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In chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure on-the-fly to increase accuracy or reduce computation time. One could, e.g., change the ensemble size, or the distribution and type of target observations. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but non intrusively as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, convolutional neural networks and long short-term memory networks. Experiments are conducted on two low-dimensional chaotic systems of ordinary differential equations, the R\"ossler and the Lorenz 63 models. We find that on average the machine learning algorithms predict the stable local Lyapunov exponent accurately, the unstable exponent reasonably accurately, and the neutral exponent only somewhat accurately. We show that greater prediction accuracy is associated with local homogeneity of the local Lyapunov exponents on the system attractor. Importantly, the situations in which (forecast) errors grow fastest are not necessarily the same as those where it is more difficult to predict local Lyapunov exponents with machine learning., Comment: 37 pages, 10 Figures

Hybrid ensemble-variational data assimilation (DA) methods have gained significant traction in recent years. These methods aim to alleviate the limitations and maximise the advantages offered by ensemble or variational methods. Most existing hybrid applications focus on the mid-latitudinal context; almost none have explored its benefits in the tropical context. In this article, hybrid ensemble-variational DA is introduced to a tropical configuration of a simplified non-hydrostatic convective-scale fluid dynamics model (the ABC model, named after its three key parameters: the pure gravity wave frequency A, the controller of the acoustic wave speed B, and the constant of proportionality between pressure and density perturbations C), and its existing variational framework, the ABC-DA system. The hybrid ensemble-variational DA algorithm is developed based on the alpha control variable approach, often used in numerical weather prediction. Aspects of the algorithm such as localisation (used to mitigate sampling error caused by finite ensemble sizes) and weighting parameters (used to weight the ensemble and climatological contributions to the background error covariance matrix) are implemented. To produce the flow-dependent error modes (ensemble perturbations) for the ensemble-variational DA algorithm, an ensemble system is also designed for the ABC model which is run alongside the hybrid DA system. A random field perturbations method is used to generate an initial ensemble which is then propagated using the ensemble bred vectors method. This setup allows the ensemble to be centred on the hybrid control analysis. Visualisation software has been developed to focus on the diagnosis of the ensemble system. To demonstrate the hybrid ensemble-variational DA in the ABC-DA system, sensitivity tests using observing system simulation experiments are conducted within a tropical framework. A 30-member ensemble was used to generate the error modes for the experiments. In general, the best performing configuration (with respect to the “truth”) for the hybrid ensemble-variational DA system used an 80%/20% weighting on the ensemble-derived/climatological background error covariance matrix contributions. For the horizontal wind variables though, full weight on the ensemble-derived background error covariance matrix (100%/0%) resulted in the smallest cycle-averaged analysis root mean square errors, mainly due to large errors in the meridional wind field when contributions from the climatological background error covariance matrix were involved, possibly related to a sub-optimal background error covariance model. The ensemble bred vectors method propagated a healthy-looking DA-centred ensemble without bimodalities or evidence of filter collapse. The ensemble was under-dispersive for some variables but for others, the ensemble spread approximately matched the corresponding root mean square errors. Reducing the number of ensemble members led to slightly larger errors across all variables due to the introduction of larger sampling errors into the system.

Infrasound waves generated by phenomena at the Earth’s surface can travel to these levels before returning to the surface and being detected. Observations like travel time, change in backazimuth angle, and trace velocity contain integrated information of all the levels the wave travelled through. These often include stratospheric and mesospheric levels which are otherwise poorly observed.In this work we take a data assimilation technique, the Modulated Ensemble Transform Kalman Filter, which is commonly used in satellite data assimilation, and illustrate how it can be readily used for infrasound data assimilation. We highlight the similarities between the two problems, and the particular challenges in extracting information from summarised quantities. To our knowledge, this is the first work doing data assimilation with a full ray-tracing model as forward operator.

