Height and Transformations

Defining the Vertical Position Vertical Datums Defining the Vertical Position h - - Geodetic Height (Height above Ellipsoid) H Orthometric Height (Height above Mean Sea Level) - N Geoid Separation { { H h { N This diagram shows how the approximate value for small h: the geodetic height is calculated from the separation, N and the geoidal height, H. The reason the equation is only approximate is that the various measurements are all measured along lines normal to different surfaces (geodetic height is measured normal to the ellipsoid, while Orthometric height is measured normal to the geoid). Since GPS receivers use geodetic height internally, NIMA recommends that weapons developers work with h whenever possible - for consistency's sake. That way, we never need to bother ourselves with elevations subject to gravity measurements. Unfortunately, anything that uses inertial measurements will be pulled around by all of those gravity variations, so Inertial Navigation Systems (INS) will always have to contain a gravity model. Further understanding the Geoid/Ellipsoid relationship will define a better vertical datum. Geoid Topo Surface Ellipsoid H is measured traditionally h is approximately = N+H N is modeled using Earth Geoid Model 96 or 99 32

ON-LINE NGS GEOID RESOURCES Official NGS Geoid Page: http://www.ngs.noaa.gov/GEOID Links to all geoid models Links to all geoid papers from NGS Lots of useful geoid information Slideshows like this one (coming soon)

Datum Transformation and Coordinate Conversion We’ve already talked at about the problems with local datums and datum mismatch. We will now talk about how we transform coordinates between different datums and coordinate systems. To change points between different datums we use a process called transformation. To change a point between different grids we use a process called conversion. Both processes are fraught with computation so DoD employs a software package called Geotrans2 to perform both transformations and conversions.

Cartesian Coordinate System Z Ellipsoid sized a & f WGS -84 is an Earth Centered Earth Fixed; Origin to with +/- 10 cm   The numbers that make transformations work right are called parameters and are specific to the datums transformed between. You will find that transformations are best described in reference to a Cartesian coordinate system. As you remember, the basis for WGS-84 is a geocentric right hand Cartesian coordinate system. If you think about it WGS-84 makes a good intermediate when transforming between different local datums. Instead of computing the transformation between multiple datums the relationship between each local datum and the WGS-84 datum can be used. WGS-84 can become a “lingua franca” between different datums. Y  X

Datums and Defining Parameters To translate one datum to another we must know the relationship between the chosen ellipsoids in terms of position and orientation. The relationship is is defined by 7 constants. A. 3 - Distance of the ellipsoid center from the center of the earth (X, Y, Z) B. 3 - Rotations around the X, Y, and Z of the Cartesian Coordinate System Axes (, , ) C. 1 - Scale change (S) of the survey control network 2 - The size and shape of the ellipsoid (semi major axis a and flattening f approximately 1/298 As already discussed the Geodetic Datum fixes the ellipsoid to the mean earth, basically we need to know the relationship between a given ellipsoid and the earth in terms of position and orientation. It is defined by 8 constants. 1. 2 - The size and shape of the ellipsoid (semi major axis and flattening) 2. 3 - The distances of the ellipsoid center from the center of the earth 3. 2 - The directions of the rotation axis with the mean rotation of the earth 4. 1 - The direction of 0 longitude with the earth’s international 0 longitude This is simply the definition of a datum restated from earlier in the lecture. 6

Differences Between Horizontal Datums The two ellipsoid centers called  X,  Y,  Z The rotation about the X,Y, and Z axes in seconds of arc The difference in size between the two ellipsoids Scale Change of the Survey Control Network S Z    System 2 NAD-27 The parameters that define the differences between 2 datums are as follows: a. The differences in meters between the two ellipsoid centers called delta x, delta y, delta z b. The rotation about the Z axis is seconds of arc between the two ellipsoids 0 longitude c. The difference in size between the two ellipsoids d. The rotations in seconds of arc about the X and Y axis, the attitude of the spin axis System 1 WGS-84 Y  Z  Y X X 8

Coordinate Conversion & Datum Transformation References: NIMA TR8350.2, TEC-SR-7, DMA TM 8358.1 Convert from Grid to Geographic Coordinates Convert from Geographic to Cartesian Coordinates Apply Datum Transformation 3 Parameter (X, Y, Z, and a & f of Ellipsoid) 7 Parameter (X , Y , Z , S, a & F of Ellipsoid) * For most uses 3 parameter shifts are acceptable Compute New Geographic Coordinates Compute New UTM, MGRS etc. Coordinates Error in transformation propagates to final coordinates These seven parameters can be derived from direct measurement but for most transformations accepted parameters have already been derived and are available from the references listed above. The general process is: Convert from grid to geographic Convert from geographic to Cartesian Apply derived parameters Compute new geographics Convert to the required grid system It is very important to note that every transformation puts error into the system. The more transformations the greater the error. It is therefore important not to chain transformations but go from the source datum to the final datum with the least amount of transformations.

Coordinate Conversion & Datum Transformation This chart shows the process for deriving and applying a datum transformation and coordinate conversion from a MGRS point in the WGS-84 datum to an Irish National Grid in the Ireland 1965 Datum. Of particular note is the use of Cartesian coordinates to apply the transformations derived from Geographic coordinates. It is beyond the scope of this lecture to delve into how these parameters are derived. I’ll you should take away from this is that the relationship between WGS-84 and many datums have already been derived at an accuracy appropriate to mapping level transformations.

Geographic Translator (GEOTRANS) Converts coordinates among a wide variety of coordinate systems, map projections, and datums. 11 different coordinate systems and map projections Over 200 different datums Choose between Ellipsoid and MSL height Program can receive coordinates from a text file, convert them, and output results in another file. Instead of “hand-jamming” the process in the last slide the preferred solution is to use the geographic translator program GEOTRANS2 to perform your transformations and conversions. The required parameters are already built into GEOTRANS2 making the process much simpler. 30

The GEOTRANS2 interface screen can be thought of in two parts: the upper portion is the input part and the lower is the output. When you perform one time transformations and conversions you change the input and output systems using the input and out menu commands. If however you are batch processing or converting more than one point at a time the input file header will make the input menu unavailable. This is because the parameters are set in the input file header. The coordinate files created by GEOTRANS are identical in format to the coordinate files that it reads. Thus, any coordinate file created by GEOTRANS may be subsequently used as an input coordinate file. The header of the output file is generated based on the output datum and coordinate system or map projection selections made in the GEOTRANS File Processing window. The number of lines in the output file header may not be identical to the number of lines in the input file header, especially if the number of map projection parameters is different. Any comment lines in the input file header are not copied to the output file header.

Here we see GEOTRANS2 set up to transform coordinates between WGS-84 and the Tokyo datum in Korea. A few important things to note about this slide: 1. The parameters for the Transformation of the Tokyo Datum to WGS-84 is different in South Korea than in Japan. This is why the datum is shown as Tokyo-B. Although the datum is the same the parameters are location dependant. 2. Approximate error estimates are shown at the bottom of the screen. CE is circular error, LE is linear error, and SE is spherical error. 3. The associated ellipsoid is shown for each datum. 4. Height can be entered and output as either ellipsoidal or geoidal heights. GEOTRANS 2 incorporates the Earth Gravitational model version