Confidence Cone Research Articles

Full vector paleomagnetic estimation from the Paraná-Etendeka Large Igneous Province, southern Brazil: Implications on the onset of Cretaceous Normal Superchron

Comprehensive paleomagnetic study from the Paraná-Etendeka Igneous Province (PEIP), Serra Geral Formation, southern Brazil (∼ 135 Ma) adds to the scarce southern hemisphere paleomagnetic database. Twelve paleomagnetic sites from the exposed stratigraphy of the PEIP were recovered from basaltic lava flows in northern Rio Grande do Sul state. These volcanic rocks erupted between 131.5 and 135.6 Ma. The reliability of the rock magnetism data was confirmed before being accepted for paleo-intensity analysis. The Thellier-Coe paleointensity protocol was applied to 109 collected specimens, and 26 individual samples from 9 sites were selected to test for consistency of the paleointensity determinations. The new high-quality mean paleointensity value of 30.6 ± 7.2μT corresponds to a virtual dipole moment (VDM) of 5.75 ± 0.49 × 1022 Am2. These values correspond to approximately 72% of the present-day Earth's magnetic field and suggest that the Earth's magnetic field strength was slightly elevated compared to the long-term average geomagnetic field that prevailed during the Cretaceous Normal Superchron. The calculated paleomagnetic pole (Plong = −27.5° and Plat = −78.6°, A95 = 3.5°) presents an angular difference of 22° with respect to the expected pole declination with an inclination of 5°. This difference may correspond to secular variations considering the confidence cones. Our results, when compared with previous paleomagnetic data, approach the accuracy of recent models for the Cretaceous interval.

A von Mises–Fisher distribution for the orbital poles of the plutinos

ABSTRACT Small Solar system bodies have widely dispersed orbital poles, posing challenges to dynamical models of Solar system origin and evolution. To characterize the orbit pole distribution of dynamical groups of small bodies it helps to have a functional form for a model of the distribution function. Previous studies have used the small-inclination approximation and adopted variations of the normal distribution to model orbital inclination dispersions. Because the orbital pole is a directional variable, its distribution can be more appropriately modelled with directional statistics. We describe the von Mises–Fisher (vMF) distribution on the surface of the unit sphere for application to small bodies’ orbital poles. We apply it to the orbit pole distribution of the observed Plutinos. We find a mean pole located at inclination i0 = 3.57° and longitude of ascending node Ω0 = 124.38° (in the J2000 reference frame), with a 99.7 per cent confidence cone of half-angle 1.68°. We also estimate a debiased mean pole located 4.6° away, at i0 = 2.26°, Ω0 = 292.69°, of similar-size confidence cone. The vMF concentration parameter of Plutino inclinations (relative to either mean pole estimate) is κ = 31.6. This resembles a Rayleigh distribution function, with width parameter σ = 10.2°. Unlike previous models, the vMF model naturally accommodates all physical inclinations (and no others), whereas Rayleigh or Gaussian models must be truncated to the physical inclination range 0–180°. Further work is needed to produce a theory for the mean pole of the Plutinos against which to compare the observational results.

Boolean implication analysis of single-cell data predicts retinal cell type markers

BackgroundThe retina is a complex tissue containing multiple cell types that are essential for vision. Understanding the gene expression patterns of various retinal cell types has potential applications in regenerative medicine. Retinal organoids (optic vesicles) derived from pluripotent stem cells have begun to yield insights into the transcriptomics of developing retinal cell types in humans through single cell RNA-sequencing studies. Previous methods of gene reporting have relied upon techniques in vivo using microarray data, or correlational and dimension reduction methods for analyzing single cell RNA-sequencing data computationally. We aimed to develop a state-of-the-art Boolean method that filtered out noise, could be applied to a wide variety of datasets and lent insight into gene expression over differentiation.ResultsHere, we present a bioinformatic approach using Boolean implication to discover genes which are retinal cell type-specific or involved in retinal cell fate. We apply this approach to previously published retina and retinal organoid datasets and improve upon previously published correlational methods. Our method improves the prediction accuracy of marker genes of retinal cell types and discovers several new high confidence cone and rod-specific genes.ConclusionsThe results of this study demonstrate the benefits of a Boolean approach that considers asymmetric relationships. We have shown a statistically significant improvement from correlational, symmetric methods in the prediction accuracy of retinal cell-type specific genes.Furthermore, our method contains no cell or tissue-specific tuning and hence could impact other areas of gene expression analyses in cancer and other human diseases.

