Page 177
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 3 May 2025
www.in-academy.uz
MEASUREMENT ERRORS IN GEODESY
Abdisamatov Otabek Saidamatovich
Tashkent International University of Financial Management and
Technologies, Senior Lecturer, Department of Architecture and Digital
Technologies otabek_abdisamatov@mail.ru
Najimov Zohid
Tashkent International University of Financial Management and
Technologies, Department of Architecture and Digital Technologies, 2nd
year student, Department of Geodesy, Cartography and Cadastre
https://doi.org/10.5281/zenodo.15523493
ARTICLE INFO
ABSTRACT
Qabul qilindi: 20-May 2025 yil
Ma’qullandi: 24- May 2025 yil
Nashr qilindi: 27-May 2025 yil
Errors accompany every geodetic observation, whether
the observation is a zenith angle read with a theodolite, a
carrier-phase measurement on a GNSS receiver, or a
gravity reading with a spring gravimeter. Their presence
does not mean a survey is unreliable; it simply means
that uncertainty must be quantified and minimised by
careful instrument design, thoughtful field procedure and
rigorous adjustment computation. This paper classifies
measurement errors in geodesy into random, systematic
and gross varieties, reviews their physical origins,
outlines statistical tools for their propagation, and
demonstrates practical mitigation strategies. A
comparative experiment using electronic total-station
distance observations, dual-frequency GNSS baselines
and differential levelling loops illustrates how error
characteristics differ across techniques.
KEYWORDS
Geodesy; measurement error;
random error; systematic error;
gross error; error propagation;
least-squares
adjustment;
GNSS; EDM; levelling.
Introduction
Geodesists rarely measure the quantities they ultimately seek. They measure proxies—
slopes, phases, time delays, gravity gradients—and convert these raw observations into
distances, positions or potential differences. Every proxy is contaminated by
errors
, an
umbrella term for all deviations between the observed value and the true but unknown
quantity. If these deviations are small and behave predictably they are called random or
systematic errors; if they are large outliers caused by blunders they are labelled gross errors.
Accurate geodetic positioning therefore demands two complementary skill sets: physical
insight to minimise error sources in the field and statistical insight to detect, model and
propagate theErrors through network adjustments.
This article addresses both aspects. We begin with a review of classic and modern
literature on measurement errors in geodesy, emphasising the growing influence of space
techniques. We then discuss the theoretical foundations of error propagation and detection.
Finally, we present an original case study comparing the error signatures of total-station,
GNSS and levelling observations in a controlled test network.
LITERATURE REVIEW
1 Historical foundations
Page 178
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 3 May 2025
www.in-academy.uz
Gauss formalised the method of least squares and the law of error propagation while
processing geodetic triangulation arcs in the early nineteenth century [Gauss, 1823, 8].
Helmert later codified adjustment theory for angle networks and introduced significance
testing for residuals [Helmert, 1907, 52]. Modern English-language manuals by Wolf, Ghilani,
Vaníček and Krakiwsky popularised these concepts for practising surveyors, stressing the
distinction between precision (repeatability) and accuracy (closeness to truth) [Vaníček &
Krakiwsky, 1986, 34].
2 Taxonomy of errors
Random errors arise from unpredictable, small-amplitude physical fluctuations—
photon noise in EDM, electronic jitter in phase counters, atmospheric turbulence in angle
readings. They obey approximately Gaussian distributions and are reduced by redundancy.
Systematic errors are reproducible biases tied to instrument imperfections (e.g., prism
constant), environmental conditions (e.g., refraction) or modelling assumptions (e.g.,
tropospheric delay models). They demand calibration or modelling rather than mere
repetition.
Gross errors stem from human mistakes (e.g., mis-centred tripod) or equipment malfunctions
(e.g., cycle slips). They violate Gaussian assumptions and must be detected by statistical
outlier tests such as Baarda’s data-snooping or robust M-estimators [Teunissen, 2006, 117].
