16.1.2 Resolution

The resolution within a region is determined by the width of the region and resolution at the bounding coordinates. Refer to Fig 16.1 as an illustration of specifying resolution within regions of longitude, latitude, and depth. Along any coordinate, if resolution at the bounding coordinates is the same, then resolution is constant across the region, otherwise it varies continuously from one boundary to the other according to an analytic function. The function describing the variation is prescribed to be a cosine. Although arbitrary, this function has two important properties: it allows the average resolution within any region to be calculated as an average of the two bounding resolutions; and it insures that the first derivative of the resolution vanishes at the region's boundaries. A vanishing first derivative allows regions to be smoothly joined. The only restriction is that there is an integral number of grid cells within a region.

To formalize these ideas, let a region be bounded along any coordinate direction (latitude, longitude, or depth) by two points $\alpha$ and $\beta$ at which resolutions are $\Delta_\alpha$ and $\Delta_\beta$. The number of discrete cells N contained between $\alpha$ and $\beta$ is given by

$$N=\frac{\vert\beta-\alpha\vert}{(\Delta_\alpha+\Delta_\beta)/2}$$ (16.1)

where N must be an integer and the resolution for any cell $\Delta_m$ is given by

$$\Delta_m = \frac{\Delta_\alpha + \Delta_\beta}{2} - \frac{\Delta_\beta -\Delta_\alpha}{2}\cos(\pi\frac{m-0.5}{N})$$ (16.2)

where $m=1\cdots N.$ As an example, if $\alpha$ and $\beta$ were longitudes, the western edge of the first cell would be a $\alpha$ and the eastern edge of cell N would be at $\beta$.