The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points, or base points for the multiquadratic basis functions, although the selection of the multiquadratic parameter also plays a nontrivial role in the process. The authors first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadratic parameter is explored. The occurrence of near multiple knots is an interesting phenomenon and the authors further extend the algorithm to include variable parameter values at the knots. Examples and results are given throughout.