The Early Cretaceous Zaduo Granite, Eastern Qiangtang Terrane (China)—An Attempt to Constrain its Paleolatitude and Tectonic Implications

The Eastern Qiangtang Terrane is an orogenic-like belt around the Eastern Himalayan syntaxis (EHS). The deformation history of this terrane must be known to understand how the EHS region responded to the Lhasa-Qiangtang collision and the closure of the Bangong-Nujiang Ocean (BNO). Here, we present a new paleomagnetic investigation on an Early Cretaceous granite (∼126 Ma) in the Zaduo area, Eastern Qiangtang Terrane. Petrographic observations reflect crystallization from primary melts with only limited subsequent alteration (chloritization of biotite). Magnetite appears to be the dominant carrier of the characteristic remanent magnetization (ChRM) based on stepwise demagnetization of the natural remanent magnetization, supplemented by detailed rock magnetic measurements, including magnetization versus temperature, and acquisition curves of the isothermal and anhysteretic remanent magnetization. End-member modeling of those acquisition curves helped to constrain the paleomagnetic analysis. The inconsistent demagnetization behavior between alternating field (AF) demagnetization at high levels and thermal demagnetization was attributed to the development of gyroremanent magnetization in the AF demagnetization generated by fine-grained single domain magnetite. The ChRM directions from 92 granite samples in geographic coordinates yield an average of declination (Dg) of 2.6° and inclination (Ig) of 38.6° (precision parameter k = 51.4, and 95% confidence cone α95 = 2.1°). The amount of tilting of the granite is poorly constrained which makes proper correction rather tedious. We compared the expected bedding attitudes (Strikeexp = 43.1°, Dipexp = 46.1°) derived from published data (Huang et al., 1992;Tong et al., 2015) with the average observed bedding attitudes (Strikeobs = 54°, Dipobs = 32°) of the Middle-Upper Jurassic sandstones of the Yanshiping Group that was intruded by the Early Cretaceous granite. The discrepancy between the expected and measured bedding attitudes implies that the strata of the Yanshiping Group in the Zaduo area were already tilted prior to the intrusion of the ∼126 Ma Zaduo granite, which was attributed to the Lhasa-Qiangtang collision and the closure of the BNO. The collision led to a series of geological events, such as the tilting of the strata, the ophiolite emplacement, the development of a peripheral foreland basin, and the magmatic activity gap. The tilting/folding of the strata was generally delayed by the layer parallel shortening processes during the early stages of the deformation, thus suggesting an older Lhasa-Qiangtang collision (i.e., >126 Ma).

Reliability of palaeomagnetic poles from sedimentary rocks

SUMMARY Palaeomagnetic poles form the building blocks of apparent polar wander paths and are used as primary input for quantitative palaeogeographic reconstructions. The calculation of such poles requires that the short-term, palaeosecular variation (PSV) of the geomagnetic field is adequately sampled and averaged by a palaeomagnetic data set. Assessing to what extent PSV is recorded is relatively straightforward for rocks that are known to provide spot readings of the geomagnetic field, such as lavas. But it is unknown whether and when palaeomagnetic directions derived from sedimentary rocks represent spot readings of the geomagnetic field and sediments are moreover suffering from inclination shallowing, making it challenging to assess the reliability of poles derived from these rocks. Here, we explore whether a widely used technique to correct for inclination shallowing, known as the elongation–inclination (E/I) method, allows us to formulate a set of quality criteria for (inclination shallowing-corrected) palaeomagnetic poles from sedimentary rocks. The E/I method explicitly assumes that a sediment-derived data set provides, besides flattening, an accurate representation of PSV. We evaluate the effect of perceived pitfalls for this assumption using a recently published data set of 1275 individual palaeomagnetic directions of a >3-km-thick succession of ∼69–41.5 Ma red beds from the Gonjo Basin (eastern Tibet), as well as synthetic data generated with the TK03.GAD field model. The inclinations derived from the uncorrected data set are significantly lower than previous estimates for the basin, obtained using coeval lavas, by correcting inclination shallowing using anisotropy-based techniques, and by predictions from tectonic reconstructions. We find that the E/I correction successfully restores the inclination to values predicted by these independent data sets if the following conditions are met: the number of directions N is at least 100, the A95 cone of confidence falls within a previously defined A95min-max reliability envelope, no negative reversal test is obtained and vertical-axis rotation differences within the data set do not exceed 15°. We propose a classification of three levels (A, B and C) that should be applied after commonly applied quality criteria for palaeomagnetic poles are met. For poles with classification ‘A’, we find no reasons to assume insufficient quality for tectonic interpretation. Poles with classification ‘B’ could be useful, but have to be carefully assessed, and poles with classification ‘C’ provide unreliable palaeolatitudes. We show that application of these criteria for data sets of other sedimentary rock types classifies data sets whose reliability is independently confirmed as ‘A’ or ‘B’, and that demonstrably unreliable data sets are classified as ‘C’, confirming that our criteria are useful, and conservative. The implication of our analysis is that sediment-based data sets of quality ‘A’ may be considered statistically equivalent to data sets of site-mean directions from rapidly cooled igneous rocks like lavas and provide high-quality palaeomagnetic poles.

