The minimum depth of Compensation of topographic masses

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1 Geophys. J. Int. (1994) 117, The minimum depth of Compensation of topographic masses ZdenEk Martinec Department of Geophysics, Faculty of Mathematics and Physics, Charles University, V HoleSoviCkbch 2, Prague 8, Czech Republic Accepted 1993 November 10. Received 1993 November 10; in original form 1991 June 3 SUMMARY Based on the recently compiled crustal-thickness data, it is shown that the Airy-Heiskanen model of compensation only partly compensates the surface topographic masses. To fit the external gravitational potential induced by the surface topography, the Pratt-Hayford concept of compensation has also to be considered. Employing the dynamical flattening of the Earth, the minimum depth of compensation has been estimated to be of the order of km. This means that the topographic masses are compensated throughout the Earth s lithosphere at least. Key words: density, gravity, topography. 1 INTRODUCTION The comparison of the Earth s observed external gravitational potential with that induced by its irregularly shaped surface (Martinec 1994) clearly indicates that the visible topographic masses are compensated by mass deficiencies in the Earth s interior. In the past, several models of compensation were proposed to cancel the effect of the topographic abundancies from surface gravity observations. Let us remind the reader of two highly idealized models. In the Pratt-Hayford model (e.g. Heiskanen & Moritz 1967) the topographic masses are compensated by the varying density distribution within the layer which under lies the mountains. The thickness of the layer is supposed to be constant and about 100 km. This implies that the higher the mountains, the less dense the underlying material. On the contrary, the Airy-Heiskanen model (Heiskanen & Moritz 1967) suggests that the topographic masses are compensated by the varying thickness of the Earth s crust with a constant mass density. The higher the mountains, the thicker the crust. Thus, root formations exist under the mountains. Both the models assume that the compensation is strictly local and confined to the uppermost regions of the Earth. Based on compiled values of crustal thickness, Cadek & Martinec (1991) have developed the model of the MohoroviEii: ( Moho ) discontinuity in terms of spherical harmonics up to degree and order 30. The degree correlation coefficients between the Earth s and the Moho topography show that the latter highly anti-correlates with the Earth s topography. Therefore, the Airy-Heiskanen model can probably be accepted as an explanation of the largest part of compensation of the topographic masses. Martinec (1994) determined the density contrast at the crust-mantle boundary coming from the Moho topography coefficients derived by Cadek & Martinec (1991). The minimization of the sum of the gravitational potentials induced by the Earth s topography and the Moho discontinuity yielded the value of 0.28 g cm-3 for the density jump at the Moho under the continental areas. In this paper, we will show that the Moho discontinuity only partly compensates the gravitational effect of the Earth s topography. Therefore, we will assume the lateral mass-density variations exist in the uppermost part of the Earth s interior and are responsible for additional compensation. In other words, we will take into consideration the Pratt-Hayford concept of compensation, as well as the Airy-Heiskanen model. The density parameters of the Pratt-Hayford compensation model will be carried out by solving the inverse gravimetric problem. Although this problem has no unique solution, we will employ the L*-minimum-norm criterion, which selects a possible solution and does not contradict the ideas of the mechanism of compensation. We shall mainly concentrate on estimating the depth of compensation, which is one of the most important parameters of the Pratt-Hayford model. Constraining the inverse problem by utilizing the dynamical flattening of the Earth, which has been determined with high accuracy by observing the precessional and nutational motion of the Earth, we shall estimate the minimum depth of compensation of the surface topography. 2 THE GRAVITATIONAL EFFECT OF A MASS DISCONTINUITY Let us first derive the external gravitational potential induced by an irregularly shaped mass-discontinuity D. Let the normal figure of D be characterized by a rotational ellipsoid r = rd,,(8) and the actual figure by the geocentric radius r=rd(q). Let the density contrast A@ at the discontinuity D be constant. The corresponding gravitational potential induced by the discontinuity D at an external point (r, Q) is expressed by the Newtonian volume 545

