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Glaciotectonic Shear Zones:Surface Sample Bias and Clast Fabric Interpretation
A Thesis Presented
by
Elliot Charles Klein
toThe Graduate School
in Partial Fulfillment of the Requirementsfor the Degree of
Master of Science
in
Department of Earth and Space Sciences
State University of New Yorkat Stony Brook
May, 2002
State University of New York
at Stony Brook
The Graduate School
Elliot Charles Klein
We, the thesis committee for the above candidate for the Master of Science degree,
hereby recommend acceptance of this thesis.
_______________________________________Daniel M. Davis, Advisor, Professor
Department of Earth and Space Sciences
_______________________________________E. Troy Rasbury, Assistant Professor
Department of Earth and Space Sciences
_______________________________________William E. Holt, Professor
Department of Earth and Space Sciences
This thesis is accepted by the Graduate School
______________________________Dean of the Graduate School
ii
Abstract of the Thesis
Glaciotectonic Shear Zones:
Surface Sample Bias and Clast Fabric Interpretation
by
Elliot Charles Klein
Master of Science
in
Department of Earth and Space Sciences
State University of New York
at Stony Brook
2002
Long Island surface geology is diverse in glacial settings and glaciotectonic
landforms. I present two models that elucidate the generation of glaciotectonic push moraines
with examples from eastern Long Island. One model, with prolonged glaciotectonic push-
from-behind, contracts glacial sediment and strata in a manner analogous to larger scale
processes in thin-skinned small-scale fold-and-thrust belts. In a prolonged glacial advance,
proglacial material shortens at the glacier margin into the form of a critical taper, a wedge
shaped packet of material containing the deformed structures in cross section. An alternative
model, with repeated glaciotectonic push-from-behind, deforms less proglacial material
since the ice, which is doing the pushing, melts back before the deforming sediment can
form a critical taper. Structures produced by repeated glaciotectonic push-from-behind are
iii
generated by seasonal or annual readvance of the glacier margin during a period of overall
glacial retreat. Two field areas located within the Ronkonkoma Moraine of Long Island are
documented to provide clear examples of glaciotectonic push-from-behind. At the Ranco
Quarry site, measured sections suggest emplacement by prolonged glaciotectonic push-
from-behind. Ground penetrating radar (GPR), seismic studies, topographic analysis, and
measured sections at the Hither Hills site indicate emplacement by repeated glaciotectonic
push-from-behind.
Quantitative clast fabric analysis, despite its limitations, is a worthwhile analytical
tool in glacial diamict studies. More robust than graphical methods, clast fabric analysis
allows quantification of otherwise descriptive three-dimensional fabrics. In conjunction
with the orientation of the long-axes, short-axes preferred direction could further establish
the nature of shear, emplacement, and deposition in glacigenic settings. Field measurement
of clast orientation, however, produces a systematic sampling bias in favor of clasts normal
to outcrop surface. This surface sampling bias is a function of the orientation of outcrop
surface to the fabric and can affect the inferred fabric strength (eigenvalues) enough to
influence interpretation. I use simple calculations and computer generated random clast
populations to quantify this bias and I find that it is greatest for those clasts best suited to
fabric analysis (those that are rod-like in shape). True fabric strengths and orientations
(eigenvectors) are misrepresented due to the surface sampling bias. Fortunately, the sampling
bias effect upon strong fabric orientation is generally small.
iv
Table of Contents
List of Figures…………………………………………………….………...…..…vi
List of Tables…………………………………………………………...……….viii
Acknowledgments……………………….…………………………..…………...ix
I. Push Moraine Glaciotectonics: Examples from Eastern Long Island 1
Introduction………………………………………………………….……………..1Setting……………………………………….……………………………………..6Rationale…………………………….……….…………………………………….9Glaciotectonic Deformation on Pleistocene Long Island….……………………..10Glacigenic Deposits of Long Island………………………….…………………...21Glaciotectonic Deformation Observed Within Eastern Long Island Moraines.….22References………………………………………………………………...……….34
II. Surface Sample Bias and Clast Fabric Interpretation 42
Abstract………………………………………………………....………………….42Clast Fabric Analysis in Glacial Sediment……………………...…………………42Surface Sample Bias………………………………………….…………………..57Quantifying the bias in limiting cases……………………….……………………61Implications for field studies……………………………….………...…………..62Conclusions………………………………………………….……………………70References………………………………………………….……………….…….71Appendix A………………………………………………….……………………74
v
List of Figures
I. Push Moraine Glaciotectonics: Examples from Eastern Long Island
Figure 1. Map of Long Island………....……...…………………………...…….….2 Figure 2. (S
1, S
3) Eigenvalue Plot…......……...……...……………………...……...4
Figure 3. Last Glacial Maximum of the Laurentide Ice Sheet….......……………...7 Figure 4. Digital Elevation Model of Long Island………….......….......………..…8 Figure 5. Critical Wedge Built by Prolonged Glaciotectonic Push-From-Behind..12 Figure 6. Moraine Formation by Seasonal Glaciotectonic Push-From-Behind......13 Figure 7. Mechanical and Glaciotectonic Parameters of Push Moraines......……..16 Figure 8A. Applied Glacial Stress: Push-From-The-Rear……....…….....……...…17 Figure 8B. Applied Glacial Stress: Gravity-Spreading.………….………....…...…17 Figure 8C. Applied Glacial Stress: Compression-From-Within…….......……...….17 Figure 8D. Applied Glacial Stress: Gravity-Sliding…...........………………......….17 Figure 9A. Schematic Illustration of the Critical Taper……......……..………...….19 Figure 9B. Interpretive Cross-Section of the Western Taiwan Fold & Thrust Belt..19 Figure 10. Exposure at Ranco Quarry…….......……….....…......…………………23 Figure 11. Montage of Aerial Photos of Hither Hills….......……...……………….24 Figure 12. Hither Hills Ridge Orientation ‘Packets’…..…....……..………………25 Figure 13. Hither Hills Ridge Elevation Transect…….........……..……………….26 Figure 14A. Glaciotectonic Folds Exposed Along the Shoreline at Hither Hills........27 Figure 14B. Seismic Reflection Section of an Anticline-Cored Hill in Hither Hills..27 Figure 15. Map of the Power Line Cut in Hither Hills…..........…………………...28 Figure 16. 50 MHz Radargram of the Power Line Cut.……....………..………….29 Figure 17. Map of Rocky Point in Hither Hills….……...………………….……...31 Figure 18. 200 MHz Radargram of the Dominant Hill Structure at Rocky Point....32 Figure 19. Three Measured Sections at Rocky Point………………...……………33
II. Surface Sample Bias and Clast Fabric Interpretation
Figure 1. Map of Long Island....…..………………...…………………...…….…44 Figure 2. A South-Facing Sea Cliff at Ditch Plains………….......……………….45 Figure 3A. Rose Diagram of 150 long-axis Clast Orientations at Ditch Plains........46 Figure 3B. Equal-Area Stereonet of the same 150 long-axis Clast Orientations......46 Figure 3C. Contoured Equal-Area Stereonet of the long-axis Clast Orientations....46 Figure 4A. Equal-Area Stereonet of short-axis Clast Orientations at Ditch Plains...49 Figure 4B. Contoured Equal-Area Stereonet of the short-axis Clast Orientations....49 Figure 5A. (S
1, S
3) Eigenvalue Plot…….....………………………………...……...51
Figure 5B. Isotropy-Elongation Ternary Diagram….....………….......……………51 Figure 6A. (S
1, S
3) Eigenvalue Plot with Isotropy-Elongation Diagram labels........53
Figure 6B. Isotropy-Elongation Ternary Diagram with (S1, S
3) Eigenvalues...........53
Figure 7A. Isotropy-Elongation Ternary Diagram divided into Four Equal Areas...54
vi
Figure 7B. Four Equal Areas Skewed on a (S1, S
3) Eigenvalue Plot…....................54
Figure 8. (S1, S
3) Eigenvalue Plot of Glacigenic Fabric Domains…....………….56
Figure 9. Cross-Section of an Ellipsoidal Clast Being Exposed With Time…..…58Figure 10A. Moraine Eroding in a Time Order Series: Erosion Begins…....……….59Figure 10B. Moraine Eroding in a Time Order Series: After time……….........…….59Figure 10C. Moraine Eroding in a Time Order Series: Erosion Continues.......…….59Figure 11. (S
1, S
3) Eigenvalue Plot of Computer Generated Clast Fabrics..............63
Figure 12. (S1, S
3) Eigenvalue Plot of Generated Clast Fabrics with a.r. = 2.5..…..64
Figure 13. Observed Eigenvectors for Strong Fabrics.............................................65Figure 14. Observed Eigenvectors for Moderate Fabrics........................................66
Figure 15. (S1, S
3) Eigenvalue Plot: Domains and Generated Clast Fabrics............69
vii
List of Tables
Table 1. Eigenanalysis Results for Clast Orientations Recorded at Ditch Plains….........52Table 2. Computer Generated Random Clast Fabric Data Sets………….......………….67
viii
Acknowledgments
This thesis would not be possible without the support of Dan Davis, my thesiscommittee, and the Department of Geosciences.
I could not thank my advisor, Dan Davis often enough for everything he has donefor me at Stony Brook. Besides making unintuitive concepts understandable, Dan guidedme with great care through the biggest transitions of my life. My background as a visualartist never impaired Dan’s view of my capabilities and he worked with me despite mydeficiencies in mathematical and scientific concepts. His lab and ideas have always beenfully open and shared with me and for this I am truly grateful. When I first approached Danfor some help on understanding the orientation tensor and its relationship to pebbleorientations, I remember Dan saying with a smile that he previously thought so little ofpebbles that he would, without much thought, gently toss them over his shoulder. Sincethat day Dan ‘Cosine is My Friend’ has continually enriched my life as well as my math andgeophysics knowledge.
I appreciate the time that Dan Davis, Bill Holt, and Troy Rasbury spent as mycommittee members reviewing my thesis. Their scientific insight and discussions werequite helpful in preparation for my defense.
I am grateful to Troy Rasbury and Bill Holt for helping me begin my graduate careerin geosciences. They have been great examples to me because of their success as scientists.Neither Troy nor Bill can stop questioning anything about geosciences or science in general.Plus they throw fun parties.
Bill Meyers allowed me to pursue undergraduate research in glacial diamict andclast fabric analysis when he knew that I lacked formal training in sedimentology. To thisday I often wonder why Bill gave me this research opportunity. I am truly indebted to him.Bill taught me how ask scientific questions with intelligence and purpose. I may neverforget some of his responses to my written words (e.g., huh!). Bill always mentioned that“good science requires a good write up”. I hope this thesis lives up to Bill’s expectations.
I give special thanks to Don Lindsley for supporting me in his experimental petrologylab during my second summer at Stony Brook where he continually encouraged me to finda way to conduct research in geosciences.
I thank Gil Hanson for sharing his general knowledge of glaciology and Long Islandgeology with me. I am also grateful to the Long Island Geologists because this organization(ran by Gil) provided me opportunities to submit papers and give oral presentation onglaciotectonics and clast orientation.
Lianxing Wen is thanked for graciously supplying me with desk, workstation, andinspiration to carry out this work.
ix
Saad Haq ‘Do you have time for a game (of wiffleball)’ has been a superb lab mateand friend. He has been generous with his advice and time. Saad has literally saved mefrom embarrassing technical disasters as I prepared graduate circus talks and otherpresentations.
I have been very fortunate to be surrounded by great characters and maturinggeoscience graduate students. In particular Lucy Flesch and Andy Winslow have alwaysmade me laugh even if I was not in the mood. I thank Wen-Che Yu, Brian Hahn, and YiWang for bringing life and culture into our geophysics group.
My friends Rob Finkenthal and Ed Keegan each deserve a hearty thank you sincethese guys always stuck by me. They have supported me through thick and thin ever sinceI was a young boy. Gail Schaefer has lifted my confidence on many occasions and I amindebted to her. Yi-Ju Chen deserves mention for constantly encouraging me.
Of course my parents George and Marcy, brothers David and Louis, sister Diane,brother-in-law Rich, grandmother Sue, grandmother Adele, great aunts Olga and Charolette,aunt Rosy, uncle Lawernce, and ‘great uncle’ Paul all merit special thanks for their belief inme and for giving me unconditional support throughout my life.
x
I. Push Moraine Glaciotectonics: Examples from Eastern Long Island
Introduction
Late Pleistocene Long Island landforms are a product of glacial sediment availabilityand glaciotectonic deformation when sea level was substantially lower (by 10’s of meters)than at present. Data obtained through geologic and geophysical studies, developed andrefined into landform evolution models, can help to elucidate the evolutionary path of LongIsland glacial landforms, and Pleistocene Long Island landforms can be compared withevolutionary path models for each distinct Quaternary glacial landform type. There are fewstudies of glacial deposits and structural features that present Long Island landform evolutionmodels, but it is clear that there is substantial local diversity in glacial settings andglaciotectonic landforms (e.g., Bernard, 1998; Meyers et al, 1998) (Fig. 1).
I have concentrated my effort in the interpretation of eastern Long Island glacigenicsediment and structure, as a first step in amassing the data necessary to develop landformevolution models. We consider the landforms at Hither Hills (Fig. 1) as a late Pleistoceneice marginal push moraine based on the results of interpreted (migrated and topographycorrected) ground penetrating radar radargrams, earlier topographic and seismic surveyanalyses (e.g., Bernard, 1998), correlation of ridges (spacing and amplitude) andglaciotectonic structural styles to modern push moraine analogs, and from detailed fieldanalysis of lateral variation in stratigraphy. There is also the evidence of syntectonicdeposition found in sediments exposed along the shoreline in sea cliffs. Future work in thissetting will, it is hoped, measure glaciotectonically-shortened features and report on theirchange in rate of contraction. The fieldwork effort that figures most prominently in thisthesis included the measurement macro-clast orientations within the stratified diamict atDitch Plains, Long Island (Fig. 1). Clast fabric analysis of the measured clasts indicates apreferred subhorizontal, slightly west of north, long-axis orientation consistent with subglacialshear due to ice advance from that direction.
Existence of potential sampling biases in clast orientation measurements collectedfrom outcrop surfaces led me to design numerical and analytic models testing the degree towhich clast axis ratios, outcrop surface orientations, and clast residence times in an erodingoutcrop influence resulting clast fabric analyses. The models incorporate uniform erosionof an outcrop from the outcrop exposure surface normal direction leading to preferentiallyover-sampled and under-sampled clast orientations. Favorably oriented clasts remainembedded in an outcrop as it erodes, producing over-sampled orientations. Meanwhile,other clasts roll out of the outcrop after a relatively short time and can not be measuredproducing under-sampled orientations, often dramatically distorting the results of clast fabricanalyses. Surface sampling bias modeling predicts that weakly anisotropic axial clast fabrics
1
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measured and quantified from surface sampled clast orientations produce large observationerrors in eigenvalue strength and eigenvector direction compared with clast fabrics calculatedfrom the entire volume of clast orientations. Volumetric clast fabric analysis would includeclast orientations within the outcrop, instead of being limited to the clasts exposed at anoutcrop surface. A highly anisotropic clast fabric represents the statistical distribution ofclasts with a very strong preference for axial orientation in close alignment with one dominantdirection.
I find that the error in preferred fabric direction for sediments with a moderate orstrong fabric due to surface observation bias is small, indeed smaller than the error anticipatedfor field measured clast orientations. Unfortunately, my results also show that clast fabriceigenanalysis of surface clasts can lead to uncertainty in eigenvalues that is much largerthan the uncertainty in eigenvalue distributions associated with other sources in fieldmeasurement of clast orientations, and compares to typical differences between sedimentsfrom different glacial environments.
Application of eigenanalysis to quantifying glacial sediment fabrics and other geophyicalproblems (e.g., Watson, 1966; Anderson and Stephens, 1972; Mark, 1973; Woodcock 1977; Woodcockand Naylor, 1983) developed into a diagnostic tool intended to deduce the genetic origin of glacialdeposits (e.g., Mark, 1974, Lawson 1979; Dowdeswell et al., 1985; Rappol, 1985; Dowdeswell andSharp, 1986; Benn, 1994, Ham and Mickelson, 1994; Hicock et al., 1996; Larsen et al., 1999; Kjaeret al., 2001) (Fig. 2) and to infer relative strain within glacigenic sediments (e.g. Hicock, 1992; Hart,1994; Benn, 1995; Benn and Evans, 1996; Rijsdijk, 2001). Large-scale glacial landforms areinterpreted by a variety of means which often includes the quantitative clast fabric analyses ofglacial sediments which are then related to emplacement by sedimentary (e.g., Johnson and Gillam,1995; Mattsson, 1997; Karlstrom, 2000; Munro-Stasiuk, 2000; Ward 2000; Hambrey et al., 2001;Kjaer and Krüger, 2001) or deformational processes (e.g., Hicock and Dreimanis, 1992; Hambreyand Huddart, 1995; Zelcs and Dreimanis, 1996; Hart, 1997, 1998; Hart and Smith, 1997; Bennett etal, 1999a; Dreimanis, 1999; Hicock and Lian, 1999; Johnson and Hansel, 1999; Blake, 2000; Evans,2000; Cofaigh and Evans, 2001; Hart and Rose, 2001; Henriksen et al., 2001). Due to the growingimportance of quantitative clast fabric analysis in landform and ice sheet reconstructions, fundamentalquestions continue to be raised about the statistical reliability and accuracy of the eigenanalysismethod (e.g., Ringrose and Benn, 1997; Kjaer and Krüger, 1998; Krüger and Kjaer, 1999; Bennettet al, 1999b; Millar and Nelson, 2001a, 2001b; Benn and Ringrose, 2001). I find that the surfacesampling bias in clast fabric analysis does not affect inferences regarding ice-flow or shear directionfor strongly oriented fabrics, but it severely limits the usefulness of the technique as an indicator ofglacial sediment genesis. Future research aimed at producing Quaternary glacial landform evolutionmodels can still integrate directional data resulting from the eigenanalysis of clast orientations, butshould do so with caution.
Detailed geologic and geophysical analyses of glacial landforms place vital constraints onspatial, temporal and climatic reconstructions of Pleistocene ice sheets and their settings. Recentstudies of contemporary glaciers worldwide has led to a more accurate understanding of thekinematics, physics, and mechanics involved in the evolution of glacial landforms by modern icesheetenvironments. Major advances in classifying sedimentary, structural, and geomorphologicalvariations at active glaciers corroborate the notion that landform assemblages are shaped by a varietyof complexly related depositional and structural processes that are dependent on the conditions thatexist during landform generation (e.g., Lawson, 1979; Bluemle and Clayton, 1984; Boulton, 1986;Hart and Boulton, 1991; Benn and Evans, 1996; Bennett et al., 2000; Boulton et al, 2001a; Bennett,2001; Houmark-Nielsen et al., 2001). The incorporation into models of what has been learned
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about landform evolution at an active glacier is a primary way to delineate processes responsible forlandform generation in the Pleistocene epoch (e.g., Hambrey and Huddart, 1995; Boulton et al.,1996a, 1996b; Hart and Watts, 1997; Dreimanis, 1999; Evans, 2000; Bennett, 2001; Boulton et al,2001b; Hart and Rose, 2001; Russell et al, 2001).
A complementary approach to field identification of the kinematics and mechanicalconditions responsible glacial landform evolution should include laboratory scale analogmodeling comparable to that commonly used in structural geology and tectonics (e.g., Daviset al., 1983; Dahlen et al., 1984; Davis and Engelder, 1985), and finite difference and/orelement modeling (e.g., Hsui et al, 1990; Wang, 1996) to compile a set of modeling predictionsfor strain patterns as a function of the governing conditions, including the shape and velocityof the advancing mass and the yield/flow criteria and thickness of sediments. In this way alink between the recent classifications of modern glacial landform assemblages and thecorresponding predicted deformation patterns should then make it possible for fieldobservations of structures and fabrics to be used more effectively to draw conclusions aboutthe geologic and climatic conditions at the time of the deformation.
Deformation imprinted on glacial landforms covers a broad range of length scales,from tiny grain size particles (e.g., van Der Meer, 1993; van Der Meer, 1997b; Menzies etal, 1997) to thousands of square kilometers (e.g., Zelcs and Dreimanis, 1997; Hart andSmith, 1997; Boulton et al., 2001b). For example, strain can be observed in the microscopicfracturing and folding of glacial sediments (e.g., Lachniet et al., 1999; Van der Wateren,1999), in cm-to-meter scale glaciotectonic structures within the mass of deformed materials(e.g., Croot, 1987; Hart, 1990; Benn, 1995; Aber and Ruszczynska-Szenajch, 1997; Kleinand Davis, 1999; Schlücther et al., 1999; Boulton et al., 2001a) as well in the macroscopicform of entire landform assemblages (e.g., Bennett et al., 1999b; Evans et al., 1999; Bennand Clapperton, 2000; Bennett et al., 2000).
Landforms resulting from Pleistocene glaciations continue to be examined bytraditional geological analyses (i.e., sedimentary, stratigraphic, structural, aerial photographic,topographic, clast fabric, and micro-morphologic) and by geophysical field surveyingmethods (i.e., seismic refraction and reflection, ground penetrating radar, satellite imagery,downhole geophysical logging, magnetometer, and resistivity-meter measurements), whichyield data that can be incorporated into landform models. Comparisons made betweenrelative strain accumulation and strain pattern in microstructures, and macroscopicdeformation features within the same glacial landform can aid in constraining a particularlandform evolution model. Pleistocene glacial landform models can be compared withmodels of landform evolution by active glaciers with similar glaciological setting (e.g., Vander Wateren, 1985; Evans et al, 1999; Bennett, 2001, Khatwa and Tulaczyk, 2001; Piotrowskiet al., 2001) and with appropriate numerical and analog models in order to synthesize acollection of consistent and reliable Quaternary glacial landform models that will aid in thereconstruction of former ice sheets.
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Setting
North America was repeatedly covered with continental glaciers in the PleistoceneEpoch. During the last glacial maximum, one of these massive ice sheets had an extendedsouthward advance from Artic North America to the southern shore of modern Long Island(e.g., Dyke, 2002) (Fig. 3). Other earlier Pleistocene ice sheets may also have extended asfar south as Long Island. Ice sheet lobes distributed glacigenic sediments and createdlandforms on Long Island before finally retreating. The erosion of bedrock in the LongIsland Sound basin, New York State, and in the southern New England region, during icesheet advances, produced and transported source materials ranging widely in particle sizeand lithology that ultimately became surface and near surface sediment deposits on theLong Island platform (e.g., Lewis and Stone, 1991) (Fig. 4).
