Many federal funding requests for more advanced computer resources assume implicitly that greater computing power creates opportunities for advancement in science and engineering. This has often been a good assumption. Given stringent pressures on the federal budget, the White House Office of Management and Budget (OMB) and Office of Science and Technology Policy (OSTP) are seeking an improved approach to the formulation and review of requests from the agencies for new computing funds. The study considered, as examples, four fields of science and engineering to determine which of their major challenges are critically dependent on high-end capability computing (HECC): atmospheric science, astrophysics, chemical separations, and evolutionary biology.While this study does identify the potential impact of HECC in these four fields, and thus implicitly identifies some potential funding opportunities, that is not the goal, and this study is no substitute for competitive review of specific proposals. Rather, the study is meant to illustrate the sort of examination that any field or federal agency could undertake in order to analyze the HECC infrastructure it needs to support progress toward its research goals, within the context of other means of attacking those goals.