Spatial analysis of the error in a model of soil nitrogen

Models of the dynamics of nitrogen in soil (soil-N) can be used to aid the fertilizer management of a crop. The predictions of soil-N models can be validated by comparison with observed data. Validation generally involves calculating non-spatial statistics of the observations and predictions, such as their means, their mean squared-difference, and their correlation. However, when the model predictions are spatially distributed across a landscape the model requires validation with spatial statistics. There are three reasons for this: (i) the model may be more or less successful at reproducing the variance of the observations at different spatial scales; (ii) the correlation of the predictions with the observations may be different at different spatial scales; (iii) the spatial pattern of model error may be informative. In this study we used a model, parameterized with spatially variable input information about the soil, to predict the mineral-N content of soil in an arable field, and compared the results with observed data. We validated the performance of the N model spatially with a linear mixed model of the observations and model predictions, estimated by residual maximum likelihood. This novel approach allowed us to describe the joint variation of the observations and predictions as: (i) independent random variation that occurred at a fine spatial scale; (ii) correlated random variation that occurred at a coarse spatial scale; (iii) systematic variation associated with a spatial trend. The linear mixed model revealed that, in general, the performance of the N model changed depending on the spatial scale of interest. At the scales associated with random variation, the N model underestimated the variance of the observations, and the predictions were correlated poorly with the observations. At the scale of the trend, the predictions and observations shared a common surface. The spatial pattern of the error of the N model suggested that the observations were affected by the local soil condition, but this was not accounted for by the N model. In summary, the N model would be well-suited to field-scale management of soil nitrogen, but suited poorly to management at finer spatial scales. This information was not apparent with a non-spatial validation. (c),2007 Elsevier B.V. All rights reserved.

The role of atmosphere feedbacks during ENSO in the CMIP3 models

Several studies using ocean-atmosphere GCMs suggest that the atmospheric component plays a dominant role in the modelled ENSO. To help elucidate these findings, the two main atmosphere feedbacks relevant to ENSO, the Bjerknes positive feedback (µ) and the heat flux negative feedback (), are analysed here in 12 coupled GCMs. We find that the models generally underestimate both feedbacks, leading to an error compensation. The strength of is inversely related to the ENSO amplitude in the models and the latent heat and shortwave flux components of this feedback dominate. Furthermore, the shortwave component could help explain the model diversity in both overall and ENSO amplitude.

Adjoint goal-based error norms for adaptive mesh ocean modelling

Flow in the world's oceans occurs at a wide range of spatial scales, from a fraction of a metre up to many thousands of kilometers. In particular, regions of intense flow are often highly localised, for example, western boundary currents, equatorial jets, overflows and convective plumes. Conventional numerical ocean models generally use static meshes. The use of dynamically-adaptive meshes has many potential advantages but needs to be guided by an error measure reflecting the underlying physics. A method of defining an error measure to guide an adaptive meshing algorithm for unstructured tetrahedral finite elements, utilizing an adjoint or goal-based method, is described here. This method is based upon a functional, encompassing important features of the flow structure. The sensitivity of this functional, with respect to the solution variables, is used as the basis from which an error measure is derived. This error measure acts to predict those areas of the domain where resolution should be changed. A barotropic wind driven gyre problem is used to demonstrate the capabilities of the method. The overall objective of this work is to develop robust error measures for use in an oceanographic context which will ensure areas of fine mesh resolution are used only where and when they are required. (c) 2006 Elsevier Ltd. All rights reserved.

