normal gravity and the nigerian height system olalekan

FS 1C - Geoid and Gravity - Modelling, Measurements and Applications 1/13 Olalekan Adekunle Isioye Normal Gravity and the Nigerian Height System FIG Congress 2010 Facing the Challenges – Building the Capacity Sydney, Australia, 11-16 April 2010

Normal Gravity and the Nigerian Height System Olalekan Adekunle ISIOYE, Nigeria

Keywords: normal height, normal gravity, normal-orthometric height, height anomaly, quasi-geoid SUMMARY The question of the system of height is intimately connected with the gravity correction to the levelling. Depending on the vertical reference surface to which the height are referred, how rigourously the gravity correction to the levelling is required to be known, numerous height systems are thus defined. This study considers the effect of normal gravity variations on levelled height differences in Nigeria. Also the normal height system which depends on actual and normal gravity for its computation and the normal-orthometric height system which depends solely on geographical latitudes values and normal gravity are discussed and compared for their distortion of the Nigerian levelling network. From the results acquired, the suitability of using normal (theoretical) gravity instead observed gravity values is evaluated and conclusions are drawn on the effect of the normal gravity on the Nigerian levelling network.


Normal Gravity and the Nigerian Height System

Olalekan Adekunle ISIOYE, Nigeria

1.0 INTRODUCTION The primary vertical control network for the country consists mainly of the geodetic levelling points and is supplemented by heights obtained through satellite positioning methods. The geodetic levelling of Nigeria started in 1955. It consists of over 250 lines totalling 20,000km, and covering fairly all parts of the country, although most of the work is concentrated on the South-western and North-western parts of the country. The heights in the geodetic levelling network are referred to the Lagos Survey Datum. The effect of gravity was neglected in computing the height values in use today. This network has formed the backbone for lower-order levelling, which, in turn, has been used for multitudes of surveying, engineering and scientific purpose in the country. According to Ebong (1981) and Ebong et al. (1991) the measures of accuracy (mean accidental error, mean accidental limiting value of the total error, mean accidental value of the systematic error) for the network have been computed by means of formulae adopted for the purpose by the IAG at the 1948 conference in Oslo and results show that the standard of the levelling is comparable with the best anywhere in the world. Uncommon with most other national level networks the observed height differences have not been corrected for the effects of variations in gravity. This discussion concentrates on this aspect of the height system, and compares alternate methods of applying this correction based on normal gravity variation. It is realized that many other aspects of the Nigerian height system, such as systematic levelling error and the definition of the mean sea level , all worthy of further investigation , are being ignored herein. 2.0 THEORETICAL (NORMAL) GRAVITY DUE TO AN ELLIPSOID

The theoretical or normal gravity, or gravity reference field, is the gravity effect due to an

equipotential ellipsoid of revolution. Approximate formulae are used widely even though we can calculate the exact theoretical gravity analytically. Somigliana (1929) developed the first rigorous formula for normal gravity (also, see Heiskanen & Moritz (1967, eq. 2.78); Torge (2001)). The theoretical gravity on the surface of the ellipsoid is given by the formula of Somigliana;

Here, the normal gravity which depends on latitude is represented by the four parameters; a and b are the semi major and semi minor axes of the ellipsoid, respectively; are the theoretical gravity at the equator and poles, respectively; and is the geodetic latitude. Also, Li


and Gotze (2001) derived a closed formula to calculate theoretical gravity at any ellipsoid height and geodetic latitude and reduced latitude can be given as;

Where

The conventionally used International Gravity Formula is obtained by substituting the parameters of the relevant reference ellipsoid into a linearised or approximate form of equation (1). Thus, rewriting equation (1) we have (see, Heiskanen and Moritz, 1967, pp.76-77, eq. 109-116 for detail derivation);

Where,

The International Gravity Formulae of 1930, 1967, and 1980, correspond respectively to the international 1924, GRS 67, and GRS 80, ellipsoids. The corresponding gravity formulae are:

1930 International Gravity Formula (IGF)


1967 Geodetic Reference System (GRS 67)

1980 Geodetic Reference System (GRS 80)

A closed form formula for the WGS 84 normal gravity is according to Meyer et al.

