Numerical Convolution Method in Time Domain and Its Application to Nonrigid Earth Nutation Theory

Abstract

AbstractWe developed a numerical method to incorporate nonrigid effects into a nutation theory of the rigid Earth. Here we assume that the nonrigid effects are based on a linear response theory and its transfer function is expressed as a rational function of frequency. The method replaces the convolution of the transfer function in the frequency domain by the corresponding integro-differential operations in the time domain numerically; namely multiplying the polynomial in the frequency domain by the numerical differentiations in the time domain and multiplying the fractions in the frequency domain by the numerical integrations with a suitable kernel in the time domain. In replacing by the integrations, the method requires the determination of the coefficients of free oscillation. This is done by a least-squares method to fit the theory incorporated with the nonrigid effects to the observational data, whose availability is also assumed. The numerical differentiation and integration are effectively computed by means of the symmetric formulas of differentiation and integration. Numerical tests showed that the method is sufficiently precise to reproduce the analytically convolved nutation at the level of 10 nano arcseconds by using the 9-point central difference formulas and the 8-point symmetric integration formula to cover the period of 15 years with 1.5-hour stepsize. Since we only require the rigid Earth nutation theory to be expressed as a numerical table of time, this method enables one to create a purely numerical theory of nutation of the nonrigid Earth.