SPATIAL UNDERSTANDING OF MATRIX INVERSION FOR INVERSE FORCE ESTIMATION

A basic approach to study a NVH problem is to break down the system in three basic elements – source, path and receiver. While the receiver (response) and the transfer path can be measured, it is difficult to measure the source (forces) acting on the system. It becomes necessary to predict these forces to know how they influence the responses. This requires inverting the transfer path. Singular Value Decomposition (SVD) method is used to decompose the transfer path matrix into its principle components which is required for the inversion. The usual approach to force prediction requires rejecting the small singular values obtained during SVD by setting a threshold, as these small values dominate the inverse matrix. This assumption of the threshold may be subjected to rejecting important singular values severely affecting force prediction. The new approach discussed in this report looks at the column space of the transfer path matrix which is the basis for the predicted response. The response participation is an indication of how the small singular values influence the force participation. The ability to accurately reconstruct the response vector is important to establish a confidence in force vector prediction. The goal of this report is to suggest a solution that is mathematically feasible, physically meaningful, and numerically more efficient through examples. This understanding adds new insight to the effects of current code and how to apply algorithms and understanding to new codes.

Spatial and temporal patterns in ungulate-ecosystem interactions

Ungulates are important components of a variety of ecosystems worldwide. This dissertation integrates aspects of ungulate and forest ecology to increase our understanding of how they work together in ways that are of interest to natural resource managers, educators, and those who are simply curious about nature. Although animal ecology and ecosystem ecology are often studied separately, one of the general goals of this dissertation is to examine how they interact across spatial and temporal scales. Forest ecosystems are heterogeneous across a range of scales. Spatial and temporal habitat use patterns of forest ungulates tend to be congregated in patches where food and/or cover are readily available. Ungulates interact with ecosystem processes by selectively foraging on plants and excreting waste products in concentrated patches. Positive feedbacks may develop where these activities increase the value of habitat through soil fertilization or the alteration of plant chemistry and architecture. Heterogeneity in ecosystem processes and plant community structure, observed at both stand and local scales, may be the integrated outcome of feedbacks between ungulate behavior and abiotic resource gradients. The first chapter of this dissertation briefly discusses pertinent background information on ungulate ecology, with a focus on white-tailed deer (Odocoileus virginianus) in the Upper Great Lakes region and moose (Alces acles) in Isle Royale National Park, Michigan, USA. The second chapter demonstrates why ecological context is important for studying ungulate ecology in forest ecosystems. Excluding deer from eastern hemlock (Tsuga canadensis) stands, which deer use primarily as winter cover, resulted in less spatial complexity in soil reactive nitrogen and greater complexity in diffuse light compared to unfenced stands. The spatial patterning of herbaceous-layer cover was more similar to nitrogen where deer were present, and was a combination of nitrogen and light within deer exclosures. This relationship depends on the seasonal timing of deer habitat use because deer fertilize the soil during winter, but leave during the growing season. The third chapter draws upon an eight-year, 39-stand data set of deer fecal pellet counts in hemlock stands to estimate the amount of nitrogen that deer are depositing in hemlock stands each winter. In stands of high winter deer use, deer-excreted nitrogen inputs consistently exceeded those of atmospheric deposition at the stand scale. At the neighborhood scale, deer-excreted nitrogen was often in excess of atmospheric deposition due to the patchy distribution of deer habitat use. Spatial patterns in habitat use were consistent over the eight-year study at both stand and neighborhood scales. The fourth chapter explores how foraging selectivity by moose interacts with an abiotic resource gradient to influence forest structure and composition. Soil depth on Isle Royale varies from east to west according to glacial history. Fir saplings growing in deeper soils on the west side are generally more palatable forage for moose (lower foliar C:N) than those growing in shallower soils on the east side. Therefore, saplings growing in better conditions are less likely to reach the canopy due to moose browsing, and fir is a smaller overstory component on the west side. Lastly, chapter five focuses on issues surrounding eastern hemlock regeneration failure, which is a habitat type that is important to many wildlife species. Increasing hemlock on the landscape is complicated by several factors including disturbance regime and climate change, in addition to the influence of deer.