2 ~ 546 Z. Martinec integral where G is the gravitational constant, Q is the abbreviation for the pair of the angular coordinates 6, q, Q,, is the full solid angle, and dq is its element. Expanding the reciprocal distance between the attracted point r, Q and the mass point r', Q' by means of the decomposition theorem (e.g. Heiskanen & Moritz 1967), 1 Ir-r'l -4ni r J=02j+1 x ($)I m=-/ i ym(q)y:n,(q'), (2) yields the external gravitational potential in the following form: where the potential coefficients are given by do) A,,, = A e 1 I Gj,,(rj Q) drdq. (4) QII r=ri+jff) We have introduced the kernel and dropped the prime with respect to the internal mass-point coordinates. The y,( Q) are fully normalized spherical harmonics, the asterisk denoting complex conjugate. The scale factors M and a, introduced in eq. (3), normalize the potential coefficients Aim; M represents the mass of the Earth and a is the mean equatorial radius of the Earth. The integration in eq. (4) with respect to r can be performed immediately, yielding 4n 1 Aim = - Ae MaJ (2j + l)(j + 3) To perform the integration over the angular coordinate Q, we will assume that the geocentric radius of the discontinuity D and its reference surface Do are represented in terms of spherical harmonics, i.e. where Djm are the spherical-harmonic coefficients of discontinuity D truncated at degree j,,,, and d is its mean radius. The coefficients Do,jm characterize the reference rotational ellipsoid Do; they can be expressed in terms of the flattening of the ellipsoid (Pi% & Martinec 1983). Using the binomial series, the kth power of r,(q), k > 0, occurring in eq. (6) can be written as (3) (7) The product of spherical harmonics, having the same angles for arguments, will be expanded by means of the Clebsch-Gordan series (Varshalovich, Moskalev & Khersonskij 1975): 11'12 Y,,m,(Q)Y,zm2(Q) = c c Q:7k112m,Ym. (9) 1=111--/z1 m The quantities Q~Tm,12mz are the Gaunt's coefficients that can be expressed in terms of the Clebsch-Gordan coefficients C:>,,zmz (Varshalovich et af. 1975) The vector addition of angular momenta j, and j2, and the algebraic addition m, + m2 define the selection rules for the summation in eq. (9); the coefficients Q{%lj2mz are zero unless Iml '1 (11) m=m,+m, jl +j2 + j is an even number. Using expansion (9) in eq. (8), the power of rlj(q) can readily be arranged to read where the power topography Coefficients Dj:) can be expressed in terms of the topography coefficients Djm (Martinec 1994). Now, we are ready to perform the integration over the full solid angle 52 in eq. (6). Inserting expansion (12) into eq. (6), and making use of the orthogonality of spherical harmonics, we arrive at (13) This basic formula is used to compute the potential coefficients of the external gravitational field induced by the Earth's topography with the density jump A$= 2.67 g cm-', and by the Moho discontinuity with the density jump AeM = 0.28 g cry3 (Martinec 1994). The respective potential coefficients, denoted by symbols Asm and A:, are 4n as A'm = (2, + j + 3) z (14) x (:T+3[M;A+3) - M(I+3) o.lm lae'. The Earth's surface topography, r = r,(q), is represented by the TUG87 model (Wieser 1987), and the Moho topography, r = rm(q), by the model derived by Cadek & Martinec (1991). Quantities R and d denote the mean radius

