The orientation of buried iceberg scours and other linear phenomena

Marine Geology, 114 (1993) 263-272

263

Elsevier Science Publishers B.V., Amsterdam

The orientation of buried iceberg scours and other linear phenomena Michael R. Gipp x

Department of Earth Sciences, Memorial University of Newfoundland, St. Johns, Nfld. A1B 3X5, Canada (Received August 5, 1992; revision accepted May 3, 1993)

ABSTRACT Gipp, M.R., 1993. The orientation of buried iceberg scours and other linear phenomena. Mar. Geol., 114: 263-272. The average orientations of surficial iceberg scours can be obtained from sidescan sonograms. Using certain assumptions, the average orientation of similar buried phenomena can be estimated from a pattern of intersecting seismic profiles. Two possible solutions are generated at each intersection of two seismic profiles, each of which is oriented at an angle ~i from the profile which encounters the fewer scours, where ~i = tan- 1[sin c~/(p,+_cos ~b)], and Pr is the ratio of the number of scours observed on the two seismic lines. Several pairs of solutions are required to determine which solution of each pair is significant. Iceberg scours observed on aerial photographs of the floor of glacial lake Agassiz demonstrate that the estimates provided by this technique correspond with the average orientation of the population. Applying this method to buried iceberg scours in Emerald Basin, Scotian Shelf, gives orientations which are parallel to paleo-bathymetric contours.

Introduction Iceberg scours are c o m m o n l y o b s e r v e d features on the seafloor o f m i d - a n d h i g h - l a t i t u d e c o n t i n e n tal shelves. T h e recent increase in e x p l o r a t i o n a n d e x p l o i t a t i o n o f offshore h y d r o c a r b o n d e p o s i t s on the G r a n d B a n k s o f N e w f o u n d l a n d a n d in the N o r t h Sea has resulted in a b e t t e r u n d e r s t a n d i n g o f iceberg trajectories, a n d their r e l a t i o n to c u r r e n t directions. Sidescan s o n a r surveys, which p r o v i d e a t w o - d i m e n s i o n a l i m a g e o f features at the seafloor, can be used to p r o d u c e ice scour mosaics, so t h a t m o d e r n a n d p a l e o c u r r e n t directions m a y be d e t e r m i n e d f r o m m o d e r n a n d fossil p o p u l a t i o n s ( T o d d et al., 1988). M o s t studies o f iceberg scours have h a d to focus on scours e x p o s e d at the seafloor, because sidescan s o n a r is u n a b l e to i m a g e objects which d o n o t o u t c r o p . H i g h - r e s o l u t i o n seismic systems, while able to detect iceberg scours at d e p t h b e l o w the seafloor (King, 1976), o n l y p r o tPresent address: Department of Geology, University of Toronto, Scarborough Campus, 1265 Military Trail, Scarborough, Ont. M1C IA4, Canada. 0025-3227/93/$06.00

duce cross-sections o f them. To d e t e r m i n e the o r i e n t a t i o n o f a linear feature, such as a fault, an iceberg scour, o r an e r o s i o n a l channel, a grid o f seismic lines m u s t be o b t a i n e d , a n d the feature traced t h r o u g h several lines. The p r o b l e m is c o m p o u n d e d w h e n several similar features are present in close p r o x i m i t y . T h e m o s t c o m m o n a p p r o a c h to solving the p r o b l e m is to a t t e m p t to c o r r e l a t e i n d i v i d u a l scours, b u t where there are m a n y scours, o r their characteristics c h a n g e over s h o r t intervals, o r where they are so s h o r t t h a t they only cross one seismic line, this m e t h o d is impossible. A m e t h o d which d e t e r m i n e s statistical p r o p e r t i e s o f the p o p u l a t i o n as a whole is b e t t e r suited in such cases. Such a m e t h o d w o u l d also allow the analysis o f p o p u l a t i o n s o f b u r i e d scours, for which sidescan s o n a r is useless, a n d f r o m which p a l e o c u r r e n t directions m a y be estimated. Such a m e t h o d is p r e s e n t e d in this p a p e r .

