Cubic spline population density functions and satellite city delimitation: The case of Barcelona

The presence of satellite cities within large metropolitan areas cannot be captured by an exponential function. Cubic spline functions seem more appropriate to depict the polycentric pattern of modern urban systems. Using data from the Barcelona Metropolitan Region, two possible population satellite city delimitation procedures using cubic spline density functions are discussed: one, taking an estimated derivative equal to zero; the other, a density gradient equal to zero. It is argued that a delimitation strategy based on derivatives is more appropriate than one based on gradients because the estimated density can be negative in sections with very low densities and few observations, leading to sudden changes in estimated gradients. It is also argued that delimiting satellite cities using a second derivative with a zero value permits the capture of a more restricted area than using a first derivative zero. This methodology can also be used for intermediate ring delimitation.