Laplace method to investigate subsurface geologic structures and its application

The ages of layered strata show a particular spatial distribution, and the bedding plane is isopotential surface of a scalar T(x, y, z) correlative to the age of a given stratigraphic horizon. The scalar T is obtained as the solution of boundary value problem of Laplace equation in power series, and the solution describes the three-dimensional geologic structures within an analytical space bounded by faults or unconformities on which the value of T jumps. The solution is termed the horizon function and consists of the datum succession and the structural part. The datum succession defines the relationship between the relative age (time-scale; T) and the thickness of strata (spatial scale; z) for undeformed horizontal strata. Geologic structures are described by the structural part of the solution composed by Fourier series, and the coefficients included there are determined by the least-square method using the dip and strike defined by particular combinations of the derivatives of T or the horizon data obtained in the area. The undersampled nature of geologic data is overcome by the selection of solution type, Eigenvalues, and boundary condition. Geological map and cross-sections are composed quantitatively and automatically by combining the spatial distribution of investigated T with the digital map defining the landform of the area. Test results were examined from this point of view. Improvement of the structural part to have the result fit completely the measured data is possible by introduction of the multiplying polynomials, although this concerns mathematical nature of the potential T.