Exercise(#2" Estimating(Strain(in(Naturally(Deformed(Rocks(with(the(Rf===Phi(Method"

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1 Exercise(#2" Estimating(Strain(in(Naturally(Deformed(Rocks(with(the(Rf===Phi(Method" Paul"Karabinos" Department"of"Geosciences,"Williams"College,"Williamstown,"MA"01267" This"exercise"is"suitable"for"an"undergraduate"structural"geology"course."It"assumes" that"students"have"read"session'5)))'practical'strain'measurement:'initially'circular' and'elliptical'markers,'from"ramsay"and"huber"(1987,"p."73ooo87)"questions"to"be" answered"are"in"blue"to"make"it"easier"to"find"them." Estimating"strain"in"naturally"deformed"rocks"in"the"field"typically"involves" using"a"ruler"and"protractor"to"measure"the"long"and"short"axes"of"deformed"pebbles" and"the"orientation"of"the"long"axis."even"though"it s"lots"of"fun,"this"approach"does" not"allow"the"geologist"to"make"a"visual"record"of"the"pebbles"measured."after"field" measurements"are"complete,"the"results"are"plotted"in"graphs"of"rf"vs."phi"to" estimate"strain"as"described"by"ramsay"and"huber"(1987)." In"this"exercise,"you"will"estimate"strain"using"a"photograph"of"a"deformed" conglomerate."you"will"use"the"photograph"as"a"background"image"to"create"ellipses" that"approximate"the"shape,"size,"and"orientation"of"naturally"deformed"pebbles."the" photograph"shows"the"basal"conglomerate"member"of"the"dalton"formation,"an" Ediacaran"to"Cambrian"unit"in"western"New"England."The"outcrop"is"along"the" Appalachian"Trail"on"Day"Mountain"in"Dalton,"Massachusetts."The"outcrop"also" features"a"famous"unconformity"between"mesoproterozoic"rocks"at"least"1179"ma" and"the"conglomerate,"which"is"approximately"550"ma."thus,"the"time"gap"is"about" 600"million"years!" As"you"digitize"pebbles,"the"points"are"automatically"added"to"the"Cartesian" or"polar"rfooophi"plots."the"best"part"is"that"you"end"up"with"a"visual"record"of"the" deformed"pebbles"used"to"estimate"strain."geoshear"can"be"used"to" undeform "the" pebbles"using"your"estimate"of"strain."this"allows"you"to"assess"how"reasonable" your"strain"calculations"are." Part(1.(Loading(the(Background(Image(and(Digitizing(Pebbles." 1) Open"GeoShear."You"will"see"the"following"resizable"window"with"a"Main"Menu, Control"Panel,"and"User"Interface: 1"

2 2) In"the"Control"Panel,"click"on"the"Edit"Mode"button,"and"then"click"the" Background"Image"button." 3) Navigate"to"the"Geoshear_2.1/Background"Images"folder,"and"select" Deformed_Conglomerate."Use"the"mouse"scroll"wheel,"the"Zoom"Slider,"and" Center"in"view"button"" at"the"top"of"the"control"panel"to"get"the" full"photo"in"view." 4) This"is"a"picture"of"a"deformed"pebble"conglomerate"from"Day"Mountain"in" Dalton,"Massachusetts." 5) Create(new(pebbles:"Because"you"needed"the"Edit"Mode"to"load"the"background" image,"you"are"ready"to"start"digitizing"pebbles"from"the"photograph."note:(you( must(be(in(edit(mode(to(add(new(pebbles."you(must(exit(edit(mode(to(use" GeoShear(to(deform(the(pebbles." a) Zoom"in"to"magnify"individual"pebbles"in"the"photo." b) Hold"down"the"SHIFT"key"or"enable"the"CAP"LOCKS"key,"leftOOOclick"on"one" end"of"a"pebble"longoooaxis,"drag"the"cursor"to"the"other"end"of"the"long"axis," hit"the"space"bar"to"accept"the"first"line,"drag"the"cursor"away"from"the"long" axis"to"define"the"short"axis,"and"hit"the"space"bar"to"accept"the"ellipse."you" must"hold"down"the"leftoooclick"button"throughout"the"process." c) It"is"easier"to"see"how"good"your"work"is"by"not"invoking"the"Fill"Pebbles" option"from"the"display"menu." d) Open"the"RfOOOPhi"Cartesian"plot"to"observe"that"points"are"added"as"pebbles" are"digitized."from"the"control"button"on"the"rfooophi"plot,"select"adaptive" Rf"Scale"so"all"pebbles"will"be"shown"as"they"are"digitized." e) Continue"digitizing"pebbles"until"you"are"confident"that"you"have"all"of"the" wellooodefined"pebbles"visible"in"the"photo." f) During"the"process"of"digitizing"pebbles,"you"might"notice"that"some"pebbles" indent"other"pebbles."this"is"evidence"for"pressure"solution." g) You"might"also"observe"that"the"average"orientation"of"the"long"axes"of" pebbles"varies"from"one"part"of"the"photo"to"another."if"you"do,"you"should" ask"whether"this"variation"reflects"differences"in"strain,"i.e."nonooo" homogeneous"strain"or"perhaps"bedoootoooobed"differences"in"preferred" orientation"of"pebble"long"axes."! 2"

