Error Propagation

Overview#

Surveyors rarely use raw measurements directly. A backsight reading, a horizontal angle, a taped distance -- these are intermediate values. The quantities we actually need are computed from them: elevations, coordinates, areas, volumes, curve data, closure ratios. Every one of these computed results inherits uncertainty from the measurements that produced it.

Error propagation is the mathematical framework for tracking that inheritance. Given the uncertainties (standard deviations) of the input measurements and the function that combines them, propagation tells us the uncertainty of the output. This is not merely an academic exercise. It is how surveyors answer practical questions:

• Will this level circuit meet second-order standards?
• How precisely must I measure angles to achieve 1:20,000 closure?
• What is the uncertainty in the area of this parcel?

"Error propagation is concerned with determining the errors in quantities computed from observed values that contain errors." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 3, p. 54

Without propagation analysis, a surveyor cannot predict whether a measurement procedure will meet specifications before going to the field. With it, surveys can be designed on paper, optimized for efficiency, and defended with rigor.

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The General Law of Error Propagation#

Suppose a computed quantity $f$ depends on $n$ independent measured values $x_1, x_2, \ldots, x_n$:

$f = f(x_1, x_2, \ldots, x_n)$

Each measured value has an associated standard deviation: $\sigma_{x_1}, \sigma_{x_2}, \ldots, \sigma_{x_n}$. The general law of propagation of variances states that the variance of the computed quantity is:

$\sigma_f^2 = \left(\frac{\partial f}{\partial x_1}\right)^2 \sigma_{x_1}^2 + \left(\frac{\partial f}{\partial x_2}\right)^2 \sigma_{x_2}^2 + \cdots + \left(\frac{\partial f}{\partial x_n}\right)^2 \sigma_{x_n}^2$

Or more compactly:

$\sigma_f^2 = \sum_{i=1}^{n} \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_{x_i}^2$

Each partial derivative $\frac{\partial f}{\partial x_i}$ acts as a sensitivity coefficient -- it quantifies how strongly a small change in $x_i$ affects the computed result $f$. A large partial derivative means the computed quantity is sensitive to errors in that particular measurement. A small one means errors in that measurement have little effect.

"The general law of error propagation... is one of the most useful and important relationships in the theory of errors." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 3, p. 55

This formula assumes that the measurements are independent (uncorrelated) and that the function $f$ is well-approximated by its first-order Taylor expansion near the measured values. Both assumptions hold well for most routine surveying computations.

The standard deviation of the computed quantity is simply the square root of the variance:

$\sigma_f = \sqrt{\sum_{i=1}^{n} \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_{x_i}^2}$

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Special Cases#

The general law simplifies into a set of rules that cover the most common operations encountered in surveying calculations.

Sum or Difference

If $Z = A + B$ or $Z = A - B$, then:

$\sigma_Z = \sqrt{\sigma_A^2 + \sigma_B^2}$

This is a critical result: errors add in quadrature regardless of whether the quantities are added or subtracted. Subtraction does not cancel errors -- it compounds them. This is why computing a small difference between two large quantities (a common surveying operation) can produce a result with poor relative precision even when the individual measurements are excellent.

For example, if $\sigma_A = \sigma_B = \pm 0.01$ ft, then $\sigma_Z = \sqrt{0.01^2 + 0.01^2} = \pm 0.014$ ft -- not $\pm 0.02$ ft, as one might naively expect from simple addition.

Multiplication by a Constant

If $Z = kA$, where $k$ is an exact (error-free) constant, then:

$\sigma_Z = |k| \cdot \sigma_A$

The error scales linearly with the constant. If you multiply a distance by 2, the error doubles. If you convert feet to meters by multiplying by 0.3048, the error also scales by 0.3048.

Product of Two Measured Quantities

If $Z = A \times B$, then the relative (fractional) errors combine in quadrature:

$\frac{\sigma_Z}{|Z|} = \sqrt{\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2}$

In other words, the relative error of the product is the root sum square of the relative errors of the factors. This rule is essential for area computations.

