ARTICLE IN PRESS
Computers & Geosciences 29 (2003) 795–799
R
= f
$
Department of Geology, Trinity College, Dublin 2, Ireland
Received24 May 2002; receivedin revisedform 21 October 2002; accepted10 November 2002
1. Introduction
The R f
= f technique for strain analysis was first described by
Ramsay (1967)
andrefinedby
Dunnet
(1969)
and
Lisle (1977)
. The R f
= f technique can potentially be usedon any deformedsuite of initially elliptical strain markers (e.g. conglomerates, oolites).
The axial ratios ð R f
Þ of typically between 50 and100 strain markers andtheir respective long axes orientations ð f Þ have to be recorded.
The R f
= f technique assumes that the suite of elliptical strain markers did not possess a preferred initial orientation prior to deformation. As these markers are deformed shapes cannot correspond to the strain ellipse.
However, it is possible to calculate the pre-deformational axial ratio ð R i
Þ andorientation ð y Þ of each marker for a particular strain. The strain ellipse ð R s
Þ can then be calculatedby selecting the particular strain which yields the most random initial distribution of marker orientations (the fundamental starting assumption). The reader is encouragedto consult a comprehensive manual on the
R f
= f technique (
Lisle, 1985
) for further details.
One of the drawbacks of the R f
= f technique is that the strain calculations are relatively time-consuming to perform manually. Strain calculation is made significantly easier by utilizing a computer-basedapproach
(e.g.
Peach andLisle, 1979
;
Mulchrone andMeere,
2001 ). The program presentedhere employs a spread-
modification of the spreadsheet structure and the ability to export andmod R computer platforms (e.g.
f
= f diagrams on a variety of
Fig. 1 ).
2. Spreadsheet structure and operation
2.1. Notes concerning the Excel implementation
This implementation of the R f
= f technique uses the
Microsoft Excel t spreadsheet. The strain calculation procedure employs macros and hence macros must be enabled(with the ensuing virus warnings) when opening the workbook Rfphi.xls
.
Rfphi.xls
is provided as an
Excel 5.0 workbook to ensure compatibility with older versions of Excel, but maybe savedin a newer format if required.
The workbook ( Rfphi.xls
) consists of two worksheets
(‘‘Enter data’’ and ‘‘Calculate R s
’’) andtwo charts (‘‘Ln
R f vs. Phi’’ and‘‘ R s vs.
w 2 ’’).
R f
= f data (see
Table 1
for nomenclature) is enteredonto the worksheet (‘‘Enter data’’) and these data are plotted on the chart (‘‘Ln R f vs. Phi’’), along with R i and y curves for the calculated strain. The secondworksheet (‘‘Calculate R s
’’) calculates the best-fit parameter ð w 2 Þ of the y -distribution test of
Lisle (1977)
over a user-specifiedstrain range. The values of the best-fit parameter over the strain range are displayed graphically on the second chart (‘‘ R s vs.
w 2 ’’).
The detailed operation of each of these modules is discussed below. Further operating instructions are suppliedin the accompanying help file ( readme.doc
).
$
Code on server at http://www.iamg.org/CGEditor/ index.htm
*Tel.: +353-1-608-1235; fax: +353-1-67111-99.
E-mail address: chewd@tcd.ie (D.M. Chew).
2.2. Worksheet ‘‘Enter data’’
The long andshort axes andorientations
ð 90 o f o 90 Þ of up to 350 markers are enteredon this worksheet. The axial ratio ð R f
Þ of each strain marker is calculated, while sample numbers can be entered into the
0098-3004/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0098-3004(03)00027-X

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ARTICLE IN PRESS
D.M. Chew / Computers & Geosciences 29 (2003) 795–799
f
-50
-30
-90
-70
30
50
-10
10
Ri =
1.25
Vector mean of
φ q = -90˚ q = -45˚ q = 0˚ q = 45˚
70 Harmonic mean of Rf
90
0
1.5
1.75 2
0.5
2.5
3 4 Ri = 6
1.5
q = 90˚
1 2
f
2.5
3 3.5
Fig. 1. Chart ‘‘Ln R f curves were labelled.
