Geostatistical Modeling

The USGS process for creating predictive spatial model accepts as input a collection of ecological attributes, such as topographic data, species data, soil characteristics, ETM+ derived vegetation indices, etc. These attributes are examined for statistically viable relationships between predictor variables and response variables. Trend surface analyses are performed, and residuals from the trend surface analyses are further analyzed for spatial structure using kriging and cokriging. The results are brought together to produce a refined spatial prediction that is accompanied by an estimate of uncertainty. The process's ability to produce both predictive maps and a maps of uncertainty significantly increases its value for decision support, since useful predictions are ultimately dependant upon a quantifiable understanding of error.

The USGS modeling framework thus combines multiple data layers, geographic information systems, image processing, and geostatistical techniques to identify patterns of biodiversity in geographic space. This is done by modeling the fieldmeasured parameters (such as native or nonnative species richness or diversity) with a set of independent variables (such as elevation, slope, aspect, soil properties, and spectral values from Landsat imagery). For example, one model for species richness could be written generally as:

where f is a linear function of variables derived from the input data sources. Within the ISFS, the function is fit using stepwise regression to identify the best combination of independent variables. Each independent variable in the model has a corresponding image or surface. The model fit with stepwise regression is then applied to the relevant input layers to produce a preliminary predicted surface. This linear model is further illustrated in the next figure, which shows a representative set of input variables.

Standard linear regression techniques require the assumption of independent error terms. However, due to the spatial nature of the phenomenon being modeled, this assumption is seldom met. This issue is addressed by first analyzing the residuals resulting from the model for spatial autocorrelation by calculating both empirical variograms and the Moran's I test statistic. If these indicate spatial structure in the residuals, kriging is applied to the residuals to create a "residual surface" that is added to the preliminary predicted surface to produce the final predicted surface.

The models are crossvalidated to assess variability in the prediction errors. Crossvalidation includes deleting one or several observations from the data set and predicting the deleted observations using the remaining observations in the data set. This process is repeated for all observations in the data set. Finally, summary statistics of the estimated values are computed for the region.

USGS predictive models provide information on largescale spatial variability that extends in crucial ways our understanding of the smallscale structure provided by field data. These techniques have been used successfully for determining potential distributions of exotic plants at the Cerro Grande Wildfire Site (shown below), in Rocky Mountain National Park, and in Grand StaircaseEscalante National Monument. They are a core component in the decision support activities of the USGS and partner federal and state agencies. In order to scale these methods to larger applications, part of the ISFS effort is to improve the computational performance of the geostatistical algorithms and move the kriging calculation to cluster computers under a separate cooperative agreement with the NASA Computational Technologies Project. We have developed an initial parallel kriging code that gives greater than 93% scaling efficiency with 32 processors. This reduction in processing time will allow the USGS to extend the techniques to larger areas and to ingest more data layers. The next major opportunity to enable an operational Invasive Species Forecasting System is to increase the use of remotely sensed data, which is the major focus of the current proposal.

References

Isaaks, E.H. and Srivastava, R.M. 1989. An Introduction to Applied Geostatistics. Oxford University Press, New York.

Stohlgren, T. J., G. W. Chong, M. A. Kalkhan, and L. D. Schell. 1997a. Rapid assessment of plant diversity patterns: A methodology for landscapes. Ecological Monitoring and Assessment, Vol. 48, pp. 25-43.

Stohlgren, T. J., G. W. Chong, M. A. Kalkhan, & L. D. Schell. 1997b. Multiscale sampling of Plant Diversity: Effects of the minimum mapping unit. Ecological Applications, Vol. 7, pp. 1064-1074.