REPRESENTING 3D FIELDS MATHEMATICALLY

One particularly interesting and challenging type of data is a three-dimensional field. I have most often dealt with this type of data when modeling contaminant transport in air, water, and porous or fractured rock (ground water). The data may be concentration of one or more contaminants or thermophysical properties (density, porosity, hydraulic conductivity, water table elevation, tortuosity, diffusion and dispersion coefficients, etc.). The data are almost never measured over a volume; rather, measurements are acquired at point locations or in a string, such as from a well, either by measurements over depth or coring.

3D interpolation is often performed using the inverse distance method or kriging. This often yields acceptable results. When these methods don't work, other mathematical approaches are available, including: relaxation (solving Laplace's Equation on a grid) and orthogonal projection (much like a Fourier Transform on a sound signal). The fields below illustrate these latter methods and the options for displaying the results.

For more details and illustrations see these books Orthogonal Functions and Plumes