The relatively warm climate of the Atlantic coastal plain, which included thelandlocked Long Island platform in the late Pleistocene, slowed the southward advance ofthe spreading ice sheet. The warmer coastal climate gradually weakened the basal couplingof the warm-based ice sheet and melted large volumes of ice. The local climate reduced thetotal glaciotectonic stress needed to overcome the basal shearing resistance, as the glaciercontinued its push forward onto the Long Island platform. The temperature became warmenough for the ablation rate at the ice margin to be nearly equal in magnitude to the iceadvance rate causing the ice sheets to nearly stall. Once the ablation rate overcame theforward advance rate of the glacier, the ice sheet melted back to the north off the LongIsland platform. Global ice melting allowed the sea level to rise slowly to its present daylevel. As a consequence, Long Island glacial sediments were exposed to direct ice contactat the glacier margin only for a limited amount of time (perhaps centuries), which stronglyinfluenced the strain histories, deformation styles, and geometric shapes of Long Islandglacial landforms. Geological dating of the Pleistocene glaciation or glaciations is not wellestablished for Long Island but deposition and deformation of the surface deposits aregenetically related to one or possibly more than one glacial cycle (e.g., Lewis and Stone,1991).
Small to moderate size temperature fluctuations at the ice sheet terminus contributedto lateral variations in stratigraphy and to complexities in glaciotectonic structure observedin Long Island moraine environments. Although Long Island was landlocked in the latePleistocene, it was still near the relatively warm waters of the Atlantic Ocean. The temperaturecontrast between the glacial ice and the ocean water caused a thermal gradient that locallyaltered ice flow and glaciation dynamics. The ice sheet terminus was influenced by themagnitude of the local temperature gradient so that prolonged periods of extreme cold andglacier advance must have been difficult to maintain and probably occurred infrequently.Therefore, the size, shape, location, and distribution of glacial lobes as well as the amountof sediment, ice, and water that these lobes carried, deposited, and deformed was primarilyfunction of local temperature variation with respect to time. The geomorphology of LongIsland was, in part, shaped by sudden surging of ice sheet advance during a period of overallglacial retreat and by relative motions between glacial lobes during a relatively short timespan. Such relatively sudden changes in ice flow dynamics probably occurred even onannual-to-decade-to-century time scales and were in direct response to change in magnitudeof the local temperature gradient.
6
Figure 3. Last Glacial Maximum of the Laurentide Ice Sheet as defined by Dykes et al. (2002). Ice sheet margins are shaded white. Ice surface contours are based predominately on direct mapping of elevations along the Last Glacial Maximum ice margin and topographic high points that were overridden by ice.
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The elevated Rokonkoma and Harbor Hill moraines are the dominant topographicfeatures on Long Island (Fig. 4). The glacial margins of the Laurentide Wisconsinan icesheet of the late Pleistocene generated the Ronkonkoma moraine of central Long Island(Fig. 1). This thin moraine trends roughly WSW-ENE for nearly the entire length of theisland as a succession of numerous interconnected kilometer-scale lobate shaped ridges.The easternmost portion of the moraine meets the Atlantic Ocean, at Montauk Point on thesouth fork of Long Island. As the Laurentide Wisconsinan ice sheet retreated from LongIsland it likely stalled temporarily to create the WSW-ENE trending Harbor Hill moraine.This moraine is another narrow elevated landform made up of many interlinked lobateshaped ridges, on the scale of a few kilometers, that traverse the entire length of LongIsland, mainly along the northern shore, reaching Orient Point on the north fork. The morainesystems each contain segments formed by the Hudson, Connecticut and Connecticut-RhodeIsland lobes. Besides the two distinct moraines, the glaciers left Long Island with outwashplains, glaciotectonic hill-hole pairs, tunnel valleys, and deltaic sequences, each with distinctstructural and depositional features. Glaciotectonically deformed strata, in exposed sectionsof Long Island moraines, contain contracted strata shortened by folding and faulting processes(e.g., Merrill, 1986; Nieter et al., 1975; Fullerton et al., 1992). Little about the subsurfacegeometry of Long Island glaciotectonized folds or faults is known, so the mechanism oftheir emplacement or formation remains unclear. Only modest effort has thus far gone intoglaciotectonic analog or numerical models or to comparing them with contemporary andPleistocene landforms. This research is needed in order to differentiate ice sheet dynamicinfluences on the generation of Long Island landforms.
Rationale
Long Island glacial sediments are well suited for geological and geophysical fieldinvestigations. In addition to being geologically and economically important, the spatialarrangement of Long Island glacial sediments and associated glaciotectonic structures playsan important role in controlling hydrologic fluid flow paths. Improved analysis of landformgeomorphology and near-surface hydrology, through the investigation of three-dimensionalheterogeneities in glacial strata, will likely influence the next generation of Long Islandgroundwater flow models. Existing groundwater flow models do not adequately accountfor lateral variability in glacial sediments and neglect the inclusion of identified regions ofglaciotectonic folding or thrusting in Long Island deposits. Glaciotectonically altered strataand abrupt changes in the sedimentology of glacial strata often redirect groundwater flow(e.g. Sminchak, 1996; Beres et al., 1999; Boyce and Eyles, 2000; Gerber et al., 2001, Regliet al., 2002). Incorporating quantitative structural and sedimentological anisotropies intofuture Long Island groundwater and contaminant flow models is absolutely necessary giventhe large population (approximately 2.7 million) who directly depend on the groundwaterpumped from fragile aquifer systems as their sole water supply.
9
Geologic studies characterizing sedimentary and structural field relationships inglacial diamict and associated sediments are fundamental in establishing glacigenic facies(e.g., Krüger and Kjaer, 1999). Measured sections at outcrops serve as ‘ground truth’ forLong Island glacigenic facies which represent the depositional and emplacement associationsfor the wide spectrum of deformed and undeformed sediment, strata, and structure that arefound within the deposits of glacigenic sediments (e.g., Meyers, 1998). Since not all ofthese glacigenic facies are documented, and others have not been correlated, Long Islandglacial strata, diamict and associated sediments, as well as glaciotectonic features shouldcontinue to be sedimentologically, structurally, and stratigraphically characterized at fieldsites. All known glacigenic facies ought to be integrated into sedimentological, glaciotectonic,and landform evolution models.
Application of geophysical instruments, imaging unconsolidated glacial sedimentson Long Island, has proven to be extremely valuable in shallow surface surveying becausethese tools provide representation of otherwise inaccessible deposits or structures (e.g.,Bernard, 1998; Davis et al., 2000). Geophysical survey methods such as seismic reflectionand refraction, ground penetrating radar (GPR), and resistivity-measurement can estimateand differentiate physical property variations in glacial sediments at a wide range of depthsand resolutions. This is especially useful in terrain that is not well exposed, providing twoor three-dimensional images of the near subsurface (e.g., Hansen et al., 1997; Ramage et al,1998; Beres et al., 1999; Penttinen et al., 1999; Gerber et al., 2001; Overgaard and Jakobsen,2001; Williams et al., 2001). Combining geophysical investigations and geologic fieldworkstudies, particularly at outcrop exposures or at excavated sites, strengthens correlationbetween ground penetrating radar, resistivity, seismic, and glacial facies (e.g., Harris et al.,1997; Davis et al., 2000; Eden and Eyles, 2001; Ékes and Hickin, 2001; Salem, 2001; Regliet al., 2002).
Glaciotectonic Deformation on Pleistocene Long Island
On Long Island, the Harbor Hill moraine ridge topography often exceeds 60 m andfrequently the Ronkonkoma moraine ridges top 90 m, attaining maximum elevation at roughly128 m above sea level (Fig. 4). Glacial erosion and transport of Long Island Sound basinbedrock material by ice sheets were the chief sedimentological processes contributing tothe deposition of the enormous supply of sediments that evolved into glaciotectonicallythickened and deformed Long Island moraine landforms. The emplacement mechanismsand landform evolution paths of moraines on Long Island, though still poorly understood,included subglacial and proglacial deposition and deformation processes. The range ofglaciotectonic structures, evident at a variety of scales, provides insight into the developmentof Long Island landforms, as well as other Quaternary landforms built by similar warmbased, weakly coupled glacier marginal systems. Therefore, documenting the distributionsand complexities of glaciotectonic structures within the landforms of Long Island is necessary.
10
At present, I believe that two principal glaciotectonic mechanisms are responsiblefor generating much of the deformed proglacial structures on Pleistocene Long Island. Thesestructures were generated glaciotectonically by either a prolonged push-from-behind, aseasonal push-from-behind, or a mix of these two mechanisms. This is consistent withmany of the commonly observed deformation features found in Long Island proglacialsediments (e.g., Meyers, 1998; Bernard, 1998; Klein and Davis; 1999). Prolongedglaciotectonic push-from-behind thin-skinned deformation shortens glacial strata byproducing fold-and-thrust structures that must be accommodated by a décollement, a weakaccommodating layer at depth in which there develops a shear zone, typically with a strongshear-related fabric (Fig. 5). The glacial sediments and structures involved in prolongedglaciotectonic push-from-behind often contract into the form of a critical taper (e.g.,Schlüchter et al., 1999; Williams et al., 2001) a wedge shaped packet of material in crosssection containing the deformed structures. This has been documented on a larger scale inthin-skinned small-scale fold-and-thrust belts (e.g., Davis et al., 1983; Dahlen et al., 1984).Seasonal glaciotectonic push-from-behind thin-skinned contraction is not capable of foldingand thrusting as much sediment since the ice, which is doing the pushing, melts back andretreats before the deforming sediment can form a critical taper (Fig. 6). The spatial andtemporal patterns associated with push-from-behind glaciotectonic deformation events thatcontracted Long Island glacial sediments are still uncertain. This is particularly true interms of discriminating deformation patterns involving newer glaciotectonic structuresoverriding previously deformed structures regardless of how any of the structures wereglaciotectonically emplaced.
The descriptive terms used to characterize proglacial glaciotectonic deformationare subdivided by size differentiations that are often arbitrarily defined (e.g., Aber et al.,1989; Hambrey and Huddart, 1995; Benn and Evans, 1998). Bennett (2001), clarifies someof the confusion in taxonomy by using the term ‘push moraine’ to define the product ofconstruction by the deformation of ice, sediment, and/or rock to produce a ridge, or ridges,oblique or transverse to the direction of ice flow at, in front of, or beneath and ice margin.The formation of a moraine by advance of the glacier margin is thus what defines pushmoraines, not whether or not the moraines were formed by seasonal or prolongedglaciotectonic push. In push moraine systems sediment displacements and dislocations canrange from a few meters to several kilometers horizontally and up to 200 m verticallyproducing larger and thicker moraine sizes with more distinctive internal tectonic style asdisplacement of pushed sediment progresses (e.g., Boulton, 1986; Hart and Boulton, 1991;Lehmann, 1993; Boulton and Caban, 1995, Boulton et al., 1999). Included in the definitionsof push moraine by Bennett (2001) are thrust moraines, thrust-block moraines, compositeridges and hill-hole pairs as long as they can be clearly linked to have occurred at, or closeto an ice margin. My thesis adopts the push moraine definition of Bennett (2001) andrecognizes that glaciotectonic push-from-behind, whether seasonal or prolonged, terrestrialor marine, generated by gravity spreading or glaciodynamic pushing forces, was the pushingmechanism which drove the deformation producing push moraine ridges, regardless ofgenerated ridge amplitude, ridge spacing, or the state (whether lithified or frozen) of thepushed sediment.
Small push moraines built by annual or seasonal push-from-behind glaciotectonicmechanisms at contemporary glacier margins develop into annual or seasonal push moraines
11
43
21
5
43
21
Fig
ure
5. I
llus
trat
ion
in ti
me
sequ
ence
of
the
grow
th o
f a
crit
ical
wed
ge in
a p
rolo
nged
gla
ciot
ecto
nic
push
-fro
m-b
ehin
d se
ttin
g.
The
you
nges
t def
orm
atio
n is
con
cent
rate
d to
war
d th
e di
stal
end
of
the
wed
ge.
Not
e th
at th
e an
ticl
ines
are
typi
call
y co
red
by
imbr
icat
e th
rust
fau
lts.
12
54
32
43
21
43
21 1
Fig
ure
6.
Thr
ee s
chem
atic
cro
ss-s
ectio
ns (
in t
ime
sequ
ence
) ill
ustr
atin
g on
e po
ssib
le m
odel
for
the
for
mat
ion
of a
sea
sona
l gl
acio
tect
onic
pus
h-fr
om-b
ehin
d m
orai
ne.
Sed
imen
ts c
an v
ary
grea
tly o
ver
shor
t di
stan
ces.
N
ote
how
sed
imen
ts f
rom
an
impo
unde
d la
ke b
etw
een
the
glac
ier
and
prev
ious
ly f
orm
ed r
idge
s ar
e em
plac
ed i
n th
e ne
xt-f
orm
ed r
idge
. I
n th
is m
odel
, ri
dges
yo
ung
tow
ard
the
ice,
in th
e op
posi
te d
irec
tion
than
in a
cri
tical
wed
ge.
13
that usually form ridges ≤ 5 m in height (e.g., Boulton, 1986; Bennett, 2001). Large pushmoraines generated by prolonged or large-scale glaciotectonic push-from-behind at glaciermargins produce ridges with heights ≥ 5 m in a sustained glacial advance often due to achange in glacier mass balance (e.g., Boulton, 1986; Bennett, 2001; Russell et al, 2001).The main distinction is not the size of the moraine, since the 5 m height is an arbitrarycutoff, but whether the moraine was produced by seasonal or annual readvance or by a moresustained advance at the glacier margin. When significant deformation has been transmittedhorizontally beyond the glacier margin, multi-crested push moraines are generated withdeformation style usually involving, multiple folds, fans of listric thrusts, fans of imbricatethrusts, or superimposed sub-horizontal nappes produced by overthrusting (e.g., Bennett,2001).
The glaciotectonic process responsible for the initiation, excavation and elevationof proglacial materials is similar for either size push moraine: the main difference betweenthe two can characterized by the amount of deformed outwash fan sediment present at theglacier margin (e.g., Boutlon, 1986; Benn and Evans, 1998; Bennett, 2001; Russell et al.,2001). Small push moraines grow by a glaciotectonic push and deformation of the proximaloutwash fan slopes of the glacier margin. Large push moraines, on the other hand, grow notonly by glaciotectonically shoving the proximal outwash fan slopes, but also by pushingmuch or the entire outwash fan. Asymmetric ridges that have steep distal and shallowproximal flanks tend to be formed in small push moraines (e.g., Sharp, 1984). Additionally,the moraine ridges may push their own pretectonic and syntectonic outwash sediment aswell as override subglacially lain tills, if there are any, during a readvance of the glaciermargin (e.g., Boulton, 1986; Bennett, 2001). Ridge amplitude is predominately controlledby sedimentological factors such as sediment character and availability as well as the durationof a glacial advance. Small glacial advances commonly produce 1 to 2 m ridge heightamplitudes such as those formed by the seasonal readvances of the Breidamerkurjökullglacier ice margins, in Iceland between 1965 and 1981 (e.g., Boulton, 1986). Small pushmoraines are associated with the formation flute and show variation in pattern of sedimentaryactivity along the ice margin (e.g., Boulton, 1986; van der Meer, 1997a; Bennett, 2001). Inthe glaciotectonic development of small push moraines, the ice advance is often annual andproceeds much like an oversized bulldozer blade plowing through loose, water saturatedsediments that deform into a series of ridges. When a glacier ablates, water, ice and debrisare sloughed off the snout to build up outwash fans and glaciofluvial streams. Duringreadvances the glacier pushes and partially overrides the fans. The forward moving glacieroversteepens the distal slopes of the fans which receive a new layer of debris when theglacier retreats. Sediment that was overridden in an advance is incorporated into thesubglacial environment where it is deformed into a thickening wedge of till beneath themargin. Both large and small push moraine systems preserve of at least 25% of theglaciotectonic structures involved in the push-from-behind deformation process and thesyntectonic plus pretectonic proglacial materials make up over 25% of a moraine systemunit area (e.g., Benn and Evans, 1998).
The extent of proglacial deformation resulting from push-from-behindglaciotectonics is highly variable, but in general is a function of time, climate, and glaciermargin environment. For example, if an advancing glacial margin begins to deform proximalmaterials but stalls after a short period (e.g. one season) then that glaciotectonic push will
14
not have deformed much material. On the other hand, if instead of stalling, the glacialmargin continued for a prolonged advance (e.g. multiple seasons) then much more proglacialmaterial will be glaciotectonically pushed and deformed. Some important parametersaffecting the deformation front and the growth of a critical taper include seasonal temperatureand moisture variation, porewater pressure gradient, coupling of basal ice with the subglacialbed, glacial lobe height and its basal area, and the rate of glacial advance and ablation.Other factors influencing the magnitude of the push-from-behind deformation are sedimentsize, state, and type, local topographic relief, friction on the décollement, and availability ofstanding water and outwash materials. Proglacial environment and climate conditions directlyaffecting physical characteristics of the glacial margin including local pore fluid pressure,evolution of drainage, sediment and glacier bed state, aspect ratio of foreland wedge, forelandrheology and strength, basal shear traction, depth and slope of the décollement (e.g., Bluemleand Clayton, 1984; van der Wateren, 1985, 1986; Boulton, 1986; Hart and Boulton, 1991;Boulton and Caban, 1995, Etzelmüller et al., 1996; Dell’Isola and Hunter, 1998; Boulton etal, 1999; Schlüchter et al., 1999; Bennett et al, 2000, 2001, Boulton et al, 2001a). (Fig. 7)
The growth of push moraines relies on the large-scale displacement of proglacialmaterials within shear zones due to stresses imposed by the gravity spreading of a glacier(Fig. 8A). The gravity spreading model demonstrates that the total glaciotectonic stressneeded to push glacial material from behind, permitting proglacial sediment failure andglaciotectonic thrusting, is obtained by the translation of compressive stress due to the weightof a spreading ice mass (e.g., van der Wateren, 1985; Aber et al., 1989; Benn and Evans,1998; Bennett, 2001). Important components of the total glaciotectonic stress field includethe glaciodynamic stress (basal shear stress) and the horizontal cumulative compressivestress transferred from the normal stress (glaciostatic stress) generated by the static weightof the ice over a given area. Failure can take place on a plane when the total glaciotectonicstress exceeds or equals the shear resistance. Push-from-the-rear, gravity sliding, andcompression-from-within models represent other mechanical ways to produce push moraines(e.g., Bennett, 2001) (Fig. 8B,C,D). Glaciotectonically deforming sediment blocks are foldedand thrust into push moraine systems by the gravity sliding of surging glacial ice movingdown a slope by the driving force of its ownweight. The laterally compressive push-from-the-rear mechanism directly shoves, foldsand thrusts sediment wedges by the forward motion of glacial ice into the foreland.Compression from within the terminal zone of the glacier occurs if there is deceleration ofice flow, strongly coupled subglacial and proglacial zones which behave as a single unitthat is deformed by listric faults, and a décollement which lies below both the glacier and itsforeland (e.g., Hart, 1990; Hambrey and Huddart, 1995; Bennett, 2001). Development ofglaciotectonic clast fabrics within internal structures of push moraines has the possibility ofshedding light on wedge propagation and paleo-ice flow directions as well as potentiallydistinguishing amongst glacial stress fields and seasonal or prolonged push deformationstyles (e.g., Sharp, 1984). The main section of this thesis concentrates on the applicabilityof clast fabric analysis as an interpretative tool in glacial settings based on clast orientationsmeasured from surfaces of outrcrop exposures.
In the glaciotectonic evolution of large push moraines, the recently deformedsediments are in contact with and are actively shoving the more distal sediments as thesystem advances forward. The geomorphology of a moraine ridge generally reflects the
15
Sche
mat
icm
odel
show
ing
how
som
epu
shm
orai
nes
may
rela
teto
sele
cted
vari
able
sus
edto
defi
nea
broa
dm
atri
x.
)
Fig
ure
7. I
mpo
rtan
t mec
hani
cal a
nd g
laci
otec
toni
c pa
ram
eter
s in
volv
ed in
the
stru
ctur
al d
evel
opm
ent o
f pu
sh m
orai
nes.
Sch
emat
ic m
odel
aft
er B
enne
tt (2
001)
.
16
Fig.
18.M
odel
sof
appl
ied
glac
ials
tres
s.F
igur
e 8.
M
odel
s of
pus
h m
orai
ne s
truc
tura
l ev
olut
ion
and
mor
phol
ogy
due
to t
he
effe
ct o
f ap
plie
d gl
acia
l st
ress
as
show
n by
Ben
nett
(200
1).
A.
Pus
h-fr
om-t
he-r
ear.
B
. G
ravi
ty-s
prea
ding
. C
. C
ompr
essi
on-f
rom
-with
in.
D.
Gra
vity
-slid
ing.
A DCB
17
shape of the thrusting glacier margin, and individual ridge crests correspond to the crests ofinternal folds (e.g., Boulton, 1986; Aber et al., 1989; Benn and Evans, 1998). One push-from-behind, glaciotectonic model for building a push moraine is the glacial analog of thethin-skinned wedge model of tectonic deformation, which produces a critical taper inmountain belts (e.g., Davis et al., 1983; Dahlen et al., 1984) (Fig. 9A,B). Glaciotectonicpush-from-behind compression that has evolved into a small-scale thin-skinned fold-and-thrust belt often exhibits multiple thrust sequences with piggyback structures being themost common of the glaciotectonic structure produced (e.g., Van der Wateren, 1985;Schlüchter et al., 1999).
In some push moraine systems the largest ridge is the most proximal to the glaciermargin and was produced first. As the ice continued to push, a new ridge formed in front ofthe previously formed ridge so that ridges are youngest in the forward direction of theadvancing ice. Newer ridges are more distal and are somewhat less elevated than thoseridges formed earlier, so ridge amplitudes decay with distance away from the glacial margin(e.g., Croot, 1987; Hambrey and Huddart, 1995; Boulton, et al., 1999). The cross-sectionaltaper of a growing wedge-shaped mass of overthrust material is dependent upon the cohesivestrength of the deforming material at the time of deformation, its thickness, and the strengthof its coupling to the base (e.g., Davis et al., 1983; Schlüchter et al., 1999; Williams et al,2001) (Fig. 5). If a glacier margin is pinned at the margin front but continues to advancefrom the rear, the shortening within the mass of sediments increases causing folds to bepushed or pinched-out. The most internally shortened structures and the shortest wavelengthsbetween ridges are closest to the glacial margin since the cumulative shortening of thepushed sediments is the furthest distance from the distal extremity of the push moraine(e.g., Boulton et al., 1999). If on Long Island the glacier moved in a sustained advance byprolonged push then one should observe diminishing ridge heights in cross-sectional shape(a critical taper) as one moves away from the suspected glacial margin. One would alsofind a décollement, a gradient in strain magnitude (more intense in the ‘hinterland’ to thenorth), a gradation in syntectonic deposition (finer, more distal facies to the south), littlelateral variation in glacial stratigraphy, and a general northward sweeping in the dips ofsediments and thrust faults.
