1. Principles
Trigonometric leveling is so named because it uses a total station instrument's (TSI) slope distance and zenith angle measurements to mathematically compute an elevation difference which, with a few more bits of information, can be used to determine a point's elevation. Using appropriate procedures, and controlling errors, elevation accuraciy can be better than 0.1 ft. Because trigonometric leveling is not limited to a horizontal line of sight, it is more flexible and provides faster elevation data collection than differential leveling.
a. Elevation determination
TSI slope reduction is discussed in the Electronic Distance Measurement topic. Slope reduction geometry is shown in Figure F1 and accomapnying Equations F1 and F2.
Figure F1 Trigonometric Relationships 
Equation F1  
Equation F2  
H: Horizontal distance Z: Zenith angle S: slope distance 
V: Vertical distance R: Earth radius k: Refraction constant 
Vertical distance, V, is the elevation difference between the TSI and the reflector. Adding V to the TSI elevation gives us the reflector elevation. What we want, however is the elevation of the ground point at the reflector location. To determine that, we need two additional pieces of information: (1) TSI elevation, and, (2) height of the reflector above the ground point. Once we have those, then the elevation of any observed point, i, is computed from:
Elev_{i} = Elev_{TSI} + V_{i } HR_{i}  Equation F3  
Elev_{TSI}: Elevation of the TSI 
b. Sideshots
Besides the way the elevations are determined, another major difference between trigonometric and differential leveling is point connectivity. In differential leveling we normally have one BS and one FS at each set up  we survey in then out of each elevation point. Trigonometric leveling is used when a number of elevations are measured from a single instrument set up. All those points are surveyed into, but not out of. Each elevation point determined by trigonometric leveling is an open link, also known as a sideshot.
In a closed differential level network, Figure F2(a), each point has a BS and FS; each is connected to another point and their elevations are based on the BM.
Figure F2(b) depicts a trigonometric network referenced to the differential network. Points B, C, and D serve as control for trigonometric leveling. The green shots at points B, C, and D are all sideshots. Because sideshots are not connected to other points their elevations cannot be checked. An elevation error could be in an individual sideshot measurement or a control point elevation error which affects all the sideshots from that point
(a) Control Network  (b) Sideshots  
Figure F2 Control Network and Sideshots 
2. Determining TSI elevation
a. TSI set up over known elevation
In this scenario the TSI is set up over a point whose elevation is known. The elevation must be transferred to the TSI by measuring the Height of the Instrument, HI, Figure F3.
Instrument Height Figure F3 
The TSI's elevation is calculated from:
Elev_{TSI} = Elev + HI  Equation F4  
Elev: Point reference elevation 
The HI is the vertical distance from the reference elevation to the TSI's Horizontal Axis (HA). Because the TSI is over the point, it is not possible to directly measure the HI. Most TSIs have an HA index mark on one or both standards which can be used to get a close value for the HI. There are two simple ways to measure the HI:
(1) Using a tape
A cloth tape or pocketrod can be used to measure the distance from the elevation reference to the HA index mark, Figure F4.
Figure F4 Measuring HI with Tape 
(2) Using a prism pole
Another method is to place the prism pole next to the instrument and raise or lower the reflector so its center lines up with the HA index mark, Figure F5. The prism pole reading is recorded as the HI.
Figure F5 Measuring HI with Prism Pole 
Each method is subject to errors which will be discussed in a following section.
b. Sight to known elevation
The TSI is set up at a location from which a reference elevation or benchmark (BM) is visible. The vertical distance is measured to a reflector held on the BM, Figure F6.
Figure F6 Using Benckmark to Establish TSI Elevation 
The elevation of the TSI is determined from:
Elev_{TSI} = Elev_{BM} + HR_{BM}V_{BM}  Equation F5 
3. Errors
Basically, the errors described in the Errors chapter of the Electronic Distance Measurement topic also affect trigonometric leveling. In addition, those errors arising from TSI zenith angle measurements will have an effect on elevation determination. Those aspects are a function of the angle measurement capacity of the TSI and will be described in the Errors chapter of the TSI topic.
Other errors by source 
(1) Personal
Most personal errors are generally in:
 determining HI or HR
 recording
 computations
(2) Instrumental
(a) Taping HI
A shown in Figure F4, measuring the HI with a tape introduces an error since the tape can't be used to measure vertically. How much error? Assuming an approximate 8 inch TSI width and a tape distance of 5.25 ft, the true HI can be computed from geometry shown in Figure F7.
In this case, an error of only 0.02 ft is introduced in the TSI elevation. That's not much although not up to diffential leveling. 