The method It is i m p o s s i b l e to recognize a detected object as p o s s i b l y being linear on the basis o f its a p p a r e n t

© 1993 - - Elsevier Science Publishers B.V. All rights reserved.

264

M.R. GIPP

shape as revealed on seismic data. It is possible to infer a linearity to a population of such features, but only if the number of features detected (referred to as "apparent density" hereafter) changes with the orientation of the seismic line. If this criterion is met, we may assume that the features are approximately straight, and have a preferred orientation. The apparent density of a randomly-oriented population of linear features should not vary systematically with the orientation of the seismic line (Fig. 1). Given such a situation, two or more seismic lines with differing orientations (AA' and BB' in Fig. 2), apparent density values (p, for AA' and Pb for BB') can be used to estimate the average

A

B (

iiiiiiiii B,

A' a A2

_ _

Fig. 2. Lines AA' and BB' separated by acute angle ~b intersect at point O and cross either population 1 (dashed lines) or population 2 (solid lines). The significance of the two populations is that the observed pattern of intersections between population 1 and lines AA' and BB' is indistinguishable from the intersections between AA', BB', and population 2.

A3

~

I'-%

B3

B1

orientation (e) and the absolute density (p) of the features at the intersection point. Absolute density, for the purposes of this paper, is defined as the apparent density along a line oriented normal to the mean trend of the population, and is a function of the total length of the scours per unit area. Apparent densities from two seismic lines do not provide sufficient information to distinguish between two possible orientations for the population of linear features (Fig. 3). Thus, two solutions, termed a "solution-pair", are determined at each intersection of two seismic lines. The two solutions, referred to as population 1 and population 2 and which are derived in Appendix 1, are: :q = tan

Fig. I. (a) When a population of linear features (solid lines) has a preferred orientation, the frequency of their detection on seismic lines (dashed lines) crossing them at different orientations varies systematically. Seismic line A1 crosses fewer features per unit of length than do lines A2 and A3. (b) If the population of linear features has a random orientation, then there is no systematic change in the frequency of occurrence with orientation of the seismic line.

-

1

[sin 49/(P, - cos ~b)]

(1)

P 1 = p,/sin ~ 1

(2)

~2 = tan - 1[sin ck/(p, + cos 4~)]

(3)

D2 =

pa/Sin O~2

(4)

where Pr = Pb/P,, the ratio of the apparent densities in the two lines; 4) is the acute angle between AA' and BB'; ~1 is the angle between AA' and the

BURIEDICEBERGSCOURSAND OTHERLINEARPHENOMENA

I~ I~A2

B2 ~"~#" I'r'""POSSIBLE SOLUTION

y

~1

FORINTERSECTION"OINTB

TRACKLINES~

b

A\ A2

pI ~ P2 POSSIBLE SOLUTIONS

B

/B1 \ B2

S

/sl \ S2

Fig. 3. As significant solutions of proximal solution-pairs are parallel, it is possible to determine the significant solutions in Fig. 3a, where seismic lines have three orientations, but not in Fig. 3b, where seismic lines have only two orientations.

average orientation of population 1, measured in the direction of the obtuse angle. Pl is the absolute density of population 1. ~2 is the angle between AA' and the average orientation of population 2, measured in the direction of the acute angle. P2 is the absolute density of population 2. Special cases: If ~b- 90 °, then cos ~b= 0, then Eqs. 1 and 3 become identical, and ~1 = %- If Pa = Pb, then, using a series of trignometric identities (Appendix 2), cq= 90 ° - ~b/2, and % = ~b/2, which means that solution 1 is the bisector of the obtuse intersection angle, and solution 2 is the bisector of the acute intersection angle. Many mathematical problems yield two or more solutions, and commonly one or more of these solutions is recognized as being trivial. If our assumptions are correct, one of the two generated solutions is real, and the second is an artifact. To

