Tailored High-Degree Geopotential Model for Egypt

Transcription

1 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 Abstract High-degree reference geopotential model tailored to the Egyptian gravity field is developed within this investigation. The window technique (Abd-Elmotaal and Kühtreiber, 23), within the remove-restore scheme, has been applied to get rid of the double consideration of the topographic-isostatic masses within the data window. The high-degree tailored reference model has been created by merging the available gravity anomalies in the area of investigation with the global gravitational data set. Such a global data set has been created by the available EGM28 reference geopotential model (complete to degree and order 36). The merged global field has been then used to estimate the harmonic coefficients of the tailored reference model by FFT and Gauss techniques. Since the low order harmonics of the anomalous gravitational potential are to a great extent due to the density disturbances in the upper mantle and even deeper sources, the lower harmonics till degree 2 have been fixed to their values as of EGM28 geopotential model. The results show that the tailored geopotential models give better residual gravity anomalies. 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 removerestore technique plays a great role in estimating the accuracy of the computed geoid. In other words, if the residual field is biased and has a high variance, then using such a biased/high variance field in the geoid computation process gives less accurate interpolated quantities, and hence worse geoid fitting to the GPS-levelling derived geoid. Practical studies so far have proved that none of the existing reference geopotential models fit the Egyptian gravity field to the desired extent. Thus, the main aim of this investigation is to have a smoothed gravity field (in terms of gravity anomalies) so that it is, more or less, unbiased and has a significantly reduced variance by using a high-degree tailored reference geopotential model. The used data sets are described first. The window technique (Abd-Elmotaal and Kühtreiber, 23) within the remove-restore technique has been outlined. The local gravity anomalies for the Egyptian data window are reduced using the remove-restore 1 Geodetic Week, Essen, Germany, October 8 1, 213

2 window technique and gridded at 3 3 grid using the Kriging interpolation technique. The local gridded data are merged with the global 3 3 gravity anomalies, computed using EGM28 till N = 36 (Pavlis et al., 28; 212), to establish the data set for computing the tailored geopotential models. The merged 3 3 global field has been then used to estimate the harmonic coefficients of the tailored reference model by an FFT technique (Abd-Elmotaal, 24) as well as by Gauss technique (Abd-Elmotaal et al., 213 a). A wide comparison among the developed tailored geopotential models and EGM28 geopotential model has been carried out. It should be noted that many scholars have tried to compute tailored geopotential models to best suit their specific areas of interest. The reader may refer, e.g., to (Wenzel, 1998; Abd-Elmotaal, 27; Abd-Elmotaal et al., 213 b) The Data Local Egyptian Free-Air Gravity Anomalies All currently available sea and land free-air gravity anomalies for Egypt and neighbouring countries 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 the current investigation. The distribution of the free-air gravity anomaly stations on land is very poor. Many areas are empty. The distribution of the data points at the Red sea is better than that at the Mediterranean sea Figure 1: Distribution of the local Egyptian free-air gravity anomalies. A total number of gravity values are available. The free-air gravity anomalies range between 21.6 mgal and mgal with an average of mgal and a 2

3 standard deviation of about 5.65 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 Models DHM s is needed. The fine EGH13S3 3 3 and the coarse EGH13S3 3 3 DHM s (Abd-Elmotaal et al., 213 c) are used for the current investigation. They cover the window 19 ϕ 35, 22 λ 4. Figure 2 illustrates the EGH13S3 fine DHM. Figure 2: The 3 3 EGH13S3 Digital Height Model (after Abd-Elmotaal et al., 213 c). Values are in meters. 3 The Window Technique Within the well-known remove-restore technique (Forsberg, 1984), the effect of the topographic-isostatic masses is removed from the source gravitational data and then restored to the resulting geoidal heights. For example, in the case of gravity data, the reduced gravity anomalies in the framework of the remove-restore technique is computed by g = g F g T C g GM, (1) where g F stands for the free-air anomalies, g T C is the terrain reduction due to the topographic-isostatic masses, and g GM is the effect of the reference field (global geopotential model) on the gravity anomalies. Thus the final computed geoid N within the remove-restore technique can be expressed by: N = N GM + N g + N T C, (2) 3

