INTERPOLATION WITH SPLINES IN TENSION - A GREENS-FUNCTION APPROACH

Interpolation and gridding of data are procedures in the physical scie nces and al-e accomplished typically using an averaging or finite diff erence scheme on an equidistant grid. Cubic splines are popular becaus e of their smooth appearances; however, these functions can have undes irable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolati on and gridding in one, two, and three dimensions using Green's functi ons for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function techniq ue is superior to conventional finite-difference methods because (1) b oth data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least- squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular gri d. We also show that the inclusion of tension greatly improves the sta bility of the method relative to gridding without tension. Moreover, t he one-dimensional situation can be extended easily to handle parametr ic curve fitting in the plane and in space. Finally, we demonstrate th e new method on both synthetic and real data and discuss the merits an d drawbacks of the Green's function technique.