Proglacial glaciotectonic deformation through seasonal or annual meters-scale glaciermargin surges or thrusts generates push moraines during a period of overall ice sheet retreat.The timing of the advance and the deformation it causes can be seasonal-to-annual, ordecadal, but if the precise time intervals of the surges are unknown then the glaciotectonicprocess is simply referred to as annual or seasonal push but probably ought to be calledrepeated push. Repeated (seasonal or annual) push glaciotectonic deformation is suspectedin the growth of push moraine structures at Hither Hills, Long Island and has likely contributedto the creation of other portions of Long Island push moraines (e.g., Klein and Davis, 1999).Local climate regime and temperature fluctuation at the glacier margin, over relatively shortperiods, can promote repeated push glaciotectonic deformation which influences pushmoraine ridge geometries and internal structures, as well as the hydrogeology of the pushmoraine foreland.
Unlike prolonged glaciotectonic push-from-behind where contractional deformationis sustained over many seasons without retreat, repeated glaciotectonic push-from-behindinvolves push from the rear during almost every ice advance season (typically, winter)
18
A B
Fig
ure
9. A
. Sch
emat
ic il
lust
ratio
n of
for
ce b
alan
ce c
alcu
latio
n us
ed in
der
ivin
g th
e cr
itica
l tap
er a
nd th
e or
ient
atio
ns o
f th
e pr
inci
pal
stre
ss a
xes
thro
ugho
ut a
wed
ge o
f m
ater
ial e
very
whe
re o
n th
e ve
rge
of f
ailu
re a
s sh
own
by D
avis
et a
l., (
1983
). A
n el
emen
t of
wed
ge
is s
ubje
ct to
str
esse
s du
e to
bod
y fo
rces
fro
m th
e si
de a
nd a
t its
bas
e, a
s w
ell a
s gr
avia
tiona
l str
esse
s .
B. I
nter
pret
ive
cros
s-se
ctio
n th
roug
h th
e fo
othi
lls o
f th
e w
este
rn T
aiw
an f
old
& th
rust
bel
t (e.
g., D
avis
et a
l., 1
983)
. N
ote
the
over
all w
edge
tape
r an
d st
acki
ng o
f th
rust
she
ets
over
the
déco
llem
ent.
19
glaciotectonically deforming the glacial margin. The seasonal or annual ice marginal advanceis immediately followed by an ice ablation season (typically, summer) where the glacialmargin retreats so this coupled with ice marginal advance and ablation form a repeatingpattern over multiple seasons (or years) deforming and elevating substantial push morainetopography. The seasonal ice advances and retreats resulting in push moraine ridges arethought to be due to thermally activated changes in ice flow dynamics that stimulate rapidsurge forward of the glacial margin that later melts back close to its original position priorto the advance. Repeated glaciotectonic push develops new push moraine ridges becausethe glacier in overall retreat deposits sediments in front of the glacier during a warm periodwhen ablation is most rapid, and then internally folds those sediments during a partial re-advance stimulated by a colder period (e.g., Boulton, 1986; Hart and Watts, 1997; Bennett,2001).
The repeated push glaciotectonic deformation process, once initiated, shoves andcontracts proximal subglacial and proglacial sediments by high-angle thrusting and foldingthat develop into a push moraine ridge during the ice advance (Fig. 6). After the first iceadvance the glacier stalls and eventually retreats depositing new sediment loads. The nextglacial advance imparts further compressive deformation to the previously formed ridgeand generates a new ridge. Over time, repeated glaciotectonic push-from-behind is capableof developing extensive push moraine ridge systems (e.g., Hart and Watts, 1997). Repeatedpush or surge of the ice sheet provides the stress necessary to shorten or extend nearbylandforms and allow the opportunity for substantial variation in depositional environments.With the seasonal ablation of the ice sheets, lowlands open up between the most recentlygenerated ridge and retreating glacial margin trapping ice, sediment, and melt-water to formproglacial lakes and outwash fan systems. If the glacier margin did propagate by repeatedglaciotectonic push-from-behind, then one would observe non-systematic variation of ridgeamplitude in cross-section throughout the push moraine ridge system. In other words, therewould be no through going décollement or obvious critical taper of the push moraine ridges.One would also expect to find asymmetric internal folding and substantial lateral variationin glacial stratigraphy and syntectonic deposition (e.g., Sharp, 1984; Boulton, 1986; Hartand Watts, 1997). Along with geologic dating techniques such as lichenometry used byHart and Watts (1997), the evaluation of clast fabrics within the internal structures of pushmoraines can, in principle, be used to distinguish seasonal or prolonged glaciotectonic push-from-behind deformation styles.
20
Glacigenic Deposits of Long Island
Classification of glacial sediments and the description of glacial facies are interpretiveand frequently controversial due to the enormous varieties of glacial deposit types and thecomplex stratigraphic and structural relationships present in glacial environments (e.g.,Dreimanis, 1989; Meyers et al., 1998, Krüger and Kjaer, 1999; Ruszczynska-Szenajch,2001). Rapid changes in deposition and deformation rates at the glacier margin cansyntectonically thicken glaciotectonic structures and drastically affect vertical and lateralstratigraphic sequences over short distances. The glacial facies system describes the productsof glaciation by classifying and organizing glacigenic sediments, spatially and temporally,at a wide range of scales, with the purpose of genetically relating glacial deposition anddeformation to glacier erosion, transport, and melt. Descriptions of glacial facies rely onprocess assemblages to reflect the variety of processes that were active in the arrangementof a particular glacial environment over a range of length scales and time spans (e.g., Bennand Evans, 1998).
Sedimentary deposits of unknown genetic origin called diamicts exist in the HarborHill and Ronkonkoma moraines and because glaciation is thought to be responsible foralmost all shallow surface sedimentation on Long Island these sediments are referred to asglacial diamicts. Diamict deposits (glacial or non-glacial) contain a broad mix of particlesvarying in shape and angularity that range in size from mud to boulder all incorporated intoa poorly sorted matrix. Important Long Island glacial diamicts, genetically known asprimary tills (or primary glacigenic deposits), can be produced by either deformation,lodgement, or melt-out processes, but most primary tills are made from a mixture of thesetill producing processes. Other genetic glacial sediment on Long Island, known as secondarytills, are sediments which have been remobilized by some form of non-glacial process thathas reworked the primary tills (e.g., Lawson, 1982). Glacial diamicts and associatedsediments produced either by subglacial, proglacial, or combined processes can not beidentified by only one diagnostic criterion so observing a set of sedimentary characteristicsmay aid in differentiating the genesis of diamict depositions (e.g., Hicock, 1990; Krügerand Kjaer, 1999; Kjaer et al., 2001).
Glacial diamict deposits include oriented clasts within stratified or massivestratigraphic units or within isolated lumps, lenses, or layers contained within a glacigenicsedimentary unit or bounded by one or more glacigenic units. Clast fabric analysis statisticallyrepresents the fabric shape, as a frequency distribution of oriented clasts by chosen axialdirection (i.e., long-axis). Eigenanalysis (Chapter 2) performed on a group of clastsquantitatively describes clast fabric shapes by normalized eigenvalues, corresponding to atleast one eigenvector, describing the likelihood that any clast axis (i.e., long-axis) from thegroup of clasts is potentially pointed in one of three mutually orthogonal minimum,intermediate, and most preferred eigenvector directions. Clast fabrics of glacigenic sedimentsmay reveal the sense of motion in a shear zone and indicate the relative strain but fails toadequately delineate types of glacial diamict.
21
Glaciotectonic Deformation Observed Within Eastern Long Island Moraines
Long Island glacigenic sedimentation processes were often complex so thatinterpreting the history of these deposits, especially for the Harbor Hill and Ronkonkomamoraines, has been problematic. Geophysical surveys and geological fieldwork studies ofthese moraines conform both the stratigraphic and structural complexities of these settings.Much of the sediment deposition and deformation occurred in front of and beneath glacierice so that most Long Island glacial sediments have under gone some glaciotectonic pushing,shearing, or folding, and may have experienced syntectonic deposition or re-deposition.
In the exposed sediments at Ranco Quarry (Fig. 1) within the Ronkonkoma moraine,coherent glaciotectonic thrust blocks have been mapped for several tens of meters aboveand hundreds of meters or more laterally from their source, with gravel-rich thrust zones asindirect evidence for the sediment having been permafrost (e.g., Meyers et al., 1998) (Fig.10). Glaciotectonic deformation studies in the Ronkonkona moraine of eastern Long Islandwere conducted in Hither Hills State Park, (Fig. 1) revealing very different sediments andstructures than found 75 km southwest at Ranco Quarry.
Sedimentary, structural, seismic, and GPR surveying at Hither Hills show evidenceof syntectonic deposition, folded strata, cm-scale faulting, and lateral variation of sedimentarylayers. Aerial photographs reveal dozens of ridges with nearly parallel strike directions(fig. 11). Topographic analysis of the Hither Hills region grouped ridges of similar height,spacing, and orientation into ‘packets’ with consistent azimuths (Fig. 12). Transects oftypical ridge successions indicate either a slight increases in maximum ridge heights fromnorth to south or no obvious ridge elevation pattern (e.g., Bernard, 1998) (Fig. 13).Photographs of sea cliff exposures and seismic data obtained from ridges in the state parkby common mid-point seismic reflection, after stacking, revealed glaciotectonic structuresin the moraine (Fig. 14A,B). After processing, seismograms depict a gently folded antiformallayer beneath a shorter and more tightly folded antiformal piggyback structure. Theseshallow surface folds are a few meters beneath the surface and have amplitudes in metersand wavelengths that are tens of meters long (e.g., Bernard, 1998) (Fig. 14B). In general,Long Island glacigenic sediments and glaciotectonicstructures are poorly exposed and are often difficult to evaluate with more traditionalgeophysical techniques such as seismic reflection surveying.
Ground penetrating radar, another geophysical tool appropriate for use in glacigenicsediment, was employed to acquire radar data at the power line cut in Hither Hills (Fig. 15).Radar data are processed and analyzed in a manner similar to seismic reflection data, allowingthe establishment of radar facies, which, like seismic facies, describe distinct structural andstratigraphic changes in depositional sequences. GPR surveys at Hither Hills reproducedthe shallow structures inferred from the seismic study of Benard (1998) but also revealmany more antiformal folds and piggyback structures at greater depths than achieved byseismic techniques (Fig. 16). The radargrams obtained with the 50, 100, and 200 MHzantennas provide complementary and far more complex images of the folded strata previouslydetected by the seismic reflection survey. The Hither Hills radargrams also provide higherresolution to a greater penetration depth (as deep as 45 m) than accomplished with theseismic reflection survey. In detail, the Hither Hills radargrams imaged many folded stratahaving an appearance resembling imbricate thrust systems. Some of these imbricate thrusts
22
Fig
ure
10.
Coh
eren
t thr
ust s
heet
s ex
pose
d in
a q
uarr
y in
cen
ral-
east
ern
Lon
g Is
land
, NY
(e.
g., M
eyer
s et
al.,
199
8).
Uni
t 4 is
a m
arin
ebe
ach/
barr
ier
sand
that
can
be
dem
onst
rate
d to
hav
e be
en tr
ansp
orte
d, a
lbei
t fol
ded
and
mic
rofr
actu
red,
ove
r a
long
dis
tanc
e. N
ote
the
clas
sic
fold
-thr
ust b
elt r
amp-
flat
geo
met
ry, t
ypic
al o
f cr
itica
l wed
ge f
old
and
thru
st b
elts
.
23
1 k
m0
km2
km
Fig
ure
11.
Mon
tage
of
aeri
al im
ages
of
Hith
er H
ills.
The
rid
ges
stri
ke r
ough
ly S
W-N
E.
The
rid
ges
do n
ot s
how
the
clea
r N
-S s
ize
prog
ress
ion
expe
cted
for
cri
tical
wed
ges.
The
re a
re tw
o si
tes
(ind
icat
ed in
red
) th
at h
ave
been
doc
umen
ted
with
GPR
. M
any
ridg
esar
e ex
pose
d by
bea
ch e
rosi
on a
long
the
nort
h sh
ore.
Pow
er L
ine
Cut
Roc
ky P
oint
24
Fig
ure
12.
Plei
stoc
ene
glac
iote
cton
ic r
idge
sys
tem
s lik
e H
ither
Hill
s in
eas
tern
Lon
g Is
land
, NY
(e.
g., K
lein
and
Dav
is, 1
999)
ca
n co
ver
seve
ral s
quar
e km
and
con
tain
larg
e nu
mbe
rs o
f pa
ralle
l rid
ges.
Ana
lysi
s of
Hith
er H
ills
ridg
e he
ight
s, s
paci
ng a
nd
orie
ntat
ions
sho
ws
that
the
ridg
es a
re g
roup
ed in
to 'p
acke
ts' o
f si
mili
ar s
ize
with
con
sist
ent m
ean
azim
uths
(ty
pica
lly ±
5 )
.
25
0
20
40
60
80
100
120
090
180
270
360
450
540
630
720
Posi
tion
Alo
ng T
rans
ect (
ft)
NS
Ele
vati
on A
long
a T
rans
ect
in H
ithe
r H
ills
050
010
0015
0020
00
Elevation (ft)
Fig
ure
13.
Typ
ical
S-N
ele
vatio
n tr
anse
ct o
f a
ridg
e su
cces
sion
at
Hith
er H
ills.
E
leva
tions
wer
e de
term
ined
on
the
basi
s of
5 f
t co
ntou
r in
terv
als
prov
ided
by
a to
pogr
aphi
c m
ap o
f th
e re
gion
(e.
g., B
erna
rd, 1
998)
.
26
Fig
ure
13 A
) G
laci
otec
toni
c fo
lds
in th
e R
onko
nkom
a m
orai
ne, e
xpos
ed a
long
the
shor
elin
e at
Hith
er H
ills,
NY
. E
ach
antic
line
corr
espo
nds
to o
ne o
f a
seri
es o
f su
bpar
alle
l rid
ges
(fig
ure
12),
and
is b
elie
ved
on th
e ba
sis
of m
appi
ng a
nd g
eoph
ysic
al d
ata
to c
onta
in a
thru
st f
ault
belo
w p
rese
nt-d
ay s
ea
leve
l (e.
g., K
lein
and
Dav
is, 1
999)
. B
) S
eism
ic r
efle
ctio
n se
ctio
n fr
om B
erna
rd e
t al.,
(19
98)
of a
n an
ticlin
e-co
red
hill
in H
ither
Hill
s, s
how
ing
a sh
allo
w r
efle
ctor
(a
fold
ed b
ed).
The
sei
smic
sur
vey
line
is o
n a
ridg
e fl
ank
that
dip
s to
the
left
(no
rth)
at 7
, so
the
bed
dips
18
to th
e ho
rizo
ntal
at l
eft
(N)
and
4 o
n th
e ri
ght (
sout
h) li
mb.
The
re is
littl
e or
no
vert
ical
exa
gger
atio
n. T
his
fold
is s
imili
ar in
wav
elen
gth,
am
plitu
de, a
nd s
hape
to th
at in
(A
).
A)
B) 4.
.
.
CD
P L
ocat
ion
(m)
Time (sec)
27
1000
fee
t
Figure 15. GPR survey (red line) conducted with 50 MHz antennas along a power line cut in Hither Hills (figure 11). Seismic survey of Bernard et al., (1998) begins at the southern end of the GPR survey profile. Note topographic contours indicate the number of feet above sea level.
power line cut power line cut
power line cutpower line cut
28
Fig
ure
16.
Topo
grap
hy c
orre
cted
and
mig
rate
d 50
MH
z ra
darg
ram
of
a 15
0 m
N-S
por
tion
of t
he p
ower
lin
e cu
t at
Hith
er
Hill
s (f
igur
e 15
). D
ashe
d pu
rple
line
s in
dica
te d
ip-d
omai
ns in
sed
imen
ts, a
nd h
eavy
red
das
hed
lines
indi
cate
pos
sibl
e fa
ults
.
Dis
tanc
e (m
)
050
100
150
0 10 403020
Depth (m)
29
core the ridges and exhibit an over-printed geometry consistent with a push-from-behindglaciotectonic mechanism being responsible for the ridge formations found at this locationin the Ronkonkoma moraine. The GPR survey using the 50 MHz antennas overlapped theseismic line of Bernard et al., (1998) along the power line cut in Hither Hills and extendedto the north for an additional 150 meters (Fig. 15). Interpreted radargrams of the HitherHills shallow subsurface indicate substantial sequences of syntectonic depositions sinceradar reflectors (dip domains) thicken away from fold hinges and the cores of the imagedridge structures are more tightly folded than is the overlaying topography (Fig. 16).
Hither Hills Radargrams processed from radar data acquired near Rocky Point (Fig.17), show the presence of complexly folded glacigenic strata of wavelength and amplitudesimilar to those seen in exposed sea cliffs along the north coast of the park (Fig. 18). Thisis consistent with my measured stratigraphic sections (e.g., Klein and Davis, 1999) alongthe north shore of Hither Hills that document lateral variation of stratigraphy on the scale of10’s to 100’s of meters (Fig. 19). Seismic reflection and refraction studies, measured sections,topographic analyses, ridge transects, and GPR surveys have led to the conclusion thatseasonal glaciotectonic push-from-behind (e.g., Boulten, 1986; Bennett, 2001) is the mostviable explanation for the development of the push moraine features observed at HitherHills.
As described early in the next chapter of this thesis, clast fabric analysis along theshoreline at Ditch Plains clearly shows a fabric consistent with glacial advance from theNNW. In conjunction with future Long Island geophysical surveys, measured sections andclast orientations should be recorded. Careful geologic and geophysical studies will continueto elucidate a more complete picture of the dynamic strain histories and complexsedimentation processes associated with glaciotectonic deformation. Long Island is currentlyloaded with a wide variety of glacial diamict deposits generated by Pleistocene continentalice sheets justifying continued emphasis on quantitative methods for establishing local iceflow direction based on clast orientation.
30
Col
umn
C. -
Sm
all A
ntic
line
Col
umn
C. -
Sm
all A
ntic
line
Col
umn
B. -
Roc
ky P
oint
Col
umn
B. -
Roc
ky P
oint
Col
umn
Col
umn
A. -
Dom
inan
t Hill
Str
uctu
re. -
Dom
inan
t Hill
Str
uctu
re 00m
0
10
Fig
ure
17.
Loc
atio
n of
a G
PR s
urve
y (r
ed li
ne)
cond
ucte
d w
ith 2
00 M
Hz
ante
nnas
acr
oss
the
dom
inan
t hill
str
uctu
re a
t Roc
ky
Poin
t, H
ither
Hill
s (f
igur
e 11
).
The
pos
ition
s of
thr
ee m
easu
red
stra
tigra
phic
sec
tions
at
sea
clif
f ex
posu
res
are
indi
cate
d. N
ote
topo
grap
hic
cont
ours
indi
cate
the
num
ber
of f
eet a
bove
sea
leve
l.
31
Fig
ure
18.
Topo
grap
hy c
orre
cted
, mig
rate
d, a
nd in
terp
rete
d 20
0 M
Hz
rada
rgra
m.
The
rad
argr
am s
ampl
ed 1
30 m
of
the
dom
inan
t hi
ll st
ruct
ure
at R
ocky
Poi
nt, H
ither
Hill
s (f
igur
e 17
).
Cha
nges
in
surv
eyin
g di
rect
ion
are
indi
cate
d in
de
gree
s. I
nter
pret
ed r
adar
fac
ies
of th
is c
ompl
icat
ed r
idge
str
uctu
re a
re s
how
n in
col
or.
32
clay
- b
ould
ercl
ay -
bou
lder
clay
- b
ould
ercl
ay -
bou
lder
clay
sand
diam
ict
cove
r
root
sof
fset
2 m
4 m
6 m
8 m
10 m
clay
- b
ould
ercl
ay -
bou
lder
Fig
ure
19.
Thr
ee m
easu
red
stra
tigra
phic
sec
tions
spa
ced
abou
t 10
0 m
(ab
out
one
ridg
e w
avel
engt
h) a
part
in
the
sea
clif
fs e
xpos
ed a
long
the
nor
thea
st c
oast
of
Hith
er H
ills
(fig
ure1
6).
Not
e th
e th
in c
lay
in c
olum
n C
. (s
mal
l ant
iclin
e) a
nd th
e gr
eat v
aria
tion
in s
edim
ento
logy
and
str
atig
raph
y be
twee
n th
e se
ctio
ns.
Col
umn
Col
umn
A. -
dom
inan
t hill
str
uctu
re. -
dom
inan
t hill
str
uctu
reC
olum
n C
. - s
mal
l ant
iclin
eC
olum
n C
. - s
mal
l ant
iclin
eC
olum
n B
. - R
ocky
Poi
ntC
olum
n B
. - R
ocky
Poi
nt
33
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ˇ
41
II. Surface Sample Bias and Clast Fabric Interpretation
Abstract
Quantitative clast fabric analysis, despite its limitations, is a useful analytical tool inglacigenic sediment studies. More powerful than graphical methods, eigenanalysis allowsquantification of otherwise descriptive three-dimensional fabrics. In conjunction with theorientation of the long-axes, short-axes preferred direction could further establish the natureof shear, emplacement, and deposition in glacigenic settings. Field measurement, however,produces a systematic sampling bias in favor of clasts normal to outcrop surface. Thissampling bias is a function of the orientation of outcrop surface to the fabric and can affectthe perceived fabric strength (eigenvalues) enough to influence interpretation. I use simplecalculations and numerically generated random clast populations to quantify this bias and Ifind that it is greatest for those clasts best suited to fabric analysis (those that are rod-like inshape). The effect of this surface bias can be mitigated with careful sampling andinterpretation. Fortunately, its effect upon strong fabric orientation (eigenvectors) is generallysmall.
Clast Fabric Analysis in Glacial Sediment
Glacial deposits typically incorporate a very broad range of particle sizes that reflectthe source and quantity of sediment supplied to the glacier. One way glacial diamict canevolve is when glacially entrained rock, bedrock fragments, and sediments are crushed,ground, deformed and mixed while being carried in the ice and particle rich basal zone of aglacier, producing deposit types which include a heterogeneous (clay to boulder) mixture ofpoorly sorted particles (e.g., Alley et al., 1999). Prolonged glacial transport modifies thetextural maturity of glacial sediments, so clast roundness and sphericity are functions of thedistance the clast has been transported (e.g. Boulton, 1978; Boulton, 1996).
