1

Error Analysis Tool for Quality Control in Cadastral Practice in Nigeria

By

Hart, Lawrence, Opuaji, T.A, Basil, D.D

Department of Surveying and Geomatics, Rivers State University of Science and

Technology, Port Harcourt, Nigeria

Abstract

Considering the increasing demand for cadastral surveying services and the pressure of

quacks and non-surveyors encroaching into our professional space, with commitment and

advocacy by the relevant stakeholders in the profession, we can create a formidable niche

and change the perception of the public and government through this quality control

mechanism in the practice.The inability of practioners to embrace quality controls in the

production of survey plans and associated spatial data of any property in the past and at

present has aggravated the situation. Therefore, the need to have geodetic perspective in

boundary determination especially in property surveys underscores the importance of error

analysis in all cadastral surveys. This work will highlight the need to show error propagation

pattern in the measurement, compute the error ellipse, indicate reliability of values of

coordinates so obtained (variance-covariance) and produce a concise quality assurance report

for each property survey carried out. This paper will use relevant mathematical models for

error propagation and method of correlates of least square adjustment as a quality tool for

specimen property surveys in parts of Port Harcourt, Rivers State. The results obtained by

this process will facilitate the quest for actual charting of plans and thereby confirm positions

of property with elimination of overlap, wrong location and inconsistent boundary.The

survey errors described must be translated into positional errors so that geoscientists can

assess the impact of those errors on their understanding of the mathematical model and

behavior of the boundary when plotted. In extreme cases, these errors, if not recognized, can

result in a misplaced boundary and the actual location will be missing completely.

(Key words: cadastral surveying, boundary, error analysis, error propagation, coordinates)

1.0 Background to the Work

The cadastral surveying practice in Nigeria has undergone several changes in terms of

instrumentation and quality of professionals. Yet the practice is plagued by the invasion of

quacks which had impacted on the quality of her product; the cadastral survey plan. The

perspective of the public toward the practice is such that anybody can be contacted to

carryout property survey, this is due to the non-entrenchment of critical quality control

mechanism that can only allow surveyors execute this important task. Although the current

laws and regulations guiding the practice of surveying have undergone a number of

amendments, there still remains a number of changes to be made. The laws guiding the

practice is still not encompassing with regards to scope and quality control. The inability of

practioners to embrace quality controls in the production of survey plans and associated

2

spatial data of any property boundaryin the past and at present has aggravated the situation.

Therefore, the need to have geodetic perspective in boundary determination especially in

property surveys underscores the importance of error analysis in all cadastral surveys.

Errors may occur in survey measurements and mapping operations for various reasons which

include; using incomplete or in accurate mathematical model, using a non-rigorous method

of adjustment, undetected systematic or gross errors in the measurement process, insufficient

observations to provide redundancy etc.

This work intends at bringing to the fore implication and dynamics of error analysis for all

survey measurements and computations with the view of generating quality geospatial data

for property boundary determination. It will further advocate the need of not just computing

coordinates for the purpose of determining the area and production of survey plans. Rather,

we have to show error propagation pattern in the measurement, compute the error ellipse

(and indicate diagrammatically), indicate reliability of values of coordinates so obtained

(variance-covariance) and produce a concise quality assurance report for each property

survey carried out. To this end, acursory analysis of all the computational approaches shall

be highlighted in this work.

1.1 Cadastral Surveys and the Surveyor; an Overview

Cadastral surveys are surveys carried out to provide detailed description of a parcel of land,

its location, extent, and value. It also serves to answer questions of the ownership of land, and

the conditions under which it is held. The (School of Military Survey, 1969) described the

properties of the cadastral surveyor as:

i. To mark on the ground with permanent beacons the limits of every land holding;

ii. To produce documentary and mathematical evidence which will describe such land

holding unambiguously and which would enable a subsequent land surveyor to re-

establish a lost boundary or mark precisely;

iii. To maintain a standard of accuracy such that, at whatever scale the initial plan is

drafted, it could stand enlargement to a scale approaching 1:1 if need be, i.e., the

survey is FINAL.

iv. To take every opportunity to apply checks to both field and office work.

v. To ensure that the work is in accordance with the requirements laid down by statute;

vi. To ensure that his work is honest and unquestionable in a court of law.

