Computers & Geosciences 29 (2003) 795–799

Short Note

An Excel spreadsheet for finite strain analysisusing the Rf=f technique$

David M. Chew*

Department of Geology, Trinity College, Dublin 2, Ireland

Received 24 May 2002; received in revised form 21 October 2002; accepted 10 November 2002

1. Introduction

The Rf=f technique for strain analysis was first

described by Ramsay (1967) and refined by Dunnet

(1969) and Lisle (1977). The Rf=f technique can

potentially be used on any deformed suite of initially

elliptical strain markers (e.g. conglomerates, oolites).

The axial ratios ðRf Þ of typically between 50 and 100

strain markers and their respective long axes orienta-

tions ðfÞ have to be recorded.

The Rf=f technique assumes that the suite of elliptical

strain markers did not possess a preferred initial

orientation prior to deformation. As these markers are

elliptical and have differing initial orientations, their

deformed shapes cannot correspond to the strain ellipse.

However, it is possible to calculate the pre-deforma-

tional axial ratio ðRiÞ and orientation ðyÞ of each markerfor a particular strain. The strain ellipse ðRsÞ can then becalculated by selecting the particular strain which yields

the most random initial distribution of marker orienta-

tions (the fundamental starting assumption). The reader

is encouraged to consult a comprehensive manual on the

Rf=f technique (Lisle, 1985) for further details.

One of the drawbacks of the Rf=f technique is that

the strain calculations are relatively time-consuming to

perform manually. Strain calculation is made signifi-

cantly easier by utilizing a computer-based approach

(e.g. Peach and Lisle, 1979; Mulchrone and Meere,

2001). The program presented here employs a spread-

sheet-based approach with the advantages of easy

modification of the spreadsheet structure and the ability

to export and modify Rf=f diagrams on a variety of

computer platforms (e.g. Fig. 1).

2. Spreadsheet structure and operation

2.1. Notes concerning the Excel implementation

This implementation of the Rf=f technique uses the

Microsoft Excelt 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 saved in a newer format if

required.

The workbook (Rfphi.xls) consists of two worksheets

(‘‘Enter data’’ and ‘‘Calculate Rs’’) and two charts (‘‘Ln

Rf vs. Phi’’ and ‘‘Rs vs. w2’’). Rf=f data (see Table 1 for

nomenclature) is entered onto the worksheet (‘‘Enter

data’’) and these data are plotted on the chart (‘‘Ln Rf

vs. Phi’’), along with Ri and y curves for the calculated

strain. The second worksheet (‘‘Calculate Rs’’) calculates

the best-fit parameter ðw2Þ of the y-distribution test of

Lisle (1977) over a user-specified strain range. The

values of the best-fit parameter over the strain range are

displayed graphically on the second chart (‘‘Rs vs. w2’’).The detailed operation of each of these modules is

discussed below. Further operating instructions are

supplied in the accompanying help file (readme.doc).

2.2. Worksheet ‘‘Enter data’’

The long and short axes and orientations

ð�90ofo90Þ of up to 350 markers are entered on this

worksheet. The axial ratio ðRf Þ of each strain marker is

calculated, while sample numbers can be entered into the

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$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).

0098-3004/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0098-3004(03)00027-X

leftmost column. The vector mean and harmonic 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 Rf=f plot

into four quadrants (defined by the vector mean and

harmonic mean of the strain markers). The index of

symmetry is calculated as follows:

ISYM ¼ 1� ðjnA � nBj þ jnC � nDjÞ=N; ð1Þ

where nA denotes the number of strain markers in quadrant

A, etc., and N equals the total number of markers. High

values of ISYM suggest the data are symmetrical, while low

values suggest that the data are markedly asymmetric

and hence the assumption of no preferred initial

orientation to the strain markers is incorrect. Critical

values for the test are listed in Lisle (1985).

2.3. Chart ‘‘Ln Rf vs. Phi’’

This chart plots lnRf against f for each strain marker

(Fig. 1). The x-axis scale ðRf Þ is logarithmic. The f data

are shifted along the y axis such that the vector mean of

the f data is equal to 0. Once the strain ðRsÞ has beendetermined (using the worksheet ‘‘Calculate Rs’’), Ri

and y curves for this calculated strain are displayed.

Eight Ri curves are plotted, with values of 1.25, 1.5, 1.75,

2, 2.5, 3, 4 and 6, respectively (Fig. 1), while the y curvesrange from �90� to 90� in increments of 9� (Fig. 1).

Lisle (1985) lists several equations relating the five

variables involved in Rf=f calculations (Rs; Rf ; Ri; fand y). Ri curves are calculated using Eq. (A1.3) of Lisle

(1985):

cosh 2ei ¼ cosh 2ef cosh 2es � cos 2f sinh 2ef sinh 2es;

ð2Þ

where ei ¼ 12ln Ri; etc. As Ri and Rs are known, the Ri

curve is calculated by looping through a range of values

for Rf and calculating the corresponding value for f:

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-90

-70

-50

-30

-10

10

30

50

70

90

0 0.5 1 1.5 2 2.5 3 3.5

Ln Rf

Ln Rf vs. Phi

Phi

1.5 1.75 2 2.5 3 4

Ri =1.25

Ri = 6

Harmonic mean of Rf

Vector mean of φ

q = 90˚

q = -45˚

q = -90˚

q = 45˚

q = 0˚

Fig. 1. Chart ‘‘LnRf vs. Phi’’. Chart was exported into Adobe Illustrator where Ri curves and the �90�, �45�, 0�, 45� and 90� ycurves were labelled.

