Representation of model error in a convective-scale ensemble prediction system

In this paper ensembles of forecasts (of up to six hours) are studied from a convection-permitting model with a representation of model error due to unresolved processes. The ensemble prediction system (EPS) used is an experimental convection-permitting version of the UK Met Office’s 24- member Global and Regional Ensemble Prediction System (MOGREPS). The method of representing model error variability, which perturbs parameters within the model’s parameterisation schemes, has been modified and we investigate the impact of applying this scheme in different ways. These are: a control ensemble where all ensemble members have the same parameter values; an ensemble where the parameters are different between members, but fixed in time; and ensembles where the parameters are updated randomly every 30 or 60 min. The choice of parameters and their ranges of variability have been determined from expert opinion and parameter sensitivity tests. A case of frontal rain over the southern UK has been chosen, which has a multi-banded rainfall structure. The consequences of including model error variability in the case studied are mixed and are summarised as follows. The multiple banding, evident in the radar, is not captured for any single member. However, the single band is positioned in some members where a secondary band is present in the radar. This is found for all ensembles studied. Adding model error variability with fixed parameters in time does increase the ensemble spread for near-surface variables like wind and temperature, but can actually decrease the spread of the rainfall. Perturbing the parameters periodically throughout the forecast does not further increase the spread and exhibits “jumpiness” in the spread at times when the parameters are perturbed. Adding model error variability gives an improvement in forecast skill after the first 2–3 h of the forecast for near-surface temperature and relative humidity. For precipitation skill scores, adding model error variability has the effect of improving the skill in the first 1–2 h of the forecast, but then of reducing the skill after that. Complementary experiments were performed where the only difference between members was the set of parameter values (i.e. no initial condition variability). The resulting spread was found to be significantly less than the spread from initial condition variability alone.

Assimilation of sea-ice concentration in a global climate model — physical and statistical aspects

We investigate the initialization of Northern-hemisphere sea ice in the global climate model ECHAM5/MPI-OM by assimilating sea-ice concentration data. The analysis updates for concentration are given by Newtonian relaxation, and we discuss different ways of specifying the analysis updates for mean thickness. Because the conservation of mean ice thickness or actual ice thickness in the analysis updates leads to poor assimilation performance, we introduce a proportional dependence between concentration and mean thickness analysis updates. Assimilation with these proportional mean-thickness analysis updates significantly reduces assimilation error both in identical-twin experiments and when assimilating sea-ice observations, reducing the concentration error by a factor of four to six, and the thickness error by a factor of two. To understand the physical aspects of assimilation errors, we construct a simple prognostic model of the sea-ice thermodynamics, and analyse its response to the assimilation. We find that the strong dependence of thermodynamic ice growth on ice concentration necessitates an adjustment of mean ice thickness in the analysis update. To understand the statistical aspects of assimilation errors, we study the model background error covariance between ice concentration and ice thickness. We find that the spatial structure of covariances is best represented by the proportional mean-thickness analysis updates. Both physical and statistical evidence supports the experimental finding that proportional mean-thickness updates are superior to the other two methods considered and enable us to assimilate sea ice in a global climate model using simple Newtonian relaxation.

Assimilation of sea-ice concentration in a global climate model — physical and statistical aspects

We investigate the initialisation of Northern Hemisphere sea ice in the global climate model ECHAM5/MPI-OM by assimilating sea-ice concentration data. The analysis updates for concentration are given by Newtonian relaxation, and we discuss different ways of specifying the analysis updates for mean thickness. Because the conservation of mean ice thickness or actual ice thickness in the analysis updates leads to poor assimilation performance, we introduce a proportional dependence between concentration and mean thickness analysis updates. Assimilation with these proportional mean-thickness analysis updates leads to good assimilation performance for sea-ice concentration and thickness, both in identical-twin experiments and when assimilating sea-ice observations. The simulation of other Arctic surface fields in the coupled model is, however, not significantly improved by the assimilation. To understand the physical aspects of assimilation errors, we construct a simple prognostic model of the sea-ice thermodynamics, and analyse its response to the assimilation. We find that an adjustment of mean ice thickness in the analysis update is essential to arrive at plausible state estimates. To understand the statistical aspects of assimilation errors, we study the model background error covariance between ice concentration and ice thickness. We find that the spatial structure of covariances is best represented by the proportional mean-thickness analysis updates. Both physical and statistical evidence supports the experimental finding that assimilation with proportional mean-thickness updates outperforms the other two methods considered. The method described here is very simple to implement, and gives results that are sufficiently good to be used for initialising sea ice in a global climate model for seasonal to decadal predictions.

