Appendix A. TRIMBLE TSC2, CU, VX/S

Appendix A

Calculations performed by the Trimble Survey Controller software

Introduction

This appendix outlines some of the calculations performed by the Trimble Survey Controller software.

•

Transformations applied to GPS positions

•

Ellipsoid calculations

•

Conventional instrument calculations

•

Area calculations

Transformations applied to GPS positions

For RTK surveying, coordinate transformations are required so that one set of coordinates (GPS positions) can be represented in terms of another (grid positions), and vice versa.

Tip - To see an example of converting measurements to grid coordinates using the Trimble Survey Controller

software, click here and then select the Calibration section.

This section provides an overview of the management and application of coordinate transformations using the

Trimble Survey Controller software. It describes how to apply a datum transformation and map projection, and horizontal and vertical adjustments.

With the Trimble Survey Controller software, the site calibration process defines transformation parameters derived from a set of points. This set of points is coordinated in terms of two systems:

•

WGS-84 Latitude, Longitude, Height (LLH) geodetic coordinates

•

A local system with project-specific Northing, Easting, Elevation (NEE) grid coordinates

The following diagram shows the order of calculations performed when a calibration is calculated.

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The formulae used in the above flow diagram are detailed below.

Transforming WGS-84 ECEF to WGS-84 LLH

When GPS signals are processed by a receiver, they yield Earth-Centered-Earth-Fixed (X, Y, Z) coordinates, which must then be transformed to more meaningful geodetic coordinates (

φ

,

λ

, H).

Here,

φ

represents the geodetic latitude,

λ

is the longitude, and H is the perpendicular height above the

WGS-84 ellipsoid.

First we define: e² = 2f - f²

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where f is the flattening value for the source ellipse and r is the semi-major axis.

The values of the ECEF coordinates are:

•

X = (N + H).cos (

φ

).cos (

λ

)

•

Y = (N + H).cos (

φ

).sin (

λ

)

•

Z = [N (1 - e²) + H].sin (

φ

)

The inverse problem (that of transforming ECEF coordinates to

φ

,

λ

, and H) is solved by using an iterative procedure. The values of e² and N now use the destination ellipsoid flattening and semi-major axis values: then iterate if 45°S <

φ

< 45°N or if

φ

> 45°N or

φ

< 45°S

Datum Transformation

A datum transformation provides the necessary parameters for converting from one geodetic coordinate system to another.

The Trimble Survey Controller software can apply a predefined three- or seven-parameter datum transformation. It can also calculate a three-parameter datum transformation given points coordinated in

WGS-84 and local L'L'H'.

X = T + kRX' where X' is a matrix of 3-D Cartesian ECEF coordinates or Cartesian local coordinates, T is a matrix of translation parameters, k is a scalar, and R is a rotation matrix. In most cases, X' is measured and T, k, and R are user-specified.

To calculate a three-parameter datum transformation, pairs of WGS-84 LLH and local L'L'H' coordinates are needed.

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In the trivial one point case, the three translation parameters are just the vector components of the ECEF vector that connect the ECEF pair derived from the WGS-84 LLH and the local L'L'H'.

In the non-trivial case, the translation parameters are the vector components of the average vector. This is represented as:

AX + W = 0 where the solution and where Xn is the value of the ECEF coordinate derived from the local L'L'H' of the nth 3-D point in the list, and X'n is the X value of the ECEF coordinate derived from the WGS-84 LLH of the nth 3-D point and is called the Molodensky matrix.

For more information, refer to GPS Satellite Surveying by A. Leick (John Wiley & Sons, 1995).

Map Projection

A map projection defines the relationship between the local ellipsoidal surface (L'L'H') and a plane.

Generally, map projection parameters are based on a local conformal mapping model.

For more information about map projections, please refer to the Map Projections--A Working Manual by J.P.

Snyder (U.S. Geological Survey Professional Paper 1295, U.S. Government Printing Office, Washington,

1987).

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For nearly all projection types used in the Trimble Survey Controller software, the height component of a projected coordinate is just the height above or below the datum at that point. For a plane projection, however, the definition includes an ellipsoidal height and a projected surface elevation at the origin point.

