A realistic estimate of the accuracy of position measurements of characteristic terrain points via the RTK-GPS method
Realna ocena natančnosti določanja karakterističnih točk terena po metodi RTK-GPS
Milivoj VuličAleš Lamot2
'University of Ljubljana, Faculty of Natural Sciences and Engineering, Department of Geo-technology and Mining Engineering, Aškerčeva cesta 12, SI-1000 Ljubljana, Slovenia;
E-mail: milivoj.vulic@ntf.uni-lj.si 2Ivan Potočnik s.p.-Drilling, Blasting and Fireworks, SI-3310 Žalec, Slovenia;
E-mail: potocnik.ognjemeti@siol.net
Received: May 9, 2007	Accepted: December 17, 2007
Abstract: Here at the Faculty of Natural Sciences and Engineering, we are developing our very own method for determining said volumes; a method that has thus far been lacking in an estimation of the accuracy of determining the position of characteristic points of the plane via the RTK-GPS method. With the aid of measurements of three different types of points, i.e. points with three different point placements, we have managed to engulf the measurements, which we then use to shape an estimation regarding the accuracy of characteristic terrain points' measurements when it comes to those points which are not materialised i.e. labelled - the third type. When measuring the positions of characteristic terrain points, what is encompassed in the procedure is both the accuracy of the GPS measuring aid as well as the accuracy regarding of the positioning of the characteristic points themselves with an average operator in mind. For the determination of the characteristic points of the plane, and with the aid of an overviewing computer programme (Microsoft Excel), a model intended for the levelling measurements of the positions of characteristic plane points and the determination of accuracy when it comes to the determination of said point positions has now been designed. The model represents an automatised process of point position measurements' levelling, and allows one to enter measuring data -the coordinates of a given point, the a priori standard deviation 60 and the correlation matrix Q as the incoming data. The outgoing data is then obtained in the form of already levelled coordinates of characteristic points, the standard deviation of the weighted standard deviation 60 a posteriori, deviations in the directions of coordinate axes 6„, 6 , the main standard
Y X z
deviations i.e. the attributed values of the tensor of deviations of point measurements and deviation pedaloid itself. In the computer programme MS Excel, a model aimed at the calculation of main deviations i.e. attributed
values of the points' deviation tensor as well as the determination of pedal-loid (ellipsoid) deviations is given.
Izvleček: Na Naravoslovnotehniški fakulteti se razvija lastna metoda za določanje prostornin, katerije do sedaj manjkala ocena natančnosti določanja položaja karakterističnih točk ploskve po metodi RTK-GPS. Z meritvami točk s tremi različnimi postavitvami na točkah, in sicer s prisilnim centriranjem, s togim grezilom na znanih točkah in s togim grezilom s simuliranjem postavitve na karakt. točkah smo zajeli meritve, s katerimi smo podali oceno natančnosti merjenj karakt. točk terena, ki niso materializirane oziroma označene. Tako je zajeta natančnost GPS izmere kot tudi natančnost postavitve na karakt. točkah terena pri povprečnem operaterju. Za določitev natančnosti karakt. točk ploskve je sedaj, s pomočjo računalniškega programa za preglednice, izdelan model za izravnavo meritev položaja karakt. točk ploskve ter določitev natančnosti določanja položaja karakt. točk ploskve. Model predstavlja avtomatiziran proces izravnave merjenj položaja točke in omogoča, da se kot vhodni podatki vstavijo podatki merjenj, in sicer koordinate točke, a priori srednji pogrešek c0 in korelacijska matrika Q. Izhodni podatki pa se dobijo kot izravnane koordinate karakt. točk, srednji pogrešek utežne enote d0 a posteriori, pogreški v smereh koordinatnih osi gy dX, dz, glavni srednji pogreški oziroma lastne vrednosti ten-zorja pogreškov merjenja točke in pedaloid pogreškov. V računalniškem programu MS Excel je podan tudi model za izračun glavnih pogreškov oziroma lastnih vrednosti tenzorja pogreškov točke in določitev pedaloidov (elipsoidov) pogreškov.
