Calculation of Gravity Field Functionals on Ellipsoidal Grids

This is an interactive Java Applet to calculate a selected gravity field functional for a set of gridded points on a reference ellipsoid. You can select one of the model files offered by this service. The computed gridfile is usually available after a few seconds or a few minutes depending on the functional, the maximum degree and the number of grid points. Calculating the functionals gravity_disturbance, gravity_anomaly, gravity_anomaly_cl and gravity needs more time than the others. A plot of the calculated grid as postscript, generated by the GMT-software will be available too. Short explanations to the selectable parameters are displayed in the yellow info line at bottom if you move the mouse pointer over them.

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See below the applet for explanation and usage info.
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Please note, that every user is restricted to have at most 3 computing processes running in parallel. Attempts to start a fourth process will result in an error message in the info line.

Short Usage Explanation

Select one modelfile and modify other parameters of your choice. Please note, that the number of gridpoints to compute in one run is limited to 300,000 for the four time-consuming functionals (gravity_anomaly, gravity_anomaly_cl, gravity_disturbance, gravity_earth) and 6,500,000 for all others. This means, you can compute the complete sphere with a gridstep of Δ φ ≥ 0.5°, resp. Δ φ ≥ 0.1°. If you want a higher resolution, you must limit the region. Please be aware that a complete 0.1°-gridfile has a size of ≈ 300Mb.
You may also select the option PS-file, then automatically a GMT-plot of your gridfile will be generated (if you additionally use the option illuminiation, the plot will use a simulated lightsource).
Then press the start computation button.
After a few seconds you will see something like 'eigen-grace02s-79.gdf' computed successfully in the yellow info line at bottom. Then you can press get grid-file to inspect and fetch your result file (or show directory to look at the result directory).

Since the grid files can be quite large, the grid directory is managed to contain at most 100 files (for all users together). Also all files older than 30 days are deleted. If you plan to make lots of computations, please load the files not too late to your computer. They could be deleted in the meantime.

Functionals of the Geopotential

keyword	explanation
height_anomaly	The so called "height anomaly" is an approximation of the geoid according to Molodensky's theory. It is equal to the geoid over sea. Here it will be calculated, as defined, on the Earth's surface approximated by Bruns’ formula on the ellipsoid plus a first order correction (eqs. 81 and 119 of STR09/02).
height_anomaly_ell	The height anomaly can be generalised to a 3-d function, (sometimes called "generalised pseudo-height-anomaly"). Here it can be calculated on (h=0) or above (h>0) the ellipsoid, approximated by Bruns’ formula (eqs. 78 and 118 of STR09/02).
geoid	The Geoid is one particular equipotential surface of the gravity potential of the Earth. Among all equipotential surfaces, the geoid is those which is equal to the undisturbed sea surface and its continuation below the continents. Here it will be approximated by the height anomaly plus a topography dependent correction term (eqs. 71 and 117 of STR09/02).
gravity_disturbance	The gravity disturbance is defined as the magnitude of the gradient of the potential at a given point minus the magnitude of the gradient of the normal potential at the same point. Here it will be calculated on the Earth's surface (eqs. 87 and 121 − 124 of STR09/02).
gravity_disturbance_sa	The gravity disturbance calculated by spherical approximation (eqs. 92 and 125 of STR09/02) on (h=0) or above (h>0) the ellipsoid.
gravity_anomaly	The gravity anomaly (according to Molodensky's theory) is defined as the magnitude of the gradient of the potential on the Earth's surface minus the magnitude of the gradient of the normal potential on the Telluroid (Earth's surface minus height anomaly) (eqs. 101 and 121 − 124 of STR09/02).
gravity_anomaly_cl	The classical gravity anomaly is defined as the magnitude of the gradient of the downward continued potential on the geoid minus the magnitude of the gradient of the normal potential on the ellipsoid (eqs. 93 and 121 − 124 of STR09/02).
gravity_anomaly_sa	The gravity anomaly calculated by spherical approximation (eqs. 100 or 104 and 126 of STR09/02). Unlike the classical gravity anomaly, the Molodensky gravity anomaly and the spherical approximation can be generalised to 3-d space, hence here it can be calculated on (h=0) or above(h>0) the ellipsoid.
gravity_anomaly_bg	The (simple) Bouguer gravity anomaly is defined by the classical gravity anomaly minus the attraction of the Bouguer plate. Here it will be calculated by the spherical approximation of the classical gravity anomaly minus  2πGρH (eqs. 107 and 126 of STR09/02). The topographic heights H(λ,φ) are calculated from the spherical harmonic model DTM2006 used up to the same maximum degree as the gravity field model. For H ≥ 0 (rock)  → ρ = 2670 kg/m3, and for H < 0 (water)  → ρ = (2670−1025) kg/m3 is used. The density contrast between ice and rock is not been taken into account ⇒ the resutls for Greenland and Antarctica are not correct.
gravity_earth	The gravity is defined as the magnitude of the gradient of the potential (including the centrifugal potential) at a given point. Here it will be calculated on the Earth's surface (eqs. 7 and 121 − 124 of STR09/02).
gravity_ell	The magnitude of the gradient of the potential calculated on or above the ellipsoid including the centrifugal potential (eqs. 7 and 121 − 124 of STR09/02).
potential_ell	The potential of the gravity field of the Earth without the centrifugal potential (better: gravitational field :-)). Here it can be calculated on or above the ellipsoid (eq. 108 of STR09/02).
gravitation_ell	The magnitude of the gradient of the potential calculated on or above the ellipsoid without the centrifugal potential (eqs. 7 and 122 of STR09/02).
second_r_derivative	The second derivative of the disturbance potential in radial direction calculated on or above the ellipsoid.
water_column	The variable thickness of a fictitious water layer which is distributed over the reference ellipsoid and produce the disturbance potential or the geoid undulations. For calculating this functional "water_column" from a gravity field model the elastic deformation of the Earth due to the load of the water layer is considered.
topography_shm	This is not a functional of the gravity field. For calculating the geoid heights from the height anomalies and for calculating the Bouguer gravity anomalies a spherical harmonic model of the Earth's topography is used, and we offer the possibility to calculate this topography separately (eq. 115 of STR09/02). The model is the spherical harmonic expansion of the (1′ × 1′) - grid of ETOPO1 (version: Ice Surface). The values are calculated, as the model itself, with respect to the geoid.
topography_grd	This is not a functional of the gravity field. Some functionals are calculated on the Earth's surface: "height_anomaly", "gravity_disturbance", "gravity_anomaly", "gravity_earth". For it, to calculate the exact coordinates, the topography of the Earth is calculated by bi-linear interpolation of the (1′ × 1′) - grid of ETOPO1. Here we offer the possibility to calculate this topography separately. The values are calculated, as the model itself, with respect to the geoid.