Moraine landforms are often composed of relatively large volumes of glacial diamictdeposits. In these settings, diamict units can be massive, stratified, laminated, imbricated,reworked, resedimented, sheared, and they can include clast pavements. Diamicts includeeither isolated or graded clasts within a matrix of finer material or an interstitial matrix offiner material within supporting clasts (e.g. Benn and Evans, 1998; Krüger and Kjaer, 1999).A clast fabric defines a distribution of clasts by the degree to which clast axes orientationsare clustered. Such fabrics range from very weak (nearly isotropic) to very strong (highly
42
directional).The manner in which glacigenic clasts are arranged provides outcrop-scale
information that may help to constrain the physical dimensions of stratigraphic units andalso provides clues to the nature and sequence of glaciotectonic and depositional processes(e.g., Hicock, 1990). A quantitative description of a clast fabric, defining the degree ofpreferred orientation observed in exposed diamict settings, is obtained by a combination offield measurement and vector analysis (e.g., Mark, 1974). The statistical description of aglacigenic sediment clast fabric reflects the effect of both depositional conditions and thesubsequent strain, in principle enabling the differentiation of the structural contrasts recordedin glaciotectonic shear zones. Published research (e.g., Lawson, 1979; Dowdeswell et al.,1985; Dowdeswell and Sharp, 1986) on a variety of glacial diamict deposits has shown abroad range of preferred clast orientations, producing fabrics of strengths that differ withdepositional setting. The glacigenic settings which produce these differing results rangefrom weakly organized clast orientations (e.g., water-lain glacigenic sediment) preservedduring outwash deposition to highly uniform lineations (e.g., subglacial melt-out till)generated by well-developed shear zones that are later preserved during subglacial melt-outdeposition from motionless glacial ice. The magnitude of clast orientation preference is ameasure of the degree to which all preserved and observed clasts measured are alignedsubparallel to a unique direction. Strongly organized, highly anisotropic clast orientationsderived from subglacial-meltout or lodgment processes are typically subparallel with localice flow direction (e.g., Lawson, 1979; Dreimanis, 1999). Fabric strength generally decreasesas water content in the sediment-ice mixture increases. Clast fabric is not uniquely indicativeof sediment genesis, but can be an indicator of relative strain within a stratigraphic unit(e.g., Bennett, et al. 1999; Karlstom, 2000).
I have studied glacigenic sediments at a site within the Ronkonkoma Moraine ofLong Island, New York. My field area, Ditch Plains, roughly 5 kilometers west of MontaukPoint on the south fork of eastern Long Island (Fig. 1), is spaced along several kilometers ofthe shoreline. Bluffs and vertical exposures outcrop at heights as great as 10 to 15 meters,as the Ronkonkoma Moraine resists erosion caused by ocean surf and other weatheringprocesses. I interpret the general stratigraphy to include two distinctive glacial diamictdeposits. A massive upper diamict unit (Dmm) overlies a stratified diamict unit (Dms) (Fig.2), (e.g., Klein, et. al., 1998, 2001). The preferred direction of the long axes for the rod-shaped clasts (subhorizontal N-S direction) is evident within the stratified diamict unit. Thenorth-south long axis orientation of most clasts in this unit is consistent with shear due toglacial advance from the north.
One way to distinguish differences in fabric strength is with a rose diagram. Thisgraphical method plots frequency distribution versus bearing. The number of clasts plottedfor a given range of bearings corresponds simply to the number of clasts with a particularaxis (e.g., long) pointed within a narrow range of bearing (commonly ±5º) of that direction.A major drawback in using rose diagrams for fabric descriptions is their inability todifferentiate between shallowly and steeply plunging clast axes, which limits the usefulnessof physical interpretation based on rose diagrams alone. Plotted on a rose diagram are thelong axis bearings of 150 clast orientations collected from the stratified diamict unit atDitch Plains (Fig. 3A). The rose petal in figure 3A, pointing to the northeast between 340ºto 350º defines the bearing direction for 23% of the measured long axes.
43
Fig
ure
1. M
ap o
f L
ong
Isla
nd.
Shad
ed in
gre
en a
re th
e H
arbo
r H
ill a
nd R
onko
nkom
a m
orai
nes.
D
itch
Plai
ns f
ield
stu
dy
loca
tion
is in
dica
ted
by th
e sm
all r
ed d
ot o
n th
e so
uth
fork
.
44
1 m
Fig
ure
2.
The
sou
th-f
acin
g se
a cl
iff
at D
itch
Plai
ns
(Fig
ure
1).
M
ost
of t
he 1
0 m
exp
osur
e co
nsis
ts o
f a
stra
tifie
d di
amic
t, te
ntat
ivel
y in
terp
erte
d as
a m
elt-
out
till.
The
top
part
of
the
expo
sure
is a
mas
sive
dia
mic
t w
hich
is
prob
ably
a f
low
till
. I
nser
t is
a t
ypic
al 1
m
squa
re c
last
ori
enta
tion
sam
plin
g ar
ea (
red
squa
re).
Dm
sD
ms
stra
tifi
ed d
iam
ict
stra
tifi
ed d
iam
ict
(mel
t-out
till
?)(m
elt-
out
till
?)
Dm
mD
mm
mas
sive
dia
mic
tm
assi
ve
dia
mic
t
(flo
w t
ill?
)(f
low
til
l?)
Cover
edC
over
ed
45
N
A
C
B
Figure 3. A. A rose diagram showing the 150 long axis clast orientations recorded in the stratified diamict unit (Dms) at Ditch Plains. The rose diagram perimeter corresponds to 23%. B. Equal-area stereonet projection of the same 150 clast orientations C. Equal-area stereonet projection of the same 150 clast orientations contoured at 2% intervals for 1% area. The contour plot of these clast orientations unambiguously suggests a subhorizontal preferred direction just west of north. Note, north is located at the top of each of the diagrams in the figure.
46
Equal-area stereonet projection is a more commonly used graphical method thatexpresses three-dimensional clustering in direction of a specified clast axis. In this projection,each point represents the bearing and plunge of one clast. In figures 3B and 3C, I presentthe same clast data from the lower stratified diamict unit projected and contoured on equal-area stereonets. The 150 long axes cluster at shallow plunge near a bearing of 345°. Unlikerose diagrams, stereonets are capable of graphically establishing a three-dimensionalmaximum preferred orientation for a single axis (e.g. long-axis) distribution where onemight also be able to infer the least and intermediate preferred directions. Eigenanalysis, amore formal quantitative technique, can statistically establish three mutually orthogonal(most, intermediate, and least) preferred directions for a single axial data set, but its advantageis greater resolution and more precision than stereonet projection.
In clast fabric analysis, each individual clast may be approximated by a triaxialellipsoid with three mutually orthogonal principal axes. It is convenient to determine suchprincipal axes for each clast by first measuring the length and orientation of the longest axisthrough the clast center. The lengths and orientations of two mutually perpendicular andorthogonal axes to the established long axis orientation can then be established and recorded.I record the orientation (bearing and plunge) of any two of these principal axes, from whichI can calculate the third. The bulk fabric information gleaned from in-situ measurement ofaxial orientations for a given number of clasts (vectors) in a clast distribution is transformableinto a three-dimensional orientation tensor (e.g., Mark, 1973; Woodcock, 1977). The elementsof the orientation tensor, defined in a 3 x 3 symmetric matrix, indicate the degree to whicha particular clast axis (this technique is usually applied to the long axis) tends to align in agiven direction. Clast shape can play a significant role in the determination of a clastorientation. Rod-like clast shapes, however, have an obvious long axis orientation, allowingthem to contribute to a more robust orientation tensor.
The analytical solution obtained by eigenanalysis determines three normalizedeigenvalues (a maximum, an intermediate, and a minimum) and assigns one eigenvalue toeach one of three mutually perpendicular preferred eigenvector directions. These eigenvectorscan be thought of as the most, intermediate, and least preferred clast-vector directions (allmutually perpendicular). The corresponding normalized eigenvalues describe the degreeof preference for each of these directions. For example, a normalized eigenvalue equal toone would mean that all axes point exactly that way, while an eigenvalue of zero wouldindicate that they are all 90° from that direction. Three-dimensional eigenvalue analysis inglacially-derived diamicts has been used by a number of researchers studying clast fabricsas a diagnostic tool intended to deduce the genetic origin of glacial deposits (e.g., Mark,1974, Lawson 1979; Dowdeswell et al., 1985; Rappol, 1985; Dowdeswell and Sharp, 1986;Benn, 1994, Ham and Mickelson, 1994; Hicock et al., 1996; Larsen et al., 1999; Kjaer et al.,2001) and to infer relative strain within glacigenic sediments (e.g., Hicock, 1992; Hart,1994; Benn, 1995; Benn and Evans, 1996; Rijsdijk, 2001). Clast shapes affect the natureand strength of the clast orientation fabric. If a is the long axis, b the intermediate axis, andc is the short axis for a particular clast, then the clast shape is defined as a rod if a>b≈c, adisc if a≈b>c, a blade if a>b>c, or a spheroid if a≈b≈c (e.g., Zingg, 1935; Sneed and Folk,1958). Spheres have infinite combinations of three mutually perpendicular axes so spheroidalclasts make a very weak fabric, have no preferred directions, and indicate little aboutemplacement or strain. Glacigenic deposits composed exclusively of sediment of extreme
47
textural maturity are therefore the poorest targets for clast fabric analysis. Rod-like clastsare often the easiest shapes in the field to measure, so they are the most ideal for thedetermining long axis orientation.
A more complete description of clast fabric includes the clast short-axis directionssince it is equally possible to study the distribution of long or short axes, although theliterature emphasizes long axis orientations, particularly for blunt rods (aspect ratio a/b ofat least 3/2) (e.g., Bennett et al., 1999; Kjaer and Krüger, 1998; Krüger and Kjaer, 1999;Millar and Nelson, 2001a,b). To this date there is limited reporting of measured short axisorientation distributions in glacigenic sediment studies. Combining long axis and shortaxis orientations of the same set of clasts may permit clearer discernment of the nature ofshear responsible for emplacement of the clasts. A clast fabric describing short axisorientations for a set of blade-like (triaxial) clasts (a>b>c) may help in identifying pureshear uniaxial shortening. For example, if a randomly oriented clast set is subjected to pureuniaxial compression, then the clasts long axes will tend to girdle in orientations radiallynormal to the compression direction and the clasts short axes orientations will be apt tocluster subparallel to the compression direction. For simple shear, long axes tend to clusterin the shear direction (e.g., Dreimanis, 1999), although some studies describe quasiperoidicbehavior (e.g., Lindsay, 1968).
The short axis direction of the clasts measured at Ditch Plains have a great propensityfor being aligned nearly vertical. This, like the long axis orientation, is consistent withemplacement and subsequent shear from the north due to glacial advance. The short axes of141 of the 150 Ditch Plains clasts cluster on a stereonet in a near-vertical orientation moretightly aligned than the long axes (Fig. 4). The distribution of the long and short axialorientations of the Ditch Plains clasts is suggestive of a well-developed fabric, enhanced byconsiderable shear.
The orientation of the long axis of a single clast is usually described in terms of avector with a given bearing and plunge. It is also possible, however, to describe that vectorin terms ofthe angle it makes with respect to each of the three coordinate axes (x, y, z) or (E,N, vertical). The three direction cosines (m, l, n) are simply the cosines of the angles fromthat pebble axis vector to the positive x, y, and z-axis directions, respectively. They can alsobe thought of as the x, y, and z components of a unit vector pointing in the same direction asthe pebble axis, with values of +1 or -1 (if pointing along a given axis) and 0 if perpendicularto that axis.
Following the example of other workers, (e.g., Mark, 1973; Woodcook, 1977), I usethe direction cosines (l, m, n) of measured pebble axis vectors to produce a symmetricmatrix that describes the direction and intensity of the overall pebble fabric. This matrix,called the orientation tensor, is the sum over all N pebbles of sets of products of the directioncosines. It is written formally as
The contribution of a single clast to the xx-component (axx) is simply the first
Σli
2
Σmili
Σnili
Σmi
2
Σlimi
Σnimi
Σni
2
Σmini
Σlini[ ]A =
1
N(1)
48
A
B
Figure 4. A. Equal-area stereonet projection of 141 short axis clast orientations recorded in the stratified diamict unit at Ditch Plains. B. Equal-area stereonet projection of the same 141 clast orientations contoured at 2% intervals for 1% area. Note, north is located at the top of each of the steronet diagrams in the figure.
49
direction cosine of the axis (usually the long axis being considered), l, multiplied by itself.Similarly, the other components along the diagonal (ayy and azz) are squares of those directioncosines. The off-diagonal terms are simply products of the corresponding pairs of directioncosines. For example, axy and ayx are each the products of the x- and y-axis directioncosines, l, and m. For this reason, the matrix is symmetric (axy = ayx, axz = azx, and ayz =azy). Such a matrix can be written for an individual clast. By summing the values of eachmatrix component over all N measured clasts (the subscript i is a counter for individualclasts) and then normalizing by that value N, one arrives at the normalized orientationmatrix A (e.g., Mark, 1973; Woodcock, 1977). This matrix contains information about thestatistics of clast axes – the frequency to which they point in any given direction.
The normalized orientation matrix A is written with respect to the x-, y-, and z-axes.The mutually orthogonal set of axes corresponding to the fabric (the directions in whichclasts most and least strongly point) can be oriented at any arbitrary angles to that (x, y, z)coordinate frame. Ideally, I would like to rewrite the normalized orientation matrix withrespect to a reference frame that has a physical meaning for the clast fabric, rather than thearbitrary (x, y, z) axis reference frame. The diagonalization of the matrix A does just that.
The eigenvectors of the matrix are simply vectors indicating the mutually orthogonalaxes that define that natural reference frame. One of these axes, written as vector V1, is themost preferred direction for clast axes. The degree of that preference is given by a scalarvalue – the greatest eigenvalue S1. The least preferred direction, V3, is an eigenvector thatis normal to the V1 axis, and its corresponding eigenvalue is called S3. A third direction,V2, is perpendicular to both the V1 and V3 directions and has an eigenvalue S2 that indicatesthe relative propensity of clasts to align in that direction. If, as in my calculations, theeigenvalues are normalized to one, I have S1+S2+S3=1, with S1 ≥ S2 ≥ S3. When the matrixhas been rewritten in diagonalized form, in which it is defined with respect to the fabricaxes (eigenvectors), the off-diagonal terms are all zero. The three remaining non-zero terms,along the diagonal, then correspond to the three eigenvalues.
Fabric strengths derived from eigenvalue analysis in the literature have been depictedgraphically in two distinct ways. Both (S1,S3) eigenvalue plots (Fig. 5A) and isotropy-elongation ternary plots (Fig. 5B) encompass the entire range of possible sets of eigenvalues,defined by the relations S1+S2+S3=0 and 0≤S3≤S2≤S1≤1 (e.g., Benn, 1994). The two typesof plots can be 1:1 mapped onto each other.
In an (S1,S3) eigenvalue plot (Fig. 5A), the horizontal axis corresponds to the magnitudeof the largest eigenvalue (S1) and the vertical axis indicates the smallest eigenvalue (S3).Each point on the plot corresponds to a unique (S1,S2,S3) set, since the third eigenvalueS2=1-S1-S3 is uniquely determined in terms of the other two. All possible (S1,S2,S3)combinations are contained within a skewed triangular region, bounded on the bottom bythe S3=0 line, at left by the S2=S1 line and at top by the S2=S3 line.
In an isotropy-elongation ternary diagram (Fig. 5B) the set of all fabric eigenvalues isplotted within an equilateral triangular region, as in any other ternary diagram. The vertical(isotropy) axis measures the ratio (S3/S1). When this ratio equals one, all three eigenvaluesare equal and the fabric is purely isotropic. When the isotropy (S3/S1) equals zero, 0=S3≤S2≤ S1, and the fabric lies somewhere between a girdle (S3=0 ; S2=S1=.5) and a cluster(S3=S2=0 ; S1=1.0). The elongation axis, 60° clockwise of the isotropy axis, measures thevalue 1-(S2/S1). The third axis value in a ternary diagram, in this case (S2-S3)/S1, is
50
.33 .40 .50 .60 .70 .80 .90 1.0
isotropipicpic
clustergird
le
S3
S1
22
SS3==
1SS
2 ==
.3333
.3303
.2202
.101
000
Figure 5. A. (S1, S3) eigenvalue plot. Note that possible eigenvalue combinations can fall only within the triangular area of the plot bounded by the lines S1=S2, S2=S3, and S3 = 0.0 with S1 eigenvalues ranging from 0.50 to 1.0 respectively. B. Isotropy-elongation ternary diagram. Ideal fabric shapes (isotropic, girdle, cluster) indicated at the apices of the useable triangle in the (S1, S3) eigenvalue plot and in the isotropy-elongation ternary diagram.
A
B
0
0.2
1.0
0.8
0.6
0.4
Elo
ngat
ion
1-
(S /S
)2
1
Isotropy (S /S )
3 1
0
0.2
1.0
0.8
0.6
0.4
cluster
isotropic
girdle
51
redundant because the three coordinate values sum to one and are thus not mutuallyindependent.
Although the mapping between the two diagrams is 1:1, their relative scaling is notuniform. The eigenvalue plot stretches the lower right-hand (cluster) area of the plot (Fig.6A,B), so that the extreme corner of the plot (where S1→1) is scaled up in area by a factorof 4.5:1 compared to the ternary plot. Likewise, the top of the ternary diagram exaggeratesthe near-isotropic region (where S3 →S1) by a factor of 6:1 compared to the eigenvalueplot. Thus, equal-area regions near the isotropic and cluster extremes of one of these diagramswill appear on the other diagram to encompass corresponding regions of greatly differentsize, differing by a factor of up to 27 at the far corners of the diagrams. Although thisdifference in scaling is less extreme for larger regions that extend away from the corners, itremains significant. For example, the ternary diagram can be divided into four equal areas,corresponding to sets of eigenvalues that tend toward being roughly isotropic, clustered, orgirdle-like, plus those that fall in between (Fig. 7A). Plotted on an eigenvalue diagram,however, those four regions are skewed and far from equal in area (Fig. 7B). The nearlyisotropic 20% of the ternary diagram (shaded dark in Fig. 7A) occupies only 1/12th of thetotal area on the eigenvalue diagram (5 times smaller than the nearly-clustered region, whichalso covers 20% of the ternary diagram). Thus it may not be surprising that relatively fewpublished observations fall in the near-isotropic region. Also, existence of naturally occurringtruly isotropic fabrics is thought to be extremely rare due to the influence of depositional ormechanical boundary conditions such as surface and stress field orientations that oftenpromote particle alignment (e.g., Benn and Ringrose, 2001). Particular care must beexercised in evaluating data points that plot close to the boundaries in eigenvalue diagrams,where at least two eigenvalues are nearly equal in strength. Such a condition can lead tomisinterpretation of the preferred orientation of a clast fabric. Random sampling can easilycause nearly equal eigenvalues to reverse their magnitude order, leading to a dramatic andspurious change (one axis for another) in eigenvector direction (e.g., Ringrose and Benn,1997; Benn and Ringrose, 2001). In fact, as I will show, there is an additional reason.There is anobservational bias that tends to produce spurious measurements away from thisregion even when the true clast fabric is indeed quite isotropic.
Various authors have attempted to associate fabric domains in (S1, S3) space withthe mode of genesis of glacial till in modern glacigenic sediments, although in relativelyfew cases are there such data where sedimentary processes are unambiguously observed(e.g., Lawson, 1979; Dowdeswell et al., 1985). The Ditch Plains sediment orientation datagive well-determined eigenvalues and eigenvectors (Table 1) that can be placed on an S3versus S1 eigenvalue plot.
Table 1. Eigenanalysis results for long and short axis clast orientations recorded at Ditch Plains.
short-axis eigenvalues short-axis eigenvectors
S1 = .911
S2 = .051
S3 = .038
V1 = (003°, 73°)
V2 = (260°, 04°)
V3 = (169°, 16°)
long-axis eigenvalues long-axis eigenvectors
S1 = .792
S2 = .150
S3 = .058
V1 = (342°, 06°)
V2 = (072°, 01°)
V3 = (178°, 84°)
52
Figure 6. Illustration of 1:1 mapping and non uniform scaling between the (S1, S3) eigenvalue plot and the isotropy-elongation ternary diagram. A. The useable triangular domain of the (S1, S3) eigenvalue plot mapped to and labeled with isotropy-elongation ternary diagram eigenvalue ratio combinations. B. Isotropy-elongation ternary diagram mapped to and labeled with (S1, S3) eigenvalue plot eigenvalue ratio combinations. Ideal fabric shapes (isotropic, girdle, cluster) indicated at the apices of both diagrams.
A
B
Elongation
1-(S
/S )
2 1
.50
.90.80
.40
.30
.20
.10
.60
.70
S 1
1.0cluster
isotropic
girdle
0
S3
isotro
pic
clustergirdle
1.0
0.90
0.80
0.700.60
0.500.400.300.200.100.01.00.900.800.700.600.50
0.400.30
0.200.10
0.0
Isotropy (S /S )3 1
53
.33 .40 .50 .60 .70 .80 .90 1.0
isotropipicpic
clustergird
le
S3
S1
222
SS
1SS
2 ==
.3333
.3303
.2202
.101
000
0
0.2
1.0
2
12
Isotropy
3 1
0
0.2
0.8
0.6
iisotropici
girdl
girdle
girdl
Figure 7. A. Isotropy-elongation ternary diagram divided into four equal area regions that correspond to general fabric shape domains. B. The equal area general fabric shape domains of the isotropy-elongation ternary diagram are skewed substantially in area on the (S1, S3) eigenvalue plot.
A
B
54
The Ditch Plains long axis stratified diamict clast fabric falls into the lower right-hand region of the plot, near where Lawson (1979) and Dowdeswell et al. (1985) plotsediments with strong fabrics, such as lodgement and subglacial melt-out tills (Fig. 8). Theresult of calculated by eigenanalysis is consistent with the rose diagram (Fig. 3A) and thelong-axis stereonet plots diagram (Fig. 3B,C) depicting the majority of the measured pebblesaligned subhorizontally with north-south direction. Each of the fabric domains describedby an ellipse on an eigenvalue plot (S3 versus S1) denotes a genetic type of glacial sedimentrecorded in-situ (Fig. 8), but glacial facies or the degree of glaciomechanical influence onfabric development should not be interpreted based on fabric
Although glacial ice might be responsible for clast fabric generation and thesubsequent deposition of lodgement, deformation, and subglacial-meltout tills from its basalzone, direct evidence for regional ice movement can not easily be delineated by fabricgeometry exclusively. For instance, ice-marginal moraines formed by many locally smalland structurally complex ice tongues may introduce so much genetic (modal) and directionalvariability that it is often not possible to establish a single local ice movement direction(e.g., Dreimanis, 1999). Inferring strain accumulation during the emplacement of glacialdiamicts is a formidable challenge. Many kilometers of ice may, before melting, shear pasta point very near the front of the moraine, leaving no further trace. An indeterminate amountof that shear may occur in the ice, as opposed to the sediments. For these reasons, absolutestrain levels may not be recorded clearly in sediments. Therefore, clast fabric analysis isbest suited as a relative (as opposed to absolute) strain indicator within a single set of glacialsediment deposits (e.g., Bennett et al., 1999).