Key words: RTK-GPS, RTK-DGPS, main deviations, accuracy
Ključne besede: RTK-GPS, RTK-DGPS, glavni pogreški, natančnost
Introduction
Focusing on realistic, on-location measurementsofcharacteristicterrainpoints and their belonging errors, rather than only those allotted by whatever programme we happen to be using to later on process said measured data, we are able to get a truer-to-life assessment of the accuracy of our measurements. The RTK-GPS (Real Time Kinematic - Global Positioning System) is one of the tools we may use to aid us in the accomplishment of this. It may actually be used for both determining the shape of our measured terrain as well as its belonging
surface and volume, which proves useful in engulfing the measurements of points of three different point placements, further categorised as we elaborate on the procedure in pages to come. Thus, we are one step closer to shaping an estimation regarding the accuracy of characteristic terrain points' measurements.
What is important is that we keep in mind both the accuracy of the GPS measuring aid as well as the accuracy of an average human operator dealing with it when it comes to the positioning of the the characteristic points. As for the actual determination of
the latter, what has now been designed with the aid of Microsoft Excel, our chosen overwieving computer tool, is a model intended for the levelling measurements of the positions of characteristic plane points and the determination of accuracy when it comes to said point positions. Here at the Faculty of Natural Sciences and Engineering, the design of such a simple and financially undemanding model - oriented at the levelling that comes with characteristic points' accuracy measurements - has still managed to enhance the accuracy estimations associated with the measurements of characteristic terrain points, as we will now proceed to make evident.
The levelling measurement of points
aimed at determining the accuracy of the characteristic points of a plane
When it comes to a derisive levelling of corellated measurements, it is necessary to determine the most probably value of the unknown vector x with the aid of
uxl
the vector of measured quantities l with
i	nuxl
the aid of a priori known corellations D,
nuxnu
and give an estimate i.e. evaluation of the accuracy of all these measuremants i.e. the sought-for quantities. When it comes to the latter, the number n is the number of measured quantities i.e. measurements, and u is the number of sought-for (unknown) quantities.
With one derisive levelling of corellated measurements, the sought-for (unknown) quantities are determined via a string of measurements, but all this under the condition that the sum of squares of their
correlated residuals is smallest:
vT v = min
(1)
where:
-	L/1/, the correlation matrix of measured
nuxnu
quantities is the matrix of cofactors of said measured quantities
-	v is the vector of residuals of the measured
nuxl
quantities.
When it comes to measuring sciences, the determination of a uniform solution is obtained through the least squares method (LS).
In the case of a derisive levelling procedure, what is determined first is the residuals' vector £ of approximate values of the unknown "Values x, and then the
uxl
residuals of measured quantities v may be
nuxl
dealt with. For n derisive measurements, the vector of residuals is labelled as v, the
nuxl
matrix of coefficients of equations of the residuals - the design matrix - as A, the
nuxu
vector of derivatives of unknowns as
uxl
and the vector of free terms with f , which
nxl
would make it:
(2)
That is how, for n derisive measurments, with the aid of with the u of unknowns is to be determined, the system of residual equations of measured quantities equals:
v = A • Ç + f
(3)
The vector of free terms f can symbolically
nuxl	J	J
be denoted as:
f = approximate value - measured value
n is the vector of free terms:
n = AT Ei i A
ux1 uxnu	unxl
(8)
f = f - 1
pribl
f = f - 1
(4)
A is the matrix of coefficients of residual
nuxu
equations - the so-called design matrix 2// is the corellation matrix of measured
nuxnu . .
quantities
For u unknowns, equations of free terms for an n number of measurements may be written as:
f1 = f (^...Su) - 1
ux1	ux1
f2 = f2 (^2^4) - 12
ux1	ux1
f3 = f3 (^"U - 13	(5)
ux1	ux1
f = f - 1
n n vn
The determination of a vector of unknowns \ 1, the calculation of a matrix of coefficients of normal equations N
*	uxu
and the determination of the vector of free terms n (Feil, 1990)
The vector of unknowns \ is begotten:
Considering that:
QÎ i = -1
uxu
(9)
Qf { is the matrix of covariances and the vector of unknowns denoted as 4 may also be written down as:
% = - QÎ f n
(10)
% = - N-1 n
UXl	uxu ux1
(6)
where:
- N is the matrix of the coefficients of
uxu
normal equations:
The correlation matrix of measured quantities L ; i (Feil, 1990)
.	nuxnu ....	1 .
