How to Perform Resection and Resection Calculations with a Total Station

Introduction to Total Station Resection

Resection is a core surveying technique that allows an operator to determine the precise coordinates of an unknown instrument station by observing two or more known control points. The total station, which integrates an electronic theodolite and an electronic distance meter (EDM), is the ideal tool for this task because it can simultaneously measure horizontal angles, vertical angles, and slope distances with high accuracy. This article explains how to perform resection with a total station, covering the underlying geometry, step-by-step field procedures, calculation methods (including least squares adjustment), common error sources, and practical tips for achieving reliable results.

Understanding Resection: Why It Matters

Resection is used when a surveyor must set up the total station at a location whose coordinates are unknown but where a clear line of sight exists to at least two established control points. This situation frequently arises on construction sites, in building surveys, and during topographic mapping when control points are distant or inaccessible from the desired setup location. Instead of occupying a known point and then traversing, the surveyor can quickly establish the instrument's position using resection, saving time and reducing the need for multiple setups.

Resection relies on the principle of intersection: the unknown point lies at the intersection of rays from the known points, with directions determined by the measured angles. When distances are also measured, the solution becomes overdetermined, allowing for error detection and adjustment. Modern total stations often include software that performs the resection calculation automatically, but understanding the manual process is essential for verifying results, diagnosing blunders, and working with legacy equipment.

Prerequisites for Resection

Before performing a resection, the surveyor must have:

Known control points: At least two points with known coordinates (e.g., from a previous survey or GPS). More points improve reliability and allow least squares adjustment.
A properly calibrated total station: The instrument must show correct compensator and collimation errors. Regular calibration checks are recommended.
Clear line of sight: The targets (prisms or reflectors) on the known points must be visible from the setup location.
Stable setup: A solid tripod on firm ground, with the instrument leveled and centered as accurately as possible. Tribrach and optical plummet adjustment help.

Field Procedure for Resection with a Total Station

Step 1: Set Up and Orient the Instrument

Place the total station on a stable tripod over the unknown point (the point whose coordinates you wish to determine). Level the instrument precisely using the plate bubble or electronic level. If the instrument has a laser plummet, check that it is directly over the mark. Some surveyors prefer to set up slightly off the mark and then compute the offset (free stationing), but for standard resection the instrument is centered over the point of interest.

Step 2: Record Instrument Height

Measure and record the height of the instrument (HI) from the ground mark to the center of the telescope trunnion axis. This value is used if you need to reduce slope distances to horizontal distances or to calculate elevations. Most total station resection routines ask for HI and target heights (HT) for each observed point.

Step 3: Sight the First Known Point

Point the telescope at the target (prism) on the first known control point. Focus carefully and ensure the crosshairs are centered on the prism. Press the appropriate measurement key to record both the horizontal and vertical angles, as well as the slope distance to that point. Some instruments allow you to store the point ID and target height before measuring.

Step 4: Sight the Second Known Point

Rotate the instrument to the second known point and repeat the measurement. For a two-point resection, this is the minimum. However, if more points are available, observe them all to create redundancy.

Step 5: Initiate the Resection Calculation

On the total station, navigate to the resection or free stationing menu. You will typically be asked to select the known points from the job list or enter their coordinates manually. The instrument then computes the coordinates of the setup point and may show residuals or standard deviations. Review these values before accepting the solution. If residuals are large (e.g., horizontal angle residual > 10 seconds, distance residual > 5 mm), re-observe the suspect point or check for target centering errors.

The Mathematics Behind Resection Calculations

Resection calculations can be performed using several methods. The most common are angle-only resection (also known as the three-point problem) and distance-angle resection (which uses both angle and distance observations). Modern total stations often use a least squares approach to handle redundant observations.

