# MATHEMATICAL MODELS

### Geodesy . . .

a branch of applied mathematics that determines the exact positions of points on large portions of the Earth’s surface; the shape and size of the Earth; and the variations of terrestrial gravity and magnetism.

Webster’s Dictionary,

The science of geodesy has been available for hundreds of years, yet many surveyors do not use this powerful mathematical model when processing their measurement data. Before the invention of the personal computer, usage of the geodetic model was limited to large-scale, high-precision surveys.

Today, anyone with a PC—and the right software—can harness the accurate 3D processing mathematics that geodetic theory provides. Whether your survey spans several thousand kilometers or only a few meters (or less), geodesy is the best mathematical model for processing field measurements and obtaining accurate coordinates.

### Columbus uses the geodetic model

With Columbus, you can perform 1D, 2D and 3D adjustments using geodetic, grid or local coordinate systems. Many 3D-type observations can be used in 2D and 1D adjustments. Columbus uses the geodetic model to eliminate mathematical distortions of both small and large projects, supporting projects that span any distance.

1D adjustments can be based on traditional leveling observations or trigonometric observations (zenith angle and slope distance). In the latter case, Columbus automatically reduces your zenith and slope distance to a height difference. This height difference reflects the curvature of the earth at each station; the zenith and slope distance standard deviation are automatically propagated to the new height difference observation.

2D adjustments are performed in 3D space and can be computed based on a mean project height or on the known height for each station. The latter case comes in handy when you are in highly-variable terrain and can approximate the height of each station within several meters. You might use the 2D approach when you don’t feel you have high quality vertical observations (zenith angles, GPS vectors, height differences, etc.) in your data set. A zenith angle that is off by one minute won’t significantly degrade the 2D results.

3D adjustments are, of course, performed in 3D space. All observation types can be used. This is often the most accurate way to process your data when 2D or 3D results are required. The underlying model is based on latitude, longitude and ellipsoidal height.

When adjusting based on orthometric height (elevation), you can apply an approximate project-wide geoid height. This geoid height is added to every orthometric height during adjustment to simulate an approximate ellipsoidal height.