For each correlated measurement l. there
/
is a belonging - on the basis of calculated variances and covariances pertaining to prior variantly known measurements -covariance matrix i.e. correlation between measured quantities L and the a priori
uxu
standard deviation of the (prior) levelling a0 . For murtually dependent measurements of varying accuracies, the matrix of cofactors of each measurement by itself is a symmetrical matrix, as seen below:
N = AT El i A
uxu uxnu
(7)
E ' i = 00, Q,
uxu
1 In an ample amount of books pertaining to this area, the vector of unknowns is denoted also as x. Due to practical reasons, however, the vector of unknowns may also be denoted with the Greek letter 4, thus making matters more concise when it is to be dealt with in actual equations.
s?;,. =cj;
uxu
<7,,			■ qxu
* xy	Hyy		*yu
* xz			tf zu
^xu	tf yu	Q ZU	Quu
(11)
(12)
S?/, =
» 2	/v 2	/v 2
q g„	q a„	q a„
J- XX 0/	Au 0/	' vr 0/
» 2	/> 2	» 2
</	^vv^o,	lyPoi
IxPoi	QzPli
(Ixu^li lyu^li q^l
qc>
Wo.


S;?« =
-//// nu xnu
^/711	^ //12	2f//lw
MX M l/XM	UXU
/12	^//22	^//2«
UX-U uxu	uxu
/In	••• 2-7/wi
(14)
Sub-matrices are of the shape depicted below:
(13)
The correlatory matrix of measured quantities L// for u unknowns and n
nuxnu
measurements is made up of the following sub-matrices:
xxi	^ xyi	^ xzi	
xyi	Z . yyi	Z . • ya	• z . yui
xzi	Z . yzi	Z . • ZZl	• z . ZUl
	Z .	Z . •	■ Z .
(15)
(16)
That is how the correlation matrix of measured values is composed of the following sub-matrices:
S	£	£	■• £
xïl			
2 xyi	2»l		1 2 yul
	2>zi	•	" 2zal
-yul
ro	0	0 ■	• 0 1
0	0	0 •	• 0
0	0	0 •	• 0
0	0	0 •	• 0
"o	0	0 ■	• 0 "
0	0	0 •	• 0
0	0	0 •	• 0
0	0	0 •	• 0
0	0	0 •	• o"
0	0	0 •	• 0
0	0	0 •	• 0
0	0	0 •	• 0
"xy2
-yyl '■•yzl
Jyul
2 , • xz2	• 2 ,1 xu 2
2^2 "	■ 27„2
2ZZ2	' 2 zia
y _ .	. y. .
0	0	0 ■	• 0~
0	0	0 •	• 0
0	0	0 •	• 0
0	0	0 •	• 0
~0	0	0 •	• o"
0	0	0 •	• 0
0	0	0 •	- 0
0	0	0 •	- 0
ro	0	0 •	• 01
0	0	0 •	• 0
0	0	0 •	• 0
0	0	0 •	• 0
S E
xun yun
(17)
which can, in short, be denoted as:
(18)
The inverse (matrix) of the matrix of measured quantities is thus denoted as:
(19)
which may, again, be shortened to:
For an individual point measurement, we are thus able to obtain the coordinates of the measured point, as well as the submatrix 2/ i.
3x3
As has already been depicted in equations (14) to (17), for the determination of a variance-covariance matrix 2/ / , it is already
nuxnu
enough when we are acquainted with the correlation matrices of an individual measurement 2/ The sub-matrices 2; ; are
3x3	3x3
symmetrical matrices, and can therefore be written as below for each of the individual measurements:
(20)
and is the row-specific matrix of the inverse correlation sub-matrices 2 ; k of individual
uxu
measured unknowns	..., ^ .
When we are dealing with the measurements of point positions via the RTK-GPS method, each measurement in itself may beget its own varance-covariance matrix 2/ i. The variance-covariance matrix 2.//
nuxnu	nuxnu
is determined on the basis of different measurement influences of the time. Sad »influences« are, in fact, the disparations of the receiver's an the satellite's timers, the atmosphere's influence upon wave expansion, a mistake pertaining to the determination of the hight of the antennae etc.
