Strain estimation from single forms of distorted fossils — A computer graphics and MATLAB approach

Most of the existing methods of strain analysis can estimate strain in a single form of distorted brachiopod, or trilobite provided independent evidence, such as the association of the fossil with cleavage and/or stretching lineation is available for
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  Strain Estimation from Single forms of Distorted Fossils – AComputer Graphics and MATLAB Approach J YOTI  S HAH 1 , D EEPAK  C. S RIVASTAVA 2 , V IPUL  R ASTOGI 3 , R AJIT  G HOSH 2  and A DITI  P AL 4 1 Schlumberger Asia Services Limited, Goregaon(E), Mumbai - 400067 2 Department of Earth Sciences, 3 Department of Physics, IIT Roorkee, Roorkee - 247 667 4 Shell Technology India, Bangalore - 560 048 Email:  dpkesfes@gmail.com Abstract:  Most of the existing methods of strain analysis can estimate strain in a single form of distorted brachiopod, ortrilobite provided independent evidence, such as the association of the fossil with cleavage and/or stretching lineation isavailable for inferring the direction of maximum principal strain. This article proposes a simple computer graphics basedmethod and its MATLAB code that determine the minimum amount of strain in a single distorted fossil form even if data for inferring the maximum principal strain direction are lacking. Our method is a rapid computer-graphics alternativeto some of the existing analytical methods.In a distorted fossil form of srcinal bilateral symmetry, the relative senses of angular shears along the hinge line andthe median line are mutually opposite to each other. It follows, therefore, that the maximum principal strain direction lieswithin the acute angle between the hinge and the median lines in the plane of the fossil. Using this principle, our methodperforms several simulations such that each simulation retrodeforms the distorted fossil by assuming a particularorientation, lying within the acute angle between the hinge line and the median line, as the potential direction of themaximum principal strain. Each simulation of retrodeformation yields a potential strain ratio. The distribution of all thepotential strain ratios, obtained by assuming different orientations as the potential directions of the maximum strain, istypically a parabola-like curve with a distinct vertex that corresponds to the minimum amount of strain in the distortedfossil. An entirely computer graphical approach is somewhat time-intensive because it involves a large number of retrodeformational simulations. We, therefore, give a MATLAB code, namely, the  Minstrain , that rapidly retrodeformsthe fossil and determines the minimum strain with precision. Keywords: Strain, Distorted fossil, Lineation, Cleavage, MATLAB. Several methods of strain estimation exist for thosesituations, where a set containing several distorted fossilforms occur on an outcrop (Wellman, 1962; De Paor, 1986a,b; Cooper, 1990; Hughes and Jell, 1992; Rushton and Smith,1993). Similarly, several methods such as, the Mohr circlemethod (Ramsay and Huber, 1983, p.134), the Breddinmethod (Breddin, 1956) and the orthographic projectionmethod (De Paor, 1986a) can provide strain estimates inthose situations where two distorted fossil forms are foundto occur in different orientations.Existing methods can estimate strain in single forms of fossils with srcinal bilateral symmetry providedindependent evidence, such as the cleavage trace or thestretching lineation, exists for assuming the maximumprincipal strain direction (Ramsay and Huber, 1983, pp.141-143; De Paor, 1986a, b; Srivastava and Shah, 2006;Shah, 2007). However, both structural geologists and INTRODUCTION Estimation of strain in distorted fossils is an importantbranch of study because it not only reveals the amount andthe nature of deformation in the earth’s crust, but also helpsrestoring the undistorted fossil shapes for the precisetaxonomic identification (Sharpe, 1847; Haughton, 1856;Fank, 1929; Lake, 1943; Sdzuy, 1966; Engelder andEngelder, 1977). Fossil invertebrates are particularly reliableindicators of homogeneous strain because they are generallysmall in size, thin in nature, and commonly