A singular vector perspective of 4D-Var: Filtering and interpolation

Four-dimensional variational data assimilation (4D-Var) combines the information from a time sequence of observations with the model dynamics and a background state to produce an analysis. In this paper, a new mathematical insight into the behaviour of 4D-Var is gained from an extension of concepts that are used to assess the qualitative information content of observations in satellite retrievals. It is shown that the 4D-Var analysis increments can be written as a linear combination of the singular vectors of a matrix which is a function of both the observational and the forecast model systems. This formulation is used to consider the filtering and interpolating aspects of 4D-Var using idealized case-studies based on a simple model of baroclinic instability. The results of the 4D-Var case-studies exhibit the reconstruction of the state in unobserved regions as a consequence of the interpolation of observations through time. The results also exhibit the filtering of components with small spatial scales that correspond to noise, and the filtering of structures in unobserved regions. The singular vector perspective gives a very clear view of this filtering and interpolating by the 4D-Var algorithm and shows that the appropriate specification of the a priori statistics is vital to extract the largest possible amount of useful information from the observations. Copyright © 2005 Royal Meteorological Society

Evaluating the ability of a numerical weather prediction model to forecast tracer concentrations during ETEX 2

In this paper the meteorological processes responsible for transporting tracer during the second ETEX (European Tracer EXperiment) release are determined using the UK Met Office Unified Model (UM). The UM predicted distribution of tracer is also compared with observations from the ETEX campaign. The dominant meteorological process is a warm conveyor belt which transports large amounts of tracer away from the surface up to a height of 4 km over a 36 h period. Convection is also an important process, transporting tracer to heights of up to 8 km. Potential sources of error when using an operational numerical weather prediction model to forecast air quality are also investigated. These potential sources of error include model dynamics, model resolution and model physics. In the UM a semi-Lagrangian monotonic advection scheme is used with cubic polynomial interpolation. This can predict unrealistic negative values of tracer which are subsequently set to zero, and hence results in an overprediction of tracer concentrations. In order to conserve mass in the UM tracer simulations it was necessary to include a flux corrected transport method. Model resolution can also affect the accuracy of predicted tracer distributions. Low resolution simulations (50 km grid length) were unable to resolve a change in wind direction observed during ETEX 2, this led to an error in the transport direction and hence an error in tracer distribution. High resolution simulations (12 km grid length) captured the change in wind direction and hence produced a tracer distribution that compared better with the observations. The representation of convective mixing was found to have a large effect on the vertical transport of tracer. Turning off the convective mixing parameterisation in the UM significantly reduced the vertical transport of tracer. Finally, air quality forecasts were found to be sensitive to the timing of synoptic scale features. Errors in the position of the cold front relative to the tracer release location of only 1 h resulted in changes in the predicted tracer concentrations that were of the same order of magnitude as the absolute tracer concentrations.

Reduced space optimal interpolation of daily rain gauge precipitation in Switzerland

The coarse spacing of automatic rain gauges complicates near-real- time spatial analyses of precipitation. We test the possibility of improving such analyses by considering, in addition to the in situ measurements, the spatial covariance structure inferred from past observations with a denser network. To this end, a statistical reconstruction technique, reduced space optimal interpolation (RSOI), is applied over Switzerland, a region of complex topography. RSOI consists of two main parts. First, principal component analysis (PCA) is applied to obtain a reduced space representation of gridded high- resolution precipitation fields available for a multiyear calibration period in the past. Second, sparse real-time rain gauge observations are used to estimate the principal component scores and to reconstruct the precipitation field. In this way, climatological information at higher resolution than the near-real-time measurements is incorporated into the spatial analysis. PCA is found to efficiently reduce the dimensionality of the calibration fields, and RSOI is successful despite the difficulties associated with the statistical distribution of daily precipitation (skewness, dry days). Examples and a systematic evaluation show substantial added value over a simple interpolation technique that uses near-real-time observations only. The benefit is particularly strong for larger- scale precipitation and prominent topographic effects. Small-scale precipitation features are reconstructed at a skill comparable to that of the simple technique. Stratifying the reconstruction method by the types of weather type classifications yields little added skill. Apart from application in near real time, RSOI may also be valuable for enhancing instrumental precipitation analyses for the historic past when direct observations were sparse.

