Physical-space Statistical Analysis System Research Articles

Variational Data Assimilation of Tides

This paper presents an incremental variational method to assimilate the observed tidal harmonic constants using a frequency domain linearized shallow water equation. A cost function was constructed with tidal boundary conditions and tidal forcing as its control (independent) variables. To minimize the cost function, optimal boundary conditions and tidal forcing were derived using a conventional dual 4-Dimensional Variational (4D-Var) Physical-space Statistical Analysis System. The tangent linear and adjoint model were solved by using a finite element method. By adapting the incremental form, the variational method streamlines the workflow to provide the incremental correction to the boundary conditions and tidal forcing of a hydrodynamic forward model. The method was tested for semi-diurnal M2 tides in a regional sea with a complex tidal system. The results demonstrate a 65–72% reduction of tidal harmonic constant vector error by assimilating the observed M2 tidal harmonic constants. In addition to improving the tides of a hydrodynamic model by optimizing boundary conditions and tidal forcing, the method computes a spatially varying uncertainty of individual tidal constituents in the model. The method provides a versatile tool for mapping the spatially continuous tides and currents in coastal and estuarine waters by assimilating the harmonic constants of individual tidal constituents of observed tides and currents.

Using multi-platform 4D-Var data assimilation to improve modeling of Adriatic Sea dynamics

A year-long Four-Dimensional Variational (4D-Var) data assimilation experiment, formulated on the physical-space statistical analysis system (PSAS) algorithm, has been applied to the Adriatic Sea for the very first time. High-resolution multi-platform observations were assimilated into the Regional Ocean Modeling System (ROMS) forced by mesoscale atmospheric model ALADIN/HR (Aire Limitée Adaptation dynamique Développement InterNational), including important freshwater river inflows and realistic open boundary conditions for the period between October 2014 and September 2015. The observations included sea surface temperature (SST) measured by satellites, in situ temperature and salinity data measured by various moving (Argo profiling floats, shipborne CTDs, sea glider, towed CTD profiler) and moored platforms, ocean current profiles measured by moored Acoustic Doppler Current Profilers (ADCPs), and 30-minute de-tided surface currents from high-frequency (HF) radars. Three model simulations were integrated: (1) a non-assimilative simulation over a year-long period — baseline simulation; (2) a non-assimilative simulation initiated by the previous assimilation 4-day cycle — background simulation, and (3) a fully assimilative simulation that used all available observations during the 4-day assimilation cycle — analysis simulation. The assimilation significantly improved the modeling system performance, especially in SST, with time average rmse equaling 0.9, 0.7 and 0.5 °C and bias equaling 0.39, 0.15, 0.01 °C for the baseline, background and analysis simulations, respectively. These reductions were mostly achieved during the wintertime outbreaks of a cold and dry bora wind, caused by an underestimation of heat and momentum fluxes by the atmospheric model. Such fine adjustments in the northern and middle Adriatic may be responsible for improved reproduction of dense water generation and the associated Adriatic thermohaline circulation.

The Inclusion of IWV Estimates from AIRS/AMSU and SSM/I Sensors into the CPTEC/INPE Global Data Assimilation System

Abstract Water vapor plays a crucial role in atmospheric processes and its distribution is associated with cloud-cover fraction and rainfall. The inclusion of integrated water vapor (IWV) estimates in numerical weather prediction improves the vertical structure of the humidity analysis and consequently contributes to obtaining a more realistic atmospheric state. Currently, satellite remote sensing is the most important source of humidity measurements in the Southern Hemisphere, providing information with good horizontal resolution and global coverage. In this study, the inclusion of IWV retrieved from the Atmospheric Infrared Sounder/Advanced Microwave Sounding Unit-A (AIRS/AMSU) and Special Sensor Microwave Imager (SSM/I) were investigated as additional information in the Physical-space Statistical Analysis System (PSAS), which is the operational data assimilation system at the Center for Weather Forecasting and Climate Studies of the Brazilian National Institute for Space Research (CPTEC/INPE). Experiments were carried out with and without the assimilation of IWV values from both sensors. Results show that, in general, the IWV assimilation reduces the error in short-range forecasts of humidity profile, particularly over tropical regions. In these experiments, an analysis of the impact of the inclusion of IWV values from SSM/I and AIRS/AMSU sensors was done. Results indicated that the impact of the SSM/I values is significant over high-latitude oceanic regions in the Southern Hemisphere, while the impact of AIRS/AMSU values is more significant over continental regions where surface measurements are scarce, such as the Amazonian region. In that area the assimilation of IWV values from the AIRS/AMSU sensor shows a tendency to reduce the overestimate of the precipitation in short-range forecasts.

