Boon S. Chua

Generalized inversion of a global numerical weather prediction model

SummaryWe construct the generalized inverse of a global numerical weather prediction (NWP) model, in order to prepare initial conditions for the model at time “t=0 hrs”. The inverse finds a weighted, least-squares best-fit to the dynamics for −24<t<0, to the previous initial condition att=−24, and to data att=−24,t=−18,t=−12 andt=0. That is, the inverse is a weak-constraint, four-dimensional variational assimilation scheme. The best-fit is found by solving the nonlinear Euler-Lagrange (EL) equations which determine the local extrema of a penalty functional. The latter is quadratic in the dynamical, initial and data residuals. The EL equations are solved using iterated representer expansions. The technique yields optimal conditioning of the very large minimization problem, which has ∼109 hydrodynamical and thermodynamical variables defined on a 4-dimensional, space-time grid.In addition to introducing the inverse NWP model, we demonstrate it on a medium-sized problem, namely, a study of the impact of reprocessed cloud track wind observations (RCTWO) from the 1990 Tropical Cyclone Motion Experiment (TCM-90). The impact is assessed in terms of the improvement of forecasts in the South China Sea att=+48 hours. The calculation shows that the computations are manageable, the iteration scheme converges, and that the RCTWO have a beneficial impact.

Generalized Inversion of Tropical Atmosphere–Ocean Data and a Coupled Model of the Tropical Pacific

Abstract It is hypothesized that the circulation of the tropical Pacific Ocean and atmosphere satisfies the equations of a simple coupled model to within errors having specified covariances, and that the Tropical Atmosphere–Ocean array (TAO) measures the circulation to within errors also having specified covariances. This hypothesis is tested by finding the circulation that is the weighted least squares best fit to the dynamics of the simple model, to its initial and boundary conditions, and to a year of monthly mean TAO data for sea surface temperature, for the depth of the 20°C isotherm, and for surface winds. The fit is defined over the entire tropical Pacific and from 1 April 1994 to 31 March 1995. The best-fit circulation or state estimate is calculated using variational methods. Posterior error covariances are estimated using statistical simulation. The best fit is also subjected to a significance test. It is found that, although the fit to data is largely within standard errors, the misfit to dynam...

Generalized Inverse of a Reduced Gravity Primitive Equation Ocean Model and Tropical Atmosphere-Ocean Data

A nonlinear 2‰-layer reduced gravity primitive equations (PE) ocean model is used to assimilate sea surface temperature (SST) data from the Tropical Atmosphere‐Ocean (TAO) moored buoys in the tropical Pacific. The aim of this project is to hindcast cool and warm events of this part of the ocean, on seasonal to interannual timescales. The work extends that of Bennett et al., who used a modified Zebiak‐Cane coupled model. They were able to fit a year of 30-day averaged TAO data to within measurement errors, albeit with significant initial and dynamical residuals. They assumed a 100-day decorrelation timescale for the dynamical residuals. This long timescale for the residuals reflects the neglect of resolvable processes in the intermediate coupled model, such as horizontal advection of momentum. However, the residuals in the nonlinear PE model should be relatively short timescale errors in parameterizations. The scales for these residuals are crudely estimated from the upper ocean turbulence studies of Peters et al. and Moum. The assimilation is performed by minimizing a weighted least squares functional expressing the misfits to the data and to the model throughout the tropical Pacific and for 18 months. It is known that the minimum lies in the ‘‘data subspace’’ of the state or solution space. The minimum is therefore sought in the data subspace, by using the representer method to solve the Euler‐Lagrange (EL) system. Although the vector space decomposition and solution method assume a linear EL system, the concept and technique are applied to the nonlinear EL system (resulting from the nonlinear PE model), by iterating with linear approximations to the nonlinear EL system. As a first step, the authors verify that sequences of solutions of linear iterates of the forward PE model do converge. The assimilation is also used as a significance test of the hypothesized means and covariances of the errors in the initial conditions, dynamics, and data. A ‘‘strong constraint’’ inverse solution is computed. However, it is outperformed by the ‘‘weak constraint’’ inverse. A cross validation by withheld data is presented, as well as an inversion with the model forced by the Florida State University winds, in place of a climatological wind forcing used in the former inversions.

Generalized Inversion of Tropical Atmosphere–Ocean (TAO) Data and a Coupled Model of the Tropical Pacific. Part II: The 1995–96 La Niña and 1997–98 El Niño

Abstract The investigation of the consequences of trying to blend tropical Pacific observations from the Tropical Atmosphere–Ocean (TAO) array into the dynamical framework of an intermediate coupled ocean–atmosphere model is continued. In a previous study it was found that the model dynamics, the prior estimates of uncertainty in the observations, and the estimates of the errors in the dynamical equations of the model could not be reconciled with data from the 1994–95 period. The error estimates and the data forced the rejection of the model physics as being unacceptably in error. In this work, data from two periods (1995–96 and 1997–98) were used when the tropical Pacific was in states very different from the previous study. The consequences of increasing the prior error estimates were explored in an effort to find out if it is possible at least to use the intermediate model physics to assist in mapping the observations into fields in a statistically consistent way. It was found that such a result is pos...

Generalized inversion of a global numerical weather prediction model, II: Analysis and implementation

SummaryThis is a sequel to Bennett, Chua and Leslie (1996), concerning weak-constraint, four-dimensional variational assimilation of reprocessed cloud-track wind observations (Velden, 1992) into a global, primitive-equation numerical weather prediction model. The assimilation is performed by solving the Euler-Lagrange equations associated with the variational principle. Bennett et al. (1996) assimilate 2436 scalar wind components into their model over a 24-hour interval, yielding a substantially improved estimate of the state of the atmosphere at the end of the interval. This improvement is still in evidence in forecasts for the next 48 hours.The model and variational equations are nonlinear, but are solved as sequence of linear equations. It is shown here that each linear solution is precisely equivalent to optimal or statistical interpolation using a background error covariance derived from the linearized dynamics, from the forcing error covariance, and from the initial error covariance. Bennett et al. (1996) control small-scale flow divergence using divergence dissipation (Talagrand, 1972). It is shown here that this approach is virtually equivalent to including a penalty, for the gradient of divergence, in the variational principle. The linearized variational equations are solved in terms of the representer functions for the wind observations. Diagonalizing the representer matrix yields rotation vectors. The rotated representers are the “array modes” of the entire system of the model, prior covariances and observations. The modes are the “observable” degrees of freedom of the atmosphere. Several leading array modes are presented here. Finally, appendices discuss a number of technical implementation issues: time convolutions, convergence in the presence of planetary shear instability, and preconditioning the essential inverse problem.

Open-Ocean Modeling as an Inverse Problem: The Primitive Equations

Abstract The ill-posedness of regional primitive equation models is examined, using a regional shallow-water model. The ill-posedness is resolved by reformulation as a least-squares inverse problem, in the sense that the Euler-Lagrange or variational boundary conditions ensure unique solutions for the linearized problem. The inverses for nonlinear problems are calculated using variants of simulated annealing and massively parallel computing. Simple experiments compare the relative merits of pointwise measurements and path-integrated measurements in compensating for bad boundary data. Error statistics are calculated, despite the large dimension of the state space.

The Inverse Ocean Modeling System. Part II: Applications

The Inverse Ocean Modeling (IOM) System is a modular system for constructing and running weakconstraint four-dimensional variational data assimilation (W4DVAR) for any linear or nonlinear functionally smooth dynamical model and observing array. The IOM has been applied to four ocean models with widely varying characteristics. The Primitive Equations Z-coordinate-Harmonic Analysis of Tides (PEZHAT) and the Regional Ocean Modeling System (ROMS) are three-dimensional, primitive equations models while the Advanced Circulation model in 2D (ADCIRC-2D) and Spectral Element Ocean Model in 2D (SEOM-2D) are shallow-water models belonging to the general finite-element family. These models, in conjunction with the IOM, have been used to investigate a wide variety of scientific phenomena including tidal, mesoscale, and wind-driven circulation. In all cases, the assimilation of data using the IOM provides a better estimate of the ocean state than the model alone.