Adjustment of Observations in Geodesy: Concept of Observation, Error and Uncertainty

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.

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