(2005) given by;

The values of derived from equations (6a-c) for every 0.5N change in latitude between

the equator and latitudes that is, the latitude range in Nigeria and also the differences in at the references ellipsoid from the three formulae are given by Osazuwa (1993 and 2007). Osazuwa (2007) attributed the large values in the difference between 1930 IGF and GRS 67 or 1930 IGF and GRS 80 to error in the Potsdam value which was partly used in establishing the 1930 IGF (see, also Li and Gotze (2001)). Similarly, the difference between the 1967 formula and 1980 formula in use today is relatively quite small. According to Osazuwa (1993) using equations (6a-c), the conversion formulae from one reference ellipsoid to the other are given as follows:

i) From IGF 30 to GRS 67;

ii) From IGF 30 to GRS 80;

iii) From GRS 67 to GRS 80;

Normal gravity used in this study is based on Geodetic Reference System 1980 (GRS

80), which was adopted at the XVII General Assembly of the IUGG in Canberra, December 1978 (Moritz, 1980). The rigorous formula for the computation of the normal gravity on the ellipsoid is based on Somigliana’s formula.

3.0 NORMAL HEIGHT SYSTEM

Main differences between the orthometric and normal height system are: the avoidance of hypotheses to determine the gravity field inside the topography, the theoretical replacement of


the Earth’s surface by telluroid and the use of reference ellipsoid with associated gravity field, see Featherstone and Kuhn (2006).

To thwart the problem of determining the integral mean value of actual gravity along the plumbline, the normal height system was introduced by M.S. Molodenski. The key differences from the orthometric height system are: the avoidance of hypotheses to determine the gravity field inside the topography; the theoretical replacement of the Earth’s surface by the telluroid; and the use of a reference ellipsoid with associated gravity field (cf. Featherstone and Kuhn, 2006).

The telluroid is an auxiliary surface obtained by the point–wise projection of points X on the Earth’s surface along the straight–line ellipsoidal normal to points Y that have the same gravity potential value in the normal gravity field as the original point X in the Earth’s gravity field , i.e., (See figure 1). As such, the telluroid is not an equipotential surface. The normal gravity field U is an approximation of the real Earth’s gravity field W and conceptually generated by masses within a reference ellipsoid.

Figure 1: The relation between the normal height, normal-orthometric height, quasi-geoid and

telluroid (after Featherstone and Kuhn, 2006)

From figure 1, the following can be defined; the normal height HN: a curved line distance reckoned along the normal gravity plumbline from the point on the surface of the reference

Y

HN‐O

HN

UY = WX

UY = WX

X

Earth’s Surface

Telluroid

Quasi‐geoid

Reference ellipsoid U0 = W0

Normal plumbline through Y Normal plumbline through X


ellipsoid to the point Y on the surface of the telluroid. The normal – orthometric height HN-O: a curved–line distance reckoned along the normal gravity plumbline from the point on the surface of the quasi-geoid to the point X on the surface of the Earth. The quasi-geoid height :

the straight-line distance reckoned along the ellipsoid surface normal from the point on the

surface of the ellipsoid to the point on the surface of the quasi-geoid has, by definition, the same height as the height anomaly : the straight–line distance reckoned along the ellipsoidal normal from the point X on the Earth’s surface to the point Y on the surface of the telluroid ( cf Featherstone and Kuhn, 2006).

Applying the general concept of a natural physical height system (geopotential number divided by a gravity value) the normal height is obtained by using the integral-mean value of normal gravity taken along the normal plumbline between the ellipsoid surface and the projected point on the telluroid. Thus;

where

where

can be solved iteratively. In practice, like orthometric heights, normal heights are computed from spirit–levelled height differences by adding the normal correction. Thus the normal height difference between points A and B is represented by

Where is the spirit levelled height increment and the normal correction is:

Where the first term is an integral along the spirit levelling path. This discretises to

In equations (11) and (12), and denote the integral mean values of normal gravity along

the plumblines that pass through points A and B and is normal gravity at 45o geodetic latitude.


3.1 Normal-Orthometric Height System The normal orthometric system uses only the normal gravity field as an approximation of

the Earth’s gravity field to derive all necessary gravity- field -related quantities, which is in contrast to orthometric and normal height systems that require actual gravity observations to be taken along the levelling traverse in order to derive the geopotential numbers (or the orthometric or normal corrections).

The geometric interpretation of the normal orthometric height is analogous with that of the normal height, except that the relations are reversed (figure 1). The height anomaly becomes the separation between the reference ellipsoid and the quasi-geoid (measured along the ellipsoid normal) and the normal–orthometric height is now the distance between the quasi geoid and point of interest (measured along the normal plumbline). Following this interpretation, the reference surface for the normal-orthometric height is the quasi-geoid and it can be seen that the normal-orthometric height follows exactly the same principles of orthometric heights, except that all quantities of the Earth’s gravity field are replaced by the corresponding quantities of the normal gravity field, hence the name normal orthometric height.