4 where N GM gives the contribution of the reference field, N g gives the contribution of the reduced gravity anomalies, and N T C gives the contribution of the topography and its compensation (the indirect effect). The traditional way of removing the effect of the topographic-isostatic masses faces a theoretical problem. A part of the influence of the topographic-isostatic masses is removed twice as it is already included in the global reference field. This leads to some double consideration of that part of the topographic-isostatic masses. Figure 3 shows schematically the traditional gravity reduction for the effect of the topographic-isostatic masses. The short-wavelength part depending on the topographic-isostatic masses is computed for a point P for the masses inside the circle (say till 167 km around the computational point P ; denoted by T C in Fig. 3). Removing the effect of the long-wavelength part by a global earth s gravitational reference field normally implies removing the influence 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 (double hatched) is then seen.. P TC EGM8 Figure 3: The traditional remove-restore technique. A possible way to overcome this difficulty in computing geoidal heights from topographic-isostatic reduced gravity anomalies is to adapt the used reference field due to the effect of the topographic-isostatic masses for a fixed local data window (denoted by the small rectangle in Fig. 4). Figure 4 shows the advantage of the window remove-restore technique. Let us concentrate on the influence of the topographic-isostatic masses in the reduction process. Consider a measurement at point P, the short-wavelength part depending on the topographic-isostatic masses is now computed by using the masses of the whole data area (the small rectangle in Fig. 4). The adapted reference field is created by subtracting the effect of the topographic-isostatic masses of the data window, in terms of potential coefficients, from the reference field coefficients (adapted reference field is then represented by the area of the big rectangle, representing the globe, minus the area of the small rectangle, representing the local data area; cf. Fig. 4). Thus, removing the long-wavelength part by using this adapted reference field does not lead to a double consideration of a part of the topographic-isostatic masses (there is no double hatched area in Fig. 4). The remove step of the window remove-restore technique can then mathematically be written as g red = g F g T C win g GM Adapt, (3) where g GM Adapt is the contribution of the adapted reference field and g T C win stands for 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 = N GM Adapt + N g + N T C win, (4) 4

5 a TC P data area. adapted EGM8 Figure 4: The window remove-restore technique. where N GM Adapt gives the contribution of the adapted reference field and N T C win gives the contribution of the topography and its compensation (the indirect effect) for the same fixed data window as used for the remove step. The reader who is interested in computing the harmonic coefficients of the topographicisostatic potential of the data window is kindly referred to (Abd-Elmotaal and Kühtreiber, 1999; 23). 4 Gravity Anomalies The SRTM 3 3 global DHM (Farr et al., 27) has been used to compute the harmonic coefficients of the global topographic-isostatic masses T T I till N = 36. Figure 5 shows the SRTM 3 3 global DHM. The heights range between 8756 m and 5719 m with an average of about 1892 m and standard deviation of 2638 m Figure 5: SRTM 3 3 global DHM. Values are in meters. The global isostatic harmonic coefficients are created by subtracting the harmonic coefficients of the global topographic-isostatic masses T T I from those of the EGM28 model (Pavlis et al., 28, 212). These global isostatic harmonic coefficients, till N = 36, are used to generate a global 3 3 field of isostatic anomalies. The available gravity anomalies within the local Egyptian window (19 ϕ 35, 22 λ 4 ) have been reduced using the window remove-restore technique and then 5

6 interpolated in 3 3 grid using Kriging interpolation technique. Figure 6 shows the local Egyptian 3 3 interpolated anomalies Figure 6: The local Egyptian 3 3 interpolated gravity anomalies. Contour interval: 1 mgal. The local Egyptian 3 3 interpolated gravity anomalies have been merged with the created 3 3 global gravity anomalies forming the data set for computing the tailored geopotential model for Egypt. Figure 7 shows that merged field Figure 7: The global 3 3 merged field including Egyptian gravity anomalies. Values are in mgal. Table 1 illustrates the statistics of the used three gravity anomalies fields. The statistics show that merging the Egyptian gravity anomalies data set has a minor effect 6

7 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 (EGM28) Local isostatic (Egypt) Merged isostatic (EGM28 + Egypt) Tailored Geopotential Models for Egypt Let us consider an analytical function f(θ, λ) defined on the unit sphere ( θ π and λ 2π). Expand f(θ, λ) in series of surface spherical harmonics (cf. Abd-Elmotaal, 24) n ( f(θ, λ)= Cnm cos mλ + S nm sin mλ ) Pnm (cos θ), (5) n= m= where C nm and S nm are the fully normalized spherical harmonic coefficients and P nm (cos θ) refers to the fully normalized associated Legendre function. Let us introduce the abbreviations R nm (θ, λ) = P nm (cos θ) cos mλ, Q nm (θ, λ) = P nm (cos θ) sin mλ. It is well known that the fully normalized harmonic coefficients are orthogonal, i.e., they satisfy the orthogonality relations R nm (θ, λ) R n m (θ, λ) d = (7) 1 4π (6) Q nm (θ, λ) Q n m (θ, λ) d = (8) R nm (θ, λ) Q nm (θ, λ) d =, (9) R nm(θ, 2 λ) = 1 4π Q 2 nm(θ, λ) = 1. (1) As a consequence of the orthogonality, the fully normalized harmonic coefficients C nm and S nm can be given by (Heiskanen and Moritz, 1967, p. 31) C nm = 1 f(θ, λ) 4π R nm (θ, λ) d, S nm = 1 f(θ, λ) 4π Q (11) nm (θ, λ) d. 7