Paleomagnetic insights into the Cambrian biogeographic conundrum: Did the North China craton link Laurentia and East Gondwana?

Abstract Redlichiid trilobite and small shelly fossils indicate strong ties of the North China craton (NCC) to Gondwana during the early Cambrian, while recent discoveries of the characteristic fossils of Laurentia in Wuliuan shales in the eastern NCC imply its possible connection with Laurentia during the middle Cambrian. Here we report a new paleomagnetic pole at 31.8°S, 140.4°E (radius of 95% confidence cone of paleomagnetic pole, A95, = 5.3°), obtained from the Wuliuan (ca. 505 Ma) Hsuchuang Formation, by averaging our new data and existing virtual geomagnetic poles acquired from different parts of the NCC. A positive regional tilt test and the presence of geomagnetic reversals demonstrate that the remanence was primary. The paleomagnetic data permit placing the NCC near 20°N between Laurentia and Australia at ca. 505 Ma, suggesting that the NCC may have played the role of biogeographic link between East Gondwana and Laurentia in the middle Cambrian. Low-latitudinal westward ocean currents may have facilitated faunal migrations from Laurentia to East Gondwanan blocks via the NCC as well as the newly formed tectono-paleogeographic archipelago, which likely further enhanced biological exchange in the late Cambrian.

Coring induced sediment fabrics at IODP Expedition 347 Sites M0061 and M0062 identified by anisotropy of magnetic susceptibility (AMS): criteria for accepting palaeomagnetic data

SUMMARY Anisotropy of magnetic susceptibility data obtained from discrete subsamples recovered from two Integrated Ocean Drilling Program sites (Expedition 347 sites M0061 and M0062 in the Baltic Sea) by an Advanced Piston Corer are compared to results obtained on subsamples recovered by replicate 6-m-long Kullenberg piston cores. Characteristic natural remanence directions were obtained from the total of 1097 subsamples using principal component analyses. The three principal anisotropy axes of subsamples taken from Advanced Piston Core liners align to the subsample axes, with the maximum axis (K1) parallel to the split core surfaces, possibly caused by outwards relaxation of the core-liners after splitting. A second anomalous anisotropy fabric is characterized by steep values of the angular difference between the inclination of the minimum anisotropy axes (K3) and that expected for horizontal bedding (90°). This fabric is confined to the upper 1–2 m of the Kullenberg cores and specific sections of the advanced piston cores, and we attribute it to conical deformation caused by either excessive penetration speeds and downwards dragging of sediment along the edge of the liner or stretching caused by undersampling. By using our data in an example, we present a protocol to accept palaeomagnetic secular variation data that uses (i) a threshold 90-K3 value of 15°, combined with a modelled, locally applicable minimum inclination of 65° and (ii) an A95 cone of confidence based on Fisher statistics applied to virtual geomagnetic pole distributions.

The First Catalog of Archaeomagnetic Directions From Israel With 4,000 Years of Geomagnetic Secular Variations

The large and well-studied archaeological record of Israel offers a unique opportunity for collecting high resolution archaeomagnetic data from the past several millennia. Here, we initiate the first catalog of archaeomagnetic directions from Israel, with data covering the past four millennia. The catalog consists of 76 directions, of which 47 fulfill quality selection criteria with Fisher precision parameter (k) ≥ 60, 95% cone of confidence (α95) < 6° and number of specimens per site (n) ≥ 8. The new catalog complements our published paleointensity data from the Levant and enables testing the hypothesis of a regional geomagnetic anomaly in the Levant during the Iron Age proposed by Shaar et al (2016, 2017). Most of the archaeomagnetic directions show <15° angular deviations from an axial dipole field. However, we observe in the 10th and 9th century BCE short intervals with field directions that are 19°-22° different from an axial dipole field and inclinations that are 20°-22° steeper than an axial dipole field. The beginning of the first millennium BCE is also characterized with fast secular variation rates. The new catalog provides additional support to the Levantine Iron Age Anomaly hypothesis.

