Satellite Gravimetry and the Solid Earth: Mathematical Foundations

Satellite Gravimetry and the Solid Earth
Copyright
Dedication
Preface
Acknowledgements
1 . Spherical harmonics and potential theory
	1.1 General solution of Laplace equation in spherical coordinates
	1.2 Solving potential from potential outside the earth
		1.2.1 Spectral solution
		1.2.2 Solving Dirichlet's problem
	1.3 Solving potential from its first-order derivatives
		1.3.1 Solving potential from its radial derivative
			1.3.1.1 Spectral solution
			1.3.1.2 Spatial solution
			1.3.1.3 Integral equation
			1.3.1.4 Solving the potential from a linear combination of the potential and its radial derivative
			1.3.1.5 Spectral solution
			1.3.1.6 Spatial solutions
			1.3.1.7 Integral equation
		1.3.2 Solving the potential from its gradients
			1.3.2.1 Vector spherical harmonics
			1.3.2.2 Spectral solutions
			1.3.2.3 Spatial solutions
			1.3.2.4 Integral equations
	1.4 Solving the potential from its second-order derivatives
		1.4.1 Tensor spherical harmonics
		1.4.2 Spectral solutions
		1.4.3 Spatial solutions
		1.4.4 Integral equations
		1.4.5 Product of a spherical harmonic series
	1.5 Spectra of the potential field
		1.5.1 Global spectra of the potential
			1.5.1.1 Global first- and second-order radial derivatives of the potential
			1.5.1.2 Global horizontal derivatives and vertical-horizontal derivatives of the potential
			1.5.1.3 Global spectra of horizontal-horizontal derivatives of the potential
		1.5.2 Local spectra of the potential
			1.5.2.1 Local spectra of the potential based on the product of spherical harmonics and the Gaunt coefficient
			1.5.2.2 Local spectra of the potential based on Sjöberg's approach
	References
2 . Satellite gravimetry observables
	2.1 Satellite orbit and the Earth's gravitational potential
	2.2 Geometry of orbit and geopotential perturbation
	2.3 Orbital elements
		2.3.1 Lagrange equations
		2.3.2 Gaussian equations
		2.3.3 Velocity and acceleration of the perturbations
	2.4 Satellite acceleration
	2.5 Satellite velocity
	2.6 Inter-satellite range rate
		2.6.1 Approach 1
		2.6.2 Approach 2
	2.7 Line-of-sight measurements
	2.8 Satellite gravity gradiometry
	2.9 Satellite altimetry data
	References
3 . Integral equations for inversion of satellite gravimetry data
	3.1 Anomalous parameters of the Earth's gravity field
		3.1.1 Normal gravity field and disturbing potential
		3.1.2 Geoid height, gravity disturbance and anomaly as anomalous quantities
		3.1.3 Deflections of the vertical
		3.1.4 From geoid height to other anomalous parameters
		3.1.5 From gravity disturbance/anomaly and deflections of the vertical to geoid height
		3.1.6 Integral relations amongst gravity disturbance/anomaly and deflections of the vertical
	3.2 Integral equations for inversion of temporal variations of orbital elements
		3.2.1 Integral equations for recovering gravity anomaly/disturbance from temporal variations of orbital elements
		3.2.2 Integral equations for inverting combination of Gaussian equations for gravity disturbance/anomaly recovery
	3.3 Integral inversion of acceleration and velocity of perturbations
	3.4 Integral equations for inversion of satellite acceleration and velocity
		3.4.1 Inversion of satellite acceleration in the terrestrial reference frame
		3.4.2 Inversion of satellite velocity in the terrestrial reference frame
	3.5 Integral equations for inversion of low-low tracking data
		3.5.1 Integral equations for inversion of range rates
		3.5.2 Integral equations for inversion of line-of-sight measurements
	3.6 Integral equations for inversion of satellite gradiometry data
		3.6.1 Integral equations for inversion of gravity gradients in the local north-oriented frame
		3.6.2 Integral equations for inversion of gravity gradients in the track-oriented frame
	3.7 Integral inversion of satellite altimetry data
