On Geophysical Application of the Separation Between the Geoid and the Quasigeoid Case Study: Four Regions in Iran

Transcription

1 Australian Journal o Basic and Applied Sciences, 5(12): , 2011 ISSN On Geophysical Application o the Separation Between the Geoid and the Quasigeoid Case Study: Four Regions in Iran 1 Mehramuz, M., 1 Zomorrodian, H. and 2 Ardalan, A.A. 1 Department o Geophysics, Science and Research Branch, Islamic Azad University, Tehran, Iran 2 Department o Surveying and Geomatics Engineering, Center o Excellence in Geomatics Engineering and Disaster Prevention, College o Engineering, University o Tehran, Tehran, Iran Abstract: In this paper the separation between the geoid and the quasi-geoid over 4 test regions, namely, Alborz and Zagros mountain ranges (as the two test areas over mountainous regions) and the coast o Caspian Sea and Khuzestan province (as the two test areas over lat regions) is computed. The result o the computations shows that average and standard deviation o the separation between the geoid and the quasi-geoid are, respectively, (mm) and (mm) over Alborz mountain, (mm) and (mm) over Zagros mountain, (mm) and (mm) over the coast o Caspian Sea, and (mm) and (mm) over the Khuzestan province. Furthermore, by using the EGM2008 geopotential model the geoidal undulations over the measured points o the aorementioned test areas are computed and by knowing the separation between the geoid and the quasi-geoid, the quasi-geoid over the test areas is derived. Based on the numerical results the separation between the geoid and the quasi-geoid in addition o its application or the transormation o normal heights to orthometric heights, and vice versa, it can be used as a tool or the study o the variation o crustal density and prevailing isostasy mechanisms rom one region to another. Key words: Separation between the geoid and the quasi-geoid, geoid, quasi-geoid, normal height, orthometric height. INTRODUCTION The main goal o this paper is to show that the separation between the geoid and the quasi-geoid besides its traditional application or the transormation o orthometric heights into the normal heights and vice-versa can be used to extract inormation about the variation o crustal density and prevailing isostasy mechanisms rom one region to another. Geoid as deined by Gauss (1828) and Listing (1873) is an equipotential surace o the Earth s gravity ield that its to Mean Sea Level (MSL) in an optimum way (i.e. least squares sense), and has been selected by the geodetic community as the reerence surace or the orthometric height system. Molodensky (1945, 1948, 1960) deined the telluroid as a mathematical surace on which the normal potential o points is equal to the actual potential o corresponding points on the surace o the earth. More precisely is shown in Fig.1 the actual potential at point P on the surace o the Earth is equal to the normal potential at point PN on the surace o the telluroid (W P =U P ). For telluroid computation we must also deine a projection scheme between the corresponding points on the surace o the earth and the telluroid. Molodensky used projection along the normal to the reerence ellipsoid. The separation between the Earth s surace and the telluroid is known as telluroidal height or height anomaly (ζ), see Fig. 1. I this separation be placed on the surace o a reerence ellipsoid we arrive at a surace known as the quasi-geoid. The quasi-geoid is the reerence surace o the normal height system. Nowadays, the height system o the countries around the world is either orthometric or normal height. Thereore it might be needed to transer the heights rom one system to another. In the next section (Section 2) we will go into details o the transormation between these two height systems. Next, in Section 3, the presented ormulas will be applied to our test areas in mountainous and lat regions within the territory o Iran. Finally, Section 4 will provide the conclusions. Corresponding Author: Mehramuz, M., Department o Geophysics, Science and Research Branch, Islamic Azad University, Tehran, Iran 127

2 2. Computation o the Separation Between the Geoid and the Quasi-geoid: To begin with the computation o the separation between the geoid and the quasi-geoid let us once again reer to Fig.1. Accordingly we can write: Fig. 1: Geoid Undulation (N), Height Anomaly (ζ), Orthometric Height (H O ) and Normal Height (H N ) O N H N H (1) or N O N H H (2) The orthometric height (H O ) and the normal height (H N ) are deined as ollows: O C H (3) g N C H (4) Where C=W o -W is the geopotential number, i.e., the geoid s potential value (W o ) minus the potential value (W) at a computational point P. g is the mean actual gravity along the plumb line rom the surace o the earth down to the geoid. is the mean normal gravity along the normal rom the telluroid to the surace o the reerence ellipsoid, or equivalently rom the surace o the earth down to the quasi-geoid. Upon the substitution or H O and H N in Eq. (2) rom Eqs. (3) and (4), respectively, we arrive at: C C N g 1 1 C( ) g g C( ) g (5) Now i we substitute or C in Eq. (5) rom Eq. (3) we have: O g N H g( ) g H O g ( ) (6) 128

