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We present a threads and halos representation for interactive volume rendering of vector-field structure and describe a number of additional components that combine to create effective visualizations of multivalued 3D scientific data. After filtering linear structures, such as flow lines, into a volume representation, we use a multilayer volume rendering approach to simultaneously display this derived volume along with other data values. We demonstrate the utility of threads and halos in clarifying depth relationships within dense renderings and we present results from two scientific applications: visualization of second-order tensor valued magnetic resonance imaging (MRI) data and simulated 3D fluid flow data. In both application areas, the interactivity of the visualizations proved to be important to the domain scientists. Finally, we describe a PC-based implementation of our framework along with domain specific transfer functions, including an exploratory data culling tool, that enable fast data exploration.