Quotient of Two Measured Quantities

If $Z = A / B$, the same relative-error formula applies:

$\frac{\sigma_Z}{|Z|} = \sqrt{\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2}$

Products and quotients propagate relative errors identically.

Power Function

If $Z = A^n$, then:

$\frac{\sigma_Z}{|Z|} = |n| \cdot \frac{\sigma_A}{|A|}$

Squaring a quantity doubles its relative error. Taking a square root halves it. This becomes relevant in computations involving squared distances or square-root expressions.

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Surveying Examples#

Elevation Difference from Leveling

In differential leveling, the elevation difference at a single setup is:

$\Delta h = BS - FS$

where $BS$ is the backsight reading and $FS$ is the foresight reading. Applying the sum/difference rule:

$\sigma_{\Delta h} = \sqrt{\sigma_{BS}^2 + \sigma_{FS}^2}$

If both rod readings have a standard deviation of $\pm 0.005$ ft:

$\sigma_{\Delta h} = \sqrt{0.005^2 + 0.005^2} = \pm 0.007 \text{ ft}$

Each setup introduces $\pm 0.007$ ft of uncertainty into the elevation, regardless of whether the instrument goes up or down.

Horizontal Distance from Stadia

The stadia formula for horizontal distance is:

$D = Ks + C$

where $K$ is the stadia interval factor (typically 100), $s$ is the stadia intercept, and $C$ is the instrument constant (often negligible). Since $K$ and $C$ are treated as constants:

$\sigma_D = K \cdot \sigma_s$

If the stadia intercept can be read to $\pm 0.003$ ft:

$\sigma_D = 100 \times 0.003 = \pm 0.3 \text{ ft}$

This immediately tells us that stadia distances are good to roughly 1 foot per 300 feet -- useful for topographic work, inadequate for boundary surveys.

Area of a Rectangular Parcel

For a rectangle with length $L$ and width $W$:

$A = L \times W$

Using the product rule for relative errors:

$\frac{\sigma_A}{A} = \sqrt{\left(\frac{\sigma_L}{L}\right)^2 + \left(\frac{\sigma_W}{W}\right)^2}$

Suppose $L = 500.00$ ft ($\sigma_L = \pm 0.05$ ft) and $W = 300.00$ ft ($\sigma_W = \pm 0.05$ ft). The area is $150{,}000$ sq ft, and:

$\frac{\sigma_A}{150{,}000} = \sqrt{\left(\frac{0.05}{500}\right)^2 + \left(\frac{0.05}{300}\right)^2} = \sqrt{(0.0001)^2 + (0.000167)^2} = 0.000194$

$\sigma_A = 150{,}000 \times 0.000194 = \pm 29.2 \text{ sq ft} \approx \pm 0.00067 \text{ acres}$

Notice that the shorter side contributes more to the area uncertainty because its relative error is larger.

Traverse Closure

In a traverse, each course contributes departures and latitudes computed from measured angles and distances. For a single course with azimuth $\alpha$ and distance $d$:

$\Delta E = d \sin \alpha, \qquad \Delta N = d \cos \alpha$

Applying the general law:

$\sigma_{\Delta E}^2 = \sin^2\!\alpha \cdot \sigma_d^2 + d^2 \cos^2\!\alpha \cdot \sigma_\alpha^2$

$\sigma_{\Delta N}^2 = \cos^2\!\alpha \cdot \sigma_d^2 + d^2 \sin^2\!\alpha \cdot \sigma_\alpha^2$

where $\sigma_\alpha$ must be in radians. For the full traverse, the closure errors in easting and northing are sums of the individual course errors, so their variances add:

$\sigma_{\Delta E_{total}}^2 = \sum_{i=1}^{n} \sigma_{\Delta E_i}^2, \qquad \sigma_{\Delta N_{total}}^2 = \sum_{i=1}^{n} \sigma_{\Delta N_i}^2$

"The standard deviation of the linear misclosure of a traverse can be computed by propagating the errors in the individual angle and distance measurements through the coordinate computations." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 3, p. 60

This analysis allows a surveyor to predict traverse closure before entering the field and to identify which measurements (angles or distances) dominate the error budget.

