ADVANCES IN SEQUENTIAL DATA ASSIMILATION AND NUMERICAL WEATHER FORECASTING: AN ENSEMBLE TRANSFORM KALMAN-BUCY FILTER, A STUDY ON CLUSTERING IN DETERMINISTIC ENSEMBLE SQUARE ROOT FILTERS, AND A TEST OF A NEW TIME STEPPING SCHEME IN AN ATMOSPHERIC MODEL

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn't represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter

Gaussian anamorphosis in the analysis step of the EnKF: a joint state-variable/observation approach

The analysis step of the (ensemble) Kalman filter is optimal when (1) the distribution of the background is Gaussian, (2) state variables and observations are related via a linear operator, and (3) the observational error is of additive nature and has Gaussian distribution. When these conditions are largely violated, a pre-processing step known as Gaussian anamorphosis (GA) can be applied. The objective of this procedure is to obtain state variables and observations that better fulfil the Gaussianity conditions in some sense. In this work we analyse GA from a joint perspective, paying attention to the effects of transformations in the joint state variable/observation space. First, we study transformations for state variables and observations that are independent from each other. Then, we introduce a targeted joint transformation with the objective to obtain joint Gaussianity in the transformed space. We focus primarily in the univariate case, and briefly comment on the multivariate one. A key point of this paper is that, when (1)-(3) are violated, using the analysis step of the EnKF will not recover the exact posterior density in spite of any transformations one may perform. These transformations, however, provide approximations of different quality to the Bayesian solution of the problem. Using an example in which the Bayesian posterior can be analytically computed, we assess the quality of the analysis distributions generated after applying the EnKF analysis step in conjunction with different GA options. The value of the targeted joint transformation is particularly clear for the case when the prior is Gaussian, the marginal density for the observations is close to Gaussian, and the likelihood is a Gaussian mixture

Validation of three new measure-correlate-predict models for the long-term prospection of the wind resource

The estimation of the long-term wind resource at a prospective site based on a relatively short on-site measurement campaign is an indispensable task in the development of a commercial wind farm. The typical industry approach is based on the measure-correlate-predict �MCP� method where a relational model between the site wind velocity data and the data obtained from a suitable reference site is built from concurrent records. In a subsequent step, a long-term prediction for the prospective site is obtained from a combination of the relational model and the historic reference data. In the present paper, a systematic study is presented where three new MCP models, together with two published reference models �a simple linear regression and the variance ratio method�, have been evaluated based on concurrent synthetic wind speed time series for two sites, simulating the prospective and the reference site. The synthetic method has the advantage of generating time series with the desired statistical properties, including Weibull scale and shape factors, required to evaluate the five methods under all plausible conditions. In this work, first a systematic discussion of the statistical fundamentals behind MCP methods is provided and three new models, one based on a nonlinear regression and two �termed kernel methods� derived from the use of conditional probability density functions, are proposed. All models are evaluated by using five metrics under a wide range of values of the correlation coefficient, the Weibull scale, and the Weibull shape factor. Only one of all models, a kernel method based on bivariate Weibull probability functions, is capable of accurately predicting all performance metrics studied

Time-correlated model error in the (ensemble) Kalman smoother

Data assimilation is often performed in a perfect-model scenario, where only errors in initial conditions and observations are considered. Errors in model equations are increasingly being included, but typically using rather ad-hoc approximations with limited understanding of how these approximations affect the solution and how these approximations interfere with approximations inherent in finite-size ensembles. We provide the first systematic evaluation of the influence of approximations to model errors within a time window of weak-constraint ensemble smoothers. In particular, we study the effects of prescribing temporal correlations in the model errors incorrectly in a Kalman Smoother, and in interaction with finite ensemble-size effects in an Ensemble Kalman Smoother. For the Kalman Smoother we find that an incorrect correlation time scale for additive model errors can have substantial negative effects on the solutions, and we find that overestimating of the correlation time scale leads to worse results than underestimating. In the Ensemble Kalman Smoother case, the resulting ensemble-based space-time gain can be written as the true gain multiplied by two factors, a linear factor containing the errors due to both time-correlation errors and finite ensemble effects, and a non-linear factor related to the inverse part of the gain. Assuming that both errors are relatively small, we are able to disentangle the contributions from the different approximations. The analysis mean is affected by the time-correlation errors, but also substantially by finite ensemble effects, which was unexpected. The analysis covariance is affected by both time-correlation errors and an in-breeding term. This first thorough analysis of the influence of time-correlation errors and finite ensemble size errors on weak-constraint ensemble smoothers will aid further development of these methods and help to make them robust for e.g. numerical weather prediction