Angle-Only Resection (Three-Point Problem)

When only angles are measured (no distances), three known points are required to uniquely solve for the unknown point. The classic solution uses the inscribed angle theorem: lines of sight from the unknown point to two known points subtend an angle that is constant along an arc of a circle. The intersection of two such circles yields the unknown position. Manual calculation involves solving two circle equations or using coordinate transformation formulas. In practice, this method is rare with total stations because distances are almost always measured.

Distance-Angle Resection

When both horizontal angles and distances to known points are measured, the problem becomes simpler. For two points, the unknown point can be computed by intersection of two circles (distance circles) and two rays (angle lines). The usual approach is to compute an initial position from the angle measurements and then refine it using the distance constraints. A robust two-point resection solution is:

Compute the distance between the two known points (AB).
From the measured horizontal angle at the unknown point (angle APB), use the Law of Sines in triangle APB to compute the distances from the unknown point to each known point.
With these computed distances and the known coordinates, apply forward intersection formulas to find the unknown point coordinates.

Suppose known points A (Ea, Na) and B (Eb, Nb), measured distances dA and dB (slope distances reduced to horizontal), and measured horizontal angle θ = angle APB. First compute distance AB. Then using Law of Sines: sin(angle PAB) / dB = sin(θ) / AB, etc. The coordinate differences can be solved via polar to rectangular conversion.

Least Squares Adjustment for Redundant Observations

When more than two points are observed, the system is overdetermined. A least squares adjustment provides the most likely coordinates of the unknown point, as well as residuals and standard deviations that help detect blunders. The observation equations relate the unknown coordinates (E, N) to the measured angles and distances. For each observed point i:

Distance observation equation: √((E - Ei)² + (N - Ni)²) = d_measured + v_d
Angle observation equation (more complex): relative bearing differences are modeled.

These are linearized and solved iteratively. Most modern total stations use a built-in least squares resection routine. The output includes the estimated coordinates and their standard deviations, which the surveyor should use as quality indicators.

Sample Calculation (Two-Point Resection)

Let us work a simplified example using plane coordinates (ignoring elevation for brevity). Known points:

Point	East (m)	North (m)
A	1000.000	2000.000
B	1500.000	2500.000

Measured from unknown point P:

Horizontal angle APB = 60°00'00"
Slope distance to A reduced to horizontal: 500.000 m
Slope distance to B reduced to horizontal: 400.000 m

First, compute distance AB: √((1500-1000)² + (2500-2000)²) = √(500² + 500²) = 707.107 m.

In triangle APB, we have side AB = 707.107 m, side PA = 500 m, side PB = 400 m, and included angle at P = 60°. We can check consistency using Law of Cosines: AB² = PA² + PB² – 2·PA·PB·cos(60°) = 250000 + 160000 – 2·500·400·0.5 = 410000 – 200000 = 210000, so AB = √210000 = 458.258 m. But we computed AB as 707.107 m – clearly the measured distances and angle are inconsistent. This illustrates that real measurements contain errors. The surveyor would re-measure or apply adjustment. For a consistent solution, the distances and angle must satisfy the triangle geometry. In practice, the instrument's software handles this by using a combination of angle and distance measurements and may give a warning if inconsistencies exceed a threshold.

For a valid dataset, the coordinates of P can be computed by first determining the azimuth from A to B: atan2(500,500)=45°. Then using known angles and distances, the azimuth from A to P or from B to P can be derived, and polar calculations yield P's coordinates.

Errors and Accuracy in Resection

Several factors affect the accuracy of a resection:

Instrument errors: Collimation error, vertical circle index error, and compensator mislevel cause systematic angle errors. Regular calibration is essential.
Target centering errors: If the prism is not exactly over the known point, both angle and distance observations will be wrong. Use plumb bobs or laser plummet centering systems.
Atmospheric conditions: Temperature, pressure, and humidity affect EDM measurements. Apply meteorological corrections if required.
Geometry: The configuration of known points relative to the unknown point influences the strength of the solution. Weak geometry occurs when points are nearly collinear with the unknown point or when the unknown point lies close to the line between two known points. A better distribution spreads the points around the instrument.
Number of points: Using only two points provides no redundancy; any undetected error becomes part of the solution. With three or more points, residuals help identify bad observations.