The strength of the clast fabric and the magnitude of bulk strain do not necessarilyhave a one-to-one relationship: quite different deformation ellipsoids (and eigenvalues) canresult, depending upon whether the clasts undergo passive (March) rotation with the matrix,or make a more independent (Jefferey) rotation in response to force couples resolved acrossthem (e.g., Jeffrey, 1922; Benn and Evans, 1996; Hooyer and Iverson, 2000). In somesediments, fabric strength may even be cyclical with strain (e.g., Lindsay, 1968). Manydifferent processes can produce similar clast fabrics (e.g., Bennett, et al., 1999; Krüger andKjaer, 1999; Karlstrom, 2000; Benn and Ringrose, 2001; Kjaer, 2001), so one must also useother sediment properties to classify till.
Reconstruction of shear as recorded in quantitative clast fabric analysis opens upthe possibility of mapping patterns of glaciomechanical strain. It is possible witheigenanalysis to map the directional changes associated with ice movements and to relatesediment emplacements to depositional processes that are evident in more traditionalexamination of the outcrop. Researchers (e.g., Hart, 1998) have sought to correlatequantitative measures of clast fabrics with shear zones.
In most subglacial tills, preferred direction is parallel with the movement of ice andthe clasts preferentially plunge upglacier (e.g., Krüger, 1970; Dreimanis, 1999). The preferredlong axis eigenvector (342°, 06°) obtained by eigenanalysis (Table 1.) for the stratifieddiamict unit at Ditch Plains suggests that the pebble orientation in that unit may be subparallelwith the Pleistocene ice flow direction believed to be from the NNW (Klein and Davis,2001). The short axes of these clasts predominately plunge steeply north (upglacier) whichsuggests that the long and short axes directions are not orthogonal. I infer that the stratifieddiamict unit originated by subglacial melt-out deposition in the local direction of glacier
55
1.0cluster
S3
Wat
erla
in G
laci
gen
ic S
edim
ent Subgla
cial
Deb
ris-
rich
Mel
t-out
Til
l
Bas
al I
ce
Flo
w T
ill
0.4
0.5
0.6
0.7
0.8
0.9
0.0
0.1
0.2
0.3
S1
girdle
Lodgem
ent
Til
l
isotropic
Fig
ure
8.
Mod
ifie
d S 1
vs
S 3 e
igen
valu
e pl
ot
of D
owde
swel
l et
al.,
(19
85)
com
pari
ng t
he s
tand
ard
devi
atio
n an
d m
ean
eige
nval
ues
of f
our
mod
ern
glac
igen
ic f
abri
c do
mai
ns w
ith d
ebri
s-ri
ch b
asal
ice
. N
ote
that
po
ssib
le S
1 vs
S3
eige
nval
ue
com
bina
tions
can
onl
y fa
ll w
ithin
the
whi
te tr
iang
ular
are
a of
the
plot
.
56
movement based on the strongly preferred orientations of the long and short axes.A clast fabric is a bulk volume property requiring careful sampling acquisition and
statistical interpretation. Randomness of the sample is assumed when performingeigenanalysis and the resultant eigenvectors obtained from the analysis are constrained tobe orthogonal although the most and least preferred directions do not need to be normal toeach other. Furthermore, because nearly equal eigenvalues yield a girdle of unresolvedvectors making impossible a distinction of orthogonal directional preference, the eigenvalueS
3 found for long axes eigenanalysis is not necessarily the preferred direction (S
1) for the
short axes.
Surface Sample Bias
Since clast fabrics are a bulk volume property, one ideally measures all clasts in adefined volume of an outcrop. This, however, is usually not possible. More commonly,clast orientations recorded from, at, and near an outcrop surface plane are assumed to representstatistically the clast fabric of an associated unit volume (e.g., Benn and Ringrose, 2001).Clast orientations more likely will be preserved in a cohesive matrix containing a highproportion of clay and silt-size particles with relatively tightly packed pore spaces. Thebest outcrops for measuring clast orientation are quite hard, making them resistant to lateslumping and facilitating accurate determination of orientations. Unfortunately, hardnessalso makes deep excavation difficult. Therefore, measurements are often constrained to atand near exposed surfaces.
Investigators commonly sample as many clast orientations as possible in order tomaximize data sets and minimize random measurement errors, but do so with limitedexcavation. Such surficial (rather than volumetric) sampling can lead to a systematic samplingbias (e.g., Millar and Nelson, 2001a,b). Furthermore, the surface sampling bias can skewby a large degree (dependent on true eigenvalue) the eigenvalues reported in eigenvaluestudies of fabric shape. Eigenvectors determined by eigenanalysis are much less susceptibleto this bias than are eigenvalues especially if S
1>>S
2. The likelihood of sampling an individual
clast depends on the angle that its long-axis makes with the plane of an exposed outcropsurface. As a diamict outcrop erodes, more and more clasts are gradually exposed (Fig. 9).The long axis of a rod-like clast aligned parallel to the strike direction of an eroding outcropsurface will fall out of an outcrop significantly sooner after first exposure than would theequivalent clast with its long-axis direction oriented normal to the exposed surface. Thedetermination of clast fabric strength from clast orientations recorded at an outcrop surfacetherefore contains an intrinsic bias, because elongated clasts aligned parallel with an erodingoutcrop surface will be undersampled and elongated clasts oriented normal to the erodingsurface will be oversampled (Fig. 10). Clast orientation sampling can be termed volumetricif outcrop material is excavated to a distance much greater than the mean particle size of theclasts being measured. Absent such excavation, the bias inherent in surface sampling will
57
X
Y
φ
φD
a
c
time
eroding outcrop surface
P (xp,yp)
N
n
Figure 9. A cross-section of an ellipsoidal clast in the (X, Y) plane being exposed with time by an eroding outcrop surface. The clast is centered at the origin of the (X, Y) plane and is positioned with its long-axis, a, parallel to the X-axis and its short-axis, c, parallel to the Y-axis. The eroding surface, a plane containing the Z-axis oriented normal to the (X, Y) plane, appears as a trace in the cross-section at some angle, φ, to
the Y-axis. As the outcrop surface begins to erode a point, P (xp,yp) is exposed on the ellipsoidal clast. The vector normal, N, and the unit vector, n, orignate at the point P (xp,yp). The length, D, is the shortest distance from the center of the ellipsoidal clast to the eroding surface. The magnitude of D, thus indicates how much erosion and time are required before the clast is half exposed. This is used an approximate indicator of the clasts exposed lifetime.
58
mor
aine
air
Cla
sts
orie
nted
pa
ralle
l to
erod
ing
surf
ace
Cla
sts
orie
nted
no
rmal
to
erod
ing
surf
ace
Surf
ace
eros
ion
begi
ns
Exp
osed
sur
face
aft
er ti
me
AB
C
Fig
ure
10.
Thr
ee c
ross
sec
tions
of
a m
orai
ne e
rodi
ng f
rom
rig
ht t
o le
ft i
n a
time
orde
red
seri
es.
As
the
outc
rop
surf
ace
grad
ually
ero
des
clas
ts o
rien
ted
norm
al t
o th
e ex
pose
d su
rfac
e ar
e ov
er-s
ampl
ed, w
hile
cla
sts
orie
nted
par
alle
l to
the
sur
face
ar
e un
der-
sam
pled
. A
. E
rosi
on b
egin
s at
the
ver
tical
sur
face
of
an o
utcr
op s
low
ly e
xpos
ing
the
clas
ts o
f a
wea
kly
orie
nted
fa
bric
tha
t is
com
pose
d of
equ
ival
ent
shap
ed c
last
s. B
. Aft
er t
ime,
an
indi
vidu
al c
last
with
a l
ong-
axis
ori
enta
tion
clos
ely
para
llel w
ith th
e er
odin
g su
rfac
e fa
lls o
ut o
f th
e ou
tcro
p w
hile
the
othe
r cl
asts
rem
ain
embe
dded
in th
e m
orai
ne C
. As
eros
ion
cont
inue
s, a
noth
er c
last
fal
ls o
ut o
f th
e ou
tcro
p ex
posu
re. T
hree
cla
sts
orie
nted
app
roxi
mat
ely
norm
al t
o th
e er
odin
g su
rfac
e re
mai
n ca
ntile
veri
ng o
ut f
rom
the
mor
aine
, and
wou
ld b
e ov
er-s
ampl
ed if
rec
orde
d. T
he tw
o cl
asts
ori
enta
tions
that
had
bee
n or
ient
ed n
earl
y pa
ralle
l to
the
outc
rop
surf
ace
wou
ld b
e un
der-
sam
pled
sin
ce th
ese
orie
ntat
ions
cou
ld n
o lo
nger
be
reco
rded
.
59
inevitably appear in the data. The magnitude of the surface sampling bias and its effect onthe apparent strength of a long-axis clast fabric is not only a function of the angle madebetween the preferred volumetric long-axis direction of the recorded clast orientations andthe orientation of the eroding outcrop surface but it is also a function of the average clastaspect ratio measured therein.
Sphere-like clasts have no bias, but they also convey no directional information.Elongate rods with aspect ratio approaching infinity (c≈b<<a) are the ideal clasts to indicatelineation fabric (e.g. Millar and Nelson, 2001b). Because of erosion, rods parallel to outcropsurface erode away almost as soon as they appear, but those with long axes normal to theoutcrop surface are preferentially exposed. Waterlain glacigenic sediments and flow tillsare deposits that are particularly vulnerable to the affect of sampling bias. These settingsproduce weak fabrics (e.g., Lawson, 1979) which could be misidentified as moderatelywell-developed fabrics if the sample of elongated clasts are limited to those visible at ornear the outcrop surface.
The probability of a clast being observed in situ depends upon the period of timeover which it is exposed at the surface. Clasts with their shortest axes normal to the erodingoutcrop surface will be removed by erosion quite soon after they are first exposed. Relativelyfew such clasts will be observed at any given time. Conversely, pebbles with their longaxes normal to the eroding outcrop surface will be exposed over an extended period timebefore they are removed. They will be relatively over-represented in field studies. I willassume that each clast is ellipsoidal and that it remains embedded in the outcrop until somefraction of its volume (here, I assume 1/2) is exposed (Fig. 9). Results seem to depend onlyweakly upon this assumption. The degree to which clasts of a given orientation are over- orunder-represented in a surficial sample can then be calculated as a function of the orientationof its axes with respect to the surface. If the erosion rate is either constant or random overtime, I can calculate this degree of preference simply by determining the projection of thepebble normal to the outcrop surface (the distance D in Figure 9).
The surface of an ellipsoidal clast is given by the equation.
[ x2/a2 + y2/b2 + z2/c2 ] = 1 (2)
I can define a function
F(x,y,z) = (1/a2) x2 + (1/b2) y2 + (1/c2) z2 - 1 (3)
such that the ellipsoid is the locus of points F(x,y,z)=0. Thus, N is a vector normal to theellipsoid at location (x,y,z).
N = (Nx, Ny, Nz) = 2 [ x /a2 , y /b2 , z /c2 ] = Kn = K( nx , ny , nz ) (4)
In general, N is not the unit normal n, but a normal vector of some length k. So
n = ( nx , ny , nz ) = 2/K [ x /a2 , y /b2 , z /c2 ] (5)
The vector P to a point on an ellipsoid of this family is
60
P(x,y,z) = (Ka2nx /2 , Kb2ny /2 , Kc2nz /2 ) (6)
Because the point falls on the ellipsoid, this must satisfy eqn. 2. Therefore,
[ (Ka2nx /2)2/a2 + (Kb2ny /2)2/b2 + (Kc2nz /2)2 /2)2/c2 ] = 1 (7)
and
4/K2 = a2nx2 + b2ny2 + c2nz2 (8)so
K = 2 [ a2nx2 + b2ny2 + c2nz2 ]-1/2 (9)
The component of P in the n direction is simply the projection D of the clast in the directionof the outcrop surface, so D = n• P = (Ka2nx2/2) + (Kb2ny2/2) + (Kc2nz2/2)
D = (K/2) [ (a2nx2) + (b2ny2) + (c2nz2) ]
Or D = [ a2nx2 + b2ny2 + c2nz2 ]1/2 (10)
Quantifying the bias in limiting cases
I use numerical methods and “Monte Carlo” calculations to quantify the samplebias due to recording clast orientations from an outcrop surface. For simplicity, it is assumedthat clasts are sufficiently widely spaced so that their interactions can be ignored. Eachclast is treated as an ellipsoid of rotation, (either b=a or b=c), in which case D (eqn. 10)gives the relative exposure ‘duration’ of a clast. Larger clasts are obviously exposed longestbut for any given size clast, shape and orientation also matter. For more nearly spheroidalclasts (a≈c), D is less strongly a function of axial orientation, but for geologically significantaspect ratios a>>c, D is a rather strong function of axial orientation leading to a significantsample bias.
In Monte Carlo calculations, I randomly ‘create’ thousands of rod-like clasts pernumerical experiment. For each experiment I assume a uniform clast size and aspect ratio(a/c), as well as a prescribed true anisotropy in clast orientation introduced by modifyingthe random statistics to produce a ‘preferred’ fabric. I then define an outcrop surfaceorientation anywhere from normal to parallel to the fabric axis. I calculate the true eigenvaluesand eigenvectors of this synthetic clast data set, as if observed by deep excavation.Simultaneously, I carry out the same calculations for the population of pebbles exposed atan outcrop surface, using D (eqn. 10) for each pebble to determine its relative likelihood of
61
appearing on the surface. For each experiment, I calculate for a population of 5000 pebblesand compare eigenvalues: the true S
1, S
2, S
3 vs. the ‘observed’ S
1', S
2', S
3'.
As a limiting case, I performed a set of numerical experiments with a true eigenvalueisotropic distribution (S
1 = S
2= S
3 = 1/3) by numerically generating clasts of aspect ratio a/c
approaching infinity (thin rods). This produced an ‘observed’ fabric of S1 = 0.5 and S
2= S
3
= 0.25. Thus, even a totally isotropic distribution composed of elongated clasts can lookanisotropic. The effect of the bias on the fabric depends on the angular separation (θ)between the most preferred eigenvector direction of the ‘true’ fabric with the orientation ofthe sampling surface. In addition, the effect is smaller with smaller a:c aspect ratio (Fig. 11)but still produces a significant sampling bias on clasts with realistic aspect ratios of a:c ≈2.5 (Fig. 12). The results of surface sampling ‘experiments’ show that the sampling biashas very little impact on ‘observed’ eigenvectors of strong fabrics (Fig. 13), which remainfairly stable even though the ‘observed’ eigenvalues are distributed widely. Moderate andweakly preferred surface sampled fabrics can have extreme observational errors in calculatedeigenvector direction (Fig. 14). These large errors are reproducible and cast serious doubton the accuracy of interpreted eigenvector directions that result from clast orientation datasets measured at poorly excavated field sites. ‘True’ eigenvalues of clast fabrics can not bereproduced from surface sampled clast orientations (Fig. 13,14, Table 2, Appendix A) but insome instances the eigenvalue ratios of both ‘true’ and ‘observed’ eigenvalues areserendipitously about equal, due to trade-offs among sets of eigenvalue combinations.
Implications for field studies
The sampling bias associated with clast axes measured in situ has very little impact onreliable eigenvector determination for strong fabrics. This bias causes deviations from truefabric directions (by a few degrees) that are smaller than the typical uncertainties due tovarious inevitable observational errors. The effect of the bias on direction is more pronouncedfor increasingly weak fabrics but is not large (tens of degrees) except for those fabricswhich very nearly isotropic. Extremely weak fabrics are inherently variable, giving littlepreferred orientation information, so they would not typically be used as directional indicatorswhen reconstructing former ice sheet movement even if they were not effected by such abias.
The sampling bias effect always skews the true eigenvalue distribution. Nearlyisotropic distributions can be falsely perceived as only a moderate lineated fabric if observedat a surface that is nearly parallel to the preferred clast long axis. Likewise, moderatelyclustered lineations can be misidentified either as nearly isotropic fabrics or as fabrics ofgreater than their true strength. Variability in flow strength and direction is undoubtedlyresponsible for the large eigenvalue range found for the flow till domain (Fig. 15). Thesurface sampling bias is by itself, capable of producing spurious results that vary by amountslarger than typically cited sizes of genetic till domain ranges (e.g., Dowdeswell et al., 1985).Figure 15, illustrates the effective eigenvalue range produced by the bias on several data
62
Fig
ure
11. (
S 1,
S 3)
eige
nval
ue p
lot
of c
ompu
ter
gene
rate
d ra
ndom
cla
st f
abri
c ei
genv
alue
dat
a se
ts f
or v
olum
e m
easu
red
fabr
ics
(sin
gle
data
poi
nts)
alo
ng w
ith s
urfa
ce s
ampl
e bi
as r
elat
ed e
igen
valu
e ob
serv
atio
n er
rors
. T
hese
eig
enva
lue
obse
rvat
ion
erro
rs a
re a
fun
ctio
n of
cla
st a
spec
t ra
tio (
a.r
= a
/c)
and
angu
lar
sepa
ratio
n, θ
, be
twee
n th
e m
ost
pref
erre
d ei
genv
ecto
r di
rect
ion
of t
he v
olum
e m
easu
red
fabr
ic a
nd t
he o
rien
tatio
n of
the
sam
plin
g su
rfac
e.
The
bia
s re
late
d ob
serv
atio
n er
rors
plo
t as
indi
vidu
al c
urve
s lin
king
eig
enva
lue
data
poi
nts
calc
ulat
ed w
ith v
alue
s of
θ =
90°,
60°,
45°,
30°,
and
0° (s
mal
l do
ts f
rom
lef
t to
rig
ht o
n ea
ch c
urve
).
The
asp
ect
ratio
for
eac
h cu
rve
link
data
poi
nts
that
are
ind
icat
ed b
y sm
all
blac
k do
ts f
or a
.r. =
5.0
and
sm
all
colo
red
dots
for
a.r.
= 2
.5.
In
addi
tion,
we
plot
com
pute
r ge
nera
ted
clas
t fa
bric
da
ta in
ord
er to
test
the
robu
stne
ss o
f fi
eld
data
obt
aine
d by
sur
face
sam
plin
g th
e st
ratif
ied
diam
ict u
nit a
t Ditc
h Pl
ains
.
63
Fig
ure
12. (
S 1,
S 3)
eige
nval
ue p
lot
pres
entin
g th
e a.
r. =
2.5
com
pute
r ge
nera
ted
rand
om c
last
fab
ric
eige
nval
ue d
ata
sets
of
figu
re 1
1, f
or v
olum
e m
easu
red
fabr
ics
(sin
gle
data
poi
nts)
alo
ng w
ith s
urfa
ce s
ampl
e bi
as r
elat
ed e
igen
valu
e ob
serv
atio
n er
rors
. T
he b
ias
rela
ted
obse
rvat
ion
erro
rs p
lot
as i
ndiv
idua
l cu
rves
lin
king
eig
enva
lue
data
poi
nts
calc
ulat
ed w
ith v
alue
s of
θ
= 90
°, 60
°, 45
°, 30
°, an
d 0°
(sm
all d
ots
from
left
to r
ight
on
each
cur
ve).
64
°
Fig
ure
13. D
etai
l of
(S1,
S 3)
eige
nval
ue p
lot s
how
n in
fig
ure
12, a
long
with
eig
enve
ctor
obs
erva
tion
erro
rs.
In
stro
ng f
abri
cs, s
urfa
ce s
ampl
ed e
igen
vect
ors
are
appr
oxim
atel
y eq
ual t
o th
e tr
ue p
opul
atio
n (v
olum
e m
easu
red)
ei
genv
ecto
rs.
Thu
s, ic
e fl
ow d
irec
tion
can
be in
ferr
ed f
rom
eig
enve
ctor
dir
ectio
ns o
f st
rong
cla
st f
abri
cs.
65
.33
.4
0
.5
0
.60
.70
S 3
S 1S 1
S 2=
.20
.10
0.0
Fig
ure
14. D
etai
l of
(S1,
S3)
eig
enva
lue
plot
sho
wn
in f
igur
e 12
, alo
ng w
ith e
igen
vect
or
obse
rvat
ion
erro
rs.
In
fab
rics
of
mod
est
stre
ngth
, su
rfac
e sa
mpl
ed e
igen
vect
ors
are
sign
ific
antly
dis
plac
ed f
rom
the
true
pop
ulat
ion
(vol
ume
mea
sure
d) e
igen
vect
or.
