This free course on Adjustment of Observations and Least Squares is designed for students and professionals in Geodesy, Surveying, and Geomatics. Each lesson includes theoretical explanations, solved examples, and practical exercises.

Lesson 02 – Types of Errors in Geodetic Observations

Objectives

Understand the different types of errors present in geodetic measurements.
Distinguish between systematic, random, and gross errors.
Identify the origin and behavior of each type of error.
Recognize the importance of error classification in the adjustment process.

1. Introduction

Every geodetic observation contains errors. Since the true value of a measured quantity is unknown, it is essential to understand the nature of these errors in order to control, model, or minimize their effects.

Errors in geodetic measurements are generally classified into three main categories:

Systematic errors.
Random errors.
Gross errors.

This classification is fundamental for the proper application of the Least Squares Method.

2. Systematic Errors

Systematic errors follow a predictable pattern and affect measurements in a consistent way.

2.1 Characteristics

Same sign and similar magnitude under the same conditions.
Caused by identifiable physical or instrumental factors.
Can be modeled and corrected.

2.2 Examples

Instrument calibration errors.
Temperature effects on distance measurements.
Atmospheric refraction.
Scale factor errors.
Incorrect prism constant.

2.3 Treatment

Systematic errors should be:

eliminated through calibration, or
modeled mathematically in the functional model.

If not treated, they affect the accuracy of the results.

3. Random Errors

Random errors are small variations caused by unpredictable factors.

3.1 Characteristics

Irregular in magnitude and sign.
Follow a normal (Gaussian) distribution.
Mean value approximately equal to zero.
Cannot be eliminated individually.

3.2 Sources

Instrument noise.
Environmental variations.
Operator limitations.
Small atmospheric fluctuations.

3.3 Treatment

Random errors are reduced by:

Repeated observations.
Redundancy.
Least Squares Adjustment.

They affect the precision of the measurements.

4. Gross Errors

Gross errors are large mistakes caused by human or operational failures.

4.1 Examples

Reading the wrong value.
Data entry mistakes.
Instrument misleveling.
Loss of GNSS signal.
Measuring the wrong point.

4.2 Characteristics

Much larger than random errors
Do not follow statistical behavior
Cannot be corrected mathematically

4.3 Treatment

Gross errors must be:

detected, and
removed before or during adjustment.

They are usually identified through:

residual analysis
statistical tests
consistency checks.

5. Importance for Adjustment of Observations

The Least Squares Method assumes that:

systematic errors have been removed or modeled
gross errors are absent
remaining errors are random and normally distributed

If gross or systematic errors remain, the adjustment results may be unreliable.

6. Solved Example

A distance was measured five times (m): 152.334; 152.331; 152.336; 152.333; 152.335.

The values show small variations around the mean and no extreme values.

6.1 Interpretation:

Errors are small and irregular.
Behavior is consistent with random errors.
Data are suitable for adjustment.

7. Proposed Example

Angle observations (seconds): 30.124; 30.118; 30.121; 30.950

7.1 Interpretation:

The value 30.950 is significantly different from the others and should be considered a gross error and investigated before adjustment.

8. Conclusion

Geodetic observations are affected by systematic, random, and gross errors. Correct identification and treatment of these errors are essential to ensure reliable and accurate results in the adjustment process.

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Lesson 01 – Concept of Observation, Error and Uncertainty

This lesson introduces the fundamental concepts required to understand the adjustment of observations in Geodesy. Every measurement performed in the field contains errors and uncertainties. Understanding their nature is essential before applying the Least Squares Method.

1. What is an Observation?

In Geodesy, an observation is the numerical result of a physical measurement obtained using an instrument such as a GNSS receiver, total station, or level. Observations may represent distances, angles, height differences, or coordinates.

Mathematically, an observation can be expressed as:

L = L_true + e

where L is the observed value, L_true is the true value (unknown), and e is the observation error.

2. Types of Errors

2.1 Systematic Errors

Systematic errors follow a predictable pattern and usually have identifiable causes. Examples include instrument calibration errors, atmospheric effects, or incorrect scale factors. These errors affect accuracy and must be modeled or corrected.

2.2 Random Errors

Random errors are unpredictable variations caused by environmental conditions, instrument noise, or observational limitations. They follow a normal distribution and have zero mean. The Least Squares Method is designed to minimize their effect.

2.3 Gross Errors

Gross errors are large mistakes caused by human or operational failures, such as incorrect readings, data entry errors, or loss of signal. These errors must be detected and removed before adjustment.

3. Precision, Accuracy and Uncertainty

Precision describes the repeatability of measurements and is related to the dispersion of the observed values. Accuracy refers to the closeness of the observations to the true value. Uncertainty quantifies the level of confidence associated with a measurement result.

It is possible to have high precision and low accuracy when systematic errors are present.

4. Importance in Geodetic Adjustment

Because the true value is unknown, the goal of adjustment is to determine the most probable value of the measured quantity. This is achieved by combining redundant observations and minimizing the effect of random errors.

5. Solved Example

A distance between two geodetic points was measured four times, producing the following values in meters:

158,327 158,321 158,332 158,326

Step 1 – Mean value:

L = 158,3265 m

Step 2 – Standard deviation:

σ ≈ 0,0043 m

Step 3 – Standard error of the mean:

m = σ / √n = 0,0021 m

Final result:

L = 158,3265 ± 0,0021 m

6. Proposed Exercise

The following distance measurements in meters were obtained:

158,406 158,403 158,410 158,405 158,402

Calculate:

a) The mean value
b) The standard deviation
c) The standard error of the mean

Expected final result:

L = 158,4052 ± 0,0030 m

7. Conclusion

All geodetic observations contain errors. Understanding the nature of systematic, random, and gross errors is essential for reliable data processing. Redundant measurements allow the estimation of the most probable value and its associated uncertainty, forming the foundation of the Least Squares Adjustment.