Preconditioning and globalizing conjugate gradients in dual space for quadratically penalized nonlinear-least squares problems

When solving nonlinear least-squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust-region Gauss-Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least-squares subproblem solved at each iteration of the method is rewritten as a quadratic minimization subject to linear equality constraints. This allows the exploitation of duality properties of the associated linearized problems. This paper considers a recent conjugate-gradient-like method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard conjugate-gradients method in the primal space. This dual algorithm is computationally interesting whenever the dimension of the dual space is significantly smaller than that of the primal space, yielding gains in terms of both memory usage and computational cost. The relation between this dual space solver and PSAS (Physical-space Statistical Analysis System), another well-known dual space technique used in data assimilation problems, is explained. The use of an effective preconditioning technique is proposed and refined convergence bounds derived, which results in a practical solution method. Finally, stopping rules adequate for a trust-region solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a Steihaug-Toint truncated conjugate-gradient method in the primal space.

The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems: Part I – System overview and formulation

The Regional Ocean Modeling System (ROMS) is one of the few community ocean general circulation models for which a 4-dimensional variational data assimilation (4D-Var) capability has been developed. The ROMS 4D-Var capability is unique in that three variants of 4D-Var are supported: a primal formulation of incremental strong constraint 4D-Var (I4D-Var), a dual formulation based on a physical-space statistical analysis system (4D-PSAS), and a dual formulation representer-based variant of 4D-Var (R4D-Var). In each case, ROMS is used in conjunction with available observations to identify a best estimate of the ocean circulation based on a set of a priori hypotheses about errors in the initial conditions, boundary conditions, surface forcing, and errors in the model in the case of 4D-PSAS and R4D-Var. In the primal formulation of I4D-Var the search for the best circulation estimate is performed in the full space of the model control vector, while for the dual formulations of 4D-PSAS and R4D-Var only the sub-space of linear functions of the model state vector spanned by the observations (i.e. the dual space) is searched. In oceanographic applications, the number of observations is typically much less than the dimension of the model control vector, so there are clear advantages to limiting the search to the space spanned by the observations. In the case of 4D-PSAS and R4D-Var, the strong constraint assumption (i.e. that the model is error free) can be relaxed leading to the so-called weak constraint formulation. This paper describes the three aforementioned variants of 4D-Var as they are implemented in ROMS. Critical components that are common to each approach are conjugate gradient descent, preconditioning, and error covariance models, which are also described. Finally, several powerful 4D-Var diagnostic tools are discussed, namely computation of posterior errors, eigenvector analysis of the posterior error covariance, observation impact, and observation sensitivity.

The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilation systems: Part II – Performance and application to the California Current System

The Regional Ocean Modeling System (ROMS) 4-dimensional variational (4D-Var) data assimilation systems have been systematically applied to the mesoscale circulation environment of the California Current to demonstrate the performance and practical utility of the various components of ROMS 4D-Var. In particular, we present a comparison of three approaches to 4D-Var, namely: the primal formulation of the incremental strong constraint approach; the dual formulation “physical-space statistical analysis system”; and the dual formulation indirect representer approach. In agreement with theoretical considerations all three approaches converge to the same ocean circulation estimate when using the same observations and prior information. However, the rate of convergence of the dual formulation was found to be inferior to that of the primal formulation. Other aspects of the 4D-Var performance that relate to the use of multiple outer-loops, preconditioning, and the weak constraint are also explored. A systematic evaluation of the impact of the various components of the 4D-Var control vector (i.e. the initial conditions, surface forcing and open boundary conditions) is also presented. It is shown that correcting for uncertainties in the model initial conditions exerts the largest influence on the ability of the model to fit the available observations. Various important diagnostics of 4D-Var are also examined, including estimates of the posterior error, the information content of the observation array, and innovation-based consistency checks on the prior error assumptions. Using these diagnostic tools, we find that more than 90% of the observations assimilated into the model provide redundant information. This is a symptom of the large percentage of satellite data that are used and to some extent the nature of the data processing employed. This is the second in a series of three papers describing the ROMS 4D-Var systems.

An observation‐space formulation of variational assimilation using a restricted preconditioned conjugate gradient algorithm

AbstractWe consider parameter estimation problems involving a set of m physical observations, where an unknown vector of n parameters is defined as the solution of a nonlinear least‐squares problem. We assume that the problem is regularized by a quadratic penalty term. When solution techniques based on successive linearization are considered, as in the incremental four‐dimensional variational (4D‐Var) techniques for data assimilation, a sequence of linear systems with particular structure has to be solved. We exhibit a subspace of dimension m that contains the solution of these linear systems, and derive a variant of the conjugate gradient algorithm that is more efficient in terms of memory and computational costs than its standard form, when m is smaller than n. The new algorithm, which we call the Restricted Preconditioned Conjugate Gradient (RPCG), can be viewed as an alternative to the so‐called Physical‐space Statistical Analysis System (PSAS) algorithm, which is another approach to solve the linear problem. In addition, we show that the non‐monotone and somehow chaotic behaviour of PSAS algorithm when viewed in the model space, experimentally reported by some authors, can be fully suppressed in RPCG.Moreover, since preconditioning and re‐orthogonalization of residuals vectors are often used in practice to accelerate convergence in high‐dimension data assimilation, we show how to reformulate these techniques within subspaces of dimension m in RPCG. Numerical experiments are reported, on an idealized data assimilation system based on the heat equation, that clearly show the effectiveness of our algorithm for large‐scale problems. Copyright © 2009 Royal Meteorological Society

