CC BY
18
УДК 528.3:528.11 Yehia Hassan Miky
Egypt
ADJUSTMENT OF GEODETIC NETWORKS USING AUTO-CORRELATION TECHNIQUE
Abstract
Control surveys networks are used to establish the locations of arbitrary points. These points may be used as reference locations for performing additional survey work. Geodetic networks are generally performed to a higher standard of accuracy than other types of surveys. This is necessary because any follow-on survey work must be able to count on the accuracy of the control points. In hybrid geodetic networks, the combination of both angular and linear measurements increases the redundant observations and hence strengthens the network. If time depending observations are given, a regression could be used as a functional models for adjustment the geodetic networks. In case of stochastic model uncorrelated observations with identical variance are assumed. Because of many observations required to achieve higher accuracy, a small time difference arises between the observations. Thus the assumed stochastic model is not suitable. The calculation has to be effected by means of auto-correlated observations. So, the main objective of the paper is to study the effect of auto-correlation technique on the accuracy of the estimated parameters and variances of geodetic networks. To achieve that task, a network of surveying over 6000 hectares in Dakahlia, Nile Delta, Egypt was chosen. The least squares adjustment technique is applied to the observations. Finally, auto-correlation adjustment technique improves the standard deviation in horizontal plane.
1. Introduction
In order to cover large areas of the earth's surface with a network of accurately determined positions (coordinates), it is necessary to have some system for extending these positions from some point of origin. Then, these widely scattered positions can be used for control surveys for such purposes. Geodetic networks usually consist of several different measurements (angles and distances) to the unknown point from several known points. The measurements are combined in a process known as network adjustment, in which existing known points are held fixed, and the position of unknown point adjusted until it best fits the measurements. Achieving the higher accuracy of the estimated coordinates in geodetic networks is the goal of the surveying work.
2. Field observations and Project Location
The project Location done over 6000 hectares in the Project (ISAWIP) Dakahlia, Nile Delta, Egypt is chosen. The ISAWIP project was a joint undertaken by the Governments of Egypt and Canada to demonstrate how an integrated approach to agriculture development can effectively increase crop production in the old lands of the Nile Delta. The topography of the project is almost flat with
elevation ranges from four meters above sea level in the western part of the area to a few centimeters below sea level in the northeastern part as shown in figure (1). A necessary complement of this large-scale program is a more local study aimed at measuring the topography of the studying area. For this purpose, and as a first step, they intend to establish a regional network (connected with the global network), to measure local position of chosen number of points in the project area. This relatively small network (up to ten stations) was measured for the first time along with the larger network. The purposed network consisting of 10 stations were observed using Sokkia Total station as shown in figure (2). A complete survey of the existing main network was to be carried out by measuring the distances and angles (30 angles and 19 distances), so as to achieve the highest practical accuracy and the point 1, 2 are fixed points.
Figure (1): Project Location
Figure (2) The Proposed Geodetic Network
3. Adjustment of parameters by Autocorrelation technique
In case of stochastic model uncorrelated observations with identical variance are assumed. Because of many observations required to achieve higher accuracy, a small time difference arises between the observations. Thus the assumed stochastic model is not suitable. The calculation has to be effected by means of auto-correlated observations. The correlation between neighbored measurements has to be taken into consideration in auto-correlation technique. In this technique, the elements of the cofactor matrix are the correlation coefficients between immediately neighbored values of observations from regression analysis. The correlation coefficient is a measure of how closely two quantity are related. It can be calculated as:
Pab ~
°ab
CT,
Unit-less.
(1)
a-ab
Where:
aa = The standard deviation of the first quantity ab = The standard deviation of the second quantity
aab = The standard deviation of the two quantities, the correlation coefficient has a limit ±1
The cofactor matrix has the (n*n) dimensions and it can be written following:
Q =
i
cib
i
f^ch
i
2)
(
Then, the weight matrix (W) can be calculated as following:
W = Q-1 (3)
This new weight matrix (W) will be reentered in the regression analysis to estimate the corrected coordinates of the networks; then the new values of parameters and its accuracy will be calculated.
