5.1 Theory of Errors, Precision vs. Accuracy, Random/Systematic Errors, and Error Propagation

Theory of Errors, Precision vs. Accuracy, Random/Systematic Errors, and Error Propagation

Every physical measurement in geodetic engineering is subject to imperfections. In geodetic surveying, no observed quantity—whether a horizontal angle, zenith angle, slope distance, or elevation difference—is exact. Understanding the fundamental nature of measurement errors, classifying their sources, quantifying precision and accuracy, and modeling how errors propagate through mathematical formulas are essential skills for passing the Geodetic Engineer Licensure Examination (GELE) and practicing professional surveying.

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1. Sources and Classification of Measurement Errors

In geodetic surveying, measurement errors originate from three primary sources:

1. Instrumental Sources: Imperfections in construction, adjustment, or calibration of instruments (e.g., optical axis misalignment, index error of a vertical circle, tape length variation from nominal standard).
2. Natural Sources: Environmental variations such as atmospheric temperature, pressure, humidity, wind, solar radiation, and atmospheric refraction.
3. Personal Sources: Limitations of human sensory perception, sight alignment, reading scales, setting up targets over stations, and reaction time.

Measurements errors are rigorously classified into three distinct categories:

Blunders (Gross Errors)

Blunders are not true statistical errors; they are human mistakes such as misreading a leveling rod by $1.0\text{ m}$, transposing digits when recording ($124.35\text{ m}$ written as $142.35\text{ m}$), targeting the wrong prism, or setting up over an incorrect monument. Blunders must be detected and removed before statistical adjustment. In modern processing, observations with residuals exceeding three standard deviations ($|v_i| > 3\sigma$) are rejected as suspected blunders.

Systematic Errors (Biases)

Systematic errors follow strict physical or mechanical relationships. If measuring conditions remain constant, systematic errors accumulate in direct proportion to the number of measurements or line length (hence termed cumulative errors). For example, a steel tape that is $0.008\text{ m}$ too long will introduce a systematic error of $-0.008\text{ m}$ per tape length measured. Key systematic error corrections include:

• Temperature Correction: $C_t = \alpha L (T - T_0)$
• Pull/Tension Correction: $C_p = \frac{(P - P_0) L}{A E}$
• Sag Correction: $C_s = -\frac{w^2 L^3}{24 P^2}$
• Curvature and Refraction in Leveling: $h_{cr} = 0.0675 , K^2$ (where $K$ is distance in km)

Because systematic errors carry consistent algebraic signs, taking the mean of multiple observations under identical conditions will not eliminate them. They must be eliminated analytically or by field procedures (such as taking Face Left and Face Right total station sightings to eliminate vertical index and horizontal collimation errors).

Random Errors (Accidental Errors)

After all blunders are eliminated and systematic errors are corrected, small discrepancies persist. These are random errors caused by micro-environmental fluctuations, tripods setting, and sensory limitations. Random errors obey the laws of probability and the Normal (Gaussian) Distribution Curve:

• Small errors occur more frequently than large errors.
• Positive and negative errors of equal magnitude are equally probable.
• Extremely large errors rarely occur and are classified as blunders.

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2. Precision versus Accuracy

In geodetic science, precision and accuracy have distinct, non-interchangeable mathematical definitions:

• Precision: Refers to the degree of refinement and repeatability among a set of independent measurements of the same quantity under identical conditions. It indicates closeness of observations to each other and is quantified by sample standard deviation ($s$) or variance ($s^2$).
• Accuracy: Refers to the degree of conformity or closeness of a measured or adjusted value to the true value (or accepted reference standard). It reflects total freedom from both systematic errors and blunders.

       High Accuracy                    Low Accuracy
     High Precision                  High Precision
       (Ideal)                    (Systematic Bias Present)
         .                              .
      .  *  .                        .     .
     .  ***  .                      .  ***  .
        .                              .
   Target Center                  Target Center

       Low Accuracy                    High Accuracy
      Low Precision                   Low Precision
    (High Random Dispersion)       (Unbiased but Noisy)
       .   *                            .
     *   .   *                       *  .  *
       .   *                           . *
   Target Center                  Target Center

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3. Statistical Measures of Precision and Central Tendency

When a line or angle is measured $n$ times under equal weighting conditions, the best estimate of the true value is the sample arithmetic mean ($\bar{x}$).

3.1 Arithmetic Mean

$\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$

3.2 Residuals ($v_i$)

The difference between the mean and an individual observation $x_i$: $v_i = x_i - \bar{x}$ A fundamental algebraic property of the arithmetic mean is that the sum of residuals is identically zero: $\sum v_i = 0$.