Interpolation of Hilbert and Sobolev spaces: quantitative estimates and counterexamples

This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the Sobolev spaces Hs(Ω) and tildeHs(Ω), for s in R and an open Ω in R^n. We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large.

Simple piecewise geodesic interpolation of simple and Jordan curves with applications

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be included in our construction. Then we present two applications of our main results: the generalized Green’s theorem and the uniqueness of signature for planar Jordan curves with finite p -variation for 1⩽p<2.

Feature tracking on the unit-sphere

Techniques used in a previous study of the objective identification and tracking of meteorological features in model data are extended to the unit sphere. An alternative feature detection scheme is described based on cubic interpolation for the sphere and local maximization. The extension of the tracking technique, used in the previous study, to the unit sphere is described. An example of the application of these techniques to a global relative vorticity field from a model integration are presented and discussed.

An improved algorithm for generating global window brightness temperatures from multiple satellite infrared imagery

An improved algorithm for the generation of gridded window brightness temperatures is presented. The primary data source is the International Satellite Cloud Climatology Project, level B3 data, covering the period from July 1983 to the present. The algorithm rakes window brightness, temperatures from multiple satellites, both geostationary and polar orbiting, which have already been navigated and normalized radiometrically to the National Oceanic and Atmospheric Administration's Advanced Very High Resolution Radiometer, and generates 3-hourly global images on a 0.5 degrees by 0.5 degrees latitude-longitude grid. The gridding uses a hierarchical scheme based on spherical kernel estimators. As part of the gridding procedure, the geostationary data are corrected for limb effects using a simple empirical correction to the radiances, from which the corrected temperatures are computed. This is in addition to the application of satellite zenith angle weighting to downweight limb pixels in preference to nearer-nadir pixels. The polar orbiter data are windowed on the target time with temporal weighting to account for the noncontemporaneous nature of the data. Large regions of missing data are interpolated from adjacent processed images using a form of motion compensated interpolation based on the estimation of motion vectors using an hierarchical block matching scheme. Examples are shown of the various stages in the process. Also shown are examples of the usefulness of this type of data in GCM validation.

A comparison of recent reanalysis datasets using objective feature tracking: Storm tracks and tropical easterly waves

Data from four recent reanalysis projects [ECMWF, NCEP-NCAR, NCEP - Department of Energy ( DOE), NASA] have been diagnosed at the scale of synoptic weather systems using an objective feature tracking method. The tracking statistics indicate that, overall, the reanalyses correspond very well in the Northern Hemisphere (NH) lower troposphere, although differences for the spatial distribution of mean intensities show that the ECMWF reanalysis is systematically stronger in the main storm track regions but weaker around major orographic features. A direct comparison of the track ensembles indicates a number of systems with a broad range of intensities that compare well among the reanalyses. In addition, a number of small-scale weak systems are found that have no correspondence among the reanalyses or that only correspond upon relaxing the matching criteria, indicating possible differences in location and/or temporal coherence. These are distributed throughout the storm tracks, particularly in the regions known for small-scale activity, such as secondary development regions and the Mediterranean. For the Southern Hemisphere (SH), agreement is found to be generally less consistent in the lower troposphere with significant differences in both track density and mean intensity. The systems that correspond between the various reanalyses are considerably reduced and those that do not match span a broad range of storm intensities. Relaxing the matching criteria indicates that there is a larger degree of uncertainty in both the location of systems and their intensities compared with the NH. At upper-tropospheric levels, significant differences in the level of activity occur between the ECMWF reanalysis and the other reanalyses in both the NH and SH winters. This occurs due to a lack of coherence in the apparent propagation of the systems in ERA15 and appears most acute above 500 hPa. This is probably due to the use of optimal interpolation data assimilation in ERA15. Also shown are results based on using the same techniques to diagnose the tropical easterly wave activity. Results indicate that the wave activity is sensitive not only to the resolution and assimilation methods used but also to the model formulation.