composed of material that has a matching competency with respect tohost rock (Ramsay, 1967, p.229).Amongst a large variety of fossil invertebrates, thosewith initial bilateral symmetry, e.g., brachiopods, or trilobitesare routinely used as strain markers. These fossils occureither as: (i) a set containing several distorted forms, orii) a pair of distorted forms, or (iii) a single distorted form. JOURNAL GEOLOGICAL SOCIETY OF INDIAVol.75, January 2010, pp.89-970016-7622/2010-75-1-89/$ 1.00 © GEOL. SOC. INDIA  JOUR.GEOL.SOC.INDIA, VOL.75,JAN.201090JYOTI SHAH AND OTHERS paleontologists working in mildly deformed sedimentaryterrains, such as the Tethyan sedimentary zone in theHimalaya or the Appalachian plateau, have observed anumber of outcrops where a single distorted trilobite orbrachiopod occurs without any association with cleavageor lineation for inferring the maximum principal straindirection. As these single forms of distorted fossilscommonly occur on flat lying bedding surfaces, they areimportant indicators of shortening along the bedding planes.A single distorted fossil form that occurs without anyevidence for inferring the principal strain direction isgenerally discarded for estimation of strain- the dataavailable from the outcrop is considered inadequate. Thisarticle proposes a simple computer graphics based methodthat determines the minimum amount of two-dimensionalfinite strain ratio in single distorted fossil forms even if thedirection of principal strain is not known. Our method is arapid alternative to the analytical solutions that are basedon the principle that the minimum strain ratio correspondsto the acute bisector of the angle between the hinge and themedian lines (De Paor, 1986a). RATIONALE AND METHODOLOGY An undistorted form of a bilaterally symmetric fossil ischaracterized by an inherent orthogonal relationship betweenthe hinge line h  and the median line m  (Fig. 1a). Thehomogeneous deformation of a bilaterally symmetric fossilinduces angular shears along the lines h  and m  and the fossilform assumes an asymmetric morphology known commonlyas the ‘oblique form‘ (Fig. 1b). Only exceptions to thisphenomenon are the ‘long‘ and ‘short‘ forms which aredistorted such that the two principal strain directions parallelthe lines h  and m , respectively. This article does not addressthese exceptional situations.Because the relative senses of angular shears along thelines h  and m  are mutually opposite to each other, themaximum principal strain direction must lie within the acuteangle θ   between the lines h  and m  (Fig.1b). Using thisprinciple, the minimum strain method tests differentorientations within the range of angle θ   as the potentialmaximum strain direction and retrodeforms the distortedfossil with respect to each orientation of the potentialmaximum strain. The retrodeformation uses a constant areapure shear deformation that restores the orthogonalrelationship between the hinge and the median line of the distorted fossil and transforms a unit reference circle,x 2 + y 2  = 1, into a potential reciprocal strain ellipse, x 2  +{y 2  / (l 2  /l 1 )} = 1, where x-y is the Cartesian reference frameand l 2  /l 1  is the axial ratio of the potential reciprocal strainellipse (Fig.2a-c).A rotation of potential reciprocal strain ellipse through90° produces potential strain ellipse. Only one of the possiblepotential strain ellipses is the actual section through thethree-dimensional strain ellipsoid. Furthermore, the majoraxis of the potential strain ellipse may or may not coincidewith the true three-dimensional principal axis of strain.The step-by-step procedure for computer graphicalapplication of the minimum strain method is as follows:(1) Import the digital image of the distorted fossil incomputer graphics software, such as CorelDraw, CorelPhoto-Paint, Smart Draw or Adobe Illustrator. (2) Mark thehinge line h  and the median line m . Draw a number of lines,say i  to n  within the angular range θ   between the lines h and m . These lines represent the potential orientations of the maximum principal strain. Also draw a small referencecircle (Fig. 2a). (3) Group all the objects, namely, the lines h  and m , the lines i  to n , the