Comparison between Local Ensemble Transform Kalman Filter and PSAS in the NASA finite volume GCM – perfect model experiments

Abstract. This paper compares the performance of the Local Ensemble Transform Kalman Filter (LETKF) with the Physical-Space Statistical Analysis System (PSAS) under a perfect model scenario. PSAS is a 3D-Var assimilation system used operationally in the Goddard Earth Observing System Data Assimilation System (GEOS-4 DAS). The comparison is carried out using simulated winds and geopotential height observations and the finite volume Global Circulation Model with 72 grid points zonally, 46 grid points meridionally and 55 vertical levels. With forty ensemble members, the LETKF obtains analyses and forecasts with significantly lower RMS errors than those from PSAS, especially over the Southern Hemisphere and oceans. This observed advantage of the LETKF over PSAS is due to the ability of the 40-member ensemble LETKF to capture flow-dependent errors and thus create a good estimate of the evolving background uncertainty. An initial decrease of the forecast errors in the Northern Hemisphere observed in the PSAS but not in the LETKF suggests that the LETKF analysis is more balanced.

Application of an error statistics estimation method to the PSAS forecast error covariance model

In atmospheric data assimilation systems, the forecast error covariance model is an important component. However, the parameters required by a forecast error covariance model are difficult to obtain due to the absence of the truth. This study applies an error statistics estimation method to the Physical-space Statistical Analysis System (PSAS) height-wind forecast error covariance model. This method consists of two components: the first component computes the error statistics by using the National Meteorological Center (NMC) method, which is a lagged-forecast difference approach, within the framework of the PSAS height-wind forecast error covariance model; the second obtains a calibration formula to rescale the error standard deviations provided by the NMC method. The calibration is against the error statistics estimated by using a maximum-likelihood estimation (MLE) with rawindsonde height observed-minus-forecast residuals. A complete set of formulas for estimating the error statistics and for the calibration is applied to a one-month-long dataset generated by a general circulation model of the Global Model and Assimilation Office (GMAO), NASA. There is a clear constant relationship between the error statistics estimates of the NMC-method and MLE. The final product provides a full set of 6-hour error statistics required by the PSAS height-wind forecast error covariance model over the globe. The features of these error statistics are examined and discussed.

The Computational Complexity and Parallel Scalability of Atmospheric Data Assimilation Algorithms

Abstract This paper quantifies the computational complexity and parallel scalability of two algorithms for four-dimensional data assimilation (4DDA) at NASA's Global Modeling and Assimilation Office (GMAO). The first, the Goddard Earth Observing System Data Assimilation System (GEOS DAS), uses an atmospheric general circulation model (GCM) and an observation-space-based analysis system, the Physical-Space Statistical Analysis System (PSAS). GEOS DAS is very similar to global meteorological weather forecasting data assimilation systems but is used at NASA for climate research. The second, the Kalman filter, uses a more consistent algorithm to determine the forecast error covariance matrix than does GEOS DAS. For atmospheric assimilation, the gridded dynamical fields typically have more than 106 variables; therefore, the full error covariance matrix may be in excess of a teraword. For the Kalman filter this problem will require petaflop s−1 computing to achieve effective throughput for scientific research.

The GEOS-3 Retrospective Data Assimilation System: The 6-Hour Lag Case

The fixed-lag Kalman smoother (FLKS) has been proposed as a framework to construct data assimilation procedures capable of producing high-quality climate research datasets. FLKS-based systems, referred to as retrospective data assimilation systems, are an extension to three-dimensional filtering procedures with the added capability of incorporating observations not only in the past and present time of the estimate, but also at future times. A variety of simplifications are necessary to render retrospective assimilation procedures practical. In this article, an FLKS-based retrospective data assimilation system implementation for the Goddard Earth Observing System Data Assimilation System is presented. The practicality of this implementation comes from the practicality of its underlying (filter) analysis system, that is, the physical-space statistical analysis system (PSAS). The behavior of two schemes is studied here. The first retrospective analysis (RA) scheme is designed simply to update the regular PSAS analyses with observations available at times ahead of the regular analysis times. Results are presented for when observations 6-h ahead of the analysis time are used to update the PSAS analyses and thereby to calculate the so-called lag-1 retrospective analyses. Consistency tests for this RA scheme show that the lag-1 retrospective analyses indeed have better 6-h predictive skill than the predictions from the regular analyses. This motivates the introduction of the second retrospective analysis scheme, which, at each analysis time, uses the 6-h retrospective analysis to create a new forecast to replace the forecast normally used in the PSAS analysis, and therefore allows the calculation of a revised (filter) PSAS analysis. This procedure is referred to as the retrospective-based iterated analysis (RIA) scheme. Results from the RIA scheme indicate its potential for improving the overall quality of the assimilation.