3 Compensation of topographic masses II E lo-? 10' \ Power spectra 1- SURFLCETDPOGWHI. T +M-SURFACE TOPOCR*PHY*MOHO. G - GEM TI e DEGREE Figure 1. The degree power spectra of the external gravitational potential coefficients induced by the Earth's surface topography with the density contrast A$ = 2.67 g (the curve 'S'), and by the Earth's topography and the Moho discontinuity with the density contrast AgM = 0.28 g cw3 (the curve 'S + M'). The observed gravitational spectrum of the model GEM-TI (Marsh et al. 1988) is represented by curve 'G'. of the Earth's surface, R=6371 km, and of the Moho discontinuity, d = 6351 km, respectively. The degree power spectra (degree variances) of the potential coefficients AS, and AS, + A; are shown in Fig. 1. The power spectrum of the observed gravitational Earth's model GEMTl (Marsh el al. 1988) is also plotted for comparison. We can see that the Moho discontinuity partly compensates the gravitational effect of the Earth's topography, but the compensation is not complete because the degree power spectrum 'S + M' is considerably greater than the observed power spectrum 'G'. It means that besides the Airy-Heiskanen type of compensation there is another mechanism compensating the topographic masses. 3 INTEGRAL CONSTRAINTS TO DENSITY The aim of the inverse gravimetric problem consists in finding the mass density distribution p(r, Q) in the Earth, if a set of measurements of its functionals are given, for instance k, Lo g(r, Q)Gjm(r, 9) dr dq =Ajm for j =0, 1,..., j, rn = -j, -j + 1,..., j, where 30 r = r,(q) is the geocentric radius of the Earth's surface, and kernel GIm(r, 52) is given by eq. (5). The observations in this case are the potential coefficients A,, of the external gravitational field cut at degree j,,,. Hereafter, the general form of the constraints (15) will be specified for the purpose of interpreting the potential coefficients ASm + A: as follows. We assume that the topographic masses are compensated by both the thickening of the Earth's crust (the Airy-Heiskanen type of compensation) and the lateral-mass heterogeneities 6e(r, Q) in the layer of compensation with the lower boundary given by a rotational ellipsoid, r=rb(6), and the upper boundary given by the Earth's surface, r = r,(q) (the Pratt-Hayford type of compensation). The density variations 6e(r, Q) are referred to the reference model Po(', 6) with the spheroidal density distribution. Nakiboglu (1982) derived such a model by applying the theory of equilibrium of a self-gravitating, slowly rotating body (Kopal 1960) to the spherically symmetric density-model PREM (Dziewonski & Anderson 1981). The mass density inside the Earth is then of the form e(r, Q) = eo(r, 6) + 6e(r, Q) for rh(@) 5 r 5 r,(q), = eo(r, 6) for 0srsrB(6). (16) Obviously, the reference density model eo(r, 6) forms the predominant part of the mass density in the Earth's interior. Inserting the density model (16) into the integral on the left-hand side of eq. (15), we can write h" I:,) W(Q),- rdq) 6p(r, Q)G,,(r, Q) dr dq Q) dr dq QI) r=o %(n) - k,l eo(r, ')Gjrn(r> Q) drd~, (17) r=rm(q) = Aim - I I edr, 6)Gjm(r, where r =rm(q) is the geocentric radius of the Moho discontinuity. Introducing the reference surfaces r = rmi,(6) and r =rs,,(6) of the Moho discontinuity and Earth's topography, respectively, the integrals on the right-hand side of eq. (17) can be readily arranged to read I"'"' 6p(r, Q)G,(r, Q) dr dq Qa r=%(b) =Aj, - I Wl)(*) Q() r=o 6) - 1 Qo 1%' r=rmll(@) - AgM jw(") r Qn r=m,)(*) ra(q) Q) drsq eo(r, 6)Gjm(r, Q) dr dq Gjm(r, 52) dr dq - Aes I I Gim(r, Q) dr dq. no r=%(l(@) Using formula (4), we arrive at I %(Q) Qo r=5(*) 6g(r, Q)Gj,(r, Q) drdq + ASm +A$

4 548 Z. Martinec where A;m and A% are the potential coefficients induced by the Earth's surface and Moho topography, respectively. Assuming that the angular part of the hydrostatic mass density distribution Po(', 8) is described by the zonal spherical harmonics with the angular degrees j = 0 and j = 2, the integral on the right-hand side of eq. (19) vanishes unless the angular indexes jm take the value j = 0 or 2 and rn = 0. Besides other conditions, the hydrostatic equilibrium body of the Earth can be constructed to fit exactly the observed value of the mass of the Earth (Nakiboglu 1982). In our notation it means that Moreover, the zonal, second-degree, hydrostatic potential coefficients approximates the observed coefficients A,,) so closely that the difference A20-At;dro is of the same small magnitude as the other observed potential coefficients A,m. Thus, we can see that the largest terms A,,,, and A,,, occurring on the right-hand side (19) are cancelled owing to the properties of the hydrostatic density model. The magnitudes of the other coefficients A,,,, are about one order smaller than those of A;m +A: (Fig. 1). Furthermore, since we are interested in interpreting the gravitational field induced by the Earth's surface and