(21)
- i , the number of the measurement The evaluations of accuracy of the
levelling of an individual quantity
The standard deviation of a given levelling procedure - a0
(22)
r = nu - u, the number of meaurements
above the required number
nu, the number of measured quantities
i.e .the number of measurements
u, the number of sought-for (unknown)
quantities
£ ; ; , the correlation matrix of measured quantities, the matrix of cofactors of measured quantities - v , the vector of residuals of measured
um,\
quantities
The standard deviation of unknown quantities - q
The medium residuals of unknowns are the functions of standard deviations of meaurements. Unknowns can thus be denoted as the functions of measurements bearing in mind that we must take into consideration the Law of Cofactor Increment. The matrix of covariances Qj { is
urn
equal to the inverse matrix of coefficients of normal equations N-1.
uxu
For three unknowns, the matrix of covariances Q{{ may be written in the form
3x3
of:
or even:
(24)
a£ =CT0
yëT
(25)
The standard deviation of levelled quantities - 6/,/,
âvn =a"o JKi n,
(26)
The necessary cofactors for an evaluation of accuracy of said levelled quantities are on the main diagonal of the matrix of cofactors of levelled quantities, £ ,
Q/,v = A Qg A
(27)
The standard deviation of residuals of measured quantities - a
1	vv
The standard deviation of residuals of measured quantities may be gotten through the following:

(28)
The necessary cofactors, intended for the estimation of the accuracy of the unknowns, are on the main diagonal of the covariance matrix (the matrix of the cofactors of the unknowns) Qf The standard deviations of
urn
the unknown quantities are therefore:
The cofactors necessary to obtain an evaluation of the accuracy of the residuals of our measured values £ , can be obtained
vvii ?
as the difference between the cofactors of measured quantities £m. and the cofactors of levelled quantities £l,l,.. .
Determination of the main standard deviations of unknowns:
During a given levelling of derisive measurements, the position of a point may be determined on the basis of certain previously measured quantities. That is how, besides levelled quantities of unknowns, L, L, L, ..., L E , one also obtains the covariance matrix, £ :
is obviously not much different:
(29)
	r			
		6:		•• CT XU
	CT xy	'I	CT y	•• a yu
	CT XZ	CT V-	CT2 • Z	•• G zu
uxu				
	CT XU	CT yu	CT ZU	CT2 u
= q;
Qxx Qxy Q x
q q
1 yz 1 Z
q q q
J- xu J- yu z
qxx qxy qx-„
Qxu
f xy	y .c
qvy~X qy:
l vv
1y
q.
qxu
= 0 (32)
The characteristic equation then goes like this:
(30)

(33)
= cfo QK
The elements on the main diagonal of the covariance matrix, £ are squares of standard deviations of the determination of the position of our given point in the direction of the coordinate axis. Many times, however, it is also necessary to determine the maximum - or minimum, depending on the case - values of said deviations.
The values of our main deviations (the maximum, minimum and binormal standard deviations) are determined with the aid of the selves' values of X . and their belonging
selves' vectors s. of the covariance matrix
i
The »self values« of the covariance matrix, Qare obtained according to the equation1::"
det(Qg - XI) = 0
- PJ&) being the characteristic polynomial
The »zeroes« of the characteristic polynomial PJX) are the self values of the covariance matrix Qg.
The self values X., of course, have belonging self vectors denoted as s, of the covariance matrix Q and the latter are defined as:
Q^ •s = X•s
(34)
(31)
and the equation for the determination of self values X. of the covariance matrix Q
Each self value of X. provides one witha system of equations with the aid of which the afore-mentioned s vectors may be determined.
Determining the solutions of higher-degree polynomials is a very expansive and demanding practice, which is why various computer programmes have been enlisted with this very purpose in mind, aiding us in this quest to determine the previously mentioned self values and vectors.
