Combination of Errors: Calculating the Propagation of Uncertainty

Overview

In surveying, we rarely measure the "final" answer directly. Instead, we measure angles and distances to calculate coordinates, areas, or volumes. Since every measurement contains a small error, these errors "propagate" through our formulas 2.

Why This Matters

Understanding error combination allows you to predict if your field methods will actually meet the project specifications before you start work. It is the difference between "guessing" accuracy and "guaranteeing" it 3.

Theory

If a derived quantity $a$ is a function of several independent variables $(x, y, z, ...)$, then a small change in those variables $(\delta x, \delta y, \delta z)$ will result in a change in $a$ $(\delta a)$ 2.

Mathematical Principles

The General Equation for Variance is the foundation of error propagation: $\sigma_a^2 = \left(\frac{\partial a}{\partial x}\right)^2 \sigma_x^2 + \left(\frac{\partial a}{\partial y}\right)^2 \sigma_y^2 + \left(\frac{\partial a}{\partial z}\right)^2 \sigma_z^2 + \dots$ 4

Where:

• $\sigma_a^2$ is the variance of the result.
• $\sigma_x, \sigma_y, \dots$ are the standard errors of the independent measurements.
• $\frac{\partial a}{\partial x}$ is the partial derivative of the function with respect to $x$.

Formula Breakdown by Operation

1. Addition and Subtraction

For $A = a + b$ or $A = a - b$: $\sigma_A = \pm \sqrt{\sigma_a^2 + \sigma_b^2}$ 4

2. Products (Area Calculation)

For $A = a \times b$: $\sigma_A = \pm ab \sqrt{\left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_b}{b}\right)^2}$ 5

3. Powers (Volume of a Cube)

For $A = a^n$: $\sigma_A = \pm (n a^{n-1} \sigma_a)$ 6

Step-by-Step Example

Problem: The three angles of a triangle are measured, each with a standard error of $\pm 2''$. What is the total error in the sum of the angles? 4

1. Formula: $T = \alpha + \beta + \gamma$
2. Standard Errors: $\sigma_\alpha = \sigma_\beta = \sigma_\gamma = 2''$
3. Calculation:$\sigma_T = \pm \sqrt{2^2 + 2^2 + 2^2} = \pm \sqrt{12}$
4. Result: $\sigma_T = \pm 3.5''$

Practical Tips

• Commensurate Accuracy: Measure your angles and distances to a similar level of precision. A rule of thumb: $1''$ of arc subtends $1 \text{ mm}$ at $200 \text{ m}$ 7.
• Relative Error: When multiplying (e.g., area), it is often easier to work with relative errors ($R_a = \sigma_a / a$) 5.

Common Mistakes

• Confusing Products with Powers: The error for $a \times a \times a$ is different from $a^3$. In $a^3$, the variables are not independent, so you cannot use the square-root-sum-of-squares rule 5.
• Ignoring Units: Ensure all standard errors are in the same units (e.g., meters or seconds) before plugging them into the variance equation 8.

Best Practices

Always describe measured data using both the Standard Deviation (S) of the set and the Standard Error ($S_{\bar{x}}$) of the mean to provide a complete picture of precision 8.

FAQ

Conclusion

Error propagation is the "quality control" of surveying. By applying the general variance equation, you can ensure that your final deliverables—whether they are property boundaries or bridge coordinates—stay within the required tolerances.

References

Schofield, W. (2001). Engineering Surveying. 5th ed. Butterworth-Heinemann. 2, 4-12.