The effects of the RAW filter on the climatology and forecast skill of the SPEEDY model

In a recent study, Williams introduced a simple modification to the widely used Robert–Asselin (RA) filter for numerical integration. The main purpose of the Robert–Asselin–Williams (RAW) filter is to avoid the undesired numerical damping of the RA filter and to increase the accuracy. In the present paper, the effects of the modification are comprehensively evaluated in the Simplified Parameterizations, Primitive Equation Dynamics (SPEEDY) atmospheric general circulation model. First, the authors search for significant changes in the monthly climatology due to the introduction of the new filter. After testing both at the local level and at the field level, no significant changes are found, which is advantageous in the sense that the new scheme does not require a retuning of the parameterized model physics. Second, the authors examine whether the new filter improves the skill of short- and medium-term forecasts. January 1982 data from the NCEP–NCAR reanalysis are used to evaluate the forecast skill. Improvements are found in all the model variables (except the relative humidity, which is hardly changed). The improvements increase with lead time and are especially evident in medium-range forecasts (96–144 h). For example, in tropical surface pressure predictions, 5-day forecasts made using the RAW filter have approximately the same skill as 4-day forecasts made using the RA filter. The results of this work are encouraging for the implementation of the RAW filter in other models currently using the RA filter

Ensemble transform Kalman-Bucy filters

Two recent works have adapted the Kalman-Bucy filter into an ensemble setting. In the first formulation, BR10, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-BGR09, the ensemble of perturbations is updated by the solution of an ordinary differential equation (ODE) in pseudo-time, while the mean is updated as in the standard KF. In the second formulation, BR10, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-time. Neither requires matrix inversions except for the frequently diagonal observation error covariance. We analyze the behavior of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background to observational error covariance, and that using the integration scheme proposed in both BGR09 and BR10 can lead to failure. An integration scheme that is both stable and is not computationally expensive is proposed. We develop transform-based alternatives for these Bucy-type approaches so that the integrations are computed in ensemble space where the variables are weights (of dimension equal to the ensemble size) rather than model variables. Finally, the performance of our ensemble transform Kalman-Bucy implementations is evaluated using three models: the 3-variable Lorenz 1963 model, the 40-variable Lorenz 1996 model, and a medium complexity atmospheric general circulation model (AGCM) known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques

Assimilating atmospheric infrasound data to constrain atmospheric winds in a two-dimensional grid

Infrasound waves travelling through atmospheric channels are affected by the conditions the encounter in their path. The shift in the backazimuth angle of a wave front detected at the reception site depends on the cross-wind it encountered. Estimating the original field from this integrated measurement is an (ill-posed) inverse problem. By using a prior, this can be converted into a Bayesian estimation problem. In this work we use the (ensemble) Kalman filter to tackle this problem. In particular, we provide an illustration of the setup and solution of the problem in a two-dimensional grid, depending on both across-track distance and height, which has not been done in previous works. We use a synthetic setup to discuss the details of the method. We show that one of the effects of along-track averaging (something done in previous studies to simplify the problem) is to overestimate the magnitudes of the analysed values, and propose that this should a source of model error. We also illustrate the process with real data corresponding to nine controlled ammunition explosions that took place in the summer of 2018. In these cases, the real infrasound waves we study seldom reach higher than 40 km in height. However, the use of covariance-based methods (e.g. the EnKF) allows for updates in higher regions where the wave did not travel, and where traditional observations are sparse. In fact, the larger impacts from observations in these cases are in the region of 40 to 60 km, agreeing with previous works. This study contributes in paving the way towards the ultimate goal of assimilating information derived from infrasound waves into operational numerical weather forecasting. More studies in quality control of the observations and proper validation of the results are urgently needed