3

The product of cadastral surveys by the surveyor is the cadastral plan. This document can

fulfil many of the functions of the large scale topographic maps for not only may they serve

such purposes as boundary control, registration of title and valuation but they may also form

a basis for planning and development (Dale, 1976). In urban areas, cadastral surveys may be

used as an inventory of resources, including details of utilities and services, census

information and State Government control while in the rural areas it may be of primary

importance in land reform.

2.0 Review of Computational Process in Cadastral Survey

In surveying, we take measurements and perform calculations to arrive at some results.

However, the result themselves are only mid the expected outcome as in most cases it

requires some estimate of the accuracy of the results. Of course, the accuracy depends on

errors due to the measurement techniques and errors in the calculation such as round-off

errors. This underscore the need to carry out further analyse adopting geodetic principles as a

basis for quality control. However, a closed traverse which predominantly is the field

approach required for boundary determination for property surveys provides an insight for

this analysis. For a polygon of n sides its internal angles sum to (2n-4) x 90o and this basis

provides angular misclosure indicator which shows the difference between the rule and the

sum of measured angles. Similarly, linear misclosure gives an indicator of the length of an

assumed missing line whose east and north components are the sums of the east and north

components of the n traverse legs. Howbeit, the accuracy of the traverse is presumed to be

the misclosure ratio. This truly is misleading as this ratio does not distinguish between

random and systematic errors and their corresponding effects. Deakin, 2012; Valentine,

1984) asserts that random errors do not accumulate in direct proportion to distance.

We will not dwell on the issues of observational techniques and computational steps in

property surveys as there are several literatures; (Dashe, 1987); (School of Military Survey,

1969); (Field, 1980). This notwithstanding, the following are some of the computational

steps involved in the determination of geospatial data required for cadastral plan production

for a closed traverse.

sinL … (1)

cosL … (2)

4

N

1tan … (3)

NNN 01 … (4)

EEE 01 … (5)

Fig 1.0 Traverse Computation Sheet

Table 1.0: List of Final Coordinates, Bearing and Distances from Bowditch Adjustment

Station ID

FINAL

DISTANCES(m)

FINAL

BEARING EASTING NORTHINGS

SC/A2000 340.825 89 ˚ 25’ 10’’ 275660.067 530546.461

SC/A2001 444.846 178˚43’ 2.73’’ 276000.875 530549.914

SC/A2002 289.578 269˚ 57’ 33’’ 276010.327 530104.987

SC/A2003 70.719 246 ̊ 41’ 11’’ 275720.727 530105.987

SC/A2004 1104.079 000˚ 1’ 53’’ 275655.804 530076.954

SC/A2005 213.723 94˚ 03’ 24’’ 275656.409 531181.078

SC/A2006 375.325 160˚ 11’ 39’’ 275869.597 531165.969

SC/A2007 262.964 179˚ 06’ 20’’ 275996.77 530812.846

5

SC/A2001 375.325 160 ̊ 11’ 55’’ 276000.785 530549.914

Fig. 2.0: Specimen Plan of the Boundary Plan of Part of RSUST

The traverse computation of the area is shown in fig 1. We are aware there are traverse

computation programs, but how accurate are the outputs. We need to have a way of

validating our results. Hence computations are still relevant. Similarly, the coordinates and

final bearings and distances of the 8 stations is as shown in table 1.

3.0 Review of Mathematical Approach for Error Analysis

This approach shall make comparison of misclosure (angular and linear) with statistical

estimates that are function of the actual traverse measurement and geometry of the traverse.