3.3 Sample Variance ($s^2$) and Standard Deviation ($s$)

The standard deviation measures the dispersion of single observations about the sample mean: $s^2 = \frac{\sum_{i=1}^n v_i^2}{n - 1}, \quad s = \sqrt{\frac{\sum_{i=1}^n v_i^2}{n - 1}}$ where $(n - 1)$ represents the degrees of freedom.

3.4 Standard Error of the Mean ($SE_{\bar{x}}$ or $\sigma_{\bar{x}}$)

Because the mean $\bar{x}$ is computed from $n$ observations, it is more precise than any single observation. Its standard error is given by: $SE_{\bar{x}} = \frac{s}{\sqrt{n}} = \sqrt{\frac{\sum v_i^2}{n(n - 1)}}$

3.5 Probable Error ($PE$) and Probability Limits

Probable error defines a symmetrical interval $\pm PE$ within which $50%$ of the observations (or means) fall: $PE = 0.6745 \cdot s \quad (\text{for a single observation})$ $PE_{\bar{x}} = 0.6745 \cdot SE_{\bar{x}} \quad (\text{for the mean})$

Standard confidence intervals for a normal distribution:

• $100(1 - \alpha) = 68.27%$ confidence limit: $\bar{x} \pm 1.000 s$
• $90.00%$ confidence limit: $\bar{x} \pm 1.6449 s$
• $95.00%$ confidence limit: $\bar{x} \pm 1.9599 s \approx \bar{x} \pm 1.96 s$
• $99.73%$ confidence limit: $\bar{x} \pm 3.000 s$

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Worked Numerical Example 1: Statistical Analysis of Distance Observations

A baseline was measured 6 times with an electronic distance meter (EDM) under identical conditions, producing the following distance readings: $x_1 = 345.820\text{ m}, x_2 = 345.826\text{ m}, x_3 = 345.814\text{ m}, x_4 = 345.822\text{ m}, x_5 = 345.818\text{ m}, x_6 = 345.820\text{ m}$.

Step 1: Compute Arithmetic Mean ($\bar{x}$) $\bar{x} = \frac{345.820 + 345.826 + 345.814 + 345.822 + 345.818 + 345.820}{6} = \frac{2074.920}{6} = 345.820\text{ m}$

Step 2: Compute Residuals ($v_i = x_i - \bar{x}$) and $v_i^2$

• $v_1 = 345.820 - 345.820 = 0.000\text{ m} \implies v_1^2 = 0.000000$
• $v_2 = 345.826 - 345.820 = +0.006\text{ m} \implies v_2^2 = 0.000036$
• $v_3 = 345.814 - 345.820 = -0.006\text{ m} \implies v_3^2 = 0.000036$
• $v_4 = 345.822 - 345.820 = +0.002\text{ m} \implies v_4^2 = 0.000004$
• $v_5 = 345.818 - 345.820 = -0.002\text{ m} \implies v_5^2 = 0.000004$
• $v_6 = 345.820 - 345.820 = 0.000\text{ m} \implies v_6^2 = 0.000000$ $\sum v_i = 0.000\text{ m}, \quad \sum v_i^2 = 0.000080\text{ m}^2$

Step 3: Compute Standard Deviation ($s$) $s = \sqrt{\frac{\sum v_i^2}{n - 1}} = \sqrt{\frac{0.000080}{6 - 1}} = \sqrt{0.000016} = \pm 0.004\text{ m} = \pm 4.0\text{ mm}$

Step 4: Compute Standard Error of the Mean ($SE_{\bar{x}}$) $SE_{\bar{x}} = \frac{s}{\sqrt{n}} = \frac{0.004}{\sqrt{6}} = \frac{0.004}{2.44949} = \pm 0.001633\text{ m} \approx \pm 1.6\text{ mm}$

Step 5: Compute Probable Error of the Mean ($PE_{\bar{x}}$) $PE_{\bar{x}} = 0.6745 \cdot SE_{\bar{x}} = 0.6745 \cdot 0.001633 = \pm 0.00110\text{ m} \approx \pm 1.1\text{ mm}$

Final Result Expression: The most probable distance is $345.8200\text{ m} \pm 0.0016\text{ m}$ (at $68.3%$ confidence level) or $345.8200\text{ m} \pm 0.0032\text{ m}$ (at $95%$ confidence level).

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4. General Law of Error Propagation

When a calculated quantity $Z$ is not measured directly but is evaluated from a function of one or more measured variables $Z = f(x_1, x_2, \dots, x_n)$, the random errors of the independent variables propagate into $Z$.

