Linear-wavelet networks

This paper proposes a nonlinear regression structure comprising a wavelet network and a linear term. The introduction of the linear term is aimed at providing a more parsimonious interpolation in high-dimensional spaces when the modelling samples are sparse. A constructive procedure for building such structures, termed linear-wavelet networks, is described. For illustration, the proposed procedure is employed in the framework of dynamic system identification. In an example involving a simulated fermentation process, it is shown that a linear-wavelet network yields a smaller approximation error when compared with a wavelet network with the same number of regressors. The proposed technique is also applied to the identification of a pressure plant from experimental data. In this case, the results show that the introduction of wavelets considerably improves the prediction ability of a linear model. Standard errors on the estimated model coefficients are also calculated to assess the numerical conditioning of the identification process.

The primary somatosensory cortex largely contributes to the early part of the cortical response elicited by nociceptive stimuli.

Research on the cortical sources of nociceptive laser-evoked brain potentials (LEPs) began almost two decades ago (Tarkka and Treede, 1993). Whereas there is a large consensus on the sources of the late part of the LEP waveform (N2 and P2 waves), the relative contribution of the primary somatosensory cortex (S1) to the early part of the LEP waveform (N1 wave) is still debated. To address this issue we recorded LEPs elicited by the stimulation of four limbs in a large population (n=35). Early LEP generators were estimated both at single-subject and group level, using three different approaches: distributed source analysis, dipolar source modeling, and probabilistic independent component analysis (ICA). We show that the scalp distribution of the earliest LEP response to hand stimulation was maximal over the central-parietal electrodes contralateral to the stimulated side, while that of the earliest LEP response to foot stimulation was maximal over the central-parietal midline electrodes. Crucially, all three approaches indicated hand and foot S1 areas as generators of the earliest LEP response. Altogether, these findings indicate that the earliest part of the scalp response elicited by a selective nociceptive stimulus is largely explained by activity in the contralateral S1, with negligible contribution from the secondary somatosensory cortex (S2).

A tutorial on pseudospectral methods for computational optimal control

[English] This paper is a tutorial introduction to pseudospectral optimal control. With pseudospectral methods, a function is approximated as a linear combination of smooth basis functions, which are often chosen to be Legendre or Chebyshev polynomials. Collocation of the differential-algebraic equations is performed at orthogonal collocation points, which are selected to yield interpolation of high accuracy. Pseudospectral methods directly discretize the original optimal control problem to recast it into a nonlinear programming format. A numerical optimizer is then employed to find approximate local optimal solutions. The paper also briefly describes the functionality and implementation of PSOPT, an open source software package written in C++ that employs pseudospectral discretization methods to solve multi-phase optimal control problems. The software implements the Legendre and Chebyshev pseudospectral methods, and it has useful features such as automatic differentiation, sparsity detection, and automatic scaling. The use of pseudospectral methods is illustrated in two problems taken from the literature on computational optimal control. [Portuguese] Este artigo e um tutorial introdutorio sobre controle otimo pseudo-espectral. Em metodos pseudo-espectrais, uma funcao e aproximada como uma combinacao linear de funcoes de base suaves, tipicamente escolhidas como polinomios de Legendre ou Chebyshev. A colocacao de equacoes algebrico-diferenciais e realizada em pontos de colocacao ortogonal, que sao selecionados de modo a minimizar o erro de interpolacao. Metodos pseudoespectrais discretizam o problema de controle otimo original de modo a converte-lo em um problema de programa cao nao-linear. Um otimizador numerico e entao empregado para obter solucoes localmente otimas. Este artigo tambem descreve sucintamente a funcionalidade e a implementacao de um pacote computacional de codigo aberto escrito em C++ chamado PSOPT. Tal pacote emprega metodos de discretizacao pseudo-spectrais para resolver problemas de controle otimo com multiplas fase. O PSOPT permite a utilizacao de metodos de Legendre ou Chebyshev, e possui caractersticas uteis tais como diferenciacao automatica, deteccao de esparsidade e escalonamento automatico. O uso de metodos pseudo-espectrais e ilustrado em dois problemas retirados da literatura de controle otimo computacional.

