On inversion for mass distribution from global (time-variable) gravity field

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1 Journal of Geodynamics 39 (2005) On inversion for mass distribution from global (time-variable) gravity field Benjamin F. Chao Space Geodesy Branch, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Received 20 July 2004; received in revised form 9 November 2004; accepted 10 November 2004 Abstract The well-known non-uniqueness of the gravitational inverse problem states that the external gravity field, even if completely and exactly known, cannot uniquely determine the density distribution of the body that produces the gravity field. In this paper, we provide conceptual insight by examining the problem in terms of spherical harmonic expansion of the global gravity field. By comparing the multipoles and the moments of the density function, we show that in 3-D the degree of knowledge deficiency in trying to inversely recover the density distribution from an external gravity field solution is (n + 1)(n + 2)/2 (2n +1)=n(n 1)/2 for each harmonic degree n. On the other hand, on a 2-D spherical shell we show via a simple relationship that the inverse solution of the surface density distribution is unique. The latter applies quite readily in the inversion of time-variable gravity signals (such as those observed by the GRACE space mission) where the sources largely come from the Earth s surface over a wide range of timescales Elsevier Ltd. All rights reserved. Keywords: Inverse problem; Solution uniqueness; Gravity; Time-variable gravity; Spherical harmonics 1. Introduction The well-known non-uniqueness of the gravitational inverse problem states the following: The external gravity field, even if completely and exactly known, cannot uniquely determine the density distribution of the body that produces the gravity field. This is an intrinsic property of a field that obeys the Laplace equation, as already treated in mathematical as well as geophysical literature (e.g., Menke, 1989; Zidarov, 1990; Parker, 1994; Backus et al., 1996). In a simple example, imagine two uniform concentric spherical address: benjamin.f.chao@nasa.gov /$ see front matter 2005 Elsevier Ltd. All rights reserved. doi: /j.jog

2 224 B.F. Chao / Journal of Geodynamics 39 (2005) shells, one with mass m and the other m, superimposing onto an arbitrary body. It is obvious that the two shells collectively have zero effect on the external gravity field, and hence their existence cannot be inferred from outside gravitationally. The same assertion applies to the time-variable gravity (TVG) produced by a mass distribution varying with time. Global TVG signals detected from space contain information about mass redistributions that occur in the Earth system. Hence, it is becoming a new data type for monitoring climatic and geophysical changes. For example, the GRACE mission (Tapley et al., 2004) is to obtain a high-precision, global gravity map every month or so, at a spatial resolution of several hundred km. The month-to-month difference then signifies mass redistributions in or on the Earth during that given month. What exactly would such TVG data tell us about mass redistribution? For one thing, the TVG is the sum of the gravitational signals originating from all geophysical sources at work at any given time. Sorting out different geophysical signals in the data is a challenge (as well as an art ), but in principle can be facilitated by recognizing the different temporal and spatial characteristics of different geophysical phenomena (e.g., Chao, 1994). This is a problem depending on the real-world geophysics it is not the subject of the present paper. What we want to examine, and we do it analytically, is something more basic conceptually the uniqueness of the general gravitational inversion in terms of spherical harmonic functions for the global field. We shall assume that we have the complete and exact knowledge of the external gravity field (which in fact only constitutes such knowledge on a closed boundary thanks to the Laplace Equation), and we will not deal with practical issues such as solution stability, error propagation, or optimization schemes. The mathematical essence of various aspects of the relevant theory has appeared in variant forms in the literature; in that sense we here cast the theory in geophysical perspective, aiming towards a comprehensive appreciation and understanding of the problem. We shall also treat some situations from a practical viewpoint of TVG, particularly if the latter originates from the surface of the Earth. 2. Gravitational inverse problem in 3-D According to Newton s gravitational law, the external gravity potential field U produced by a (finite) body of internal density distribution ρ over volume is given by the volume integral: ρ(r 0 ) U(r) = G r r 0 d (1) where G is the universal gravitational constant. In the spherical coordinates, r = (radius r, co-latitude θ, longitude λ)=(r, solid angle Ω), if we expand into spherical harmonic functions by the addition theorem the function 1/ r r 0 = n n=0 m= n (2n + 1) 1 (r0 n/rn+1 )Ynm (Ω)Y nm(ω 0 ), we get the so-called multipole expansion of U (e.g., Jackson, 1975): U(r) = G n n=0 m= n 1 (2n + 1)r n+1 [ ρ(r 0 )r n 0 Y nm(ω 0 )d ] Ynm (Ω) (2) for a singly connected, where Y nm is the 4 -normalized surface spherical harmonic function of degree n and order m, and the volume integral in the bracket is the corresponding (complex-valued) multipole of the density distribution ρ(r) where dv =r 2 dω dr =r 2 sinθ dθ dλ dr.