4.1 General Multivariate Error Propagation Formula

For any non-linear function $Z = f(x_1, x_2, \dots, x_n)$ where $x_i$ are independent, uncorrelated measurements with standard errors $\sigma_{x_i}$, the variance $\sigma_Z^2$ is determined using a first-order Taylor series expansion: $\sigma_Z^2 = \left(\frac{\partial f}{\partial x_1}\right)^2 \sigma_{x_1}^2 + \left(\frac{\partial f}{\partial x_2}\right)^2 \sigma_{x_2}^2 + \dots + \left(\frac{\partial f}{\partial x_n}\right)^2 \sigma_{x_n}^2$ $\sigma_Z = \sqrt{ \sum_{i=1}^n \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_{x_i}^2 }$

If variables are correlated, the covariance term $2 \sum \sum \left(\frac{\partial f}{\partial x_i}\right) \left(\frac{\partial f}{\partial x_j}\right) \sigma_{x_i x_j}$ must be added inside the square root.

4.2 Special Cases of Error Propagation

1. Sum or Difference ($Z = a A + b B - c C$): $\sigma_Z = \sqrt{ a^2 \sigma_A^2 + b^2 \sigma_B^2 + c^2 \sigma_C^2 }$

2. Series of $n$ Identical Measurements ($L_{total} = l_1 + l_2 + \dots + l_n$ with standard error $\sigma_l$): $\sigma_{total} = \sqrt{ \sigma_l^2 + \sigma_l^2 + \dots + \sigma_l^2 } = \sqrt{n} \cdot \sigma_l$ (Note: The total error grows with the square root of line segments, not linearly!)

3. Product / Area ($Z = A \cdot B$): $\frac{\partial Z}{\partial A} = B, \quad \frac{\partial Z}{\partial B} = A \implies \sigma_Z = \sqrt{ B^2 \sigma_A^2 + A^2 \sigma_B^2 }$ Dividing by $Z^2 = A^2 B^2$ yields the relative error relationship: $\left(\frac{\sigma_Z}{Z}\right)^2 = \left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2$

4. Product with Constant ($Z = k \cdot A$): $\sigma_Z = |k| \cdot \sigma_A$

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Worked Numerical Example 2: Reduction of Slope Distance to Horizontal Distance

In total station surveying, a horizontal distance $H$ is derived from slope distance $S$ and zenith angle $Z$: $H = S \cdot \sin(Z)$

Given:

• Measured slope distance $S = 520.450\text{ m}$ with standard error $\sigma_S = \pm 0.012\text{ m}$.
• Measured zenith angle $Z = 82^\circ 15' 20''$ with standard error $\sigma_Z = \pm 15''$.

Step 1: Compute Horizontal Distance ($H$) $Z = 82^\circ 15' 20'' = 82 + \frac{15}{60} + \frac{20}{3600} = 82.255556^\circ$ $H = 520.450 \cdot \sin(82.255556^\circ) = 520.450 \cdot 0.9908687 = 515.7026\text{ m}$

Step 2: Convert $\sigma_Z$ from arcseconds to radians $\sigma_Z = \frac{15''}{206,264.806''/\text{rad}} = 7.2722 \times 10^{-5}\text{ rad}$

Step 3: Evaluate Partial Derivatives $\frac{\partial H}{\partial S} = \sin(Z) = \sin(82^\circ 15' 20'') = 0.9908687$ $\frac{\partial H}{\partial Z} = S \cdot \cos(Z) = 520.450 \cdot \cos(82^\circ 15' 20'') = 520.450 \cdot 0.134830 = 70.1723\text{ m/rad}$

Step 4: Calculate Component Variance Contributions

• Contribution from distance error: $\left(\frac{\partial H}{\partial S} \sigma_S\right)^2 = (0.9908687 \cdot 0.012)^2 = (0.011890)^2 = 0.00014138\text{ m}^2$
• Contribution from zenith angle error: $\left(\frac{\partial H}{\partial Z} \sigma_Z\right)^2 = (70.1723 \cdot 7.2722 \times 10^{-5})^2 = (0.005103)^2 = 0.00002604\text{ m}^2$

Step 5: Compute Total Standard Error of Horizontal Distance (\sigma_H) $\sigma_H = \sqrt{0.00014138 + 0.00002604} = \sqrt{0.00016742} = \pm 0.01294\text{ m} \approx \pm 0.013\text{ m} = \pm 13\text{ mm}$

This demonstrates that distance error dominates the variance budget in this specific geometric setup, contributing over $84%$ of the total variance in the computed horizontal distance.