EDM Slope Reduction

[Eddie W. Shore]

The program EDMSLOPE calculates the following:

* curvature correction due to the Earth
* horizontal distance from observer to reflector at both elevation of the observer and at sea level
* change of in elevation from observer to reflector

Given:

* slope distance between observer to reflector
* height of both the observer and reflector’s instruments
* the elevation the observer
* zenith angle: angle from directly above to the slope distance line

Notes:

All measurements are given in feet. Assume that the correction factor due to light refraction is negligible. The radius of Earth is approximately 20,902,231 feet.

Formulas (see diagram above)

C = asin((90* sin Z * S)/(π * radius))

H = (S * sin(Z – C))/cos(Z)

L = H * (radius / (radius + D))

E = (S * cos Z / cos C) + M – F

HP Prime Program EDMSLOPE

Code:

EXPORT EDMSLOPE()
BEGIN
// 2017-07-26 EWS
// in feet
// Radius of Earth:
// based off 6371 km, to nearest
// integer
LOCAL R:=20902231;

// Degrees
HAngle:=1;

LOCAL Z,S,M,D,F;
INPUT({Z,S,M,D,F},"EDM Slope
Reduction",{},{"Zenith Angle",
"Slope Distance (ft)",
"Instrument Height (ft)",
"Elevation (ft)",
"Reflector Height (ft)"});

// Curvature correction
LOCAL C:=ASIN((90*SIN(Z)*S)/(π*R));

// Horizontal dist-elevation
LOCAL H:=(S*SIN(Z-C))/COS(C);

// Horizontal dist-Sea level
LOCAL L:=H*R/(R+D);

// Elevation change
LOCAL E:=S*COS(Z)/COS(C)+M-F;

PRINT();
PRINT("Curvature correction");
PRINT(C);
PRINT("Horizontal dist-elevation");
PRINT(H);
PRINT("Horizontal dist-Sea level");
PRINT(L);
PRINT("Elevation change");
PRINT(E);

END;

Example

Observer: D = 1,238.32 feet, instrument height = M = 3.5 feet
Slope distance: S = 1,474 feet, at a zenith angle of 86.11°
Reflector height = F = 3 feet

Results:
Curvature correction = C = 0.1154831691
Horizontal Distance-Elevation = H = 1,470.40255 ft
Horizontal Distance-Sea Level = L = 1,470.315444 ft
Elevation Change = E = 100.4980742 ft

Sources:

TI Programmable 58/59 Surveying – Texas Instruments 1977