There is the unlikelihood to observe a perfect linear relationship, this is as a result of

observational and computational error and the relationship of the variables may not be

exactly linear. We shall consider the least square techniques, propagation of variance and

error ellipse display.

3.1Least Square Adjustment (LSA)

The least square approach is a tool that helps you analyse and adjust the random errors in

your survey. It computes adjusted coordinate position using estimated precision of observed

coordinates to reconcile between observation and the inverse to their adjusted coordinates.

Least squares reports the statistics of the adjustment indicating strength of the computed

position.

The generalized linear mathematical model of the least square observation equation is given

by (Ayeni, 1980; Cross, 1993; Ghilani and Wolf, 2006) as:

6

V = Ax-b … (6)

aa fL (Mathematical model for Observation equation) … (6.1)

0aLF (Mathematical model for condition equation) … (6.2)

The least square solution can be basically by two methods; method of variation of parameters

and method of correlates or condition equation. Solution of the unknown parameters or the

estimate of the correction to approximate parameter vector [x] is given by (Kouba, 1974;

Uotila, 1974; Ezeigbo, 1990; Mikhail and Anderson, 1998; Ayeni, 2001; Halleck, 2001;

Marzooki et al, 2005; Nwilo et al, 2006; Ghilani and Wolf, 2006; Ojigi, 2010)

PLAPAAX TT 1

… (7)

Where PAAT the normal equation coefficient matrix which is non-singular, L is non-linear

matrix and PLAT is the normal equation constant (or absolute) term vector. We will use the

method of condition equation method for the analysis.

3.2Traverse Adjustment by Condition Equation Approach

In Ayeni, 1980 the following are basic mathematical expressions for least square analysis by

method of correlates:

WMK 1 … (8)

KBPV T1 … (9)

VLL ba … (10)

11112

0

BPMBPPL Ta … (11)

r

PVV T

2

0 … (12)

Where M is normal equation, W- misclosure vector, P- unit weight, B-design matrix, K-

vector of Lagranges multiplier, V- residual vector, La-adjusted observation,∑La- variance-

covariance of adjusted observation, σ2-aposteriori of unit weight matrix, r- number of

conditions.

In this work we used the boundary survey of part of Rivers State University of Science and

Technology to demonstrate the process of error analysis in geodetic perspective. One of the

tools will be the least square adjustment approach by the method of correlates thatrequires

the forming of conditions. Hence, we define three (3) conditions viz

7

0

0

1

TBBPM

WBM … (13)

Condition 1: Azimuth condition

01801

b

n

j

jc n … (14)

Where χc,χb- initial and closing azimuth of the traverse; β- observed angles; n- number of

observed angles.

Condition 2 & 3 are coordinate conditions given in equations 12& 13

01

m

i

bci XXx … (15)

01

bc

m

i

i YYy … (16)

Where Δxi = di sin αi

Δyi = di cos αi

m= number of traverse legs

di = measured distance i

αi= azimuth of line i

Reference to equations 8-10, we developed the design and misclosure matrices B and W, with

other associated matrices. The weight matrix P is assumed to be unity. Using the matlab

applications we developed a program for the adjustment. The specimen results of the design

matrix B, misclosure vector matrix W and residual matrix Vare shown below.

B=

0 0 0

26.9995- 25.4032- 1.9100-

0.0516- 9.1473 26.9322-

0 0 0 0 27.0000 27.0000 27.0000 27.0000 27.0000 27.0000 27.0000 27.0000

26.9973 10.6863- 0.0208- 26.9930- 0 0.0005 0.0172 0.0459 0.0459 0.0378 0.0012- 0.0001

0.0140 24.7941- 26.9973- 0.6048 0 0.0351- 0.0810- 0.0837- 0.0621 0.0594 0.0594 0.0000-

W=

0.0014

0.4280

0.2440-

8

V

0.0040

0.0048

0.0029-

0.0040-

0.0013-

0.0032-

0.0041

0.0000-

0.0000-

0.0000-

0.0000-

0.0000-

0.0000-

0.0000

0.0000-

We can see that from the adjusted observation as per equation 10, the values are as shown in

table 2.0. Similarly, the variance-covariance of the adjusted observation as shown in table 5.0

depicts the accuracy of the adjusted observables (i.e. bearing and distances) which will give

values using equations 1&2, 4&5 to obtained adjusted coordinates.

