Isostasy Geodesy Lars E. Sjöberg a * and Mohammad Bagherbandi a,b a Division of Geodesy and Satellite Positioning, Royal Institute of Technology, Stockholm, Sweden b Department of Industrial Development, IT and Land Management, University of G avle, G avle, Sweden Definition Isostasy (Greek isos equal, stasis stand still ) is a term in geology, geophysics, and geodesy to describe the state of mass balance (equilibrium) between the Earth s crust and upper mantle. It describes a condition to which the mantle tends to balance the mass of the crust in the absence of external forces. Introduction The term isostasy was proposed in 1889 by the American geologist C. Dutton, but the first idea of mass balancing of the Earth s upper layer goes back to Leonardo da Vinci (1452 1519). The term means that the Earth s topographic mass is balanced (mass conservation) in one way or another, so that at a certain depth the pressure is hydrostatic. Isostasy is an alternative view of Archimedes principle of hydrostatic equilibrium. According to this principle, a floating body displaces its own weight. A light mountain chain can be compared with an iceberg or a cork floating in water or in proper term floating on the denser underlying mantle. When a certain area of the crust reaches the state of isostasy, it is said to be in isostatic equilibrium (balance), and the depth at which isostatic equilibrium prevails is called the compensation depth (Heiskanen and Vening Meinesz, 1958, chapter 5). Although it is generally accepted that the Earth is a dynamic system that responds to loads in many different ways, isostasy still provides an important view of the on-going processes. Usually topographic loads of wavelengths below 50 km are supported by the underlying lithosphere and are therefore not isostatically compensated; for wavelengths from 50 to 500 km, the topography is typically compensated by elastic flexure in the upper lithosphere, and for longer wavelengths, it is generally in isostatic balance, except in the very long wavelengths, which are mainly due to dynamic processes in the interior of the mantle. The isostatic effect was first documented by observational evidence in the late 1850s by Sir G Everest in connection with his leadership of the great surveys of India. In this work, isostatic compensation could explain while the large topographic masses of the Himalayas caused much less deflections of the vertical than expected on the closures of the geodetic triangles. The observational results were theoretically explained by Airy (1855) and Pratt (1855) and (1859). For instance, at one trigonometric station, Pratt estimated that the surrounding mountain masses should cause a deflection of the vertical (i.e., the amount the actual plumb line deviates from the direction of the local normal) of 28 00, but the actual deflection determined by the geodetic/astronomic method was only 5 00. In a similar way, isostasy can explain why the observed vertical gravity component in mountain areas is usually much less than one would expect when considering the mass of the mountains. In other words, Bouguer gravity anomalies (which can be described as gravity residuals with the topographic attraction removed) are systematically negative in mountainous areas and decreasing with elevation. Hence, this observation is another evidence that there is some kind of compensation. *Email: sjoberg@infra.kth.se Page 1 of 9
The Mohorovičić discontinuity (Moho), a seismic velocity and density contrast at the bottom of the crust, was first identified in 1909 by the Croatian seismologist A. Mohorovičić, when he observed that seismic waves from shallow-focus earthquakes refracted by a high-velocity medium, the crust-mantle boundary. Later a man-made seismic reflection technique was developed as a technical method to directly measure the Moho depth. Isostasy has played and still plays an important role in geodesy. J. F. Hayford applied Pratt s isostatic model to compute the global reference ellipsoid that was adopted as the international ellipsoid by the meeting of International Association of Geodesy in Stockholm in 1924, and W. A. Heiskanen used Airy s model to derive the International Gravity Formula (1930). In physical geodesy, the isostatic gravity anomaly has always been regarded as the most suitable anomaly for various applications, mainly due to its smooth behavior. See also Applications below. In geophysics, isostasy is essential mainly for studying geodynamic processes in the crust and upper mantle (see Applications below), and in geology it helps in explaining various topographic and geologic features around the world. Classical Isostatic Models The original principles of isostasy by Airy (1855) and Pratt (1855) are based on local compensation mechanisms by assuming that the density of a unit prism of the Earth s crust times its volume is constant, i.e., equal-pressure and equal-mass hypotheses at the compensation depth (that varies between the models). These models assume that the topographic mass compensation is uniformly distributed vertically and directly compensates the topographic masses along the vertical (local compensation), i.e., the reciprocal forces from the mantle compensates for the pressure of the topographic masses. As the Earth s crust is very complicated, some approximations must be considered for compensating the topographic masses. For example, using a constant density for different layers of the topographic masses is such an approximation. Different hypotheses have been presented based on this principle. The ideal model should be realistic and easy to apply, and it may involve either or both a variable compensation depth or/and a variable crustal density. Airy s Mountain Root Model The model of Airy was mathematically described and extensively applied by Heiskanen (1924) and (1938). According to Airy-Heiskanen s isostatic hypothesis, the Earth s crust with a constant density r 0 (usually set to 2.67 g/cm 3 ) floats on the mantle, the denser layer under the crust with a density r 1 > r 0 (frequently set to 3.27 g/cm 3 ), exactly as an iceberg in the ocean. This implies that there are light roots under the mountains and heavy anti-roots under the oceans (see Fig. 1). In continental areas, the isostatic hypothesis implies that the mass surplus/unit area above sea level equals the mass deficit/unit area below sea level, which can be expressed mathematically as hr 0 ¼ tdr with Dr ¼ r 1 r 0 ; (1) where h is the topographic height and t is the mountain root below the normal Moho depth (T 0 = 30 km in Fig. 1). Using the above standard density values in Eq. 1 yields t = 4.45 h. Note that for h = 0, there is no mountain root. In ocean regions, the mass deficiency/unit area of the ocean mass, ðr 0 r w Þh 0, where h 0 is the ocean depth, is compensated by a mass surplus t 0 Dr between the normal Moho depth and sea bottom, where t 0 is the anti-root, yielding the equation of mass balance Page 2 of 9
Fig. 1 Airy s model. The solid Earth topographic features are compensated by roots (t) over land and anti-roots (t 0 ) over ocean areas. The nominal compensation depth (T 0 ) is set to 30 km ðr 0 r w Þh 0 ¼ t 0 Dr ) t 0 ¼ r 0 r w h 0 ; (2) Dr where r w ð1:03g=cm 3 Þis the density of ocean water. For standard density values, the anti-root becomes 2.73 h 0. Pratt s Dough Model According to the model outlined by Pratt and put into mathematical form by Hayford (1909), the isostatic compensation depth (D), where equilibrium prevails, is constant, but the crust behaves like a dough with a density (r) that varies such that under the mountains and oceans, it is smaller and larger than under the flat regions, respectively (see Fig. 2). Mathematically the equilibrium condition can be expressed in continental and ocean regions as ðd þ hþr ¼ Dr 0 ) r ¼ D D þ h r 0 (3a) and respectively. Usually D is set to 100 km. ðd h 0 Þr þ h 0 r w ¼ Dr 0 ) r ¼ r 0 þ d ð D d r 0 r w Þ; (3b) Page 3 of 9
Fig. 2 Pratt s model. The solid Earth crust variable elevation/depth (h/h 0 ) is compensated by a variable density above the compensation depth D = 100 km Vening Meinesz Regional Model The Airy and Pratt isostatic models are very much oversimplified in the sense that they only assume a local compensation along the normal sea level. Already Gilbert (1889) and Barell (1914) suggested some kind of flexure of the lithosphere with a regional compensation of topography to explain isostasy, but it was mainly Vening Meinesz that later explored this idea. Similar to Airy, Vening Meinesz (1931) (see also Heiskanen and Vening Meinesz, 1958, pp. 137 147) uses a flat Earth model with the crust being a load on the mantle, leading to crustal roots of compensation, but here the surface of the mantle is elastic and unbroken. Due to the load, the surface bends as an elastic plate. Vening Meinesz thus assumed that the crust is a homogenous elastic plate floating on a viscous mantle (see Fig. 3). Seismology shows that the crust is thicker (30 90 km) underneath continents and mountains, but is thinner (5 15 km) under oceans. This variable depth of the Moho and the general agreement between the thicknesses of the crust estimated from the seismic and gravity methods frequently support Airy- Heiskanen s and Vening Meinesz hypotheses. While in Airy-Heiskanen s theory there is no correlation between neighboring crustal columns, in reality there is due to the elasticity of the Earth. The difference between the Airy-Heiskanen and the Vening Meinesz hypotheses is thus a matter of assuming local versus regional mechanisms of topographic mass compensation. Page 4 of 9
