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In this chapter I apply tensor fields in an application area where generally only scalar fields have been used, to improve the agreement of parameters measured in with different types of experiments. The transformation of scalar measured quantities in well logs to tensors allows for an improved description of the medium. The primary example I show here is the prediction of surface seismic velocities from well logs.
Measurements carried out at different scales may not agree in their observations. An equivalent medium approach can link measurements carried out at different scales. I apply high frequency (Dix) and low frequency (Schoenberg&Muir) averages to a well log and compare the results to a conventional velocity analysis of surface seismic data.
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 for any given point in space. The equation describing the wave 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 infinitesimal 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 wavelength. 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. [(Folstad and Schoenberg,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).