		3.7.1 Determination of gravity anomaly/disturbance from altimetry geoid heights and deflections of the vertical
		3.7.2 Determination of gravity anomaly/disturbance from altimetry-derived deflections of the vertical
		3.7.3 Integral inversion of geoid and deflections of the vertical
			3.7.3.1 Integral equations for inversion of altimetry geoid heights
			3.7.3.2 Integral equations for inversion of altimetry deflections of the vertical
	References
4 . Numerical inversion of satellite gravimetry data
	4.1 Discretisation of integral formulae
		4.1.1 Numerical solution of integrals
		4.1.2 Kernels of integrals
			4.1.2.1 Well-behaving kernels
			4.1.2.2 Bell-shaped kernels
		4.1.3 Numerical inverse solution of integrals
	4.2 Handling spatial truncation error
		4.2.1 Estimation of spatial truncation error based on the integral and spherical harmonics
		4.2.2 Estimation of spatial truncation error based on spherical harmonics
		4.2.3 Size of the inversion area and spatial truncation error
		4.2.4 Example: spatial truncation error of satellite gradiometry data
		4.2.5 Example: spatial truncation error in the results of inversion of satellite inter-satellite tracking data
		4.2.6 Example: inversion of Trr to Δg
		4.2.7 Example: inversion of orbital data to gravity anomaly
	4.3 Regularisation methods
		4.3.1 Gauss-Markov model
		4.3.2 A conceptual overview of regularisation methods
			4.3.2.1 Iterative methods
				4.3.2.1.1 Classical iterative methods
					4.3.2.1.1.1 The ν method
					4.3.2.1.1.2 Algebraic reconstruction technique
				4.3.2.1.2 Krylov subspaces-based methods
					4.3.2.1.2.1 Range-restricted generalised minimum residual method
					4.3.2.1.2.2 Conjugate gradient
			4.3.2.2 Estimation of the optimal iteration number by L-curve
			4.3.2.3 Example: application of the iterative methods for determining equivalent water height from the GRACE mission
		4.3.3 Direct methods
			4.3.3.1 Truncated singular value decomposition
			4.3.3.2 Tikhonov regularisation
			4.3.3.3 Generalised cross validation
			4.3.3.4 Example: application of the TSVD and Tikhonov regularisation methods for determining equivalent water height from the GRACE ...
			4.3.3.5 Example: inversion of on-orbit satellite gradiometry data in the local north-oriented frame and orbital reference frame
			4.3.3.6 Example for bias corrections and estimation of a posteriori variance factor
	4.4 Sequential Tikhonov regularisation
		4.4.1 Example: application of sequential Tikhonov regularisation
	4.5 Variance component estimation in ill-conditioned systems
		4.5.1 Best quadratic unbiased estimator of variance component in ordinary systems
		4.5.2 Best quadratic unbiased estimator of variance component in a system solved by truncated singular value decomposition
		4.5.3 Best quadratic unbiased estimator of the variance components in systems solved by Tikhonov regularisation
		4.5.4 Example: variance component estimation for inverting satellite gravity gradients
	4.6 Quality of integral inversion in the presence of spatial truncation error
		4.6.1 Reduction of spatial truncation error from the a posteriori variance factor
		4.6.2 Reduction of spatial truncation error from variance components
	References
5 . The effect of mass heterogeneities and structures on satellite gravimetry data
	5.1 Gravitational potential of topographic and bathymetric masses
		5.1.1 Gravitational potential of topographic masses
			5.1.1.1 Approach 1 to consider lateral density variation of topographic masses
			5.1.1.2 Approach 2 to consider lateral density variation of topographic masses
			5.1.1.3 Gravitational potential of the bathymetric masses (water)
	5.2 Gravitational potential of crustal layers based on CRUST1.0
	5.3 Gravitational potential of sediments
		5.3.1 Gravitational potential of sediments based on the CRUST1.0 model
		5.3.2 Gravitational potential of sediments based on density models