Fig. 3 General schematic structure of the Earth s crust according to Airy s and Vening Meinesz models. h is the solid Earth topographic height, and T 0 is the nominal crustal thickness Fig. 4 General schematic structure of the Earth s crust according to the Vening Meinesz- Moritz model. h and h 0 are the solid Earth topographic height and depth below the geoid (sea level), T 0 is the nominal crustal thickness, and R is mean Earth radius Recent Considerations Vening Meinesz-Moritz Model Moritz (1990, section 8.3.2) generalized Vening Meinesz method from a regional, isostatic compensation with a flat Earth approximation of sea level to that of a global compensation with a spherical sea level approximation (see Fig. 4). Sjöberg (2009) expressed the Vening Meinesz-Moritz problem in a mathematical form as a nonlinear Fredholm integral equation of the first kind and presented a second-order solution. Later refinements of this solution (Tenzer and Bagherbandi, 2012; Sjöberg, 2013; Tenzer et al. 2015) imply (a) a better reduction of the topographic signal in the gravimetric observation (which could either be the so-called no-topography gravity anomaly or the Bouguer gravity disturbance) and (b) removal of non-isostatic effects in the mantle and deep Earth interior by comparisons of the gravimetric-isostatic and Page 5 of 9
seismic-derived Moho depths. Then, starting from the isostatic condition that the isostatic gravity disturbance dg I vanishes at each point on the mean Earth sphere S of radius R, i.e., dg I ¼ 0onS; (4) a global gravity inversion leads to the following second-order solution for the Moho depth T P ¼ ðt 1 Þ P þ T 2 1 P R þ 1 ðð 8R 3 S T 2 1 P T 2 1 sin 3 c Q ds Q ; (5) where P and Q are the computation and integration points, respectively, T 1 is the first-order solution, which can be determined from the Bouguer gravity disturbance, c is the geocentric angel between P and Q, and ds is the area integration element on the sphere. Non-Isostatic Effects Ideally the isostatic gravity disturbance of Eq. 4 implies that the attractions of all topographic masses, including those of ice sheets and sedimentary basins, must be removed and compensated. This requires that not only the topographic elevations must be known but also their density distributions. However, Airy s and Vening Meinsez models are usually approximated to constant densities of the crust and upper mantle, which causes a non-isostatic effect in the computed isostatic anomaly (Bagherbandi and Sjöberg, 2012b). Other non-isostatic effects are typically of dynamic origin. J Barrell (1914) divided the Earth s volume into the mechanical units lithosphere and asthenosphere, but it was only after the birth of the theory of plate tectonics in 1931 that these terms became widely used among geoscientists. The Earth s lithosphere includes the crust and the uppermost mantle, which constitute the hard and rigid outer layer of the Earth. The lithosphere is broken into tectonic plates underlain by the asthenosphere, which is the weaker, hotter, and deeper part of the upper mantle. In 1934 and later, J. A. Walcott published three papers on the flexural rigidity of the lithosphere, and he extended the work of Vening Meinesz. In 1970 he showed that the elastic thickness of the lithosphere (as determined from its flexural rigidity) in regions with large loads ranges between 5 km and 114 km, and the thickness is approximately inversely related with the age of the load. This experience is closely related with mantle convection, cooling of the oceanic lithosphere, and seafloor spreading. Plate tectonics describes the large-scale motions of the Earth s lithosphere, and tectonic plates are able to move because the Earth s lithosphere has a higher strength and lower density than the underlying masses. Heating from the Earth s core drives convection in the upper mantle, which compensates for some parts of the topographic potential. The convection is extremely slow, and the speed with which material in the Earth s crust spreads from the mid-ocean ridges is of the order of several cm per year. Warming of the slabs and cooling of the surrounding mantle would decrease the density excess (e.g., Kaban et al., 1999). The crustal plates on adjacent sides of the upwelling hot mantle rock beneath the ocean ridges move laterally away from each other. As the melted rock cools down by the seawater, the plate attains higher density. pffiffiffiffiffiffiffi Generally, the crustal plates sink and ocean depths thicken with age away from the mid-ocean ridges up to 70 80 my, and further away the thicknesses are rather constant (e.g., Casenave, 1994). Another large-scale dynamic effect that leaves large portions of the Earth in isostatic imbalance is caused by glacial isostatic adjustment (GIA) of the Earth s surface and upper mantle from previous glaciation followed by deglaciation. For instance, during the last ice age, the crusts in Fennoscandia and Laurentia were considerably depressed, yielding today s regional gravity lows of the order of 10 20 Page 6 of 9