265 determine which of the solutions is real, several pairs must be generated from a collection of profiles run in at least three directions (Fig. 3). If the average orientation of the real population of scours is constant through neighbouring solution-pairs, the real solutions will be parallel, whereas the trivial ones will not. An orthogonal grid over a pattern of oriented features will produce a group of identical solution-pairs, making it impossible to deduce the significant solution (Fig. 3b). Thus, seismic surveys should be based on a trigonal grid, because the data obtained from an orthogonal grid is insufficient to use this technique. Errors result because the method assumes that the phenomena are parallel, whereas populations of linear features in the real world are subparallel at best. The estimated orientation of a populations of linear features distributed symmetrically around a mean will have smaller errors than such an estimate for a population of linear features distributed in a skewed manner. Although a population of randomly oriented linear features will have similar apparent densities on every profile, so that all of the solution-pairs will bisect the angles between intersecting seismic profiles, equal apparent densities on two seismic profiles is insufficient to conclude that the population is randomly distributed. If several seismic lines, in at least three different directions, encounter equal apparent densities, then the distribution is probably random. Field test: Exposed scours of glacial Lake Agassiz Aerial photographs of the exposed floor of extinct glacial lake Agassiz commonly show straight to curvilinear features a few tens of metres wide and hundreds of metres to a few kilometres long (Mollard, 1983). Although of low relief, these features are recognizable at ground level as a series of broad, low-relief ridges (Woodworth-Lynas and Guignr, 1990). These lineations, interpreted as ice scours, commonly show a preferred orientation (Dredge, 1982). For this reason, an aerial photograph along the trans-Canada highway, approximately 50 km southeast of Winnipeg, Manitoba, has been selected to test the geometrical method detailed above (Fig. 4). The test consisted of running different "surveys"

266

M.R. GIPP

T1 2

T3 O1

b

o3

LI

Fig. 4. Interpretation of iceberg scours of extinct glacial lake Agassiz, exposed in south east Manitoba, near Winnipeg. Large box is the outline of the study area of Woodworth-Lynas and Guign~ (1990). The road is the Trans-Canada Highway, Numbers represent points which were studied in more detail. on the i n t e r p r e t e d scours. T h e surveys included (1) three two k i l o m e t r e lines, intersecting at a c o m m o n point, s e p a r a t e d by an angle o f 60 ~ (2) three two kilometre lines, all intersecting at one point, s e p a r a t e d by angles o f 45 °, 45 ° a n d 9if' and (3) three two k i l o m e t r e lines, s e p a r a t e d by angles o f 47 °, 61 ° a n d 72 °, b u t n o t all intersecting at the same p o i n t (Fig. 5). T h r e e p o i n t s on the interpreted scour m o s a i c were selected, the m e a n o r i e n t a t i o n a n d linear density o f the scours within two kilometres o f the p o i n t s calculated by direct m e a s u r e m e n t , a n d the surveys " r u n " over the point. All scours crossed by one o f the "seismic lines" were counted, a n d Eqs. 1 - 4 were used to calculate solution-pairs, from which the significant solutions were chosen. F o r reference, the m e a n o r i e n t a t i o n o f the scours within one k i l o m e t r e o f each p o i n t was calculated by the v e c t o r - s u m m e t h o d . Scour density was m e a s u r e d as the length o f scours per unit area. The results, s u m m a r i s e d in Table l a - c , show that

Fig. 5. Hypothetical seismic grids used to analyze the scours in Fig. 4, with the scours around point 1. The method can thus be illustrated. Line 7"1 (in a) crosses 9 scours, T2 crosses 16. Since ~b=60°, Eqs. 1-4 yield ~ =34.1 ~', Pl =8.03 km -1, ~2 = 20.8, P2- 12.7 km ~. Since p~
the significant solutions from all three types o f surveys agree with the o r i e n t a t i o n s derived from direct m e a s u r e m e n t s . T h e best results were o b t a i n e d f r o m the trigonal survey (Fig. 5a). The estimates for density are usually p o o r e r t h a n the estimates o f average orientation.