Truncation and Gaussian Filtering

Truncation

With the parameter 'max_used_degree' the model can be truncated at the given degree. If it is zero or greater than the maximum degree of the model, the model will not be truncated. If 'startgentlecut' is used, the coefficients from 'startgenlecut' up to 'max_used_degree' will be multiplied (damped) by a continuous function decreasing from 1 for degree   n = 'startgenlecut' to zero for n = 'max_used_degree'. More details can be found here (pdf-file, 4 pages).

Gaussian Filtering

Alternatively it is possible to apply a Gaussian filtering. Filtering the resulting grid in the spatial domain by an averaging Gaussian bell shaped function (which is generated by rotating the Gaussian bell curve) is equivalent to multiplying the spherical harmonic coefficients (i.e. in the frequency domain) with an appropriate Gaussian function depending on the spherical harmonic degree. With the parameter 'filterlength_degree' or 'filterlength_meter' the width of the gaussian bell can be chosen in the spatial domain, where it is measured in meters ('filterlength_meter') or in angular units (degrees) ('filterlength_degree').
Usually the averaging function in the spatial domain, also called Impulse Response R(x), is written as: R(x) = e^{−1/2 (x/σ)2}. It can be shown that the corresponding Transfer Function T_f in the frequency domain is: T_f = e^−2(πσf)2. Consequently, the Transfer Function T_λ depending on the wavelength  λ = 1/f  is: T_λ = e^{−2(πσ/λ)2} and the Transfer Function depending on the degree n of the spherical harmonics, with  λ= 2π/n , is: T_n = e^{−1/2 (σn)2}.
The parameter σ defines the width of the Gaussian bell, hence, the filter width (or filter length), which will be denoted here by the symbol Φ, can be set by choosing the value for σ. However, there are different definitions Φ=Φ(σ) of what is called "filter length", i.e. it is to be defined, between which two points of the Gaussian bell curve the width is measured.

With the parameter 'flength_definition', we offer the following 3 definitions for Φ which are commonly in use:

(1) flength_definition = "6sigma"
       Definition: Φ = 6σ

(2) flength_definition = "halftransfer"
       Definition: T_λ(λ=Φ)  =  ½

Illustration: the Transfer function T_f (or T_λ) shows how an input pulse is deformed (damped) in the frequency domain. This definition results in a damping factor of ½ for a wavelength of λ = Φ (with a factor of 1 for λ = ∞)

(3) flength_definition = "halfresponse"
       Definition: R(x=Φ)  =  ½

The following table summarises the characteristics of the 3 filterlength definitions:

flength_definition	Φ = Φ(σ)	R(x) = e−1/2 (x/σ)2 ↓ R(x=Φ) = ?	Tλ = e−2(πσ/λ)2 ↓ Tλ(λ=Φ) = ?	Tn  = e−1/2 (σn)2 e.g.: Φ[rad] = 5° · (π/180°) ↓ Tn(n=?) = 1/2
6sigma	Φ=6σ	R(x) = e−18(x/Φ)2 ↓ R(x=Φ) = 1.523 ·10−8	Tλ = e−(1/18) (πΦ/λ)2 ↓ Tλ(λ=Φ) = 0.578	Tn = e−(1/72) (Φn)2 ↓ Tn(n=81) = 1/2
halftransfer	Φ=√(2/ln2)πσ Φ=5.336σ	R(x) = e−(π2/ln2) (x/Φ)2 ↓ R(x=Φ) = 6.549 ·10−7	Tλ = eln2(Φ/λ)2 ↓ Tλ(λ=Φ) = 1/2	Tn = e−[ln2/(2π)2] (Φn)2 ↓ Tn(n=72) = 1/2
halfresponse	Φ=√(2ln2)σ Φ=1.178σ	R(x) = e−ln2 (x/Φ)2 ↓ R(x=Φ) = 1/2	Tλ = e−(π2/ln2) (x/λ)2 ↓ Tλ(λ=Φ) = 6.549 ·10−7	Tn = e−[1/(4ln2)] (Φn)2 ↓ Tn(n=16) = 1/2

Troubleshooting

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