3 B.F. Chao / Journal of Geodynamics 39 (2005) At the same time, because U satisfies the Laplace equation exterior of, it has a closed-form solution customarily expressed as (e.g., Kaula, 1966): U(r, θ, λ) = GM n ( a ) n+1pnm (cos θ)(c nm cos mλ + S nm sin mλ) (3) a r n=0 m=0 where M is the Earth s mass, P nm is the 4 -normalized Legendre function: Y nm (Ω)=P nm (cosθ) exp(imλ). The dimensionless coefficients C nm and S nm are known as the (normalized) Stokes coefficients of degree n (= 0,1,2,..., ) and order m (= 0,1,2,...,n); they constitute the values given to specify a gravity field model. The value of the Stokes coefficients actually refers to the value of some convenient length parameter a; in the case of the Earth a is often chosen to be the mean radius of the Earth along the Equator. It really does not matter what a is as long as r = a is conveniently exterior of the body, because once the Stokes coefficients are known with respect to the chosen a, the entire external gravity field is completely determined all one needs to do is to upward or downward continue, as the case may be, via the radial transfer function (a/r) n +1 for each degree n according to Eq. (3). Comparing Eqs. (2) and (3), one gets 1 C nm + is nm = ρ(r)r n Y (2n + 1)Ma n nm (Ω)dV (4) (where for brevity the subscript 0 in the integrand is omitted). That is, the Stokes coefficients are simply normalized multipoles of the density distribution of the body (e.g., Chao and Gross, 1987; Chao, 1994). The gravitational inverse problem in the present context, Eq. (4), is a canonical linear inversion: given a complete and exact knowledge of the gravity field in the form of the (infinite) set of Stokes coefficients, how much can we learn about ρ(r) in by way of Eq. (4)? All we know are the multipoles, i.e., the projections of ρ(r) onto the solid spherical harmonics functions r n Y nm (Ω) in a Hilbert space H that contains all possible ρ(r) functions. However, the (infinite) set of solid spherical harmonics does not form a complete set of basis function for H it only spans the harmonic sub-space of H, leaving an orthogonal or null sub-space wherefrom any element when added to any solution forms a new solution but having the same Stokes coefficients. To proceed we note that in the Cartesian coordinates one can invoke another expansion for 1/ r r 0, namely the Taylor expansion: 1 r r 0 = ( 1) n [ x 0 n! x + y 0 y + z 0 z n=0 ] n ( ) 1 r Now invoking the multinomial theorem on Eq. (5) and substituting it into Eq. (1), we get another expression for the solution of U (Morse and Feshbach, 1953, pp ): U(r) = G n=0 α+β+γ=n α,β,γ 0 ( 1) n [ α!β!γ! x α 0 yβ 0 zγ 0 ρ(r 0)d ] n ( ) 1 x α y β z γ r We shall call Eq. (6) the moment expansion for U, because the quantity in the bracket is recognized as the (3-D) moment of order α + β + γ = n for the density distribution ρ(r) in. The multipole expansion (2) and the moment expansion (6) are, of course, equivalent. Equating the degree of multipole with the order of moment (we have actually used the same symbol (5) (6)

4 226 B.F. Chao / Journal of Geodynamics 39 (2005) n), we note the following: For each n, there are 2n + 1 independent multipoles while there are (n + 1)(n + 2)/2 independent moments. [The latter is because the form x α y β z γ where α + β + γ = n has (n +1)=(n + 1)(n + 2)/2 terms.] This seeming paradox can be resolved by realizing that certain (in fact the majority of combination of) terms in (6) actually have zero contribution to U (see Morse and Feshbach, 1953, pp ); that is, they belong to the null sub-space in H with respect to U. On the other hand, it is of central importance in the present discussion that, as far as ρ(r) is concerned, the moments uniquely determine ρ(r). In other words, knowing the infinite set of moments of ρ(r) is equivalent to knowing ρ(r) itself. This is known as the Moment Theorem in statistics literature (e.g., Papoulis, 1965, p. 214), and is closely related to the Weierstrass Approximation Theorem (e.g., Backus et al., 1996). In the present context, the moments constitute of projections onto a complete basis function set that spans H. Thus, in trying to determine the moments but only knowing the multipoles, our degree of deficiency in knowledge is (n + 1)(n + 2)/2 (2n +1)=n(n 1)/2 for each n, where n = 0,1,2,...,. Conceptually, one can decompose the Hilbert space H into disjoint sub-spaces, H = H n, each one having a null sub-space of dimension n(n 1)/2. For n = 0, there is only one term in both multipole and moment expansions, i.e., the monopole = zeroth moment, corresponding to the total mass of the Earth. A complete determination is also assured for n = 1, where there are three terms for both expansions: the three degree-1, or dipole, Stokes