Lecture 2 Hard

Trigonometric Leveling

• Trigonometric leveling determines elevation differences by measuring slope distance (S or SD) or horizontal distance (H or HD) and zenith angle (Z) or altitude angle (α).
• Zenith angle is the vertical angle measured from the horizontal to the target.
• As the instrument rotates to sight the prism, the zenith angle increases in size.
• A zenith angle below 90° indicates an uphill sight; 90° indicates a level line of sight; above 90° indicates a downhill sight.
• The formula for calculating the difference in elevation (ΔH) using zenith angle is: ΔH = S cos Z, where S is the slope distance and Z is the zenith angle in decimal degrees.

Elevation Calculation with Zenith Angles

• Convert zenith angles from degrees, minutes, and seconds to decimal degrees.
• If instrument height (hi) and prism height (r) are equal, the difference in elevation (ΔH) is calculated by using the zenith angle (Z) with the slope distance formula ΔH = (S) (cos Z). Round to the nearest whole seconds.
• Ensure to carry decimal places to at least 8th place during calculations to avoid errors.

Correction for Curvature and Refraction

• For distances greater than 1000 feet, correct for curvature and refraction using this equation: ΔH = (SD) (cos Z) + (0.0206)[(SD/1000)(sin Z)²]
• where:
  • ΔH = difference in elevation, corrected for curvature and refraction
  • SD = measured slope distance
  • Z = zenith angle (converted to decimal degrees)
• To improve accuracy, measure from lower to higher points and vice-versa, averaging the results.

Altitude Angles

• The difference in elevation (ΔH) using altitude angles is: ΔH = (H) (tan α)
• where H is the horizontal distance and α is the altitude angle (in decimal degrees).
• Instrument and rod heights should be the same to eliminate complex calculations.

Level Rods and Tripods

• Philadelphia rods are common for leveling. They are graduated in hundredths of a foot with sliding sections that can be adjusted.
• Chicago rods are independently-sectioned, often used in construction surveys.
• Tripods provide a sturdy support for leveling instruments.

Level Vials

• Tube levels and bull's-eye levels are two types of level vials.
• The radius of curvature of the level vial's curvature affects its sensitivity.
• Centering the bubble creates a horizontal line of sight.

Parallax

• Parallax creates a false reading if not avoided by ensuring crosshairs are clear and distinct against a background when viewing through the eyepiece.

Test your knowledge of trigonometric leveling and elevation calculations using zenith angles. This quiz covers concepts like the zenith angle, slope distance, and the formulas used to determine elevation differences. Challenge yourself with practical questions on this essential surveying technique.