Field application: Buried scours in Emerald Basin, Scotian Shelf E m e r a l d Basin is a glacially o v e r d e e p e n e d basin on the central N o v a Scotian Shelf ( K i n g a n d M a c L e a n , 1976) a b o u t 80 k m s o u t h - s o u t h e a s t o f Halifax, N o v a Scotia, C a n a d a (Fig. 6). Pleistocene sediments in E m e r a l d Basin are up to 100 m thick and are c o m p o s e d o f three acoustically distinct

BURIED |CEBERG SCOURS AND OTHER LINEAR PHENOMENA

267

TABLE 1

Comparison of average orientations and scour densities calculated by the geometrical method at points 1 (a), 2 (b) and 3 (c) on Fig. 4, using each of the grids depicted in Fig. 5 AA'

BB'

Pa Pb (km - x)

(a) Point 1. Mean trend." 146-326. Mean density: 7.7kin TI × T2 000-180 060-240 4.5 8.0 TI × T3 120-300 000-180 4.0 4.5 T2 x T3 120-300 060-180 4.0 8.0 Best triplet: 146-326, 148-328, 150-330. Mean trend: Ol x 0 2 000-180 045-225 4.5 6.5 O1 x 03 000-180 090-270 4.5 6.5 02 x 03 045-225 090-270 6.5 6.5 Best triplet: 136-316, 145-325, 167-347. Mean trend: L1 x L2 000-180 047-227 4.5 7.5 Ll × L3 119-299 000-180 3.0 4.5 L2 x L3 119 299 047-227 3.0 7.0 Best triplet: 143-323, 143-323, 143-323. Mean trend:

~

~1 ~2 (degrees)

-1 60 34.1 20.8 146-326 60 54.2 28.1 066-246 60 30.0 19.1 150-330 148-328. Mean density: 8.2 k m 45 43.8 18.2 136-316 90 34.7 34.7 145-325 45 67.5 22.5 167-347 150-330. Mean density: 7.2 k m 47 36.6 17.3 143-323 61 40.7 23.8 078-258 72 23.5 18.7 143-323 143-323. Mean density: 7.5 km

(b) Point 2. Mean trend." 161-341. Mean density: 6.6 kin- 1 T1 x T2 000-180 060 240 2.0 5.0 60 23.4 16.1 157-337 T1 × T3 000-180 120-300 2.0 3.5 60 34.7 21.0 035-215 T2 x T3 120-300 060-240 3.5 5.0 60 43.0 24.2 163-343 Best triplet: 157-337, 159-339, 163-343. Mean trend: 160-340. Mean density: 5.3 km -~ O1 × 0 2 000-180 045 225 2.0 6.0 45 17.1 10.8 163-343 O1 x 03 000-180 090-270 2.0 5.5 90 20.0 20.0 160-340 0 2 x 03 090 270 045 225 5.5 6.0 45 61.5 21.5 152-332 Best triplet: 163-343, 160-340, 152-332. Mean trend: 158-338. Mean density: 6.3 km -1 L1 × L2 000-180 047-227 2.0 5.5 47 19.4 11.9 161-341 L1 x L3 000-180 119-299 2.0 4.5 61 30.4 17.9 030-210 L2 × L3 119-299 047-227 4.5 5.5 72 46.2 31.8 165-345 Best triplet: 161-341, 162-342, 165-345. Mean trend: 162-342. Mean density: 6.3 k m (c) Point 3. Mean trend." 161-341. Mean density: 7.4 k m - 1 T1 ×T2 000-180 060-240 3.5 6.5 60 32.5 20.2 148-328 T1 × T3 000-180 120-300 3.5 5.5 60 38.9 22.7 039-219 T2 x T3 120-300 060-240 5.5 6.5 60 51.8 27.2 172-352 Best triplet: 148-328, 157-337, 172-352. Mean trend: 159-339. Mean density: 7.5 km -1 O1 x 0 2 000-180 045 225 3.5 5.5 45 39.3 17.2 141-321 O1 x 03 000-180 090-270 3.5 5.5 90 32.5 32.5 148-328 0 2 x 03 045-225 090-270 5.5 5.5 45 67.5 22.5 157-337 Best triplet: 141-321, 148-328, 157-337. Mean trend: 149-329. Mean density: 6.0km -1 L1 × L2 000-180 047-227 3.0 6.5 47 26.1 14.3 164-344 L1 × L3 000-180 119-299 3.0 5.0 61 36.5 22.3 037-217 L2 x L3 119-299 047-227 5.0 6.5 72 43.8 30.6 162-342 Best triplet: 164-344, 168-348, 162-342. Mean trend: 165-345. Mean density: 7.3 km -1

units; a basal till or diamict sheet, glaciomarine muds, and Holocene muds (King and Fader, 1986). High resolution seismic profiles have been obtained using both the Huntec deep-tow seismic system (Hutchins et al., 1976), and the Nova Scotia Research Foundation Corporation V-fin (Bidgood, 1974) using a sweep-rate of either 250 or 500 ms. Buried erosion features 2-5 m deep and 30-120 m

