Basic operators and adjoints

Nearest-neighbor coordinates

In describing physical processes, we often either specify models as values given on a uniform mesh or we record data on a uniform mesh. Typically, we have a function $f$ of time $t$ or depth $z$, and we represent it by f(iz) corresponding to $f(z_i)$ for $i=1,2,3,\ldots, n_z$ where $z_i = z_0+ (i-1)\Delta z$. We sometimes need to handle depth as an integer counting variable $i$, and we sometimes need to handle it as a floating-point variable $z$. Conversion from the counting variable to the floating-point variable is exact and is often seen in a computer idiom, such as either of

            for (iz=0; iz < nz; nz++) {   z = z0 + iz * dz; 
            for (i3=0, i3 < n3; i3++) {  x3 = o3 + i3 * d3;

The reverse conversion from the floating-point variable to the counting variable is inexact. The easiest thing is to place it at the nearest neighbor. Solve for iz; add one half; and round down to the nearest integer. The familiar computer idioms are:

        iz = 0.5 + ( z - z0) / dz;
        i3 = 0.5 + (x3 - o3) / d3;

A small warning is in order: People generally use positive counting variables. If you also include negative ones, then to get the nearest integer, you should do your rounding with the C function round().

Basic operators and adjoints