Surveyors should always check their resection result by sighting a check point (another known point not used in the calculation). If the computed coordinates of that check point match the known coordinates within acceptable tolerances, the resection is validated.

Free Stationing vs. Resection

The terms resection and free stationing are often used interchangeably, but there is a subtle difference. In classic resection, the total station is set up directly over the point whose coordinates are to be determined. In free stationing, the instrument may be set up at an arbitrary location (not necessarily over a marked point) and its position is determined by resection. The coordinates of the setup are then used as a temporary station for subsequent measurements. Free stationing is extremely useful on construction sites where control points are visible but cannot be occupied (e.g., because they are in a hazardous location or obstructed). Many total station resection routines are actually free stationing routines because they do not require the instrument to be precisely over a mark.

Practical Tips for Reliable Resection

Use three or more points: Even if the software allows two points, adding a third provides a check and improves accuracy. Ideally, the points should surround the setup.
Check prism constants: Ensure the prism constant is correctly entered in the total station. A mismatch introduces a systematic offset in distances.
Record target heights: Enter the correct height of each prism (HT) in the instrument. If horizontal distances are needed, ensure the reduction mode is set to horizontal or slope with appropriate corrections.
Set the correct coordinate system: Ensure the total station is using the correct projection or local coordinate system. If working in a local system, define it clearly.
Observe in direct and reverse faces: For critical work, measure angles in both face positions to eliminate collimation and vertical index errors. The software may average them automatically.
Apply atmospheric corrections: Input current temperature and pressure (or use the average for the area) to correct EDM measurements.

Alternative Methods for Determining Instrument Position

Resection is not the only way to determine a setup point. Alternatives include:

Traverse: Starting from a known point, measure angles and distances along a path to the setup point. This is slower but allows for error propagation control.
GNSS (GPS): Using a GNSS receiver to get coordinates of the setup point, which can then be used as a base for the total station. Accuracy depends on equipment and corrections (RTK).
Intersection: If two known points are occupied and the unknown point is targeted, forward intersection calculates its coordinates. This is the reverse of resection.

Each method has advantages; resection is usually the fastest when good control points are visible.

Software and Automation in Modern Total Stations

Leading total station manufacturers such as Leica, Trimble, Topcon, and Sokkia provide embedded resection routines. The user selects the known points from a list, measures them, and the instrument displays the computed coordinates along with quality indicators like standard deviation of the point or RMS of residuals. For example, Leica Captivate software includes a "Resection" application that supports both angle-only and distance-angle methods and allows the user to choose a least squares solution. Trimble Access has a "Resection" screen that guides the operator through observing up to 10 points and displays the residual graph.

These software tools greatly reduce manual computation time, but the surveyor must still understand the principles to recognize erroneous results. Always check that the instrument's calculation method (e.g., free stationing vs. fixed point) matches your needs.

Case Study: Resection for Building Layout

Consider a construction site where two control points (A and B) were established using GPS. They are located on opposite sides of a building excavation. The surveyor needs to set up the total station near the center of the excavation to stake out column locations. She sets up over a temporary nail, measures to points A and B (both with known coordinates), and adds a third check point C. The resection software computes the instrument position with horizontal standard deviation of 2 mm. She then sights point C, whose computed coordinates match the known values within 4 mm – well within the project tolerance of 10 mm. She proceeds with staking out columns. This workflow saved time compared to running a traverse from one control point.

Conclusion

Resection with a total station is a powerful and efficient technique for establishing instrument position using known control points. By understanding the geometry, field procedures, calculation methods, and error sources, surveyors can achieve consistent and accurate results. While modern instruments automate the computation, a solid grasp of the underlying principles allows for validation and troubleshooting. Always use redundant observations, check residuals, and verify against an independent check point. Practice with real data and software to build confidence in applying resection in diverse surveying scenarios.