coefficients correspond to the three first moments of ρ(r) which give the three components of the position vector of the body s center of mass, or the so-called geocenter in the case of the Earth. For n = 2, we have five degree-2, or quadrupole, Stokes coefficients but six second moments, i.e. the elements of the inertia tensor. The multipoles are no longer sufficient to determine the corresponding moments (e.g., Chao and Gross, 1987). The degree of deficiency grows rapidly as n increases to infinity; its numerability provides a quantitative assessment of how grossly non-unique the 3-D gravitational inversion is. The situation is summarized in Table 1. This degree of deficiency is associated with the freedom in determining the radial dependence of ρ(r), as will be shown in the next section. We should mention that, the multipoles (Eq. (2)) and the moments (Eq. (6)) being linear in ρ(r), conceptually we can replace ρ(r) by just the lateral density anomaly, i.e., the laterally heterogeneous part of ρ(r) after removing the dominant, spherically symmetric part (which only translates into the n =0 term). This scheme is useful in practical applications, because then under certain extra constraints the inversion of Eq. (4) can become unique so as to yield unique solutions for ρ(r). For example, applying Table 1 The degree of deficiency as a function of spherical harmonic degree n in the 3-D gravitational inversion Degree n No. of multipoles (2n + 1) No. of moments (n + 1)(n + 2)/2 Degree of deficiency n(n 1)/2 0 1 (monopole) 1 (total mass) (dipole) 3 (center of mass) (quadrupole) 6 (inertia tensor) (octupole) 10 (3rd moment)

5 B.F. Chao / Journal of Geodynamics 39 (2005) the physical condition of minimum shear energy in the mantle, Kaula (1963) determined a unique lateral density distribution for the Earth. Rubincam (1982) achieved the unique solution by employing the mathematical constraint of maximum entropy. Likewise, it can be shown that minimizing the norm-2 variance for the lateral distribution also leads to unique solutions. However, the variability of the lateral density obtained in the latter two cases is highly concentrated towards the surface as a consequence of the imposed mathematical conditions that do not necessarily reflect the reality. 3. Gravitational inverse problem on a 2-D spherical shell Anticipating the TVG case below, let us now consider a special case a gravitating body that is a spherical shell with infinitesimal thickness, called S 0. Let the (2-D) surface mass density distribution over S 0 be σ(ω). How much of the σ(ω) function can we recover given a complete knowledge of external Stokes coefficients as above? Intuitively, we no longer need to recover quite as much as in the 3-D case, because no radial dependence exists. We now show that such σ(ω) can be uniquely determined, and that the above non-uniqueness in 3-D is thus associated with the radial dependence. The proof of this uniqueness for the gravitational inversion over S 0 is rather simple, as follows. Eq. (4) reduces to C nm + is nm = a 2 (2n + 1)M S 0 σ(ω)y nm (Ω)dΩ (7) on S 0, where we have let a simply be the radius of the shell. Now note that the integral on the right side of Eq. (7), thanks to the orthogonality of spherical harmonics, is simply the coefficient of the (n,m)th harmonic component of σ(ω), that is, the σ nm (apart from a multiplicative constant) in the spherical harmonic expansion: σ(ω) = Σ n,m σ nm Ynm (Ω). Eq. (7) can then be written as σ nm = (2n + 1)M 4πa 2 (C nm + is nm ) (8) Eq. (8) states that each of the harmonic components of σ(ω) is determined by the corresponding harmonic component of the external gravity field (i.e. Stokes coefficient), by way of the transfer function 2n + 1 (and a multiplicative constant), which increases with increasing n. This seemingly trivial fact that Eq. (7) is invertible into (8) is actually quite profound: knowing the set of the gravitational Stokes coefficients is equivalent to knowing the spherical harmonic expansion of σ(ω), which in turn is equivalent to knowing σ(ω) itself (because the spherical harmonic functions form a complete set for S 0 ). An equivalent way of stating the above is that the surface spherical harmonics span the Hilbert space containing all 2-D σ(ω) functions defined on S 0. Alternatively, one can state that on a 2-D spherical surface the moment expansion has 2n + 1 terms, the same as the number of independent multipoles for any given n. Therefore, the 2-D spherical σ(ω) can be uniquely determined gravitationally. A corollary is that one can always construct a σ(ω)ons 0 to mimic the external field created by some arbitrary mass distribution in (e.g., Menke, 1989, p. 270). We shall elaborate on this point when discussing the TVG below.