reference circle and the imageof the distorted fossil. (4) Rotate the grouped objects suchthat line i  becomes vertical (Fig. 2b). Using the “pick tool” of the graphics software, select the grouped objects to displayeight dragging handles ( 1  to 8   in Fig.2b). (5) Drag the handle mh h (a) (b)  m    Fig.1. (a)  Orthogonal relationship between the hinge line h  and the median line m in a bilaterally symmetric undistorted brachiopod.Reference circle is shown in black. (b)  Oblique form of brachiopod resulting from distortion of the fossil in (a) by a homogeneousdeformation. It is noteworthy that the lines h  and m  are no longer perpendicular to each other and the relative senses of shearalong these two lines are dextral and sinistral, respectively. The major axis of the finite strain ellipse (black) lies within theangular range θ  .  JOUR.GEOL.SOC.INDIA, VOL.75,JAN.2010STRAIN ESTIMATION FROM SINGLE FORMS OF DISTORTED FOSSILS – A MATLAB APPROACH91 Fig.2. (a) Imported image of the oblique form of a distorted brachiopod. Lines i  to n  are assumed as potential orientations of themaximum principal strain that must lie within the angular range θ   . h  and m  are the hinge line and median line, respectively. A unitcircle (black) is drawn for reference. (b)  The fossil image in (a) is rotated such that line i  assumes the vertical position. 1  to 8   arethe dragging handles displayed by selecting the objects by “pick tool“ of graphics software (in this example- CorelDRAW).  x-y  is the Cartesian reference frame. (c) Retrodeformed image of the distorted form in (b). The retrodeformation, obtained bydragging the handle 4  in (b) to the right, restores the orthogonal relationship between the lines h  and m  and, therefore, thebilateral symmetry of the fossil. The unit reference circle in (b) transforms into the potential reciprocal strain ellipse of axial ratio l 2  /l 1 . (d)  Distribution of the potential strain ratios with respect to orientation of different lines, i  to n , that are assumed as thepotential direction of maximum stretching in successive simulations of retrodeformation. The minimum strain ratio is marked byvertex of the parabola-like distribution. h i ii n Potential reciprocal strain ellipseof axial ratio (l /l ) 21 m m 90  (b)17 8 2 6 35 4 hm i ii n  x y hi ii m   (a)(d)c  ( ) h i ii n m 10  ° 4.0 Minimum strain ratio Angle from median line Potentialstrain ratio 3.0 2.0 i  ii....................................................................................... n1.0 0.0 20  ° 30  ° 40  ° 50  ° 60  °            v     e      r       t        i     c      a        l unit circleunit circle  JOUR.GEOL.SOC.INDIA, VOL.75,JAN.201092JYOTI SHAH AND OTHERS 4  to the right until the line h  becomes perpendicular to theline m  (Fig. 2c). This step retrodeforms the distorted fossiland it also transforms the reference circle into the potentialreciprocal strain ellipse, which can easily be converted intothe potential strain ellipse. (6) By successively bringingeach line, ii  to n , into a vertical position, retrodeform thedistorted fossil by the procedure outlined in step 4. Eachretrodeformation yields a potential strain ellipse. At the endof all the retrodeformations, there are n  potential strainellipses corresponding to n  potential directions of themaximum principal strain.The Cartesian plot of the axial ratios of the potentialstrain ellipses versus the orientation of the correspondinglines i  to n  with respect to the median line shows a typicallyparabola-like distribution with a distinct minimum at thevertex (Fig.2d). The strain ratio that corresponds to the vertexof the distribution pattern represents the minimum amountof strain in the distorted fossil form. EXAMPLES We test the validity of the above method on a few singleforms of distorted trilobites that are associated with evidencefor inferring the maximum principal strain direction, namely,the stretching lineation (Fig.3a-d, after Ramsay, 1967, p.233;van der Pluijm and Marshak, 1997, p. 53). Such a choice of distorted fossils enables a comparison between the resultsof the minimum strain method and those obtained by theother methods, namely, the Mohr circle method (Ramsayand Huber, 1983, p. 