Cyclic loss of open solar flux since 1868: the link to heliospheric current sheet tilt and implications for the Maunder Minimum

Open solar flux (OSF) variations can be described by the imbalance between source and loss terms. We use spacecraft and geomagnetic observations of OSF from 1868 to present and assume the OSF source, S, varies with the observed sunspot number, R. Computing the required fractional OSF loss, χ, reveals a clear solar cycle variation, in approximate phase with R. While peak R varies significantly from cycle to cycle, χ is surprisingly constant in both amplitude and waveform. Comparisons of χ with measures of heliospheric current sheet (HCS) orientation reveal a strong correlation. The cyclic nature of χ is exploited to reconstruct OSF back to the start of sunspot records in 1610. This agrees well with the available spacecraft, geomagnetic, and cosmogenic isotope observations. Assuming S is proportional to R yields near-zero OSF throughout the Maunder Minimum. However, χ becomes negative during periods of low R, particularly the most recent solar minimum, meaning OSF production is underestimated. This is related to continued coronal mass ejection (CME) activity, and therefore OSF production, throughout solar minimum, despite R falling to zero. Correcting S for this produces a better match to the recent solar minimum OSF observations. It also results in a cycling, nonzero OSF during the Maunder Minimum, in agreement with cosmogenic isotope observations. These results suggest that during the Maunder Minimum, HCS tilt cycled as over recent solar cycles, and the CME rate was roughly constant at the levels measured during the most recent two solar minima.

Integration of a 3D Variational data assimilation scheme with a coastal area morphodynamic model of Morecambe Bay

This paper describes the implementation of a 3D variational (3D-Var) data assimilation scheme for a morphodynamic model applied to Morecambe Bay, UK. A simple decoupled hydrodynamic and sediment transport model is combined with a data assimilation scheme to investigate the ability of such methods to improve the accuracy of the predicted bathymetry. The inverse forecast error covariance matrix is modelled using a Laplacian approximation which is calibrated for the length scale parameter required. Calibration is also performed for the Soulsby-van Rijn sediment transport equations. The data used for assimilation purposes comprises waterlines derived from SAR imagery covering the entire period of the model run, and swath bathymetry data collected by a ship-borne survey for one date towards the end of the model run. A LiDAR survey of the entire bay carried out in November 2005 is used for validation purposes. The comparison of the predictive ability of the model alone with the model-forecast-assimilation system demonstrates that using data assimilation significantly improves the forecast skill. An investigation of the assimilation of the swath bathymetry as well as the waterlines demonstrates that the overall improvement is initially large, but decreases over time as the bathymetry evolves away from that observed by the survey. The result of combining the calibration runs into a pseudo-ensemble provides a higher skill score than for a single optimized model run. A brief comparison of the Optimal Interpolation assimilation method with the 3D-Var method shows that the two schemes give similar results.

Controlling the computational modes of the arbitrarily structured C-grid

The arbitrarily structured C-grid, TRiSK (Thuburn, Ringler, Skamarock and Klemp, 2009, 2010) is being used in the ``Model for Prediction Across Scales'' (MPAS) and is being considered by the UK Met Office for their next dynamical core. However the hexagonal C-grid supports a branch of spurious Rossby modes which lead to erroneous grid-scale oscillations of potential vorticity (PV). It is shown how these modes can be harmlessly controlled by using upwind-biased interpolation schemes for PV. A number of existing advection schemes for PV are tested, including that used in MPAS, and none are found to give adequate results for all grids and all cases. Therefore a new scheme is proposed; continuous, linear-upwind stabilised transport (CLUST), a blend between centred and linear-upwind with the blend dependent on the flow direction with respect to the cell edge. A diagnostic of grid-scale oscillations is proposed which gives further discrimination between schemes than using potential enstrophy alone and indeed some schemes are found to destroy potential enstrophy while grid-scale oscillations grow. CLUST performs well on hexagonal-icosahedral grids and unrotated skipped latitude-longitude grids of the sphere for various shallow water test cases. Despite the computational modes, the hexagonal icosahedral grid performs well since these modes are easy and harmless to filter. As a result TRiSK appears to perform better than a spectral shallow water model.

Femtosecond pulse shaping using differential evolutionary algorithm and wavelet operators

Evolutionary meta-algorithms for pulse shaping of broadband femtosecond duration laser pulses are proposed. The genetic algorithm searching the evolutionary landscape for desired pulse shapes consists of a population of waveforms (genes), each made from two concatenated vectors, specifying phases and magnitudes, respectively, over a range of frequencies. Frequency domain operators such as mutation, two-point crossover average crossover, polynomial phase mutation, creep and three-point smoothing as well as a time-domain crossover are combined to produce fitter offsprings at each iteration step. The algorithm applies roulette wheel selection; elitists and linear fitness scaling to the gene population. A differential evolution (DE) operator that provides a source of directed mutation and new wavelet operators are proposed. Using properly tuned parameters for DE, the meta-algorithm is used to solve a waveform matching problem. Tuning allows either a greedy directed search near the best known solution or a robust search across the entire parameter space.

