ParametrizationGravity

This class gives a parametrization of the time depending gravity field. Together with the class oberservation it will be used to set up the design matrix in a least squares adjustment. If multiple parametrizations are given the coefficients in the parameter vector are sequently appended.

SphericalHarmonics

The potential $V$ is parametrized by a expansion of (fully normalized) spherical harmonics \[ V(\lambda,\vartheta,r) = \frac{GM}{R}\sum_{n=0}^\infty \sum_{m=0}^n \left(\frac{R}{r}\right)^{n+1} \left(c_{nm} C_{nm}(\lambda,\vartheta) + s_{nm} S_{nm}(\lambda,\vartheta)\right). \]You can set the range of degree $n$ with minDegree and maxDegree. The sorting sequence of the potential coefficients in the parameter vector can be defined by numbering.

The total count of parameters is $(n_{max}+1)^2-n_{min}^2$ and the parameter names are

• *:sphericalHarmonics.c_<degree>_<order>:*:*,
• *:sphericalHarmonics.s_<degree>_<order>:*:*.

Name	Type	Annotation
minDegree	uint
maxDegree	uint
GM	double	Geocentric gravitational constant
R	double	reference radius
numbering	sphericalHarmonicsNumbering	numbering scheme

RadialBasis

The potential $V$ is represented by a sum of space localizing basis functions \[ V(\M x) = \sum_i a_i \Phi(\M x, \M x_i) \]where $a_i$ the coefficients which has to be estimated and $\Phi$ are the basis functions given by isotropic radial kernel functions \[ \Phi(\cos\psi,r,R) = \sum_n \left(\frac{R}{r}\right)^{n+1} k_n\sqrt{2n+1}\bar{P}_n(\cos\psi). \]The basis functions are located on a grid $\M x_i$ given by grid. This class can also be used to estimate point masses if kernel is set to density.

The parameter names are *:radialBasis.<index>.<total count>:*:*.

Name	Type	Annotation
kernel	kernel	shape of the radial basis function
grid	grid	nodal point distribution

Temporal

The time variable potential is given by \[ V(\M x,t) = \sum_i V_i(\M x)\Psi_i(t), \]wehre $V_i(\M x)$ is the spatial parametrization of the gravity field and can be choosen with parametrizationGravity. The parametrization in time domain $\Psi_i(t)$ is selected by parametrizationTemporal. The total parameter count is the parameter count of parametrizationTemporal times the parameter count of parametrizationGravity.

Name	Type	Annotation
parametrizationGravity	parametrizationGravity
parametrizationTemporal	parametrizationTemporal

LinearTransformation

Parametrization of the gravity field on the basis of a linear transformation of a source parametrization. The linear transformation changes the original solution space represented by pararametrizationGravitySource from \[ \mathbf{l} = \mathbf{A}\mathbf{x} + \mathbf{e} \]to \[ \mathbf{l} = \mathbf{A}\mathbf{F}\mathbf{y} + \mathbf{e} \]through the linear transformation $\mathbf{x}=\mathbf{F}\mathbf{y}$. It follows that the rows of the matrix $\mathbf{F}$ in inputfileTransformationMatrix coincides with the number of parameters in pararametrizationGravitySource. The new parameter count is given by the number of columns in $\mathbf{F}$ and may be smaller, equal or larger than the original parameter count.

The parameter names are *:transformedParameter.<index>.<total count>:*:*.

Name	Type	Annotation
parametrizationGravitySource	parametrizationGravity
inputfileTransformationMatrix	filename	transformation matrix from target to source parametrization (rows of this matrix must coincide with the parameter count of the source parametrization)

EarthquakeOscillation

This class is used to estimate the earthquake oscillation function parameters, i.e. $C_{nlm}$, $\omega_{nlm}$, and $P_{nlm}$. The results describes the variation in the gravitational potential field caused by large earthquakes. \[ C_{lm}(\M t) = \sum_{n=0}^NC_{nlm}(1-\cos(\omega_{nlm}d\M t)\exp(P_{nlm}\omega_{nlm}d\M t)), \]with $\omega_{nlm}=\frac{2\pi}{T_{nlm}}$ and $P_{nlm}=\frac{-1}{2Q_{nlm}}$ . In this equation, $Q_{nlm}$ is the attenuation factor, $n$ is the overtone factor, $m$ is degree, $l$ is order, and $t$ is time after earthquake in second.

The parameter names are

• *:earthquakeParameter.c_<degree>_<order>_A:*:*,
• *:earthquakeParameter.s_<degree>_<order>_A:*:*,
• *:earthquakeParameter.c_<degree>_<order>_W:*:*,
• *:earthquakeParameter.s_<degree>_<order>_W:*:*,
• *:earthquakeParameter.c_<degree>_<order>_P:*:*,
• *:earthquakeParameter.s_<degree>_<order>_P:*:*.

Name	Type	Annotation
inputInitialCoefficient	filename	initial values for oscillation parameters
time0	time	the time earthquake happened
minDegree	uint
maxDegree	uint
GM	double	Geocentric gravitational constant
R	double	reference radius
numbering	sphericalHarmonicsNumbering	numbering scheme