Figure F7 HI Measurement Error 
(b) Prism pole reading
Because a prism pole is used extensively in trigonometric leveling we need to understand some potential errors associated with it and how they can be compensated.
If the prism pole is set to 5.70 ft, is it really 5.70 ft from the bottom of the rod to the center of the reflector, Figure F8?
Figure F8 Prism Pole Uncertainty 
There are a few reasons the prism pole reading may be incorrect:
How the reflector is mounted. Most screw onto the top of the pole and are held in place using a lock nut. Depending on how far down the reflector is screwed will affect its height above the ground.


Figure F9 Prism Mounting Height Difference 

The type of reflector used. Different reflector designs, single versus multiple reflectors, have different center locations.


Figure F10 Prism Type Height Difference 

Pole tip. A regular pointed tip can sink in soft ground changing the reflector height. A topo shoe, having a blunt tip, is often used instead of the pointed tip. The blunt tip prevents the pole sinking in the ground maintaing a consistent HR regardless the terrain. But... The height of the topo shoe may be different than the pointed tip which could affect the pole height reading accuracy.


Figure F11 Pole Bottom Height Difference 
If a tape or pocketrod is used to measure HI, it should also be used to set the HR.
For example, assume the HI measured with a tape is 5.35 ft, Figure F12(a). With the reflector set to the same height using the tape, the prism pole reading is 5.52 ft, Figure F12(b). Had the HR been set using a 5.35 ft prism pole reading, there would be almost 0.2 ft error in each measured elevation. The difference between the HI and the reading on the prism pole is a systematic error.
(a) Taped Height  (b) Prism Height 
Figure F12 Prism Height Systematic Error 
If the prism pole is used to measure the HI, then as long as the same pole is used to measure points any height error is compensated. That's because the error is added in the HI and subtracted in the HR to determine point elevations.
However, an HI error can be introduced if the if the ground isn't flat, Figure F13, making the pole bottom higher or lower than the point.
Figure F13 Uneven Ground Slope 
In those situations a tape should be used to determine the HI and to set the reflector height.
If sighting a reference BM to establish the TSI elevation, Figure F6, then pole reading error is automatically compensated as long as the same pole is used throughout. The pole height error is added to obtain the TSI elevation, then subtracted for point elevations. However, the TSI elevation may be incorrect subject to the pole reading error in the HR_{BM}.
(c) Rod bubble
The prism pole not being truly vertical has a greater effect on horizontal distance than vertical distance measurement. That's not to say a maladjusted bubble should be used for trigonometric leveling because that operation is generally combined with horizontal angles and distances to perform three dimensional surveys. The rod bubble on the prism pole should be checked and adjusted as described in the Errors chapter of Electronic Distance Measurement.
(3) Natural
The same environmental conditons affecting distance measurement and angle measurement also affect trigonometric leveling.
4. Summary
Trigonometric leveling is a fast efficient way to measure many elevations from a single set up. Being based on a TSI, the process is more flexible than differential leveling.
Whereas in differential leveling the primary instrument is relatively simple, trigonometric leveling depends on a TSI which integrates more measurement types and hence more errors to control. In trigonometric leveling, the surveyor has the option to trade accuracy for productivity depending on the project at hand. Trigonometric leveling does not replace differential leveling but augments it. Differential leveling can be used to build a vertical control network for a project; trigonometric leveling can use that control to collect additional elevation data for design or quantities determination.
Combining trigonometric leveling with horizontal coordinate geometry allows the surveyor to work, design, and implement, in threedimensions.