The MADEIRA project (Middle Atmosphere Dynamics: Exploiting Infrasound Using a Multidisciplinary Approach at High Latitudes) is a four-year basic research project finishing in the spring 2023, funded by the Research Council of Norway. Its primary objective has been in elucidating the 30-60 km altitude range over large spatial scales using wind and temperature constraints from infrasound data collected at Arctic stations. Wave propagation modelling and infrasound interpretation from well-characterized sources provide remote atmospheric sensing. These data are more continuous in space and time than from many other direct measurement techniques. An aim has been to constrain high-top atmospheric models and explore stratosphere-mesosphere coupling with meteor radar wind measurements sampling the 70-100 km altitude range in combination with the infrasonic data. Another ambition has been to develop real-time diagnostic tools for the stratospheric polar vortex circulation and extreme events like Sudden Stratospheric Warmings. Thanks to this project, the research teams have got the opportunity to explore several aspects and building blocks related to infrasound-based middle atmospheric probing and to work towards an assimilation of such datasets into atmospheric models. This paper reviews key research output from the project and highlights accomplishments in the domains of, e.g.: Tropospheric and stratospheric cross-wind estimation using infrasound from explosions; Assimilation of atmospheric infrasound data to constrain tropospheric and stratospheric winds; Atmospheric wind and temperature profile inversion in an ensemble model context; Microbarom radiation and propagation model benchmarking; Speeding up infrasound transmission loss estimation using deep learning; Probing internal middle atmospheric gravity waves; Using a machine learning and stochastics-founded model to provide near real-time stratospheric polar vortex diagnostics. This project has included several high risk / high gain components and we highlight results that maybe could be labelled as unexpected successes, but we also discuss challenging research obstacles that occurred in our journey.

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.

When an infrasound wave travels through the atmosphere, it is affected by the atmospheric variables it encounters (e.g. temperature and winds) in its path. When the wave is detected, the integrated effect of these variables along the trajectory of the wave affects measured quantities such as apparent velocity, backazimuth angle and travel time. Data assimilation combines background atmospheric information with observations to get a better estimate (analysis) of atmospheric variables. In this work, we use the ensemble Kalman filter --with the 10-member ERA-5 reanalysis as background-- to assimilate integrated infrasound observations from the Hukkakero explosions detected by the ARCES array. This process helps get better estimates of atmospheric variables, specially in the stratosphere and lower mesosphere. For each explosion, this process has three steps: (i) the mapping of each of the 10 atmospheric profiles into observation space using the Infra-GA wave propagation model, (ii) the application of the filer equations in observation space, and (c) the mapping back to the space of model variables. The results of these experiments are compared to the Copernicus Artic Regional Reanalysis Service.

Study of cause and effect relationships – causality - is central to identifying mechanisms that cause the phenomena we observe. And in non-linear, dynamical systems, we wish to understand these mechanisms unfolding over time. In areas within physical sciences like geosciences, astrophysics, etc. there are numerous competing causes that drive the system in complicated ways that are hard to disentangle. Hence, it is important to demonstrate how causal attribution works with relatively simpler systems where we have a physical intuition. Furthermore, in earth and atmospheric sciences or meteorology, we have a plethora of observations that are used in both understanding the underlying science beneath the phenomena as well as forecasting. However in order to do this, optimally combining the models (theoretical/numerical) with the observations through data assimilation is a challenging, computationally intensive task. Therefore, understanding the impact of observations and the required cadence is very useful. Here, we present experiments in causal inference and attribution with the Lorenz 63 system – a system studied for a long time. We first test the causal relations between the variables characterising the model. And then we simulate observations using perturbed versions of the model to test the impact of the cadence of observations of each combination of the 3 variables.