S2
3=
S
Obs
erva
tion
erro
rO
bser
vatio
n er
ror
Out
crop
ori
enta
tion
Obs
erva
tion
erro
rO
bser
vatio
n er
ror
1°±1
°O
utcr
op o
rien
tatio
n
Gen
erat
ed d
ata
(vol
ume
mea
sure
d fa
bric
)
8484°±
3°90
°
90°
60°
60°
45°
45°
30°
30°
0°
0°
41 41°±
3°28 28
°±2° 1919
°±2°
0°±1
°
0°±0
°5°
±1°
8°±1
°9°
±0°
66
Table 2. Computer generated random clast fabric data sets for volume measured fabrics (eigenvaluesand eigenvectors) along with observational errors and standard deviations due to the surface samplingbias. Surface measured eigenvalues and observation errors are based on clast aspect ratio, a/c, (a.r.= 2.5 and a.r. = 5.0) and the angular separation, θ, (0°, 30°, 45°, 60°, and 90°) between the preferredeigenvector direction of the volume measured fabric and the orientation of the outcrop surfaceresponsible for creating the sample bias.
s1 s3 s1 s3 θ (°) AVG (˚) ST DEV (˚) AVG (˚) ST DEV (˚)0.38 0.30 0.47 0.26 0 1.2 1.2 1.2 1.00.38 0.30 0.46 0.27 30 18.7 2.4 21.6 1.90.38 0.30 0.44 0.27 45 29.9 2.6 32.8 2.70.38 0.30 0.42 0.27 60 42.4 2.9 47.7 4.00.38 0.30 0.40 0.27 90 85.1 3.5 85.7 3.40.40 0.24 0.49 0.21 0 3.9 2.0 1.8 3.70.40 0.24 0.48 0.21 30 24.1 5.1 12.3 6.40.40 0.24 0.47 0.21 45 36.7 4.8 18.8 4.50.40 0.24 0.46 0.21 60 52.4 5.6 23.1 4.40.40 0.24 0.45 0.21 90 84.0 4.8 12.8 10.00.40 0.20 0.49 0.18 0 1.1 2.2 4.1 3.50.40 0.20 0.47 0.19 30 8.1 1.7 25.8 5.80.40 0.20 0.45 0.20 45 13.5 5.9 40.5 5.00.40 0.20 0.42 0.23 60 15.1 4.3 55.7 6.40.40 0.20 0.37 0.27 90 7.2 19.9 84.7 3.80.45 0.11 0.54 0.10 0 39.7 26.1 42.5 29.10.45 0.11 0.54 0.10 30 46.3 28.3 51.9 24.90.45 0.11 0.54 0.10 45 50.8 25.6 43.8 24.20.45 0.11 0.54 0.10 60 37.7 26.6 42.9 29.20.45 0.11 0.54 0.10 90 51.9 23.4 32.8 21.90.50 0.20 0.59 0.17 0 0.3 0.5 0.5 0.50.50 0.20 0.57 0.17 30 11.4 0.5 14.1 0.90.50 0.20 0.54 0.17 45 17.2 0.7 21.8 0.60.50 0.20 0.51 0.18 60 22.4 1.0 31.7 1.20.50 0.20 0.43 0.19 90 4.4 2.7 84.0 4.20.50 0.15 0.59 0.15 0 0.2 0.4 0.3 0.50.50 0.15 0.57 0.15 30 4.5 0.5 6.2 0.40.50 0.15 0.55 0.15 45 6.7 0.5 9.4 0.60.50 0.15 0.52 0.15 60 7.6 0.6 12.4 0.50.50 0.15 0.46 0.15 90 0.6 0.5 0.7 0.80.50 0.10 0.59 0.09 0 1.2 1.1 1.5 0.80.50 0.10 0.58 0.09 30 18.7 1.8 21.4 1.80.50 0.10 0.56 0.09 45 28.3 1.9 32.6 2.50.50 0.10 0.53 0.09 60 41.2 3.1 47.4 2.60.50 0.10 0.49 0.09 90 84.2 3.1 86.3 2.80.51 0.00 0.60 0.00 0 39.5 26.5 49.3 26.10.51 0.00 0.60 0.00 30 35.3 25.8 46.3 25.80.51 0.00 0.60 0.00 45 53.4 24.8 54.9 28.50.51 0.00 0.60 0.00 60 46.1 25.5 52.9 24.10.51 0.00 0.60 0.00 90 43.9 28.2 43.7 24.10.55 0.05 0.64 0.04 0 0.9 0.9 0.9 0.70.55 0.05 0.62 0.05 30 15.6 1.0 18.2 1.10.55 0.05 0.60 0.05 45 24.5 1.7 28.8 1.30.55 0.05 0.57 0.05 60 33.9 2.1 42.2 2.30.55 0.05 0.50 0.05 90 83.1 6.1 87.2 2.10.60 0.20 0.68 0.16 0 0.0 0.0 0.1 0.30.60 0.20 0.66 0.17 30 5.4 0.5 6.7 0.50.60 0.20 0.63 0.17 45 7.7 0.5 10.5 0.50.60 0.20 0.60 0.18 60 9.1 0.3 14.5 0.60.60 0.20 0.54 0.19 90 0.4 0.5 1.1 0.90.60 0.15 0.68 0.15 0 0.0 0.0 0.2 0.40.60 0.15 0.66 0.15 30 3.8 0.4 5.0 0.20.60 0.15 0.64 0.15 45 5.6 0.5 7.9 0.30.60 0.15 0.60 0.15 60 6.7 0.5 10.4 0.50.60 0.15 0.54 0.15 90 0.2 0.4 0.7 0.60.60 0.10 0.68 0.08 0 0.2 0.4 0.3 0.40.60 0.10 0.66 0.09 30 9.1 0.4 10.8 0.40.60 0.10 0.63 0.09 45 13.4 0.5 17.4 0.60.60 0.10 0.59 0.09 60 16.7 0.9 24.7 1.00.60 0.10 0.51 0.10 90 1.4 1.0 15.2 11.60.65 0.15 0.72 0.15 0 0.3 0.4 0.2 0.40.65 0.15 0.71 0.15 30 4.8 0.4 6.0 0.40.65 0.15 0.68 0.15 45 6.9 0.3 9.5 0.50.65 0.15 0.64 0.15 60 8.2 0.4 12.9 0.50.65 0.15 0.58 0.15 90 0.4 0.5 1.0 0.6
volume surface obs. error (a.r. = 2.5) obs. error (a.r. = 5.0)
67
s1 s3 s1 s3 θ (°) AVG (˚) ST DEV (˚) AVG (˚) ST DEV (˚)0.65 0.05 0.73 0.05 0 0.0 0.0 0.0 0.00.65 0.05 0.72 0.05 30 1.0 0.0 1.3 0.40.65 0.05 0.70 0.05 45 1.8 0.4 2.2 0.40.65 0.05 0.68 0.05 60 2.0 0.0 3.2 0.40.65 0.05 0.63 0.05 90 0.1 0.3 0.3 0.40.70 0.15 0.77 0.11 0 0.0 0.0 0.1 0.20.70 0.15 0.75 0.12 30 3.3 0.4 4.0 0.30.70 0.15 0.73 0.13 45 4.7 0.5 6.6 0.50.70 0.15 0.70 0.14 60 5.7 0.5 8.7 0.50.70 0.15 0.64 0.15 90 0.1 0.3 0.4 0.60.70 0.10 0.77 0.08 0 0.2 0.4 0.1 0.30.70 0.10 0.75 0.08 30 4.3 0.5 5.8 0.40.70 0.10 0.73 0.08 45 6.8 0.4 8.9 0.40.70 0.10 0.69 0.09 60 8.0 0.4 12.4 0.60.70 0.10 0.62 0.10 90 0.2 0.4 0.9 0.40.70 0.06 0.77 0.05 0 0.2 0.4 0.3 0.50.70 0.06 0.75 0.05 30 5.7 0.5 7.1 0.20.70 0.06 0.72 0.05 45 8.2 0.4 11.0 0.40.70 0.06 0.68 0.06 60 9.8 0.6 15.1 0.60.70 0.06 0.61 0.06 90 0.3 0.5 1.2 0.80.70 0.00 0.77 0.00 0 0.2 0.4 0.5 0.50.70 0.00 0.75 0.00 30 7.7 0.5 9.4 0.60.70 0.00 0.72 0.00 45 11.1 0.3 14.6 0.50.70 0.00 0.68 0.00 60 13.8 0.6 20.4 0.60.70 0.00 0.60 0.00 90 0.8 0.8 3.5 2.70.75 0.05 0.81 0.04 0 0.0 0.0 0.0 0.00.75 0.05 0.80 0.04 30 1.0 0.0 1.2 0.40.75 0.05 0.79 0.05 45 1.4 0.5 2.0 0.00.75 0.05 0.77 0.05 60 1.9 0.4 3.0 0.00.75 0.05 0.72 0.07 90 0.0 0.0 0.1 0.20.79 0.06 0.84 0.04 0 0.2 0.4 0.1 0.30.79 0.06 0.83 0.05 30 2.9 0.3 3.9 0.40.79 0.06 0.81 0.05 45 4.1 0.3 5.8 0.40.79 0.06 0.78 0.05 60 5.0 0.3 7.7 0.50.79 0.06 0.72 0.06 90 0.2 0.4 0.6 0.50.80 0.10 0.85 0.07 0 0.1 0.2 0.1 0.20.80 0.10 0.84 0.08 30 2.0 0.2 2.3 0.50.80 0.10 0.82 0.08 45 2.9 0.4 3.8 0.40.80 0.10 0.80 0.09 60 3.1 0.2 5.1 0.20.80 0.10 0.75 0.11 90 0.1 0.2 0.5 0.50.80 0.03 0.85 0.03 0 0.0 0.0 0.0 0.00.80 0.03 0.83 0.03 30 3.1 0.3 4.0 0.30.80 0.03 0.81 0.03 45 4.6 0.5 6.2 0.40.80 0.03 0.78 0.03 60 5.7 0.5 8.7 0.50.80 0.03 0.72 0.04 90 0.3 0.5 0.8 0.40.90 0.05 0.93 0.04 0 0.0 0.0 0.0 0.00.90 0.05 0.92 0.04 30 1.0 0.0 1.0 0.00.90 0.05 0.91 0.04 45 1.0 0.0 1.9 0.40.90 0.05 0.90 0.05 60 1.5 0.5 2.1 0.20.90 0.05 0.87 0.06 90 0.0 0.0 0.2 0.40.90 0.02 0.93 0.01 0 0.1 0.2 0.0 0.00.90 0.02 0.92 0.01 30 1.2 0.4 1.9 0.30.90 0.02 0.91 0.02 45 2.0 0.0 2.7 0.50.90 0.02 0.89 0.02 60 2.4 0.5 3.7 0.50.90 0.02 0.85 0.02 90 0.0 0.0 0.4 0.50.94 0.03 0.96 0.02 0 0.0 0.0 0.0 0.00.94 0.03 0.95 0.02 30 0.5 0.5 0.5 0.50.94 0.03 0.95 0.02 45 1.0 0.2 1.0 0.00.94 0.03 0.94 0.03 60 1.0 0.0 1.0 0.00.94 0.03 0.92 0.03 90 0.0 0.0 0.1 0.20.98 0.01 0.98 0.01 0 0.0 0.0 0.0 0.00.98 0.01 0.98 0.01 30 0.0 0.0 0.0 0.00.98 0.01 0.98 0.01 45 0.0 0.0 0.0 0.00.98 0.01 0.98 0.01 60 0.0 0.0 0.1 0.30.98 0.01 0.97 0.01 90 0.0 0.0 0.0 0.0
obs. error (a.r. = 5.0)volume surface obs. error (a.r. = 2.5)
68
Fig
ure
15. (
S 1, S
3) e
igen
valu
e pl
ot o
f gl
acig
enic
dia
mic
t fa
bric
dom
ains
by
stan
dard
dev
iatio
n el
lipse
s (e
.g.,
Dow
desw
ell
et a
l., 1
985)
. I
n ad
ditio
n, t
he p
lot
cont
ains
the
a.r.
= 2
.5 c
ompu
ter
gene
rate
d ra
ndom
cla
st f
abri
c ei
genv
alue
dat
a se
ts
show
n in
fig
ure
12,
for
volu
me
mea
sure
d fa
bric
s (s
ingl
e da
ta p
oint
s) a
long
with
sur
face
sam
ple
bias
rel
ated
eig
enva
lue
obse
rvat
ion
erro
rs.
The
sur
face
sam
ple
bias
ob
serv
atio
n er
rors
plo
t as
ind
ivid
ual
curv
es
with
ran
ge i
n S 1
and
S3
mag
nitu
des
prod
uced
by
the
sam
ple
bias
that
oft
en e
xcee
ds th
e st
anda
rd c
ited
size
s of
gla
cige
nic
diam
ict f
abri
c do
mai
ns.
69
points with aspect ratio a:c = 2.5 and how the magnitude of the eigenvalue bias is largerthan the standard deviation ellipses of genetic glacial diamict domains. Thus, ‘observed’eigenvalues vary greatly because of the surface sampling bias and cannot be relied upon todistinguish genetic glacial diamicts.
Conclusions
Studying glacigenic sediments using clast fabric analysis allows quantification ofotherwise descriptive fabrics. Eigenanalysis results of short-axis and long-axis orientationsobtained from the same set of clasts may establish a more complete picture of strain inglacigenic settings. Clast measurement in a poorly excavated outcrop produces a samplingbias in favor of clasts oriented normal to outcrop surface. The bias increases in severity forthose rod-like clasts which are the best suited to field analysis. This sampling bias is afunction of the orientation of the outcrop surface with respect to the true fabric direction. Ihave shown, through simple calculation and randomly generated clast populations, the degreeto which the surface sampling bias effects the true fabric. Such a bias can affect the perceivedfabric strength enough to distort interpretation dramatically. Fortunately, the effect uponfabric orientation for strong fabrics is generally smaller than random field measurementerrors and therefore can often be ignored. Orientation preferences of strong fabrics arereproducible despite the effect of the surface sampling, but interpretation of glacial settingsor the discrimination of genetic till-type using any set of surface observed eigenvalues shouldbe avoided. The robust long-axis eigenvector orientation found for the recorded clastswithin the stratified diamict unit of Ditch Plains is suggestive of well-developed shear fromthe NNW, but the mode of emplacement for the diamict still remains speculative. Under thebest of circumstance, distinguishing genetic till types by glacial diamict fabric ‘domains’should only be done under limited, local conditions (e.g., Kjaer and Krüger, 1998; Bennettet al., 1999). My results restrict the utility of such analyses even further. The effects of thesurface sampling bias are averted with careful sampling and interpretation.
70
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73
74
App
endi
x A
. C
ompi
latio
n of
all
com
pute
r ge
nera
ted
rand
om c
last
fab
ric
data
set
s fo
r vo
lum
e m
easu
red
fabr
ics
(eig
enva
lues
and
eig
enve
ctor
s)
alon
g w
ith
obse
rvat
iona
l er
rors
(an
gula
r di
spla
cem
ents
fro
m v
olum
e pr
efer
red
eige
nvec
tor
orie
ntat
ions
), s
tand
ard
devi
atio
ns,
max
imum
ob
serv
atio
n er
rors
, oth
er r
elat
ed p
aram
eter
s du
e to
the
sur
face
sam
plin
g bi
as.
Surf
ace
mea
sure
d ei
genv
alue
com
bina
tions
and
obs
erva
tion
erro
rs
are
base
d on
cla
st a
spec
t ra
tio,
a/c,
(a.
r. =
2.5
and
a.r.
= 5
.0)
and
the
angu
lar
sepa
ratio
n, θ
, (0
°, 30
°, 45
°, 60
°, an
d 9
0°)
betw
een
the
volu
me
mea
sure
d pr
efer
red
eige
nvec
tor
dire
ctio
n an
d th
e or
ient
atio
n of
the
outc
rop
surf
ace
resp
onsi
ble
for
crea
ting
the
sam
ple
bias
.
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)2.
50
0.38
10.
314
0.30
50.
470
0.26
80.
262
1.25
1.80
1.44
0.50
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2.4
1.2
1.2
2.5
300.
382
0.31
30.
304
0.45
80.
276
0.26
61.
261.
731.
370.
507
2321
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42.
545
0.38
20.
313
0.30
50.
443
0.28
90.
269
1.26
1.65
1.31
0.50
735
32.5
29.9
2.6
2.5
600.
382
0.31
30.
304
0.42
50.
306
0.27
01.
261.
581.
250.
507
4845
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92.
590
0.38
30.
313
0.30
40.
395
0.33
30.
272
1.26
1.46
1.16
0.50
790
88.5
85.1
3.5
2.5
00.
379
0.37
00.
251
0.46
30.
319
0.21
81.
512.
121.
400.
596
8772
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530
0.38
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369
0.25
10.
465
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219
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2.13
1.40
0.59
590
66.3
36.4
29.9
2.5
450.
379
0.36
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252
0.46
30.
317
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502.
111.
400.
594
8773
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560
0.37
70.
370
0.25
30.
463
0.31
70.
220
1.49
2.10
1.41
0.59
488
70.7
43.0
27.7
2.5
900.
379
0.37
00.
250
0.46
50.
317
0.21
81.
522.
131.
410.
597
7959
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50
0.40
10.
358
0.24
10.
490
0.30
20.
208
1.66
2.35
1.41
0.59
88
5.8
3.9
2.0
2.5
300.
400
0.35
80.
242
0.48
10.
309
0.21
01.
652.
291.
390.
597
3629
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12.
545
0.40
00.
360
0.24
00.
472
0.31
80.
209
1.67
2.26
1.35
0.60
047
41.4
36.7
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2.5
600.
399
0.35
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243
0.46
00.
328
0.21
21.
652.
171.
320.
596
6658
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62.
590
0.40
00.
358
0.24
20.
447
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213
1.65
2.10
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0.59
790
88.7
84.0
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2.5
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405
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203
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333
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72.
002.
771.
380.
660
103.
31.
12.
22.
530
0.40
30.
393
0.20
40.
473
0.33
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190
1.98
2.50
1.26
0.65
910
9.8
8.1
1.7
2.5
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404
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202
0.45
10.
345
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002.
221.
110.
662
3819
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92.
560
0.40
30.
393
0.20
40.
418
0.35
40.
228
1.98
1.83
0.93
0.65
925
19.3
15.1
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403
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30.
203
0.36
90.
360
0.27
21.
981.
360.
680.
659
9027
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219
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50
0.44
00.
284
0.27
60.
529
0.23
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232
1.60
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1.43
0.50
81
0.8
0.3
0.5
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441
0.28
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275
0.51
20.
250
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602.
151.
340.
508
1413
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02.
545
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284
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50.
489
0.27
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241
1.61
2.03
1.26
0.50
822
19.9
18.8
1.1
2.5
600.
441
0.28
40.
275
0.45
80.
296
0.24
71.
611.
851.
160.
508
2825
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32.
590
0.44
10.
283
0.27
60.
386
0.36
10.
253
1.60
1.53
0.95
0.50
725
15.4
7.9
7.5
2.5
00.
450
0.43
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111
0.53
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363
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055.
491.
350.
798
7765
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530
0.45
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440
0.11
10.
538
0.36
40.
098
4.06
5.49
1.35
0.79
990
74.6
46.3
28.3
2.5
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449
0.43
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113
0.53
70.
364
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995.
401.
350.
796
8376
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560
0.44
90.
438
0.11
20.
539
0.36
10.
099
4.00
5.44
1.36
0.79
689
64.3
37.7
26.6
2.5
900.
449
0.43
80.
114
0.53
60.
364
0.10
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965.
341.
350.
794
8875
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50
0.45
10.
441
0.10
80.
541
0.36
40.
095
4.18
5.67
1.36
0.80
483
84.4
37.6
26.8
2.5
300.
450
0.44
10.
109
0.54
00.
363
0.09
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135.
611.
360.
802
9067
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545
0.45
20.
441
0.10
70.
541
0.36
40.
095
4.22
5.70
1.35
0.80
582
67.9
44.6
23.4
2.5
600.
450
0.44
10.
109
0.53
90.
365
0.09
64.
155.
611.
350.
802
8578
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590
0.45
10.
441
0.10
80.
539
0.36
50.
096
4.19
5.64
1.35
0.80
388
76.3
50.7
25.7
volu
me
surf
ace
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)2.
50
0.46
50.
302
0.23
30.
554
0.21
30.
233
1.99
2.83
1.42
0.56
51
0.5
0.2
0.4
2.5
300.
465
0.30
10.
234
0.53
50.
232
0.23
41.
992.
541.
280.
563
109.
48.
80.
62.
545
0.46
40.
302
0.23
40.
509
0.25
70.
234
1.99
2.23
1.12
0.56
414
13.4
12.8
0.6
2.5
600.
465
0.30
10.
234
0.47
50.
291
0.23
41.
981.
850.
930.
562
1716
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52.
590
0.46
50.
300
0.23
50.
414
0.35
10.
235
1.98
1.51
0.76
0.56
03
2.1
1.3
0.8
2.5
00.
483
0.26
20.
255
0.57
10.
217
0.21
21.
902.
701.
420.
507
10.
80.
40.
52.
530
0.48
10.
263
0.25
60.
550
0.23
00.
220
1.88
2.50
1.33
0.50
611
10.2
9.6
0.6
2.5
450.
481
0.26
30.
256
0.52
40.
251
0.22
41.
882.
341.
240.
507
1515
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62.
560
0.48
20.
262
0.25
60.
491
0.28
00.
229
1.89
2.14
1.14
0.50
619
18.8
18.1
0.7
2.5
900.
482
0.26
30.
255
0.42
40.
339
0.23
71.
891.
790.
940.
508
42.
31.
41.
02.
50
0.50
10.
300
0.19
80.
589
0.24
60.
166
2.53
3.56
1.41
0.60
21
0.8
0.3
0.5
2.5
300.
501
0.30
10.
198
0.56
90.
262
0.16
92.
533.
371.
330.
603
1211
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52.
545
0.50
10.
299
0.20
00.
543
0.28
30.
174
2.50
3.12
1.25
0.59
919
17.9
17.2
0.7
2.5
600.
500
0.30
00.
200
0.50
80.
313
0.17
92.
502.
851.
140.
600
2523
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02.
590
0.50
00.
298
0.20
20.
431
0.38
30.
185
2.48
2.33
0.94
0.59
611
7.0
4.4
2.7
2.5
00.
499
0.34
90.
152
0.58
80.
260
0.15
23.
284.
571.
390.
696
10.
50.
20.
42.
530
0.49
80.
350
0.15
10.
570
0.27
80.
151
3.29
4.12
1.25
0.69
85
5.0
4.5
0.5
2.5
450.
497
0.35
20.
151
0.54
60.
303
0.15
13.
303.
621.
100.
700
77.
16.
70.
52.
560
0.49
90.
351
0.15
10.
515
0.33
40.
151
3.31
3.02
0.91
0.69
99
8.2
7.6
0.6
2.5
900.
499
0.35
00.
151
0.46
10.
388
0.15
13.
302.
230.
680.
698
11.
10.
60.
52.
50
0.50
30.
399
0.09
80.
594
0.32
10.
085
5.12
6.98
1.36
0.80
24
2.3
1.2
1.1
2.5
300.
503
0.39
70.
100
0.57
80.
335
0.08
75.
056.
641.
310.
800
2220
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82.
545
0.50
30.
397
0.10
00.
561
0.35
10.
088
5.04
6.36
1.26
0.79
932
30.1
28.3
1.9
2.5
600.
503
0.39
60.
101
0.53
40.
375
0.09
14.
995.
901.
180.
797
4844
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.23.
12.
590
0.50
30.
398
0.09
90.
494
0.41
60.
090
5.09
5.47
1.08
0.80
190
87.2
84.2
3.1
2.5
00.
507
0.49
30.
000
0.60
10.
399
0.00
01.
7E+
162.
5E+
161.
461.
000
8666
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.52.
530
0.50
60.
494
0.00
00.
601
0.39
90.
000
1.7E
+16
2.5E
+16
1.47
1.00
080
61.1
35.3
25.8
2.5
450.
