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# Introduction

Each physical experiment of finite size and duration carried out in nature implies a certain scale over which the measurement takes place. The actual value of the scale is determined by the discrete sampling of a natural phenomenon. For surface seismic measurements, the time scale is determined by the length of recording time. The frequency scale is determined by the temporal sampling interval. Both scales are related and dual to each other. They serve as an upper limit of observable scales. What scale is actually observed depends on the experiment itself; for example one factor determining the experiment is the frequency content of a seismic source radiating energy into the subsurface. These scale dimensions can vary to a limited degree, depending on the type of seismic experiment. Given that the subsurface structure can be heterogeneous on any scale, the spatial correlation of material properties is directly related to the scale of heterogeneities. When elastic waves propagate in the subsurface, the subsurface velocity and the given temporal frequency determine the spatial wave number locally at any given point in space. The wave equation describing the propagation, couples spatial and temporal resolution.

Imagine a high frequency seismic surface source which radiates energy into the subsurface. Let frequency and bandwidth go to infinity and let the spatial sampling interval shrink infinitely small, then reflected energy could be recorded and resolved from infinitely small scale obstacles. Lowering the frequency content of the source and increasing spatial sampling intervals only enables us to record wave propagation effects (reflection, diffraction) at a larger scale. Thus we can only hope to resolve obstacles (layers, diffractors) properly, that are of the order of that minimum/dominant wave length. However, if there are material discontinuities, which are much smaller in scale than our minimum/dominant wavelength, then a wave, while propagating through a medium, will ``feel'' some average effect. Folstad et al. 1992 showed synthetic comparisons of fine and large scale wave propagation effects, using a Backus averaging of elastic constants. Thus, for a given finite resolution, we can only measure the properties of the ``equivalent medium'' at that scale. Small scale variations still influence effects at a larger scale, but in some averaged sense. I introduce one application of equivalent medium theory in form of homogeneous overburden replacement. Using such a group-theoretical approach, I find NMO (or focusing) velocities by calculating an anisotropic elastic replacement medium and carrying out small offset approximations along the vertical and horizontal axes. I compare this approach (low frequency) to a Dix rms velocity calculation (high frequency).

Next: THE EFFECTIVE MEDIUM Up: Karrenbach: Relating seismic measurements Previous: Karrenbach: Relating seismic measurements
Stanford Exploration Project
11/17/1997