vs. Phi’’. Chart was exportedinto Adobe Illustrator where R i curves andthe 90 , 45 , 0 , 45 and90 y
Table 1
R f
= f nomenclature
R f f
R y
R i s
Measuredellipticity of strain marker
Angle between marker long axis andmaximum extension direction
Initial ellipticity of strain marker in undeformed state
Initial angle between the undeformed marker long axis andthe maximum extension direction
Axial ratio of the strain ellipse leftmost column. The vector mean andharmonic mean of the strain markers are displayed in the rightmost box
(
Fig. 2 ). The spreadsheet performs the index of
symmetry test of
Lisle (1985)
by dividing the R f
= f plot into four quadrants (defined by the vector mean and harmonic mean of the strain markers). The index of symmetry is calculatedas follows:
I
SYM
¼ 1 ðj n
A n
B j þ j n
C n
D jÞ = N ; ð 1 Þ where n
A denotes the number of strain markers in quadrant
A, etc., and N equals the total number of markers. High values of I
SYM suggest the data are symmetrical, while low values suggest that the data are markedly asymmetric andhence the assumption of no preferredinitial orientation to the strain markers is incorrect. Critical values for the test are listedin
Lisle (1985) .
2.3. Chart ‘‘Ln R f vs. Phi’’
This chart plots ln R f against
(
Fig. 1 ). The
x -axis scale ð R f
Þ f for each strain marker is logarithmic. The f data are shiftedalong the y axis such that the vector mean of the f data is equal to 0. Once the strain ð R s
Þ has been determined (using the worksheet ‘‘Calculate R s
’’), R i and y curves for this calculated strain are displayed.
Eight R i curves are plotted, with values of 1.25, 1.5, 1.75,
2, 2.5, 3, 4 and6, respectively (
Fig. 1
), while the y curves range from 90 to 90 in increments of 9 (
Fig. 1 ).
Lisle (1985)
lists several equations relating the five variables involvedin R f and y ).
R i
= f calculations ( R s
; R f
; R i
; f curves are calculatedusing Eq. (A1.3) of
Lisle
(1985) :
cosh 2 e i
¼ cosh 2 e f cosh 2 e s cos 2 f sinh 2 e f sinh 2 e s
;
ð 2 Þ where e i
¼ 1
2 ln R i
; etc. As R i and R s are known, the R curve is calculatedby looping through a range of values i for R f andcalculating the corresponding value for f :

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D.M. Chew / Computers & Geosciences 29 (2003) 795–799 797
Fig. 2. Worksheet ‘‘Enter data’’, where R f
= f data is input.
The y curves are calculatedusing the following equation of
Lisle (1977)
:
R f
¼
" tan 2 y ð R 2 s tan 2 f Þ 2 R s tan f tan 2 y ð 1 R 2 s tan 2 f Þ 2 R s tan f
#
1 = 2
ð 3 Þ
As y and R s are known, the y curve is calculatedby looping through a range of values for f andcalculating the corresponding values for R f
:
The R f
= f chart is easily exportedinto computer drafting programs (e.g. Adobe Illustrator, Canvas) by copying the chart andpasting it into the relevant package.
2.4. Worksheet ‘‘calculate R s
’’
This worksheet calculates the best-fit parameter w 2
(e.g.
Borradaile, 1976 ) for the
y -distribution test of
Lisle
(1977)
over a range of user-specifiedstrains. Low values for w 2 indicate that the initial distribution of marker orientations ð y Þ is uniform (random). The strain ellipse
ð R s
Þ is then calculatedby selecting the particular strain which yields the most random initial distribution of marker orientations. Critical values for the test are given in
Lisle (1985)
.
The user inputs an initial strain, the number of steps
(up to a maximum of 75) anda step increment into the box on the left of the worksheet (
Fig. 3 ). Clicking the
‘‘Calculate’’ button on the right of the worksheet initiates a Visual BASIC macro which calculates the best-fit parameter over the specifiedstrain range. The results are displayed graphically on the chart ‘‘
(
Fig. 4 ).
R s vs.
w 2 ’’
The strain value corresponding to the minimum value for w 2 is automatically enteredinto the rightmost box
(‘‘ R s value’’).
R i chart ‘‘Ln R f and y curves are then createdon the vs. Phi’’ for this particular strain ratio.
2.5. Chart ‘‘R s vs.
w 2 ’’
This chart displays how the best-fit parameter ð w 2 Þ varies with strain ð R s
Þ over the user-specifiedstrain range. The harmonic mean of the displayed.
R f data is also
3. Spreadsheet verification
Firstly, the sample dataset ( s1a.rfp
) of
Mulchrone and
Meere (2001)
was enteredinto the spread

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ARTICLE IN PRESS
D.M. Chew / Computers & Geosciences 29 (2003) 795–799
Fig. 3. Worksheet ‘‘Calculate R s
’’. Clicking on the ‘‘Calculate’’ button initiates a macro which calculates the best-fit parameter ð w
2
Þ over the specifiedstrain range.