Global snow mass measurements and the effect of stratigraphic detail on inversion of microwave brightness temperatures

Snow provides large seasonal storage of freshwater, and information about the distribution of snow mass as Snow Water Equivalent (SWE) is important for hydrological planning and detecting climate change impacts. Large regional disagreements remain between estimates from reanalyses, remote sensing and modelling. Assimilating passive microwave information improves SWE estimates in many regions but the assimilation must account for how microwave scattering depends on snow stratigraphy. Physical snow models can estimate snow stratigraphy, but users must consider the computational expense of model complexity versus acceptable errors. Using data from the National Aeronautics and Space Administration Cold Land Processes Experiment (NASA CLPX) and the Helsinki University of Technology (HUT) microwave emission model of layered snowpacks, it is shown that simulations of the brightness temperature difference between 19 GHz and 37 GHz vertically polarised microwaves are consistent with Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E) and Special Sensor Microwave Imager (SSM/I) retrievals once known stratigraphic information is used. Simulated brightness temperature differences for an individual snow profile depend on the provided stratigraphic detail. Relative to a profile defined at the 10 cm resolution of density and temperature measurements, the error introduced by simplification to a single layer of average properties increases approximately linearly with snow mass. If this brightness temperature error is converted into SWE using a traditional retrieval method then it is equivalent to ±13 mm SWE (7% of total) at a depth of 100 cm. This error is reduced to ±5.6 mm SWE (3 % of total) for a two-layer model.

Systematic model forecast error in Rossby wave structure

Diabatic processes can alter Rossby wave structure; consequently errors arising from model processes propagate downstream. However, the chaotic spread of forecasts from initial condition uncertainty renders it difficult to trace back from root mean square forecast errors to model errors. Here diagnostics unaffected by phase errors are used, enabling investigation of systematic errors in Rossby waves in winter-season forecasts from three operational centers. Tropopause sharpness adjacent to ridges decreases with forecast lead time. It depends strongly on model resolution, even though models are examined on a common grid. Rossby wave amplitude reduces with lead time up to about five days, consistent with under-representation of diabatic modification and transport of air from the lower troposphere into upper-tropospheric ridges, and with too weak humidity gradients across the tropopause. However, amplitude also decreases when resolution is decreased. Further work is necessary to isolate the contribution from errors in the representation of diabatic processes.

Ocean heat uptake processes: a model intercomparison

We compare the quasi-equilibrium heat balances, as well as their responses to 4×CO2 perturbation, among three global climate models with the aim to identify and explain inter-model differences in ocean heat uptake (OHU) processes. We find that, in quasi-equilibrium, convective and mixed layer processes, as well as eddy-related processes, cause cooling of the subsurface ocean. The cooling is balanced by warming caused by advective and diapycnally diffusive processes. We also find that in the CO2-perturbed climates the largest contribution to OHU comes from changes in vertical mixing processes and the mean circulation, particularly in the extra-tropics, caused both by changes in wind forcing, and by changes in high-latitude buoyancy forcing. There is a substantial warming in the tropics, a significant part of which occurs because of changes in horizontal advection in extra-tropics. Diapycnal diffusion makes only a weak contribution to the OHU, mainly in the tropics, due to increased stratification. There are important qualitative differences in the contribution of eddy-induced advection and isopycnal diffusion to the OHU among the models. The former is related to the different values of the coefficients used in the corresponding scheme. The latter is related to the different tapering formulations of the isopycnal diffusion scheme. These differences affect the OHU in the deep ocean, which is substantial in two of the models, the dominant region of deep warming being the Southern Ocean. However, most of the OHU takes place above 2000 m, and the three models are quantitatively similar in their global OHU efficiency and its breakdown among processes and as a function of latitude.