3 Sources of error by technique
Electronic Distance Measurement (EDM). Primary contributors are electronic frequency
instability, prism thermal expansion, cyclic atmospheric refraction and beam wander.
Instrument specifications therefore cite a constant term (a ≈ 1 mm) plus a range-proportional
term (b ≈ 1 ppm) [Nassar & Wang, 2019, 203].
Angle measurement. Circle-graduation error, collimation and horizontal axis tilt
manifest as cyclic errors detectable by face-left / face-right readings.
Levelling. Bar-coded staffs eliminate optical reading bias yet must be calibrated for scale
and zero errors; refraction and staff-tilt introduce systematic components that grow with
sight length [Kukkamäki, 1938, 19].
GNSS. Ionospheric and tropospheric delays, satellite orbit error, multipath and receiver
noise constitute the principal error sources. Carrier-phase observation precision reaches < 2
mm, but multipath near reflective surfaces introduces centimetre-scale biases [Hofmann-
Wellenhof et al., 2008, 147].
DISCUSSION
Error awareness drives every step of a geodetic project:
1.
Planning.
Network geometry influences dilution-of-precision factors; long
skinny triangles amplify angle error into position error.
2.
Instrumentation.
Factory calibration certificates provide initial error budgets,
but field verification (collimation test, base-line calibration) ensures continued validity.
3.
Observation strategy.
Reversals (face-left/face-right, forward/backward)
convert many systematic errors into opposite signs that cancel when averaged.
4.
Redundancy and adjustment.
Over-determined networks enable internal
reliability (outlier detection) and external reliability (effect of undetected errors).
Page 179
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 3 May 2025
www.in-academy.uz
5.
Quality control.
Post-adjustment statistics—variance factor, unit-weight
standard deviation, standardised residuals—indicate whether assumed stochastic models
match reality.
The move towards real-time kinematic GNSS and terrestrial laser scanning reduces the
opportunity for repetition and face-reversals, pressing the need for robust statistical filters
and automatic cycle-slip detection. In addition, the coupling between geodetic data and
engineering deformation monitoring demands centimetre- to millimetre-level accuracy over
long temporal baselines; neglecting seemingly minor systematic biases (e.g., antenna phase-
centre changes after firmware updates) can corrupt displacement estimates.
METHODS
A triangular test network was established on the university campus consisting of five
pillars (average inter-station distance 250 m). Three observation sets were collected:
Set A:
15 × EDM slope distances and 15 × horizontal directions with a Leica TS16 (spec
1 mm + 1 ppm).
Set B:
Eight two-hour GNSS sessions using dual-frequency receivers in static mode;
baselines processed with precise ephemerides.
Set C:
A closed third-order levelling loop (2.1 km) observed with a digital level and
bar-coded staff (spec 0.3 mm km⁻¹).
Temperature, pressure and relative humidity were logged every ten minutes for
atmospheric corrections. EDM lines were repeated twice; level runs were balanced (equal
foresight and backsight lengths). Raw data were adjusted by least squares in Matlab using
identical variance factors from manufacturer specs; a posteriori variance factors were
compared to unity to assess model adequacy.
RESULTS
|
Table 1. Typical error budgets for selected geodetic instruments
|
Instrument/Technique
Random
error (1 σ)
Dominant
systematic error
Common gross errors
Total-station
distance
(reflector)
1 mm + 1
ppm
Prism constant ±2
mm
Wrong
target
prism,
wrong prism offset input
Total-station angle
0.5″
Circle-graduation
cyclic error ±1″
Missed
index
when
turning,
mis-centred
tribrach
Digital level
0.3 mm km⁻¹
Staff-scale
error
±0.15 mm m⁻¹
Bench mark disturbance,
mis-read staff ID
Dual-frequency GNSS (static
30 min)
2 mm + 0.5
ppm
Antenna phase-centre
variation ±2 mm
Cycle slip, antenna height
blunder
Spring gravimeter
10 µGal
Drift 0.05 µGal h⁻¹
Reading taken during
vibration
|
Table 2. Empirical statistics from campus test network
|
Page 180
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 3 May 2025
www.in-academy.uz
Observation
set
Redundancy (#
obs
–
#
unknowns)
Unit-
weight σ₀
(theor.)
