In geodetic adjustment, the reliability of results depends not only on measurement precision but also on the amount of available information. When the number of observations exceeds the number of unknowns, redundancy is introduced into the system. This redundancy allows error detection, quality control, and the application of the Least Squares Method.

Lesson 05 – Redundancy and Degrees of Freedom

Objectives

Understand the concept of redundancy in geodetic observations.
Define degrees of freedom.
Interpret the relationship between observations and unknowns.
Recognize the importance of redundancy for adjustment and quality control.
Apply the concept in simple practical situations.

1. Observations and Unknowns

In a geodetic problem:

n = number of observations
u = number of unknown parameters

Three situations may occur:

Case	Condition	Interpretation
Underdetermined	n < u	Not enough information
Determined	n = u	Unique solution, no redundancy
Redundant	n > u	Extra information available

In practice, geodetic networks are designed so that:

2. Concept of Redundancy

Redundancy represents the excess of observations relative to the number of unknowns:

Where:

r = redundancy (degrees of freedom)

This extra information allows:

detection of gross errors
reliability assessment
statistical testing
improved precision through adjustment

3. Degrees of Freedom

Degrees of freedom indicate how many independent residuals remain after adjustment.

Interpretation:

Higher r → better reliability and control
r = 0 → no redundancy, no statistical control

In Least Squares Adjustment, degrees of freedom are essential for:

variance estimation
quality tests
reliability analysis

4. Example of Redundancy

A distance is measured four times to determine one unknown value.

The system has three degrees of freedom, allowing statistical evaluation of the measurements.

5. Importance in Geodesy

Redundancy is fundamental in:

geodetic networks
leveling lines
traverse adjustment
GNSS processing

Without redundancy:

gross errors cannot be detected
precision cannot be evaluated
Least Squares cannot estimate variance

For this reason, redundancy is intentionally introduced during network design.

6. Solved Example

A leveling section includes:

6 height difference observations
2 unknown elevations

6.1 Interpretation:

The system is redundant
Four degrees of freedom are available for statistical analysis.

7. Proposed Exercise

A geodetic problem contains:

8 observations
3 unknown parameters

Determine:

a) Whether the system is redundant

b) The degrees of freedom

7.1 Answer

The system is redundant with five degrees of freedom.

8. Conclusion

Redundancy and degrees of freedom quantify the amount of extra information available in a geodetic adjustment. Systems with ( n > u ) allow error detection, precision evaluation, and reliable application of the Least Squares Method.

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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.