A revised implicit equal-weights particle filter

Particle filters are fully non-linear data assimilation methods and as such are highly relevant. While the standard particle filter degenerates for high-dimensional systems, recent developments have opened the way for new particle filters that can be used in such systems. The implicit equal-weights particle filter (IEWPF) is an efficient approach which avoids filter degeneracy because it gives equal particle weights by construction. The method uses implicit sampling whereby auxiliary vectors drawn from a proposal distribution undergo a transformation before they are added to each particle. In the original formulation of the IEWPF, the proposal distribution has a gap causing all but one particle to have an inaccessible region in state space. We show that this leads to a systematic bias in the predictions and we modify the proposal distribution to eliminate the gap. We achieved this by using a two-stage proposal method, where a single variance parameter is tuned to obtain adequate statistical coverage properties of the predictive distribution. We discuss properties of the implicit mapping from an auxiliary random vector to the state vector, keeping in mind the aim of avoiding particle resampling. The revised filter is tested on linear and weakly nonlinear dynamical models in low-dimensional and moderately high-dimensional settings, demonstrating the suiccess of the new methodology in removing the bias

Comparing hybrid data assimilation methods on the Lorenz 1963 model with increasing nonlinearity

We systematically compare the performance of ETKF-4DVAR, 4DVAR-BEN and 4DENVAR with respect to two traditional methods (4DVAR and ETKF) and an ensemble transform Kalman smoother (ETKS) on the Lorenz 1963 model. We specifically investigated this performance with increasing nonlinearity and using a quasi-static variational assimilation algorithm as a comparison. Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering (1) assimilation window length and observation interval size and (2) ensemble size to investigate the influence of hybrid background error covariance matrices and nonlinearity on the performance of the methods. For short assimilation windows with close to linear dynamics, it has been shown that all hybrid methods show an improvement in RMSE compared to the traditional methods. For long assimilation window lengths in which nonlinear dynamics are substantial, the variational framework can have diffculties fnding the global minimum of the cost function, so we explore a quasi-static variational assimilation (QSVA) framework. Of the hybrid methods, it is seen that under certain parameters, hybrid methods which do not use a climatological background error covariance do not need QSVA to perform accurately. Generally, results show that the ETKS and hybrid methods that do not use a climatological background error covariance matrix with QSVA outperform all other methods due to the full flow dependency of the background error covariance matrix which also allows for the most nonlinearity

Drawings of twogermans in Spain: Otto Schubert and Oskar Jürgens

[ES] En las primeras décadas del siglo XX se produce la curiosa coincidencia de dos arquitectos alemanes que se interesan por la arquitectura y las ciudades de España, efectuando un intenso trabajo de campo en, es de suponer, precarias condiciones de idioma y recursos. Lo más sorprendente de su labor consiste en la aportación gráfica que ambos realizan como estrategia metodológica, desarrollando una sistemática específica de dibujos en función de sus estudios que, en algunos aspectos, no ha sido superada al cabo de un siglo. Se intenta así realzar sus logros, analizando y valorando los aspectos gráficos de sus obras[EN] In the first decades of the 20th century occurs the curious coincidence of two German architects who are interested in architecture and the cities of Spain, carrying out an intense field work, presumably, at precarious conditions of language and resources. The most surprising aspect of their work is the graphical contribution that both perform as a methodological strategy, developing a specific classification system of drawings based on his studies that, in some respects, has not been exceeded after a century. We try thus to highlight their achievements, analyzing and evaluating the graphic aspects of their worksOrtega Vidal, J.; Amezcua Pajares, V. (2014). Dibujos de dos alemanes en España: Otto Schubert y Oskar Jürgens. EGA. Revista de Expresión Gráfica Arquitectónica. 19(24):106-115. doi:10.4995/ega.2014.3093.SWORD106115192