Combining satellite observations and reanalysis energy transports to estimate global net surface energy fluxes 1985-2012

Two methods are developed to estimate net surface energy fluxes based upon satellite-based reconstructions of radiative fluxes at the top of atmosphere and the atmospheric energy tendencies and transports from the ERA-Interim reanalysis. Method 1 applies the mass adjusted energy divergence from ERA-Interim while method 2 estimates energy divergence based upon the net energy difference at the top of atmosphere and the surface from ERA-Interim. To optimise the surface flux and its variability over ocean, the divergences over land are constrained to match the monthly area mean surface net energy flux variability derived from a simple relationship between the surface net energy flux and the surface temperature change. The energy divergences over the oceans are then adjusted to remove an unphysical residual global mean atmospheric energy divergence. The estimated net surface energy fluxes are compared with other data sets from reanalysis and atmospheric model simulations. The spatial correlation coefficients of multi-annual means between the estimations made here and other data sets are all around 0.9. There are good agreements in area mean anomaly variability over the global ocean, but discrepancies in the trend over the eastern Pacific are apparent.

Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.

Data Assimilation and Inverse Methods in Terms of a Probabilistic Formulation

The weak-constraint inverse for nonlinear dynamical models is discussed and derived in terms of a probabilistic formulation. The well-known result that for Gaussian error statistics the minimum of the weak-constraint inverse is equal to the maximum-likelihood estimate is rederived. Then several methods based on ensemble statistics that can be used to find the smoother (as opposed to the filter) solution are introduced and compared to traditional methods. A strong point of the new methods is that they avoid the integration of adjoint equations, which is a complex task for real oceanographic or atmospheric applications. they also avoid iterative searches in a Hilbert space, and error estimates can be obtained without much additional computational effort. the feasibility of the new methods is illustrated in a two-layer quasigeostrophic model.

Diagnosing observation error correlations for Doppler radar radial winds in the Met Office UKV model using observation-minus-background and observation-minus-analysis statistics

With the development of convection-permitting numerical weather prediction the efficient use of high resolution observations in data assimilation is becoming increasingly important. The operational assimilation of these observations, such as Dopplerradar radial winds, is now common, though to avoid violating the assumption of un- correlated observation errors the observation density is severely reduced. To improve the quantity of observations used and the impact that they have on the forecast will require the introduction of the full, potentially correlated, error statistics. In this work, observation error statistics are calculated for the Doppler radar radial winds that are assimilated into the Met Office high resolution UK model using a diagnostic that makes use of statistical averages of observation-minus-background and observation-minus-analysis residuals. This is the first in-depth study using the diagnostic to estimate both horizontal and along-beam correlated observation errors. By considering the new results obtained it is found that the Doppler radar radial wind error standard deviations are similar to those used operationally and increase as the observation height increases. Surprisingly the estimated observation error correlation length scales are longer than the operational thinning distance. They are dependent on both the height of the observation and on the distance of the observation away from the radar. Further tests show that the long correlations cannot be attributed to the use of superobservations or the background error covariance matrix used in the assimilation. The large horizontal correlation length scales are, however, in part, a result of using a simplified observation operator.

Uncertainties in tidally adjusted estimates of sea level rise flooding (bathtub model) for the Greater London

Sea-level rise (SLR) from global warming may have severe consequences for coastal cities, particularly when combined with predicted increases in the strength of tidal surges. Predicting the regional impact of SLR flooding is strongly dependent on the modelling approach and accuracy of topographic data. Here, the areas under risk of sea water flooding for London boroughs were quantified based on the projected SLR scenarios reported in Intergovernmental Panel on Climate Change (IPCC) fifth assessment report (AR5) and UK climatic projections 2009 (UKCP09) using a tidally-adjusted bathtub modelling approach. Medium- to very high-resolution digital elevation models (DEMs) are used to evaluate inundation extents as well as uncertainties. Depending on the SLR scenario and DEMs used, it is estimated that 3%–8% of the area of Greater London could be inundated by 2100. The boroughs with the largest areas at risk of flooding are Newham, Southwark, and Greenwich. The differences in inundation areas estimated from a digital terrain model and a digital surface model are much greater than the root mean square error differences observed between the two data types, which may be attributed to processing levels. Flood models from SRTM data underestimate the inundation extent, so their results may not be reliable for constructing flood risk maps. This analysis provides a broad-scale estimate of the potential consequences of SLR and uncertainties in the DEM-based bathtub type flood inundation modelling for London boroughs.

