Adjustment of Observations in Geodesy: Redundancy and Degrees of Freedom.

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.

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Lesson 05 – Redundancy and Degrees of Freedom

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Objectives

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

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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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