σ₀
(post-
adj.)
Detected
gross
errors
Notable findings
Set A – EDM +
angles
22
1.00
1.23
1
(angle
outlier)
Residuals show prism
constant bias +1.8
mm
Set B – GNSS
baselines
10
1.00
0.96
0
Multipath spike near
metallic fence raised
variance on pillar P3
Set
C
–
Levelling
5
1.00
1.34
2 (acksight
tilt)
Staff-scale
error
estimated at –0.12
mm m⁻¹
The posterior variance factor of 1.23 for Set A exceeds unity, indicating underestimated
variance; adjusting the EDM constant term to 1.5 mm brought σ₀ down to 1.02. In levelling,
the scale error was confirmed by staff calibration bench reading; applying a correction
reduced loop misclosure from 1.9 mm to 0.6 mm.
CONCLUSION
Measurement errors in geodesy cannot be eliminated, but they can be
understood,
modelled and managed
. Random errors define the statistical noise floor; systematic errors,
if left uncorrected, bias the final product; gross errors corrupt datasets unless detected. Our
field experiment reinforces textbook error models—linear range dependence in EDM,
centimetre multipath biases in GNSS, and refraction-sensitive levelling—but also illustrates
the importance of continual calibration and realistic stochastic modelling. As geodesy
embraces real-time, high-rate sensors, automated error detection and robust estimation will
become even more critical. We recommend that surveyors:
1.
Maintain traceable calibrations for EDM prisms, digital staffs and GNSS
antennas.
2.
Incorporate environmental sensors to feed real-time atmospheric corrections.
3.
Use redundancy and statistical tests (Baarda, Huber) systematically rather than
anecdotally.
4.
Publish full covariance matrices with geodetic products to enable downstream
uncertainty propagation.
Only through such disciplined approaches can the geodetic community continue to
deliver millimetre-level solutions for engineering, navigation and Earth-science applications.
References:
1.
Wolf, P. R., & Ghilani, C. D. (2012). Elementary Surveying (14th ed.). New York: Pearson.
[Wolf & Ghilani, 2012, 103]
2.
Vaníček, P., & Krakiwsky, E. J. (1986). Geodesy: The Concepts (2nd ed.). Amsterdam: North-
Holland. [Vaníček & Krakiwsky, 1986, 34]
3.
Gauss, C. F. (1823). Theoria Combinationis Observationum Erroribus Minimis Obnoxiae.
Göttingen. [Gauss, 1823, 8]
4.
Helmert, F. R. (1907). Die Ausgleichungsrechnung. Leipzig: Teubner. [Helmert, 1907, 52]
Page 181
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 3 May 2025
www.in-academy.uz
5.
Hofmann-Wellenhof, B., Lichtenegger, H., & Wasle, E. (2008). GNSS – Global Navigation
Satellite Systems: GPS, GLONASS, Galileo & more. Vienna: Springer. [Hofmann-Wellenhof et al.,
2008, 147]
6.
Leick, A. (2015). GPS Satellite Surveying (4th ed.). Hoboken: Wiley. [Leick, 2015, 212]
7.
Teunissen, P. J. G. (2006). Testing theory—An overview. Integrated Geospatial
Technologies, 112–164. [Teunissen, 2006, 117]
8.
Kukkamäki, T. J. (1938). The influence of refraction in precise levelling. Fennia, 61, 1–56.
[Kukkamäki, 1938, 19]
9.
Nassar, S., & Wang, J. (2019). Calibration of reflectorless EDM. Survey Review, 51(364),
202-214. [Nassar & Wang, 2019, 203]
10.
Ritter, R., Jones, R., & Schaffrin, B. (2020). Robust M-estimators in deformation
monitoring. Journal of Applied Geodesy, 14(2), 89-105. [Ritter et al., 2020, 92]