Metrics for solar wind prediction models: Comparison of empirical, hybrid, and physics-based schemes with 8 years of L1 observations

Space weather effects on technological systems originate with energy carried from the Sun to the terrestrial environment by the solar wind. In this study, we present results of modeling of solar corona-heliosphere processes to predict solar wind conditions at the L1 Lagrangian point upstream of Earth. In particular we calculate performance metrics for (1) empirical, (2) hybrid empirical/physics-based, and (3) full physics-based coupled corona-heliosphere models over an 8-year period (1995–2002). L1 measurements of the radial solar wind speed are the primary basis for validation of the coronal and heliosphere models studied, though other solar wind parameters are also considered. The models are from the Center for Integrated Space-Weather Modeling (CISM) which has developed a coupled model of the whole Sun-to-Earth system, from the solar photosphere to the terrestrial thermosphere. Simple point-by-point analysis techniques, such as mean-square-error and correlation coefficients, indicate that the empirical coronal-heliosphere model currently gives the best forecast of solar wind speed at 1 AU. A more detailed analysis shows that errors in the physics-based models are predominately the result of small timing offsets to solar wind structures and that the large-scale features of the solar wind are actually well modeled. We suggest that additional “tuning” of the coupling between the coronal and heliosphere models could lead to a significant improvement of their accuracy. Furthermore, we note that the physics-based models accurately capture dynamic effects at solar wind stream interaction regions, such as magnetic field compression, flow deflection, and density buildup, which the empirical scheme cannot.

Calibrating flood inundation models: identifying and addressing uncertainty in satellite observed flood extents

Improvements in the resolution of satellite imagery have enabled extraction of water surface elevations at the margins of the flood. Comparison between modelled and observed water surface elevations provides a new means for calibrating and validating flood inundation models, however the uncertainty in this observed data has yet to be addressed. Here a flood inundation model is calibrated using a probabilistic treatment of the observed data. A LiDAR guided snake algorithm is used to determine an outline of a flood event in 2006 on the River Dee, North Wales, UK, using a 12.5m ERS-1 image. Points at approximately 100m intervals along this outline are selected, and the water surface elevation recorded as the LiDAR DEM elevation at each point. With a planar water surface from the gauged upstream to downstream water elevations as an approximation, the water surface elevations at points along this flooded extent are compared to their ‘expected’ value. The pattern of errors between the two show a roughly normal distribution, however when plotted against coordinates there is obvious spatial autocorrelation. The source of this spatial dependency is investigated by comparing errors to the slope gradient and aspect of the LiDAR DEM. A LISFLOOD-FP model of the flood event is set-up to investigate the effect of observed data uncertainty on the calibration of flood inundation models. Multiple simulations are run using different combinations of friction parameters, from which the optimum parameter set will be selected. For each simulation a T-test is used to quantify the fit between modelled and observed water surface elevations. The points chosen for use in this T-test are selected based on their error. The criteria for selection enables evaluation of the sensitivity of the choice of optimum parameter set to uncertainty in the observed data. This work explores the observed data in detail and highlights possible causes of error. The identification of significant error (RMSE = 0.8m) between approximate expected and actual observed elevations from the remotely sensed data emphasises the limitations of using this data in a deterministic manner within the calibration process. These limitations are addressed by developing a new probabilistic approach to using the observed data.

A comparison of three canopy interception models for a leafless mixed deciduous forest stand in the eastern United States

Canopy interception of incident precipitation is a critical component of the forest water balance during each of the four seasons. Models have been developed to predict precipitation interception from standard meteorological variables because of acknowledged difficulty in extrapolating direct measurements of interception loss from forest to forest. No known study has compared and validated canopy interception models for a leafless deciduous forest stand in the eastern United States. Interception measurements from an experimental plot in a leafless deciduous forest in northeastern Maryland (39°42'N, 75°5'W) for 11 rainstorms in winter and early spring 2004/05 were compared to predictions from three models. The Mulder model maintains a moist canopy between storms. The Gash model requires few input variables and is formulated for a sparse canopy. The WiMo model optimizes the canopy storage capacity for the maximum wind speed during each storm. All models showed marked underestimates and overestimates for individual storms when the measured ratio of interception to gross precipitation was far more or less, respectively, than the specified fraction of canopy cover. The models predicted the percentage of total gross precipitation (PG) intercepted to within the probable standard error (8.1%) of the measured value: the Mulder model overestimated the measured value by 0.1% of PG; the WiMo model underestimated by 0.6% of PG; and the Gash model underestimated by 1.1% of PG. The WiMo model’s advantage over the Gash model indicates that the canopy storage capacity increases logarithmically with the maximum wind speed. This study has demonstrated that dormant-season precipitation interception in a leafless deciduous forest may be satisfactorily predicted by existing canopy interception models.

