Reply to comment by Shlomo P. Neuman on ‘‘Spatial correlation of permeability in cross-stratified sediment with hierarchical architecture’’

[1] We appreciate the commentary by Neuman [2006] and the opportunity for discussion and further clarification of our article [Ritzi et al., 2004]. In his commentary, Neuman expounds upon his derivation of a model for the permeability semivariogram [Neuman, 2003, equation (53)] and describes aquifer architecture that would be represented by such a model. As requested by the Associate Editor, our reply focuses on illustrating the distinctions between the types of architecture represented by Neuman [2006] and Ritzi et al. [2004]. [2] Ritzi et al. [2004] considered the structure of permeability semivariograms arising in unconsolidated sedimentary deposits, the most common type of aquifer. Sedimentologists conventionally describe such deposits using a hierarchy of stratal unit types. Each of these stratal unit types is designated based on distinguishable characteristics, such as texture and geometry. A unit type defined at one hierarchical level is composed of smaller-scale unit types defined at the next lower level. Our approach makes a link between this hierarchical stratal architecture and the structure of the permeability semivariogram. To more clearly make this point we refer to Figure 1, which illustrates stratal architecture typically found in the sedimentological literature [e.g., Bridge, 2003]. Figure 1 shows a hierarchy of unit types designated within a sedimentary deposit formed as a braid bar within a river. The braid bar is composed of smaller-scale unit types such as large-scale inclined strata and cross-bar channel fills. In this example we refer to these as level II unit types within the hierarchy. Each of these level II unit types is, in turn, composed of smaller-scale level I unit types. For example, one large-scale inclined stratum (level II) is shown to bemade up of level I units including small-scale cross strata composed of fine sand, planar strata composed of medium sand, and medium-scale cross strata composed of sandy gravel. [3] An expanded hierarchy could be created by including unit types that exist at still larger or smaller scales, with any number of levels. For example, we could add a higher level in the hierarchy by representing larger-scale unit types such as point bars and major-channel fills, which occur with braid bars in channel belts. We could add a lower level in the hierarchy by representing smaller-scale stratal unit types, such as avalanche beds and interbeds, which occur within a set of cross strata [see Ritzi et al., 2004]. Dai et al. [2005] showed that hierarchies may not be unique and that a unique hierarchal designation is not required for the approach presented by Ritzi et al. [2004]. [4] Ritzi et al. [2004] showed that the composite permeability semivariogram is related to a stratal hierarchy through their equation (1), in which the stratal architecture is represented using the conventional indicator geostatistics. Here we write a form of that equation for the two-level hierarchy illustrated in Figure 1. Consider two points x and x0 which are separated by a lag vector h. Consider that location x is within level I unit type k which, in turn, is within level II unit type i, and that x0 is within level I unit type m which is within level II unit type j. Thus, the tail of the lag vector at x is in region type ik, and the head at x0 is in region type jm. The composite permeability semivariogram then can be written as: