Inversion of a seismic transmission response

Abstract

Traveling waves are used not only in exploration geophysics but also in other disciplines faced with remote detection problems. A physical system may be described in terms of the input (the source), the medium, and the output (the received signal). The received signal can be made up of either transmitted waves or reflected waves. Two types of inverse problems can be considered, namely, the inverse source problem and the inverse medium problem. In the inverse source problem, the objective is to determine the source. In the inverse medium problem, the objective is to determine the medium. Thus, in terms of this general classification, four types of problems can be encountered, namely, an inverse source problem with transmitted waves, an inverse source problem with reflected waves, an inverse medium problem with transmitted waves, and an inverse medium problem with reflected waves. Let us look at nature. Twinkle, twinkle, little star. The transmission of starlight though the atmosphere makes the star twinkle. A better image of the star can be obtained by solving an inverse source problem using the transmitted starlight. In the typical inverse source problem, the source of energy is remote, the medium transmits the source signal, and the received data are the transmitted waves. Examples are classical earthquake seismology, radio transmission, and passive sonar. Shakespeare said; “For the eye sees not by itself, but by reflection.” Thus the miracle of eyesight solves an inverse medium problem that uses reflected waves. In the typical inverse medium problem, the source of energy is local and often man‐made, the medium reflects the source signal, and the received data are the reflected waves. Examples are reflection seismology, radar, and active sonar. Thus, the two principle types of inverse problems encountered in nature are the inverse source problem with transmitted waves and the inverse medium problem with reflected waves.