On the determination of atmospheric water vapor from GPS measurements

Surface-based GPS measurements of zenith path delay (ZPD) can be used to derive vertically integrated water vapor (IWV) of the atmosphere. ZPD data are collected in a global network presently consisting of 160 stations as part of the International GPS Service. In the present study, ZPD data from this network are converted into IWV using observed surface pressure and mean atmospheric water vapor column temperature obtained from the European Centre for Medium-Range Weather Forecasts' (ECMWF) operational analyses (OA). For the 4 months of January/July 2000/2001, the GPS-derived IWV values are compared to the IWV from the ECMWF OA, with a special focus on the monthly averaged difference (bias) and the standard deviation of daily differences. This comparison shows that the GPS-derived IWV values are well suited for the validation of OA of IWV. For most GPS stations, the IWV data agree quite well with the analyzed data indicating that they are both correct at these locations. Larger differences for individual days are interpreted as errors in the analyses. A dry bias in the winter is found over central United States, Canada, and central Siberia, suggesting a systematic analysis error. Larger differences were mainly found in mountain areas. These were related to representation problems and interpolation difficulties between model height and station height. In addition, the IWV comparison can be used to identify errors or problems in the observations of ZPD. This includes errors in the data itself, e.g., erroneous outlier in the measured time series, as well as systematic errors that affect all IWV values at a specific station. Such stations were excluded from the intercomparison. Finally, long-term requirements for a GPS-based water vapor monitoring system are discussed.

On the use of a time sequence of surface pressures in four-dimensional data assimilation

The possibility of using a time sequence of surface pressure observations in four-dimensional data assimilation is being investigated. It is shown that a linear multilevel quasi-geostrophic model can be updated successfully with surface data alone, provided the number of time levels are at least as many as the number of vertical levels. It is further demonstrated that current statistical analysis procedures are very inefficient to assimilate surface observations, and it is shown by numerical experiments that the vertical interpolation must be carried out using the structure of the most dominating baroclinic mode in order to obtain a satisfactory updating. Different possible ways towards finding a practical solution are being discussed.

An experiment in the Assimilation of Data in Dynamical Analysis

A system for continuous data assimilation is presented and discussed. To simulate the dynamical development a channel version of a balanced barotropic model is used and geopotential (height) data are assimilated into the models computations as data become available. In the first experiment the updating is performed every 24th, 12th and 6th hours with a given network. The stations are distributed at random in 4 groups in order to simulate 4 areas with different density of stations. Optimum interpolation is performed for the difference between the forecast and the valid observations. The RMS-error of the analyses is reduced in time, and the error being smaller the more frequent the updating is performed. The updating every 6th hour yields an error in the analysis less than the RMS-error of the observation. In a second experiment the updating is performed by data from a moving satellite with a side-scan capability of about 15°. If the satellite data are analysed at every time step before they are introduced into the system the error of the analysis is reduced to a value below the RMS-error of the observation already after 24 hours and yields as a whole a better result than updating from a fixed network. If the satellite data are introduced without any modification the error of the analysis is reduced much slower and it takes about 4 days to reach a comparable result to the one where the data have been analysed.

Problems in four-dimensional data assimilation

With the introduction of new observing systems based on asynoptic observations, the analysis problem has changed in character. In the near future we may expect that a considerable part of meteorological observations will be unevenly distributed in four dimensions, i.e. three dimensions in space and one in time. The term analysis, or objective analysis in meteorology, means the process of interpolating observed meteorological observations from unevenly distributed locations to a network of regularly spaced grid points. Necessitated by the requirement of numerical weather prediction models to solve the governing finite difference equations on such a grid lattice, the objective analysis is a three-dimensional (or mostly two-dimensional) interpolation technique. As a consequence of the structure of the conventional synoptic network with separated data-sparse and data-dense areas, four-dimensional analysis has in fact been intensively used for many years. Weather services have thus based their analysis not only on synoptic data at the time of the analysis and climatology, but also on the fields predicted from the previous observation hour and valid at the time of the analysis. The inclusion of the time dimension in objective analysis will be called four-dimensional data assimilation. From one point of view it seems possible to apply the conventional technique on the new data sources by simply reducing the time interval in the analysis-forecasting cycle. This could in fact be justified also for the conventional observations. We have a fairly good coverage of surface observations 8 times a day and several upper air stations are making radiosonde and radiowind observations 4 times a day. If we have a 3-hour step in the analysis-forecasting cycle instead of 12 hours, which is applied most often, we may without any difficulties treat all observations as synoptic. No observation would thus be more than 90 minutes off time and the observations even during strong transient motion would fall within a horizontal mesh of 500 km * 500 km.

A low-order model investigation of the analysis of gravity waves in the ensemble Kalman filter.

