Seismic upscaling of layered anisotropic media

All measurements of anisotropic coefficients in exploration seismology are made over finite volumes of inevitably<br>heterogeneous rocks and, therefore, represent some effective values. As a result, a geophysicist often needs to estimate<br>how unaccounted heterogeneity influences (or biases) the estimates of anisotropy. We show that this influence is<br>usually weaker than one might think.<br>In particular, we demonstrate that the difference between any effective quantity me and its arithmetic average ¯m<br>over a given volume is always quadratic in the local fluctuations ˜m, i.e., me ¡ ¯m = O( ˜m2). We prove this statement<br>under quite general conditions that have mathematical nature rather than express specifics of a given quantity m.<br>To verify our theoretical findings, we perform Backus- and Dix-type averages of a typical well log intentionally<br>“contaminated” by moderate anisotropy.