameters are chosen for calibration. The method performs very well in the setup where a perfect initial condition is used, as well as for the combined parameter and initial condition estimation. A relatively well calibrated initial condition contributes to accurate parameter estimations. Not accounting for the wrongly assigned initial condition leads to incorrectly calibrated parameters.

Tangent linear approximation based observation data impact estimation in 4D‐Var

AbstractIn variational data assimilation systems (variational DASs), the effect of each assimilated observation dataset on an analysis field is one of the main factors in determining analysis and succeeding forecast accuracy. We call the effects ‘observation impacts’. However, the estimation of the observation impacts is difficult because variational DASs do not construct the Kalman gain that fully determines them. Therefore, all known methods for the estimation have some shortcomings. We construct a new method for the observation impact estimation. The method estimates the observation impacts as a partial analysis increment vector (PIV) that is generated by each observation dataset. For real calculation of the PIVs, we introduce partial departure vectors. The method enables us to see how the Kalman gain transforms information of observations into analysis increments. The method can also estimate the observation impacts on a forecast field while the tangent linear approximation of a forecast model is valid. Comparisons of formulations of three types of the known methods with the new method clarify the origins of nonlinear interactions between observation impacts in the known methods. We perform a preliminary experiment of one analysis cycle on the global variational DAS of the Japan Meteorological Agency to demonstrate the validity and performance of the method. Observation data are divided into satellite radiance and other conventional data, and the latter are further divided into rawinsondes, satellite winds, and the others. The experimental results are as follows: (i) the method can identify the observation impacts of each observation dataset on an analysis and a forecast field in the operational variational DAS, (ii) there are negative correlations between an integrated background error vector and dominant PIVs, and (iii) the nonlinear interactions between observation impacts in the known methods can be estimated quantitatively. The method is useful for observational data impact estimation in variational DASs. Copyright © 2011 Royal Meteorological Society

Model-reduced gradient-based history matching

Gradient-based history matching algorithms can be used to adapt the uncertain parameters in a reservoir model using production data. They require, however, the implementation of an adjoint model to compute the gradients, which is usually an enormous programming effort. We propose a new approach to gradient-based history matching which is based on model reduction, where the original (nonlinear and high-order) forward model is replaced by a linear reduced-order forward model and, consequently, the adjoint of the tangent linear approximation of the original forward model is replaced by the adjoint of a linear reduced-order forward model. The reducedorder model is constructed with the aid of the proper orthogonal decomposition method. Due to the linear character of the reduced model, the corresponding adjoint model is easily obtained. The gradient of the objective function is approximated, and the minimization problem is solved in the reduced space; the procedure is iterated with the updated estimate of the parameters if necessary. The proposed approach is adjointfree and can be used with any reservoir simulator. The method was evaluated for a waterflood reservoir with channelized permeability field. A comparison with an adjoint-based history matching procedure shows that the model-reduced approach gives a comparable quality of history matches and predictions. The computational efficiency of the model-reduced approach is lower than of an adjoint-based approach, but higher than of an approach where the gradients are obtained with simple finite differences.

An economical approach to four-dimensional variational data assimilation

Four-dimensional variational data assimilation (4DVar) is one of the most promising methods to provide optimal analysis for numerical weather prediction (NWP). Five national NWP centers in the world have successfully applied 4DVar methods in their global NWPs, thanks to the increment method and adjoint technique. However, the application of 4DVar is still limited by the computer resources available at many NWP centers and research institutes. It is essential, therefore, to further reduce the computational cost of 4DVar. Here, an economical approach to implement 4DVar is proposed, using the technique of dimensionreduced projection (DRP), which is called “DRP-4DVar.” The proposed approach is based on dimension reduction using an ensemble of historical samples to define a subspace. It directly obtains an optimal solution in the reduced space by fitting observations with historical time series generated by the model to form consistent forecast states, and therefore does not require implementation of the adjoint of tangent linear approximation.