3 The relation shown in Eq. (6) in some literatures is reerred to as C2-Term (Featherstone and Kirby 1998,Nahavandchi 2002). In the case o the normal height computation, the normal ield is generated by the Somigliana-Pizzetti reerence ield (Pizzetti 1894, Somigliana 1930). The term g is mainly due to the masses above the geoid, known as the topographic masses. I we approximate the eect o topographical masses by the Bouguer anomaly, g can be approximated by gb (see Appendix or the proo) and Eq. (6) can be written as ollows: g O N ( B ) H (7) An approximate ormula or the computation o according to Homann-Wellenho and Moritz (2006) is as ollows: H H a a O O 2 2 [1 (1 m 2 sin ) ] 2 Where γ is the normal gravity o the Somigliana-Pizzetti type on the surace o the reerence ellipsoid, is the geometric lattening, m is the Clairaut s constant, a is the semi major axis o the reerence ellipsoid and n is the geodetic latitude o the computational point. Having computed N-ζ (the C2-Term ) by knowing the orthometric height H O, the normal height H N can be derived rom Eq. (1) as ollows: (8) N O H H ( N ) (9) Or again by using Eq. (1) knowing the C2-Term and the geoidal undulation N, ζ can be derived as ollows: O N N ( H H ) (10) Having urnished the deinition and ormulas related to the separation between the geoid and the quasigeoid we will proceed to the next section where we will present our test computations and will discuss on the numerical results. 3. Test Computations and Analysis o Results: In this section, we will present the results o our test computation or derivation o the separation between the geoid and the quasi-geoid within our test regions. Besides, using the equations (9) and (10) the normal heights and the height anomalies, respectively, are computed or the our test regions. These test regions are located at Alborz mountain range ( , ) ( , ) and Zagros mountain range as highland areas with the heights more than 800 m and two areas at the coast o Caspian Sea (48 50, ) ( , ) and the Khuzestan province as lowland areas with the heights less than 400 m. The data used or the our test computations are comprised o gravity and orthometric height over 1471 points in Alborz, 1224 points in Zagros, 267 points in Caspian Sea coast and 552 points in Khuzestan province. Table 1, provides some statistical inormation about the height variations in the test areas. These data are provided by the kind grace o Iranian National Cartographic Center (NCC). Table 1: Summary o statistical inormation about the height variations within the test regions Test regions Heights (m) Max Min Mean Alborz mountain Zagros mountain Caspian Sea coast Khuzestan Province

4 For the computation o the separation between the geoid and the quasi-geoid according to ormula (7) in the mountainous test regions due to the rough topography, the complete Bouguer anomaly, i.e., the Bouguer plate plus the topographic correction, is used while or the Caspian Sea coast and Khuzestan province test regions the simple Bouguer anomaly, i.e. Bouguer plate, is considered.it is due to the acts that contribution o the topographic corrections to the separation between the geoid and the quasi-geoid is estimated to be 2-3 cm in very rugged areas(flury and Rummel 2009). We also used the constant value density o the crust in the computations o the Bouguer anomaly ( g B ) (gr / cm ) or the The separation between the geoid and the quasi-geoid as well as height anomalies and normal heights derived or the test areas are shown in Fig. 2 Fig. 7 or the Alborz Mountains and Khuzestan Province as samples o our results or highland and lowland areas, respectively. The summary statistical inormation o the computed separation between the geoid and quasi-geoid within the our test areas are given in Table 2. Fig. 2: The separation between the geoid and the quasi-geoid within Alborz Mountain test area in millimeter (contour intervals 100 mm) Fig. 3: Height anomalies within Alborz Mountain test area in meters (contour intervals 0.5 m) Fig. 4: Normal heights within Alborz Mountain test area in meters (contour intervals 150 m) 130

