ABSTRACT
A 4-point bi-linear interpolation method can be used to obtain contour levels for each pixel within a box, which can be chosen to coincide with a modelling grid cell (sec Figure 4(a)). This algorithm is easy to implement into the computer code and any dry or land cell can easily be taken into account, provided that the concentration values at the four corners are known. If these four corners have the same contour band levels there is no need to interpolate for pixels inside this box. However, it is worth noting that to obtain concentration values at these four corners may result in some approximations, since steep gradients may be smoothed out by the averaging process (assuming all cells are wet):-
(8)
Alternatively the interpolation box can be chosen to be at the centre of four adjacent grid cells (sec Figure 4(b)) in order to avoid smoothing. The bi-linear interpolation formula can be used for this purpose, giving:-
(9)
where 0 $ e $ tl.X and 0 $ '1 $ fl.}' are the local co-ordinates in the X and Y directions respectively, as shown in Figure 4(b ), and
Figure 5: Comparison of different contouring algorithms
A 9-point bi-quadratic interpolation algorithm can also he applied to obtain contour level for each pixel (see Figure 4(c)) using the following formula:-
where -~aX $ { $ ~aX and -~~}' $ 11 $ !AY are the local coordinates in the X and Y directions respectively as shown in Figure 4(c), and
in which k = -1,0, 1. Equations (9 )-( 12) are 1·elatively straight forward and efficient to im-
plement into computer codes since double loops for e and 11 are required to complete the interpolation process and the calculations of Q~c or L~c can be placed outside the inner loop for 1J so that substantial computation time can be saved.