Table 2.0: Adjusted Coordinates after Least Square Adjustment

Station ID

ADJUSTED

DISTANCES(m)

FINAL

BEARING

ADJUSTED

COORDINATES

EASTING

ADJUSTED

COORDINATES

NORTHINGS

SC/A2000 340.825 89 ˚ 25’ 10’’ 275660.067 530546.461

SC/A2001 444.850 178˚43’ 2.73’’ 276000.875 530549.914

SC/A2002 289.583 269˚ 57’ 33’’ 276010.316 530104.939

SC/A2003 70.722 246 ̊ 41’ 11’’ 275720.733 530105.987

SC/A2004 1104.013 000˚ 1’ 47’’ 275655.787 530076.947

SC/A2005 213.714 94˚ 03’ 24’’ 275656.344 531180.959

SC/A2006 375.346 160˚ 11’ 39’’ 275869.523 531165.841

SC/A2007 263.003 179˚ 06’ 20’’ 275996.694 530812.695

SC/A2001 375.325 160 ̊ 11’ 55’’ 276000.785 530549.914

Table 3.0: Difference between Adjusted and Computed Distances and Bearings

Station ID

Computed

Distances (m)

Adjusted

Distances

(m)

Difference

in Distances Computed

Bearings Adjusted Bearings

Difference

in Bearings

SC/A2000 340.825 340.825 0.0000 89 ˚ 25’ 10’’ 89 ˚ 25’ 10’’ 00.”00

SC/A2001 444.846 444.850 0.0004 178˚43’ 2.73’’ 178˚43’ 2.73’’ 00.”00

SC/A2002 289.578 289.583 0.0050 269˚ 57’ 33’’ 269˚ 57’ 33’’ 00.”00

SC/A2003 70.719 70.722 0.0030 246 ̊ 41’ 11’’ 246 ̊ 41’ 11’’ 00.”00

SC/A2004 1104.079 1104.013 -0.0660 000˚ 1’ 53’’ 000˚ 1’ 47’’ -06”

SC/A2005 213.723 213.714 -0.0090 94˚ 03’ 24’’ 94˚ 03’ 24’’ 00.”00

SC/A2006 375.325 375.346 0.0210 160˚ 11’ 39’’ 160˚ 11’ 39’’ 00.”00

SC/A2007 262.964 263.003 0.0390 179˚ 06’ 20’’ 179˚ 06’ 20’’ 00.”00

SC/A2001 375.325 375.325 0.0000 160 ̊ 11’ 55’’ 160 ̊ 11’ 55’’ 00.”00

9

Table 4.0: Difference between Adjusted and Computed Coordinates

Station ID

Eastings (m)

(Computed)

Eastings (m)

Adjusted

Difference

(m)

Northings

(m)

(Computed

Northings

(m)

Adjusted Difference

SC/A2000 275660.067 275660.067 0.0000 530546.461 530546.461 0.0000

SC/A2001 276000.875 276000.875 0.0000 530549.914 530549.914 0.0000

SC/A2002 276010.327 276010.316 -0.011 530104.987 530104.939 -0.048

SC/A2003 275720.727 275720.733 0.006 530105.987 530105.987 0.0000

SC/A2004 275655.804 275655.787 0.017 530076.954 530076.947 -0.007

SC/A2005 275656.409 275656.344 -0.065 531181.078 531180.959 -0.119

SC/A2006 275869.597 275869.523 -0.074 531165.969 531165.841 -0.128

SC/A2007 275996.77 275996.694 -0.076 530812.846 530812.695 -0.151

SC/A2001 276000.785 276000.785 0.0000 530549.914 530549.914 0.0000

Table 5.0: Specimen of Variance-Covariance Matrix of Adjusted Observation

Table 6.0: Adjusted Distances and Bearings with Standard Errors

Station ID

Adjusted

Distances

(m)