Representation of paleomagnetic data in virtual globes: A case study from the Pyrenees

Virtual globes allow geo-referencing and visualization of diverse geologic datasets. A vertical axis paleomagnetic rotation study in the Southern Pyrenees, Spain, is used to illustrate the potential of virtual globes for representing paleomagnetic data. A macro-enabled workbook that we call P2K, allows KML files to be generated from conventional paleomagnetic datasets. Cones and arch models are used to represent the paleomagnetic vector, and the rotation with regard to the local reference direction, respectively. This visualization provides simultaneous representation of local magnetic declination, inclination and precise confidence cones, shown in their geographic position from diverse perspectives and scales.

Propagating and incorporating the error in anisotropy-based inclination corrections

SUMMARY Sedimentary rock palaeomagnetic inclinations that are too shallow with respect to the ambient field inclination may be restored using anisotropy-based inclination corrections or techniques that rely on models of the past geomagnetic field. One advantage of the anisotropy technique is that it relies on measured parameters (declinations, inclinations, bulk rock magnetic fabrics and particle magnetic anisotropy) that have measurement errors associated with them, rather than relying on a geomagnetic field model and statistical treatment of the data. Sofar,however,theerrorassociatedwiththemeasurementshasnotbeenpropagatedthrough the corrections and the reported uncertainties are simply the α95 95 per cent confidence circles of the corrected directions. In this paper we outline different methodologies of propagating the error using bootstrap statistics and analytic approximations using the case example of the Shepody Formation inclinationcorrection.Bothtechniquesareingoodagreementandindicateamoderate, ∼15per cent, uncertainty in the determination of the flattening factor (f) used in the correction. Such uncertainty corresponds to an ∼0.31 ◦ increase of the confidence cone and a bias that steepens the mean inclination by 0.32 ◦ . For other haematite-bearing formations realistic uncertainties for f ranging from 0 and 30 per cent were used (together with an intermediate value of 15 per cent) yielding a maximum expected increase in the confidence cones and steepening of the inclinationsof ∼1 ◦ .Suchresultsindicatethatformoderateerrorsoff theinclinationcorrection itself does not substantially alter the uncertainty of a typical palaeomagnetic study. We also compare the uncertainties resulting from anisotropy-based corrections to those resulting from the elongation/inclination (E/I) technique. Uncertainties are comparable for studies with a large sample number (>100), otherwise the anisotropy-based technique gives smaller uncertainties. Whenanisotropydataarenotavailable,itispossibletoestimateacorrectionusingflattening factors (f) obtained from the literature. A range of flattening factors has been observed for both magnetite and haematite-bearing rocks (0.4 ≤ f ≤ 1 for haematite and 0.54 ≤ f ≤ 1f or magnetite), but the exact value is specific to the anisotropy of the formation. To evaluate the maximum effects of inclination shallowing, the smallest f (for magnetite or haematite) should be used.

Archeomagnetism of Piton de la Fournaise: Bearing on volcanic activity at La Réunion Island and geomagnetic secular variation in Southern Indian Ocean

Historical and other recent lavas from Piton de la Fournaise (La Réunion Island) are studied using the large sample archeomagnetic method which provides here paleodirections of the Earth Magnetic Field with confidence cones between 0.9 and 2.5°, thus offering a precise record of the geomagnetic field and potential dating constraints. Ages of the lavas are known thanks to historical chronicles (from 1708) or 14C dating and their analysis give information about the directional secular variation (SV) in a region where observatory measurements are scarce and not available before the end of the 19th century. For the past 250yr we find a high value of inclination (−50 to −55°) with respect to the geographic latitude (21°S), connected with a very restricted directional SV. Conversely, the older volcanic units present a larger SV with magnetic inclinations of between −31° and −46°, and declinations from 16°W to 10°E. These results, which are in reasonable agreement with instrumental measurements made on ships in the vicinity of La Réunion, allow us to infer that undated lavas of the northern part of the caldera emanated from eruptions during the second half of the 1700s. Other volcanic products (e.g. Mare Longue flow, Piton Chisny cone and flows and Pointe Langevin), despite their fresh morphology, are necessarily older than about 1500AD. Further knowledge of the path of the SV in the more distant past, and therefore further archeomagnetic dating, is hampered by the lack of a precise chronology through radiometric or other similar methods. Available models of the geomagnetic field during the last millennium, which suffer from the scarcity of data in the Southern Hemisphere, are discussed in the light of our results.