The self values of X. and their belonging
self vectors s of the covariance matrix
i
Qff determine the quantitative values and
directions of our main deviations. The quantities of main standard deviations of unknowns are thus determined with the aid of the following equations:
CM	eigen	
¿2	eigen	
¿3	eigen	
(35)
=do \Ik

X2' X3'
X the self values of the
covariance matrix
uxu ÇÇ
ou eigen, the main deviation of unknowns in the direction of itselfs coordinate system plane (of a pedalloid as well as ellipoid)
O0, the standard deviation of the levellin of the measured quantities
(37)
are the equations for the determination of main deviations of the position of the point. The main standard deviations obinor, Ominor, amajor represent the values of the half-axes a, b, c of the deviations' ellipsoid. The directions and quantitative values of the half-axes a, b, c of the deviations' ellipsoid may be obtained with the aid of vectors sx
sy s^z, as in:
s„v = a
s - =g
I x/Vï
,VV!
(38)
The main standard deviations and the pedalloid (ellipsoid) of deviations
The directions of mai deviation unknowns
O1_eigen O2_eigen O3_eigen Ou_eigen are determined
by the self vectors s.. That is how the sm vectors, which obviously determine the directions and quantities of main standard
1 • , '	A	A	A	A
deviations, O^ge^ O2_eigen O3_eigen	Ou_eigen ,
of unknowns, may be written down as:


-cf0 Jx^ ■
(36)
s- = o,
JK
If the main standard deviations are written down for three particular coordinates X, Y, Z, then:
a realistic evaluation of the accuracy of the measurements of the position of a plane's characteristic points
The characteristic points of a plane
The data in connection to the positions of characteristic points of a given plane have been obtained on-terrain, via the methods of RTK-GPS for three different types of point placement, namely via forced centering (type 01), a lame plumb line upon the known points (type 02), or a lame plumb line and a simulated placement upon the characteristic points (type 03). Even the levelling and evaluation of measurements according to the RTK-GPS (Real Time Kinematic - Global Positioning System) method has been developed in accordance with all three types of point placement, which enables a comparison of
the measurement results and their accuracy for all three types. The points are sorted as the figure directly below depicts.
For each type of point (type 01 - type 03), four points have been measured with 25 separate measurements, and four points with five measurements per point. That is how a number of measurements above the required number has been secured, and the procedure of levelling and accuracy evaluation may thus commence. The label of a given point represents its type and how many times exactly it has been measured. Points with an odd first digit have been measured 25 times, and those with an even first digit have had their positions measured five times. The second digit pertaining to the label on each individual point, on the
other hand, denotes the type of positioning upon the point, i.e. the numbers 1, 2 and 3 inform us whether is is a (type 01), (type 02) or (type 03), respectively. The palcement upon a point which is labelled as belonging to (type 01) is a placement upon a known point on the ground, with forced centering via a tripod and antenna latched onto said tripod, whereas a (type 02) informs us that teh positioning has been done with a lame plumb line upon a known point on the ground and (type 03) with the same sort of lame plumb line upon a point, except that here it is on the ground within the area which has been labelled via circularly distributed spray paint with a circumference of approximately 1m. These circles have not been drawn with the utmost precision, but via free-hand, and
Figure 1. The point arrangement and labels Slika 1. Razporeditev in oznake točk
Table 1. Maximum deviations in the direction of coordinate axes and main deviations (type 01) Tabela 1. Maksimalni pogreški v smeri koordinatnih osi in glavni pogreški (tip 01)
Tripod points (type 01):
ct
		Maximum deviation of the coordinate axes		
No. of measurements	Point	Z	X	Y
25	11	0,0020	0,0010	0,0008
25	31	0,0017	0,0008	0,0006
25	51	0,0016	0,0007	0,0006
25	71	0,0019	0,0009	0,0007
5	21	0,0037	0,0018	0,0014
5	41	0,0036	0,0017	0,0013
5	61	0,0040	0,0019	0,0015
5	81	0,0045	0,0021	0,0017
ct
	Main deviations		
Point	Z	\	V
11	0,0020	0,0010	0,0008
31	0,0017	0,0008	0,0006
51	0,0016	0,0007	0,0006
71	0,0019	0,0009	0,0007
21	0,0037	0,0018	0,0014
41	0,0036	0,0017	0,0013
61	0,0041	0,0019	0,0015
81	0,0045	0,0021	0,0017
x1
maxi
were consequently less like circles than ellipses. The figurant had tried to place the plumb line on the same place every time, but only with the aid of a visual evaluation of where the pervieved centre of the circle
The results of the levelling of characteristic point measurements
In the comparative tables below, the results of said levelling of characteristic point measurements are presented. The said results have shown that the folowing for point of forced centering (type 01):
-	Deviations in the direction of coordinate axis Z do not exceed 5 mm (the largest one would be 4.5 mm), even for the smallest number of measurements (5).
