Point Scale Factor

Scale Factor

A scale factor accounts for a true horizontal ground distance on a surface of the Earth and the same distance when projected onto a mapping plane or grid. The point scale factor gives a linear distortion at the specific location on the specified map projection. It is fact of life that the projected coordinate systems are distort and cannot be eliminated.

Look at the diagram, In summary, there are four surfaces: the topography (terrain), Geoid, Ellipsoid and the Grid plane. When surveyors measure the angles and distances of the features on the ground terrain. In case of distances it has to reduce the distance to the Grid plane which is based on projected coordinate system. Measurements of lines must always be done on the topography of the earth, and not on the ellipsoid. Therefore the first step in deriving a grid distance must be moving a measured line form the earth to the ellipsoid. To move from the surface of the earth down to the ellipsoid we need to have what’s known as an elevation scale factor (ESF). The second step we need grid scale factor (GSF), The grid scale factor is therefore ratio and can be used as multiplier to convert ellipsoidal length (also known as geodetic distance) to length on the map projection surface, which are also known as grid distance. The multiplication between elevation scale factor with the grid scale factor, the result is known as the combined scale factor (CSF). The combined factor will move the line from the topographic surface of the earth to the grid plane.

When you start the Surveyor Pocket Tools you will see the Point Scale Factor in the main window.

Double click to run and the screen will be displayed as the screenshot below.

The selection of coordinate systems are on the top of the section. It is common that many tools in Surveyor Pocket Tools are involved with the coordinate conversion and transformation. The middle section are coordinate definitions that consists of Point name, Grid Northing, Grid Easting. These coordinates will be transformed to WGS84 to use later for compute geoid height.

The next definition is the selection of “Vertical References” that are available two choices: “Orthometric height (H)” or “Ellipsoidal height (h)“. If your selection is “Orthometric height(H)” the screen will be shown as above. If your selection is “Ellipsoidal height(h)” the screen will be changed to as below. The reason that the calculation of elevation scale factor need the height above ellipsoid.

Notice the formula:

h = H + N; h – Ellipsoidal height, H – Orthometric height, N – Geoid height

ESF = R / (R + h); R – Mean Radius of Earth, h – Ellipsoidal height

If you provide Orthometric height (H) it need the geoid height (N) to calculate the ellipsoidal height (h) with the above formula. The geoid height can be calculated by using the EGM2008 model with the geodetic coordinate on WGS84 as aforementioned. In addition, the formula of elevation scale factor need the mean radius of earth (R), therefore the tool also calculate the mean radius of earth using geodetic coordinate on WGS84 as well.

The another four small buttons are function to:

• The button use to store/save point to database.
• The  button use to restore point from database.
• The  button use to pin location to Google maps.
• The  button use to save KML file and display location on Google Earth.

Working with Coordinate Reference Systems (CRS)

To specify the given coordinate reference system definition you can do easily by the steps that provided by Surveyor Pocket Tools. Please follow the tutorial:

How to select a Coordinate System?

Point Scale Factor

The Point Scale Factor can calculate the Elevation scale factor (ESF), Grid scale factor (GSF) and the Combined scale factor (CSF) that derived from multiplied by ESF and GSF.

Example 1 – Test on Indian 1975 / UTM zone 47N

Given : Projected coordinate system is “Indian 1975 / UTM zone 47N” (Thailand), Point name “0006-029“, Grid Northing = 1597821.4998 m, Grid Easting = 703926.8705 m, Elevation = 11.046 m (Orthometric height). Calculate the scale factor: ESF, GSF and CSF.

Select the coordinate system, enter the coordinate values, select the “Vertical Reference” to “Orthometric height (H)” and enter the value 11.046.

Click  to compute the scale factors. The results will be calculated and displayed as the screenshot below. The Elevation scale factor = 1.0000031162, the Grid scale factor =  1.0001145096 and the Combined scale factor = 1.0001150582

Example 2 – Test on State Plane Coordinate System (SPC)

Given: The survey control point PID = DI7082, Colorado North, SPC, NAD83(NSRS2007), Grid Northing = 379289.251m, Grid Easting = 934837.709m, Ellipsoidal height = 1624.117m

Select the coordinate system and enter the coordinate values. Select the “Vertical Reference” to “Ellipsoidal height(h)“, enter the ellipsoidal height value.

Click  button to calculate.

Check the results against the NGS datasheet.

Conclusion

The Point Scale Factor is handy tool enable to calculate the scale factor in one step.

Limitation of the tool

The tool is not yet supported another geoid models at this time. But the plans will be implemented USA geoid grids from 1999, 2003, 2009 and 2012.