An X-ray computed tomography study of inclusion trail orientations in multiple porphyroblasts from a single sample

Inclusion trails in 58 garnet porphyroblasts in a single sample from the Cram Hill Formation in southeast Vermont were imaged using high-resolution X-ray computed tomography. Texturally the porphyroblasts have a large core with a sub-vertical foliation that has a well defined geometry with a mean orientation of 113/72E for a 95% confidence cone semi-angle of 5.3°. This steep foliation curves into a sub-horizontal foliation. The foliation intersection/inflection axes (FIAs) defined by hinge lines for this curvature are tightly clustered with a mean plunge of 11° towards 201° with a 95% confidence cone semi-angle of 4.1°. Eigenvalue analysis indicates that the clustered distribution of the foliation and FIA data are unlikely to be the result of a random event. There is evidence in the specimen for multiple foliation-forming events subsequent to garnet nucleation, and the preservation of these clustered distributions in their wake strongly suggests that the porphyroblasts have not rotated with respect to a geographic reference frame. FIAs represent a measurable structural element that can provide information on tectonic events at the time the porphyroblasts grew. A comparison of FIA data collected using high-resolution X-ray computed tomography with data collected using the asymmetry method demonstrates that the asymmetry method is a valid technique for defining the mean FIA orientations in a sample.

Palaeomagnetism of Cretaceous Pacific Seamounts*

Summary. The palaeomagnetism of Cretaceous Pacific seamounts, as deduced from their magnetic anomalies, is reviewed. Results from 35 seamounts are reported here for the first time. Of these 35 results, 15 are of high enough quality to be useful. This has resulted in a considerable increase in the available information concerning the palaeomagnetism of the Pacific plate. Using only the most reliable results, we calculate a palaeomagnetic pole for the upper Cretaceous at 58° North, 350° East, with a 95 per cent cone of confidence of only 5°. This pole is considerably more accurate than the one previously reported and is farther to the west. The previously reported difference between the mean pole position from seamounts close to Japan and the mean pole position from seamounts close to Hawaii is not real, although one group of seamounts close to Japan gives an anomalous pole position. Several mechanisms have been suggested whereby errors might be introduced in the determinations of pole positions from seamounts. No error is introduced in the pole position because of the restricted extent of surveys and the consequent inaccuracy in the determination of the regional field. The demagnetization effect, which can produce differences between the observed direction of magnetization and the direction of the magnetizing field in an elongated body, is unimportant for seamounts. The new pole position is sufficiently accurate for use in many types of calculations in which it is desired to locate the Pacific plate with respect to the Earth's spin axis.

Comment on “Recent eruptive history of Stromboli (Aeolian Islands, Italy) determined from high‐accuracy archeomagnetic dating” by S. Arrighi et al.

[1] Arrighi et al. [2004] (hereinafter referred to as AR2004) recently reported paleomagnetic directions gathered from Stromboli volcano by using the unconventional ‘‘large sample method’’ (LSM). These directions are partly different and have smaller confidence cones than those previously reported by Speranza et al. [2004] (hereinafter referred to as SP2004) for the same spatter lavas by using the traditional (core-drilling and systematic stepwise demagnetization) paleomagnetic technique. Hence AR2004 conclude that ‘‘traditional paleomagnetic sampling cannot yield sufficient precision’’, and is therefore unsuitable to study the paleosecular variation of the geomagnetic field recorded in volcanics, and to use it as a dating tool. Here we contend that the laboratory procedures and analysis methods employed by AR2004 yield a fictitious improvement in statistical uncertainty. We conclude that the traditional paleomagnetic techniques provide a more realistic estimate of archeomagnetic dating and related uncertainties.