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Error Propagation in Series#

A particularly important special case arises when the same measurement is repeated $n$ times in series, and the results are summed. This occurs constantly in surveying:

• A level circuit with $n$ instrument setups
• A distance measured by laying out $n$ tape lengths end-to-end
• A traverse with $n$ courses

If each individual measurement has the same standard deviation $\sigma_0$, and the total quantity is the sum of all $n$ measurements, then by the sum rule:

$\sigma_{total} = \sqrt{\sigma_0^2 + \sigma_0^2 + \cdots + \sigma_0^2} = \sigma_0\sqrt{n}$

This is the square root of $n$ law, and it has profound practical consequences:

• Errors grow with $\sqrt{n}$, not with $n$. Doubling the number of setups does not double the error -- it increases it by a factor of $\sqrt{2} \approx 1.41$.
• A 10-setup level run has $\sqrt{10} \approx 3.16$ times the error of a single setup, not 10 times.
• A 100-meter distance measured with 10 tape lengths of 10 m each has $\sqrt{10}$ times the error of one tape length.

"The standard deviation of a sum of $n$ independently observed quantities, each having the same standard deviation $\sigma$, is $\sigma\sqrt{n}$." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 3, p. 56

This law is the basis for leveling standards. For example, if the standard deviation per setup is $\pm 1$ mm and a circuit involves 16 setups:

$\sigma_{circuit} = 1 \times \sqrt{16} = \pm 4 \text{ mm}$

Leveling specifications are often stated in the form $\sigma = c\sqrt{K}$ where $K$ is the one-way distance in kilometers and $c$ is a constant that defines the order of accuracy. The square-root relationship appears directly from propagation theory applied to the series of setups along the route.

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Correlated Measurements#

The general law presented above assumes that all measured quantities are independent. When measurements are correlated -- meaning that an error in one tends to accompany an error in another -- the formula must be extended to include covariance terms:

$\sigma_f^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_j} \, \sigma_{x_i x_j}$

where $\sigma_{x_i x_j}$ is the covariance between $x_i$ and $x_j$. When $i = j$, $\sigma_{x_i x_j} = \sigma_{x_i}^2$ (the variance), and the formula reduces to the standard form. When $i \neq j$, the cross terms can either increase or decrease the total variance depending on the sign of the correlation.

In matrix notation, this is expressed compactly as:

$\sigma_f^2 = \mathbf{J} \, \boldsymbol{\Sigma} \, \mathbf{J}^T$

where $\mathbf{J}$ is the Jacobian matrix of partial derivatives and $\boldsymbol{\Sigma}$ is the variance-covariance matrix of the measurements.

In many surveying operations, measurements can reasonably be treated as independent:

• Separate instrument setups with independent centering, leveling, and reading errors
• Rod readings at different stations
• GPS baselines observed in separate sessions

However, some common situations introduce correlation:

• Instrument calibration: All distances measured with the same EDM share a systematic scale error. An EDM that reads 2 ppm long makes every distance long by the same proportion.
• Atmospheric conditions: All measurements taken during the same session share the same refraction and atmospheric correction errors.
• Datum and control: All coordinates referenced to the same control point share the error in that point's position.
• Computed quantities: Coordinates derived from the same traverse are correlated because they share common angle and distance measurements.

"When observations are correlated, the covariances between all pairs of observations must be included in the error propagation formula." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 3, p. 62

For rigorous work -- particularly GPS network adjustments and high-order control surveys -- the full variance-covariance matrix must be propagated. For routine boundary and topographic surveys, treating measurements as independent is usually an acceptable and conservative simplification (it tends to overestimate the total error slightly).