Horizontal Adjustment

It may be necessary to minimize the discrepancy between local fixed control coordinates (NEcontrol) and projected grid coordinates (N'E'). The horizontal adjustment solves parameter translations in north and east (

∆

N,

∆

E), a rotation

φ

, and a scale factor k using two different sets of plane coordinates--one converted from measurements in the field, and the other from a control list. Where the Trimble Survey Controller software generates its own three-parameter datum transformation, it is necessary to provide for scaling and rotation.

This is done by a horizontal adjustment.

The following diagram shows the transformation between two coordinate systems.

Coordinate systems for a horizontal adjustment

The Trimble Survey Controller software minimizes the discrepancy between local NE control and NE values derived using GPS observations and a datum transformation and map projection. It does this by performing a plane horizontal least-squares adjustment with no weighting.

In the trivial one point case, the translation parameters are just the Northing and Easting components of the vector between the two coordinated values. The scale factor is one, and the rotation value is zero.

For two or more points, calculation of horizontal adjustment uses a simple four-parameter transformation.

This solves for two translations (

∆

N,

∆

E), a rotation (

φ

), and a scale factor (k) between coordinate pairs.

The geometry between the two coordinate systems results in two transformation equations:

•

N' = aN + bE +

∆

N

•

E' = -bN + aE +

∆

E where a = k cos

φ

and b = k sin

φ

are used to simplify the matrix representation, and

∆

N and

∆

E represent the shifts of the N and E axis in the N' and E' system.

Common points in both coordinate systems are used in a least-squares adjustment to solve the four unknown parameters (a, b,

∆

E, and

∆

N).

Once estimates of a and b are determined, the rotation and scale between the two systems is calculated by:

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and

For more information about horizontal adjustment, please refer to Observations and Least Squares by E.

Mikhail (John Wiley & Sons, 1982).

Vertical Adjustment

The Trimble Survey Controller software determines a vertical adjustment using least-squares with no weighting. This adjustment requires measured WGS-84 heights and control elevations.

In the trivial one-point case, the adjustment consists of a constant height shift only. For two or more points, a tilt in north and east are also computed.

The inclined plane parameters are determined by solving the matrix equation:

AX = B where the solution the components being the constant height shift and tilt east and north (in terms of a height shift per unit distance east or north), and the design matrix where En Nn are the coordinates of the nth point as derived from the WGS-84 data set.

E

1

N

1

are the coordinates of the origin point of the adjustment. (The origin point can be any one of the n points.) where H'n - Hn is the difference in elevation between the keyed in value for the nth point and the value derived from the WGS-84 data set.

Ground Scale Factor

In the Trimble Survey Controller software, you can define a Project height when you create a No

438

projection/no datum job. The project height is used after you perform a site calibration to compute a scale factor for the projection, so that ground coordinates are computed at elevation.

The projection scale factor is calculated as follows: and where:

SF scale factor of projection

R ellipsoidal radius h average elevation (reference elevation) a semimajor axis e² eccentricity²

φ origin latitude of projection

Geoid Model

The Trimble Survey Controller software can use a geoid model to provide orthometric heights from measured

(by GPS) WGS-84 heights.

Geoid model is one of the options in the Vertical adjustment field. (The other options in this field are No

adjustment, Inclined plane, Geoid model/Inclined plane. )

If you select Geoid model and you do not perform an in-field calibration, the elevation values displayed by the Trimble Survey Controller software represent the unadjusted elevation above the defined geoid using the relationship: hgeoid = H - N where: hgeoid the unadjusted elevation above the geoid

H the measured GPS height above the ellipsoid

N the geoid ellipsoidal separation, derived from a geoid model

If you select Geoid model and then perform an in-field calibration, the Trimble Survey Controller software computes the calibration parameters using hcontrol and hgeoid as inputs, so that the geoid model is inclined to fit the local control elevations. The method of vertical adjustment becomes Geoid/Inclined plane.

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Ellipsoid Calculations

The ground and ellipsoid distances in the Trimble Survey Controller software are calculated parallel to the ellipsoid. The equations used for these calculations are based on the Robbins ellipsoid geometry formulas.