5 Fig. 5: The separation between the geoid and the quasi-geoid within Khuzestan Province test area in millimeter (contour intervals 2 mm) Fig. 6: Height anomalies within Khuzestan Province test area in meters (contour intervals 0.25 m) Table 2: Summary o statistical inormation o the separation between the geoid and the quasi-geoid within the our test regions Test Area Max. (mm) Min. (mm) Mean (mm) STD (mm) Alborz Mountains (Highland) Zagros Mountains (Highland) Caspian Sea Coast (Lowland) Khuzestan Province (Lowland)

6 According to Table 2, the separation between the geoid and the quasi-geoid is most pronounced in mountainous areas (compare the maximum separation o m and m in Alborz and in Zagros Mountains, respectively, with the maximum values o mm and mm in Caspian Sea coast and Khuzestan province, respectively). The negative values o the separation between the geoid and the quasi-geoid within the Zagros Mountain test area indicate that the quasi-geoid is higher than the geoid in that region. Besides, the computed separations within the Alborz Mountain, Caspian Sea coast and Khuzestan province are 99.8%, 27.3% and 95.7% negative, respectively. Furthermore, in Fig. 8 and Fig. 9 we have examined the correlation o the computed separation o the geoid and the quasi-geoid with the heights within the our test regions. Fig. 7: Normal heights within Khuzestan Province test area in meters (contour intervals 30 m) (a) (b) Fig. 8: The Pearson Correlation Coeicient o the separation between the geoid and the quasi-geoid and the heights within (a) Alborz Mountains and (b) Zagros Mountains as our highland test areas 132

7 (a) (b) Fig. 9: The Pearson Correlation Coeicient o the separation between the geoid and the quasi-geoid and the heights within the (a) Caspian Sea coast and (b) Khuzestan province A glance to Fig. 8 and Fig. 9 shows that the separation between the geoid and the quasi-geoid is reduced by increasing the topographic heights. Indeed, the absolute value o the separation between the geoid and the quasi-geoid is increased by increasing the heights. Besides, rom Fig. 8 and Fig. 9 we can clearly see that the rate o change o the separation between the geoid and the quasi-geoid in the highland test regions is much greater than the lowlands. In other words the geoid and quasi-geoid in lowland regions are closer to each other as compared to the highland test regions. The same conclusion can be drawn rom Fig. 2 - Fig. 4 corresponding to the highland test region and Fig. 5 Fig 7 which are corresponding to lowland region. As a matter o act, in the places with higher slope rates, we see sharper variations in the changes o the height anomalies as well as the separations between the geoid and the quasi-geoid. On the other hand, the comparison o the our graphs shown in Fig. 8 and Fig 9 suggests that even though the separation between the geoid and the quasi-geoid is highly correlated with the heights, however, this correlation can vary rom one region to another (compare the correlation coeicient o associated to the Caspian Sea coast, one o our lowland test regions, with the value o , which corresponds to Khuzestan province as the other test region at the lowland region in our test computations). It is important to note that the heights within these two regions are below 400 m and the two regions are having almost the same topography. This can also be seen in the results o the highland test regions as well. To explain the dierence among the computed correlation coeicients we must once again reer to Eq. (7). According to this equation the separation between the geoid and the quasi-geoid is related not only to the orthometric heights but also to the Bouguer anomalies, which in terns are also aected by the applied hypothetical value or the crustal density. Thereore, it can be concluded that the computation o the separation between the geoid and the quasi-geoid not only provides a tool or the transormation rom normal heights to orthometric heights and vice versa, but also can be used as a means or getting inormation about the crustal density variation rom one region to another. Besides, the dierence among the calculated correlation coeicients can be partially due to the dierence in the prevailing isostasy mechanisms. For example, here the distinct dierence between the correlation coeicients o the separation between the geoid and the quasi-geoid with orthometric heights can be associated with the dierent geological structure o the two regions, which has already been reported by Dehghani and Makris (1983), Shahabpour (1999) and Mokhtari et al. (2004). 133