Improving the GNSS positioning stochastic model in the presence of ionospheric scintillation

Ionospheric scintillations are caused by time-varying electron density irregularities in the ionosphere, occurring more often at equatorial and high latitudes. This paper focuses exclusively on experiments undertaken in Europe, at geographic latitudes between similar to 50 degrees N and similar to 80 degrees N, where a network of GPS receivers capable of monitoring Total Electron Content and ionospheric scintillation parameters was deployed. The widely used ionospheric scintillation indices S4 and sigma(phi) represent a practical measure of the intensity of amplitude and phase scintillation affecting GNSS receivers. However, they do not provide sufficient information regarding the actual tracking errors that degrade GNSS receiver performance. Suitable receiver tracking models, sensitive to ionospheric scintillation, allow the computation of the variance of the output error of the receiver PLL (Phase Locked Loop) and DLL (Delay Locked Loop), which expresses the quality of the range measurements used by the receiver to calculate user position. The ability of such models of incorporating phase and amplitude scintillation effects into the variance of these tracking errors underpins our proposed method of applying relative weights to measurements from different satellites. That gives the least squares stochastic model used for position computation a more realistic representation, vis-a-vis the otherwise 'equal weights' model. For pseudorange processing, relative weights were computed, so that a 'scintillation-mitigated' solution could be performed and compared to the (non-mitigated) 'equal weights' solution. An improvement between 17 and 38% in height accuracy was achieved when an epoch by epoch differential solution was computed over baselines ranging from 1 to 750 km. The method was then compared with alternative approaches that can be used to improve the least squares stochastic model such as weighting according to satellite elevation angle and by the inverse of the square of the standard deviation of the code/carrier divergence (sigma CCDiv). The influence of multipath effects on the proposed mitigation approach is also discussed. With the use of high rate scintillation data in addition to the scintillation indices a carrier phase based mitigated solution was also implemented and compared with the conventional solution. During a period of occurrence of high phase scintillation it was observed that problems related to ambiguity resolution can be reduced by the use of the proposed mitigated solution.

Electromagnetic structure and weak decay of pseudoscalar mesons in a light-front QCD-inspired model

We study the scaling of the S-3(1)-S-1(0) meson mass splitting and the pseudoscalar weak-decay constants with the mass of the meson, as seen in the available experimental data. We use an effective light-front QCD-inspired dynamical model regulated at short distances to describe the valence component of the pseudoscalar mesons. The experimentally known values of the mass splitting, decay constants (from global lattice-QCD averages) and the pion charge form factor up to 4 [GeV/c](2) are reasonably described by the model.

Global optimization for the ℋ∞-norm model reduction problem

A branch and bound algorithm is proposed to solve the [image omitted]-norm model reduction problem for continuous and discrete-time linear systems, with convergence to the global optimum in a finite time. The lower and upper bounds in the optimization procedure are described by linear matrix inequalities (LMI). Also proposed are two methods with which to reduce the convergence time of the branch and bound algorithm: the first one uses the Hankel singular values as a sufficient condition to stop the algorithm, providing to the method a fast convergence to the global optimum. The second one assumes that the reduced model is in the controllable or observable canonical form. The [image omitted]-norm of the error between the original model and the reduced model is considered. Examples illustrate the application of the proposed method.

Modifying the stochastic model to mitigate GPS systematic errors in relative positioning

The GPS observables are subject to several errors. Among them, the systematic ones have great impact, because they degrade the accuracy of the accomplished positioning. These errors are those related, mainly, to GPS satellites orbits, multipath and atmospheric effects. Lately, a method has been suggested to mitigate these errors: the semiparametric model and the penalised least squares technique (PLS). In this method, the errors are modeled as functions varying smoothly in time. It is like to change the stochastic model, in which the errors functions are incorporated, the results obtained are similar to those in which the functional model is changed. As a result, the ambiguities and the station coordinates are estimated with better reliability and accuracy than the conventional least square method (CLS). In general, the solution requires a shorter data interval, minimizing costs. The method performance was analyzed in two experiments, using data from single frequency receivers. The first one was accomplished with a short baseline, where the main error was the multipath. In the second experiment, a baseline of 102 km was used. In this case, the predominant errors were due to the ionosphere and troposphere refraction. In the first experiment, using 5 minutes of data collection, the largest coordinates discrepancies in relation to the ground truth reached 1.6 cm and 3.3 cm in h coordinate for PLS and the CLS, respectively, in the second one, also using 5 minutes of data, the discrepancies were 27 cm in h for the PLS and 175 cm in h for the CLS. In these tests, it was also possible to verify a considerable improvement in the ambiguities resolution using the PLS in relation to the CLS, with a reduced data collection time interval. © Springer-Verlag Berlin Heidelberg 2007.