This paper aims at combining the geodetic gravimetric inverse problem with the geophysical one to do ‘local gravity field modeling’ and ‘topographic mass-density anomaly determination’ simultaneously. For this purpose, it recalls the basic theories of the two types of gravimetric inverse problems and finds certain relations between their corresponding unknown parameters. The current research proposes a method, which is based on certain anharmonic Radial Basis Functions (RBFs) as well as Generalized Tikhonov regularization method, for modeling the gravity field at the Earth’s surface. Then, it shows that the proposed method can provide certain formulae, based on a combination of the Newton’s IE and the Poisson’s PDE, for extracting the mass-density anomalies from land-based gravity data. Finally, the proposed method is numerically examined by a case study at which the proposed method has been applied to 6350 gravity stations scattered over the coastal Fars of Iran in the north of the Persian Gulf. The case study numerically indicates the possibility of the main idea of the research for simultaneously solving the two types of gravimetry inverse problem.
**Gravimetric inverse problems**
From point of view of mathematics, a geophysical gravimetric inverse problem can be defined by the following equation:
where the mass-density anomaly is considered as the unknown and defined by subtracting the mass-density functional from its prior knowledge ; is the Newton integral operator; is the gravity anomaly yielded by applying the gravity reduction to land-based gravity observations. In a local coordinate system which z-axis is a tangent to the plumb line and outwards from the Earth’s surface, the above geophysical gravimetric inverse problem (i.e. Eq.) can be re-written as follows:
The above equation is a type of the Newton’s IE at which is the position vector, is the topographic space of the region of the interest, and is the differential volume element at the position .
On the other hand, the above definition is so similar to the definition of the gravimetric inverse problem associated with the Earth’s gravity field modeling in physical Geodesy. The Earth’s gravity field modeling in physical Geodesy is often involved with various types of gravimetric inverse problem such as downward continuation, at which the incremental gravity observations such as gravity anomaly are also placed at the right side of corresponding equations and the unknowns are also placed under the mathematical operators. Land-based, airborne, ship borne and satellite gravity data, as well as astronomical, leveling and tidal observations are counted as common observations of physical Geodesy, which are usually used for modeling the Earth’s gravity field. In this way, the unknowns are usually constituted by coefficients of certain algebraic expansions which enable us to model the Earth’s gravity potential as well as its derivatives such as the gravity vector and gravity gradient tensor. These algebraic expansions are commonly linear and obtained by discretizing linear integral equations achieved by Geodetic Boundary Value Problems (GBVP). Furthermore, these expansions can also be obtained by certain basis functions such as spherical harmonic functions, radial basis functions (RBFs), Slepian and wavelet basis functions. Hence, the current research considers the following equation as the general form of the gravimetric inverse problem associated with the subject of the gravity field modeling in physical Geodesy:
where is an operator which depends on either the corresponding integral equations or the expansions of the basis functions; and the vector comprises of the unknowns which model the Earth’s gravity potential and its derivatives. Now, if the Earth’ gravity potential is able to be expanded by a set of basis functions such as , the above general form of the gravimetric inverse problem (i.e. Eq.) will have the following form in the above-mentioned local coordinate system.
where and .
This paper comprises of an idea of correlating the concepts of the abovementioned gravimetric inverse problems with each other. Hence, first it analyzes each one of the abovementioned gravimetric inverse problems separately and then combines them with each other by finding mathematical relations between their unknowns. For this purpose, the concept of the Poisson’s PDE is used for converting the unknowns of Eq. (i.e. ) into the unknown of the Newton’s IE (i.e. the mass-density anomaly ). Indeed, the theories discussed by the paper will also examined by certain numerical experiments in a real case study based on 6350 gravity stations scattered over a part of the north coasts of the Persian Gulf. |