3 6) To"save"your"work"go"to"Main"Menu/File/Save"Current"State"and"give"the".tab"file an"eastoootooooremember"name. Part(2.(Estimating(Strain(with(the(Cartesian(Rf===Phi(plot." 1. Using"the"RfOOOPhi"plot"as"described"by"Ramsay"and"Huber"(1987)"to estimate"strain"requires"identification"of"the"maximum"and"minimum pebble"axial"ratios,"rfmax"and"rfmin,"respectively. 2. The"axial"ratios"of"specific"pebbles"can"be"found"quite"easily"in"the"plot"by clicking"on"the"corresponding"points"and"reading"the"values"that"appear"on the"screen."the"difficulty"in"finding"rfmax"and"rfmin"in"a"real"data"set"is identifying"outliers. A"few"anomalously"elliptical"pebbles"in"the"original"conglomerate"can"lead"to an"overestimate"of"strain."in"exercise"1,"you"learned"that"the"rfooophi method 1) assumes"a"nonooopreferred"initial"orientation"of"pebble"long"axes,"and"2) that"the(reliability(of(the(method(depends(critically(on(a(non= ==preferred(and(wide(initial(distribution(of(the(most(elliptical(pebbles." GeoShear"does"not"have"a"builtOOOin"algorithm"for"finding"outliers"in"the" population." 3. The"RfOOOPhi"plot"shows"a"closed"envelope"of"points;"this"indicates"that"the strain"ellipse"ratio"(rs)"is"greater"than"the"maximum"initial'ellipse"axial"ratio (Rimax). 4. Use" all( points( in" the" Cartesian" RfOOOPhi" plot," and" the" equations" for" a closed" envelope" (Ramsay" and" Huber," 1987)," shown" below" to" calculate the" strain" ellipse" axial" ratio," Rs," and" the" maximum" initial" pebble" ratio, Rimax. a. Rs"="(Rfmax*Rfmin) 1/2" = b. Rimax"="(Rfmax/Rfmin)" 1/2" = 5. Eliminate(obvious(outliers(in"the"Cartesian"RfOOOPhi"plot,"and"the"equations for"a"closed"envelope"(ramsay"and"huber,"1987),"shown"below"to"calculate the"strain"ellipse"axial"ratio,"rs,"and"the"maximum"initial"pebble"ratio,"rimax. a. Rs"="(Rfmax*Rfmin) 1/2" = b. Rimax"="(Rfmax/Rfmin)" 1/2" = Part(3.(Using(GeoShear(to( Undeform (the(pebbles." 1) Using"the"Cartesian"RfOOOPhi"plot"to"find"Rfmax"and"Rfmin,"as"described above,"does"not"allow"us"to"assess"visually"how"reasonable"the"strain estimate"is."here"is"a"simple"way"to"estimate"strain"that" undeforms "the pebbles,"yields"the"reciprocal"strain"ellipse,"and"gives"us"a"visual representation"of"the"pebbles"before"deformation. 2) If"necessary,"disable"Edit"Mode"by"clicking"on"the"button. 3) Click"in"the"deg."box"under"rotation."Use"the"upOOOarrow"or"downOOOarrow"on 3"

