Frequency Content of Randomly Scattered Signals I

Signals reflected from a randomly layered medium are nonstationary stochastic processes that can have very complicated structure. In [l, 21 we considered pulse reflection from a randomly layered half space when the pulse width is large compared to the size of the layers but small compared to the large-scale variations of the layering. Then the reflected signals have a particular structure: they are locally stationary Gaussian processes with a local power spectral density that depends only on the large-scale properties of the layered half space. Local means on the scale of the pulse width. In this paper we present a comprehensive set of numerical simulations that articulate the scope of the theory. In the paper following this one [3], which we refer to as Part II, we show how the theory can be used to solve inverse problems: the recovery of large-scale properties of the medium from the noisy reflected signals. In both papers we restrict attention to plane wave pulses that are normally incident on the randomly layered half space, as we did in [l, 21. In [4] the theory is extended to reflection of a pulse from a point source, incident on a randomly layered half space. The associated numerical simulations and inversion problems are being worked out at present. Our theory of reflected signals from randomly layered media can be interpreted and extended using the localization length associated with time-harmonic waves. This was done in [5,6] where results from numerical simulations less extensive than the present ones are shown. An introduction and review of how localization length ideas can be used in the study of pulse reflection is given in [7]. Randomly layered media are rather special media where wave propagation phenomena can be analyzed extensively. They are of considerable interest in modeling wave propagation in the earth [8]. Our own study of pulse propagation in randomly layered media was stimulated by the numerical simulations of Richards and Menke [9, lo]. In the geophysical context inhomogeneous layers, and inhomogeneities in general, come in many different sizes and it is difficult to distinguish what is a large scale variation from a small scale one. The assumption we make that small scale and large scale variations are well separated, which is the basis of our theory, seems somewhat restrictive. It has to be tested against experimental data.