Table 1

Rf=f nomenclature

Rf Measured ellipticity of strain marker

f Angle between marker long axis and maximum extension

direction

Ri Initial ellipticity of strain marker in undeformed state

y Initial angle between the undeformed marker long axis

and the maximum extension direction

Rs Axial ratio of the strain ellipse

D.M. Chew / Computers & Geosciences 29 (2003) 795–799796

The y curves are calculated using the following

equation of Lisle (1977):

Rf ¼tan 2yðR2

s � tan2 fÞ � 2Rs tan f

tan 2yð1� R2s tan

2 fÞ � 2Rs tan f

" #1=2ð3Þ

As y and Rs are known, the y curve is calculated by

looping through a range of values for f and calculating

the corresponding values for Rf :The Rf=f chart is easily exported into computer

drafting programs (e.g. Adobe Illustrator, Canvas) by

copying the chart and pasting it into the relevant

package.

2.4. Worksheet ‘‘calculate Rs’’

This worksheet calculates the best-fit parameter w2

(e.g. Borradaile, 1976) for the y-distribution test of Lisle

(1977) over a range of user-specified strains. Low values

for w2 indicate that the initial distribution of marker

orientations ðyÞ is uniform (random). The strain ellipse

ðRsÞ is then calculated by 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) and a 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 specified strain range. The

results are displayed graphically on the chart ‘‘Rs vs. w2’’(Fig. 4).

The strain value corresponding to the minimum value

for w2 is automatically entered into the rightmost box

(‘‘Rs value’’). Ri and y curves are then created on the

chart ‘‘LnRf vs. Phi’’ for this particular strain ratio.

2.5. Chart ‘‘Rs vs. w2’’

This chart displays how the best-fit parameter ðw2Þvaries with strain ðRsÞ over the user-specified strain

range. The harmonic mean of the Rf data is also

displayed.

3. Spreadsheet verification

Firstly, the sample dataset (s1a.rfp) of Mulchrone and

Meere (2001) was entered into the spreadsheet, and

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Fig. 2. Worksheet ‘‘Enter data’’, where Rf=f data is input.

D.M. Chew / Computers & Geosciences 29 (2003) 795–799 797

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0

10

20

30

40

50

60

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

c2

Rs vs. c2

Rs

Rs = 1.47 from minimum value for c2

Harmonic mean of Rf

Fig. 4. Chart ‘‘Rs vs. w2’’ illustrating how the best-fit parameter ðw2Þ of y-distribution test varies with strain ðRsÞ:

Fig. 3. Worksheet ‘‘Calculate Rs’’. Clicking on the ‘‘Calculate’’ button initiates a macro which calculates the best-fit parameter ðw2Þover the specified strain range.

D.M. Chew / Computers & Geosciences 29 (2003) 795–799798

yielded a strain of 2.11 comparing favourably with the

published value of 2.1 (Mulchrone and Meere, 2001).

Secondly, a set of synthetically deformed strain

markers was generated using a separate spreadsheet

(synthetic.xls), using the equations listed in Lisle (1985).

The markers have a random initial ellipticity, Ri,

(bounded by a specified maximum), and a random

initial orientation, y. Ten synthetic datasets for a

specified strain were fed into the Rfphi.xls workbook,

and the procedure repeated for a range of strains. The

difference between the calculated strain and the strain

used to generate the synthetic data is minimal (Table 2).

Finally, raw Rf=f data from a previously published

source (Johnston, 1993) was entered into the spread-

sheet. The published strain estimates of 1.5 for both

samples compare favourably with the spreadsheet-

calculated strains of 1.53 and 1.48.

4. Discussion and suggestions to users

A spreadsheet-based approach to Rf=f strain analysis

significantly reduces the time-consuming calculations

involved in estimating strain ratios, and can be used on

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 Rf=f plot and the initial orientation of the strain

markers. Resultant Rf=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 and accompanying 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 and orientation in relation

to cleavage from the Aberystwyth Grits, Wales. Tectono-

physics 39, 381–385.

Lisle, R.J., 1985. Geological Strain Analysis: A Manual for the

Rf/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.

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Table 2

Calculated strains from a synthetically strained suite of elliptical markers

Strain used to generate data Strains calculated using spreadsheet Average Standard deviation

1.25 1.23 1.24 1.22 1.22 1.24 1.25 1.29 1.28 1.3 1.25 1.252 0.029

1.5 1.5 1.54 1.48 1.48 1.49 1.5 1.49 1.5 1.52 1.51 1.501 0.019

1.75 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 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 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 4.01 4.06 4.07 3.98 4.08 4.02 3.92 3.91 3.94 3.999 0.061

5 5.01 5.05 5.03 5.01 5.06 5.06 4.95 5.11 5.12 5.13 5.053 0.056

10 9.81 9.74 9.92 10.24 10.12 10 9.97 10.22 10.01 10.17 10.02 0.169

15 14.85 14.64 14.92 15.09 14.96 14.94 15.2 14.83 15.19 15.12 14.97 0.178

20 20.23 20.27 19.87 20.38 19.86 20.35 19.73 20.09 20.23 20.57 20.16 0.266

D.M. Chew / Computers & Geosciences 29 (2003) 795–799 799