Large–Scale Computer Models for Environmental Systems
January - June 2003

Modeling of complex environmental systems is a major area of involvement of statisticians and applied mathematicians with disciplinary scientists. Yet it is also a prime example of the differing emphases of the two groups, applied mathematicians focusing on deterministic modeling of the systems and statisticians on utilizing the (often extensive) data for development and analysis of more generic stochastic models. The prime objective of the SAMSI program on environmental systems is to pursue the synthesis between the statistics and applied mathematics approaches, interactions of this nature being one major institutional goals of SAMSI. The program will focus on two areas of major interest and scientific importance, Large-Scale Atmospheric Models and Flows in Porous Media.

Large-Scale Atmospheric Models: Much contemporary work in atmospheric science revolves around large-scale models such as NCAR’s Community Climate System Model (which includes atmosphere, ocean, land and ice), mesoscale models such as the Penn-State/NCAR MM5 model, and the multi-scale air quality model (Models-3) developed by the EPA. Such models are generally deterministic, but their formulation and use involve a variety of sources of uncertainty, such as unknown initial and boundary conditions; "model parameterizations" (the treatment of relevant physical phenomena varying at scales smaller than the grid size of the model, e.g., clouds); and numerical stability issues. Some climatologists have begun work on "stochastic parameterizations." Other basic issues central to these models include the need for an improved scientific understanding of unresolvable, subgrid-scale phenomena, and large-scale (chaotic) non-determinism.

The objectives of this part of the program will be to combine the expertise of applied mathematicians and statisticians, (1) to improve computer models through better integration with data, (2) to introduce stochastic components into models, and (3) to link models while maintaining uncertainty management. Novel approaches to these issues will extend current “data assimilation” methods. The program will also focus on developing a framework for "ensemble forecasting," combining statistical principles associated with Monte Carlo experimentation and mathematical procedures for the optimal selection of ensemble members. For the air quality models, specific objectives include the quantification of model sensitivity due to uncertainty in input values.

Flows In Porous Media: Porous medium dynamics is an active and increasingly interdisciplinary research area with applications from a diverse set of fields including applied sciences (such as environmental studies, geology, hydrology, petroleum engineering, civil engineering and soil physics), and basic sciences (such as physics, chemistry and mathematics). However, there have been few collaborative efforts that have equal footing in mathematics, disciplinary science, and statistics, so that a program in this direction will be distinctive and of high impact. The program will be built around four major themes: model formulation, parameter estimation, numerical methods, and design and optimization.

Model formulation in porous media, flow and transport has traditionally been approached phenomenologically, but this is giving way to first-principle conservation approaches, which appear to resolve some of the troubling aspects of traditional approaches. However, considerable work is still needed on the derivation of non-traditional closure relations, which raise many computational and statistical problems. It is important to integrate the derivation of appropriate mathematical models with statistical considerations of what can be estimated in practice.

Parameter estimation is important because relevant parameters change substantially over short distances and involve multiple scales of correlation. Laboratory experiments may be used for individual parameter values but lead to very sparse data sets when applied over field scales. Some work has shown the potential for Markov chain Monte Carlo methods for estimating permeability, but much remains to be done.

Numerical methods traditionally use deterministic solutions. However, the existence of multiple phases, multiple species, complex reactions, and widely varying model parameters makes efficient construction of such solutions a challenging matter. Important issues include spatial and temporal discretizations, nonlinear solution algorithms, efficient parallel solvers for the resultant linear algebra problems, and solutions of stochastic differential equations.

Design and optimization issues arise in practice, for example, in deciding where to drill new wells in an oilfield – a fantastically expensive operation that is initially subject to very high uncertainty. Application of statistical design principles to high-dimensional and data-poor arenas, such as the drilling of wells under severe economic and environmental constraints, raises a host of challenging issues.

The program will last from January to June 2003, and will include long-term visitors including one SAMSI Fellow, postdocs and graduate students, as well as faculty from the Research Triangle universities. There will be weekly seminars and meetings of principal participants, and two graduate-level courses will be taught in conjunction with the program. In addition, workshops will provide an opportunity for the national statistical and applied mathematical sciences community to participate.

The program will include a six-day workshop on Multi-Scale Modeling, bringing together both atmospheric and porous media scientists, as well as theoretical applied mathematicians, numerical analysts and statisticians. This workshop will be divided among three principal sub-themes:

• Micro-scale phenomena, such as boundary layer problems in meteorology, cloud parameterizations in climate models, canopy models for the biosphere-atmosphere interaction, pollution in groundwater;
  
• Large-scale models: global and regional climate modeling, large-scale porous media phenomena and land-surface fluxes;
  
• Data analysis and statistics: e.g. validation of climate models from long-term data sources, data assimilation in meteorological and air pollution models.