129-130), the numerical method 10  °   20  °   30  °   40  °   50  °   60  °  2.0 2.0 2.0 3.0 3.0 10  °   20  °   30  °   40  °  hm L (a)(e)(g)(f)(b) c  hm L (d)( ) 10  °   20  °   30  °   40  °   50  °   60  °   Angle from medi an line 1.2 1.0 1.0 1.0 10  °   20  °   30  °   40  °   50  °   60  °   70  °  Potentialstrain ratioPotentialstrain ratioPotentialstrain ratioPotentialstrain ratio  Angle from median lineMinimum strain ratioMinimum strain ratioMinimum strain ratioMinimum strain ratio Angle from median line Angle from median line 1.41.6 1.8 2.0 (h) m mm m  L Fig.3. (a)-(d)  Four examples of single forms of distorted trilobites ( after   Ramsay, 1967, p. 233; van der Pluijm and Marshak, 1997,p. 53) that are associated with a stretching lineation (L). ( e)-(h)  Results of analyses of the fossil forms in (a)-(d) by entirelygraphical adoption of the minimum strain method. m - median line; h - hinge line. For comparison, these forms are also analyzedby several other methods (Table 1). The parabola-like curves are fitted visually through the points representing potential strainratios. The minimum on each curve is also estimated visually.  JOUR.GEOL.SOC.INDIA, VOL.75,JAN.2010STRAIN ESTIMATION FROM SINGLE FORMS OF DISTORTED FOSSILS – A MATLAB APPROACH93 (Ramsay, 1967, p.234) and, the digital method (Srivastavaand Shah, 2006). The application of the minimum strainmethod on all four distorted fossils confirms the parabola-like distribution of the potential strain ratios (Fig.3e-h). Theresults further show that the axial ratio of the minimum strainellipse is always an underestimate of the actual strain ratio(Table 1).Having established the validity of the minimum strainmethod, we illustrate its practical application on fourdistorted fossil forms that are not associated with anylineation/cleavage for assuming the maximum principalstrain direction (Fig.4a-d taken from Zen et al. 1952; Valdiya,1998; Paleontological Museum, Oslo). In each of the fourexamples, the distribution of potential strain ratios is aparabola-like curve with a distinct minimum at the vertex.The vertex corresponds to the minimum amount of strainin each fossil (Fig. 4e-h and Table 2). ‘  Minstrain’ –  THE MATLAB CODE Several examples of the results obtained by purelycomputer graphics approach are shown in Fig.3a-d andFig.4a-d. Such a purely computer graphics approach fordetermination of minimum strain is time-intensive becauseit requires: (i) many simulations of retrodeformation, eachcorresponding to a potential maximum strain direction thatlies between the hinge line and the median line, (ii) plotting Table 1. Comparative results of strain analyses on the single forms of distorted fossils shown in Fig.3a-d. Figs.3a-c are trilobite  Angelina  (Fig. 5.54 in Ramsay, 1967) from the O rdovician slatesof Lleyn, North Wales whereas Fig.3d is Trilobite  Angelinasedgwicki  (Fig. 4.1 in van der Pluijm and Marshak, 1997) fromCambrian slates, WalesFig.MethodsMohrNumericalDigitalMinimum  Minstrain circlestrain(MATLAB) (Graphical)3a1.801.781.771.501.203b1.721.721.731.331.243c2.182.162.151.601.363d1.752.151.931.201.20 Fig.4. (a)-(d)  Natural examples of four distorted fossils, three brachiopods (a-c) and, one trilobite (d), that occur without any associationwith a cleavage or lineation ( after   Zen et al. 1952; Valdiya, 1998; Paleontolgoical museum, Oslo). (e)-(h)  Results of the analysesof the fossils in (a)-(d) by an entirely graphical adoption of the minimum strain method (Table 2). The parabola-like curves arefitted visually through the points representing potential strain ratios. The minimum on each curve is also estimated visually. m 5  °  10  °  15  °  20  °  25  °  30  °  35  °  40  °  1.20 1.28 1.36 2.0 1.0  m 10  °  20  °  30  °  40  °  1.60 1.40 1.20 1.00 m5  °  10  °  15  °  20  °  25  °  30  °  35  °  40  °  hmmh (b) hm (c) hm (d) m 5  °  10  °  15  °  20  °  25  °  30  °  35  °  1.2 1.31.4 Angle from median lineMinimum strain ratioMinimum strain ratioMinimum strain ratioMinimum strain ratio Angle from median line Angle from median line Angle from median line 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(a)(e)(f)(g)(h)
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