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Underestimation of scale lengths in stochastic fields and their seismic response: a quantification exercise



The possibility that parts of the Earth's continental lower crust can be described with stochastic geological models has been suggested for some time. Recent studies of deeper well logs also indicate a possible stochastic structure at mid-crustal levels. This motivates a closer examination of the relation between the statistics of reflection wavefields and that of the lower crust. Such a relation can put important constraints on possible lower crustal models. This study follows up earlier efforts to quantify the statistics of both stochastic lower crustal models and the reflected wavefield. Since modelling of the seismic response of stochastic (von Karman) fields implies the usage of the impedance contrast field of the latter, we wish to compare the second-order statistics of both types of fields (velocity and impedance). This study concludes that the vertical derivative operator on a von Karman velocity field, implicitly present in the impedance contrast field, alters the second-order horizontal von Karman statistics of the velocity fields in a profound way. Wavefield effects, undoubtedly present in observed seismic data, which have earlier been proposed as possible causes for the aforementioned change, seem to play a secondary role. The vertical derivative operation, inherent in the impedance contrast field, reduces the estimated horizontal scale length and Hurst number by a factor of 2–22 and 1–3, respectively. Original vertical scale length and Hurst number of the velocity fields have a (quasi-)linear influence on this underestimation. Horizontal scale lengths and Hurst numbers were also estimated from the seismic response (Primary Reflectivity Section) of the von Karman fields. The values obtained are close to those obtained from the causative impedance contrast fields, and are similarly underestimated. This suggests a dominant role for the vertical derivative operator in the underestimation of horizontal scale length and Hurst number. This attempt to quantify the relation between the horizontal spatial statistics of von Karman fields and the estimates derived from their seismic response, may be useful in upscaling the latter.