8 In fact, (11) cannot be used in practice to compute the harmonic coefficients simply because the analytical function f(θ, λ) is generally unavailable. Only a finite set of noisy measurements f(θ i, λ j ), covering the whole sphere, might be available. Discretizing (11) on an equal angular grid covering the whole sphere gives the following quadratures formula Ĉ nm = 1 N 1 4π i= Ŝ nm = 1 N 1 4π i= 2N 1 j= 2N 1 j= f(θ i, λ j ) R nm (θ i, λ j ) ij, f(θ i, λ j ) Q nm (θ i, λ j ) ij, where Ĉnm and Ŝnm are the estimate of Cnm and S nm, respectively, ij indicates the segment 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 by a set of point values f(θ i, λ j ), and is known as the Gauss technique. It should be noted that (12) is usually only an approximation due to the discretization effect of f(θ, λ). If all Cnm and S nm are known till degree and order N max, one can compute f(θ i, λ j ) as follows: f(θ i, λ j ) = N max n= (12) n ( Cnm cos mλ j + S ) nm sin mλ j Pnm (cos θ i ), (13) m= which can be regarded as an approximation to f(θ, λ) at point (θ i, λ j ). Expression (13) defines the object of spherical harmonic synthesis: given the coefficients, it is required to estimate the function. The double summation appearing in (12) for harmonic analysis as well as in (13) for spherical harmonic synthesis are computed using FFT. Colombo (1981) has written two subroutines for this purpose, called HARMIN and SSYNTH. Abd-Elmotaal (24) has written a program called HRCOFITR, which uses Colombo s subroutines, after great modifications, in an iteration process to obtain the best coefficients accuracy and minimum residual filed either the given filed is on sphere or on ellipsoid. The merged 3 3 global field has been used to estimate the harmonic coefficients of the tailored reference model by HRCOFITR Program (Abd-Elmotaal, 24), called FFT technique. Also the Gauss technique (12) has been used to estimate the harmonic coefficients of the tailored model using the merged 3 3 global field as input. Since the low order harmonics of the anomalous gravitational potential are to a great extent due to the density disturbances in the upper mantle and even deeper sources, the lower harmonics till N = 2 have been fixed to their values as of EGM28 geopotential model. Figure 8 shows the tailored geopotential model using the FFT harmonic analysis technique compared to the EGM28 goepotential model. To improve the estimation of the harmonic coefficients using the Gauss harmonic analysis technique, an iterative process takes place. Figure 9 shows the tailored geopotential model using the Gauss harmonic analysis technique after 4 iterations. Comparing Figs. 8 and 9 shows that both harmonic analysis techniques give similar harmonic coefficients. Figure 1 shows the differences at the grid points between the Egyptian 3 3 interpolated gravity anomalies and the computed gravity anomalies using the tailored geopotential model computed by the FFT harmonic analysis technique. Figure 11 shows the same differences using the Gauss harmonic analysis technique. The white areas indicate differences less than 1 mgal in magnitude. Both figures show good matching especially in the data areas. 8

9 Figure 8: The tailored geopotential model using the FFT harmonic analysis technique. Figure 9: The tailored geopotential model using the Gauss harmonic analysis technique after 4 iterations. 9

10 Figure 1: Differences between Egyptian 3 3 interpolated gravity anomalies and the computed gravity anomalies using the FFT tailored geopotential model. Contour interval: 1 mgal. Table 2: Statistics of the differences between the Egyptian gridded gravity anomalies and those computed by the tailored geopotential models statistical parameters anomalies difference min. max. range average std mgal mgal mgal mgal mgal Data - EGM Data - FFT tailored model Data - Gauss tailored model Table 2 illustrates the statistics of the differences between the gridded Egyptian gravity anomalies data and those computed by the tailored geopotential models created within this investigation. Table 2 shows that both developed geopotential models give practically the same results. The tailored geopotential models created in this investigation give better residual gravity anomalies (unbiased and have much less variance than that of EGM28). The variance has dropped by its one third. The range has dropped by about 2 mgal. 6 Conclusions Two high-degree tailored reference geopotential models for Egypt have been developed in this investigation. The window remove-restore technique has been applied to get rid of the double consideration of the topographic-isostatic masses within the data window. 1