Finally, normal-orthometric heights can be computed in practice from spirit levelled height differences using the normal–orthometric correction. Thus we have;

Where is the spirit levelled height increment and the normal orthometric correction is:

after some manipulation the normal orthometric correction between two successive levelling points A and B is simplified to

where 4.0 COMPUTATIONS AND ANALYSIS

The normal and normal-orthometric heights as described in the preceding section were

computed for the Nigerian levelling network (see appendices (1)), in practice, some issues were addressed which include the appropriateness of adopting normal gravity at 45o latitude. It was suggested that the adoption of this value leads to systematically large errors in areas located at a significant separation from mid-latitudes. In this paper Normal and Normal-orthometric height differences are computed by adopting mean latitude value 90 which is the mean latitude value for Nigeria and the adoption of this value has helped to cushion the effect of the problem raised above. Also, normal gravity values were computed based on GRS 80 using Somigliana rigorous formula, the predicted values of observed gravity for the benchmark under investigation was


done using multiquadric method. See Isioye (2008) for the values of the least squares free net adjusted Normal and Normal-orthometric height differences for Nigeria. The sum of the normal and normal- orthometric height corrections are 0.04030m and -0.14670m, respectively.

To investigate the relationship between least squares adjusted heights(uncorrected) for gravity and the normal gravity measurement, a scatter plot was drawn using SPSS and the chart gives an insight into the relationship that exists between the variables as shown in figure 2.

Figure2 : Regression analysis (Normal Gravity Vs Adjusted Uncorrected Heights)

The above chart does not seem to suggest that there is a relationship between the height system and the normal gravity measurement. The line of best fit (regression line) was fitted into the graph and report s that there is a relationship which it puts at 0.07 between the two variables. Since the scatter plot is just a suggestion, the Pearson moment correlation coefficient was carried out to confirm if the relationship between these two variables is significant or not. The hypotheses in this case are; Ho: There is no significant relationship between adjusted height and normal gravity measurement. H1: There is a significant relationship between adjusted height and normal gravity

measurement. The result is displayed in Table (3):


Table 3: Results of Correlation Test between Normal Gravity and Adjusted Heights Normal Gravity

(mgal) Adjusted Heights

Normal Gravity (mgal) Pearson Correlation Sig. (2-tailed) N

1.000 - 162

.189*

.023 146

Adjusted Height Pearson Correlation Sig. (2-tailed) N

.189*

.023 146

1.000 - 162

* Correlation is significant at the 0.05 level (2-tailed)

Comparing the value of the r-cal which is 0.189 and the r-tab which is 0.023, it is evident that r-cal is greater than r-tab; hence we reject the null hypothesis and accept the alternative hypothesis which means that there is significant relationship between adjusted height and the normal gravity measurement.

The same analysis is carried out to measure the level of relationship between adjusted height and the predicted gravity measurement. At first the scatter plot was drawn and the resulted chart is displayed in Figure (3) below:

Figure 3: Regression analysis (Predicted Gravity Vs Adjusted Uncorrected Heights)

The above chart shows that there appears to be a cluster of values within a specific range of figures; the regression line shows that the relationship is very weak which it puts at 0.06, the next question to be asked now is “Is this relationship Significant?” To answer this question, the Pearson moment correlation coefficient was again employed and the result is displayed in table 4:


Table 4: Results of Correlation Test between Normal Gravity and Adjusted Heights

Normal Gravity (mgal)

Adjusted Heights

Normal Gravity (mgal) Pearson Correlation Sig. (2-tailed) N

1.000 - 162

.157

.058 146

Adjusted Height Pearson Correlation Sig. (2-tailed) N

.157

.058 146

1.000 - 162

Comparing the value of the r-cal which is 0.157 and the r-tab which is 0.058, it is evident that r-cal is greater than r-tab, hence we reject the null hypothesis and accept the alternative hypothesis which means that there is significant relationship between adjusted height and the predicted gravity measurement.

Wanting to know if normal gravity differs significantly from the predicted gravity, the descriptive were first considered and the table of descriptive is displayed in Table (5):

Table 5: Group Statistics (Normal Vs Predicted Gravity)

Gravity Type N Mean Std. Deviation Std Error of Mean

VARIABLE Predicted Gravity Normal Gravity

162 162

978089.3 978078.0

47.4423 24.9792

3.7274 1.9626

Taking a good look at the mean measure of both methods, it could almost be concluded

that there is no difference in the measurement of both methods. The t-test was carried out as a confirmatory statistical tool and the result is displayed in Table (6) :

Table 6: t-test result showing significant difference between normal gravity and predicted gravity

t-test for Equality of Means

T df Sig. (2-tailed)

VARIABLE Equal

variances assumed

2.68 322 0.008


Comparing the value of t-cal (2.68) and that of t-tab (0.008), it can be seen that t-cal is greater than t-tab, under this situation we reject the null hypothesis and accept the alternative hypothesis which states that there is significant difference between normal gravity measurement and predicted gravity measurement.

