Comparison of structural and least-squares lines for estimating geologic relations

Two different goals in fitting straight lines to data are to estimate a true linear relation (physical law) and to predict values of the dependent variable with the smallest possible error. Regarding the first goal, a Monte Carlo study indicated that the structural-analysis (SA) method of fitting straight lines to data is superior to the ordinary least-squares (OLS) method for estimating true straight-line relations. Number of data points, slope and intercept of the true relation, and variances of the errors associated with the independent (X) and dependent (Y) variables influence the degree of agreement. For example, differences between the two line-fitting methods decrease as error in X becomes small relative to error in Y. Regarding the second goal—predicting the dependent variable—OLS is better than SA. Again, the difference diminishes as X takes on less error relative to Y. With respect to estimation of slope and intercept and prediction of Y, agreement between Monte Carlo results and large-sample theory was very good for sample sizes of 100, and fair to good for sample sizes of 20. The procedures and error measures are illustrated with two geologic examples.     

Genesis of monazite and Y zoning in garnet from the Black Hills, South Dakota

The paragenesis of monazite in metapelitic rocks from the contact aureole of the Harney Peak Granite, Black Hills, South Dakota, was investigated using zoning patterns of monazite and garnet, electron microprobe dating of monazite, bulk-rock compositions, and major phase mineral equilibria. The area is characterized by low-pressure and high-temperature metamorphism with metamorphic zones ranging from garnet to sillimanite zones. Garnet porphyroblasts containing euhedral Y annuli are observed from the garnet to sillimanite zones. Although major phase mineral equilibria predict resorption of garnet at the staurolite isograd and regrowth at the andalusite isograd, textural and mass balance analyses suggest that the formation of the Y annuli is not related to the resorption-and-regrowth of garnet having formed instead during garnet growth in the garnet zone. Monazite grains in Black Hills pelites were divided into two generations on the basis of zoning patterns of Y and U: monazite 1 with low-Y and -U and monazite 2 with high-Y and -U. Monazite 1 occurs in the garnet zone and persists into the sillimanite zone as cores shielded by monazite 2 which starts to form in the andalusite zone. Pelites containing garnet porphyroblasts with Y annuli and monazite 1 with patchy Th zoning are more calcic than those with garnet with no Y annuli and monazite with concentric Th zoning. Monazite 1 is attributed to breakdown of allanite in the garnet zone, additionally giving rise to the Y annuli observed in garnet. Monazite 2 grows in the andalusite zone, probably at the expense of garnet and monazite 1 in the andalusite and sillimanite zones. The ages of the two different generations of monazite are within the precision of chemical dating of electron microprobe. The electron microprobe ages of all monazites from the Black Hills show a single ca. 1713 Ma population, close to the intrusion age of the Harney Peak Granite (1715 Ma). This study demonstrates that Y zoning in garnet and monazite are critical to the interpretation of monazite petrogenesis and therefore monazite ages.