508
0.49
20.
000
0.59
80.
402
0.00
01.
7E+
162.
4E+
161.
451.
000
9078
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560
0.50
70.
493
0.00
00.
599
0.40
10.
000
1.7E
+16
2.4E
+16
1.46
1.00
087
71.6
46.1
25.5
2.5
900.
506
0.49
40.
000
0.60
00.
400
0.00
01.
7E+
162.
5E+
161.
471.
000
8972
.143
.928
.22.
50
0.55
00.
318
0.13
20.
636
0.25
40.
110
4.16
5.77
1.39
0.70
61
0.8
0.4
0.5
2.5
300.
548
0.31
80.
134
0.61
40.
273
0.11
44.
115.
391.
310.
704
1211
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.00.
52.
545
0.54
80.
318
0.13
40.
587
0.29
60.
117
4.09
5.00
1.22
0.70
317
17.0
16.4
0.6
2.5
600.
548
0.31
80.
134
0.54
80.
332
0.12
14.
114.
541.
110.
704
2322
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.70.
82.
590
0.55
00.
316
0.13
40.
468
0.40
70.
125
4.12
3.74
0.91
0.70
37
4.2
2.4
1.9
volu
me
surf
ace
75
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)2.
50
0.55
10.
398
0.05
10.
641
0.31
50.
044
10.8
214
.52
1.34
0.88
63
1.7
0.9
0.9
2.5
300.
552
0.39
70.
051
0.62
30.
332
0.04
510
.77
13.8
11.
280.
885
1716
.615
.61.
02.
545
0.55
30.
396
0.05
10.
599
0.35
50.
046
10.9
713
.18
1.20
0.88
728
26.2
24.5
1.7
2.5
600.
554
0.39
50.
051
0.56
70.
386
0.04
610
.98
12.2
11.
110.
887
3936
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.92.
12.
590
0.55
10.
398
0.05
10.
499
0.45
30.
048
10.7
810
.36
0.96
0.88
690
89.1
83.1
6.1
2.5
00.
557
0.22
50.
218
0.64
00.
183
0.17
72.
563.
621.
410.
508
10.
50.
20.
42.
530
0.55
90.
224
0.21
70.
622
0.19
40.
184
2.57
3.39
1.32
0.50
77
6.9
6.4
0.5
2.5
450.
559
0.22
40.
217
0.59
70.
213
0.19
02.
583.
131.
220.
508
109.
99.
40.
52.
560
0.55
80.
224
0.21
80.
559
0.24
10.
200
2.57
2.80
1.09
0.50
712
11.7
11.1
0.6
2.5
900.
557
0.22
40.
219
0.49
40.
296
0.21
02.
542.
350.
920.
506
21.
30.
60.
72.
50
0.56
80.
344
0.08
80.
653
0.27
30.
074
6.44
8.79
1.37
0.79
61
0.7
0.3
0.4
2.5
300.
567
0.34
40.
088
0.63
20.
292
0.07
66.
448.
321.
290.
796
1312
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.70.
62.
545
0.56
70.
343
0.08
90.
605
0.31
60.
079
6.35
7.65
1.20
0.79
320
18.6
17.9
0.7
2.5
600.
568
0.34
50.
088
0.56
50.
355
0.08
06.
497.
071.
090.
797
2725
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.21.
42.
590
0.56
80.
344
0.08
80.
477
0.43
90.
083
6.46
5.75
0.89
0.79
617
11.1
6.1
5.1
2.5
00.
597
0.20
50.
198
0.67
70.
164
0.15
93.
014.
251.
410.
508
00.
00.
00.
02.
530
0.60
00.
203
0.19
70.
660
0.17
40.
166
3.05
3.99
1.31
0.50
86
5.8
5.4
0.5
2.5
450.
599
0.20
40.
197
0.63
50.
193
0.17
23.
053.
691.
210.
509
88.
17.
70.
52.
560
0.60
00.
203
0.19
70.
599
0.22
00.
181
3.05
3.30
1.09
0.50
810
9.4
9.1
0.3
2.5
900.
601
0.20
20.
197
0.53
60.
272
0.19
33.
052.
780.
910.
507
10.
90.
40.
52.
50
0.59
80.
250
0.15
10.
678
0.17
10.
151
3.95
5.51
1.39
0.62
30
0.0
0.0
0.0
2.5
300.
597
0.25
20.
151
0.66
00.
189
0.15
13.
964.
971.
260.
625
44.
23.
80.
42.
545
0.59
70.
251
0.15
20.
638
0.21
00.
152
3.93
4.30
1.10
0.62
26
6.1
5.6
0.5
2.5
600.
598
0.25
00.
152
0.60
50.
243
0.15
23.
923.
550.
910.
621
77.
26.
70.
52.
590
0.59
60.
252
0.15
20.
545
0.30
30.
152
3.92
2.60
0.66
0.62
31
0.6
0.2
0.4
2.5
00.
599
0.30
00.
101
0.68
00.
237
0.08
35.
938.
161.
380.
748
10.
60.
20.
42.
530
0.60
00.
300
0.10
10.
659
0.25
50.
086
5.95
7.69
1.29
0.74
810
9.4
9.1
0.4
2.5
450.
599
0.30
10.
100
0.63
00.
282
0.08
85.
977.
141.
200.
750
1413
.913
.40.
52.
560
0.60
10.
299
0.10
10.
591
0.31
60.
092
5.97
6.40
1.07
0.74
818
17.6
16.7
0.9
2.5
900.
602
0.29
90.
099
0.51
30.
392
0.09
56.
095.
380.
880.
752
42.
41.
41.
02.
50
0.59
90.
327
0.07
40.
681
0.25
80.
062
8.14
11.0
81.
360.
816
10.
50.
20.
42.
530
0.60
00.
326
0.07
40.
660
0.27
60.
064
8.10
10.3
51.
280.
814
1110
.710
.10.
62.
545
0.59
70.
329
0.07
50.
628
0.30
50.
066
7.98
9.47
1.19
0.81
517
16.5
15.7
0.9
2.5
600.
598
0.32
80.
074
0.59
00.
342
0.06
88.
118.
681.
070.
816
2120
.720
.00.
72.
590
0.59
70.
329
0.07
50.
503
0.42
50.
072
8.01
6.99
0.87
0.81
55
3.5
2.0
1.6
volu
me
surf
ace
76
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)2.
50
0.65
30.
195
0.15
20.
725
0.12
30.
152
4.29
5.98
1.39
0.56
21
0.7
0.3
0.4
2.5
300.
651
0.19
60.
152
0.70
60.
141
0.15
24.
275.
621.
320.
563
55.
24.
80.
42.
545
0.65
20.
197
0.15
10.
682
0.16
70.
151
4.33
5.25
1.21
0.56
67
7.2
6.9
0.3
2.5
600.
649
0.19
70.
154
0.64
20.
203
0.15
44.
214.
561.
080.
561
98.
68.
20.
42.
590
0.65
10.
196
0.15
20.
580
0.26
80.
152
4.28
3.84
0.90
0.56
31
0.8
0.4
0.5
2.5
00.
651
0.29
90.
051
0.72
60.
223
0.05
112
.87
17.4
11.
350.
855
00.
00.
00.
02.
530
0.65
20.
298
0.05
10.
717
0.23
30.
051
12.8
816
.02
1.24
0.85
51
1.0
1.0
0.0
2.5
450.
651
0.29
90.
051
0.70
10.
248
0.05
112
.86
14.1
61.
100.
855
22.
21.
80.
42.
560
0.65
00.
299
0.05
10.
677
0.27
20.
051
12.8
111
.94
0.93
0.85
52
2.0
2.0
0.0
2.5
900.
650
0.29
90.
050
0.62
90.
321
0.05
012
.92
8.80
0.68
0.85
61
0.4
0.1
0.3
2.5
00.
664
0.17
10.
165
0.73
40.
135
0.13
14.
015.
611.
400.
509
10.
80.
30.
52.
530
0.66
30.
172
0.16
50.
717
0.14
50.
138
4.01
5.21
1.30
0.51
04
4.2
4.0
0.2
2.5
450.
662
0.17
20.
166
0.69
30.
162
0.14
43.
984.
801.
210.
509
66.
15.
60.
52.
560
0.66
20.
172
0.16
60.
659
0.18
60.
155
3.98
4.25
1.07
0.50
87
7.1
6.6
0.5
2.5
900.
663
0.17
20.
165
0.59
70.
235
0.16
84.
013.
570.
890.
509
10.
80.
40.
52.
50
0.70
30.
151
0.14
60.
767
0.11
80.
115
4.80
6.69
1.39
0.50
80
0.0
0.0
0.0
2.5
300.
704
0.15
10.
146
0.75
30.
128
0.11
94.
836.
321.
310.
509
43.
73.
30.
42.
545
0.70
00.
153
0.14
70.
729
0.14
30.
128
4.75
5.69
1.20
0.50
95
5.2
4.7
0.5
2.5
600.
703
0.15
10.
146
0.69
90.
165
0.13
64.
825.
151.
070.
509
66.
15.
70.
52.
590
0.70
20.
152
0.14
60.
638
0.21
00.
152
4.81
4.20
0.87
0.50
91
0.4
0.1
0.3
2.5
00.
700
0.20
20.
098
0.76
60.
156
0.07
87.
149.
841.
380.
673
10.
50.
20.
42.
530
0.70
00.
201
0.09
90.
748
0.17
00.
082
7.04
9.11
1.29
0.66
95
4.8
4.3
0.5
2.5
450.
702
0.20
10.
097
0.72
50.
190
0.08
47.
258.
601.
190.
674
77.
26.
80.
42.
560
0.70
10.
201
0.09
90.
687
0.22
10.
092
7.10
7.50
1.06
0.67
09
8.3
8.0
0.4
2.5
900.
700
0.20
10.
098
0.61
90.
280
0.10
17.
126.
150.
860.
672
10.
60.
20.
42.
50
0.70
10.
239
0.06
00.
768
0.18
40.
048
11.6
415
.87
1.36
0.79
81
0.6
0.2
0.4
2.5
300.
698
0.24
00.
062
0.74
60.
202
0.05
211
.33
14.4
71.
280.
796
66.
15.
70.
52.
545
0.70
10.
238
0.06
10.
722
0.22
40.
054
11.5
513
.47
1.17
0.79
79
8.5
8.2
0.4
2.5
600.
700
0.23
90.
061
0.68
20.
261
0.05
811
.45
11.8
01.
030.
796
1110
.49.
80.
62.
590
0.70
10.
238
0.06
10.
610
0.32
70.
063
11.5
99.
750.
840.
798
10.
80.
30.
52.
50
0.70
00.
300
0.00
00.
770
0.23
00.
000
3.8E
+16
5.5E
+16
1.43
1.00
01
0.6
0.2
0.4
2.5
300.
700
0.30
00.
000
0.74
80.
252
0.00
03.
8E+
164.
9E+
161.
281.
000
88.
27.
70.
52.
545
0.70
10.
299
0.00
00.
720
0.28
00.
000
3.8E
+16
4.2E
+16
1.10
1.00
012
11.4
11.1
0.3
2.5
600.
700
0.30
00.
000
0.67
80.
322
0.00
03.
8E+
163.
4E+
160.
901.
000
1514
.413
.80.
62.
590
0.70
00.
300
0.00
00.
595
0.40
50.
000
3.8E
+16
2.4E
+16
0.63
1.00
03
1.6
0.8
0.8
volu
me
surf
ace
77
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)2.
50
0.75
00.
199
0.05
10.
808
0.15
20.
040
14.8
020
.24
1.37
0.79
70
0.0
0.0
0.0
2.5
300.
749
0.20
10.
050
0.79
90.
158
0.04
315
.00
18.7
71.
250.
800
11.
01.
00.
02.
545
0.75
10.
199
0.05
10.
788
0.16
40.
048
14.8
216
.43
1.11
0.79
72
1.8
1.4
0.5
2.5
600.
751
0.19
90.
050
0.76
70.
178
0.05
515
.08
13.9
80.
930.
800
22.
21.
90.
42.
590
0.74
90.
200
0.05
10.
721
0.20
60.
073
14.7
69.
850.
670.
797
00.
00.
00.
02.
50
0.76
20.
121
0.11
70.
816
0.09
30.
090
6.54
9.05
1.38
0.51
01
0.3
0.1
0.2
2.5
300.
762
0.12
10.
116
0.80
40.
101
0.09
56.
558.
451.
290.
511
32.
52.
20.
42.
545
0.76
00.
122
0.11
70.
785
0.11
40.
101
6.47
7.74
1.20
0.51
04
4.0
3.5
0.5
2.5
600.
760
0.12
20.
118
0.75
60.
132
0.11
16.
466.
811.
060.
509
54.
34.
10.
22.
590
0.76
20.
122
0.11
70.
702
0.17
30.
125
6.53
5.60
0.86
0.51
01
0.6
0.2
0.4
2.5
00.
793
0.15
10.
057
0.84
10.
114
0.04
413
.96
19.1
21.
370.
726
10.
50.
20.
42.
530
0.79
30.
150
0.05
80.
828
0.12
50.
047
13.7
417
.62
1.28
0.72
23
3.2
2.9
0.3
2.5
450.
793
0.15
00.
057
0.80
90.
141
0.05
013
.96
16.3
51.
170.
724
54.
44.
10.
32.
560
0.79
20.
150
0.05
70.
777
0.16
80.
055
13.8
514
.27
1.03
0.72
46
5.3
5.0
0.3
2.5
900.
793
0.15
00.
057
0.71
80.
220
0.06
214
.03
11.5
20.
820.
727
10.
50.
20.
42.
50
0.80
10.
102
0.09
70.
847
0.07
80.
075
8.22
11.3
41.
380.
511
10.
30.
10.
22.
530
0.80
00.
102
0.09
80.
836
0.08
40.
080
8.15
10.5
21.
290.
510
22.
22.
00.
22.
545
0.80
00.
102
0.09
80.
821
0.09
40.
085
8.18
9.69
1.18
0.51
13
3.2
2.9
0.4
2.5
600.
800
0.10
20.
098
0.79
50.
113
0.09
28.
158.
621.
060.
510
43.
33.
10.
22.
590
0.80
20.
101
0.09
70.
747
0.14
60.
106
8.31
7.03
0.85
0.51
11
0.3
0.1
0.2
2.5
00.
802
0.16
40.
034
0.84
90.
124
0.02
623
.61
32.1
51.
360.
828
00.
00.
00.
02.
530
0.80
20.
165
0.03
40.
835
0.13
70.
028
23.8
330
.15
1.27
0.83
04
3.4
3.1
0.3
2.5
450.
802
0.16
40.
034
0.81
50.
155
0.03
023
.39
26.8
81.
150.
827
55.
14.
60.
52.
560
0.80
20.
164
0.03
30.
783
0.18
40.
032
24.1
324
.18
1.00
0.83
16
6.2
5.7
0.5
2.5
900.
803
0.16
40.
034
0.72
20.
240
0.03
823
.90
19.0
10.
800.
830
10.
80.
30.
52.
50
0.79
90.
168
0.03
30.
847
0.12
00.
033
23.9
832
.54
1.36
0.83
40
0.0
0.0
0.0
2.5
300.
799
0.16
70.
034
0.84
10.
125
0.03
423
.88
29.4
91.
240.
833
11.
20.
80.
42.
545
0.79
80.
169
0.03
40.
830
0.13
60.
034
23.6
426
.33
1.11
0.83
31
1.0
1.0
0.0
2.5
600.
799
0.16
80.
033
0.81
50.
152
0.03
324
.03
22.6
50.
940.
835
11.
01.
00.
02.
590
0.79
90.
168
0.03
40.
777
0.18
90.
034
23.8
616
.08
0.67
0.83
30
0.0
0.0
0.0
2.5
00.
833
0.13
50.
032
0.87
40.
102
0.02
525
.73
35.1
31.
370.
806
10.
30.
10.
22.
530
0.83
40.
134
0.03
20.
862
0.11
20.
026
26.1
833
.18
1.27
0.80
83
3.0
2.5
0.5
2.5
450.
834
0.13
50.
031
0.84
50.
128
0.02
826
.56
30.6
71.
150.
811
44.
13.
60.
52.
560
0.83
40.
134
0.03
20.
817
0.15
20.
031
26.0
326
.26
1.01
0.80
75
4.5
4.2
0.4
2.5
900.
832
0.13
50.
032
0.75
90.
203
0.03
725
.85
20.3
60.
790.
808
10.
70.
30.
4
volu
me
surf
ace
78
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)2.
50
0.86
10.
072
0.06
80.
895
0.05
40.
051
12.6
917
.39
1.37
0.51
30
0.0
0.0
0.0
2.5
300.
860
0.07
10.
068
0.88
70.
058
0.05
512
.61
16.2
21.
290.
511
11.
01.
00.
02.
545
0.85
90.
072
0.06
90.
875
0.06
60.
059
12.4
814
.75
1.18
0.51
22
2.0
2.0
0.0
2.5
600.
861
0.07
10.
068
0.85
80.
077
0.06
412
.74
13.3
31.
050.
512
22.
02.
00.
02.
590
0.86
00.
072
0.06
80.
815
0.10
70.
079
12.5
710
.36
0.82
0.51
10
0.0
0.0
0.0
2.5
00.
903
0.05
00.
047
0.92
70.
037
0.03
619
.14
26.1
51.
370.
515
00.
00.
00.
02.
530
0.90
10.
051
0.04
80.
921
0.04
10.
038
18.7
123
.93
1.28
0.51
31
1.0
1.0
0.0
2.5
450.
902
0.05
10.
047
0.91
30.
046
0.04
119
.10
22.5
61.
180.
516
11.
01.
00.
02.
560
0.90
10.
051
0.04
80.
898
0.05
60.
046
18.6
719
.40
1.04
0.51
42
2.0
1.5
0.5
2.5
900.
902
0.05
00.
048
0.86
70.
077
0.05
618
.82
15.4
00.
820.
512
00.
00.
00.
02.
50
0.90
00.
080
0.02
00.
925
0.06
00.
015
44.5
660
.89
1.37
0.79
80
0.0
0.0
0.0
2.5
300.
901
0.07
90.
020
0.91
80.
066
0.01
645
.05
57.3
01.
270.
798
21.
71.
30.
42.
545
0.90
00.
080
0.02
00.
907
0.07
50.
018
44.9
551
.86
1.15
0.79
93
2.3
2.1
0.2
2.5
600.
900
0.08
00.
020
0.88
80.
092
0.02
044
.22
44.2
71.
000.
796
32.
62.
20.
42.
590
0.90
00.
080
0.02
00.
848
0.12
80.
025
45.2
834
.39
0.76
0.80
11
0.3
0.1
0.2
2.5
00.
901
0.08
20.
017
0.92
60.
061
0.01
352
.52
71.4
91.
360.
827
10.
30.
10.
22.
530
0.90
00.
082
0.01
70.
918
0.06
80.
014
52.0
866
.09
1.27
0.82
62
1.6
1.2
0.4
2.5
450.
900
0.08
30.
018
0.90
60.
078
0.01
551
.27
59.1
41.
150.
824
22.
02.
00.
02.
560
0.90
10.
082
0.01
70.
888
0.09
40.
017
51.8
251
.34
0.99
0.82
53
2.8
2.4
0.5
2.5
900.
901
0.08
20.
017
0.84
70.
131
0.02
252
.07
38.9
60.
750.
825
00.
00.
00.
02.
50
0.94
10.
030
0.02
80.
956
0.02
20.
021
33.1
344
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1.36
0.51
70
0.0
0.0
0.0
2.5
300.
941
0.03
10.
028
0.95
30.
025
0.02
333
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42.2
01.
270.
520
11.
00.
50.
52.
545
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10.
031
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80.
948
0.02
80.
024
33.3
039
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1.2
1.0
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600.
942
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00.
028
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00.
033
0.02
733
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34.7
01.
040.
518
11.
01.
00.
02.
590
0.94
10.
030
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80.
918
0.04
70.
035
33.3
126
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0.80
0.51
70
0.0
0.0
0.0
2.5
00.
959
0.03
60.
005
0.97
00.
027
0.00
419
9.66
269.
841.
360.
881
00.
00.
00.
02.
530
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80.
037
0.00
50.
966
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00.
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198.
6624
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0.88
31
1.2
0.8
0.4
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959
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60.
005
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10.
035
0.00
419
6.98
222.
151.
130.
881
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01.
00.
02.
560
0.95
80.
037
0.00
50.
952
0.04
30.
005
191.
3318
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0.96
0.87
91
1.0
1.0
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959
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005
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20.
062
0.00
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8.46
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580.
700.
881
10.
40.
10.
32.
50
0.97
90.
011
0.01
00.
985
0.00
80.
007
100.
3013
5.56
1.35
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10
0.0
0.0
0.0
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300.
979
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10.
010
0.98
40.
009
0.00
810
2.00
128.
541.
260.
533
00.
00.
00.
02.
545
0.97
90.
011
0.01
00.
982
0.01
00.
008
101.
7311
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30
0.0
0.0
0.0
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600.
979
0.01
10.
010
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80.
012
0.01
098
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99.6
61.
010.
531
00.
00.
00.
02.
590
0.97
90.
011
0.01
00.
970
0.01
70.
012
102.
5780
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0.78
0.53
30
0.0
0.0
0.0
volu
me
surf
ace
79
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)5.
00
0.38
30.
313
0.30
40.
515
0.24
60.
239
1.26
2.15
1.71
0.50
84
2.2
1.2
1.0
5.0
300.
382
0.31
40.
305
0.50
00.
255
0.24
51.
252.
051.
630.
507
2623
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9
5.0
450.
382
0.31
40.
304
0.48
60.
267
0.24
71.
261.
961.
560.
508
3935
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7
5.0
600.
382
0.31
20.
305
0.46
60.
282
0.25
21.
251.
851.
480.
506
5551
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0
5.0
900.
382
0.31
30.
305
0.44
10.
305
0.25
41.
251.
741.
390.
506
9089
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4
5.0
00.
378
0.36
80.
254
0.50
60.
289
0.20
51.
492.
471.
660.
592
8376
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5.0
300.
378
0.36
90.
253
0.50
70.
289
0.20
41.
502.
481.
660.
593
8873
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5.0
450.
379
0.36
80.
253
0.50
70.
289
0.20
41.
502.
481.
650.
593
8572
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5.0
600.
378
0.36
90.
253
0.50
60.
289
0.20
51.
502.
471.
650.
593
7570
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5.0
900.
380
0.37
00.
251
0.50
90.
289
0.20
21.
522.
521.
660.
596
8859
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5.0
00.
404
0.39
30.
203
0.53
30.
302
0.16
61.
993.