mgal. In these regions, the crusts still experience GIAs of the orders of several mm/year, followed by mantle flow of magma. All such effects must be corrected to validate Eq. 4 as the gravimetric-isostatic basis for gravity inversion in Moho determination (Bagherbandi and Sjöberg, 2013). Applications Geodetic applications of isostasy are wide and only a few applications are presented here. One application of the isostatic Moho model is the construction of a synthetic Earth gravity model (SEGM), which can be used to fill the gaps of regions where gravity data are unavailable or sparse, to validate gravity field recovery techniques, to test gravity products like geoid models, and to extend Earth gravitational models (EGMs) to higher degrees. A main idea to apply topographic-isostatic harmonic coefficients is to combine them with a low- and medium-degree EGM for creating a high-degree EGM (e.g., Haagmans, 2000; Bagherbandi and Sjöberg, 2012a). As the isostatic gravity anomaly, isostatic disturbance, and gravity gradient data are smooth and independent on topography (Heiskanen and Moritz, 1967, p. 152; Martinec, 1998, chapter 3), they are also very suitable for interpolation, extrapolation, and integration, e.g., in determining the geoid and deflections of the vertical and for stabilizing the downward continuation of satellite and airborne gravity and gravity gradient data to ground and sea level. The classical definition of isostasy assumes that loads at the Earth surface are compensated by the density variations down to a certain level of compensation. As presented above, much of the Earth s crust is not isostatically compensated, and therefore non-isostatic effects must also be considered. The isostatic balance is affected by several geological and geophysical phenomena (Watts, 2001; Tenzer and Bagherbandi, 2012), implying that isostatic gravity anomalies can be used as a tool in geology and geodynamic interpretations. One geodynamic application is to use the isostatic gravity anomaly in determining the upper mantle viscosity, which is crucial in geodynamic processes, e.g., in studying GIA, mantle convection, and plate motion. Postglacial rebound data provide important data about the properties of the Earth s mantle and its lower layers. Land uplift and the Earth s gravity field, including its temporal change, provide important signatures of mantle convection and viscosity. Isostatic gravity anomalies are also a versatile tool in interpreting geology features of the crust. Clearly, any crustal density structures will influence the isostatic system. For instance, isostatic density modeling can explain the lacking root of a mountain chain (Ebbing, 2007; Bagherbandi et al., 2015), and it can explain why topographic masses are not compensated. Sometimes a high mountain correlates with a Bouguer gravity low, indicating isostatic compensation (e.g., in the Scandes, the Scandinavian mountain chain). It is also possible to study flexural forces within the elastic lithosphere to study to what extent the lithosphere responds to loading. The flexural rigidity characterizes the apparent strength of the lithosphere (elasticity), which acts against the forces induced by loading (Ebbing, 2007; Bagherbandi et al., 2015). Here satellite gravity data and isostatic assumptions contribute to the modeling of the lithosphere and geological structures, for example, in refining regional heat-flow patterns and sedimentary thickness models (Mckenzie, 1967). Page 7 of 9
Summary Isostasy is a term in geology, geophysics, and geodesy to describe the state of mass balance between the Earth s crust and upper mantle. The term means that the topographic mass is balanced in one way below the crust. In classical models, one assumes that the mass along a vertical column above the so-called compensation depth in the upper mantle is constant, implying that at a certain depth, the pressure is hydrostatic in accord with Archimedes principle of hydrostatic equilibrium. More realistic models assume a regional or even global compensation. Isostasy directly affects the Earth s gravity field in such a way that the attractions of the mass surplus of mountains and deficit in the oceans are much reduced. Isostasy is a versatile tool in the interpretations