1988_Shape of the rose.pdf

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Sedimentary Geology, 59 (1988) 149-152 149Elsevier Science Publishers B.V., Amste rdam - Printed in The Nethe rlan ds

S h o r t N o t e

T h e s h a p e o f t h e ro s eW . N E M E C

Geological Institute (A), University of Bergen , N-5007 Bergen (Norway)Received April 20, 1988; revised and accepted June 28, 1988

AbstractNemec, W., 1988. The shape of the rose. Sediment. Geol., 59: 149-152.

The vast majority of rose diagrams that appear in the sedimentological literature are constructed incorrectly, withthe use of a linear frequency scale. This leads to gross distortions of the shape of the rose. Correct construction of arose diagram requires non-linear frequency scale to be used. The correct lengths (radii) of the diagram's sectors can bederived numerically, or determined more rapidly by plotting the frequencies directly on an appropriate net.

T h e c i r cu l a r h is t o g r am , m o r e o f t e n r e f e r r e d t oas rose diagram b y g e o l o g i s t s ( P o t t e r a n d P e t t i -j o h n , 1 9 6 3) , i s t h e m o s t c o m m o n l y u s e d g r a p h i c a lm e a n s o f p r e s e n ti n g t h e f r e q u e n c y d i s tr i b u t io n o ft w o - d i m e n s i o n a l o r i e n t a t i o n ( .v e c to r ia l ) d a t a . Al a r g e n u m b e r o f s e d i m e n t a r y r o c k a t t r i b u t e s , p a r -t i c u l a r l y t h e d i r e c t i o n a l s e d i m e n t a r y f e a t u r e s u s e da s p a l a e o c u r r e n t i n d i c a to r s , a re m e a s u r e d i n t e r m so f t h e i r a z i m u t h s a n d s u b s e q u e n t l y a n a l y s e dg r a p h i c a l l y w i t h t h e u s e o f a r o se d i a g r a m ( P o t t e ra n d P e t t i j o h n , 1 9 6 3 ) . S u c h a p l o t i s r e l a t i v e l y e a s yt o c o n s t r u c t a n d g i v e s a g o o d v i s u a l i m p r e s s i o n o ft h e d a t a d e n s i t y v a r i a t io n s .

A c c o r d i n g l y , r o s e d i a g r a m s h a v e g a i n e d m u c hp o p u l a r i t y a m o n g s e d i m e n t a r y g e o l o g i s t s a n d a r ef o u n d i n n e a r l y a l l o f t h e n u m e r o u s p u b l i c a t i o n st h a t d e a l w i t h p a l a e o c u r r e n t d a t a . I t i s s i m p l yu s e f u l t o p l o t t h e t w o - d i m e n s i o n a l v e c t o r i a l d a t ai n a r o s e - d i ag r a m f o r m b e f o r e a n e s t i m a t i o n o fs t a ti s t ic a l p a r a m e t e r s ( s u c h a s a v e c t o r m e a n , i t sm a g n i t u d e , c i r c u l a r v a r i a n c e , e t c .) is a t t e m p t e d .M o r e o v e r , r o s e d i a g r a m s a r e o f t e n u s e d b y r e -s e a r c h e r s a s t h e o n l y m e a n s o f s u m m a r i s i n g t h e i rd a t a s et s, e s p e c ia l l y w h e n t h e f r e q u e n c y d i s t r i b u -t i o n a p p e a r s t o b e n o n - u n i m o d a l a n d s t a t i s t i c sc a n n o t r e a d i l y b e c a l c u l a t e d . I n o t h e r w o r d s ,0037-0 738/8 8/$03. 50 1988 Elsevier Science Publis hers B.V.

s e d i m e n t o l o g i c a l p a l a e o c u r r e n t i n f e r e n c e s a r eb a s e d , p a r t l y o r e n t i r e l y , o n t h e a c t u a l shape o ft h e r o s e d i a g r a m .

I t i s , t h e r e f o r e , r a t h e r s u r p r i s i n g a n d q u i t e u n -f o r t u n a t e t h a t t h e v a s t m a j o r i t y o f t h e r o s e d i -a g r a m s t h a t a p p e a r i n t h e s e d i m e n t o l o g i c a l l i te r a -t u r e , a n d a l s o i n m a n y t e x t b o o k s , a r e incorrect.S i m p l y , t h e w a y i n w h i c h s u c h d i a g r a m s a r e c o n -s t r u c t e d b y m o s t g e o l o g i s t s i s , f r o m a m e t h o d -o l o g i c a l p o i n t o f v i e w , w r o n g . T h i s p a p e r i s i n -t e n d e d t o m a k e t h e g e o l o g i s t s a w a r e o f t h e e r r o ra n d t o p r e v e n t i t s c o n t i n u a t i o n i n t h e l i t e r a t u r e .