		5.3.3 Sediments' gravitational potential based on the exponential density contrast model
			5.3.3.1 Approach 1
			5.3.3.2 Approach 2
		5.3.4 Sediments' gravitational potential based on the hyperbolic density contrast model
		5.3.5 Sediments' gravitational potential based on the exponential compaction density model
		5.3.6 Example: the effect of upper sediments based on different density models on satellite gradiometry data
	5.4 Gravitational potential of atmospheric masses
		5.4.1 Gravitational potential of atmospheric masses according to the exponential density model
		5.4.2 Gravitational potential of atmospheric masses according to the power density model
		5.4.3 Gravitational potential of atmospheric masses according to the polynomial density model
		5.4.4 Gravitational potential of atmospheric masses according to a combination of the polynomial and power density models
		5.4.5 Example: atmospheric effect on satellite gradiometry data
	5.5 Remove-compute-restore model of topographic and atmospheric masses
		5.5.1 Restoring the topographic effect
		5.5.2 Restoring the atmospheric effect
	5.6 Topographic and atmospheric bias
	References
6 . Isostasy
	6.1 Isostatic equilibrium
	6.2 Pratt-Hayford isostasy model
		6.2.1 Pratt-Hayford isostatic model based on gravitational potential in the spherical domain
		6.2.2 Approximate solution in spherical harmonics
	6.3 Airy-Heiskanen model
		6.3.1 Airy-Heiskanen isostatic model based on gravitational potential in the spherical domain
		6.3.2 Solutions to the Airy-Heiskanen model
			6.3.2.1 Linear approximation of the binomial term
			6.3.2.2 Approximation of the binomial term to second order
			6.3.2.3 Iterative solution
			6.3.2.4 Solving a non-linear integral equation
		6.3.3 Gravimetric isostasy
			6.3.3.1 Approach 1
		6.3.4 Linear approximation
		6.3.5 Second-order approximation
			6.3.5.1 Iterative solution
			6.3.5.2 Non-linear inversion
		6.3.6 Approach 2
			6.3.6.1 Linear approximation
			6.3.6.2 Second-order approximation
			6.3.6.3 Iterative solution
			6.3.6.4 Non-linear inversion
		6.3.7 A numerical example: Moho model based on approach 1 over the Tibet Plateau
	6.4 Flexure isostasy and the Vening Meinesz principle
		6.4.1 Simple flexure model
		6.4.2 Flexural model considering membrane stress
		6.4.3 A numerical example: Moho model based on flexure isostasy over Tibet Plateau
		6.4.4 Combination of gravimetric and flexural isostasy
		6.4.5 Flexural convolution in the spherical domain
	6.5 The effect of sediment, ice and crustal masses in isostasy
	6.6 Non-isostatic equilibriums
	References
7 . Satellite gravimetry and isostasy
	7.1 Smoothing satellite gravimetry data
		7.1.1 Reductions based on combination of flexure and gravimetric isostatic theories
		7.1.2 Example: smoothing the gravitational potential tensor
		7.1.3 Example: removing the effects of mass density and structure heterogeneities from second-order radial derivatives measured b ...
	7.2 Determination of the product of Moho depth and density contrast
		7.2.1 Product of Moho depth variation and density contrast from satellite altimetry data
		7.2.2 Product of Moho depth variation and density contrast from satellite gravity gradiometry data
			7.2.2.1 Product of Moho depth variation and density contrast from inversion satellite gradiometry data and effects of topographic/b ...
			7.2.2.2 Contribution of satellite gradiometry data to the product of Moho depth variation and density contrast
		7.2.3 Product of Moho depth variation and density contrast from inter-satellite range rates
			7.2.3.1 Simultaneous inversion of inter-satellite range rates and effects of topographic/bathymetric, sediment, crustal crystalline ...
			7.2.3.2 Contribution of inter-satellite range rates to the product of Moho depth variation and density contrast
		7.2.4 Simultaneous inversion of inter-satellite line-of-sight and effects of topographic/bathymetric, sediment, crustal crystalli ...
