Kernel

Kernel defines harmonic isotropic integral kernels $K$. \[ T(P) = \frac{1}{4\pi}\int_\Omega K(P,Q)\cdot f(Q)\,d\Omega(Q), \]where $T$ is the (disturbance)potential and $f$ is a functional on the spherical surface $\Omega$. The Kernel can be exapanded into a series of (fully normalized) legendre polynomials \[\label{eq.kernel} K(\cos\psi,r,R) = \sum_n \left(\frac{R}{r}\right)^{n+1} k_n\sqrt{2n+1}\bar{P}_n(\cos\psi). \]On the one hand the kernel defines the type of the functionals $f$ that are measured or have to be computed, e.g. gravity anomalies given by the Stokes-kernel. On the other hand the kernel functions can be used as basis functions to represent the gravity field, e.g. as spline functions or wavelets.

GeoidHeight

The geoid height is defined by Bruns formula \[ N = \frac{1}{\gamma}T \]with $T$ the disturbance potential and the normal gravity \[\label{normalgravity} \gamma = \gamma_0 - 0.30877\cdot 10^{-5}/s^2(1-0.00142\sin^2(B))h \]and \[ \gamma_0 = 9.780327\,m/s^2(1+0.0053024\sin^2(B)-0.0000058\sin^2(2B)) \]where $h$ is the ellipsoidal height in meter and $B$ the longitude.

The kernel is given by \[ K(\cos\psi,r,R) = \gamma\frac{R(r^2-R^2)}{l^3}, \]and the coefficients in \eqref{eq.kernel} are \[ k_n = \gamma. \]

Anomalies

Gravity anomalies in linearized form are defined by \[ \Delta g = -\frac{\partial T}{\partial r}-\frac{2}{r}T. \]The Stokes kernel is given by \[ K(\cos\psi,r,R) = \frac{2R^2}{l}-3\frac{Rl}{r^2}-\frac{R^2}{r^2}\cos\psi \left(5+3\ln\frac{l+r-R\cos\psi}{2r}\right), \]and the coefficients in \eqref{eq.kernel} are \[ k_n = \frac{R}{n-1}. \]

Disturbance

Gravity disturbances in linearized form are defined by \[ \delta g = -\frac{dT}{dr}. \]The Hotine kernel is given by \[ K(\cos\psi,r,R) = \frac{2R^2}{l}-R\ln\frac{l+R-r\cos\psi}{r(1-\cos\psi)}, \]and the coefficients in \eqref{eq.kernel} are \[ k_n = \frac{R}{n+1}. \]

Potential

The Abel-Poisson kernel is given by \[ K(\cos\psi,r,R) = \frac{R(r^2-R^2)}{l^3}, \]and the coefficients in \eqref{eq.kernel} are \[ k_n = 1. \]

Density

This kernel defines a point mass or mass on a single layer ($1/l$-kernel) taking the effect of the loading into account.

The coefficients of the kernel defined in \eqref{eq.kernel} are \[ k_n = 4\pi G R\frac{1+k_n'}{2n+1}, \]where $G$ is the gravitational constant and $k_n'$ are the load Love numbers.

Name	Type	Annotation
inputfileLoadingLoveNumber	filename

WaterHeight

Height of equivalent water columns taking the effect of the loading into account.

The coefficients of the kernel defined in \eqref{eq.kernel} are \[ k_n = 4\pi G \rho R\frac{1+k_n'}{2n+1}, \]where $G$ is the gravitational constant, $\rho$ is the density of water and $k_n'$ are the load Love numbers.

Name	Type	Annotation
density	double	[kg/m**3]
inputfileLoadingLoveNumber	filename

BottomPressure

Ocean bottom pressure caused by water and atmosphere masses columns taking the effect of the loading into account.

The coefficients of the kernel defined in \eqref{eq.kernel} are \[ k_n = \frac{4\pi G R }{\gamma}\frac{1+k_n'}{2n+1}, \]where $G$ is the gravitational constant, $\gamma$ is the normal gravity and $k_n'$ are the load Love numbers.

Name	Type	Annotation
inputfileLoadingLoveNumber	filename

Deformation

Computes the radial deformation caused by loading.

The coefficients of the kernel defined in \eqref{eq.kernel} are \[ k_n = \gamma\frac{1+k_n'}{h_n'}, \]where $\gamma$ is the normal gravity defined in \eqref{normalgravity}, $h_n'$ and $k_n'$ are the load Love numbers and the load deformation Love numbers.

Name	Type	Annotation
inputfileDeformationLoadLoveNumber	filename
inputfilePotentialLoadLoveNumber	filename	if full potential is given and not only loading potential

RadialGradient

This kernel defines the second radial derivative of the (disturbance) potential. \[ T_{rr} = \frac{\partial^2 T}{\partial r^2}. \]The coefficients of the kernel defined in \eqref{eq.kernel} are \[ k_n = \frac{r^2}{(n+1)(n+2)}. \]

Coefficients

The kernel is defined by the coefficients $k_n$ given by file.

Name	Type	Annotation
inputfileCoefficients	filename

FilterGauss

Another kernel is smoothed by a gauss filter which is defined by \[ F(\cos\psi) = \frac{b\cdot e^{-b(1-\cos\psi)}}{1-e^{-2b}} \]with $b = \frac{ln(2)}{1-\cos(r/R)}$ where $r$ is the given smoothing radius in km and $R=6378.1366$ km is the Earth radius. The coefficients $k_n$ of the kernel are multiplicated by \[ f_n = \frac{1}{2n+1} \int_{-1}^1 F(t)\cdot \bar{P}_n(t)\,dt. \]

Name	Type	Annotation
kernel	kernel
radius	double	filter radius [km]

BlackmanLowpass

Another kernel is smoothed by a Blackman low-pass filter. The filter is defined through the beginning and end of the transition from pass-band to stop-band. This transition band is specified by startDegreeTransition ($n_1$) and stopDegreeTransition ($n_2$).

The coefficients of this kernel are defined as \[ \begin{cases} 1 & \text{for } n < n_1 \\ A_n^2 & \text{for } n_1\leq n \leq n_2 \\ 0 & \text{for } n > n_2 \\ \end{cases} \]with \[ A_n = 0.42 + 0.5\cos(\pi \frac{n-n_1}{n_2-n_1}) + 0.08 \cos(2\pi\frac{n-n_1}{n_2-n_1}). \]

Name	Type	Annotation
kernel	kernel
startDegreeTransition	uint	minimum degree in transition band
stopDegreeTransition	uint	maximum degree in transition band

Truncation

Another kernel is truncated before minDegree and after maxDegree. The coefficients of this kernel are defined as \[ k_n = \begin{cases} 1 & \text{for } n_{\text{minDegree}} \leq n \leq n_{\text{maxDegree}}\\ 0 & \text{else.} \\ \end{cases} \]

Name	Type	Annotation
kernel	kernel
minDegree	uint	truncate before minDegree
maxDegree	uint	truncate after maxDegree

SelenoidHeight

The selenoid height is defined by Bruns formula \[ N = \frac{1}{\gamma}T \]with $T$ the disturbance potential and the normal gravity $\gamma=\frac{GM}{R^2}$ of the moon.

The kernel is given by \[ K(\cos\psi,r,R) = \gamma\frac{R(r^2-R^2)}{l^3}, \]and the coefficients in \eqref{eq.kernel} are \[ k_n = \gamma. \]