221.
620.
659
165.
51.
83.
7
5.0
300.
404
0.39
30.
203
0.51
10.
310
0.17
91.
992.
861.
440.
659
3218
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4
5.0
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404
0.39
30.
204
0.48
60.
316
0.19
81.
982.
451.
240.
658
3023
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5
5.0
600.
405
0.39
20.
202
0.44
70.
325
0.22
82.
011.
960.
980.
660
3227
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4
5.0
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403
0.39
30.
203
0.34
40.
336
0.32
01.
981.
080.
540.
659
3422
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5.0
00.
399
0.35
70.
244
0.53
20.
274
0.19
41.
642.
741.
670.
594
117.
54.
13.
5
5.0
300.
401
0.35
80.
242
0.52
30.
282
0.19
51.
662.
691.
620.
597
3631
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8
5.0
450.
399
0.35
80.
243
0.51
30.
289
0.19
71.
642.
601.
580.
596
5445
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0
5.0
600.
398
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90.
243
0.50
40.
299
0.19
71.
642.
551.
560.
597
6562
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4
5.0
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400
0.35
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241
0.49
40.
309
0.19
71.
662.
501.
510.
599
8988
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8
5.0
00.
441
0.28
40.
275
0.56
90.
218
0.21
21.
602.
681.
670.
508
11.
10.
60.
5
5.0
300.
441
0.28
40.
275
0.55
00.
231
0.21
91.
602.
521.
570.
508
1615
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8
5.0
450.
441
0.28
40.
276
0.52
50.
251
0.22
41.
602.
341.
460.
507
2524
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0
5.0
600.
440
0.28
40.
276
0.48
80.
281
0.23
11.
602.
111.
320.
507
3735
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9
5.0
900.
441
0.28
30.
275
0.41
10.
352
0.23
71.
601.
731.
080.
507
8987
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6
5.0
00.
451
0.43
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112
0.57
90.
328
0.09
34.
036.
211.
540.
796
8871
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5.0
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450
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112
0.57
80.
328
0.09
44.
026.
151.
530.
796
8876
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5.0
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449
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112
0.58
00.
327
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201.
540.
797
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5.0
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449
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112
0.57
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327
0.09
34.
016.
201.
540.
797
8972
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900.
450
0.43
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113
0.58
10.
324
0.09
43.
976.
161.
550.
794
8654
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5.0
00.
452
0.44
10.
107
0.58
30.
328
0.09
04.
236.
491.
540.
805
8974
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5.0
300.
452
0.44
00.
109
0.58
10.
328
0.09
14.
166.
431.
540.
802
8773
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5.0
450.
451
0.44
10.
109
0.58
30.
326
0.09
14.
156.
431.
550.
802
6852
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5.0
600.
451
0.43
90.
109
0.58
00.
329
0.09
14.
136.
341.
540.
801
8877
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090
0.45
30.
439
0.10
80.
580
0.33
00.
090
4.20
6.43
1.53
0.80
390
80.6
54.2
26.4
volu
me
surf
ace
80
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)5.
00
0.46
50.
301
0.23
40.
592
0.22
80.
181
1.99
3.27
1.64
0.56
31
0.6
0.2
0.4
5.0
300.
465
0.30
10.
234
0.57
10.
233
0.19
61.
992.
921.
470.
563
1211
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6
5.0
450.
465
0.30
20.
233
0.54
30.
241
0.21
72.
002.
501.
250.
565
1817
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7
5.0
600.
465
0.30
10.
234
0.49
80.
255
0.24
71.
992.
021.
020.
562
2625
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1
5.0
900.
465
0.30
20.
233
0.38
20.
356
0.26
22.
001.
460.
730.
565
3417
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28.
1
5.0
00.
481
0.26
30.
256
0.60
50.
200
0.19
51.
883.
111.
650.
507
10.
80.
40.
5
5.0
300.
480
0.26
30.
257
0.58
40.
214
0.20
11.
872.
901.
550.
506
1313
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7
5.0
450.
482
0.26
20.
255
0.55
80.
236
0.20
61.
892.
711.
430.
506
2019
.418
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7
5.0
600.
481
0.26
40.
255
0.51
30.
271
0.21
61.
882.
381.
260.
508
3027
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0
5.0
900.
479
0.26
40.
257
0.39
30.
380
0.22
71.
861.
730.
930.
507
8976
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5.0
00.
500
0.30
00.
200
0.62
30.
223
0.15
42.
504.
051.
620.
600
11.
00.
50.
5
5.0
300.
499
0.30
00.
200
0.60
10.
241
0.15
82.
493.
801.
520.
600
1615
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9
5.0
450.
500
0.30
00.
200
0.57
50.
263
0.16
22.
513.
551.
420.
601
2322
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6
5.0
600.
500
0.30
00.
200
0.53
30.
300
0.16
82.
503.
181.
270.
601
3432
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2
5.0
900.
497
0.30
10.
202
0.43
80.
386
0.17
62.
462.
481.
010.
599
9088
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2
5.0
00.
498
0.35
20.
150
0.62
20.
259
0.11
93.
325.
251.
580.
701
10.
80.
30.
5
5.0
300.
497
0.35
10.
152
0.60
50.
265
0.13
13.
274.
621.
410.
698
76.
56.
20.
4
5.0
450.
499
0.34
90.
152
0.57
90.
273
0.14
83.
283.
931.
200.
697
109.
99.
40.
6
5.0
600.
497
0.35
20.
151
0.53
50.
290
0.17
53.
293.
050.
930.
700
1312
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5
5.0
900.
495
0.35
30.
152
0.42
50.
320
0.25
53.
271.
670.
510.
700
31.
50.
70.
8
5.0
00.
503
0.39
80.
099
0.63
00.
289
0.08
15.
067.
791.
540.
800
32.
31.
50.
8
5.0
300.
502
0.39
80.
100
0.61
40.
304
0.08
25.
047.
481.
480.
800
2523
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8
5.0
450.
501
0.39
90.
100
0.59
60.
320
0.08
45.
027.
141.
420.
800
4035
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5
5.0
600.
501
0.39
90.
100
0.57
20.
342
0.08
55.
006.
711.
340.
799
5249
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6
5.0
900.
501
0.39
90.
100
0.54
10.
372
0.08
75.
026.
221.
240.
800
9089
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8
5.0
00.
505
0.49
50.
000
0.64
00.
360
0.00
01.
7E+
162.
9E+
161.
741.
000
9075
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5.0
300.
507
0.49
30.
000
0.64
00.
360
0.00
01.
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162.
9E+
161.
731.
000
8472
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5.0
450.
506
0.49
40.
000
0.63
90.
361
0.00
01.
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162.
9E+
161.
731.
000
8983
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5.0
600.
506
0.49
40.
000
0.63
80.
362
0.00
01.
7E+
162.
9E+
161.
731.
000
8977
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5.0
900.
507
0.49
30.
000
0.64
10.
359
0.00
01.
7E+
162.
9E+
161.
741.
000
8467
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5.0
00.
550
0.31
80.
133
0.66
70.
230
0.10
24.
156.
531.
580.
705
10.
80.
30.
5
5.0
300.
552
0.31
60.
132
0.64
70.
248
0.10
54.
186.
171.
480.
705
1413
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9
5.0
450.
550
0.31
70.
133
0.61
60.
275
0.10
94.
155.
641.
360.
705
2322
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2
5.0
600.
550
0.31
80.
133
0.57
20.
314
0.11
44.
145.
021.
210.
705
3130
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75.
090
0.55
00.
318
0.13
30.
460
0.41
90.
120
4.14
3.83
0.92
0.70
590
88.9
83.7
5.2
volu
me
surf
ace
81
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)5.
00
0.55
20.
397
0.05
10.
675
0.28
40.
041
10.9
216
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1.50
0.88
72
1.5
0.9
0.7
5.0
300.
551
0.39
80.
051
0.65
50.
302
0.04
310
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15.3
71.
430.
886
2119
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1
5.0
450.
552
0.39
70.
052
0.63
20.
324
0.04
410
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14.3
11.
330.
885
3130
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3
5.0
600.
551
0.39
80.
051
0.59
90.
355
0.04
510
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13.2
51.
230.
886
4644
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3
5.0
900.
552
0.39
70.
051
0.54
40.
408
0.04
710
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11.5
31.
070.
886
9089
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1
5.0
00.
557
0.22
50.
218
0.67
10.
167
0.16
22.
554.
131.
620.
508
10.
70.
30.
4
5.0
300.
559
0.22
40.
217
0.65
20.
180
0.16
82.
583.
871.
500.
508
98.
78.
30.
4
5.0
450.
558
0.22
40.
218
0.62
20.
202
0.17
62.
563.
541.
380.
507
1413
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6
5.0
600.
561
0.22
30.
216
0.57
60.
240
0.18
42.
593.
121.
200.
508
1918
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6
5.0
900.
560
0.22
30.
217
0.45
10.
346
0.20
22.
582.
230.
870.
507
42.
71.
80.
9
5.0
00.
567
0.34
50.
088
0.68
40.
247
0.06
96.
479.
921.
530.
797
21.
20.
60.
6
5.0
300.
567
0.34
50.
088
0.66
20.
267
0.07
16.
459.
311.
440.
797
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6
5.0
450.
564
0.34
70.
089
0.63
20.
294
0.07
56.
338.
481.
340.
795
2623
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2
5.0
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565
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70.
089
0.58
90.
334
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76.
387.
631.
200.
797
3534
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0
5.0
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40.
089
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940.
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8988
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6
5.0
00.
599
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30.
198
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149
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034.
871.
610.
507
10.
40.
10.
3
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30.
197
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80.
162
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054.
571.
500.
508
77.
26.
70.
5
5.0
450.
599
0.20
40.
198
0.65
70.
183
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93.
034.
121.
360.
507
1111
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5
5.0
600.
598
0.20
40.
198
0.60
90.
221
0.17
13.
023.
571.
180.
508
1615
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6
5.0
900.
599
0.20
40.
198
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50.
325
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032.
550.
840.
507
31.
91.
10.
9
5.0
00.
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152
0.70
60.
180
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43.
936.
201.
580.
620
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50.
20.
4
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300.
598
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151
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80.
187
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43.
965.
541.
400.
624
55.
25.
00.
2
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00.
152
0.66
30.
195
0.14
23.
934.
671.
190.
621
88.
27.
90.
3
5.0
600.
598
0.25
00.
152
0.61
70.
210
0.17
33.
933.
570.
910.
622
1110
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5
5.0
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596
0.25
20.
151
0.50
00.
260
0.23
93.
942.
090.
530.
625
21.
30.
70.
6
5.0
00.
601
0.29
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100
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00.
214
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76.
009.
281.
550.
749
10.
70.
30.
4
5.0
300.
600
0.30
00.
100
0.68
70.
233
0.07
95.
988.
651.
450.
749
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4
5.0
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601
0.29
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101
0.65
60.
261
0.08
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957.
851.
320.
747
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6
5.0
600.
602
0.29
90.
100
0.60
60.
307
0.08
76.
056.
941.
150.
750
2725
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0
5.0
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601
0.29
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100
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30.
442
0.09
56.
014.
880.
810.
749
3926
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5.0
00.
597
0.32
80.
074
0.70
90.
233
0.05
88.
0512
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1.53
0.81
61
0.4
0.1
0.3
5.0
300.
596
0.32
90.
075
0.68
50.
254
0.06
17.
9011
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1.43
0.81
313
13.0
12.4
0.7
5.0
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597
0.32
90.
075
0.65
50.
283
0.06
38.
0010
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1.31
0.81
522
20.9
20.1
0.8
5.0
600.
600
0.32
60.
074
0.60
80.
326
0.06
68.
069.
191.
140.
814
3028
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15.
090
0.59
70.
328
0.07
50.
478
0.45
10.
071
7.96
6.75
0.85
0.81
490
88.4
81.3
7.2
volu
me
surf
ace
82
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)5.
00
0.65
10.
197
0.15
20.
748
0.14
10.
111
4.28
6.75
1.58
0.56
41
0.5
0.2
0.4
5.0
300.
651
0.19
60.
153
0.72
90.
156
0.11
64.
266.
311.
480.
561
76.
36.
00.
4
5.0
450.
653
0.19
70.
151
0.70
20.
178
0.12
04.
335.
841.
350.
567
1010
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50.
5
5.0
600.
652
0.19
60.
152
0.65
30.
216
0.13
14.
294.
971.
160.
563
1413
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.90.
5
5.0
900.
653
0.19
70.
150
0.52
50.
325
0.15
04.
353.
510.
810.
568
21.
61.
00.
6
5.0
00.
652
0.29
80.
050
0.75
10.
210
0.03
813
.06
19.5
81.
500.
857
00.
00.
00.
0
5.0
300.
650
0.29
90.
051
0.74
30.
215
0.04
312
.78
17.4
81.
370.
855
21.
71.
30.
4
5.0
450.
652
0.29
80.
051
0.73
00.
222
0.04
812
.85
15.3
01.
190.
854
32.
62.
20.
4
5.0
600.
649
0.30
10.
051
0.70
00.
241
0.05
912
.80
11.9
70.
930.
856
43.
53.
20.
4
5.0
900.
648
0.30
20.
051
0.59
50.
306
0.10
012
.80
5.98
0.47
0.85
61
0.7
0.3
0.4
5.0
00.
664
0.17
10.
166
0.75
80.
123
0.11
94.
016.
341.
580.
508
10.
30.
10.
2
5.0
300.
661
0.17
20.
167
0.73
90.
135
0.12
63.
975.
841.
470.
508
55.
25.
00.
2
5.0
450.
664
0.17
10.
165
0.71
40.
154
0.13
14.
045.
431.
350.
510
98.
37.
80.
5
5.0
600.
664
0.17
10.
165
0.66
80.
188
0.14
44.
014.
631.
160.
508
1111
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5
5.0
900.
662
0.17
20.
166
0.54
30.
287
0.17
03.
993.
210.
800.
508
11.
10.
60.
5
5.0
00.
703
0.15
10.
145
0.78
80.
108
0.10
44.
847.
571.
560.
510
10.
30.
10.
2
5.0
300.
702
0.15
10.
147
0.77
20.
118
0.11
04.
797.
011.
460.
508
54.
34.
00.
3
5.0
450.
700
0.15
20.
147
0.74
60.
136
0.11
84.
756.
331.
330.
508
77.
16.
60.
5
5.0
600.
702
0.15
10.
146
0.70
50.
167
0.12
84.
815.
501.
140.
509
99.
18.
70.
5
5.0
900.
703
0.15
10.
146
0.58
20.
262
0.15
64.
823.
730.
770.
508
21.
00.
40.
6
5.0
00.
700
0.20
00.
100
0.78
70.
141
0.07
27.
0210
.90
1.55
0.66
71
0.4
0.1
0.3
5.0
300.
701
0.20
10.
098
0.76
90.
156
0.07
57.
1710
.31
1.44
0.67
26
6.2
5.8
0.4
5.0
450.
701
0.20
10.
098
0.74
00.
181
0.08
07.
149.
271.
300.
672
109.
38.
90.
4
5.0
600.
700
0.20
10.
099
0.69
00.
222
0.08
87.
117.
881.
110.
671
1313
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.40.
6
5.0
900.
700
0.20
20.
098
0.55
80.
339
0.10
37.
135.
400.
760.
673
11.
20.
90.
4
5.0
00.
702
0.23
70.
060
0.79
00.
166
0.04
511
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17.6
41.
520.
797
10.
80.
30.
5
5.0
300.
701
0.23
90.
061
0.76
80.
185
0.04
711
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16.2
81.
410.
797
87.
37.
10.
2
5.0
450.
701
0.23
90.
060
0.73
80.
212
0.05
011
.65
14.7
61.
270.
799
1211
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4
5.0
600.
700
0.23
90.
061
0.68
50.
259
0.05
611
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12.2
71.
070.
797
1615
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.10.
6
5.0
900.
700
0.23
90.
061
0.54
80.
387
0.06
511
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8.45
0.74
0.79
63
2.0
1.2
0.8
5.0
00.
701
0.29
90.
000
0.79
30.
207
0.00
03.
8E+
166.
2E+
161.
631.
000
11.
00.
50.
5
5.0
300.
700
0.30
00.
000
0.77
00.
230
0.00
03.
8E+
165.
5E+
161.
431.
000
1010
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40.
6
5.0
450.
700
0.30
00.
000
0.73
70.
263
0.00
03.
8E+
164.
6E+
161.
201.
000
1515
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.60.
5
5.0
600.
700
0.30
00.
000
0.68
60.
314
0.00
03.
8E+
163.
6E+
160.
931.
000
2121
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.40.
65.
090
0.70
00.
300
0.00
00.
534
0.46
60.
000
3.8E
+16
1.9E
+16
0.49
1.00
012
6.2
3.5
2.7
volu
me
surf
ace
83
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)5.
00
0.75
00.
199
0.05
10.
825
0.13
80.
037
14.7
122
.23
1.51
0.79
60
0.0
0.0
0.0
5.0
300.
747
0.20
20.
051
0.81
60.
143
0.04
114
.65
20.1
01.
370.
798
21.
51.
20.
4
5.0
450.
750
0.20
00.
050
0.80
60.
149
0.04
514
.91
17.7
21.
190.
799
22.
02.
00.
0
5.0
600.
750
0.19
90.
051
0.78
10.
162
0.05
714
.76
13.6
70.
930.
797
33.
03.
00.
0
5.0
900.
749
0.20
00.
050
0.68
00.
217
0.10
314
.85
6.62
0.45
0.79
91
0.3
0.1
0.2
5.0
00.
760
0.12
20.
118
0.83
10.
086
0.08
36.
449.
981.
550.
508
10.
30.
10.
2
5.0
300.
762
0.12
20.
117
0.82
00.
094
0.08
76.
539.
451.
450.
511
43.
33.
00.
3
5.0
450.
762
0.12
10.
116
0.80
00.
108
0.09
26.
558.
661.
320.
511
55.
24.
80.
4
5.0
600.
761
0.12
20.
117
0.76
10.
135
0.10
46.
497.
341.
130.
510
77.
06.
50.
5
5.0
900.
761
0.12
20.
117
0.64
10.
225
0.13
46.
494.
790.
740.
509
11.
10.
60.
5
5.0
00.
794
0.15
00.
057
0.85
60.
103
0.04
014
.06
21.3
01.
520.
726
10.
40.
10.
3
5.0
300.
793
0.15
00.
057
0.84
10.
116
0.04
313
.92
19.6
91.
410.
725
44.
23.
90.
4
5.0
450.
793
0.15
10.
057
0.81
80.
135
0.04
613
.96
17.7
31.
270.
726
66.
25.
80.
4
5.0
600.
793
0.14
90.
058
0.77
80.
168
0.05
313
.69
14.6
11.
070.
720
88.
27.
70.
5
5.0
900.
793
0.15
00.
056
0.65
20.
280
0.06
814
.07
9.54
0.68
0.72
71
1.1
0.6
0.5
5.0
00.
800
0.10
20.
098
0.86
10.
071
0.06
88.
2112
.60
1.53
0.51
21
0.3
0.1
0.2
5.0
300.
798
0.10
30.
099
0.84
80.
078
0.07
38.
0811
.59
1.43
0.51
03
2.8
2.3
0.5
5.0
450.
800
0.10
20.
098
0.83
30.
090
0.07
88.
1810
.75
1.31
0.51
04
4.2
3.8
0.4
5.0
600.
801
0.10
20.
098
0.80
00.
113
0.08
78.
209.
211.
120.
510
65.
35.
10.
2
5.0
900.
800
0.10
20.
098
0.68
50.
197
0.11
98.
175.
770.
710.
511
11.
00.
50.
5
5.0
00.
801
0.16
50.
034
0.86
20.
113
0.02
423
.54
35.4
61.
510.
828
00.
00.
00.
0
5.0
300.
801
0.16
50.
033
0.84
80.
127
0.02
523
.98
33.3
41.
390.
831
54.
34.
00.
3
5.0
450.
801
0.16
50.
034
0.82
30.
149
0.02
823
.76
29.4
51.
240.
830
76.
66.
20.
4
5.0
600.
800
0.16
60.
034
0.78
00.
188
0.03
323
.37
23.9
81.
030.
829
99.
18.
70.
5
5.0
900.
803
0.16
40.
034
0.65
50.
303
0.04
223
.84
15.5
70.
650.
829
11.
20.
80.
4
5.0
00.
799
0.16
70.
034
0.86
10.
115
0.02
423
.65
35.3
91.
500.
831
00.
00.
00.
0
5.0
300.
798
0.16
80.
034
0.85
50.
118
0.02
723
.64
32.3
01.
370.
833
11.
01.
00.
0
5.0
450.
797
0.17
00.
034
0.84
60.
124
0.03
023
.75
28.1
81.
190.
835
21.
31.
10.
2
5.0
600.
798
0.16
80.
034
0.82
90.
134
0.03
723
.84
22.4
30.
940.
834
22.
21.
90.
3
5.0
900.
799
0.16
80.
033
0.74
10.
188
0.07
123
.90
10.4
60.
440.
833
00.
00.
00.
0
5.0
00.
833
0.13
50.
031
0.88
50.
093
0.02
226
.53
39.6
41.
490.
812
10.
40.
10.
3
5.0
300.
833
0.13
40.
032
0.87
20.
104
0.02
425
.83
35.9
21.
390.
806
33.
03.
00.
0
5.0
450.
832
0.13
60.
032
0.85
10.
122
0.02
725
.92
32.1
51.
240.
808
55.
24.
90.
3
5.0
600.
833
0.13
40.
033
0.81
50.
154
0.03
125
.38
26.0
91.
030.
803
77.
26.
70.
55.
090
0.83
30.
135
0.03
20.
692
0.26
60.
042
26.2
116
.44
0.63
0.80
91
1.1
0.6
0.5
volu
me
surf
ace
84
aspe
ctvo
lsu
r su
r(s1
/s3)
/ol
d K
ang.
dis.
AV
G +
ang.
dis.
ang.
dis.
ratio
( a
/c)
θ (°
) s1
s2s3
s1s2
s3 (
s1/s
3)(s
1/s3
)vo
l(s1
/s3)
s2/
s2+s
3M
AX
(˚)
ST D
EV
(˚)
AV
G (
˚)ST
DE
V (
˚)5.
00
0.86
10.
071
0.06
80.
905
0.04
90.
047
12.7
319
.34
1.52
0.51
20
0.0
0.0
0.0
5.0
300.
861
0.07
10.
068
0.89
70.
053
0.05
012
.72
18.1
11.
420.
513
22.
11.
60.
5
5.0
450.
860
0.07
10.
068
0.88
40.
062
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volu
me
surf
ace
85