			7.2.4.1 Inversion of the effects of the crustal mass and structure heterogeneities and line-of-sight measurements
			7.2.4.2 Inversion of inter-satellite line-of-sight measurements to the product of Moho depth variation and density contrast
		7.2.5 Example: oceanic Moho model computed based on gravity model derived from satellite altimetry data
		7.2.6 Example: Moho determination over the Indo-Pak region from GOCE data
		7.2.7 Example: Moho model of Iran from GOCE gradiometry data
	7.3 Determination of density contrast
		7.3.1 Determination of density contrast from satellite altimetry data
		7.3.2 Determination of density contrast from satellite gradiometry data
		7.3.3 Determination of density contrast from inter-satellite range rates
		7.3.4 Determination of density contrast from the combination of inter-satellite line-of-sight measurements
		7.3.5 Example: Moho density contrast from CryoSat-2 and Jason-1 marine gravity model
		7.3.6 Example: density contrast determination over central Eurasia from GOCE data
	7.4 Determination of lithospheric elastic thickness and rigidity
		7.4.1 The mathematical foundation
		7.4.2 Determination of elastic thickness from satellite altimetry, gradiometry and inter-satellite data
		7.4.3 Example: determination of effective elastic thickness from GOCE gradiometry data over Africa
	7.5 Determination of oceanic bathymetry
		7.5.1 Mathematical foundations
			7.5.1.1 Direct linear estimation of bathymetry depth
			7.5.1.2 Constrained solution to mean depth
		7.5.2 Bathymetry from satellite altimetry, gradiometry and inter-satellite data
	7.6 Continental ice thickness determination
		7.6.1 Mathematical foundation
		7.6.2 Continental ice from satellite gradiometry and inter-satellite data
	7.7 Sediment basement determination
		7.7.1 Mathematical foundation
		7.7.2 Sediment thickness from satellite altimetry, gradiometry and inter-satellite measurements
	References
8 . Gravity field and lithospheric stress
	8.1 Runcorn's theory for sub-lithospheric stress modelling
		8.1.1 Poloidal and toroidal flows
		8.1.2 Navier-Stokes equation
		8.1.3 Gravity and sub-lithospheric stress caused by mantle convection
	8.2 Hager and O'Connell theory for sub-lithospheric stress modelling
	8.3 Stress propagation from sub-lithosphere to lithosphere
		8.3.1 Partial differential equation of elasticity for a spherical shell
		8.3.2 Displacement and the gravity field
		8.3.3 Displacement, strain and stress
		8.3.4 Stress and gravity field
		8.3.5 Boundary-values and their role
		8.3.6 Application: global subcrustal stress
	Acknowledgements
	References
9 . Satellite gravimetry and lithospheric stress
	9.1 Mathematical foundation based on Runcorn's formula
	9.2 Sub-lithospheric shear stresses from satellite gradiometry data
	9.3 Sub-lithospheric stress from vertical-horizontal satellite gravity gradients
	9.4 Example: application of Gravity Field and Ocean Circulation Explorer data for determining sub-lithospheric shear stresses i ...
	9.5 Example: application of Gravity Field and Ocean Circulation Explorer and seismic data for sub-lithospheric stress modelling ...
	9.6 Example: considering lithospheric mass and structure heterogeneities to determine sub-lithospheric shear stress
		9.6.1 Data and area
		9.6.2 The effect of topographic-bathymetric, sediment and consolidated crustal masses on the on-orbit Gravity Field and Ocean Cir ...
		9.6.3 Sub-lithospheric shear stresses from Gravity Field and Ocean Circulation Explorer data with and without considering lithosp ...
		9.6.4 Sub-lithospheric shear stresses from Gravity Field and Ocean Circulation Explorer data corrected for crust density heteroge ...
	9.7 Satellite gradiometry data and lithospheric stress tensor
		9.7.1 Diagonal elements of stress and gravitation tensors
		9.7.2 Off-diagonal elements of stress and gravitation tensors
		9.7.3 Simple application of integral equations for inverting real on-orbit Gravity Field and Ocean Circulation Explorer data to l ...
			9.7.3.1 Gravity Field and Ocean Circulation Explorer data and coverage
			9.7.3.2 Recovery of lithospheric stress from on-orbit Gravity Field and Ocean Circulation Explorer data
	9.8 Inter-satellite tracking data and stress
		9.8.1 Inter-satellite low-low range rates and sub-lithospheric shear stresses
		9.8.2 Inter-satellite line-of-sight measurements and sub-lithospheric shear stresses
		9.8.3 Example: application of Gravity Recovery and Climate Experiment-type data for recovering sub-lithospheric shear stresses
	9.9 Determination of lithospheric stress tensor from inter-satellite tracking data
		9.9.1 Determination of elements of lithospheric stress tensor from low-low inter-satellite range-rates
		9.9.2 Determination of elements of stress tensor from inter-satellite line-of-sight measurements
	Acknowledgements
	References
10 . Satellite gravimetry and applications of temporal changes of gravity field
	10.1 Time-variable gravity field
	10.2 Hydrological effects and equivalent water height from time-variable gravity field
		10.2.1 Gravitational potential of a surface mass
		10.2.2 Effect of hydrological masses
		10.2.3 Determination of hydrological parameters from satellite gravimetry
		10.2.4 Earthquake monitoring and time-variable gravity field
	10.3 Surface mass changes over ocean and satellite gravimetry
	10.4 Determination of land uplift caused by postglacial rebound
	10.5 Determination of upper mantle viscosity
	10.6 Gravity strain tensor and epicentre points of shallow earthquakes
	Acknowledgements
	References
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	K
	L
	M
	N
	O
	P
	R
	S
	T
	U
	V
	W