"Investigation of Population Density Gradients Using Trend Surface Analysis": Reply

Abstract

In studying urban population density gradients the assumption is usually made that population density decreases at a constant rate as one moves in any direction from the central business district (CBD); that is, the relationship between distance and density is given by a negative exponential function. The use of the negative exponential function implies that density isopleths are circles with the center at the CBD. Such functions, therefore, do not allow for differences in the rate of change in density as one moves in different directions from the CBD or for the possible existence of secondary density centers. The purpose of this paper is to explore the use of an alternative technique for studying density gradients that allows for gradients to be direction specific and for the existence of secondary centers. In density gradient studies, density is measured for sub-areas, such as census tracts, and distance is measured from the center of the sub-area to the CBD. Thus, in estimating the density gradient the investigator is interpolating the data spatially. Since the estimation of density gradients can be considered a form of spatial interpolation, or isopleth construction, we utilize a technique that has been developed to construct isoplethstrend surface analysis. Trend surface analysis (TSA) is a statistical method that is used for smoothing and analyzing spatial data. Geologists are probably the most extensive users of this technique while geographers have more recently begun using it. Since it appears, however, that few economists are aware of this t chnique, we present a pedagogical discussion of TSA. We then utilize this technique to study density gradients within the Atlanta, Georgia, SMSA using 1970 Census data with the objective of determining whether gradients are direction specific and whether secondary centers exist. The findings are then compared with those obtained from use of exponential functions and with other previous studies of density gradients. In the first section we outline the method used in TSA to accomplish spatial data analysis. The second section is devoted to a brief review of the more