Precise Local Geoid

Using Surface Fitting Methods

Mahmoud El-Mewafi Associate Professor, Public Works Dept.,

Faculty of Engineering, Mansoura University, Egypt

أعطي االنتشار السريع الستخدام األقمار الصناعية في تحديد إحداثيات النقط :الملخصأعطي اهتماما كبيرا لحساب سطح الجيويد مما يزيد من دقة حساب مناسيب النقط من

. أرصاد األقمار الصناعية كبديل سريع ألعمال الميزانيات في األعمال المساحيةوالهدف من هذا البحث هو إيجاد طريقة رياضية مثلي وبسيطة لحساب سطح الجيويد

ولذلك قدم البحث األسس الرياضية لثالث طرق لتوقيع األسطح بدقة عالية . في مصر(. HARN)مع دراسة تطبيقها علي الشبكة الجيودسية المصرية عالية الدقة

وقد أوضحت النتائج إمكانية استخدام الطرق الرياضية الثالث لتوقيع سطح الجيويد في

مصر بدقة عالية إال أن اعلي دقة أمكن الحصول عليها كانت من طريقة اقل تحدب كما تم الحصول علي دقة عالية . مقارنة بالطرق العالمية والمحلية (MCS )لألسطح

.عند تطبيق هذه الطريقة عمليا في منطقة بحيرة ناصر

ABSTRACT

According to wide spread use of satellite based positioning techniques, especially GPS

(Global Positioning System), a greater attention has started to be paid to precise

determination of geoid models with an aim to replace the geometric leveling

measurements with GPS measurements during geodetic and surveying works. This paper

briefly discusses the data and methodology used in the establishment of an Egyptian local

geoid with high accuracy . Numerical solution of geoid modeling determination applying

Surface fitting ; regression models; least square collocation with linear basis functions;

and minimum curvature surface (MCS) with Laplace equation are presented.

In addition a comparative study of these techniques, mainly it is going to emphasize the

applicability of interpolation techniques as a tool for modeling the geoid in a local area

precisely using GPS/ Levelling data to serve practical applications of geodesy. Also, The

obtained results revealed that, the applicability of MCS technique as a tool for

determining the Egyptian Local Geoid precisely with law distortion at common points.

1. INTRODUCTION

A precise geoid model is an important part of the geodetic infrastructure. Geodetic

infrastructure includes whole of the geodetic networks, which realize the coordinate

systems and datum definitions as the bases of geodetic works. The heights of points on

land are traditionally determined by leveling technique. However, GPS derived heights

are referred to WGS-84 ellipsoid, while the leveled heights are referred to geoid. Their

difference is called geoidal height or geoid undulation. In practices, most of survey

projects require leveled heights, for they have physical meaning. Therefore precise geoid

must be available to transform GPS derived heights to leveled heights. The fundamental

expression of relationship between ellipsoidal heights obtained from GPS measurements

and heights with respect to a vertical geodetic datum established from leveling data with

gravimetric corrections is given in [Moritz, 1980 ].

The transformation of ellipsoidal heights to orthometric ones requires that the geoid

height must refer to the same reference ellipsoid (to the WGS-84 ellipsoid). With the

spreading of the GPS measurements the need for a good geoid model has been increased.

The geoid can be calculated from different types of input data. The simplest method is to

use GPS and leveling points, where both the ellipsoidal and orthometric heights are given.

Another possibility and the most commonly used technique for precise determination of

geoid is using the gravimetric solution. In this paper, geoid modeling is evaluated as a

surface fitting problem according to GPS and Leveling data and the focus is modeling the

geoid of a local area as an analytical surface to serve practical geodetic applications.

During implementation of the subject, three different interpolation methods are

considered. The presented interpolation methods were programmed as to be computation

modules of a geoid modelling software and the results of the computation algorithms are

going to be mentioned. A comparative study between surface fitting methods and both of

OSU91 and EGM96 are also presented.

2. Egyptian Geoid Determination

In Egypt considerable investigations are in progress for the determination of the

separation of the geoid: GPS/leveling data geoid solution [Nassar et al., 1993], [ El-

Tokhey and Maghraby, 1995], [Shaker et al., 1997], and [Amin et al., 2005]. This

solution was based on GPS and leveling observations data, and the EGM96 geopotential

model. Gravimetric solution based on terrestrial gravity data was carried out by many

authors: [ Saad et al., 2002], [ Dawod, 2005] . Previous studies have shown that the

accuracy of GPS/leveling data geoid solution model over Egypt is almost 1.5 meter

accuracy [Saad and Dawod, 2002 ]. Even the new global models, such as EGM96,

PGM2000, and UCPH2002, do not represent the gravitational field over Egypt better than

0.80 meter[ Dawod, 2005]. Therefore to improve the estimation of the geoid with high

accuracy we had to look for a new method. Surface fitting methods may be give a high

accuracy geoid determination in Egypt.

