Tailored High-Degree Geopotential Model for Egypt

Hussein A. Abd-Elmotaal and Mostafa Abd-Elbaky

Civil Engineering Department, Faculty of Engineering,

Minia University, Minia 61111, Egypt

abdelmotaal@lycos.com

Abstract

High-degree reference geopotential model tailored to the Egyptian gravity fieldis developed within this investigation. The window technique (Abd-Elmotaal andKühtreiber, 2003), within the remove-restore scheme, has been applied to get ridof the double consideration of the topographic-isostatic masses within the datawindow. The high-degree tailored reference model has been created by merging theavailable gravity anomalies in the area of investigation with the global gravitationaldata set. Such a global data set has been created by the available EGM2008reference geopotential model (complete to degree and order 360). The mergedglobal field has been then used to estimate the harmonic coefficients of the tailoredreference model by FFT and Gauss techniques. Since the low order harmonics ofthe anomalous gravitational potential are to a great extent due to the densitydisturbances in the upper mantle and even deeper sources, the lower harmonicstill degree 20 have been fixed to their values as of EGM2008 geopotential model.The results show that the tailored geopotential models give better residual gravityanomalies. The variance has dropped by its one third.

Keywords. geopotential models, harmonic analysis, window technique, Egypt,geoid determination.

1 Introduction

The quality of the reference geopotential model used in the framework of the remove-restore technique plays a great role in estimating the accuracy of the computed geoid. Inother words, if the residual field is biased and has a high variance, then using such a bi-ased/high variance field in the geoid computation process gives less accurate interpolatedquantities, and hence worse geoid fitting to the GPS-levelling derived geoid. Practicalstudies so far have proved that none of the existing reference geopotential models fit theEgyptian gravity field to the desired extent. Thus, the main aim of this investigation isto have a smoothed gravity field (in terms of gravity anomalies) so that it is, more orless, unbiased and has a significantly reduced variance by using a high-degree tailoredreference geopotential model.

The used data sets are described first. The window technique (Abd-Elmotaal andKühtreiber, 2003) within the remove-restore technique has been outlined. The localgravity anomalies for the Egyptian data window are reduced using the remove-restore

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Geodetic Week, Essen, Germany, October 8–10, 2013

window technique and gridded at 30′×30′ grid using the Kriging interpolation technique.The local gridded data are merged with the global 30′×30′ gravity anomalies, computedusing EGM2008 till N = 360 (Pavlis et al., 2008; 2012), to establish the data set forcomputing the tailored geopotential models. The merged 30′ × 30′ global field has beenthen used to estimate the harmonic coefficients of the tailored reference model by anFFT technique (Abd-Elmotaal, 2004) as well as by Gauss technique (Abd-Elmotaal etal., 2013 a). A wide comparison among the developed tailored geopotential models andEGM2008 geopotential model has been carried out.

It should be noted that many scholars have tried to compute tailored geopotentialmodels to best suit their specific areas of interest. The reader may refer, e.g., to (Wenzel,1998; Abd-Elmotaal, 2007; Abd-Elmotaal et al., 2013 b).

2 The Data

2.1 Local Egyptian Free-Air Gravity Anomalies

All currently available sea and land free-air gravity anomalies for Egypt and neighbouringcountries have been merged. A scheme for gross-error detection has been carried out.Figure 1 shows the distribution of the free-air gravity anomalies for Egypt used for thecurrent investigation. The distribution of the free-air gravity anomaly stations on landis very poor. Many areas are empty. The distribution of the data points at the Red seais better than that at the Mediterranean sea.

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Figure 1: Distribution of the local Egyptian free-air gravity anomalies.

A total number of 102418 gravity values are available. The free-air gravity anomaliesrange between −210.60 mgal and 314.99 mgal with an average of −27.58 mgal and a

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standard deviation of about 50.65 mgal. Highest values are in sea region.

2.2 Digital Height Models

For the terrain reduction computation, a set of fine and coarse Digital Height ModelsDHM’s is needed. The fine EGH13S03 3

′′ × 3′′ and the coarse EGH13S30 30′′ × 30′′DHM’s (Abd-Elmotaal et al., 2013 c) are used for the current investigation. They coverthe window 19◦ ≤ ϕ ≤ 35◦, 22◦ ≤ λ ≤ 40◦. Figure 2 illustrates the EGH13S03 fineDHM.

Figure 2: The 3′′ × 3′′ EGH13S03 Digital Height Model (after Abd-Elmotaal et al.,

2013 c). Values are in meters.

