# Details: Maps of Geographic Patterns in Population Trends

The design of the BBS allows us to conduct an analysis that directly uses geographic information to illustrate geographic patterns in population change. Route locations can be incorporated into a geographic information system (GIS), and population change can be displayed directly for each route. Geographic patterns can be summarized by 2-dimensional smoothing procedures such as Kriging or inverse distancing (Cressie 1991) to display regions of increasing and declining populations. Temporal patterns can be displayed using population change estimates on individual routes for periods of interest.

We used the geographic location of the starting points of survey routes summarized in latitude and longitude as the index to route locations. The route locations were projected as a series of point locations (a point coverage) in an ARC/Info GIS (Environmental Systems Research Institute 1993). Using the GIS, data for individual species can be associated with the point locations for analysis and display.

Because most BBS routes have some missing data and >1 observer during the survey period, simple use of yearly counts or average counts would lead to spurious geographic patterns. Therefore, we summarized the data before entering it into the GIS. In particular, we estimated the population trend (b) for individual routes using the estimating equations estimator of Link and Sauer (1994). On our maps, we transform this proportional change to a %/year for ease in interpretation.

To accommodate the obvious differences in quality of information among routes, we weighted the trend estimates by an estimate of the variance of trends from individual routes. This variance is estimated using a model-based variance from the estimating equations, corrected for a common overdispersion (W. A. Link, Personal Communication). We note that this variance estimate is proportional to the precision weights used in the estimation of the regional mean trends (Geissler and Sauer 1990).

### Smoothing Procedure

Although simply displaying the trend estimates for individual routes provides insight into regional patterns, some kind of summary of the data is needed to assist in evaluating regional consistency in trends. We contoured the route trend estimates using a procedure called inverse distancing. See the discussion related to the abundance maps for a description of how inverse distancing was applied to the route data, and a rationale for the use of inverse distancing relative to other procedures such as Kriging. As noted in the discussion regarding relative abundance maps, there are many technical details for the smoothing, and many features that can create bias and imprecision in the smoothed maps Because of the complexity of the modeling, different investigators tend to choose slightly different methods, all of which lead to slightly different smooths (e.g., Englund 1990, Weber and Englund 1992).

Because no information exists for areas outside the range of the species, we truncated these maps at the edge of the species' range, as estimated from BBS data. Also, the "range," as indicated by the trend maps is somewhat smaller than the range from the relative abundance maps, because many of the marginal routes for species do not provide sufficient data for estimation of population trends.

### Literature Cited

Cressie, N. 1992. Statistics for spatial data. Wiley, New York. 900pp. Englund, E. J. 1990. A variance of geostatisticians. Math. Geol. 22:417-456. Environmental Systems Research Institute. 1991. Surface Modeling with TIN. Environmental Systems Research Institute, Inc., Redlands, CA. Environmental Systems Research Institute. 1993. Understanding GIS-The Arc/Info Method. Environmental Systems Research Institute, Inc., Redlands, CA. Geissler, P.H., and J.R. Sauer. 1990. Topics in route- regression analysis. Pp. 54-57 in J.R. Sauer and S. Droege (eds.) Survey designs and statistical methods for the estimation of avian population trends. U.S. Fish Wildl. Serv., Biol. Rep. 90(1). Isaaks, E. H., and R. M. Srivastava. 1989. An introduction to applied geostatistics. Oxford University Press, New York. 561pp. Link, W. A., and J. R. Sauer. 1994. Estimating equations estimates of trend. Bird Populations 2:23-32. Weber, D., and E. Englund. 1992. Evaluation and comparison of spatial interpolators. Math. Geol. 24:381-391.