-	Deviations in the direction of coordinate axis Y do not surpass 2 mm (the largest one would be 1.7 mm).
-	Deviations in the direction of coordinate axis X do not surpass 2.5 mm (the largest one would be 2.1 mm).
This is all depicted in the table 1.
Results of the levelling of characteristic
point measurements with a lame plumb
line upon known points show that:
-	Deviations in the direction of coordinate axis Z do not exceed 8 mm (the largest one would be 7.4 mm), even when it comes to the smallest number of measurements (3).
-	Deviations in the direction of coordinate axis Y do not exceed 3 mm (the largest one would be 2.5 mm).
-	Deviations in the direction of coordinate axis X do not exceed 3 mm (the largest one would be 3.0 mm).
Again, this is all depicted in the table 2.
Table 2. Maximum deviations in the direction of coordinate axes and main deviations (type 02) Tabela 2.Maksimalni pogreški v smeri koordinatnih osi in glavni pogreški (tip 02)
Lame plumb line points - determined in the ground (type 02):
ct
		Maximum deviation of the coordinate axes		
No. of measurements	Point	Z	X	Y
19	12	0,0026	0,0012	0,0010
19	32	0,0027	0,0013	0,0010
19	52	0,0025	0,0011	0,0009
21	72	0,0022	0,0010	0,0008
3	22	0,0037	0,0016	0,0013
3	42	0,0074	0,0030	0,0025
4	62	0,0053	0,0022	0,0018
4	82	0,0049	0,0023	0,0018
a
	Main deviations		
Point	Z	X	Y
12	0,0027	0,0012	0,0010
32	0,0027	0,0013	0,0010
52	0,0025	0,0011	0,0009
72	0,0022	0,0010	0,0008
22	0,0037	0,0016	0,0013
42	0,0074	0,0030	0,0025
62	0,0053	0,0023	0,0018
82	0,0049	0,0023	0,0018
x2
Table 3. Maximum deviations in the direction of coordinate axes and main deviations (type 03) Tabela 3. Maksimalni pogreški v smeri koordinatnih osi in glavni pogreški (tip 03)
Lame plumb line points - not determined in the ground (type 03):
ct
		Maximum deviation of the coordinate axes		
No. of measurements	Point	Z	X	Y
20	13	0,0045	0,0022	0,0017
20	33	0,0036	0,0017	0,0014
20	53	0,0038	0,0018	0,0014
20	73	0,0039	0,0018	0,0014
5	23	0,0072	0,0035	0,0028
5	43	0,0037	0,0020	0,0016
5	63	0,0099	0,0047	0,0039
5	83	0,0099	0,0045	0,0037
ct
	Main deviations		
Point	Z	X	Y
13	0,0045	0,0022	0,0017
33	0,0036	0,0017	0,0014
53	0,0038	0,0018	0,0014
73	0,0039	0,0018	0,0014
23	0,0072	0,0035	0,0027
43	0,0038	0,0020	0,0015
63	0,0099	0,0047	0,0038
83	0,0099	0,0045	0,0037
x3
max3
Results of the levelling of characteristic point measurements with a lame plumb line via simulating the approximate positioning upon the characteristic points (type 03) have shown that:
-	Deviations in the direction of coordinate axis Z do not exceed 1 cm (the largest one would be 9.9 mm), even when it comes to the smallest number of measurements (5).
-	Deviations in the direction of coordinate axis Y do not exceed 4 mm (the largest one would be 3.9 mm).
-	Deviations in the direction of coordinate axis X do not exceed 5 mm (the largest one would be 4.7 mm).
Once again, this is all depicted in the table 3.
On the following picture, a pedalloid and ellipsoid of deviations of measurement point 13 are shown:
Conclusions
With the on-terrain measurements, we have managed to obtain a realistic estimate i.e. evaluation of the accuracy of the characteristic point meaurements on our terrain. With the measurements of three different types of points, i.e. points with three different varieaties of on-point placement - namely with forced centering (type 01), a lame plumb line on known points (type 02) and a lame plumb line and simulated placement on characteristic points (type 03) - we have obtained the measurements necessary for us to provide an evaluation of the accuracy of said characteristic point measurements,
which have not been materialised i.e. labelled (type 03). When measuring these characteristic point upon our terrain, the accuracy of GPS measurements has thus also been captured, as well as the accuracy of average operator placement of said characteristic terrain points.
