Journal of Seismology 2: 179–182, 1998. 179 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. Short note Using fast Hartley transform to study the free oscillations of the earth Kamal 1 , Pratibha 2 & Ashis Chakravarty 1 1 Department of Earthsciences, University of Roorkee, Roorkee, India 247 667. 2 Department of Mathematics, University of Roorkee, Roorkee, India 247 667 Received 4 July 1997; accepted in revised form 2 February 1998 Abstract In recent years, Hartley Transform (HT) as a substitute of much widely used Fourier Transform (FT) has been practised in science and industries. The advantage of faster computation of HT is enormous when one is dealing with very long data sets. One such application arises in computation of parameters of Free Oscillations of the Earth (FOE), where one needs to study very long period vibrations of the earth, excited after a large earthquake. We demonstrate here an application of HT to determine the parameters of these normal modes of the earth after the Minahasa Peninsula earthquake of 18 April 1990 (M s = 7.5). Introduction Fast computing of transforms makes a lot of prob- lems in science and industry tractable which once were thought to be unsolvable. The Fourier transform is still one of the most used transform for the spec- tral analysis in almost all the technical areas. Lately, scientists and technologists have turned towards such transforms which are computationally faster than the traditionally used Fast Fourier Transform (FFT). One such transform is the Hartley transform. The Hartley transform was first introduced by Hartley (1942) but did not receive a good response until a Fast Hartley Tansform (FHT) was developed by Bracewell (1984) on the same line as the FFT. Its main advantage over the FT is that HT is a real transform and does not invite diverse mode of representation (FT does). The defini- tions of the forward and inverse HT are same and the algorithms are claimed to be faster than those of FT due to its realistic nature. We examine the application of the HT in long period seismology, where one carries out spectral analyses of very long data sets to determine the modal parameters of the earth to restrain the fine structure of the earth. The transform The Hartley transform is a real valued transform related to the Fourier transform of a real valued function. The HT pair of a real function , is given by d d where the function is defined as cos sin There is apparently not much difference between the FT integrals and the pair of HT integrals, but they are very different in practice. First thing, the HT is real, not complex as the case will be with the FT. Secondly, and more importantly, the integrals for the direct and inverse transformation are precisely the same. How- ever, is not the familiar Fourier transform and the reader is referred to the text by Bracewell (1986) for HT’s unfamiliar properties and behavior. One can directly obtain the from the HT amplitude spectrum 2 2 . There are several algorithms available to compute discrete HT, claimed to be faster than the correspond- ing discrete FT. Some of these algorithms are given by