3. Surface Fitting Methods

Surface fitting is a method that uses the difference in position at each common point to

in effect fit a surface, which can then be used to interpolate the change for other points.

This means that there will be a surface for the change in latitude, and another for the

change in longitude. Examples of this type of transformation are the Minimum Curvature

method used in the USA; the Multiple Regression method used in Canada; and the

Collocation method used in Australia.

3.1 POLYNOMIAL FITTING

The mathematical formula of the polynomial model fitting was used to approximate the

geoid heights as a function of geographic coordinates latitude Φ and longitude λ. The

formula of the used 6th order fitting polynomial is the following [Erol, B., and Çelik, R.,

2004] :

N= A0+A1λ+A2Φ+A3 λ2+A4 λ Φ +A5Φ

2 + A6 λ

3+A7 λ

2 Φ + A8 λ Φ

2 + A9Φ

3 + A10

λ4+A11 λ

3 Φ + A12 λ

2Φ

2 + A13 λ Φ

3 + A14 Φ

4 + A15 λ

5+A16 λ

4 Φ + A17 λ

3 Φ

2 + A18 λ

2Φ

3

+ A19 λ Φ4 + A20 Φ

5 + A21 λ

6+A22 λ

5 Φ + A23 λ

4 Φ

2 + A24 λ

3Φ

3 + A25 λ

2 Φ

4 + A26 λ Φ

5

+ A27 Φ6 …………(1)

Where N is the geoid height and Ai are the coefficients of the polynomial.

Using general least square technique to solve this system for every point altogether and

notice that, the coordinates (Φ,λ) will be considered as observations and the unknowns

are the A0,A1,A2,A3,A4,A5,A6,……and An. The maximum accuracy in this study resulted

by applying 2nd

order polynomial.

3.2 Least - Squares Collocation Fitting

Least - squares collocation can be applied on the interpolation of the geoid undulation

depending only on the GPS/Leveling observations at common points. When predicting

(interpolating) the value of geoid undulation and the distortion at each grid node, the

geoid undulation and the distortion at neighboring data points is considered. The grid

spacing used in this paper for the Egyptian model is ∆Φ = 0.50o and Δλ = 0.50

o and was

computed from points which have leveling and GPS data. The contribution to the

predicted value of each data point is determined from a weighting (covariance) function.

Therefore, the covariance matrix must be constructed from the distances (r) between

points and coordinate differences as follows, [Cross, 1983]:

1. Calculation of the height differences.

The difference between datum heights at stations

N = h – H ………………………………….. (2)

Where: h is the ellipsoidal height, H is an orthometric height, and N is the geoid

undulation

2. Estimation of a priori variance-covariance matrix (Cij) using the general form of least-

squares technique.

(ij

brea

ijC

) ……………………….. (3)

Where:

(a, b) are unknown parameters, (rij) is the distance between each pair of common points

and Cij are observations of geoid undulation. In this case the covariance equation will be:

Cij = 1.00000000008583 e - 0 .99999999983886 r

ij

3. Once, a priori variance-covariance matrix is built, the predicted values (geoid

undulation ) of the computed new points at the grid nodes can be obtained by least

squares collocation as, [Mikhail, 1976] and [Cross, 1983]:

bsCsCCN

BCssCsCCsBsC

bsCsCCsBS

NNNN

NN

T

NN

T

1

11

1

11

1

11

)(ˆ

)(ˆ

)(ˆ

------------------------------ (4)

Where:

b is the constant vector of the system (n,1).

CNN is the covariance matrix of observations (geoid undulation).

11sCs is a covariance matrix of data points and it is a square matrix of (n,n)

dimension (n is the number of data points).

Cs is the covariance matrix between data points and the computed new points

N is the noise, and S is known as the signal

3.3 Minimum Curvature Surface (MCS) Fitting

Minimum curvature surface (MCS) is a method of converting coordinates/ or heights

from one datum to another by interpolation within a mesh (grids) of points. In computing

geiodel heights for Egypt, a regularly spaced grid is derived from randomly distributed

data. The contribution to the predicted value of each data point is determined from a

weighting (covariance) function. Nearby data points have more influence on the predicted

value than those which are farther away [Junkins and Erickson, 1996].

