Here, P, is the population density at a distanced from the cluster centre. In is the total population of the duster, and tin, is the outer adios. Clearly. in all the foregoing models, the surfaces are assumed to be symmetrical about a cen• tral point. However, Casetti and Semple (1969) have shown how even simple models of this kind may be extended to portray actual density surbees. Using logarithmic or reciprocal transformations of the distance terms in the conical model, a cone is fitted by regression analysis to an optimum origin (de) in the study city. This origin is located by trial-and-error procedures to maximize the correlations between measured (P,) and estimated (P,) population densities. Values of P, about the optimal centre are regarded as the regional trend, while residuals from this trend (P, – P,) yield a new spatial series (d. the 'trend' and 'residual' maps in Figure 6.1). By 16Cating a new spatial origin within the residual surface, a second trend and residual sada= may be computed, and so on. See the description of Figure 12.4 in Section 12.2.3 for a discussion of the estimation problems involved. Paralleling the search for appropriate isotropic, single or multiple-centred models are the anisot topic models in which population density is modelled in