Simulation of crop yields using ERA-40: Limits to skill and nonstationarity in weather-yield relationships

Reanalysis data provide an excellent test bed for impacts prediction systems. because they represent an upper limit on the skill of climate models. Indian groundnut (Arachis hypogaea L.) yields have been simulated using the General Large-Area Model (GLAM) for annual crops and the European Centre for Medium-Range Weather Forecasts (ECMWF) 40-yr reanalysis (ERA-40). The ability of ERA-40 to represent the Indian summer monsoon has been examined. The ability of GLAM. when driven with daily ERA-40 data, to model both observed yields and observed relationships between subseasonal weather and yield has been assessed. Mean yields "were simulated well across much of India. Correlations between observed and modeled yields, where these are significant. are comparable to correlations between observed yields and ERA-40 rainfall. Uncertainties due to the input planting window, crop duration, and weather data have been examined. A reduction in the root-mean-square error of simulated yields was achieved by applying bias correction techniques to the precipitation. The stability of the relationship between weather and yield over time has been examined. Weather-yield correlations vary on decadal time scales. and this has direct implications for the accuracy of yield simulations. Analysis of the skewness of both detrended yields and precipitation suggest that nonclimatic factors are partly responsible for this nonstationarity. Evidence from other studies, including data on cereal and pulse yields, indicates that this result is not particular to groundnut yield. The detection and modeling of nonstationary weather-yield relationships emerges from this study as an important part of the process of understanding and predicting the impacts of climate variability and change on crop yields.

Phylogenetic mixture models can reduce node-density artifacts

We investigate the performance of phylogenetic mixture models in reducing a well-known and pervasive artifact of phylogenetic inference known as the node-density effect, comparing them to partitioned analyses of the same data. The node-density effect refers to the tendency for the amount of evolutionary change in longer branches of phylogenies to be underestimated compared to that in regions of the tree where there are more nodes and thus branches are typically shorter. Mixture models allow more than one model of sequence evolution to describe the sites in an alignment without prior knowledge of the evolutionary processes that characterize the data or how they correspond to different sites. If multiple evolutionary patterns are common in sequence evolution, mixture models may be capable of reducing node-density effects by characterizing the evolutionary processes more accurately. In gene-sequence alignments simulated to have heterogeneous patterns of evolution, we find that mixture models can reduce node-density effects to negligible levels or remove them altogether, performing as well as partitioned analyses based on the known simulated patterns. The mixture models achieve this without knowledge of the patterns that generated the data and even in some cases without specifying the full or true model of sequence evolution known to underlie the data. The latter result is especially important in real applications, as the true model of evolution is seldom known. We find the same patterns of results for two real data sets with evidence of complex patterns of sequence evolution: mixture models substantially reduced node-density effects and returned better likelihoods compared to partitioning models specifically fitted to these data. We suggest that the presence of more than one pattern of evolution in the data is a common source of error in phylogenetic inference and that mixture models can often detect these patterns even without prior knowledge of their presence in the data. Routine use of mixture models alongside other approaches to phylogenetic inference may often reveal hidden or unexpected patterns of sequence evolution and can improve phylogenetic inference.

Nonlinear system identification using particle swarm optimisation tuned radial basis function models

A novel particle swarm optimisation (PSO) tuned radial basis function (RBF) network model is proposed for identification of non-linear systems. At each stage of orthogonal forward regression (OFR) model construction process, PSO is adopted to tune one RBF unit's centre vector and diagonal covariance matrix by minimising the leave-one-out (LOO) mean square error (MSE). This PSO aided OFR automatically determines how many tunable RBF nodes are sufficient for modelling. Compared with the-state-of-the-art local regularisation assisted orthogonal least squares algorithm based on the LOO MSE criterion for constructing fixed-node RBF network models, the PSO tuned RBF model construction produces more parsimonious RBF models with better generalisation performance and is often more efficient in model construction. The effectiveness of the proposed PSO aided OFR algorithm for constructing tunable node RBF models is demonstrated using three real data sets.