3 The Window Technique

Within the well-known remove-restore technique (Forsberg, 1984), the effect of thetopographic-isostatic masses is removed from the source gravitational data and thenrestored to the resulting geoidal heights. For example, in the case of gravity data, thereduced gravity anomalies in the framework of the remove-restore technique is computedby

∆g = ∆gF −∆gTC −∆gGM , (1)where ∆gF stands for the free-air anomalies, ∆gTC is the terrain reduction due to thetopographic-isostatic masses, and ∆gGM is the effect of the reference field (global geopo-tential model) on the gravity anomalies. Thus the final computed geoid N within theremove-restore technique can be expressed by:

N = NGM +N∆g +NTC , (2)

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where NGM gives the contribution of the reference field, N∆g gives the contribution ofthe reduced gravity anomalies, and NTC gives the contribution of the topography andits compensation (the indirect effect).

The traditional way of removing the effect of the topographic-isostatic masses facesa theoretical problem. A part of the influence of the topographic-isostatic masses isremoved twice as it is already included in the global reference field. This leads to somedouble consideration of that part of the topographic-isostatic masses. Figure 3 showsschematically the traditional gravity reduction for the effect of the topographic-isostaticmasses. The short-wavelength part depending on the topographic-isostatic masses iscomputed for a point P for the masses inside the circle (say till 167 km around the com-putational point P ; denoted by TC in Fig. 3). Removing the effect of the long-wavelengthpart by a global earth’s gravitational reference field normally implies removing the influ-ence of the global topographic-isostatic masses (shown as a the full rectangle in Fig. 3).The double consideration of the topographic-isostatic masses inside the circle (doublehatched) is then seen.

P

EGM08

TC

.

Figure 3: The traditional remove-restore technique.

A possible way to overcome this difficulty in computing geoidal heights from topo-graphic-isostatic reduced gravity anomalies is to adapt the used reference field due to theeffect of the topographic-isostatic masses for a fixed local data window (denoted by thesmall rectangle in Fig. 4). Figure 4 shows the advantage of the window remove-restoretechnique. Let us concentrate on the influence of the topographic-isostatic masses inthe reduction process. Consider a measurement at point P , the short-wavelength partdepending on the topographic-isostatic masses is now computed by using the masses ofthe whole data area (the small rectangle in Fig. 4). The adapted reference field is createdby subtracting the effect of the topographic-isostatic masses of the data window, in termsof potential coefficients, from the reference field coefficients (adapted reference field isthen represented by the area of the big rectangle, representing the globe, minus the areaof the small rectangle, representing the local data area; cf. Fig. 4). Thus, removing thelong-wavelength part by using this adapted reference field does not lead to a doubleconsideration of a part of the topographic-isostatic masses (there is no double hatchedarea in Fig. 4).

The remove step of the window remove-restore technique can then mathematicallybe written as

∆gred = ∆gF −∆gTC win −∆gGM Adapt , (3)where ∆gGM Adapt is the contribution of the adapted reference field and ∆gTC win standsfor the terrain reduction of the topographic-isostatic masses for a fixed data window.The restore step of the window remove-restore technique can be written as

N = NGM Adapt +N∆g +NTC win , (4)

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a

adapted EGM08

P

data area

TC.

Figure 4: The window remove-restore technique.

where NGM Adapt gives the contribution of the adapted reference field and NTC win givesthe contribution of the topography and its compensation (the indirect effect) for thesame fixed data window as used for the remove step.

The reader who is interested in computing the harmonic coefficients of the topographic-isostatic potential of the data window is kindly referred to (Abd-Elmotaal and Kühtreiber,1999; 2003).

4 Gravity Anomalies

The SRTM 30′ × 30′ global DHM (Farr et al., 2007) has been used to compute the

harmonic coefficients of the global topographic-isostatic masses TTI tillN =360. Figure 5shows the SRTM 30

′×30′ global DHM. The heights range between −8756 m and 5719 mwith an average of about −1892 m and standard deviation of 2638 m.

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Figure 5: SRTM 30′ × 30′ global DHM. Values are in meters.

The global isostatic harmonic coefficients are created by subtracting the harmoniccoefficients of the global topographic-isostatic masses TTI from those of the EGM2008model (Pavlis et al., 2008, 2012). These global isostatic harmonic coefficients, till N =360, are used to generate a global 30

′ × 30′ field of isostatic anomalies.The available gravity anomalies within the local Egyptian window (19◦ ≤ ϕ ≤ 35◦,

22◦ ≤ λ ≤ 40◦) have been reduced using the window remove-restore technique and then

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interpolated in 30′ × 30′ grid using Kriging interpolation technique. Figure 6 shows the

local Egyptian 30′ × 30′ interpolated anomalies.

