Ar-40/Ar-39 Evidence for Middle Proterozoic (1300-1500 Ma) Slow Cooling of the Southern Black Hills, South Dakota, Midcontinent, North America: Implications for Early Proterozoic P-T Evolution and Posttectonic Magmatism

Ar-40/Ar-39 total gas and plateau dates from muscovite and biotite in the southern Black Hills, South Dakota, provide evidence for a period of Middle Proterozoic slow cooling. Early Proterozoic (1600-1650 Ma) mica dates were obtained from metasedimentary rocks located in a synformal structure between the Harney Peak and Bear Mountain domes and also south of Bear Mountain. Metamorphic rocks from the dome areas and undeformed samples of the similar to 1710 Ma Harney Peak Granite (HPG) yield Middle Proterozoic mica dates (similar to 1270-1500 Ma). Two samples collected between the synform and Bear Mountain dome yield intermediate total gas mica dates of similar to 1550 Ma. We suggest two end-member interpretations to explain the map pattern of cooling ages: (1) subhorizontal slow cooling of an area which exhibits variation in mica Ar retention intervals or (2) mild folding of a Middle Proterozoic (similar to 1500 Ma) similar to 300 degrees C isotherm. According to the second interpretation, the preservation of older dates between the domes may reflect reactivation of a preexisting synformal structure (and downwarping of relatively cold rocks) during a period of approximately east-west contraction and slow uplift during the Middle Proterozoic. The mica data, together with hornblende data from the Black Hills published elsewhere, indicate that the ambient country-rock temperature at the 3-4 kbar depth of emplacement of the HPG was between 350 degrees C and 500 degrees C, suggesting that the average upper crustal geothermal gradient was 25 degrees-40 degrees C/km prior to intrusion. The thermochronologic data suggest HPG emplacement was followed by a similar to 200 m.y. period of stability and tectonic quiescence with little uplift. We propose that crust thickened during the Early Proterozoic was uplifted and erosionally(?) thinned prior to similar to 1710 Ma and that the HPG magma was emplaced into isostatically stable crust of relatively normal thickness. We speculate that uplift and crustal thinning prior to HPG intrusion was the result of differential thinning of the subcrustal lithosphere beneath the Black Hills. If so, this process would have also caused an increase in mantle heat flux across the Moho and triggered vapor-absent melting of biotite to produce the HPG magma. This scenario for posttectonic granite generation is supported, in part, by the fact that in the whole of the Black Hills, the HPG is spatially associated with the deepest exposed Early Proterozoic country rock.

A Mathematical Model for Simplifying Representations of Objects in a Geographic Information System

The study of operations on representations of objects is well documented in the realm of spatial engineering. However, the mathematical structure and formal proof of these operational phenomena are not thoroughly explored. Other works have often focused on query-based models that seek to order classes and instances of objects in the form of semantic hierarchies or graphs. In some models, nodes of graphs represent objects and are connected by edges that represent different types of coarsening operators. This work, however, studies how the coarsening operator "simplification" can manipulate partitions of finite sets, independent from objects and their attributes. Partitions that are "simplified first have a collection of elements filtered (removed), and then the remaining partition is amalgamated (some sub-collections are unified). Simplification has many interesting mathematical properties. A finite composition of simplifications can also be accomplished with some single simplification. Also, if one partition is a simplification of the other, the simplified partition is defined to be less than the other partition according to the simp relation. This relation is shown to be a partial-order relation based on simplification. Collections of partitions can not only be proven to have a partial- order structure, but also have a lattice structure and are complete. In regard to a geographic information system (GIs), partitions related to subsets of attribute domains for objects are called views. Objects belong to different views based whether or not their attribute values lie in the underlying view domain. Given a particular view, objects with their attribute n-tuple codings contained in the view are part of the actualization set on views, and objects are labeled according to the particular subset of the view in which their coding lies. Though the scope of the work does not mainly focus on queries related directly to geographic objects, it provides verification for the existence of particular views in a system with this underlying structure. Given a finite attribute domain, one can say with mathematical certainty that different views of objects are partially ordered by simplification, and every collection of views has a greatest lower bound and least upper bound, which provides the validity for exploring queries in this regard.