Weather and climate are well known exemplars of chaotic systems exhibiting extreme sensitivity to initial conditions. Initial condition errors are subject to exponential growth on average, but the rate and the characteristic of such growth is highly state dependent. In an ideal setting where the degree of predictability of the system is known in real-time, it may be possible and beneficial to take adaptive measures. For instance a local decrease of predictability may be counteracted by increasing the time- or space-resolution of the model computation or the ensemble size in the context of ensemble-based data assimilation or probabilistic forecasting.Local Lyapunov exponents (LLEs) describe growth rates along a finite-time section of a system trajectory. This makes the LLEs the ideal quantities to measure the local degree of predictability, yet a main bottleneck for their real-time use in operational scenarios is the huge computational cost. Calculating LLEs involves computing a long trajectory of the system, propagating perturbations with the tangent linear model, and repeatedly orthogonalising them. We investigate if machine learning (ML) methods can estimate the LLEs based only on information from the system’s solution, thus avoiding the need to evolve perturbations via the tangent linear model. We test the ability of four algorithms (regression tree, multilayer perceptron, convolutional neural network and long short-term memory network) to perform this task in two prototypical low dimensional chaotic dynamical systems. Our results suggest that the accuracy of the ML predictions is highly dependent upon the nature of the distribution of the LLE values in phase space: large prediction errors occur in regions of the attractor where the LLE values are highly non-smooth. In line with classical dynamical systems studies, the neutral LLE is more difficult to predict. We show that a comparatively simple regression tree can achieve performance that is similar to sophisticated neural networks, and that the success of ML strategies for exploiting the temporal structure of data depends on the system dynamics.

This data assimilation study exploits infrasound from explosions to probe an atmospheric wind component from the ground up to stratospheric altitudes. Planned explosions of old ammunition in Finland generate transient infrasound waves that travel through the atmosphere. These waves are partially reflected back towards the ground from stratospheric levels, and are detected at a receiver station located in northern Norway at 178 km almost due North from the explosion site. The difference between the true horizontal direction towards the source and the backazimuth direction (the horizontal direction of arrival) of the incoming infrasound wave-fronts, in combination with the pulse propagation time, are exploited to provide an estimate of the average cross-wind component in the penetrated atmosphere. We perform offline assimilation experiments with an ensemble Kalman filter and these observations, using the ERA5 ensemble reanalysis atmospheric product as background (prior) for the wind at different vertical levels. We demonstrate that information from both sources can be combined to obtain analysis (posterior) estimates of cross-winds at different vertical levels of the atmospheric slice between the explosion site and the recording station. The assimilation makes greatest impact at the 12-60 km levels, with some changes with respect to the prior of the order of 0.1-1.0 m/s, which is a magnitude larger than the typical standard deviation of the ERA5 background. The reduction of background variance in the higher levels often reached 2-5%. This is the first published study demonstrating techniques to implement assimilation of infrasound data into atmospheric models. It paves the way for further exploration in the use of infrasound observations - especially natural and continuous sources - to probe the middle atmospheric dynamics and to assimilate these data into atmospheric model products., Comment: Manuscript submitted to the Quarterly Journal of the Royal Meteorological Society. The current document is an e-print typeset by the authors

Data assimilation is often performed under the perfect model assumption. Although there is an increasing amount of research accounting for model errors in data assimilation, the impact of an incorrect specification of the model errors on the data assimilation results has not been thoroughly assessed. We investigate the effect that an inaccurate time correlation in the model error description can have on data assimilation results, deriving analytical results using a Kalman Smoother for a one‐dimensional system. The analytical results are evaluated numerically to generate useful illustrations. For a higher‐dimensional system, we use an ensemble Kalman Smoother. Strong dependence on observation density is found. For a single observation at the end of the window, the posterior variance is a concave function of the guessed decorrelation time‐scale used in the data assimilation process. This is due to an increasing prior variance with that time‐scale, combined with a decreasing tendency from larger observation influence. With an increasing number of observations, the posterior variance decreases with increasing guessed decorrelation time‐scale because the prior variance effect becomes less important. On the other hand, the posterior mean‐square error has a convex shape as a function of the guessed time‐scale with a minimum where the guessed time‐scale is equal to the real decorrelation time‐scale. With more observations, the impact of the difference between two decorrelation time‐scales on the posterior mean‐square error reduces. Furthermore, we show that the correct model error decorrelation time‐scale can be estimated over several time windows using state augmentation in the ensemble Kalman Smoother. Since model errors are significant and significantly time correlated in real geophysical systems such as the atmosphere, this contribution opens up a next step in improving prediction of these systems., Variable x in the Lorenz 1963 model with no model error, time‐independent random error, and strongly auto‐correlated error.

This work shows how one can use iterative ensemble smoothers to effectively estimate parameters of an SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters leading to model predictions in close agreement with the observations. The updated ensemble of model simulations have predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development given knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries with vastly different developments of the epidemic, and we can accurately model the development of the COVID-19 outbreak in these countries. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.

Data assimilation has been often performed under the perfect model assumption known as the strong-constraint setting. There is an increasing number of researches accounting for the model errors, the weak-constrain setting, but often with different degrees of approximation or simplification without knowing their impact on the data assimilation results. We investigate what effect inaccurate model errors, in particular, the an inaccurate time correlation, can have on data assimilation results, with a Kalman Smoother and the Ensemble Kalman Smoother.We choose a linear auto-regressive model for the experiment. We assume the true state of the system has the correct and fixed correlation time-scale ωr in the model errors, and the prior or the background generated by the model contains the model error with the fixed, guessed time-scale ωg which differs from the correct one and is also used in the data assimilation process. There are 10 variables in the system and we separate the simulation period into multiple time-windows. And we use a fairly large ensemble size (up to 200 ensemble members) to improve the accuracy of the data assimilation results. In order to evaluate the performance of the EnKS with auto-correlated model errors, we calculate the ratio of root-mean-square error over the spread of all ensemble members.The results with a single observation at the end of the simulation time-window show that, using an underestimated correlation time-scale leads to overestimated spread of the ensemble, and with an overestimated time-scale, the results show underestimation in the ensemble spread. However, with very dense observation frequency, observing every time-step for instance, the results are completely opposite to the results with a single observation. In order to understand the results, we derive the expression for the true posterior state covariance and the posterior covariance using the incorrect decorrelation time-scale. We do this for a Kalman Smoother to avoid the sampling uncertainties. The results are richer than expected and highly dependent on the observation frequency. From the analytical solution of the analysis, we find that the RMSE is a function of both ωr and ωg, and the spread or the variance only depends on ωg. We also find that the analyzed variance is not always a monotonically increasing function of ωg, and it also depends on the observation frequency. In general, the results show the effect of the correlated model error and the incorrect correlation time-scale on data assimilation result, which is also affected by the observation frequency.

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.

A new method for the long-term prediction of the wind resource based on the concept of statistical-dynamical downscaling is presented. This new approach uses mean sea level pressure maps from global reanalysis data (National Centers for Environmental Prediction Department of Energy Atmospheric Model Intercomparison Project (NCEP-DOE AMIP-II)) and image processing techniques to identify a synthetic reference period which optimally matches the corresponding long-term maps. Four different image processing techniques, averaged into one image similarity error index, are used to evaluate image similarity. A representative set of days is selected by requiring the error index to be minimal. Validation of representativeness in terms of the wind resource for the Iberian domain is performed against 10 years of measured wind data from Navarra (Spain), as well as mesoscale simulations of the Iberian Peninsula. The new approach is shown to outperform not only the industry-standard method but also other recently proposed methods in its capability to achieve mesoscale level representativeness. A particular advantage of the new method is its capability of simultaneously providing a representative period for all potential wind farm sites located within large regional domains without requiring re-running of the method for different candidate sites.

Encephalomyopathic mitochondrial DNA (mtDNA) depletion syndrome 13 (MTDPS13) is a rare genetic disorder caused by defects in F-box leucine-rich repeat protein 4 (FBXL4). Although FBXL4 is essential for the bioenergetic homeostasis of the cell, the precise role of the protein remains unknown. In this study, we report two cases of unrelated patients presenting in the neonatal period with hyperlactacidemia and generalized hypotonia. Severe mtDNA depletion was detected in muscle biopsy in both patients. Genetic analysis showed one patient as having in compound heterozygosis a splice site variant c.858+5G>C and a missense variant c.1510T>C (p.Cys504Arg) in FBXL4. The second patient harbored a frameshift novel variant c.851delC (p.Pro284LeufsTer7) in homozygosis. To validate the pathogenicity of these variants, molecular and biochemical analyses were performed using skin-derived fibroblasts. We observed that the mtDNA depletion was less severe in fibroblasts than in muscle. Interestingly, the cells harboring a nonsense variant in homozygosis showed normal mtDNA copy number. Both patient fibroblasts, however, demonstrated reduced mitochondrial transcript quantity leading to diminished steady state levels of respiratory complex subunits, decreased respiratory complex IV (CIV) activity, and finally, low mitochondrial ATP levels. Both patients also revealed citrate synthase deficiency. Genetic complementation assays established that the deficient phenotype was rescued by the canonical version of FBXL4, confirming the pathological nature of the variants. Further analysis of fibroblasts allowed to establish that increased mitochondrial mass, mitochondrial fragmentation, and augmented autophagy are associated with FBXL4 deficiency in cells, but are probably secondary to a primary metabolic defect affecting oxidative phosphorylation.

Particle filters are fully nonlinear 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 that 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 the 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 success of the new methodology in removing the bias. This is the peer reviewed version of an article, which has been published in final form at [https://doi.org/10.1002/qj.3506]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. "

This work demonstrates the efficiency of using iterative ensemble smoothers to estimate the parameters of an SEIR model. We have extended a standard SEIR model with age-classes and compartments of sick, hospitalized, and dead. The data conditioned on are the daily numbers of accumulated deaths and the number of hospitalized. Also, it is possible to condition the model on the number of cases obtained from testing. We start from a wide prior distribution for the model parameters; then, the ensemble conditioning leads to a posterior ensemble of estimated parameters yielding model predictions in close agreement with the observations. The updated ensemble of model simulations has predictive capabilities and include uncertainty estimates. In particular, we estimate the effective reproductive number as a function of time, and we can assess the impact of different intervention measures. By starting from the updated set of model parameters, we can make accurate short-term predictions of the epidemic development assuming knowledge of the future effective reproductive number. Also, the model system allows for the computation of long-term scenarios of the epidemic under different assumptions. We have applied the model system on data sets from several countries, i.e., the four European countries Norway, England, The Netherlands, and France; the province of Quebec in Canada; the South American countries Argentina and Brazil; and the four US states Alabama, North Carolina, California, and New York. These countries and states all have vastly different developments of the epidemic, and we could accurately model the SARS-CoV-2 outbreak in all of them. We realize that more complex models, e.g., with regional compartments, may be desirable, and we suggest that the approach used here should be applicable also for these models.

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.

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 non-linearity 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 non-linearity 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 non-linear dynamics are substantial, the variational framework can have difficulties finding 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 non-linearity. Keywords: data assimilation, hybrid methods, flow dependence (Published: 26 May 2015) Citation: Tellus A 2015, 67, 26928, http://dx.doi.org/10.3402/tellusa.v67.26928

In recent years, hybrid data assimilation methods which avoid computation of tangent linear and adjoint models by using ensemble 4-dimensional cross-time covariances have become a popular topic in Numerical Weather Prediction. 4DEnsembleVar is one such method. In spite of its capabilities, its application can sometimes become problematic due to the not-trivial task of localising cross-time covariances. In this work we propose a formulation that helps to alleviate such issues by exploiting the presence of model error, i.e a weak-constraint 4DEnsembleVar. We compare the weak-constraint 4DEnsenbleVar to that of other data assimilation methods. \ud \ud This is part II of a two-part paper. In part I, we describe the 4DEnsembleVar framework and problems with localised temporal cross-covariances associated with this method are discussed and illustrated on the Korteweg de Vries model. We also introduce our Weak-Constraint 4DEnsembleVar formulation and show how it can alleviate --at least partially-- the problem of having low-quality time cross-covariances.\ud \ud The second part of this paper deals with experiments on larger and more complicated models, namely the Lorenz 1996 model and a modified shallow water model with simulated convection, both of them under the presence of model error. We investigate the performance of weak-constraint 4DEnsembleVar against strong-constraint 4DEnsembleVar (both with and without localisation) and other traditional methods (4DVar and the Local Ensemble Transform Kalman Smoother). Using the analysis root mean square error (RMSE) as a metric, these methods have been compared considering observation density (in time and space), observation period, ensemble sizes and assimilation window length. In this part we also explain how to perform outer loops in the EnVar methods. We show that their use can be counter-productive if the presence of model error is ignored by the assimilation method. \ud \ud We show that the addition of a weak-constraint generally improves the RMSE of 4DEnVar in cases where model error has time to develop, especially in cases with long assimilation windows and infrequent observations. We have assumed good knowledge of the statistics of this model error.

The time discretization in weather and climate models introduces truncation errors that limit the accuracy of the simulations. Recent work has yielded a method for reducing the amplitude errors in leapfrog integrations from first-order to fifth-order. This improvement is achieved by replacing the Robert--Asselin filter with the RAW filter and using a linear combination of the unfiltered and filtered states to compute the tendency term. The purpose of the present paper is to apply the composite-tendency RAW-filtered leapfrog scheme to semi-implicit integrations. A theoretical analysis shows that the stability and accuracy are unaffected by the introduction of the implicitly treated mode. The scheme is tested in semi-implicit numerical integrations in both a simple nonlinear stiff system and a medium-complexity atmospheric general circulation model, and yields substantial improvements in both cases. We conclude that the composite-tendency RAW-filtered leapfrog scheme is suitable for use in semi-implicit integrations.

Two recent works have adapted the Kalman–Bucy filter into an ensemble setting. In the first formulation, 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 Kalman filter. In the second formulation, 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 analyse the behaviour of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background error to observational error variance, and that using the integration scheme proposed in both formulations can lead to failure. A numerical 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 known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques.

Ensemble clustering (EC) can arise in data assimilation with ensemble square root filters (EnSRFs) using non-linear models: an M -member ensemble splits into a single outlier and a cluster of M −1 members. The stochastic Ensemble Kalman Filter does not present this problem. Modifications to the EnSRFs by a periodic resampling of the ensemble through random rotations have been proposed to address it. We introduce a metric to quantify the presence of EC and present evidence to dispel the notion that EC leads to filter failure. Starting from a univariate model, we show that EC is not a permanent but transient phenomenon; it occurs intermittently in non-linear models. We perform a series of data assimilation experiments using a standard EnSRF and a modified EnSRF by a resampling though random rotations. The modified EnSRF thus alleviates issues associated with EC at the cost of traceability of individual ensemble trajectories and cannot use some of algorithms that enhance performance of standard EnSRF. In the non-linear regimes of low-dimensional models, the analysis root mean square error of the standard EnSRF slowly grows with ensemble size if the size is larger than the dimension of the model state. However, we do not observe this problem in a more complex model that uses an ensemble size much smaller than the dimension of the model state, along with inflation and localisation. Overall, we find that transient EC does not handicap the performance of the standard EnSRF. Keywords: non-linear models, ensemble square root filters, non-Gaussianity (Published: 23 July 2012) Citation: Tellus A 2012, 64 , 18039, http://dx.doi.org/10.3402/tellusa.v64i0.18039

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.

It has been described that 70% of all maternal deaths are provoked by obstetrical hemorrhage, infections, abortions, hypertension and delivery dystocies. Poverty, social exclusion, low level education and violence are important causes of maternal mortality.To establish the changes in the maternal mortality in a term of 15 years in a hospital of assistance obstetrical complicated.A retrospective and descriptive study, in which the number and causes of obstetrical death was analyzed, occurred from 1991 to 2005. The comparison was done by five-year periods using descriptive statistics to analyze frequency of results.The number of maternal deaths was 105, 97 and 42 by each one of the three five-year periods, the mortality rate x 10,000 decreased from 28.7 to 16.4 in the last quinquennium and was found from 6.1 just including the last year. In the first and second quinquennia the eclampsia occupied the first place as cause of death, followed by the hemorrhage and the infections. In the third quinquennium the eclampsia also occupied the first place with a rate of 8.6, followed by the cardiopathy (2.3) and the infections (1.9), but the hemorrhage with a rate of 1.5 was displaced to the fourth place.The maternal mortality has diminished in a general way; the eclampsia has occupied the first place as cause of death from 1991 to 2005. The death by obstetrical hemorrhage has diminished in important form, possibly due to the specific groups of medical attention by modules, which has also helps the decrease of mortality by other causes. The increment of the deaths by cardiopathy should be considered as a possibility of risk, associate undoubtedly to the present style of life from our society.

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. Keywords: Gaussian anamorphosis, ensemble Kalman filter, nonlinearity, joint transformations, non-Gaussianity (Published: 26 September 2014) Citation: Tellus A 2014, 66 , 23493, http://dx.doi.org/10.3402/tellusa.v66.23493

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.

A systematic evaluation of agricultural factors affecting the adaptation of the tropical oil plant Jatropha curcas L. to the semi-arid subtropical climate in Northeastern Mexico has been conducted. The factors studied include plant density and topology, as well as fungi and virus abundances. A multiple regression analysis shows that total fruit production can be well predicted by the area per plant and the total presence of fungi. Four common herbicides and a mechanical weed control measure were established at a dedicated test array and their impact on plant productivity was assessed.

The collection of wind speed time series by means of digital data loggers occurs in many domains, including civil engineering, environmental sciences and wind turbine technology. Since averaging intervals are often significantly larger than typical system time scales, the information lost has to be recovered in order to reconstruct the true dynamics of the system. In the present work we present a simple algorithm capable of generating a real-time wind speed time series from data logger records containing the average, maximum, and minimum values of the wind speed in a fixed interval, as well as the standard deviation. The signal is generated from a generalized random Fourier series. The spectrum can be matched to any desired theoretical or measured frequency distribution. Extreme values are specified through a postprocessing step based on the concept of constrained simulation. Applications of the algorithm to 10-min wind speed records logged at a test site at 60 m height above the ground show that the recorded 10-min values can be reproduced by the simulated time series to a high degree of accuracy.

4DEnsembleVar is a hybrid data assimilation method which purpose is not only to use ensemble flow-dependent covariance information in a variational setting, but to altogether avoid the computation of tangent linear and adjoint models. This formulation has been explored in the context of perfect models. In this setting, all information from observations has to be brought back to the start of the assimilation window using the space-time covariances of the ensemble. In large models, localisation of these covariances is essential, but the standard time-independent localisation leads to serious problems when advection is strong. This is because observation information is advected out of the localisation area, having no influence on the update.\ud \ud This is part I of a two-part paper in which we develop a weak-constraint formulation in which updates are allowed at observational times. This partially alleviates the time-localisation problem. Furthermore, we provide --for the first time-- a detailed description of strong- and weak-constraint 4DEnVar, including implementation details for the incremental form. \ud \ud The merits of our new weak-constraint formulation are illustrated using the Korteweg-de-Vries equation (propagation of a soliton). The second part of this paper deals with experiments in larger and more complicated models, namely the Lorenz 1996 model and a shallow water equations model with simulated convection.