4. Result Analysis of the proposed techniques
The adjusted coordinates and its associated surveying accuracy of each point in the network are calculated by using the least square theory, and then the network will be resolved by applying autocorrelation adjustment technique. This will be done by computing the correlation coefficients between adjusted observations from regression analysis, and then calculate the form of cofactor matrix (Q). The final
adjusted coordinates from auto-correlation technique and its standard deviations are shown in Table (1).
Table (1): Corrected Coordinates of the Horizontal Control Network stations with
Auto-correlation
No Corrected Coorc inates Error Propagation
Station Northing (m) Easting (m) Standard Deviations Error Ellipses Confidence Region = 95%
N (m) E (m) SemiMajor Axis Semi-Minor Axis Azimuth of Major Axis
1 2000 933987.590 668820.780 0.000 0.000 0.0000 0.0000 0 o 00'
2 1000 932176.501 670555.140 0.000 0.000 0.0000 0.0000 0 00
3 6 934223.050 672549.700 0.010 0.011 0.0236 0.0067 85 03
4 5 937059.330 671551.440 0.012 0.014 0.0260 0.0240 35 16
5 4 938696.902 672424.289 0.015 0.016 0.0353 0.0273 11 11
6 3 936333.857 676951.575 0.020 0.019 0.4400 0.0370 172 43
7 2 940524.743 675694.286 0.021 0.021 0.0516 0.0423 161 22
8 116 936727.445 679474.546 0.022 0.02 0.0606 0.0480 142 24
9 117 941065.077 679971.970 0.023 0.021 0.0563 0.0496 131 04
10 118 937263.377 682712.403 0.023 0.022 0.0526 0.0050 100 49
The comparison between the resulted standard deviations of the coordinates from regression analysis by using autocorrelation and without autocorrelation can be indicated as shown in figures (3 and 4) and table (2)
Table (2) Comparison between regression analysis with and without AutoCorrelation for all network stations
Station Standard Deviation in X-direction (m) Standard Deviation in Y-direction (m)
Without Autocorrelation With Autocorrelation Without Autocorrelation With Autocorrelation
1 0.0 0.00 0.0 0.00
2 0.0 0.00 0.0 0.00
3 0.035937 0.011 0.031959 0.010
4 0.046368 0.014 0.040620 0.012
5 0.044624 0.016 0.043995 0.015
6 0.056449 0.019 0.054440 0.02
7 0.063336 0.021 0.062958 0.021
8 0.053080 0.020 0.057794 0.022
9 0.063945 0.021 0.062514 0.023
10 0.063338 0.022 0.067781 0.023
Standard deviation in X direction
Value (mm)
80 60 40 H 20 0
LL
tL
LL
LL
tL
2 3 4 5 6 Network points
11
h
1
7
8
□ aX From regression analysis □ aX From Auto-correlation
Figure (3): Comparison between regression analysis with and without AutoCorrelation for aX
Standard deviation in Y direction
80
Value (mm)
60 -40 -20 -0
Ik
tL
hlh.
1---n--^^
12345678
Network points
□ aY from regression analysis □ aY from Auto-Correlation
Figure (4): Comparison between regression analysis with and without AutoCorrelation for aY
As indicated in figures (3 and 4), the standard deviation in X direction (oX ) was reduced by using autocorrelation technique about 0.3 to 0.36 of the original value - without autocorrelation-, the standard deviation in Y direction (oY) was reduced also about 0.295 to 0.38 of the original value. The greatest reduction in X-direction occurs at point 4 and lowest at point 5, but in Y-direction, the greatest reduction at point 4 and the lowest reduction at point 8.
5. Conclusions
From the previous analysis and numerical results obtained, the following conclusions can be deduced as Auto-correlation adjustment technique depends mainly on the variance covariance matrix of observations. It improves the standard deviation in horizontal plane (X and Y).
© Yehia Hassan Miky, 2007