CONCLUDING REMARKS This paper has reviewed and compared two height systems (normal and normal-orthometric heights) which have very strong dependence on normal gravity values in an attempt to answer the question of advisability of using normal instead of observed gravity for correcting height values in Nigeria. The results obtained indicate a strong correlation between normal gravity values and height values in the country, ironically normal gravity values and gravity values predicted from actual gravity observation did not show any significant relationship. In view of the variation observed , the sum normal and normal-orthometric corrections (0.04030m and -0.14670m, respectively)were compared to those of the different variants of orthometric heights(Helmert, Baranov, Vignal, Ramsayer and Ledersteger’s with respective sums of 0.01674m, -0.07596m, -0.04177m, 0.04665m, 0.11015m) computed for the Nigerian levelling by Isioye(2008) show no remarkable improvement or degradation on the national levelling network . The benefits of the choice of the normal-orthometric height system includes the fact that it is simple to compute (no hypotheses about topographic density or terrain correction is required), and also it geometric relationship with the quasi geoid; quasi geoid models are generally easier to compute than geoid models, normal-orthometric heights are compatible with GPS when derived from quasi geoid.

REFERENCES

(1) Ebong, M. B. (1981). A Study and Analysis of the Geodetic Levelling Of Nigeria. PhD

Thesis, University of Newcastle upon Tyne.

(2) Ebong, M. B., Adaminda, I. J. K., Osazuwa, I. B. (1991). Some Aspects of the Geodetic Networks in Nigeria. Allgemeine Vermessungs- Nachrichten, vol. 8, pp. 16-26.

(3) Featherstone, W. E., and Kuhn, M. (2006). Height Systems and Vertical Datums: A Review

in the Australian Context, Journal of Spatial Sciences, 51(11), pp.21-42. (4) Heiskanen, W. A., and Moritz, H. (1967). Physical Geodesy. Freeman and company, San

Francisco, 364 pages. (5) Isioye, O. A. (2008). Options for the Definition of the Nigerian Height Datum With a View to

Its Unification in a Global Datum. M.Sc Thesis, Department of Surveying and Geo-Informatics, University of Lagos, Lagos-Nigeria, 217pages.


(6) Li. X., and Gotze, H. J. (2001). Ellipsoid, Geoid, Gravity, Geodesy, and Geophysics. Geophysics, vol. 66, no. 6, pp.1160-1668.

(7) Meyer, T. H., Roman, D.R., and Zilkoski, D.B. (2005). What Does Height Really Mean? Part

I: Introduction. Surveying and Land Information Science, 64(4), pp. 223-234.

(8) Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Geodesique, 54, pp.395-405. (9) Osazuwa, I. B. (1993). An Evaluation Study of the Correct Computational Approach for the

Conversion of Gravimetric Data from One Reference Datum to Another. Survey Review, 32, 249, pp.167-174.

(10) Osazuwa, I. B. (2007). The Geodetic Reference Systems and the Need for a Common

Reference Datum for Gravimetric Studies in Nigeria. A Paper Presented at the Workshop on the Relevance of Gravity Measurement to Geodesy and Geodynamics, Toro, Bauchi-Nigeria, November 19-20, 2007.

(11) Torge, W. (2001). Geodesy, Third Edition, Waller The Gruyter, Berlin, 416pages. BIOGRAPHICAL NOTE ISIOYE Olalekan Adekunle (B.Sc, M.Sc.) is a lecturer at the Ahmadu Bello University, Nigeria, in the Faculty of Engineering. He is a member of Nigerian Association of Geodesy (NAG) and also an associate member of the Nigerian Institution of Surveyors (NIS). His research interests include GNSS science and application, vertical reference frame studies and geodynamic. CONTACTS Olalekan Adekunle ISIOYE Department of Geomatics Engineering, Faculty of Engineering Ahmadu Bello University, Zaria-Nigeria Email: [email protected], Mobile: +234-8036055824


Appendix (1): Distribution of First Order Geodetic Levelling Benchmarks in Nigeria

normal gravity and the nigerian height system olalekan

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