4 the"keyboard"to"rotate"the"pebbles"until"the"yellow"horizontal"line"coincides" with" 4"

5 Phi"="0."The"arrow"keys"give"one"degree"increments;"you"can"click"in"the"box" and"type"in"more"precise"degree"values"to"help"center"the"yellow"line"at"phi"=" 0. This"yellow"line"is"the"vector"mean"of"the"pebble"long"axes."Click"Apply"in the"control"panel." 4) Now"that"the"vector"mean"coincides"with"Phi"="0,"we"can"use"pure"shear"to undeform "the"pebbles"in"the"xooodirection."click"in"the"x"box"under"pure shear"and"use"the"upoooarrow"on"the"keyboard"to"shorten"the"pebbles"in"the xooo"direction."continue"shortening"in"the"xooodirection"until"the"points"cover the"full"range"of"phi"values"(ooo90"to"90);"there(shouldn t(be(any(large concentrations(of(points(or(gaps(over(the(range(of(phi,"and"collectively the"range(of(rf(values(should(be(minimized."this"approach"attempts"to recreate"an"idealized"initially"random"distribution"of"pebble"long"axes. 5) Note"that"outliers"now"stick"out"like"sore"thumbs"that"you"just"hit"with"a"rock hammer. 6) Click"Apply"to"lock"this"increment"of"deformation"in."Then"go"to"the"Main Menu/File/Save"All"History"to"save"the"original"pebble"data"and"the"two transformations"you"used"to"undeform"the"pebbles."give"the"file"an"easyoootoo OO"remember"name."If"you"want"to"export"information"for"transformed"pebble data,"use"the"save"current"state"option"in"the"file"menu."this"will"give"the"x,"y coordinates,"long"and"short"axis"lengths,"and"long"axis"orientations"of"pebbles after"they"have"been"transformed"to"analyze"in"a"spreadsheet"or"in"another program. 7) To"review"the"transformations"you"just"applied,"use"the"forward"and backward"arrows"on"the"control"panel"" to"cycle"through the"steps."you"can"also"review"the"transformation"matrices"by"opening"the Deformation"Matrix"window. 8) The"strain"ellipse"shape"estimated"using"this"approach"is"given"by"the"ratio"of the"reciprocal(strain(ellipse."this"is"simply"the"rs"value"below"the"black matrix"in"the"control"panel. 9) The"Phi"value"of"the"strain"ellipse"is"given"by"the"rotation"you"applied"in"step 3"above"to"get"the"vector"mean"to"coincide"with"Phi"="0. 10) What"is"the"axial"ratio"of"the"strain"ellipse"determined"by" undeforming "the pebbles? 11) How"does"this"estimate"compare"with"the"one"determined"in"Part"2? 12) As"a"bonus,"this"flexible"approach"to"estimating"strain"by" undeforming "the pebbles"makes"it"possible"to"evaluate"visually"the"results."the"deformed background"image"produced"above"is"really"a"picture"of"what"the"pebbles should"have"looked"like"before"strain"if"this"estimate"is"valid. 13) Does"the" undeformed "pebble"population"look"like"a"naturally"deposited conglomerate"to"you? 14) Explain"your"answer"above: 5"

6 15) This"visual"approach"of" undeforming "pebbles"can"be"accomplished"with polar"rfooophi"plots;"in"fact"it"is"arguably"easier."follow"the"same"procedure you"used"with"the"cartesian"rfooophi"plot:"click"in"the"deg."box"under"rotation and"rotate"the"pebbles"until"the"yellow"line"coincides"with"phi"="0,"then"click in"the"x"box"under"pure"shear"and"shorten"the"pebbles"in"the"xooodirection until"the"pebbles"are"clustered"in"the"center"of"the"polar"plot. 16) What"is"the"axial"ratio"of"the"strain"ellipse"determined"by" undeforming "the pebbles"using"the"polar"rfooophi"plot? 17) Which"plot"do"you"think"does"a"better"job"of"handling"outliers"during"the undeforming "simulation? 18) Explain"your"answer"to"17"above. 6"