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Figure 6: The local Egyptian 30′ × 30′ interpolated gravity anomalies. Contour interval:

10 mgal.

The local Egyptian 30′ × 30′ interpolated gravity anomalies have been merged with

the created 30′ × 30′ global gravity anomalies forming the data set for computing the

tailored geopotential model for Egypt. Figure 7 shows that merged field.

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Figure 7: The global 30′ ×30′ merged field including Egyptian gravity anomalies. Values

are in mgal.

Table 1 illustrates the statistics of the used three gravity anomalies fields. Thestatistics show that merging the Egyptian gravity anomalies data set has a minor effect

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on the global gravity field.

Table 1: Statistics of the used three gravity anomalies fields

statistical parameters

gravity anomalies type min. max. average st. dev.

mgal mgal mgal mgal

Global isostatic (EGM2008) −337.0 273.2 −0.85 27.2Local isostatic (Egypt) −190.0 202.0 0.25 36.7

Merged isostatic (EGM2008 + Egypt) −337.0 273.2 −0.86 27.3

5 Tailored Geopotential Models for Egypt

Let us consider an analytical function f(θ, λ) defined on the unit sphere (0 ≤ θ ≤ π and0 ≤ λ ≤ 2π). Expand f(θ, λ) in series of surface spherical harmonics (cf. Abd-Elmotaal,2004)

f(θ, λ)=∞∑n=0

n∑m=0

(C̄nm cosmλ+ S̄nm sinmλ

)P̄nm(cos θ) , (5)

where C̄nm and S̄nm are the fully normalized spherical harmonic coefficients and P̄nm(cos θ)refers to the fully normalized associated Legendre function. Let us introduce the abbre-viations

R̄nm(θ, λ) = P̄nm(cos θ) cosmλ ,

Q̄nm(θ, λ) = P̄nm(cos θ) sinmλ .(6)

It is well known that the fully normalized harmonic coefficients are orthogonal, i.e.,they satisfy the orthogonality relations∫∫

σ

R̄nm(θ, λ) R̄n′m′(θ, λ) dσ = 0 (7)

∫∫σ

Q̄nm(θ, λ) Q̄n′m′(θ, λ) dσ = 0 (8)

∫∫σ

R̄nm(θ, λ) Q̄nm(θ, λ) dσ = 0 , (9)

1

4π

∫∫σ

R̄2nm(θ, λ) =1

4π

∫∫σ

Q̄2nm(θ, λ) = 1 . (10)

As a consequence of the orthogonality, the fully normalized harmonic coefficients C̄nmand S̄nm can be given by (Heiskanen and Moritz, 1967, p. 31)

C̄nm =1

4π

∫∫σ

f(θ, λ) R̄nm(θ, λ) dσ ,

S̄nm =1

4π

∫∫σ

f(θ, λ) Q̄nm(θ, λ) dσ .(11)

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In fact, (11) cannot be used in practice to compute the harmonic coefficients simplybecause the analytical function f(θ, λ) is generally unavailable. Only a finite set of noisymeasurements f(θi, λj), covering the whole sphere, might be available. Discretizing(11) on an equal angular grid covering the whole sphere gives the following quadraturesformula

Ĉnm=1

4π

N−1∑i=0

2N−1∑j=0

f(θi, λj) R̄nm(θi, λj)∆ij ,

Ŝnm=1

4π

N−1∑i=0

2N−1∑j=0

f(θi, λj) Q̄nm(θi, λj)∆ij ,

(12)

where Ĉnm and Ŝnm are the estimate of C̄nm and S̄nm, respectively, ∆ij indicates thesegment area and N is the number of grid cells in the latitude direction. Expression (12)is used to compute the harmonic coefficients if the available data field is represented bya set of point values f(θi, λj), and is known as the Gauss technique. It should be notedthat (12) is usually only an approximation due to the discretization effect of f(θ, λ).

If all C̄nm and S̄nm are known till degree and order Nmax, one can compute f̃(θi, λj)as follows:

f̃(θi, λj) =Nmax∑n=0

n∑m=0

(C̄nm cosmλj + S̄nm sinmλj

)P̄nm(cos θi) , (13)

which can be regarded as an approximation to f(θ, λ) at point (θi, λj). Expression (13)defines the object of spherical harmonic synthesis: given the coefficients, it is required toestimate the function.

