Abstract Characterizing the uncertainty and extracting reliable visual information from ensemble data have been persistent challenges in various disciplines, specifically in simulation sciences. Many ensemble analysis and visualization techniques take a probabilistic approach to this problem with the assumption that the ensemble size is large enough to extract reliable statistical or probabilistic summaries. However, many real‐life ensembles are rather limited in size, with only a handful of members, due to various restrictions such as storage, computational power, or sampling limitations. As a result, probabilistic inference is subject to imprecision and can potentially result in untrustworthy information in the presence of a limited sample‐size ensemble. In this case, a more reliable approach is to fuse the information present in an ensemble with a limited number of members with minimal assumptions. In this paper, we propose a technique to construct a representative consensus that is particularly suited for ensembles of a relatively small size. The proposed technique casts the problem as an ordering problem in which at each point in the domain, the ensemble members are ranked based on the local neighborhood. This local approach allows us to provide shape and irregularity sensitivity. The local order statistics will then be fused to construct a global consensus using a Bayesian approach to ensure spatial coherency of the local information. We demonstrate the utility of the proposed technique using a synthetic and two real‐life examples.

This article surveys the history and current state of the art of visualization in meteorology, focusing on visualization techniques and tools used for meteorological data analysis. We examine characteristics of meteorological data and analysis tasks, describe the development of computer graphics methods for visualization in meteorology from the 1960s to today, and visit the state of the art of visualization techniques and tools in operational weather forecasting and atmospheric research. We approach the topic from both the visualization and the meteorological side, showing visualization techniques commonly used in meteorological practice, and surveying recent studies in visualization research aimed at meteorological applications. Our overview covers visualization techniques from the fields of display design, 3D visualization, flow dynamics, feature-based visualization, comparative visualization and data fusion, uncertainty and ensemble visualization, interactive visual analysis, efficient rendering, and scalability and reproducibility. We discuss demands and challenges for visualization research targeting meteorological data analysis, highlighting aspects in demonstration of benefit, interactive visual analysis, seamless visualization, ensemble visualization, 3D visualization, and technical issues.

Abstract The U.S. National Hurricane Center (NHC) issues advisories every six hours during the life of a hurricane. These advisories describe the current state of the storm, and its predicted path, size, and wind speed over the next five days. However, from these data alone, the question “What is the likelihood that the storm will hit Houston with hurricane strength winds between 12:00 and 14:00 on Saturday?” cannot be directly answered. To address this issue, the NHC has recently begun making an ensemble of potential storm paths available as part of each storm advisory. Since each path is parameterized by time, predicted values such as wind speed associated with the path can be inferred for a specific time period by analyzing the statistics of the ensemble. This paper proposes an approach for generating smooth scalar fields from such a predicted storm path ensemble, allowing the user to examine the predicted state of the storm at any chosen time. As a demonstration task, we show how our approach can be used to support a visualization tool, allowing the user to display predicted storm position – including its uncertainty – at any time in the forecast. In our approach, we estimate the likelihood of hurricane risk for a fixed time at any geospatial location by interpolatingsimplicial depthvalues in the path ensemble. Adaptivelysizedradial basis functionsare used to carry out the interpolation. Finally, geometric fitting is used to produce a simple graphical visualization of this likelihood. We also employ a non‐linear filter, in time, to assure frame‐to‐frame coherency in the visualization as the prediction time is advanced. We explain the underlying algorithm and definitions, and give a number of examples of how our algorithm performs for several different storm predictions, and for two different sources of predicted path ensembles.

In simulation science, computational scientists often study the behavior of their simulations by repeated solutions with variations in parameters and/or boundary values or initial conditions. Through such simulation ensembles, one can try to understand or quantify the variability or uncertainty in a solution as a function of the various inputs or model assumptions. In response to a growing interest in simulation ensembles, the visualization community has developed a suite of methods for allowing users to observe and understand the properties of these ensembles in an efficient and effective manner. An important aspect of visualizing simulations is the analysis of derived features, often represented as points, surfaces, or curves. In this paper, we present a novel, nonparametric method for summarizing ensembles of 2D and 3D curves. We propose an extension of a method from descriptive statistics, data depth, to curves. We also demonstrate a set of rendering and visualization strategies for showing rank statistics of an ensemble of curves, which is a generalization of traditional whisker plots or boxplots to multidimensional curves. Results are presented for applications in neuroimaging, hurricane forecasting and fluid dynamics.

Ensembles of numerical simulations are used in a variety of applications, such as meteorology or computational solid mechanics, in order to quantify the uncertainty or possible error in a model or simulation. Deriving robust statistics and visualizing the variability of an ensemble is a challenging task and is usually accomplished through direct visualization of ensemble members or by providing aggregate representations such as an average or pointwise probabilities. In many cases, the interesting quantities in a simulation are not dense fields, but are sets of features that are often represented as thresholds on physical or derived quantities. In this paper, we introduce a generalization of boxplots, called contour boxplots, for visualization and exploration of ensembles of contours or level sets of functions. Conventional boxplots have been widely used as an exploratory or communicative tool for data analysis, and they typically show the median, mean, confidence intervals, and outliers of a population. The proposed contour boxplots are a generalization of functional boxplots, which build on the notion of data depth. Data depth approximates the extent to which a particular sample is centrally located within its density function. This produces a center-outward ordering that gives rise to the statistical quantities that are essential to boxplots. Here we present a generalization of functional data depth to contours and demonstrate methods for displaying the resulting boxplots for two-dimensional simulation data in weather forecasting and computational fluid dynamics.

We present a quasi interpolation framework that attains the optimal approximation-order of Voronoi splines for reconstruction of volumetric data sampled on general lattices. The quasi interpolation framework of Voronoi splines provides an unbiased reconstruction method across various lattices. Therefore this framework allows us to analyze and contrast the sampling-theoretic performance of general lattices, using signal reconstruction, in an unbiased manner. Our quasi interpolation methodology is implemented as an efficient FIR filter that can be applied online or as a preprocessing step. We present visual and numerical experiments that demonstrate the improved accuracy of reconstruction across lattices, using the quasi interpolation framework.

Abstract This paper introduces a quasi‐interpolation method for reconstruction of data sampled on the Body Centered Cubic (BCC) lattice. The reconstructions based on this quasi‐interpolation achieve the optimal approximation order offered by the shifts of the quintic box spline on the BCC lattice. We also present a local FIR filter that is used to filter the data for quasi‐interpolation. We document the improved quality and fidelity of reconstructions after employing the introduced quasi‐interpolation method. Finally the resulting quasi‐interpolation on the BCC sampled data are compared to the corresponding quasi‐interpolation method on the Cartesian sampled data.