6 228 B.F. Chao / Journal of Geodynamics 39 (2005) Time-variable gravity problem: 3-D body versus 2-D spherical shell When the mass distribution in the body changes with time, the external gravity field would change accordingly. This is the forward problem. The inverse problem is to recover to the extent possible the mass redistribution in from the observed TVG field based on Eqs. (4) and (7). There are two ways to describe a mass transport in a continuum: the Eulerian approach and the Lagrangian approach. They are conceptually equivalent to each other. Here for the sake of argument, we shall adopt the Eulerian description because of its simpler mathematics. In the Eulerian description for a 3-D body, the TVG is expressed in terms of time-variable Stokes coefficients (e.g., Chao, 1994): 1 C nm (t) + i S nm (t) = ρ(r; t)r n Y (2n + 1)Ma n nm (Ω)dV (9) The corresponding 2-D spherical shell case reduces to a 2 C nm (t) + i S nm (t) = σ(ω; t)y nm (Ω)dΩ (10) (2n + 1)M S 0 or equivalently for σ(ω; t) = Σ n,m σ nm (t)ynm (Ω), as Eq. (8), (2n + 1)M σ nm (t) = [ C 4πa 2 nm (t) + i S nm (t)] (11a) As before, Eq. (9) is not invertible and cannot uniquely determine the bodily density change ρ(r;t)in. In contrast, without the radial dependence Eq. (10) is completely invertible to yield a unique surface density change σ(ω;t)ons 0 as specified in Eqs. (11a) or (11b) below. A caveat should be injected at this point. In applying Eq. (11a), one should ascertain that σ truly represents the surface value. If, for example, σ is evaluated from an apparent surface water load in terms of water depth, then σ will be accompanied by an additional (and in general non-superficial) deformation in the solid Earth due to sold Earth s elastic yielding, which is proportional to the load itself. The factor of the proportionality, different for each n, is a negative fractional number known as the load Love number k n. The net effect is that σ is somewhat reduced by a factor of (1 + k n ), about 0.70 for n = 2 and monotonically approaching 1 as n increases. Thus, to undo this loading effect, the apparent σ becomes (2n + 1)M σ nm (t) = 4πa 2 (1 + k n )[ C nm(t) + i S nm (t)] (11b) This formula has been employed by, e.g., Chao et al. (1987), Chao (1994), and Wahr et al. (1998). Even though only applicable to a 2-D spherical shell, this invertibility is actually quite useful with respect to our real Earth. To be more specific, suppose we observed the TVG in terms of a set of changing Stokes coefficients. We can convert the latter readily into σ(ω) onr = a according to Eqs. (11a) or (11b) whichever is appropriate. In doing so, we construct a σ(ω) ons 0 that mimics the external field produced by the original mass redistributions, whether occurring on r = a or elsewhere. In principle that constitutes the entire information about the density redistribution that can be recovered from the Stokes coefficients alone, while nothing more can be inferred about the depth or altitude (i.e., radial distance) at which the mass redistributions actually take place. However, for the Earth, this is already a fairly accurate representation, because a majority of the important mass transports do take place near the Earth s surface.