The double summation appearing in (12) for harmonic analysis as well as in (13) forspherical harmonic synthesis are computed using FFT. Colombo (1981) has written twosubroutines for this purpose, called HARMIN and SSYNTH. Abd-Elmotaal (2004) haswritten a program called HRCOFITR, which uses Colombo’s subroutines, after greatmodifications, in an iteration process to obtain the best coefficients accuracy and mini-mum residual filed either the given filed is on sphere or on ellipsoid.

The merged 30′× 30′ global field has been used to estimate the harmonic coefficientsof the tailored reference model by HRCOFITR Program (Abd-Elmotaal, 2004), calledFFT technique. Also the Gauss technique (12) has been used to estimate the harmoniccoefficients of the tailored model using the merged 30′ × 30′ global field as input. Sincethe low order harmonics of the anomalous gravitational potential are to a great extentdue to the density disturbances in the upper mantle and even deeper sources, the lowerharmonics till N = 20 have been fixed to their values as of EGM2008 geopotential model.

Figure 8 shows the tailored geopotential model using the FFT harmonic analysistechnique compared to the EGM2008 goepotential model. To improve the estimationof the harmonic coefficients using the Gauss harmonic analysis technique, an iterativeprocess takes place. Figure 9 shows the tailored geopotential model using the Gaussharmonic analysis technique after 40 iterations. Comparing Figs. 8 and 9 shows thatboth harmonic analysis techniques give similar harmonic coefficients.

Figure 10 shows the differences at the grid points between the Egyptian 30′ × 30′

interpolated gravity anomalies and the computed gravity anomalies using the tailoredgeopotential model computed by the FFT harmonic analysis technique. Figure 11 showsthe same differences using the Gauss harmonic analysis technique. The white areasindicate differences less than 10 mgal in magnitude. Both figures show good matchingespecially in the data areas.

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Figure 8: The tailored geopotential model using the FFT harmonic analysis technique.

Figure 9: The tailored geopotential model using the Gauss harmonic analysis techniqueafter 40 iterations.

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Figure 10: Differences between Egyptian 30′ × 30′ interpolated gravity anomalies and

the computed gravity anomalies using the FFT tailored geopotential model. Contourinterval: 10 mgal.

Table 2: Statistics of the differences between the Egyptian gridded gravity anomaliesand those computed by the tailored geopotential models

statistical parameters

anomalies difference min. max. range average std

mgal mgal mgal mgal mgal

Data - EGM2008 −115.66 71.15 168.81 0.63 17.17Data - FFT tailored model −100.71 68.53 169.24 1.20 13.55Data - Gauss tailored model −100.75 68.58 169.33 1.20 13.55

Table 2 illustrates the statistics of the differences between the gridded Egyptiangravity anomalies data and those computed by the tailored geopotential models createdwithin this investigation. Table 2 shows that both developed geopotential models givepractically the same results. The tailored geopotential models created in this investiga-tion give better residual gravity anomalies (unbiased and have much less variance thanthat of EGM2008). The variance has dropped by its one third. The range has droppedby about 20 mgal.

6 Conclusions

Two high-degree tailored reference geopotential models for Egypt have been developedin this investigation. The window remove-restore technique has been applied to get ridof the double consideration of the topographic-isostatic masses within the data window.

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Figure 11: Differences between Egyptian 30′ × 30′ interpolated gravity anomalies and

the computed gravity anomalies using the Gauss tailored geopotential model. Contourinterval: 10 mgal.

The FFT technique using an iteration process has been used to recover the harmoniccoefficients to enhance the accuracy of the obtained harmonic coefficients and to minimizethe residual field. The Gauss harmonic analysis technique with an iterative process toimprove the quality of the obtained harmonic coefficients has also been used to estimatethe tailored geopotential model for Egypt. The lower harmonic coefficients till N = 20have been fixed to their values of EGM2008 as they are to a great extent due to thedensity disturbances in the upper mantle and even deeper sources. Both FFT and Gaussharmonic analysis techniques give nearly the same results in the space and frequencydomains. The tailored geopotential models created in this investigation give betterresidual gravity anomalies (unbiased and have much less variance). Hence they are moresuitable for gravity interpolation and geoid determination for Egypt. The variance hasdropped by its one third. The range has dropped by about 20 mgal.

Acknowledgements

This project was supported financially by the Science and Technology Development Fund(STDF), Egypt, Grant No. 366.

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