These formulas (by Dr A.R. Robbins) are in the Empire Survey Review No. 125, 1962. They are accurate to better than 20 mm over distances of 1,500 km. The errors can reach 16 meters at 4,500 km and more than

2,000 meters at 9,000 km.

Conventional Instrument Calculations

The following diagram labels the observations and corrections that are applied when you use the Trimble

Survey Controller software with a conventional instrument.

Note - Interim corrections are not shown in the diagram. They are enclosed in square brackets in the table that follows.

[VA

3

]

VA

4

SD

R

[SD

1

]

[SD

2

]

SD

3

HD

1

VD

1

H

I

H

T

PC

Correction variables used in conventional instrument calculations

Where:

VA

2

Vertical angle from conventional instrument.

The Trimble Survey Controller software assumes that the conventional instrument applies any corrections for collimation and tilt.

Vertical angle corrected for curvature and refraction

Vertical angle corrected for curvature and refraction, instrument and target heights

Slope distance from EDM

Slope distance corrected for prism constant (PC)

Slope distance corrected for prism constant and PPM

Slope distance corrected for prism constant, PPM, instrument and target heights

Horizontal distance between instrument point and target point

Vertical distance between instrument point and target point

Height of instrument

Height of target

Prism constant

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Prism Constant Correction

The prism constant is applied to all slope distances. It is usually negative, but can be positive.

SD

1

= SD

R

+ PC where:

SD

R

SD

1 measured (raw) slope distance resultant slope distance

PC prism constant

PPM Correction

The parts per million (PPM) correction is applied to the slope distance after being corrected for the prism constant (see above). PPM depends on pressure and temperature.

where:

P

T air pressure in millibars temperature in °C

J &N constants supplied by the EDM manufacturer

The following table lists some manufacturers of conventional instruments, and the J and N constants that the

Trimble Survey Controller software uses to compute the PPM correction for those instruments.

PPM Constants for Conventional Instruments

Manufacturer of conventional instruments

Trimble VX/S Series

Trimble 5600

J constant N constant from instrument from instrument

274.41

79.39

Trimble 3300/3600

Trimble TTS300/500

Sokkia SET

278.77

270.0

279.0

80.653

79.167

79.400

Topcon

Geotronics 400/600

Leica

Zeiss Elta2/Elta3/Elta4

Zeiss Elta C

Pentax

279.7

275.0

282.0

255.0

281.8

279.0

79.600

79.550

79.400

79.100

79.391

79.400

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Nikon 275.0

79.5065

Note - The Trimble VX/S Series instrument J and N constants that are read from the instrument can be viewed in an exported JobXML file.

Note - The J constant value is the refractive index of the instrument. The N constant value is for use with pressure measurements in millibars. In the .dc file, the value is converted to one that can be used with pressure measurements in mmHg.

Curvature and Refraction Correction

The curvature and refraction correction is applied to vertical angles according to the coefficient of refraction that you set.

(Curvature and refraction correction) where: k

R

SD

3

VA

2

VA

3 coefficient of terrestrial refraction, selected in the Corrections field in the survey style approximate spheroid radius = 6378137m. (WGS-84 semi-major axis) slope distance, from equation vertical angle, from instrument corrected vertical angle

Point to point slope distance

Instrument and Target Height Reduction

The corrected vertical angle (VA

4

) from the instrument to the target is: where:

H

I

H

T

SD

2

VA

3

VA

4 instrument height target height slope distance vertical angle, from equation -

Curvature and refraction correction

corrected vertical angle

The slope distance from source point to target point (SD

3

) is given by the following:

(Point to point slope distance)

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Face 1/Face 2 Determination

This section describes how the Trimble Survey Controller software reduces Face 2 readings to Face 1 readings in order to perform calculations. It does this automatically.

The observed raw vertical angle is used to determine whether an observation is Face 1 or Face 2:

•

If the vertical angle is not present the observation is assumed to be Face 1.

•

If the vertical angle is in the range 0° to 180°, the observation is Face 1.

•

If the vertical angle is in the range 180°-360°, the observation is Face 2.