11 Figure 11: Differences between Egyptian 3 3 interpolated gravity anomalies and the computed gravity anomalies using the Gauss tailored geopotential model. Contour interval: 1 mgal. The FFT technique using an iteration process has been used to recover the harmonic coefficients to enhance the accuracy of the obtained harmonic coefficients and to minimize the residual field. The Gauss harmonic analysis technique with an iterative process to improve the quality of the obtained harmonic coefficients has also been used to estimate the tailored geopotential model for Egypt. The lower harmonic coefficients till N = 2 have been fixed to their values of EGM28 as they are to a great extent due to the density disturbances in the upper mantle and even deeper sources. Both FFT and Gauss harmonic analysis techniques give nearly the same results in the space and frequency domains. The tailored geopotential models created in this investigation give better residual gravity anomalies (unbiased and have much less variance). Hence they are more suitable for gravity interpolation and geoid determination for Egypt. The variance has dropped by its one third. The range has dropped by about 2 mgal. Acknowledgements This project was supported financially by the Science and Technology Development Fund (STDF), Egypt, Grant No References Abd-Elmotaal, H. (24) An Efficient Technique for Harmonic Analysis on a Spheroid (Ellipsoid and Sphere), Österreichische Zeitschrift für Vermessung & Geoinformation, 3+4,

12 Abd-Elmotaal, H. (27) Reference Geopotential Models Tailored to the Egyptian Gravity Field, Bollettino di Geodesia e Scienze Affini, 66 (3), Abd-Elmotaal, H. and Khtreiber, N. (1999) Improving the Geoid Accuracy by Adapting the Reference Field, Physics and Chemistry of the Earth, 24 (1), 53 59, DOI: 1.116/S (98)1-6. Abd-Elmotaal, H. and Kühtreiber, N. (23) Geoid Determination Using Adapted Reference Field, Seismic Moho Depths and Variable Density Contrast, Journal of Geodesy, 77, Abd-Elmotaal, H., Seitz, K., Abd-Elbaky, M. and Heck, B. (213 a) Comparison among Three Harmonic Analysis Techniques on the Sphere and the Ellipsoid, Journal of Applied Geodesy, 8 (1), 1 19, DOI: /jag Abd-Elmotaal, H., Seitz, K., Abd-Elbaky, M. and Heck, B. (213 b) Tailored Reference Geopotential Model for Africa, Scientific Assembly of the International Association of Geodesy, Potsdam, Germany, September 1-6, 213. Abd-Elmotaal, H., Abd-Elbaky, M. and Ashry, M. (213 c) 3 Meters Digital Height Model for Egypt, VIII Hotine-Marussi Symposium, Rome, Italy, June 17 22, 213. Colombo, O. (1981) Numerical Methods for Harmonic Analysis on the Sphere, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 31. Farr, T.G., Rosen, P.A., Caro, E., Crippen, R., Duren, R., Hensley, S., Kobrick, M., Paller, M., Rodriguez, E., Roth, L., Seal, D., Shaffer, S., Shimada, J., Umland, J., Werner, M., Oskin, M., Burbank, D. and Alsdorf, D. (27) The Shuttle Radar Topography Mission, Reviews of Geophysics, 45, RG24, doi:1.129/25rg183. Forsberg, R. (1984) A Study of Terrain Reductions, Density Anomalies and Geophysical Inversion Methods in Gravity Field Modelling, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 355. Heiskanen, W.A., and Moritz, H. (1967) Physical Geodesy. Freeman, San Francisco. Pavlis, N.K., Holmes, S.A., Kenyon, S.C., and Factor, J.K. (28) An Earth Gravitational Model to Degree 216: EGM28, General Assembly of the European Geosciences Union, Vienna, Austria, April 13 18, 28. Pavlis, N.K., Holmes S.A., Kenyon S.C., and Factor J.K. (212) The development and evaluation of the Earth Gravitational Model 28 (EGM28). Journal of Geophysical Research 117, B446, doi:1.129/211jb8916. Wenzel, H.G. (1998) Ultra High Degree Geopotential Models GPM98A, B and C to Degree 18 Tailored to Europe, In Vermeer, M. and Ádám, J., eds. (1998) Second Continental Workshop on the Geoid in Europe, Budapest, Hungary, March 1 14, 1998, Reports of the Finnish Geodetic Institute, 98 (4),