Standard

Error (m)

Adjusted Bearings

Standard

Error

SC/A2000 340.825 0.0000 89 ˚ 25’ 10’’ 00.”1

SC/A2001 444.850 ±0.0045 178˚43’ 2.73’’ 00.”1

SC/A2002 289.583 ±0.0040 269˚ 57’ 33’’ 00.”1

SC/A2003 70.722 ±0.00457 246 ̊ 41’ 11’’ 00.”1

SC/A2004 1104.013 ±0.00480 000˚ 1’ 47’’ 00.”1

SC/A2005 213.714 ±0.0045 94˚ 03’ 24’’ 00.”1

SC/A2006 375.346 ±0.00045 160˚ 11’ 39’’ 00.”1

SC/A2007 263.003 ±0.0001 179˚ 06’ 20’’ 00.”1

SC/A2001 375.325 0.0000 160 ̊ 11’ 55’’ 00.”1

Fig. 3.0: Specimen of Differences in Eastings and Northings of Adjusted and Computed Coordinates

10

3.3 Error Ellipse:

This tool of error analysis gives an indication of the direction and circle of uncertainty in

error propagation in any survey measurements. The understated mathematical equations will

enable us determine the standard error in each station with respect to size and dimension.

LSinEE ASCASC 2001/2002/ … (17)

LCosNN ASCASC 2001/2002/ … (18)

222222

2002/ CosLSin LE ASC … (19)

222222

2002/ SinLCos LN ASC … (20)

222

LCosSin LEN … (21)

22

1 2tan

2

1

EN

ENt

… (22)

Where σ2- is the variances, σEN- is the co-variances, t- is the angle of rotation of the ellipse.

3.5 Survey Report

This is a clear documentation of the process and outcome of all field and office survey

operation carried out for property boundary determination. In this paper we will mention the

contents. The content should include the title of project, location, purpose of survey, pre-

analysis and optimization of survey job, field procedures (in line with survey standard),

computational analysis, error analysis, statistical indicators, validation of spatial information.

Your data will not be valuable to others unless you too prepare a data quality report.

4.0 Discussion on Results

It is important to state that result from Least Squares technique is dependent of the quality of

the observations as well as the model. In carrying out the field observation, the total station

survey equipment with a least count of 0.1sec was deployed. The directions and distances

were observed on four zeros to give the mean of observation. From fig 1.0, the linear

accuracy of the traverse is 1/15,000, however, this ratio does not give an indication of the

actual precision of the boundary stations. This is the same for the angular misclosure. They

are quite misleading. We have shown from the relevant tables and figures in this work that

11

error analysis can be carried out in our quest for boundary determination in Cadastral

surveys. The length and direction of each traverse line has the accuracy to which it was

measured (see table 6.0). This analysis further revealed that the least count of the equipment

is the limit of accuracy you can achieve in any field measurement.From table 4.0, the errors

are prominent in the northings coordinates particularly in stations SC/A2005-2007. This

therefore gives an indication of the quality of the station coordinates.The error ellipse

provides information on the precision of the adjusted positions of the stations, it will further

compare the relative precision between any station by looking at the shape, size and

orientation of the ellipse. We can further subject the output to other statistical testing, but for

this purpose of this work we are considering the LSA and error ellipse.

Recommendation:

We make the following recommendations in view of the urgent need to bring quality control

in our cadastral practice in Nigeria:

1. That error analysis using the least square adjustment and error ellipse technique be

deployed to ascertain the quality of our observables and final coordinates that defines

property boundary.

2. That the enabling law for cadastral practice in Nigeria be reviewed immediately to

accommodate this important scope of the work of the surveyor.

