| Abstract: | The potential distribution and the wave propagation in a horizontally stratified earth is considered and the analogy of the mathematical expression for seismic transfer function, electromagnetic and electric kernel functions, and magnetotelluric input impedance is discussed. Although these specific functions are conveniently treated by a separate expression in each method, it is indicated that the function for seismic and electromagnetic methods is mathematically the same with a change in the physical meaning of the variables from one method to the other. Similarly, the identity of the mathematical expressions of the resistivity kernel function and magnetotelluric input impedance is noticed. In each method a specific geophysical function depends on the thickness and the physical properties of the various layers. Every specific function involves two interdependent fundamental functions, that is Pn and Qn, or Pn and P*n, having different physical meaning for different methods. Specific functions are expressible as a ratio Pn/Qn or P*n/Pn. Fundamental functions may be reduced to polynomials. The fundamental polynomials Q*n and P*n describing the horizontally stratified media are a system of polynomials orthogonal on the unit circle, of first and second order, respectively. The interpretation of geophysical problems corresponds to the identification of the parameters of a system of fundamental orthogonal polynomials. The theorems of orthogonal polynomials are applied to the solution of identification problems. A formula for calculating theoretical curves and direct resistivity interpretation is proposed for the case of arbitrary resistivity of the substratum. The basic equation for synthetic seismograms is reformulated in appendix A. In appendix B a method is indicated for the conversion of the seismic transfer function from arbitrary to perfectly reflective substratum. |