As mentioned above, the grid spacing used in this paper for the Egyptian model is ∆Φ =

0.50o and Δλ = 0.50

o and was computed from points having leveling and GPS data.

Minimum curvature surface (MCS) creates a surface from three-dimensional point

elements, the nodes and edges that have X, Y and Z coordinate fields for each node from

common point's record. Minimum curvature surface (MCS) algorithm works in two

stages, initialization and iteration. In the initialization stage, Minimum curvature surface

(MCS) method applies a Laplacian power or, an alternative function (Poissan), satisfies

the bi-harmonic to the set of input coordinates values and fits a smooth surface to the

elevation values [Nikos, 1997].

D

dxdyyxf2

2 ),( …………… (5)

For the general case, equation (5) can be rearranged (when the four arms about the nods

may be not completed) referring fig. (1), to give the final form of the observation

equation as:

(6) Poisson )

0,

0(2h

Laplace ,0

)

21

2

21

2(

)21

(2

42

)21

(1

2

)21

(2

2

)21

(1

2

yxfkkllllllll

b

kkk

c

kkk

a

where k1,k2,l are the ratios from complete grid arm h as shown in fig. (1),

fo is the function of unknown value Φo و

2

2

2

2)2

(

2 c,

2

2

2

2)1

(

1 a

ox

hk

oxhko

ox

hk

oxhko

2

2

2

2 4,

2

2

2

2)(

b oy

hoy

hoando

y

hLoy

hLo

Once initialization is completed, iterations of the algorithm adjust output values so that a

smooth, uniformly representative surface is produced. A Special program using c-

language was written [Delores, 1996] . The program automatically calculate the effect of

stations position on the neighbor nodes by asking the user about the number of unknown

nodes and the number of stations. The program automatically apply the unified Least

l

h

K2 K1

b

a c

4

2

1 3

Figure (1): Boundary coordinates

computation

Squares to equation (6) for every node in the grid of the study area (between (22o<

latitude < 32o) and (25

o < longitude < 37

o)).

Finally, the solution can be given in the form of the adjusted geoid undulation at common

stations on both two datums, and the residuals between two datums at every node on the

grid.

The study has been carried out using HARN network GPS /Levelling data [Powel, S.

1997] see fig. (2). For modeling the geoid in Egypt as an analytical surface, three

interpolation methods were applied. Mathematical algorithms of these three interpolation

methods have been programmed as to comprise the computation modules of a Local

Geoid Modeling software. In this study, 17 of totally 31 GPS/ Levelling points were

decided to be used as reference points (modelling benchmarks), which contributes the

computation of new points according to appropriate interpolation method, and the other

rest 10 GPS/ Leveling points (testing benchmarks) were used for testing the developed

interpolation algorithms.

Figure (2): Recent precise GPS geodetic control networks in Egypt

4. Analysis of Results

A comparison was made between the proposed MCS and both of LSC and the use of

regression model. Based on the results of the determined geoid, as outlined by Tables (1)

and shown by Figures (3) and (4), it is clear that the accuracy obtained by the MCS

technique in the considered area, in terms of the standard deviation of the differences that

amounts to 0.10 m, and with max. distortion 0.035 m at check points are considered

rather satisfactory.

0

0.5

1

1.5

2

2.5

N7R5Y5P4A4E5B3S2A2L2F1

the check points

dis

tort

ion

(m

)

regression MCS lsc

Table (1) : Distortion and standard deviation values at check points

Techniques MCS LSC regression

models

Point Distortion (m) Distortion (m) 10.1682

N7 0.0305 2.0758 0.9590

R5 0.0087 1.4635 0.0510

Y5 0.0081 1.4195 0.6673

P4 0.0095 0.47619 1.3051

A4 0.0169 0.1763 0.8551

E5 0.0046 0.8075 0.7136

B3 0.0198 0.8474 3.1268

S2 0.0173 0.8528 3.4524

A2 0.0294 1.6018 5.0175

L2 0.0230 1.2113 3.6636

F1 0.0358 2.0592 10.1682

Max.distortion 0.0358 2.0758 0.0510

Min.distortion 0.0046 0.1763 2.7254

Average 0.0185 1.1810 2.9375

STDEV 0.0103 0.6124 10.1682

Range 0.0311 1.8995 0.9590

Figure (3): Check points distortion by different surface fitting methods

Figure (4) Contour line of the geoid undulation by using MCS model

The second test was carried out by analyzing the difference between computed geoid

heights from the MCS model and geoid heights from most common geoid models in the

world EGM96 using Geomatics program , and OSU91A as shown in (Scott, 1997). As

the result of these analysis as shown in table (2) and figure (5), it was seen that the model

fitted the data with ±1.032 cm root means square error and max. distortion 3.58 cm at

check points. This is also an indicator for usability of the model for computing the new

points. At the end of the analysis, it was decided that both interpolation method gave

reasonable results for The Egypt and they are applicable for modelling the geoid of this

area as an analytical surface and to serve practical geodetic applications.