Orientation Correction

To orient circle readings so that they become azimuths, an orientation correction is applied. The orientation correction is the difference between the backsight circle reading and the backsight azimuth. This term is applied to all the other observations (circle readings) at a station.

The formula is:

(Orientation correction) where:

Az

x

HA

x

Az

B

HA

B azimuth to any point X horizontal observation to any point X actual backsight azimuth (`reference azimuth') observed backsight circle reading

Slope Reduction

The horizontal and vertical components of an observation (HD

1

and VD

1

) are found from the vertical angle and slope distances by: where:

HD

1

VD

1

VA

4

SD

3 horizontal distance vertical distance zenith angle slope distance

Coordinate Calculation

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Coordinates of a target point are calculated from observations and the coordinates of the instrument point using: where:

N

1

, E

1

, Z

1

N

2

, E

2

, Z

2

HD

1

VD

1

Az

1

Northing, Easting, Elevation of instrument point

Northing, Easting, Elevation of target point horizontal distance vertical distance from equation -

Orientation correction.

Mean Turned Angle Calculations

When the software calculates the mean turned angle and the mean distance for a point, it also calculates standard errors as follows:

For the angles, the standard error of the mean of a set of measurements formula is used:

For distances, the standard error of the set of measurements formula is used:

Resection Calculation

The resection calculation is a least-squares calculation that uses all available data.

Observations to the same point taken on different faces are treated as separate observations. However, the results are the same as those gained from using meaned (averaged) observations.

The residuals are given for each point , not each observation.

The formula used for the standard errors is as follows:

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Traverse Calculation

This section shows the formulae that the software uses when it calculates a traverse.

Compass adjustment

The Compass adjustment distributes the error in proportion to the length of the traverse lines. The formulae are as follows:

Northing adjustment x Northing misclosure where:

D horizontal distance sum of the horizontal distances in the traverse

Easting adjustment x Easting misclosure where:

D horizontal distance sum of the horizontal distances in the traverse

Transit adjustment

The Transit adjustment distributes the error in proportion to the northing and easting of each traverse point.

Northing adjustment x Northing misclosure where: change in northing for the traverse line sum of the changes in northings of all the traverse lines

x Easting misclosure Easting adjustment where: change in easting for the traverse line sum of the changes in eastings of all the traverse lines

Angular adjustments

There are three options for distributing the angular error in a traverse:

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•

Proportional to distance-the error is distributed among the angles based on the sum of the inverses of the forward and back traverse distances for each point. The formula used is as follows: where:

A

a

A

m

Angular adjustment

Angular misclose

•

Equal proportions--the error is distributed evenly among the angles in the traverse.

•

None--the error is not distributed.

Elevation adjustments

There are three options for distributing the elevation error in a traverse:

•

Proportional to distance--the error is distributed in proportion to the length of the traverse line to the

• point.

Equal proportions--the error is distributed evenly among the traverse lines.

•

None--the error is not distributed.

Standard Errors Recorded With Conventional Observations

Each conventional observation that is recorded in the job file has standard errors associated with it. The standard errors that are recorded with the observations are determined as follow:

•

For a single observation to a point (for example, Measure topo) the standard error values assigned to the observation are the instrument standard error values (a-priori values). This is because there is

• nothing else that the standard errors can be determined from.

For an averaged observation, the standard deviations that are reported during the measurement process indicate the precision of the instrument, but do not indicate its accuracy. If the standard deviation of any of the measurement components (a-posteriori value) exceeds the equivalent instrument standard error value, then this standard deviation value is recorded with the observation.

However, any standard deviation values that are below the instrument standard error value are replaced by the instrument standard error (a-priori) value when the observation is stored. This ensures that the recorded standard errors indicate the instrument accuracy rather than the precision of

• a given observation.

For rounds measurements (Rounds, Resection, and Station Setup Plus), the MTA (Mean Turned

Angle ) observations record the a-posteriori standard errors computed from the individual observations that contribute to the MTA.

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Area Calculations

The area in Cogo / Compute area, is calculated at the average elevation of all the points when the Distances field is set to Ground. Average elevation is calculated as follows: calculate then

×

area at sea level where:

N number of stations average elevation

R ellipsoidal radius

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