Abstract: The cosine integral of the transient decay voltage in a horizontally receiving loop, for central loop and coincident loop systems, placed on a horizontally stratified earth model, and driven by a step （on-off） current source, has been deduced in this thesis. An adaptive linear digital filtering method is used in the article. The forward transient response of a 1D-layered earth model can be computed. Finally, a discussion is given on some major results and problems which deserve to be noticed.