Abstract. The primary drawback of forward gravity field modelling is that the Earth's density distribu- tion must be known. Nowadays, increasingly more.
On the Optimal Spatial Resolution of Crustal Mass Distributions for Forward Gravity Field Modelling M. Kuhn, W.E. Featherstone Western Australian Centre for Geodesy, Curtin University of Technology, GPO Box U1987, Perth WA 6845, Australia Abstract. The primary drawback of forward gravity field modelling is that the Earth’s density distribution must be known. Nowadays, increasingly more information on the Earth’s mass distribution is available, such as high-resolution digital elevation models, models of the crustal mass distribution as well as of deeper masses. This paper studies the spatial resolution of crustal mass distributions required in forward gravity field modelling using spherical harmonic expansion of global data and power spectral density functions of local data. The spectral sensitivity of different gravity field parameter is examined by means of degree variances and analytical degree variances. Numerical results for the geoid height and gravity disturbance are given globally and in a test area over part of Australia. Keywords. Forward gravity field modelling, spatial resolution, crustal mass distribution
1 Introduction Several authors have studied the spectral properties of different parameters regarding the Earth’s gravity field parameter (e.g., Tscherning and Rapp 1974, Schwarz 1984). These studies are empirical, being based on observations of the Earth’s gravity field. However, all the data types used in practical gravity field modelling are band-limited, i.e., they are only sensitive to specific spectral bands. To resolve the complete spectrum, a combination of these different gravity field parameters is necessary (e.g. Schwarz 1984). Instead of relying solely upon observations, the Earth’s gravity field can, in principle, be completely described by forward gravity field modelling. Using Newton’s law, the gravitational potential and its derivatives in arbitrary directions can be uniquely determined from the Earth’s mass distribution. This ‘direct’ determination of the gravity field, known as the forward problem or forward gravity field modelling in physical geodesy, has the primary drawback that accurate knowledge of the Earth’s mass-density
distribution is not completely known, and is unlikely to be so in the near future. Nowadays, however, increasingly more information on the Earth’s internal mass distribution is available from geodetic and geophysical measurements. These include high-resolution digital elevation models (DEMs), as well as models of the crustal mass distribution (e.g., Tanimoto 1995, Mooney et al. 1998) and deeper masses (e.g., Olafur and Sambridge 1998). With the availability of such information, the theoretical as well as practical question is: to what spatial resolution must these mass distributions be known to generate a forward gravity field model of a prescribed accuracy? In order to answer this question, global spherical harmonic expansions of Newton’s gravitational potential and locally estimated power spectral density (PSD) functions will be used. The spectral sensitivity of two different gravity field parameters is then examined by means of empirical and analytical degree variance models. In particular, the minimum spatial resolution of these data when used in a forward model is quantified with the view of determining the geoid height and gravity disturbance to a given accuracy. This includes the spatial resolution of topographic masses by way of a DEM and deeper compensation masses by way of a model of the Earth’s crust.
2 Gravitational Effects of Different Mass Distributions in Spherical Harmonics 2.1 Effect of a Spherical Shell In a spherical approximation, all mass-density distributions can be expressed as deviations above t ( + ) (Ω) and below t ( − ) (Ω) a mean sphere at depth L , i.e., with the radius RL = R − L (Fig. 1). Here Ω = (λ ,θ ) denotes the coordinate pair of spherical longitude λ and co-latitude (polar distance) θ . Throughout this paper, the density distribution ρ L (Ω) within a single shell is assumed to only vary laterally, which in turn can be used to replicate a 3D
structure for sufficiently thin shells. In this paper, however, only the spectral behaviour is studied, and thus an exact determination of the gravity field lies beyond the focus of this paper.
Γ(n + 3) (±) , κ pnm p ∈ N + (5) p!Γ(n + 4 − p) with the fully normalized spherical harmonic coef(± ) ficients κ p nm of the surface density functions * κ pnm
(±)
= (±1) p
p
κp ( ± ) = ρ ( ± )
nt Contine
Ocean
t
t
ρ (θ,λ)
h (θ,λ) c
ρ (θ,λ)
o
d (θ,λ)
R
geoid ρo(θ,λ)
Crust
(+)
t (θ,λ) (+) ρ (θ,λ) ) ρ(θ,λ ρ(-)(θ,λ) t ( ) (θ,λ)
L
mass layer
RL
ρc (θ,λ)
m
ρ (θ,λ)
Mantle Fig. 1: Representation of a mass layer by deviations above (+) and below (-) the mean sphere of radius RL.
The gravitational potential V at point P(Ω, r ) outside the shell is given by Newton’s volume integral ρ ( ± ) (Ω ' ) V ( ± ) (Ω, r ) = G ∫∫∫ dv' (1) v l (Ω, r , Ω' , r ' ) for the radial distance r , the spherical volume element dv = r 2 sin(θ )dλdθdr , the gravitational constant G , and the direct distance l (Ω, r , Ω' , r ' ) between the computation point P(Ω, r ) outside the masses and the volume element dv at the source point P(Ω' , r ' ) . In all of the following, the superscripts “+/-” indicate gravity field values caused by mass distributions above/below the sphere R L , respectively. Replacing the inverse distance by a series of Legendre polynomials (e.g., Heiskanen and Moritz 1967), Newton’s integral is expressed by n +1
R V (Ω, r ) = ∑ ∑ Vnm( ± )Ynm (Ω) (2) n =0 r m=− n with the fully normalised spherical harmonic coefficients Vnm(± ) , the surface spherical harmonics N max
n
P (cos θ ) cos mλ m≥0 , (3) Ynm (Ω) = nm m