2
60
50 Harmonic mean of R f
40
2
30
20
10
0
1 1.1
R s
= 1.47 from minimum value for c
2
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Fig. 4. Chart ‘‘ R s vs.
w 2 ’’ illustrating how the best-fit parameter ð w 2 Þ of y -distribution test varies with strain ð R s
Þ :
1.9

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D.M. Chew / Computers & Geosciences 29 (2003) 795–799 799
5
10
15
20
3
4
1.25
1.5
1.75
2
Table 2
Calculatedstrains from a synthetically strainedsuite of elliptical markers
Strain used to generate data Strains calculated using spreadsheet Average Standard deviation
1.23
1.24
1.22
1.22
1.24
1.25
1.29
1.28
1.3
1.5
1.54
1.48
1.48
1.49
1.5
1.49
1.5
1.52
1.25
1.51
1.252
1.501
0.029
0.019
1.67
1.72
1.74
1.78
1.78
1.76
1.78
1.75
1.76
1.75
1.749
0.034
2.02
1.99
1.95
2.02
2.04
1.94
1.96
2.04
1.99
1.99
1.994
0.036
3.06
3.06
2.94
3.18
3.12
2.99
2.91
2.92
3.05
3.09
3.032
0.090
4 4.01
4.06
4.07
3.98
4.08
4.02
3.92
3.91
3.94
3.999
0.061
5.01
5.05
5.03
5.01
5.06
5.06
4.95
5.11
5.12
5.13
5.053
0.056
9.81
9.74
9.92 10.24 10.12 10 9.97 10.22 10.01 10.17 10.02
0.169
14.85 14.64 14.92 15.09 14.96 14.94 15.2
14.83 15.19 15.12 14.97
20.23 20.27 19.87 20.38 19.86 20.35 19.73 20.09 20.23 20.57 20.16
0.178
0.266
yielded a strain of 2.11 comparing favourably with the
publishedvalue of 2.1 ( Mulchrone andMeere, 2001 ).
Secondly, a set of synthetically deformed strain
( synthetic.xls
), using the equations listedin
Lisle (1985) .
The markers have a random initial ellipticity, R i
,
(bounded by a specified maximum), and a random initial orientation, y . Ten synthetic datasets for a specifiedstrain were fedinto the Rfphi.xls
workbook, andthe procedure repeatedfor a range of strains. The difference between the calculated strain and the strain usedto generate the synthetic data is minimal (
Table 2 ).
Finally, raw R f
= f data from a previously published source (
Johnston, 1993 ) was enteredinto the spread
sheet. The publishedstrain estimates of 1.5 for both samples compare favourably with the spreadsheetcalculatedstrains of 1.53 and1.48.
4. Discussion and suggestions to users
A spreadsheet-based approach to R f
= f strain analysis significantly reduces the time-consuming calculations involvedin estimating strain ratios, andcan be usedon any computer platform that supports Excel. The user can at any stage check the strain ratio along with statistical parameters which describe the symmetry of the R f
= f plot andthe initial orientation of the strain markers. Resultant R f
= f diagrams are easily exported into a variety of computer drawing packages.
References
Borradaile, G.J., 1976. A strain study of a granite-gneiss transition andaccompanying schistosity formation in the
Betic orogenic zone, SE Spain. Journal of the Geological
Society, London 134, 417–428.
Dunnet, D., 1969. A technique of finite strain analysis using elliptical particles. Tectonophysics 7, 117–136.
Johnston, J.D., 1993. Ice wedge clasts in the Dalradian of south
Donegal—evidence for subaerial exposure of the boulder bed. Irish Journal of Earth Sciences 12, 13–26.
Lisle, R.J., 1977. Clastic grain shape andorientation in relation to cleavage from the Aberystwyth Grits, Wales. Tectonophysics 39, 381–385.
Lisle, R.J., 1985. Geological Strain Analysis: A Manual for the
R f
/ f Technique. Pergamon Press, Oxford, 99pp.
Mulchrone, K.F., Meere, P.A., 2001. A Windows program for the analysis of tectonic strain using deformed elliptical markers. Computers & Geosciences 27 (10), 1251–1255.
Peach, C.J., Lisle, R.J., 1979. A Fortran IV program for the analysis of tectonic strain using deformed elliptical markers.
Computers & Geosciences 5 (3–4), 325–334.
Ramsay, J.G., 1967. Folding and Fracturing of Rocks.
McGraw-Hill, New York, 531pp.