Moho discontinuity only, we will drop the observed coefficients Ajm in eq. (19). We are thus looking for the lateral density variations 6e(r, Q) in the layer of compensation which are constrained by It should be noted that to interpret the observed quantities A, in terms of a density distribution, i.e. to solve the inverse gravimetric problem for Aim, is more problematic than making it for the coefficients A;,,, +A:, because the coefficients Aim are supposed to be induced by the mass distribution in the whole Earth's body, while the gravitational field of Asm + A: is reasonable to assume that it is compensated in the uppermost part of the Earth's only. Therefore, the non-uniqueness in the radial direction for the latter problem is not so strong as in the first case. 4 THE DYNAMICAL FLATTENING Another constraint imposed on the mass density distribution follows from the dynamical flattening H defined in terms of the principal moments of inertia A, B and C as (BurSa 1982) The value of H is known with high accuracy from astronomical observations of the nutational and precessional motion of the Earth. Instead of the dynamical flattening H, it is more suitable to formulate the constraint on the density model in terms of the trace of the inertia tensor, TrZ = A + B + C, (24) which is an invariant quantity. Nakiboglu (1982) and also PEE & Martinec (1984) showed that the trace of the inertia tensor can be expressed as where A2() is the second-degree zonal Stokes' coefficient. Evaluating TrZ in terms of the density (Pi% & Martinec 1984) yields Inserting the density model (16) into the integral on the left-hand side of eq. (26), we can write s(q),f,/ r=ro(e) where 6e(r, Q)r4 dr dq = Tr Z - Z, (27) By analogy with arrangement in eq. (18), expression (28) can be written in the form Z = / /?'(",eo(r, 8)r4 dr dq R,1 r=o + AeM / /m(q) r4 dr dq QI r.=w,,(b) Besides the condition on the mass, eq. (20), the hydrostatic equilibrium body of the Earth can be constructed to fit the observed value of the trace of the inertia tensor (Nakiboglu 1982). It then follows that and the integral constraint (27) on the density variations 6e(r, Q) takes the form The integration on the right-hand side of the last relation can be performed by making use of the spherical harmonic expansion (12) for the fourth power of the Earth's surface and Moho topography radii: / / 6e(r, Q)r4drdQ R(I r=rh(-'>)

5 Compensation of topographic masses INVERSION AND PARAMETRIZATION Up to the spectral degree j,,, the integral constraints (22) and (32) do not determine the lateral density 6p(r, 52) uniquely; the problem as a whole represents an underdetermined inverse problem (Tarantola & Valette 1982) for the lateral density variation 6p(r, 52). To fix one solution, an additional criterion has to be added. Sansb, Barzaghi & Tscherning (1986) discussed several different criteria, e.g. the minimum-energy solution, minimum-norm solution, etc., which can be used to render the solution of inverse gravimetric problem tractable. Here we will choose the minimum-norm solution, which means that the lateraldensity variation 6p(r, &) differs minimally (in the sense of the L minimum norm) from the rotationally symmetric, reference, density model po(r, 6). Sansb et al. (1986) have shown that the L -minimum-norm criterion, applied to integral constraint (22), only generates a harmonic-density model which attains extreme values at the boundaries of the region under study. This mini-max property of the L2-minimum-norm criterion will be perturbed in our case when the density variation 6p(r, Q) is further constrained by the dynamical flattening restriction (32). Thus, we want to determine density 6p(r, Q) so that represents the anharmonic part of the density variations &(r, Q) and will play an important role in determining the thickness of the layer of compensation. The parametrization (36) will be considered in terms of a slightly different normalization, (37) where a is the mean equatorial radius of the Earth. Inserting (37) into integral constraints (22) and performing the integration, we arrive at the system of linear algebraic equations: (33) under integral constraints (22) and (32); here dt = r2 dr dq. The problem can be transformed to the variational form as for j = 0, 1,..., j,,,, rn = -1, -j + 1,..., j; Bfi) are the harmonic coefficients