COMPARISON OF FIVE LEAST-SQUARES INVERSION TECHNIQUES IN RESISTIVITY SOUNDING*

A brief history of the development of the inverse problem in resistivity sounding is presented with the development of the equations governing the least-squares inverse. Five algorithms for finding the minimum of the least-square problem are described and their speed of convergence is compared on data from two planar earth models. Of the five algorithms studied, the ridge-regression algorithm required the fewest numbers of forward problem evaluations to reach a desired minimum. Solution space statistics, including (1) parameter-standard errors, (2) parameter correlation coefficients, (3) model parameter eigenvectors, and (4) data eigenvectors are discussed. The type of weighting applied to the data affects these statistical parameters. Weighting the data by taking log₁₀ of the observed and calculated values is comparable to weighting by the inverse of a constant data error. The most reliable parameter standard errors are obtained by weighting by the inverse of observed data errors. All other solution statistics, such as dataparameter eigenvector pairs, have more physical significance when inverse data error weighting is used.     

RESISTIVITY MODELLING FOR ARBITRARILY SHAPED TWO-DIMENSIONAL STRUCTURES*

A numerical technique is developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations by using point- as well as area-discretization of the subsurface. Potential distributions at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct explicit matrix inversion techniques. An empirical mixed boundary condition is used at the “infinitely distant” edges of the lower half-space. Accurate solutions using area-discretization method are obtained with significantly less attendant computational costs than with the relaxation, finite-element, or network solution techniques for models of comparable dimensions.     

A 19<Superscript>th</Superscript> Century Baseline Temperature Estimate for Peninsular India from Combined Analysis of Geothermal and Meteorological Records

Direct information about climate change from meteorological surface air temperature records are available in India only since 1901 A.D. Meteorological surface air temperature (SAT) data for the period 1901–2006 from 49 sites in peninsular India have been combined with the geothermal data from 146 sites to extract a baseline (or pre-observational mean, POM) surface temperature prior to the existence of the observational record in the region. Periodicities of 5, 11 and 22 years in the SAT time series have little influence on the combined analysis to infer long-term climate change. The best estimate of the long-term average temperature for the 19^th Century is 0.7 °C lower than the 1961–1990 mean temperature. Considering the additional warming of 0.38°C relative to the 1961–1990 mean over a 10-year window centred on the year 2000, the hybrid POM-SAT method suggests that the total surface warming in peninsular India from mid-1800s to early- 2000s is about 1.1 °C. The study provides new evidence for significant warming prior to the establishment of widespread meteorological stations in peninsular India.     

Early Neoarchaean A-type granitic magmatism by crustal reworking in Singhbhum craton: Evidence from Pala Lahara area,Orissa

Several volumetrically minor \(\sim \)2.8 Ga anorogenic granites and rhyolites occur along the marginal part of the Singhbhum craton whose origin and role in crustal evolution are poorly constrained. This contribution presents petrographic, geochemical, zircon U–Pb and trace element, and mineral chemical data on such granites exposed in the Pala Lahara area to understand their petrogenesis and tectonic setting. The Pala Lahara granites are calc-alkaline, high-silica rocks and define a zircon U–Pb age of 2.79 Ga. These granites are ferroan, weakly metaluminous, depleted in Al, Ca and Mg and rich in LILE and HFSE. They are classified as A2-type granites with high Y/Nb ratios. Geochemical characteristics (high \(\hbox {SiO}_{2}\) and \(\hbox {K}_{2}\hbox {O}\), very low MgO, Mg#, Cr, Ni and V, negative Eu anomaly, flat HREE and low Sr/Y) and comparison with melts reported by published experimental studies suggest an origin through high-temperature, shallow crustal melting of tonalitic/granodioritic source similar to the \(\sim \)3.3 Ga Singhbhum Granite. Intrusion of the Pala Lahara granites was coeval with prominent mafic magmatism in the Singhbhum craton (e.g., the Dhanjori mafic volcanic rocks and NNE–SSW trending mafic dyke swarm). It is suggested that the \(\sim \)2.8 Ga A-type granites in the Singhbhum craton mark a significant crustal reworking event attendant to mantle-derived mafic magmatism in an extensional tectonic setting.     

Palaeoweathering,composition and tectonics of provenance of the Proterozoic intracratonic Kaladgi–Badami basin,Karnataka, southern India: Evidence from sandstone petrography and geochemistry

Petrographic and geochemical data on the sandstones of the Proterozoic intracratonic Kaladgi–Badami basin, southern India are presented to elucidate the palaeoweathering pattern, and composition and tectonics of their provenance. The Kaladgi–Badami basin, hosting the Kaladgi Supergroup, occupies an E–W trending area. The Supergroup unconformably overlies Archaean basement TTG gneisses, granites and greenstones, comprises a cyclic arenite–pelite–carbonate association and is divided into the Bagalkot and Badami Groups. The immature arkosic character of the basal Saundatti Quartzite Member (Bagalkot Group) containing fresh and angular feldspars, along the northern margin of the basin, suggests that during the initial stage of deposition, this part of the basin received sediments from a restricted, uplifted and less weathered source dominated by K-rich granites occurring to the north. In contrast, the Saundatti Quartzite along the southern margin displays a mostly mature, quartz-rich character with less abundant but severely weathered feldspars, and higher SiO₂ and CIA but lower Al₂O₃, TiO₂, Rb, Sr, Ba, K₂O, K₂O/Na₂O, Zr/Ni and Zr/Cr. This is interpreted in terms of a tectonically stable, considerably weathered mixed source (Archaean gneisses, granites and greenstones) along the southern fringe of the basin. The highly mature (quartz arenite) Muchkundi Quartzite Member (also of the Bagalkot Group), occurring higher up in the succession, exhibits minor but severely altered feldspars, and higher SiO₂, Na₂O, CIA, Cr and Ni with lower K₂O, Al₂O₃, TiO₂ and K₂O/Na₂O. This reflects that with the passage of time the source evolved to a uniform, extensively weathered, tectonically stable peneplained provenance which consisted of less evolved TTG gneisses and greenstones. This was followed by closure, deformation and upliftment of the basin hosting the Bagalkot Group and subsequent deposition of the Badami Group. Sandstone Members of this younger Group (Cave-Temple Arenite and Belikhindi Arenite) range widely in mineralogy (quartz arenite to arkose) and chemistry (including CIA), and point to a source that varied from uplifted, less weathered K-rich granites to less evolved, peneplained TTG gneisses and greenstones or even Bagalkot sediments. Variable alteration of feldspars in the Kaladgi sandstones and severe depletion of Ca, Na and Sr in the associated shales indicate a humid tropical (tropical and subtropical) climate facilitating chemical weathering.