Stochastic parameterization: uncertainties from convection

In 2005, the ECMWF held a workshop on stochastic parameterisation, at which the convection was seen as being a key issue. That much is clear from the working group reports and particularly the statement from working group 1 that “it is clear that a stochastic convection scheme is desirable”. The present note aims to consider our current status in comparison with some of the issues raised and hopes expressed in that working group report.

On the equivalence between radiance and retrieval assimilation

The need for consistent assimilation of satellite measurements for numerical weather prediction led operational meteorological centers to assimilate satellite radiances directly using variational data assimilation systems. More recently there has been a renewed interest in assimilating satellite retrievals (e.g., to avoid the use of relatively complicated radiative transfer models as observation operators for data assimilation). The aim of this paper is to provide a rigorous and comprehensive discussion of the conditions for the equivalence between radiance and retrieval assimilation. It is shown that two requirements need to be satisfied for the equivalence: (i) the radiance observation operator needs to be approximately linear in a region of the state space centered at the retrieval and with a radius of the order of the retrieval error; and (ii) any prior information used to constrain the retrieval should not underrepresent the variability of the state, so as to retain the information content of the measurements. Both these requirements can be tested in practice. When these requirements are met, retrievals can be transformed so as to represent only the portion of the state that is well constrained by the original radiance measurements and can be assimilated in a consistent and optimal way, by means of an appropriate observation operator and a unit matrix as error covariance. Finally, specific cases when retrieval assimilation can be more advantageous (e.g., when the estimate sought by the operational assimilation system depends on the first guess) are discussed.

A study of the arrival over the United Kingdom in April 2010 of the Eyjafjallajökull ash cloud using ground-based lidar and numerical simulations

We make a qualitative and quantitative comparison of numericalsimulations of the ashcloud generated by the eruption of Eyjafjallajökull in April2010 with ground-basedlidar measurements at Exeter and Cardington in southern England. The numericalsimulations are performed using the Met Office’s dispersion model, NAME (Numerical Atmospheric-dispersion Modelling Environment). The results show that NAME captures many of the features of the observed ashcloud. The comparison enables us to estimate the fraction of material which survives the near-source fallout processes and enters into the distal plume. A number of simulations are performed which show that both the structure of the ashcloudover southern England and the concentration of ash within it are particularly sensitive to the height of the eruption column (and the consequent estimated mass emission rate), to the shape of the vertical source profile and the level of prescribed ‘turbulent diffusion’ (representing the mixing by the unresolved eddies) in the free troposphere with less sensitivity to the timing of the start of the eruption and the sedimentation of particulates in the distal plume.

Linear-wavelet networks

This paper proposes a nonlinear regression structure comprising a wavelet network and a linear term. The introduction of the linear term is aimed at providing a more parsimonious interpolation in high-dimensional spaces when the modelling samples are sparse. A constructive procedure for building such structures, termed linear-wavelet networks, is described. For illustration, the proposed procedure is employed in the framework of dynamic system identification. In an example involving a simulated fermentation process, it is shown that a linear-wavelet network yields a smaller approximation error when compared with a wavelet network with the same number of regressors. The proposed technique is also applied to the identification of a pressure plant from experimental data. In this case, the results show that the introduction of wavelets considerably improves the prediction ability of a linear model. Standard errors on the estimated model coefficients are also calculated to assess the numerical conditioning of the identification process.

Numerical convergence of the Block-Maxima approach to the generalized extreme value distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.

Ensemble data assimilation in the presence of cloud

In numerical weather prediction (NWP) data assimilation (DA) methods are used to combine available observations with numerical model estimates. This is done by minimising measures of error on both observations and model estimates with more weight given to data that can be more trusted. For any DA method an estimate of the initial forecast error covariance matrix is required. For convective scale data assimilation, however, the properties of the error covariances are not well understood. An effective way to investigate covariance properties in the presence of convection is to use an ensemble-based method for which an estimate of the error covariance is readily available at each time step. In this work, we investigate the performance of the ensemble square root filter (EnSRF) in the presence of cloud growth applied to an idealised 1D convective column model of the atmosphere. We show that the EnSRF performs well in capturing cloud growth, but the ensemble does not cope well with discontinuities introduced into the system by parameterised rain. The state estimates lose accuracy, and more importantly the ensemble is unable to capture the spread (variance) of the estimates correctly. We also find, counter-intuitively, that by reducing the spatial frequency of observations and/or the accuracy of the observations, the ensemble is able to capture the states and their variability successfully across all regimes.

Adjoint methods in data assimilation for estimating model error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure.

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

Smoothed histogram modification for image processing

The technique of constructing a transformation, or regrading, of a discrete data set such that the histogram of the transformed data matches a given reference histogram is commonly known as histogram modification. The technique is widely used for image enhancement and normalization. A method which has been previously derived for producing such a regrading is shown to be “best” in the sense that it minimizes the error between the cumulative histogram of the transformed data and that of the given reference function, over all single-valued, monotone, discrete transformations of the data. Techniques for smoothed regrading, which provide a means of balancing the error in matching a given reference histogram against the information lost with respect to a linear transformation are also examined. The smoothed regradings are shown to optimize certain cost functionals. Numerical algorithms for generating the smoothed regradings, which are simple and efficient to implement, are described, and practical applications to the processing of LANDSAT image data are discussed.

Equidistributing Meshes with Constraints

Adaptive methods which “equidistribute” a given positive weight function are now used fairly widely for selecting discrete meshes. The disadvantage of such schemes is that the resulting mesh may not be smoothly varying. In this paper a technique is developed for equidistributing a function subject to constraints on the ratios of adjacent steps in the mesh. Given a weight function $f \geqq 0$ on an interval $[a,b]$ and constants $c$ and $K$, the method produces a mesh with points $x_0 = a,x_{j + 1} = x_j + h_j ,j = 0,1, \cdots ,n - 1$ and $x_n = b$ such that\[ \int_{xj}^{x_{j + 1} } {f \leqq c\quad {\text{and}}\quad \frac{1} {K}} \leqq \frac{{h_{j + 1} }} {{h_j }} \leqq K\quad {\text{for}}\, j = 0,1, \cdots ,n - 1 . \] A theoretical analysis of the procedure is presented, and numerical algorithms for implementing the method are given. Examples show that the procedure is effective in practice. Other types of constraints on equidistributing meshes are also discussed. The principal application of the procedure is to the solution of boundary value problems, where the weight function is generally some error indicator, and accuracy and convergence properties may depend on the smoothness of the mesh. Other practical applications include the regrading of statistical data.

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.