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

Abstract

S U M M A R YThe 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.

Geostatistical inversion of seismic and ground‐penetrating radar reflection images: What can we actually resolve?

“…Our numerical tests also confirm the analytical finding that the second-order statistics of a seismic or GPR reflection image should be sensitive to the power spectral slope of the velocity heterogeneity. However, the available empirical evidence regarding this result is still too thin and inconsistent to either corroborate or challenge it [e.g., Carpentier and Roy-Chowdhury, 2007;Scholer et al, 2010]. We therefore see this as an important topic for future research.…”

“…This means that we estimate parameters describing the geostatistical nature of the heterogeneity, rather than a detailed distribution of material properties. To this end, estimation of the geostatistical properties of subsurface velocity heterogeneity from surface-based seismic and ground-penetrating radar (GPR) reflection images has been a long-standing problem of significant interest [e.g., Holliger et al, 1992;Hurich, 1996;Pullammanappallil et al, 1997;Rea and Knight, 1998;Bean et al, 1999;Poppeliers and Levander, 2004;Carpentier and Roy-Chowdhury, 2007;Knight et al, 2007]. Of particular interest has been the estimation of the lateral statistics of a subsurface velocity field from those of the corresponding reflection image, as this information cannot be obtained from borehole log or core analysis.…”

Estimation of the lateral correlation structure of subsurface water content from surface‐based ground‐penetrating radar reflection images

“…Work by Holliger et al [1994] and Pullammanappallil et al [1997] initially suggested that the average lateral correlation structures of both of these fields should be equivalent. However, the more recent results of Bean et al [1999] and Carpentier and Roy-Chowdhury [2007] have pointed out the fundamental dependence of the lateral correlation properties of a seismic image on bandwidth and on the vertical derivative operator that acts to create reflection coefficients from an impedance field, respectively. Clearly, such details concerning the physics of the experiment must be considered in any strategy to estimate the correlation structure of subsurface velocity from seismic reflection data.…”

“…[3] A number of studies have explored the relationship between the lateral correlation structure of seismic and radar velocity fields and those of the corresponding seismic and GPR reflection data [e.g., Gibson, 1991;Hurich, 1996;Pullammanappallil et al, 1997;Line et al, 1998;Rea and Knight, 1998;Bean et al, 1999;Knight et al, 2004;Oldenborger et al, 2004;Dafflon et al, 2006;Carpentier and Roy-Chowdhury, 2007;Knight et al, 2007]. Although the seismic and GPR communities have to date worked largely independently of one another, it is important to note that the results obtained are mutually transferrable because of the strong mathematical analogies that exist between seismic and electromagnetic wave propagation [e.g., Carcione and Cavallini, 1995;Belina et al, 2009].…”

Stochastic Heterogeneity Mapping around a Mediterranean salt lens

“…This method of statistically analyzing the reflection field of seismic data has been used extensively to study complex acoustic impedance variability in the solid earth (e.g. Goff et al, 1994;Holliger et al, 1994;Hurich and Kocurko, 2000;Carpentier and Roy-Chowdhury, 2007) even in areas where there is a predominance of diffuse reflectivity (as opposed to specular reflectivity) observed. This research represents the first application of Stochastic Heterogeneity Mapping to the ocean.…”