of the kth power of the rotational ellipsoid radius r = rb(6), which describes the lower boundary of the compensation layer, and 6 is its mean radius. Gaunt s coefficients Q~ml12m2, occurring in eq. (38), are defined by eq. (10). Similarly, constraint (32) on the dynarnical flattening of the model leads to the equation where MaJ dim = -(2j + 1) - 4n +A:), (35) In eq. (34) symbol 6 denotes the variation of the functional in brackets with respect to the density anomalies 6p(r, Q); aim and B denote Lagrange s multipliers. The solution of (34) is given by imax i 6p(r, Q) = j=om=-j 2 C aimriym(q) + Br2KX)(Q). (36) The first term on the right-hand side appeared due to the L -minimization with respect to constraints (22) on the external gravitational potential coefficients A& + AE; this minimization leads to the well-known fact (Sansb et af. 1986) that the parametrization is set up by means of harmonic functions. The second term reflects the restrictions due to the dynamical flattening of the model. In fact, this term 6 NUMERICAL RESULTS (39) Equations (38) and (39) represent a system of linear algebraic equations in the unknown coefficients ejm, j = 0, 1,..., jmax, rn = -j, -j + 1,..., j, and /3. The system is exactly determined since the number of unknowns, (jmax + 1) + 1, is equal to the number of equations. The system was solved for degree j, = 30 which means that both the input Earth s surface (Wieser 1987) and Moho (Cadek & Martinec 1991) topographies were truncated at the same degree 30. The numerical test was intended to estimate the role of constraint (32) on the dynamical flattening of the density model 6g(r, a). For this reason, the thickness of the layer of compensation was changed by moving its lower boundary

6 Martinec Correction to PREM i.e. the quantities b(km) Figure 2. Corrections Atr(a) to the PREM model. If the thickness of the compensation layer is smaller than 100km, the density corrections A,, increase enormously and create a double layer with opposite signs of anomalous masses. (the mean radius of which was denoted by symbol b in the text) from the core-mantle boundary position (b = 3481 km) towards the Earth's surface. The system of eqs (38) and (39) was solved for each position of b, and the corrections to the reference, rotationally symmetric, density model eo(r, 6), 90 le (T Ao(r)=-+fi -, Density map (I00 km) were evaluated. The result of the test, shown in Fig. 2 for r =u, can be interpreted as follows: if the mean radius b of the lower boundary of the compensation layer is at depths below km, the corrections A. are small and, thus, the density variations 6e(r, 9) represent small corrections to the reference model po(r, 6). This means that any radius r,(b) below the depth of km is acceptable and it cannot be uniquely decided where the lower boundary of compensation lies. If the thickness of the compensation layer is smaller less than a value of about 100 km, corrections A. become unacceptably large. Such large contributions to the reference rotationally symmetric model are unrealistic and cannot be accepted, owing to the high internal precision of the PREM model. It should be noted that if the dynamical flattening constraint is removed, the lower boundary of the compensating layer can be moved up until it reaches the Earth's surface. However, if we retain this important constraint, the level of compensation needs to be at a depth of at least 100 to 150 km and cannot be put at depths of only tens of kilometres. For example, the Earth's topographic irregularities cannot be explained solely by the lateral density variations in the Earth's crust. Table 1 gives the coefficients ajm for the case when the lower boundary of the compensation layer is put at a depth of 150 km. The largest terms qm and fi, whose role for estimating the minimum depth of compensation was tested in the preceeding paragraph, are given in the caption. The harmonic representation of the density variations " Figure 3. The representation of the lateral density variation Q(r, Q) by the spherical-harmonic expansion (37) (j,,, = 30) at the depth of 100km. The contour interval is 5 kgm-', the solid and dashed contours are for regions of positive and negative density deviations, respectively. The minimum is -53 kg m-3 under the Himalayas, the maximum is +34 kg m--3 under Australia.