The third test were carried out in a local area of " Naser lack", to verify the MCS model

(NMCS) and available data from the NRI(Nobs). As shown in table (3) the different in the

geoid undulation at this zone varied between maximum different (0.833 cm) and

minimum different (0.798 cm) with ± 0.012 cm root means square error. This is also an

indicator for usability of the model for computing the new points of any local area in

Egypt.

Table (2): The comparison between different geoid model at check points

point OSU-91A EGM-96 MCS

N7 2.21 0.827 0.030

R5 0.549 0.376 0.008

Y5 0.669 0.561 0.008

P4 0.708 0.928 0.009

A4 0.387 0.581 0.016

E5 0.701 0.149 0.004

B3 0.453 0.05 0.019

S2 0.226 0.322 0.017

A2 0.456 0.024 0.029

L2 0.835 0.278 0.023

F1 1.068 0.163 0.035

max 2.21 0.928 0.035

min 0.226 0.024 0.004

STD 0.535 0.303 0.010

average 0.751 0.387 0.018

The third test were carried out in a local area of " Naser lack", to verify the MCS model

(NMCS) and available data from the NRI(Nobs). As shown in table (3) the different in the

geoid undulation at this zone varied between maximum different (0.833 cm) and

minimum different (0.798 cm) with ± 0.012 cm root means square error. This is also an

indicator for usability of the model for computing the new points of any local area in

Egypt.

Table (3) The comparison between the geoid undulation ( Nobs and NMCS.)

5. Conclusions and Recommendations

It is obvious that after GPS techniques have become widespread in geodetic practice,

geoid model determination for especially use in practical applications of geodesy has an

increased importance. Hence, the aim of this study was to produce some solutions to

practical applications, such as large scale map production, engineering surveying

applications etc. especially at a national level. Therefore, to test the performances and

applicability of surface fitting methods in geoid modeling using GPS/ Leveling

measurements in a local area, this study was carried out. In the study, three interpolation

techniques, ; regression models; least square collocation with linear basis functions; and

minimum curvature surface (MCS) were applied separately. In the results, it has been

seen that the interpolation methods are applicable for modelling the geoid of study area as

an analytical surface; however, MCS method fits the data better than both method

according to tests criteria.

point latitu. longitu. h H Nobs

(m)

NMCS

(m)

Diff.

dam13 24.032368 32.863689 131.84 121.298 10.5414 10.5497 0.0083 dam12 24.030022 32.860706 153.51 142.927 10.5826 10.5909 0.0083 dam11 24.025503 32.858181 151.8 141.204 10.596 10.6042 0.0082 dam10 24.018234 32.857872 178.13 167.582 10.5476 10.5558 0.0082 dam09 24.014378 32.853733 184.23 173.624 10.6066 10.6147 0.0081 dam08 24.004421 32.852749 186.38 175.894 10.4857 10.4938 0.0081 dam07 23.994453 32.854034 192.19 181.611 10.5789 10.5869 0.0080 dam06 23.989676 32.853607 196.69 186.176 10.5138 10.5218 0.0080 dam04 23.981094 32.853863 200.32 189.819 10.5002 10.5081 0.0079 dam03 23.978214 32.855679 196.4 185.910 10.49 10.4979 0.0079 dam02 23.977167 32.860516 197.81 187.299 10.5101 10.5180 0.0079 dam01 23.976591 32.864426 202.73 191.912 10.8177 10.8257 0.0080 dam00 23.970802 32.868248 200.76 190.209 10.551 10.5589 0.0079

Max. dist. 0.0083

Min. dist. 0.0079

Average 0.0081

STDEV 0.0001

Therefore, the obtained geoid using MCS is highly recommended to be used for any

future rigorous geodetic computation in the newly developed sector of the any part of

Egypt. It is also recommended to be utilized for densifying leveling networks of lower

order through rigorous GPS observations, especially in remote areas of the western or

eastern desert of that part of Egypt. Consequently, appropriate future planning for

rigorous GPS measurements in these areas is highly recommended.

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