Barnes interpolation

For a given grid point i, j the interpolated function g(x_i, y_i) is first approximated by the inverse weighting of the data points. To do this as weighting values is assigned to each Gaussian for each grid point, such that

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

where ${\displaystyle \kappa }$ is a falloff parameter that controls the width of the Gaussian function. This parameter is controlled by the characteristic data spacing, for a fixed Gaussian cutoff radius w_ij = e⁻¹ giving Δn such that:

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

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

${\displaystyle g_{0}(x_{i},y_{j})={\frac {\displaystyle \sum _{k}w_{ij}f_{k}(x,y)}{\displaystyle \sum _{k}w_{ij}}}.}$

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:[1]

${\displaystyle g_{1}(x_{i},y_{j})=g_{0}(x_{i},y_{j})+\sum (f(x,y)-g_{0}(x,y))\exp \left(-{\frac {r_{m}^{2}}{\gamma \kappa }}\right).\,}$

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.