With the measurements of said terrain points, the designing of a simple and financially undemanding model intended for the levelling of the afore-mentioned measurements, we have, at the Faculty of Natural Sciences and Engineering, managed to complement the accuracy estimations pertaining to the measurement of characteristic terrain points.
The characteristic points of the terrain which have not been materialised i.e. labelled (type 03) can be detected and measured with an accuracy where the standard deviation in the direction of coordinate axis Z does not, even with the smallest given number of measurements (5) surpass 1 cm (the largest having been 9.9 mm), the standard deviations in the direction of coordinate axis Y do not surpass 4 mm (the largest having been 3.9 mm), and the deviations in the direction of coordinate axis X do not surpass 5 mm (the largest having been 4.7 mm).
The comparison of measurements of all three types of points that have been measured has shown us that the designed model has provided us with realistic results. We are thus able to claim that the accuracy of the determination of characteristic terrain points via the RTK-GPS method is 1 cm or less.
0.02 -0.02
Figure 2. The pedalloid and ellipsoid of deviations of point 13 Slika 2. Pedaloid in elipsoid pogreškov točke 13
Povzetek
Realna ocena natančnosti določanja karakterističnih točk terena po metodi RTK-GPS
Z meritvami na terenu smo dobili realno oceno natančnosti meritev položaja karakterističnih točk terena. Z meritvami treh različnih tipov točk oz. točk s tremi različnimi postavitvami na točkah, in sicer s prisilnim centriranjem (tip01), s togim grezilom na znanih točkah (tip02) in s togim grezilom s simuliranjem postavitve na karakterističnih točkah (tip03) smo zajeli meritve, s katerimi smo podali oceno natančnosti merjenj karakterističnih točk terena, ki niso materializirane oziroma označene (tip03). Pri merjenju položaja karakterističnih točk terena je tako zajeta natančnost GPS izmere kot tudi natančnost postavitve na karakterističnih točkah terena pri povprečnem operaterju.
Z meritvami položaja karakterističnih točk terena, izdelavo enostavnega in cenovno ugodnega modela za izravnavo merjenj in podanimi ocenami natančnosti merjenj karakterističnih točk, pri povprečnem operaterju smo lastno metodo za določanje prostornin, ki se razvija na NTF, dopolnili z oceno natančnosti merjenj karakterističnih točk terena.
Karakteristične točke terena, ki niso materializirane oziroma označene (tip03) lahko zaznamo in izmerimo z natančnostjo
kjer srednji pogreški v smeri koordinatne osi Z tudi pri manjšem številu merjenj (5) ne presegajo 1 cm (največji je 9,9 mm), pogreški v smeri koordinatne osi Y ne presegajo 4 mm (največji je 3,9 mm) in pogreški v smeri koordinatne osi X ne presegajo 5 mm (največji je 4,7 mm).
Primerjava meritev vseh treh omenjenih tipov točk je pokazala, da je izdelani model podal realne rezultate. Trdimo lahko, da je natančnost določitve karakterističnih točk terena po metodi RTK-GPS 1cm ali manjša.
References
Durgutovic, A. (2005): Izmera in določanje prostornin nepravilnih teles z uporabo metode 'RTK-GPS': diplomsko delo. Naravoslovno-teh-niška fakulteta, 51 p. Feil, L. (1990): Teorija pogrešaka i račun izjednačenja. Part II, Sveučilište u Zagrebu, Zagreb, 202 p. Lamot, A. (2007): Priloga k določanju natančnosti karakterističnih točk terena za potrebe RTK-GPS metode: diplomsko delo. Naravoslovnoteh-niška fakulteta, 84 p. Strang, G. (1997): Linear Algebra, Geodesy, and GPS. Wellesley-Cam-bridge Press, 624 p. Wolf, R.P., Ghilani D.C. (2005): Elementary Surveying: An Introduction to Geomatics. XI Edition, 916 p.