Pl (km - 1)

P2

021-201 148-328 101-281

8.03 4.93 8.00

12.70 8.49 12.20

018-198 035-215 068-248

6.50 7.90 7.04

14.41 7.90 16.99

017-197 143-323 100-280

7.55 4.60 7.52

15.13 7.43 9.36

016-196 159-339 096-276

5.04 3.51 5.13

7.21 5.58 8.54

012-192 020-200 069-249

6.80 5.84 6.26

10.67 5.84 15.01

012-192 162-342 087-267

6.02 3.95 6.23

9.70 6.51 8.54

020-200 157-337 093-273

6.51 5.57 7.00

10.14 9.07 12.03

017-197 033-213 068-248

5.53 6.51 5.95

11.84 6.51 13.07

014-194 168-348 088-268

6.82 5.04 7.22

12.14 7.91 9.82

Azimuths

wide are observed in the lower part of the glaciomarine muds (Fig. 7), between reflections dated at 17.5 ka and 15 ka (Gipp and Piper, 1989). On the basis of morphology, and occurrence relative to sediment type, paleobathymetry, and seismic line orientation (Fig. 8), these erosion features are identified as iceberg scours exhibiting a preferred orientation.

268

M.R. ~wP

NEW BRUNSWICK

• :'.'~

E~BAIERALD SIN

44N

68W

64W

60W

EMERALD BANK

10

20 km B

Fig. 6. Map of the central Scotian Shelf, showing the location of Emerald Basin.

water depth

. . . . (metres)

175 200 225

i 0

125

I

I

O

lOre

250 m

I

Fig. 7. Interpreted NSRFC deep-towed sparker record, depicting buried iceberg scours in Emerald Silt.

As the buried scours are restricted to the basin flanks, all of the solution-pairs are generated from the seismic intersections along the southeastern flank of Emerald Basin. From a plot of the solution-pairs (Fig. 8), it is possible to determine the significant solutions. In area A (Fig. 8b), three solution-pairs are depicted, all within two kilometres of one another. Because the three seismic lines that comprise the survey in this area have different orientations, the significant solutions are determined to be solutions AI, B2 and PI. The average scour orientation is thus considered to be approximately N S. Similarily, the significant solutions can be determined for most solution-pairs in the basin (Fig. 9). Iceberg scours in Emerald Basin appear to be oriented roughly parallel to paleobathymetric contours. Such a result is consistent with iceberg scours observed on Saglek Bank, Labrador Shelf (Todd et al., 1988). Where possible, absolute densities have been estimated. There is no statistical relationship

269

BURIED ICEBERG SCOURS AND OTHER LINEAR PHENOMENA

63°W

62030 '

63"W

62030 '

Fig. 8. (a) Paleobathymetric plot showing the location of buried scours as tick marks drawn perpendicular to the seismic line on which they were observed. Contours are in milliseconds two-way travel time (twt) below a datum plane (110 m below sea level). (b) Plot of solution-pairs on a paleobathymetric map o f Emerald Basin. (c) Enlargement of solution-pairs along the eastern flank, suggesting that the scours are oriented approximately N - S , and is analagous to Fig. 3a. (d) Enlargement of solution-pairs near the eastern channel, showing that the scours are likely oriented N E - S W .

between the paleodepth of the main scoured surface and the density of iceberg scours, although the density appears to increase with decreasing paleodepth (Fig. 10), as observed in modern populations (Todd et al., 1988). At the shallowest paleodepths, scour density decreases dramatically, possibly due to deflection of the icebergs away from topographic highs.