3. That an increased advocacy to enlighten the public of the need to have the quality

control for all property boundary survey services for which the surveyor will rise to

the occasion.

4. In the same vein, costing of cadastral survey jobs will be properly situated to reflect

the input of the professional surveyor.

Conclusion

The ability to statistically reach a conclusion as to the quality of observation within a survey

and test to see if these measurements meet the minimum standard of the profession is what

Least Square Adjustment can provide. The least squares method does not guarantee that the

solution is always a good one. However, if care is taken in collecting our observations and all

systematic errors are removed from the observations, it will always yield the most-probable

solution for any given set of data.

12

We have shown that error analysis as a tool for quality control is possible for cadastral

practice in Nigeria. There may be initial opposition considering the challenge of

computational analysis, but the quality of our product coordinates, plans and such like must

be validated based on the quality. The issue of quacks will naturally be phased out as the

public and government will look for the professionals to get quality jobs.

Reference

Ayeni O.O (1981): Statistical Adjustment and Analysis of Data (with application in Geodetic

Surveying and Photogrammetry).

Dashe, J.D.J (1987): Cadastral Surveying Practice in Nigeria.

Deakin R.E (2012): Traverse Analysis. Paper presented at Geospatial Science Research

Conference Melbourne, Australia, 10-12 December 2012.

Ezeigbo, C. U. (2004): “Integrating Nigerian Geodetic Network into AFREF”, What are the

Issues? “Technical Proceedings, Annual General Meeting of the NigerianInstitution of

Surveyors, Port Harcourt 19-24May, 2004.

Ghilani, Charles D and Wolf, Paul R. (2006); Adjustment Computations: Spatial Data

Analysis, 4th Edition Pennsylvania State Univ. and Emeritus, Univ. of Wisconsin-Madison

ISBN: 978-0-471-69728-2.

Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A (1986). Robust Statistics. The

Appraoch Based on Influence Functions. New York: John Wiley and Sons.

Hart, L (2015). Development of Datum Transformation Procedure for Nigeria Based on

National Transformation Version 2 (NTv2) Model. An Unpublished Ph.D Thesis, submitted

to the Department of Geoinformatics and Surveying, University of Nigeria, Enugu Campus,

Nigeria.

Nwilo P.C.; Fajemirokun, F.A.; Ezeigbo, C.U.; Oyewusi A.M. &Ayodele E.G. (2006):

“Comparison of Some Transformation Procedures” A paper presented at General

Assembly Conference of the Nigeria Association of Geodesy, University of Lagos. 23rd –

25th August 2006,to appear in African Geodetic Journal, 2007.

Obenson G. &Fajemirokun F.A. (1979): “UNIDOP- A Program to Strengthen (Through

Doppler Observations) and Adjust Rigorously the Nigerian Geodetic Control Network”

Proceedings of the 2nd International Geodetic symposium on Doppler Positioning, Austin,

Texas. Vol.2, Jan 22-26.

13

Ojigi, M.L and Dodo J.D (2014). A modified version of a paper presented at the Technical

Meeting on Adoption of Sustainable Transformation Parameters for Nigeria (WGS84-

Minna Datums), held at the Office of the Surveyor General of the Federation (OSGof),

Abuja; 2-3 December 2013.

Okeke F.I, & Hart L. (2011). Determination of Magnetic Declination and Convergence for

Nigerian Map Sheet Centers. The Tropical Environment, Journal of the Faculty of

Environmental Studies, University of Nigeria, Enugu Campus, vol. 10, No. 1, pp 27-40 ISSN

0189-4293.

United States (US) Army Engineer School (2001): "Surveying III (Topographic and

Geodetic Surveys)”, Sub-course EN0593, Edition A, October 2001. Fort Leonard Wood,

MO 65473.

Uotila U.A. (1967) Introduction to Adjustment Computations with Matrices; Lecture notes

(unpublished), Geodetic Science Department, the Ohio State University.