A Method of Successive Corrections of the Control Subspace in the Reduced-Order Variational Data Assimilation*

Abstract A version of the reduced control space four-dimensional variational method (R4DVAR) of data assimilation into numerical models is proposed. In contrast to the conventional 4DVAR schemes, the method does not require development of the tangent linear and adjoint codes for implementation. The proposed R4DVAR technique is based on minimization of the cost function in a sequence of low-dimensional subspaces of the control space. Performance of the method is demonstrated in a series of twin-data assimilation experiments into a nonlinear quasigeostrophic model utilized as a strong constraint. When the adjoint code is stable, R4DVAR’s convergence rate is comparable to that of the standard 4DVAR algorithm. In the presence of strong instabilities in the direct model, R4DVAR works better than 4DVAR whose performance is deteriorated because of the breakdown of the tangent linear approximation. Comparison of the 4DVAR and R4DVAR also shows that R4DVAR becomes advantageous when observations are sparse and noisy.

SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

Tests of an Adjoint Mesoscale Model with Explicit Moist Physics on the Cloud Scale

Abstract An adjoint modeling system based upon the Naval Research Laboratory’s Coupled Ocean–Atmosphere Mesoscale Prediction System’s atmospheric component has been developed. The system includes the adjoint model of the explicit moist physics parameterization, which allows for gradients with respect to the initial hydrometeor concentrations to be calculated. This work focuses on the ability of the system to calculate evolved perturbations and gradients for the hydrometeor variables. Tests of the tangent linear and adjoint models for an idealized convective case at high model resolution (4-km horizontal grid spacing) are presented in this study. The tangent linear approximation is shown to be acceptable for all model variables (including the hydrometeors) with sizable perturbations for forecasts of 1 h. The adjoint model was utilized with the same convective case to demonstrate its applicability in four-dimensional variational data assimilation experiments. Identical twin experiments were conducted where the adjoint model produced gradients for all model variables, leading to improved analyses and forecasts. The best agreement between model forecasts and simulated observations occurred when information on all model variables was assimilated. In the case where only conventional data were assimilated, the agreement was not as good in the early forecast period. However, the hydrometeor values spun up quickly, and at later times, the forecast performed almost as well as when all data were assimilated.

Sensitivity analysis of mountain waves using an adjoint model

An adjoint modeling system developed for the COAMPS nonhydrostatic model is used to explore the sensitivity of lee-side winds to the upstream atmospheric conditions for flow over a two-dimensional obstacle. For relatively small hills in the hydrostatic wave regime, the sensitivity patterns exhibit a dual lobed structure that is a manifestation of a superposition of internal gravity waves with downward-directed energy propagation. Nonhydrostatic waves generated by small terrain have corresponding sensitivities that are tilted vertically against the shear, which when introduced into the flow as perturbations, evolve into structures that resemble vertically decaying evanescent waves. Flow over higher obstacles near the gravity wave breaking threshold exhibits complex sensitivity patterns characterized by a wave-like packet of maxima and minima upstream of the middle- and upper-tropospheric region of wave breaking. In general, as the mountain height is increased, the tangent linear approximation becomes less accurate. However, the strongest nonlinearity occurs for flows very near the wave breaking threshold, rather than fully within the wave breaking regime forced by higher terrain.

Further investigation on adjoint sensitivity of surface precipitation to initial conditions

The adjoint version of the Canadian global numerical weather prediction model including moist physical parameterizations is used to investigate the sensitivity of surface precipitation to initial conditions. This study complements results shown in a previous paper by MAHFOUF and BILODEAU (2007). The importance of the dynamics is examined by considering the sensitivity in terms of streamfunction and velocity potential. The contribution of the divergence field to the sensitivity produced by the dynamics dominates at the beginning of the adjoint integration. The adjoint sensitivity of surface precipitation is compared to the sensitivity of a diagnostic function defined by the integrated water vapour in rainy areas. This comparison provides useful information on the relevance of the 1D+4D-Var methodology for the variational assimilation of rainfall observations. Finally, a set of 8 additional case studies is examined in order to increase the confidence in our previous conclusions obtained for two meteorological systems. The validity of the tangent-linear approximation is case dependent but in average is better for mid-latitude than for tropical regions. The dominance of the initial temperature field on surface precipitation is confirmed for all systems.