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Design of Surveys#

One of the most powerful applications of error propagation is working the problem in reverse. Instead of asking "given these measurement errors, what is the error in my result?" the surveyor asks: "given a required accuracy for my result, what measurement accuracy do I need?"

This is the essence of survey design. Consider a practical example:

Problem: A rectangular parcel must have its area determined to within $\pm 50$ sq ft (at the 95% confidence level). The parcel is approximately $400 \times 250$ ft. What distance measurement accuracy is needed?

Solution: The 95% confidence level corresponds to approximately $2\sigma$, so we need $\sigma_A \leq 25$ sq ft. Using the area propagation formula and assuming equal precision in both directions ($\sigma_L = \sigma_W = \sigma_d$):

$\sigma_A = A \sqrt{\left(\frac{\sigma_d}{L}\right)^2 + \left(\frac{\sigma_d}{W}\right)^2} = 100{,}000 \sqrt{\frac{\sigma_d^2}{400^2} + \frac{\sigma_d^2}{250^2}}$

Setting $\sigma_A = 25$ sq ft and solving for $\sigma_d$:

$25 = 100{,}000 \cdot \sigma_d \sqrt{\frac{1}{160{,}000} + \frac{1}{62{,}500}}$

$25 = 100{,}000 \cdot \sigma_d \cdot \sqrt{0.00000625 + 0.000016} = 100{,}000 \cdot \sigma_d \cdot 0.004744$

$\sigma_d = \frac{25}{474.4} = \pm 0.053 \text{ ft}$

The surveyor now knows that distance measurements accurate to approximately $\pm 0.05$ ft will meet the specification -- well within the capability of an EDM or a properly used steel tape, but beyond the capability of stadia or GPS with poor PDOP.

This same approach applies to:

• Determining the required angular precision for a traverse to meet a closure specification
• Specifying leveling procedures (number of setups, rod reading precision) to achieve a target elevation accuracy
• Choosing between measurement methods (total station vs. GPS vs. tape) based on the accuracy they deliver relative to project requirements

"The concept of designing surveys so that specified tolerances will not be exceeded is an important application of error propagation." -- Ghilani & Wolf, Elementary Surveying (13th Ed.), Ch. 3, p. 63

By running propagation calculations during the planning phase, a surveyor avoids two costly mistakes: over-measuring (wasting time and money on precision that is not needed) and under-measuring (discovering in the office that the work does not meet standards and must be repeated).

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Key Takeaways#

1. The general law of propagation of variances is the foundational tool. Every special case -- sum, product, constant multiplier -- derives from it through partial differentiation.

2. Errors add in quadrature, not linearly. The combined error from independent sources is always less than the arithmetic sum of the individual errors.

3. Subtraction does not cancel errors. When computing a difference between two quantities (elevation difference, coordinate difference), the errors combine the same way as in addition.

4. Series measurements grow with $\sqrt{n}$. Ten tape lengths do not produce ten times the error of one tape length -- they produce $\sqrt{10} \approx 3.16$ times the error.

5. Sensitivity coefficients matter. The partial derivatives tell you which measurements have the greatest impact on the final result. Focus your precision efforts there.

6. Propagation works in reverse for survey design. Start with the required output accuracy and solve for the input measurement accuracy. This is how professionals design measurement procedures before going to the field.

7. Independence is usually a safe assumption for routine surveying, but correlated errors (common instrument, common atmosphere, common control) must be considered in rigorous adjustments.

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References#

• Ghilani, C. D., & Wolf, P. R. (2012). Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson. Chapters 3 and 4.
• Mikhail, E. M., & Gracie, G. (1981). Analysis and Adjustment of Survey Measurements. Van Nostrand Reinhold.
• Leick, A., Rapoport, L., & Tatarnikov, D. (2015). GPS Satellite Surveying (4th Ed.). Wiley. Chapter 2.
• FGCS (1984). Standards and Specifications for Geodetic Control Networks. Federal Geodetic Control Subcommittee.