The International Urban Energy Balance Models Comparison Project: First Results from Phase 1

A large number of urban surface energy balance models now exist with different assumptions about the important features of the surface and exchange processes that need to be incorporated. To date, no com- parison of these models has been conducted; in contrast, models for natural surfaces have been compared extensively as part of the Project for Intercomparison of Land-surface Parameterization Schemes. Here, the methods and first results from an extensive international comparison of 33 models are presented. The aim of the comparison overall is to understand the complexity required to model energy and water exchanges in urban areas. The degree of complexity included in the models is outlined and impacts on model performance are discussed. During the comparison there have been significant developments in the models with resulting improvements in performance (root-mean-square error falling by up to two-thirds). Evaluation is based on a dataset containing net all-wave radiation, sensible heat, and latent heat flux observations for an industrial area in Vancouver, British Columbia, Canada. The aim of the comparison is twofold: to identify those modeling ap- proaches that minimize the errors in the simulated fluxes of the urban energy balance and to determine the degree of model complexity required for accurate simulations. There is evidence that some classes of models perform better for individual fluxes but no model performs best or worst for all fluxes. In general, the simpler models perform as well as the more complex models based on all statistical measures. Generally the schemes have best overall capability to model net all-wave radiation and least capability to model latent heat flux.

Linear-wavelet models for system identification

A model structure comprising a wavelet network and a linear term is proposed for nonlinear system identification. It is shown that under certain conditions wavelets are orthogonal to linear functions and, as a result, the two parts of the model can be identified separately. The linear-wavelet model is compared to a standard wavelet network using data from a simulated fermentation process. The results show that the linear-wavelet model yields a smaller modelling error when compared to a wavelet network using the same number of regressors.

Generic unified modelling process for developing semantically rich, dynamic and temporal models

Models play a vital role in supporting a range of activities in numerous domains. We rely on models to support the design, visualisation, analysis and representation of parts of the world around us, and as such significant research effort has been invested into numerous areas of modelling; including support for model semantics, dynamic states and behaviour, temporal data storage and visualisation. Whilst these efforts have increased our capabilities and allowed us to create increasingly powerful software-based models, the process of developing models, supporting tools and /or data structures remains difficult, expensive and error-prone. In this paper we define from literature the key factors in assessing a model’s quality and usefulness: semantic richness, support for dynamic states and object behaviour, temporal data storage and visualisation. We also identify a number of shortcomings in both existing modelling standards and model development processes and propose a unified generic process to guide users through the development of semantically rich, dynamic and temporal models.

Estimation of systematic error in an equatorial ocean model using data assimilation

Assimilation of temperature observations into an ocean model near the equator often results in a dynamically unbalanced state with unrealistic overturning circulations. The way in which these circulations arise from systematic errors in the model or its forcing is discussed. A scheme is proposed, based on the theory of state augmentation, which uses the departures of the model state from the observations to update slowly evolving bias fields. Results are summarized from an experiment applying this bias correction scheme to an ocean general circulation model. They show that the method produces more balanced analyses and a better fit to the temperature observations.

Adjoint methods in data assimilation for estimating model error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure.

Bayesian estimation of the spatial Durbin error model with an application to voter turnout in the 2004 presidential election

The potential for spatial dependence in models of voter turnout, although plausible from a theoretical perspective, has not been adequately addressed in the literature. Using recent advances in Bayesian computation, we formulate and estimate the previously unutilized spatial Durbin error model and apply this model to the question of whether spillovers and unobserved spatial dependence in voter turnout matters from an empirical perspective. Formal Bayesian model comparison techniques are employed to compare the normal linear model, the spatially lagged X model (SLX), the spatial Durbin model, and the spatial Durbin error model. The results overwhelmingly support the spatial Durbin error model as the appropriate empirical model.