Barnes interpolation

Method

Here we describe the method of interpolation used in a multi-pass Barnes interpolation.

First pass

At a given grid point i, j the interpolated function g(x_i, y_i) is approximated based on a weighted sum of the data points. The weighting is defined by

${\displaystyle w_{ij}=\exp \left(-\frac{r_m^2}{\kappa }\right),}$

where ${\displaystyle r_m}$ is the Euclidean distance from a given data point to ${\displaystyle (x_i,y_i)}$. Here, ${\displaystyle \kappa }$ may be thought of as a squared influence radius. Indeed, if ${\displaystyle \mathrm{\Delta }n}$ is taken to be the typical data spacing, it has been suggested to use

${\displaystyle {\kappa }_0=5.052{\left(\frac{2\mathrm{\Delta }n}{\pi }\right)}^2}$

as an estimate for ${\displaystyle \kappa }$.

The initial interpolation for the function from the measured values ${\displaystyle f_k(x,y)}$ then becomes:

${\displaystyle g_0(x_i,y_j)=\frac{{\displaystyle \sum _k{w}_{ij}{f}_k(x,y)}}{{\displaystyle \sum _k{w}_{ij}}}.}$

Second pass

The correction for the next pass then utilises the difference between the observed field and the interpolated values at the measurement points to optimise the result:

${\displaystyle g_1(x_i,y_j)=g_0(x_i,y_j)+\sum (f(x,y)-g_0(x,y))\exp \left(-\frac{r_m^2}{\gamma \kappa }\right).}$

It is worth to note that successive correction steps can be used in order to achieve better agreement between the interpolated function and the measured values at the experimental points.

Parameter selection

Although described as an objective method, there are many parameters which control the interpolated field. The choice of Δn, grid spacing Δx and ${\displaystyle \gamma }$ as well influence the final result. Guidelines for the selection of these parameters have been suggested, however the final values used are free to be chosen within these guidelines.

The data spacing used in the analysis, Δn may be chosen either by calculating the true experimental data inter-point spacing, or by the use of a complete spatial randomness assumption, depending upon the degree of clustering in the observed data. The smoothing parameter ${\displaystyle \gamma }$ is constrained to be between 0.2 and 1.0. For reasons of interpolation integrity, Δx is argued to be constrained between 0.3 and 0.5.