The behavior of the ensemble Kalman filter (EnKF) is examined in the context of a model that exhibits a nonlinear chaotic (slow) vortical mode coupled to a linear (fast) gravity wave of a given amplitude and frequency. It is shown that accurate recovery of both modes is enhanced when covariances between fast and slow normal-mode variables (which reflect the slaving relations inherent in balanced dynamics) are modeled correctly. More ensemble members are needed to recover the fast, linear gravity wave than the slow, vortical motion. Although the EnKF tends to diverge in the analysis of the gravity wave, the filter divergence is stable and does not lead to a great loss of accuracy. Consequently, provided the ensemble is large enough and observations are made that reflect both time scales, the EnKF is able to recover both time scales more accurately than optimal interpolation (OI), which uses a static error covariance matrix. For OI it is also found to be problematic to observe the state at a frequency that is a subharmonic of the gravity wave frequency, a problem that is in part overcome by the EnKF.However, error in themodeled gravity wave parameters can be detrimental to the performance of the EnKF and remove its implied advantages, suggesting that a modified algorithm or a method for accounting for model error is needed.

Conditioning model output statistics of regional climate model precipitation on circulation patterns

Dynamical downscaling of Global Climate Models (GCMs) through regional climate models (RCMs) potentially improves the usability of the output for hydrological impact studies. However, a further downscaling or interpolation of precipitation from RCMs is often needed to match the precipitation characteristics at the local scale. This study analysed three Model Output Statistics (MOS) techniques to adjust RCM precipitation; (1) a simple direct method (DM), (2) quantile-quantile mapping (QM) and (3) a distribution-based scaling (DBS) approach. The modelled precipitation was daily means from 16 RCMs driven by ERA40 reanalysis data over the 1961–2000 provided by the ENSEMBLES (ENSEMBLE-based Predictions of Climate Changes and their Impacts) project over a small catchment located in the Midlands, UK. All methods were conditioned on the entire time series, separate months and using an objective classification of Lamb's weather types. The performance of the MOS techniques were assessed regarding temporal and spatial characteristics of the precipitation fields, as well as modelled runoff using the HBV rainfall-runoff model. The results indicate that the DBS conditioned on classification patterns performed better than the other methods, however an ensemble approach in terms of both climate models and downscaling methods is recommended to account for uncertainties in the MOS methods.

Global-scale climate impact functions: the relationship between climate forcing and impact

Although there is a strong policy interest in the impacts of climate change corresponding to different degrees of climate change, there is so far little consistent empirical evidence of the relationship between climate forcing and impact. This is because the vast majority of impact assessments use emissions-based scenarios with associated socio-economic assumptions, and it is not feasible to infer impacts at other temperature changes by interpolation. This paper presents an assessment of the global-scale impacts of climate change in 2050 corresponding to defined increases in global mean temperature, using spatially-explicit impacts models representing impacts in the water resources, river flooding, coastal, agriculture, ecosystem and built environment sectors. Pattern-scaling is used to construct climate scenarios associated with specific changes in global mean surface temperature, and a relationship between temperature and sea level used to construct sea level rise scenarios. Climate scenarios are constructed from 21 climate models to give an indication of the uncertainty between forcing and response. The analysis shows that there is considerable uncertainty in the impacts associated with a given increase in global mean temperature, due largely to uncertainty in the projected regional change in precipitation. This has important policy implications. There is evidence for some sectors of a non-linear relationship between global mean temperature change and impact, due to the changing relative importance of temperature and precipitation change. In the socio-economic sectors considered here, the relationships are reasonably consistent between socio-economic scenarios if impacts are expressed in proportional terms, but there can be large differences in absolute terms. There are a number of caveats with the approach, including the use of pattern-scaling to construct scenarios, the use of one impacts model per sector, and the sensitivity of the shape of the relationships between forcing and response to the definition of the impact indicator.

An investigation into the impact of using various techniques to estimate Arctic surface air temperature anomalies

Time series of global and regional mean Surface Air Temperature (SAT) anomalies are a common metric used to estimate recent climate change. Various techniques can be used to create these time series from meteorological station data. The degree of difference arising from using five different techniques, based on existing temperature anomaly dataset techniques, to estimate Arctic SAT anomalies over land and sea ice were investigated using reanalysis data as a testbed. Techniques which interpolated anomalies were found to result in smaller errors than non-interpolating techniques relative to the reanalysis reference. Kriging techniques provided the smallest errors in estimates of Arctic anomalies and Simple Kriging was often the best kriging method in this study, especially over sea ice. A linear interpolation technique had, on average, Root Mean Square Errors (RMSEs) up to 0.55 K larger than the two kriging techniques tested. Non-interpolating techniques provided the least representative anomaly estimates. Nonetheless, they serve as useful checks for confirming whether estimates from interpolating techniques are reasonable. The interaction of meteorological station coverage with estimation techniques between 1850 and 2011 was simulated using an ensemble dataset comprising repeated individual years (1979-2011). All techniques were found to have larger RMSEs for earlier station coverages. This supports calls for increased data sharing and data rescue, especially in sparsely observed regions such as the Arctic.