Simple Layer Potential

A method developed by Karl-Rudolf Koch in the 1970s is referred to as the potential of the simple layer , with which the calculation of the earth's gravity field can be simplified or refined as required by introducing surface coverages . The method does have - in contrast to the harmonious models using Legendre - and spherical harmonics - in the approximation of the gravity potential a discontinuous characteristic of the model boundaries, but requires only a fraction of the computing time and a half as large system of equations like the classic Neumann's Method .

Geoid, interference potential and surface occupancy

The gravity potential and its functionals - the most important are the geoid , the vertical deviations and the gravity anomalies - are largely caused by the gravitation of the earth's body and the centrifugal force of the earth's rotation . However, all irregularities of the earth's surface ( terrain , different densities ) and the geological structure of the earth's crust cause deviations from the normal field , which is equated to the theoretical gravity field of a mean earth ellipsoid .

Each additional mass or mass missing compared to the earth ellipsoid - which is referred to as a mass distribution in its entirety - changes the gravity potential of the earth, the more the closer the anomaly is to the respective measuring point. The change in potential is referred to as the interference potential and can be expressed by a scalar (a local difference in the potential energy ).

A mountain massively increases z. B. the earth potential by a few tenths of a per thousand , which results in increased gravity and a slight bulge of the level surfaces and the geoid. This effect is better conceivable if one looks at the perpendicular direction .

The mountain range pulls a freely hanging plumb line towards it, which also changes the local horizontal line that is perpendicular to it (a parallel to the geoid). The resulting bulge is called geoid undulation . On the other side of the mountain, the effect is the opposite. In the corresponding calculations, which are carried out with rock cubes or prisms , the immediate surroundings must be taken into account precisely, but a relatively rough model is sufficient for the more distant terrain. The vertical deviations are a maximum of 50 ″ (0.015 °) in the Alps, about a tenth of that in the hill country, but they must be taken into account for every precise survey (see topographical reduction ).

The area coverage method can simplify these calculations by modeling the mountains using mass coverage . These are infinitely thin plates that can be laid one on top of the other and whose (fictitious) masses correspond to the rock. A mass deficit (e.g. a deep valley) is modeled by negative masses.

The disturbance potential and its functionals

To calculate the interference potential at a certain point, theoretically all interference masses (i.e. the deviations of the earth's figure from an ideal ellipsoid) must be taken into account. In fact, however, an exact digital terrain model is only required for the immediate vicinity , while a very large-meshed model is sufficient for distances over 50 km. ${\ displaystyle T}$

The total gravitational potential at a point on the earth's surface (the so-called reference point ) with the Cartesian coordinates can be written as a volume integral over the entire earth's mass by adding up the potential of all the earth's mass points . These mass points with the volume element have the individual density (rocks 2.5–3.3 g / cm³, earth's mantle 4-6 g / cm, earth core ~ 10 g / cm³), which depends on their location in the earth's body . In the denominator there is the vector between the starting point and the respective mass point : ${\ displaystyle W}$ ${\ displaystyle P}$ ${\ displaystyle X, Y, Z}$ ${\ displaystyle P}$ ${\ displaystyle \ mathrm {d} V}$ ${\ displaystyle \ rho}$ ${\ displaystyle (x, y, z)}$ ${\ displaystyle \ mathbf {r}}$ ${\ displaystyle P}$ ${\ displaystyle \ rho \ mathrm {d} V}$

{\ displaystyle W (X, Y, Z) = \ int _ {\ text {Earth}} {\ frac {\ rho (x, y, z) \ mathrm {d} V} {| \ mathbf {r} | }}}

.

Numerically, an exact solution is not possible because the earth would have to be modeled in a trillion parts if it were broken down into 1 km³ "point masses". In addition, the course of the density in the deeper underground is not known exactly enough. The practical calculation of such potentials must therefore be content with approximate solutions and regional estimates.

A significant simplification results if the above equation is restricted to so-called perturbation masses , which represent the deviation from the earth's ellipsoid. The interior of the earth is recorded with theories such as equilibrium figures, which is possible today to an accuracy of a few millionths. The associated theoretical potential is denoted by ${\ displaystyle U}$

The interference potential thus results in ${\ displaystyle T}$

{\ displaystyle T (X, Y, Z): = W (X, Y, Z) -U (X, Y, Z) = \ int _ {\ text {Disturbing masses}} {\ frac {\ rho (x, y, z) \ mathrm {d} V} {| \ mathbf {r} |}}}

,

which is already accessible to an approximately practicable modeling.

The other variables of interest of the gravity field are functionals of this interference potential , with the ellipsoidal normal gravity at the starting point and the mean earth radius . The Cartesian coordinate system is replaced by a local system , in which points in the direction of the local vertical, north and east: ${\ displaystyle T}$ ${\ displaystyle \ gamma}$ ${\ displaystyle P}$ ${\ displaystyle R}$ ${\ displaystyle x, y, z}$ ${\ displaystyle u, v, w}$ ${\ displaystyle w}$ ${\ displaystyle u}$ ${\ displaystyle v}$

Geoid undulation

{\ displaystyle \ zeta = T / \ gamma}

Gravity anomaly

{\ displaystyle \ Delta g = {\ frac {\ partial T} {\ partial w}} - T {\ frac {2} {R}}}

Solder deviation components ${\ displaystyle {\ begin {aligned} \ xi & = - {\ frac {1} {\ gamma}} {\ frac {\ partial T} {\ partial u}} \\\ eta & = - {\ frac { 1} {\ gamma}} {\ frac {\ partial T} {\ partial v}} \ end {aligned}}}$

Approximation of the interference potential with area coverage

The modeling of the earth's gravity by means of potential of the single layer ( potential of a simple layer ). Several of these thin layers, afflicted with concrete masses, are fictitiously spread out on the earth's surface and can at best overlap. Their densities are as unknown set and by means of the given data by gravity compensation calculation determined by least squares.

In the first version of the method (Koch 1970) 192 surface elements were defined whose masses were adapted to a harmonic potential model ( spherical functions up to degree and order 15) of satellite geodesy , as well as a large data set of terrestrial gravity anomalies .

The density of these spherical caps was based on a reference ellipsoid , which has the same flattening as an earth's body in hydrostatic equilibrium. The results obtained in this way are therefore also related to geophysical issues.

In later applications (Koch 1975f) additional combination solutions between gravity anomalies and satellite altimetry were carried out. These height measurements were already so precise in the mid-1970s that a joint modeling with the orbital disturbances could be attempted.

The density values of the surface elements to be determined are assumed to be constant, but the resolution of the model can be adapted to the quality of the gravity data by changing their number .

If the altimeter or gravity data cover the earth very densely or are of high accuracy , the method can be designed very flexibly through a finer grid of potential layers .

When calculating these surface elements, the adjustment can be supplemented by additional transition or buffer zones and can be split into smaller, independent subsystems to solve very large systems of equations .
A transition to collocation methods is also possible.

Literature and Sources

K. Ledersteger, Handbook of Surveying ( JEK ) Volume 5
KH Koch: Journal of Surveying 1975
Surface Density Values for the Earth from Satellite and Gravity Observations, Karl-Rudolf Koch 1970, Geoph. J. Int. Vol. 21/1 p. 1–12 doi : 10.1111 / j.1365-246X.1970.tb01763.x
Karl-Rudolf Koch: Processing of altimetry data , Journ. Of Geodesy 49/1, March 1975, doi : 10.1007 / BF02523941