8 Conclusions: In this paper we oered the detailed computational procedure or calculation o the separation between the geoid and the quasi-geoid. Besides, we showed that this separation is highly correlated with topography. However, since the separation between these two suraces is due to the masses between the geoid and the surace o the Earth, the computed separation is inluenced by mixed eect o the topographic heights, mass density variations within the topographic layers, and the isostasy mechanism. Thereore, the computation o the separation between the geoid and the quasi-geoid besides its application or the transormation o the normal heights to the orthometric heights and vice versa,it also can be used or obtaining inormation about the variation o the geophysical and the geological structures o the crust rom one region to another. REFERENCES Dehghani, G.A. and J. Makris, The gravity ield and crustal structure o Iran. In: Geodynamics project (geotraverse) in Iran. Geological Survey o Iran, Rep., 51: Featherstone, W.E. and J. Kirby, Estimates o the separation between the geoid and the quasi-geoid over Australia. Geomatics Research Australasia, 68: Flury, J. and R. Rummel, On the geoid-quasigeoid separation in mountain areas. Journal o Geodesy, (83): Gauss, C.F., Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona. Vandenhoek und Ruprech, Göttingen. Homann-Wellenho, B. and H. Moritz, Physical geodesy. Springer-verlag, Wien. Listing, J.B Über unsere jetzige Kenntnis der Gestalt und Größe der Erde. Dietrichsche Verlagsbuchhandlung, Göttingen. Mokhtari, M., A.M. Farahbod, C. Lindholm, M. Alahyarkhani and H. Bungum, An Approach to a Comprehensive Moho Depth Map and Crust and Upper Mantle Velocity Model or Iran. Iranian Int. J. Sci., (5): Molodensky, M.S., Main problem o geodetic gravimetry. Trans. centr. res. Inst., G., A. & C. 42. Molodensky, M.S., External gravitational ield and the igure o the Earth s physical surace. Inormation o the U.S.S.R. Academy o Sciences, Geographical and Geophysical Series., 13: 3. Molodensky, M.S., V.F. Eremeev and M.I. Yurkina, Methods or study o the external gravitational ield and igure o the Earth. Translated rom Russian by Israel Program or Scientiic Translations or the Oice o Technical Services, Department o Commerce, Washington, D.C., U.S.A., Nahavandchi, H., Two dierent methods o geoidal height determinations using a spherical harmonic representation o the geopotential, topographic corrections and the height anomaly-geoidal height dierence. Journal o Geodesy, (76): Pizzetti, P., Geodesia--Sulla espressione della gravita alla supericie del geoide, supposto ellissoidico. Atti Reale Accademia dei Lincei, (3): Shahabpour, J., The role o deep structures in the distribution o some major ore deposits in Iran, NE o the Zagros thrust zone. Journal o Geodynamics, (28): Somigliana, C., Geoisica--Sul campo gravitazionale esterno del geoide ellissoidico. Atti della Reale Academia Nazionale dei Lincei Rendiconti., (6): Appendix: Approximation o the eect o topographic masses ( g ) by ( g B ). (The proo) The observed gravity value at the geoid s level as ollows: g g g 2 g Geoid obs B gobs on the surace o the Earth can be reduced to the gravity value Where and g are ree-air and Bouguer reductions, respectively. On the other hand according to g Prey-Poincare reduction, the mean gravity value be derived as ollows: B g 2 obs g g g g g Geoid g 2 2 obs obs B ggeoid g rom the surace o the Earth down to the geoid s level can 134

9 Here the variations o the gravity value rom the surace o the Earth down to the geoid s level is considered linear. Substituting the g B in the above ormula by 13, we have: 1 1 g gobs g g 2 3 Now the normal gravity on the surace o the telluroid, knowing the normal height HN and the tell g normal gravity value on the surace o the reerence ellipsoid 0, can be computed rom the linear part o the Taylor expansion o the normal gravity on the surace o the telluroid, as ollows: N O tell 0 H H 0 g h In the above ormula we have also approximated the normal height H N by the orthometric height H O. Finally, the mean normal gravity rom the surace o the telluroid to the surace o the reerence ellipsoid, by assuming the linearity o the vertical variations o the normal gravity, can be computed as ollows: 0 0 tell 0 g 1 0 g Thereore, g can be derived as ollows: g gobs g g 0 g gobs g g 0 3 gobs g gb 0 g and inally we obtain: B g g B 135