of geologic and geodynamic processes as well as in physical geodetic operations. Cross-References Glacial Isostatic Adjustment Gravity Inversion References and Reading Airy, G. B., 1855. On the computations of the effect of the attraction of the mountain masses as disturbing the apparent astronomical latitude of stations in geodetic surveys. Philosophical Transactions of the Royal Society of London, Series B, 145, 101 104. Bagherbandi, M., and Sjöberg, L. E., 2012a. A synthetic Earth gravity model based on a topographicisostatic model. Journal of Studia Geophysica and Geodetica, 56(2012). doi:10.1007/s11200-011- 9045-1. Bagherbandi, M., and Sjöberg, L. E., 2012b. Non-isostatic effects on crustal thickness: a study using CRUST2.0 in Fennoscandia. Physics of the Earth and Planetary Interiors, 200 201, 37 44, doi:10.1016/j.pepi.2012.04.001. Bagherbandi, M., and Sjöberg, L. E., 2013. Improving gravimetric-isostatic models of crustal depth by correcting for non-isostatic effects and using CRUST 2.0. Earth Science Review, 117, 29 39, doi:10.1016/j.earscirev.2012.12.002. Bagherbandi M., Sjöberg L. E., Tenzer R., and Abrehdary, M., 2015. A new Fennoscandian crustal thickness model based on CRUST1.0 and gravimetric isostatic approach. Earth Science Review, 145, 132 145, doi:10.1016/j.earscirev.2015.03.003. Barrel, J., 1914. The strength of the Earth s crust I. Geologic tests of the limits of strength. Journal of Geology, 22, 28 48. Casenave, A., 1994. The geoid and oceanic lithosphere. In Vanicek, P., and Christou, N. T. (eds.), Geoid and its Geophysical Interpretation. Boca Raton: CRC Press, p. 13. Ebbing, J., 2007. Isostatic density modelling explains the missing root of the Scandes. Norwegian Journal of Geology, 87, 13 20. Gilbert, G. K., 1889. The strength of the Earth s crust. Bulletin of the Geological Society of America, 1, 23 27. Haagmans, R., 2000. A synthetic earth for use in geodesy. Journal of Geodesy, 74, 503 511. Hayford, J. F., 1909. The Figure of the Earth and Isostasy from Measurements in the United States. Washington, D.C.: GPO. Page 8 of 9
Heiskanen, W. A., 1924. Untersuchungen ueber Schwerkraft und Isostasie. Finn. Geod. Inst. Publ. No. 4, Helsinki. Heiskanen, W. A., 1938. New isostatic tables for the reduction of the gravity values calculated on the basis of Airy s hypothesis. Isostat. Inst. of IAG Publ. No. 2. Finnish Geodetic Institution, Helsinki. Heiskanen, W. A., and Moritz, H., 1967. Physical Geodesy. New York: W.H. Freeman. Heiskanen, W. A., and Vening Meinesz, F. A., 1958. The Earth and its Gravity Field. New York: McGraw-Hill. Kaban, M. K., Schwintzer, P., and Tikhotsky, S. A., 1999. A global isostatic gravity model of the Earth. Geophysical Journal International, 136, 519 536. Martinec, Z., 1998. Boundary-Value Problems for Gravimetric Determination of a Precise Geoid. Berlin/ Heidelberg/New York: Springer. Lecture Notes in Earth Sciences, Vol. 73. Mckenzie, D. P., 1967. Some remarks on heat flow and gravity anomalies. Journal of Geophysical Research, 72, 61 71. Moritz, H., 1990. The Figure of the Earth. Karlsruhe: H Wichmann. Pratt, J. H., 1855. On the attraction of the Himalaya mountains, and on the elevated regions beyond; upon the plumb line in India. Philosophical Transactions of the Royal Society of London, 145, 53 100. Pratt, J. H., 1859. On the deflection of the plumb-line in India, caused by the attraction of the Himalaya Mountains and of the elevated regions beyond, and its modification by the compensating effect of a deficiency of matter below the mountain mass. Philosophical Transactions of the Royal Society of London, 149, 745 778. Sjöberg, L. E., 2009. Solving Vening Meinesz-Moritz inverse problem in isostasy. Geophysical Journal International, 179(3), 1527 1536, doi:10.1111/j.1365-246x.2009.04397.x. Sjöberg, L. E., 2013. On the isostatic gravity anomaly and disturbance and their applications to Vening Meinesz-Moritz inverse problem of isostasy. Geophysical Journal International, 193, 1277 128. Tenzer, R., and Bagherbandi, M., 2012. Reformulation of the Vening-Meinesz Moritz inverse problem of isostasy for isostatic gravity disturbances. International Journal of Geosciences, 3, 918 929, doi:10.4236/ijg.2012.325094. Tenzer, R., Chen, W., Tsoulis, D., Bagherbandi, M., Sjöberg, L. E., Novák, P., and Jin, S., 2015. Analysis of the refined CRUST1.0 crustal model and its gravity field. Surveys in Geophysics, 36, 139 165. Vening Meinesz, F. A., 1931. Une nouvelle méthodepour la réduction isostatique régionale de l intensité de la pesanteur. Bulletin Géodésique, 29, 33 51. Watts, A. B., 2001. Isostasy and Flexure of the Lithosphere. Cambridge, UK: Cambridge University Press. Page 9 of 9
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