The orientation distribution of single joint sets

Abstract Poles from line samples of systematic joint sets scatter about a mean pole because joints are neither perfectly planar nor parallel, and because measurement instruments are imprecise. Definition of a single joint set can be based solely on its orientation distribution and this distribution is assessed using two statistical parameters: square root of the circular variance (approximately equal to the standard deviation σ for two-dimensional (2D) data) and cone of confidence (α 95 for 3D data). The distribution for joints generated in the absence of tectonic deformation is well clustered with σ = 1.7° and α 95 = 0.48° based on a bootstrap sample of 50. Jointing associated with various fold styles show less clustering: the kink of a fault-bend fold (σ = 6.1° and α 95 = 1.7°), basement-cored anticline (σ = 3.5° and α 95 = 1.5°), regional joint set transected by a basement-cored anticline (σ = 5.2° and α 95 = 1.8°) and a buttress anticline (σ = 4.3° and α 95 = 1.7°). Jointing associated with local faulting tends to show even less clustering: a Cretaceous marl (σ = 8.3° and α 95 = 2.4°) and a glauconitic sandstone (σ = 8.6° and α 95 = 2.2°). The latter sample was drawn from two overlapping joint sets, indicating that distribution data greater than α 95 = 2.2° may signal overlapping joint sets.

Archeomagnetic dating of Mediterranean volcanics of the last 2100 years: validity and limits

Archeomagnetic dating developed at St. Maur laboratory has been applied to the Mediterranean volcanoes Etna, Vesuvius and Ischia. The method involves samples from lava flows or high temperature emplaced pyroclasts (welded scoriae, pumice, etc.) weighing 0.5–1 kg each, that allows reaching a precision of a few tenths of a degree on the direction of their thermoremanent magnetization, and hence a semi-angle of the Fisher 95% confidence cone between 0.6 and 1.8° for every volcanic unit. Among the factors reducing precision on the mean magnetic direction, the most important appears to be a distortion of the ambient field induced by magnetization of the cooling lava, which means that a number of samples should be collected over a large area. Age determination is based upon similarity between variation curves of the Direction of Earth’s Magnetic Field (DEMF) reconstructed in France from 120 well-dated archeological sites, and on Italian volcanoes from historically dated eruptions. A total of 63 lava flows and pyroclastic units, such as cinder cones or nuée ardente deposits, are shown to be dated with an overall precision of ±40 years for the last 1500 years, and ±50 to ±100 years from AD 500 to 150 BC, this lesser precision resulting from both an increasing uncertainty on the shape of the DEMF curve and a smaller variation of the DEMF itself. This irregularity of the DEMF path plus an increasing number of ambiguities, related to similarity of the DEMF at different times further into the past, are the most serious limitations of the method. Though well-dated eruptions are known for the last two millennia, retrieval of their products is often misleading and about 50% of volcanics presumed of known date prior to the 17th century are in fact of older age, discrepancies usually reaching several hundreds of years. Owing to good agreement between the DEMF curves of France and southern Italy, the method may confidently be extended to volcanic materials from the whole of Mediterranean Europe, provided there are firm constraints that they were erupted within the last 2100 years.

Easy handling of tectonic data: the programs TectonicVB for Mac and TectonicsFP for Windows™

TectonicVB for Macintosh and TectonicsFP for Windows TM operating systems are two menu–driven computer programs which allow the shared use of data on these environments. The programs can produce stereographic plots of orientation data (great circles, poles, lineations). Frequently used statistical procedures like calculation of eigenvalues and eigenvectors, calculation of mean vector with concentration parameters and confidence cone can be easily performed. Fault data can be plotted in stereographic projection (Angelier and Hoeppener plots). Sorting of datasets into homogeneous subsets and rotation of tectonic data can be performed in interactive two-diagram windows. The paleostress tensor can be calculated from fault data sets using graphical (calculation of kinematic axes and right dihedra method) or mathematical methods (direct inversion or numerical dynamical analysis). The calculations can be checked in dimensionless Mohr diagrams and fluctuation histograms.