7 Table 1. The coefficients a,,,, and 0 of the lateral-density variation 6e(r, a) (in kg m- '), a,k,/& = and B = Q jm j m... j m jm a go lo a os OO * lo a go Compensation of topographic masses 551 Q

8 552 Z. Martinec I ll I lo lo

9 * * , lo I El ll

10 554 Z. Martinec &(r, Q) at the depth of 100 km is plotted in Fig. 3. The negative anomalies (lighter material) are associated with high mountains, Himalayas (-53 kg m- ), the Rocky Mountains (-35 kg m- ), Andes (-28 kg m- ), Antarctica (-35 kg m- )), Greenland (-30 kg m- ), and the southeastern part of Africa (-30 kg m- ). The greatest positive anomalies are located under Australia (+34 kg rn- ), the western part of the Eurasian plate (+30kgm- ) and the Hudson Bay territory (+20 kg m- ). 7 CONCLUSION Large sources of gravitational attraction such as the Earth s surface with a large density contrast are only weakly exhibited in the external gravitational field. Therefore, the Earth s topography has to be gravitationally compensated. If the Airy-Heiskanen type of compensation is taken into account, the compensation of the Earth s topography is not sufficient enough, namely the external gravitational field produced by such a compensation model is still too large compared with observations. Thus, besides the Airy- Heiskanen compensation model, lateral-density variations in the uppermost part of the Earth s interior must exist. This concept of compensation of the topographic masses originates with Pratt and with Hayford. Based on the gravimetric inverse problem, the parameters of this latter model of compensation were estimated. Since determining the radial distribution of the lateral density variations within the layer of compensation still belongs to the class of highly non-unique inverse problems, we have only concentrated on estimating the thickness of the compensation layer. When the dynamical flattening of the Earth is used to constrain the density model, the minimal thickness of the compensation layer is about km. This value is in good agreement with the value of 100 km originally suggested by Pratt and by Hayford (Heiskanen & Moritz 1967). This also means that the topographic masses are not only compensated in the Earth s crust, but also throughout the whole lithosphere. The test with the depth of Compensation clearly indicates that the constraint on the density model due to the dynamical flattening of the Earth must not be omitted in the solution of the inverse gravimetric problem, e.g. as in the formulations by Tscherning & Sunkel (1981), Sans6 et al. (1986), or VaniEek & Kleusberg (1985). Although this restriction does not uniquely determine the radial distribution of the density variations, it does not permit fitting the external gravitational field by a single-layer material surface because unrealistically large density corrections are presented. ACKNOWLEDGMENT I would like to thank C. Matyska for inspiring suggestions concerning the parametrization of the lateral-density variations. I also wish to acknowledge A.K. Goodacre who carefully reviewed the manuscript. REFERENCES BurSa, M., Luni-solar precession constant and accuracy of the Earth s principal moments of inertia, Stud. Geophys. Geod., 26, cadek, 0. & Martinec, Z., Spherical harmonic expansion of the Earth s crustal thickness up to degree and order 30, Stud. Geophys. Geod., 35, Dziewonski, A.M. & Anderson, D.L., Preliminary Reference Earth Model, Phys. Earth planet. Infer., 25, Heiskanen, W.A. & Moritz, H., Physical Geodesy, Freeman and Co., San Francisco, CA. Kopal, A., Figures of Equilibrium of Celestial Bodies, University of Wisconsin Press, Madison, WI. 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