I t i s t h e p r i n c i p l e o f h i s t o g r a m c o n s t r u c t i o nt h a t t h e area o f e a c h c o l u m n , o r c i r c u l a r s e c t o r , o ft h e d i a g r a m s h o u l d b e m a d e p r o p o r t i o n a l t o t h ec l as s f r e q u e n c y ( d e n s i t y ) . I n s t e a d , t h e c i r c u l a rh i s t o g r a m i s o f t e n t h o u g h t l e s s l y c o n s t r u c t e d w i t ht h e radius o f t h e s e c t o r p r o p o r t i o n a l t o t h e c l a s sf r e q u e n c y . T h i s c o m m o n m i s t a k e a p p a r e n t l y d e -r iv e s f r o m a m i s u n d e r s t a n d i n g o f t h e n a t u r e o f ah i s t o g r a m , w i t h a f a l s e ly d r a w n a n a l o g y b e t w e e n as i m p l e , c o l u m n a r h i s t o g r a m a n d a c i r c u l a r h i s t o -g r a m ( s ee F ig ~ 1 a n d e n c l o s e d c o m m e n t ) . T h ec o l u m n a r h i s t o g r a m s u s u a l l y h a v e l i n e a r h o r i z o n -t a l s c a le s ( c o n s t a n t c l a s s w i d t h s ) , s o t h a t t h e h e i g h t so f t h e c o l u m n s a r e m a d e l i n e a r l y p r o p o r t i o n a l t o


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/ , r e . . . . . , , . / /| ' i . . . . . . . . . I - , \ ~ / 1 \~ i /

Fig. 1. Simple columnar histogram (A) and its circular equiv-alent (B). The diagrams, and their equivalent columns (A) andsectors (B), have the same areas. Note that the frequency scalebecomes non-linear when histogram A is squeezed into acircular form B.C o m m e n t : In the columnar histogram (A), the horizontal scaleis divided into classes and the class widths remain constant forthe entire height of the columns. The area of the column isproportional to the number of data in that class and representsthe class frequency (density); note carefully that it is the a r e a ,not the height, of the column. The vertical (frequency) scalehere is linear because the horizontal scale is finear and invaria-ble. In the circular histogram (B), the classes are the same as in(A), but their linear scale (actual width) varies along thediagram radius; in such case, the frequency scale can no longerbe linear.

the class frequencies; for example, a column of20% frequency is made twice as high as the col-umn of 10% (Fig. 1A). This makes the "areal"

A

S A M P L E S T A T I S T I C SM e a n v e c t o r : i ~ = 8 V e c t o r m a g n i tu d e ( l e n g t h ) : R = 0 . 2 7 ( 2 7 % )V e c t o r s t r e n g t h : A = 0 . 0 0 6B e t a c h e l e t ~ s c i r c u l a r s t a n d a r d d e v i a t i o n : s = 81

R A Y L E I G H S T E S TC r i t i c a l v a l u e s : S o . g 1 = 0 . 3 2 , R o . o s = 0 . 2 6 .R e s u l t : n o p r e f e r r e d o r i e n t a t io n a t 1% significance( r i s k - o f - e r r o r ) l e v e l ; a n o r t h e r ly t r e n d c a n b e i n f e r re d a t 1 0 %o r h i g h e r l e v e l , b u t i s s t i ll d o u b t f u l a t 5 % l e v e l ( s i n c e R ~ o . o s )

F i g . 2 . A z i m u t h s o f r i p p l e f o r e s e t s f r o m a n e a r l y C r e t a c e o u sbay-fill sequence (Helvetiafjellet Formation, Svalbard) plottedin a rose diagram; the number of data is n = 44. In diagram A,a correct frequency scale is used and the rose shows littlepreferred orientation; no preferred orientation at a= 0.01significance level is revealed by statistical test. In diagram B, alinear (incorrect) frequency scale is used and the rose shows anapparent "strongly preferred" orientation (false shape).

aspect of the histogram less apparent, and leads toa serious error once the same linear approach ismechanica lly used to construct a circular histo-gram.

Few geologists seem to be aware that the use ofa linear (arithmetic) frequency scale in circularhistograms leads to gross distortions of the di-agram's shape (false relative densities are pro-duced). This is il lustrated by the example shownin Fig. 2. When the diagram in Fig. 2B is consid-ered, the eye is likely to be undu ly impressed bythe dom in at ing size of the modal sector, com-pletely out of proportion to its true significance.The degree of preferred orientation is greatly ex-aggerated by this kind of plot (Fig. 2B), and thegeologist is then easily misled by the shape of therose.

The correct radius of each sector (or frequencycircle) of a rose diagram can be calculated in atleast two ways:

(a) Th e radiu s, r 1


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Fig. 3. A frequency net prod uced to facilitate rapid con struction of circular histograms. The net (frequency circles) can b e extendedto higher values, if needed, with the use of form ula (1) given in tex t; rl~ = 8 mm in the p resent case. The % symbols have to b eomitted if pure number frequencies (not percentages) are plotted.

c m 2 ) a n d w = 0. 5 2 3 r a d ( 3 0 ) , t h e n t h e r a d i u s o ff = 2 5 fr e q u e n c y s e c t o r i s:r25 = ~ ( 2 - 3 0 0 0 . 2 5 ) / ( 6 0 - 0 . 5 2 3 ) = 6 9 m m .

S o m e c o m p u t a t i o n a l e f f o r t i s t h u s n e c e s s a r y t oc o n s t r u c t t h e c i r c u l a r h i s t o g r a m . A l t h o u g h t h ec o m p u t a t i o n i t s e l f i s s i m p l e a n d r e l a t i v e l y r a p i d ,m a n y g e o l o g i s t s m a y f i n d i t s o m e w h a t d i s c o u r a g -i n g o r i n c o n v e n i e n t t o u s e a n u m e r i c a l p r o c e d u r ef o r p u r e l y g r a p h i c a l p u r p o s e s , e s p e c i a l l y w h e n al a r g e n u m b e r o f d i a g r a m s a r e to b e c o n s t r u c te d . I ti s , t h e r e f o r e , m o r e p r a c t i c a l t o p r o d u c e a s t a n d a r df r e q u e n c y n e t ( F i g . 3 ) f o r p l o t t i n g t h e d a t a . W i t ht h e u s e o f s u c h a n e t , t h e c o n s t r u c t i o n o f a r o s ed i a g r a m b e c o m e s a s r a p i d a s t h e c o n s t r u c t i o n o f as i m p l e , c o l u m n a r h i s t o g r a m .

F i n a l ly , i t i s w o r t h w h i l e r e m i n d i n g r e s e a r c h e r st h a t a c i r c u l a r h i s to g r a m , e s p e c i a ll y w h e n p u b -l i s h e d , s h o u l d n e c e s s a r i l y p r o v i d e t h e f o l l o w i n g

i n f o r m a t i o n : t h e t y p e o f f r e q u e n c y e x p r e s s io n u s e d( w h e t h e r p u r e n u m b e r o r n u m b e r p e r c e n t ) , t h ef r e q u e n c y s c a l e u s e d , a n d t h e n u m b e r o f d a t a o nw h i c h t h e d i a g r a m i s b a s e d . O t h e r w i s e , r o s e d i -a g r a m s m a y n o t b e c o m p a r e d a n d e v e n t h e ir v a li d -i t y (c o r r e c t n es s ) c a n n o t b e e v a l u a t e d b y t h e v i e w e r.T h e u s e o f s t a t i s t i c s a n d a p p r o p r i a t e s t a t i s t i c a lt e s t s ( C u r r a y , 1 9 5 6 ; P o t t e r a n d P e t t i j o h n , 1 9 6 3 ;M a r d i a , 1 9 7 2; K r a u s e a n d G e i j e r , 19 8 7) is s tr o n g l yr e c o m m e n d e d , w h e n e v e r t h e d a t a s e t c a n a t l e a s tb e a s s u m e d t o d e ri v e f r o m a u n i m o d a l p o p u l a t i o nw i t h c i r c u l a r - n o r m a l p r o p e r t i e s .Acknowledgements

T h e m a n u s c r i p t w a s t y p e d b y A . P e d e r s en a n dc r i t i c a l l y r e v i e w e d b y K . L . K l e i n s p e h n ( M i n n e a p -o l i s ) , M . R . T a l b o t ( B e r g e n ) , a n d J . C . T i p p e r( C a n b e r r a ) .


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Ref erence sC u r r a y , J . R . , 1 9 5 6. T h e a n a l y s i s o f t w o - d i m e n s i o n a l o r i e n t a -

t i o n d a t a . J . G eo l . , 6 4 : 1 1 7 - 1 3 1 .K r a u s e , R . G . F . a n d G e i j e r , T . A . M . , 1 9 87 . A n i m p r o v e d m e t h o d

f o r c a l c u l a t i n g t h e s t a n d a r d d e v i a t i o n a n d v a r i a n c e o fp a leo cu r r en t d a t a . J . S ed imen t . P e t r o l . , 5 7 : 7 7 9 - 7 8 0 .

M ar d ia , K . V . , 1 9 72 . S t a t i s t i c s o f D i r ec t io n a l D a ta . A cad e micP r e s s , L o n d o n , 2 3 7 p p .

P o t t e r , P . E . an d P e t t i j o h n , F . J . , 19 6 3. P a l eo cu r r en t s an d B as inA n a l y s i s . A c a d e m i c P r e s s , N e w Y o r k , N . Y . , 2 9 6 p p .

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