7 B.F. Chao / Journal of Geodynamics 39 (2005) Notable departures from a spherical surface arise from the ellipticity of solid Earth s shape (less than half of 1/300a or 10 km), land topography (within a few km), atmospheric thickness (scale height about 10 km), depth of the oceans (within about 5 km), or crustal thickness (within say, 30 km). Mass transports do take place in the deeper interior of the Earth, but have typically much longer timescales which make the signal separation possible. A final point to make: suppose there is a near-surface 2-D mass redistributions, which occurs on a surface at some radial distance from r = a. How do we map the nominal density redistribution σ on r = a obtained above into its true value at its true location and depth? In the trivial case where the new surface is simply another concentric sphere, all one needs to do is to upward or downward continue σ(ω) by multiplying with the radial transfer function (a/r) n+1 just as one would for the gravity field as mentioned earlier (Eq. (2)). For an analytical departure such as the elliptical shape of the Earth, it is conceivable that some analytical expressions involving Wigner 3-j symbols can be obtained for the mapping, at least to first order of small amplitude of the radial departure. If so, then the mapping for an arbitrary bumpy surface slightly deviated from a sphere, whose amplitude can be expanded into spherical harmonics, can also be worked out. This is beyond our present interest, however. 5. Discussions and conclusions We examined the uniqueness problem of the gravitational inversion in terms of spherical harmonics. We showed that the reason for the non-uniqueness in the 3-D inversion is because the set of 2n + 1 multipoles are inadequate to recover the (n + 1)(n + 2)/2 moments for the (bodily) density function, and hence the degree of deficiency in knowledge is n(n 1)/2, which grows rapidly with the degree n. In complete contrast, in the case of a 2-D spherical shell without the radial dependence, the gravitational inversion for the (surface) density function proves to be unique. This has important implications for actual geophysical situations where the time-variable gravity signals often do originate from Earth s surface which can be approximated by a spherical shell. The simplicity in the 2-D inversion (8), (11a) and (11b) owes its existence to our use of the spherical harmonics as basis functions. The reason is traced to the fact that the solid spherical harmonics are themselves solutions to the Laplace equation which the external gravity field obeys. A consequence is the following. Since the spherical harmonics are global by definition and hence inefficient in expanding regional gravity or density variabilities, there have been efforts in adopting other types of localized basis function sets such as spatial wavelets. However, although potentially more efficient in numerical representations, these basis functions in general do not enjoy the simple relationship between the (timevariable) gravity and the density (re)distribution, as do spherical harmonics in Eqs. (8), (11a) and (11b). Hence their applicability to the gravitational inversion, particularly for the 2-D spherical shell, may not be straightforward. This of course awaits further investigations. Another advantage of treating the problem in spherical harmonics lies in the natural wavelength scale of the spherical harmonics depending on the degree n. In this paper, we have made our theoretical formulas as if we had complete knowledge of the gravity field with n going to infinity. In practice there is always a limit as to how high n can be, and one simply truncates the expansions at a certain maximum degree N that corresponds to some highest possible spatial resolution. For example, in space gravity missions, a rule of thumb is that the nominal spatial resolution 20,000/N km is comparable to the spacecraft altitude from which the spacecraft senses the gravity. Within N, the entire argument presented in the above holds

8 230 B.F. Chao / Journal of Geodynamics 39 (2005) valid perfectly. We can truncate at any N; the lack of information beyond N simply need not enter into the consideration. Acknowledgments This study is supported by NASA s Solid Earth and Natural Hazards Program. References Backus, G., Parker, R., Constable, C., Foundations of Geomagnetism. Cambridge University Press, New York. Chao, B.F., Gross, R.S., Changes in the Earth s rotation and low-degree gravitational field induced by earthquakes. Geophys. J. Roy. Astron. Soc. 91, Chao, B.F., O Connor, W.P., Chang, A.T.C., Hall, D.K., Foster, J.L., Snow-load effect on the Earth s rotation and gravitational field, J. Geophys. Res. 92, Chao, B.F., The geoid and Earth rotation. In: Vanicek, P., Christou, N. (Eds.), Geophysical Interpretations of Geoid. CRC Press, Boca Raton. Jackson, J.D., Classical Electrodynamics, 2nd ed. Wiley, New York. Kaula, W.M., Elastic models of the mantle corresponding to variations in the external gravity field. J. Geophys. Res. 68, Kaula, W.M., Theory of Satellite Geodesy. Blaisdell, Waltham, MA. Menke, W., Geophysical Data Analysis Discrete Inverse Theory. Academic Press, San Diego. Morse, P.M., Feshbach, H., Methods of Theoretical Physics. McGraw-Hill, New York. Papoulis, A., Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New York. Parker, R.L., Geophysical Inverse Theory. Princeton University Press, Princeton, NJ. Rubincam, D.P., Information theory lateral density distribution for Earth inferred from global gravity field. J. Geophys. Res. 87, Tapley, B.D., Battadpur, S., Watkins, M., Reigber, C., The gravity and climate recovery experiment: mission overview and early results. Geophys. Res. Lett. 31, L09607, doi: /2004gl Wahr, J., Molenaar, M., Bryan, F., Time variability of the Earth s gravity field: Hydrological and oceanic effects and their possible detection using GRACE. J. Geophys. Res. 103, Zidarov, D., Inverse Gravimetric Problem in Geoprospecting and Geodesy. Elsevier, New York.