Proposed research

1. Two tasks will be addressed here. The first is the extension of the current software of Meiring to permit the fitting of our spatial deformation models utilizing using thin-plate splines mapping the the geographic plane into a D-space of dimension three, rather than just . This will provide the greater flexibility that seems necessary for modeling the nonstationary correlation structure over the entire San Joaquin Valley. Second, we will further develop an approach to temporally asymmetric space-time correlation structure incorporating periodic (24-hour) structure. We are currently using a ``multivariate'' geostatistical approach based on the linear model of coregionalization treating hours of the day as separate variables. An extension to incorporate asymmetry in both spatial and spatio-temporal cross-covariances will build on the cross-variogram approach introduced by Grzebyk and Wackernagel (1994).

2. We will study the statistical properties of lifting as an estimator of trend in space-time data. In particular we will develop estimates of the variability of the trend estimate, and study the relationship between lifting and other schemes used to detrend irregularly spaced space-time data. More conventional wavelet schemes may be appropriate for trend estimation on gridded model output data.

3. Preliminary demonstrations of the calculation of grid cell estimates were demonstrated in Meiring's thesis (1995) for a subset of the monitoring data of the SARMAP field study. We will utilize the space-time correlation models fitted under task 1 to compute grid cell estimates for the entire SAQM domain for purposes of assessment of the model predictions for the 5-day episode simulated. Grid cell estimates will be based on currently computed geostatistical trend estimates, but the effect of other trend estimates, including wavelets, may also be considered.

4. We will develop methods for fitting our spatial deformation model (task 1) to gridded model output data. This will involve a hierarchical analysis to assess the spatial resolution necessary for accurate representation of this correlation structure.

5. In cooperation with Dr. LeDuc and other EPA scientists, these methods will be applied to air quality and/or deposition data in order to assist in the assessment of model predictions provided by the ``Models-3'' system. We expect to obtain both monitoring data and model output for an entire season with a resolution of 36 km grid cells over most of the Eastern U.S. Analyses for smaller domains or urban areas covered with model predictions on 12 or 4 km grid cells may also be carried out. A particular focus of this application will be the comparison of spatio-temporal correlation structures derived from the field monitoring data (task 1 above) and the gridded model output (task 4). Other particular diagnostic methods suggested in the NRCSE proposal (involving cross-covariances between species, and decompositions of error fields) may also be demonstrated.