Magnetic fabrics and petrofabrics: their orientation distributions and anisotropies

Magnetic-fabric and other petrofabric anisotropies may be described by second-rank tensors represented by ellipsoids. For a homogeneous petrofabric that is adequately sampled, a stereoplot of the orientation-distribution of the tensors’ principal axes (maximum, intermediate and minimum) should show three orthogonal concentrations. The concentrations form some combination of shapes from circular clusters through partial girdles to full girdles. The concentrations’ elliptical eccentricities are constrained by the symmetry of the sample-orientation-distribution (i.e. L, L>S, L=S etc.) as well as the individual sample-anisotropies. The mean orientations of principal axes must be orthogonal, just as with individual sample-tensors. This requires tensor-statistics for their calculation (Jelinek, 1978). Furthermore, elliptical confidence cones for the means should parallel principal planes, preserving overall orthorhombic symmetry. However, in practice, sub-orthorhombic symmetry may arise from unrepresentative sampling but it may also be a useful indicator of multiple or heterogeneous petrofabrics. In the case of magnetic fabrics, the wide range in average susceptibility values and variation in magnetic mineralogy permit small numbers of high-susceptibility samples to deflect the orientation of the tensor-mean away from the majority of samples. Normalizing the samples by their bulk susceptibility overcomes this, but the orientation of high-susceptibility outliers may signify an event or subfabric of importance that we should not discard. Therefore, stereoplots of both normalized and non-normalized orientation-distributions should be compared, preferably also identifying the outliers. It is important to distinguish the shape of the orientation distribution ellipsoid from the shape of the individual magnetic fabric ellipsoids. (The qualitative L–S nomenclature is best replaced by Tj where Tj=+1=oblate; Tj=−1=prolate (Jelinek, 1981).) Invariably, the orientation distribution is described by an ellipsoid whose shape is more spherical than that of the individual sample-anisotropy ellipsoids because the latter have scattered orientations. Furthermore, the shape of the orientation-distribution ellipsoid need bear no relation to the shape of individual sample-ellipsoid anisotropies. The concepts are illustrated with 1119 measurements of anisotropy of magnetic susceptibility (AMS) from seven areas and with 188 measurements of anisotropy of anhysteretic remanence (AARM) from two areas.

Experimental designs for determination of the anisotropy of remanence-test of the efficiency of least-square and bootstrap methods applied to metamorohic rocks from southern Poland

The collection of samples from two drill cores (serpentinite/ peridotite and gabbro/ lamprophyre) from the Fore-Sudetic Block (SW Poland) has been used to test several experimental designs for determination of anisotropy of anhysteretic remanence. Ellipsoids of anisotropy for ARM were obtained for batches of samples from four levels of drill cores. Standard least - square methods for 3, 9, 12 and 15 independent measurements of ARM were used to determine the anisotropy of remanence ellipsoid for each single sample and to evaluate the measurement errors of parameters of the ellipsoid. The distribution of anisotropy parameters (axes orientation and anisotropy ratios) for batches of samples from four core levels was also evaluated for each of the ARM determination methods. Bootstrap, Jelinek (1978) and Hext (1963) methods were applied to calculate confidence cones for mean anisotropy axes (Constable and Tauxe, 1990) for four tested batches of samples.For highly-anisotropic, low-coercive serpentinite/peridotite rocks time-efficient 3-orientations procedures (Stephenson et al. 1986) may lead to accurate determination of ARM ellipsoid for the tested sample and a batch of samples. The differences of anisotropy axes directions determined by different methods are here of order 3–5°. The anisotropy parameters Pj-T of anisotropy ellipsoid are practically the same for 6- and 15-positions methods and slightly different for 3- positions method. For pyrrhotite bearing gabbros 3-positions method produces the AARM ellipsoids (for single samples and the collection) that is 5–10° apart from results of the other methods and overestimates the anisotropy ratio Pj. That effect is enhanced for lamprophyres. On the contrary to the 6- and 15-positions method (Pj<1.5, weakly grouped anisotropy axes), the 3-positions method gives here the well-grouped, differently oriented axes with Pj of 2–3.5! The correction for suspected GRM effect (Stephenson, 1993) shifts the AARM axes to be in agreement with results of the remaining methods. However, the high anisotropy still persists after GRM correction.

Multiresponse Process Optimization via Constrained Confidence Regions

This paper presents a new methodology for analyzing multiresponse experiments. The methodology consists of computing confidence regions for the stationary points of quadratic responses and confidence cones for the direction of maximum improvement for linear responses. The stationary points are constrained to lie within the region of experimentation. It is shown that the confidence regions depend on the value of the Lagrange multiplier of the region's constraint. The value of the Lagrange multiplier is found by solving the Karush-Kuhn-Tucker optimality conditions. Then, nonlinear optimization problems are set up and solved for obtaining experimental points that lie inside all the confidence regions, cones and constraints. Robust process design examples illustrate the methods proposed. The examples address the “target is best” and “larger the better” cases.