Atmospheric Predictability at Synoptic Versus Cloud-Resolving Scales

The limited atmospheric predictability has been addressed by the development of ensemble prediction systems (EPS) that are now routinely applied for medium-range synoptic-scale numerical weather prediction (NWP). With the increase of computational power, interest is growing in the design of high-resolution (cloud resolving) NWP models and their associated short-range EPS. This development raises a series of fundamental questions, especially concerning the type of error growth and the validity of the tangent-linear approximation. To address these issues, a comparison between perturbed medium-range (10 day) synoptic-scale integrations (taken from the operational ECMWF EPS with a horizontal resolution of about 80 km) and short-range (1 day) high-resolution simulations (based on the Lokal Modell of the Consortium for Small-Scale Modeling with a grid spacing of 2.2 km) is conducted. The differences between the two systems are interpreted in a nondimensional sense and illustrated with the help of the Lorenz att...

Adjoint Sensitivity of Surface Precipitation to Initial Conditions

Abstract The adjoint version of the Global Environmental Multiscale model including a comprehensive package of simplified and linearized physical processes (large-scale condensation, deep moist convection, vertical diffusion, and subgrid-scale orographic effects) is used to evaluate the sensitivity of surface precipitation to initial conditions for up to 24 h for two meteorological systems: a midlatitude front and a tropical cyclone. Such diagnostics are useful to improve the understanding on variational assimilation of precipitation data. In agreement with a similar study, the largest sensitivity is found with respect to the temperature field for both stratiform and convective precipitation. Close to the observation time and for stratiform precipitation, the sensitivity with respect to specific humidity is rather large, which corroborates conclusions from previous one-dimensional variational data assimilation experimentations. The sensitivity is then reduced significantly after the observation time. The sensitivities of surface precipitation to the wind components and to specific humidity are comparable and are at a maximum at the observation time. The sensitivity to the surface pressure is always much smaller than the sensitivity to the other variables. In general, sensitivities are largest at the observation time and then decrease. However, for the midlatitude perturbation, the sensitivity is enhanced after 12 h for stratiform precipitation and also for convective precipitation using a scheme based on the moisture convergence closure. This results from a dynamical coupling upstream of the area of interest through baroclinic instability as evidenced by vertically backward-tilted sensitivities. Such enhancement is not observed for the tropical case. The tangent-linear approximation remains acceptable for accumulated precipitation up to 24 h but is rather poor for instantaneous rain rates. The variational assimilation of accumulated precipitation should thus be favored.

Assimilation of the Surface Precipitation Data with JNoVA using 2-ice Bulk Microphysics Scheme

The assimilation of surface precipitation data with the ‘JMA nonhydrostatic model(JMA-NHM)’-based four-dimensional variational data assimilation system (JNoVA) is presented. The tangent linear and the adjoint codes of the 2-ice bulk microphysics scheme of Rutledge and Hobbs (1983) have been recently developed in nearly straightforward manner and the accuracy of the tangent linear model has been investigated. We succeed to assimilate the Radar-Raingauge analyzed precipitation data with the JNoVA using this cloud microphysics with horizontal grid spacing of 10 km and improve the precipitation of the analysis.The same precipitation data is successfully assimilated with the JNoVA using the simplified moist physics. The comparison of both analyses revealed that the analysis by the JNoVA using the 2-ice bulk scheme is not optimized as much as that by the JNoVA using the simplified physics. From the investigation of the cost functions, the tangent linear approximation of 2-ice bulk scheme is not accurate enough to use in the variational data assimilation system with the horizontal grid spacing of 10 km.