An Ensemble Smoother with Error Estimates

A smoother introduced earlier by van Leeuwen and Evensen is applied to a problem in which real obser vations are used in an area with strongly nonlinear dynamics. The derivation is new , but it resembles an earlier derivation by van Leeuwen and Evensen. Again a Bayesian view is taken in which the prior probability density of the model and the probability density of the obser vations are combined to for m a posterior density . The mean and the covariance of this density give the variance-minimizing model evolution and its errors. The assumption is made that the prior probability density is a Gaussian, leading to a linear update equation. Critical evaluation shows when the assumption is justified. This also sheds light on why Kalman filters, in which the same ap- proximation is made, work for nonlinear models. By reference to the derivation, the impact of model and obser vational biases on the equations is discussed, and it is shown that Bayes’ s for mulation can still be used. A practical advantage of the ensemble smoother is that no adjoint equations have to be integrated and that error estimates are easily obtained. The present application shows that for process studies a smoother will give superior results compared to a filter , not only owing to the smooth transitions at obser vation points, but also because the origin of features can be followed back in time. Also its preference over a strong-constraint method is highlighted. Further more, it is argued that the proposed smoother is more efficient than gradient descent methods or than the representer method when error estimates are taken into account

Analysis Scheme in the Ensemble Kalman Filter

This paper discusses an important issue related to the implementation and interpretation of the analysis scheme in the ensemble Kalman filter . I t i s shown that the obser vations must be treated as random variables at the analysis steps. That is, one should add random perturbations with the correct statistics to the obser vations and generate an ensemble of obser vations that then is used in updating the ensemble of model states. T raditionally , this has not been done in previous applications of the ensemble Kalman filter and, as will be shown, this has resulted in an updated ensemble with a variance that is too low . This simple modification of the analysis scheme results in a completely consistent approach if the covariance of the ensemble of model states is interpreted as the prediction error covariance, and there are no further requirements on the ensemble Kalman filter method, except for the use of an ensemble of sufficient size. Thus, there is a unique correspondence between the error statistics from the ensemble Kalman filter and the standard Kalman filter approach

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.

CO2 emission estimation in the urban environment: measurement of the CO2 storage term

Eddy covariance has been used in urban areas to evaluate the net exchange of CO2 between the surface and the atmosphere. Typically, only the vertical flux is measured at a height 2–3 times that of the local roughness elements; however, under conditions of relatively low instability, CO2 may accumulate in the airspace below the measurement height. This can result in inaccurate emissions estimates if the accumulated CO2 drains away or is flushed upwards during thermal expansion of the boundary layer. Some studies apply a single height storage correction; however, this requires the assumption that the response of the CO2 concentration profile to forcing is constant with height. Here a full seasonal cycle (7th June 2012 to 3rd June 2013) of single height CO2 storage data calculated from concentrations measured at 10 Hz by open path gas analyser are compared to a data set calculated from a concurrent switched vertical profile measured (2 Hz, closed path gas analyser) at 10 heights within and above a street canyon in central London. The assumption required for the former storage determination is shown to be invalid. For approximately regular street canyons at least one other measurement is required. Continuous measurements at fewer locations are shown to be preferable to a spatially dense, switched profile, as temporal interpolation is ineffective. The majority of the spectral energy of the CO2 storage time series was found to be between 0.001 and 0.2 Hz (500 and 5 s respectively); however, sampling frequencies of 2 Hz and below still result in significantly lower CO2 storage values. An empirical method of correcting CO2 storage values from under-sampled time series is proposed.

Seasonal surface urban energy balance and wintertime stability simulated using three land-surface models in the high-latitude city Helsinki

The performance of three urban land surface models, run in offline mode, with their default external parameters, is evaluated for two distinctly different sites in Helsinki: Torni and Kumpula. The former is a dense city centre site with 22% vegetation, while the latter is a suburban site with over 50% vegetation. At both locations the models are compared against sensible and latent heat fluxes measured using the eddy covariance technique, along with snow depth observations. The cold climate experienced by the city causes strong seasonal variations that include snow cover and stable atmospheric conditions. Most of the time the three models are able to account for the differences between the study areas as well as the seasonal and diurnal variability of the energy balance components. However, the performances are not systematic across the modelled components, season and surface type. The net all-wave radiation is well simulated, with the greatest uncertainties related to snowmelt timing, when the fraction of snow cover has a key role, particularly in determining the surface albedo. For the turbulent fluxes, more variation between the models is seen which can partly be explained by the different methods in their calculation and partly by surface parameter values. For the sensible heat flux, simulation of wintertime values was the main problem, which also leads to issues in predicting near-surface stabilities particularly at the dense city centre site. All models have the most difficulties in simulating latent heat flux. This study particularly emphasizes that improvements are needed in the parameterization of anthropogenic heat flux and thermal parameters in winter, snow cover in spring and evapotranspiration in order to improve the surface energy balance modelling in cold climate cities.

Implicit equal-weights particle filter

Filter degeneracy is the main obstacle for the implementation of particle filter in non-linear high-dimensional models. A new scheme, the implicit equal-weights particle filter (IEWPF), is introduced. In this scheme samples are drawn implicitly from proposal densities with a different covariance for each particle, such that all particle weights are equal by construction. We test and explore the properties of the new scheme using a 1,000-dimensional simple linear model, and the 1,000-dimensional non-linear Lorenz96 model, and compare the performance of the scheme to a Local Ensemble Kalman Filter. The experiments show that the new scheme can easily be implemented in high-dimensional systems and is never degenerate, with good convergence properties in both systems.

Relational architecture and relational capability: organisational levers to support strategic supplier relationships

The resource based view of strategy suggests that competitiveness in part derives from a firms ability to collaborate with a subset of its supply network to co-create highly valued products and services. This relational capability relies on a foundational intra and inter-organisational architecture, the manifestation of strategic, people, and process decisions facilitating the interface between the firm and its strategic suppliers. Using covariance-based structural equation modelling we examine the relationships between internal and external features of relational architecture, and their relationship with relational capability and relational quality. This is undertaken on data collected by mail survey. We find significant relationships between both internal and external relational architecture and relational capability and between relational capability and relational quality. Novel constructs for internal and external elements of relational architecture are specified to demonstrate their positive influence on relational capability and relationship quality.

Surface Urban Energy and Water Balance Scheme (SUEWS): development and evaluation at two UK sites

The Surface Urban Energy and Water Balance Scheme (SUEWS) is evaluated at two locations in the UK: a dense urban site in the centre of London and a residential suburban site in Swindon. Eddy covariance observations of the turbulent fluxes are used to assess model performance over a twoyear period (2011-2013). The distinct characteristics of the sites mean their surface energy exchanges differ considerably. The model suggests the largest differences can be attributed to surface cover (notably the proportion of vegetated versus impervious area) and the additional energy supplied by human activities. SUEWS performs better in summer than winter, and better at the suburban site than the dense urban site. One reason for this is the bias towards suburban summer field campaigns in observational data used to parameterise this (and other) model(s). The suitability of model parameters (such as albedo, energy use and water use) for the UK sites is considered and, where appropriate, alternative values are suggested. An alternative parameterisation for the surface conductance is implemented, which permits greater soil moisture deficits before evaporation is restricted at non-irrigated sites. Accounting for seasonal variation in the estimation of storage heat flux is necessary to obtain realistic wintertime fluxes.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.