PSAS and 4D-var data assimilation for chemical state analysis by urban and rural observation sites

Advanced data assimilation can be applied to retrieve as much information as possible from photooxidant measurements in the trophosphere in order to analyze the chemical state. Traditional assimilation methods have proved to be unsatisfactory due to their inability (i) to account for locally varying chemical scenarios, (ii) to exploit informations inherent in time series evolution, and (iii) to use observations measured at different times. The Physical—space Statistical Analysis System (PSAS) and a space—time data assimilation analysis are proposed aiming at the combination of both the four dimensional variational calculus (4D—var) and inhomogeneous error covariances allowing to focus on regional differences in emission rates an surface conditions. The University of Cologne, Eulerian mesoscale—α Chemistry Transport Model EURAD is part of a comprehensive 4D—var chemical data assimilation system which makes best use of available observations to provide skillful analyses. Special emphasis is placed on the locally varying representativeness of individual measurement sites.

Assessing the Effects of Data Selection with the DAO Physical-Space Statistical Analysis System*

Conventional optimal interpolation (OI) analysis systems solve the standard statistical analysis equations approximately, by invoking a local approximation and a data selection procedure. Although solution of the analysis equations is essentially exact in the recent generation of global spectral variational analysis systems, these new systems also include substantial changes in error covariance modeling, making it difficult to discern whether improvements in analysis and forecast quality are due to exact, global solution of the analysis equations, or to changes in error covariance modeling. The formulation and implementation of a new type of global analysis system at the Data Assimilation Office, termed the Physical-space Statistical Analysis System (PSAS), is described in this article. Since this system operates directly in physical space, it is capable of employing error covariance models identical to those of the predecessor OI system, as well as more advanced models. To focus strictly on the effect of global versus local solution of the analysis equations, a comparison between PSAS and OI analyses is carried out with both systems using identical error covariance models and identical data. Spectral decomposition of the analysis increments reveals that, relative to the PSAS increments, the OI increments have too little power at large horizontal scales and excessive power at small horizontal scales. The OI increments also display an unrealistically large ratio of divergence to vorticity. Dynamical imbalances in the OI-analyzed state can therefore be attributed in part to the approximate local method of solution, and are not entirely due to the simple geostrophic constraint built into the forecast error covariance model. Root-mean-square observation minus 6-h forecast errors in the zonal wind component are substantially smaller for the PSAS system than for the OI system.

Dual formulation of four-dimensional variational assimilation

A duality between two formulations of variational assimilation, three-dimensionsal (3D-Var) and the Physicalspace Statistical Analysis System (PSAS), is presented. It is shown that their conditioning is identical. the temporal extension of 3D-Var leads to 4D-Var. the temporal extension of PSAS, 4D-PSAS, is achieved using an algorithm inspired by the representer technique but without the explicit computation of the representers. Assuming the model is perfect, both 4D-Var and 4D-PSAS are equivalent in terms of results produced and cost. Assuming the model is not perfect, the equivalence is preserved, but in 4D-Var it is necessary to increase the size of the control variable while 4D-PSAS remains almost unchanged. In duality, 4D-Var remains almost unchanged by an increase in the number of observations used, whereas the size of the control variable in PSAS depends directly on this number.

Dual formulation of four‐dimensional variational assimilation

AbstractA duality between two formulations of variational assimilation, three‐dimensionsal (3D‐Var) and the Physicalspace Statistical Analysis System (PSAS), is presented. It is shown that their conditioning is identical. the temporal extension of 3D‐Var leads to 4D‐Var. the temporal extension of PSAS, 4D‐PSAS, is achieved using an algorithm inspired by the representer technique but without the explicit computation of the representers. Assuming the model is perfect, both 4D‐Var and 4D‐PSAS are equivalent in terms of results produced and cost. Assuming the model is not perfect, the equivalence is preserved, but in 4D‐Var it is necessary to increase the size of the control variable while 4D‐PSAS remains almost unchanged. In duality, 4D‐Var remains almost unchanged by an increase in the number of observations used, whereas the size of the control variable in PSAS depends directly on this number.