Geophysical Geodesy Scientific Computation

Topographic Masses Effects for Physical Geodesy

The concept of topographic masses effects of anomalous gravity field elements have 3 essential key factors, which are the adjustment manner of the topographic or crustal masses, the type of the anomalous field elements affected, and location of the computed point relative to the masses.

According to different masses adjustment manners, the topographic effects commonly include: local terrain effect, crustal isostasy effect, topographic Bouguer masses effects, sea water Bouguer condensation effect, topographic Helmert condensation, residual topographic masses effect and so on.

Similar to the remove-restore scheme using an earth geopotential model, a remove-restore scheme using the topographic effects can be adopted to improve the local gravity field approach performance and algorithm stability when the topographic effects outside the earth are harmonic.

Principles of Selecting Topographic Effects

In the local gravity field approach, the topographic effects computation has two basic purposes: one is to grid the discrete anomalous field elements for numerical integral operations, and the other is to separate the ultrashort wave components for the integral of anomalous gravity field.

In order to improve the gridding accuracy of the discrete field elements, it is required that the discrete field elements becomes smoother after removing the topographic effects. In this case, the criteria for selecting some a type of topographic effects is that after the topographic effects are removed, the standard deviation of the discrete field elements becomes smaller.

When the field elements are integrated, it is required that the topographic effects only have the ultrashort wave components. In this case, the criteria for selecting some a type of topographic effect is that after the topographic effects are removed, the standard deviation of the field elements are smaller, and the average value of the topographic effects is also small in the range of more than tens of kilometers.

The uniqueness of the geoid and its realization

The Stokes boundary value problem requires that there is no masses outside the geoid, and the terrain masses should be compressed into the geoid under the condition of keeping the disturbing geopotentials outside the earth. There is a way to compress the terrain masses that the disturbing geopotentials between ground and geoid are equal to the analytical continuation value of the disturbing geopotentials outside the ground, thereby the corresponding geoidal height is the analytical continuation solution of the geoid.

The topographic effects of gravity disturbances and the topographic effects of height anomalies computed by PALGrav4.0 satisfy Hotine integral formula. For example, the topographic Helmert condensation of gravity disturbances (direct effects) and the Helmert condensation of height anomalies (indirect effects) satisfy the Hotine integral formula. Therefore, no matter whether you choose local terrain effect, topographic Helmert condensation, or residual topographic effect, the regional geoid refined by the PALGrav4.0 programs with the topographic effects remove-restore scheme is the analytical continuation solution of geoid.

Obviously, the geoid height determined by the satellite gravity field or directly calculated by a geopotential coefficients model are all an analytical continuation solution of the geoid. By maintaining the uniqueness of the geoid solution, PALGrav4.0 can deeply integrate satellite gravity, geopotential coefficients model and regional gravity field data, and then theoretically strictly approach the local gravity field and geoid.

Uniqueness of integral solutions and its roles

The integral solutions of local gravity field are unique, which can guarantee the harmonic invariance of the local gravity field and the analytical invariance of the field spatial signature. The numerical integral solution of the local gravity field have the same properties and functions on the regional scale as the earth geopotential coefficients model on the global scale.

Since the numerical integral solutions are unique and harmonic, the remove-restore schemes with the previous numerical integral solutions as the reference field can be constructed and then play a scale controlling role to approach the current local gravity field and to refine a very small area geoid. So that the seamless splicing problem between the adjacent geoid models will become disappeared, and the current geoid model can be rigorously compatible with the previous geoid model.

Geopotential, Geoid and Height Datum

The gravimetric geoid, which is the solution of the boundary value problem, whose geopotential is a constant which is equal to the normal geopotential of the ellipsoidal surface which is the starting surface of the geoidal height.

Global geopotential W₀ is an appoint geopotential for the global height datum in IERS numerical standards, which can be calculated by a latest geopotential model and sea surface heights according to the Gaussian geoid appoint define.

The zero normal height surface always coincides with the zero orthometric height surface everywhere, namely whether the orthometric or normal height system, the zero-height surface is the only one whose geopotential is constant.

The appoint geopotential W₀ has no direct geodetic relationship with the geopotential of the gravimetric geoid namely normal geopotential of the ellipsoidal surface.

Analytical orthometric system is more suitable

Let the gravity value of the moving point between the ground and the geoid be equal to the gravity value analytically continued to the moving point by the outer gravity field, and the resulting orthometric height is called the analytical orthometric height. The difference between the analytical orthometric height and the Helmert orthometric height on global ground is no more than 10cm.

The geoidal height is the ellipsoidal height of the geoid, which is the solution of the Stokes boundary value problem in the earth coordinate system. So its measurement scale is the geometric scale of the earth coordinate system. The measurement scale of the analytical orthometric height difference in the vertical direction is also strictly expressed by the geometric scale. It can be found that the analytical orthometric height is consistent with the geometric scales of the earth coordinate system and the geoid height.

The analytical orthometric height is not directly related to the terrain density, and can be continuously refined with the latest gravity field data. On the view of uniqueness, repeatability and measurability of geodetic datum, analytical orthometric height is more suitable for height datum purpose than other types of orthometric height.