Data assimilation for a coastal area morphodynamic model: Morecambe Bay

This paper investigates the use of data assimilation in coastal area morphodynamic modelling using Morecambe Bay as a study site. A simple model of the bay has been enhanced with a data assimilation scheme to better predict large-scale changes in bathymetry observed in the bay over a 3-year period. The 2DH decoupled morphodynamic model developed for the work is described, as is the optimal interpolation scheme used to assimilate waterline observations into the model run. Each waterline was acquired from a SAR satellite image and is essentially a contour of the bathymetry at some level within the inter-tidal zone of the bay. For model parameters calibrated against validation observations, model performance is good, even without data assimilation. However the use of data assimilation successfully compensates for a particular failing of the model, and helps to keep the model bathymetry on track. It also improves the ability of the model to predict future bathymetry. Although the benefits of data assimilation are demonstrated using waterline observations, any observations of morphology could potentially be used. These results suggest that data assimilation should be considered for use in future coastal area morphodynamic models.

Data assimilation: From photon counts to Earth System forecasts

Data assimilation – the set of techniques whereby information from observing systems and models is combined optimally – is rapidly becoming prominent in endeavours to exploit Earth Observation for Earth sciences, including climate prediction. This paper explains the broad principles of data assimilation, outlining different approaches (optimal interpolation, three-dimensional and four-dimensional variational methods, the Kalman Filter), together with the approximations that are often necessary to make them practicable. After pointing out a variety of benefits of data assimilation, the paper then outlines some practical applications of the exploitation of Earth Observation by data assimilation in the areas of operational oceanography, chemical weather forecasting and carbon cycle modelling. Finally, some challenges for the future are noted.

A new set of orthogonal patterns in weather and climate: Optimally interpolated patterns.

A new spectral-based approach is presented to find orthogonal patterns from gridded weather/climate data. The method is based on optimizing the interpolation error variance. The optimally interpolated patterns (OIP) are then given by the eigenvectors of the interpolation error covariance matrix, obtained using the cross-spectral matrix. The formulation of the approach is presented, and the application to low-dimension stochastic toy models and to various reanalyses datasets is performed. In particular, it is found that the lowest-frequency patterns correspond to largest eigenvalues, that is, variances, of the interpolation error matrix. The approach has been applied to the Northern Hemispheric (NH) and tropical sea level pressure (SLP) and to the Indian Ocean sea surface temperature (SST). Two main OIP patterns are found for the NH SLP representing respectively the North Atlantic Oscillation and the North Pacific pattern. The leading tropical SLP OIP represents the Southern Oscillation. For the Indian Ocean SST, the leading OIP pattern shows a tripole-like structure having one sign over the eastern and north- and southwestern parts and an opposite sign in the remaining parts of the basin. The pattern is also found to have a high lagged correlation with the Niño-3 index with 6-months lag.

Estimating detailed soil water profile records from point measurements

Temporal and spatial patterns of soil water content affect many soil processes including evaporation, infiltration, ground water recharge, erosion and vegetation distribution. This paper describes the analysis of a soil moisture dataset comprising a combination of continuous time series of measurements at a few depths and locations, and occasional roving measurements at a large number of depths and locations. The objectives of the paper are: (i) to develop a technique for combining continuous measurements of soil water contents at a limited number of depths within a soil profile with occasional measurements at a large number of depths, to enable accurate estimation of the soil moisture vertical pattern and the integrated profile water content; and (ii) to estimate time series of soil moisture content at locations where there are just occasional soil water measurements available and some continuous records from nearby locations. The vertical interpolation technique presented here can strongly reduce errors in the estimation of profile soil water and its changes with time. On the other hand, the temporal interpolation technique is tested for different sampling strategies in space and time, and the errors generated in each case are compared.

Wiggle-match dating of tree-ring sequences

Given the non-monotonic form of the radiocarbon calibration curve, the precision of single C-14 dates on the calendar timescale will always be limited. One way around this limitation is through comparison of time-series, which should exhibit the same irregular patterning as the calibration curve. This approach can be employed most directly in the case of wood samples with many years growth present (but not able to be dated by dendrochronology), where the tree-ring series of unknown date can be compared against the similarly constructed C-14 calibration curve built from known-age wood. This process of curve-fitting has come to be called "wiggle-matching." In this paper, we look at the requirements for getting good precision by this method: sequence length, sampling frequency, and measurement precision. We also look at 3 case studies: one a piece of wood which has been independently dendrochronologically dated, and two others of unknown age relating to archaeological activity at Silchester, UK (Roman) and Miletos, Anatolia (relating to the volcanic eruption at Thera).