Conclusions The apparent linear densities of a population of buried linear features with a preferred orientation, observed on reflection seismic profiles or other geophysical profile data, may be used to estimate the mean orientation of the hidden features. From the variation in apparent densities observed on two intersecting lines, the average orientation of the features is found to be one of two possible solutions. The significant solution can be determined when the following conditions are satisfied: (1) the seismic survey consists of a trigonal grid, or an orthogonal grid with additional lines run at

an angle of 300-60 ° to the grid lines; (2) the apparent density of the scours is observed to vary systematically with the orientation of the seismic lines. Buried iceberg scours have been observed by the author in seismic data from many areas off the east coast of North America. Buried scours have been reported by S. McLaren (pers. commun., 1989) and King (1976). It is likely that such features are also widespread on the Norwegian Shelf, and continental shelves around the North Sea. Application of this method on the existing data base may provide insight into ice drift directions during the late Pleistocene. This technique may be applied to other linear features, such as meltwater channels and linear moraines on seismic records, and dikes detected on magnetic or electromagnetic surveys.

Acknowledgements The author wishes to thank A.E. Aksu, D.J.W. Piper, R.N. Hiscott, G. Quinlan, C.M.T.

270

M.R. GIPP

63ow

62030 '

B

A

44°N

(I)

43o30 '

Fig. 9. Plot of all known significant solutions on paleobathymetry as in figure 8. Flow directions are along the southeastern flank of Emerald Basin and the channel east of the basin, while flow on the eastern flank of the basin is N - S . Storm-driven currents would be expected to flow counter-clockwise, with the east channel serving as a conduit between Emerald Basin and the Atlantic Ocean.

10

48 E v 6

+

.=>,

%+++

to t-

[~2

ireld 1

B' Fig. I1. Lines AA' and BB', separated by a known angle qS, intersect at point O over two possible populations of linear features, oriented perpendicularly to OP~ and OPa, respectively. Apparent densities p, and Pb are observed on AA' and BB" respectively. Absolute densities p~ and Pz would be observed along OP~ and OP 2. ~1, ~2, hi, ha, Pl and ,02 are derived in the appendix.

Woodworth-Lynas and C.F.M. Lewis for providing useful discussion. This paper has benefitted from seismic data collected by D.E.T. Bidgood, R. Boyd, A.G. McKay, L.H. King and D.J.W. Piper. Funding for this project has been supplied from the School of Graduate Studies, Memorial University of Newfoundland, and by NSERC operating grants to A.E. Aksu and D.J.W. Piper. Thanks are also due to the Huntec and NSRFC technicians, and the officers and crew of the CSS Hudson for their expertise at sea.

4-

o

-o ,--i 4 o o 09

++

+

+

2

0

+

Appendix 1: Derivation o f equations

-12o

-16o

+

-$o

Jo

Paleodepth relative to datum (ms twt) Fig. 10. Plot of absolute density against paleodepth below a d a t u m plane 110 m b.s.1. Scour densities are not simply related to depth.

Given two seismic lines AA' and BB' (Fig. 11), separated by a known acute angle ~b, from which linear density values of Pa and Pb have been respectively determined, there are two possible populations of linear features with absolute density values of Pl and P2, separated from line AA' by angles of ~1 and ~2- To find ~1, we construct line OP1, normal to the trend of the first possible population of features, for which the apparent density will be Pl. As the angle between OP~ and

271

BURIED ICEBERG SCOURS AND OTHER LINEAR PHENOMENA

A A ' is fl~, then the a p p a r e n t density on line A A ' (pa) will be:

and

Pa = Pl "cos fll

and

(A1)

N o t e that ~1 + fl~ = 90 °. As the angle between OPa and BB' is ~b- fl~, the a p p a r e n t density on line BB' (Pb) will be:

tan a2 = sin c~/(pr + cos q~)

P2 = p j s i n 0~2

(A4)

(A5)

Angle a2 will be measured f r o m line AA', in the direction o f the acute angle.

pb=pl " cos(~-flx) Appendix 2: Identities used in the text Both p. and Pb are k n o w n a p p a r e n t densities. Therefore: Pb __ P l " C O S ( O - - / ~ I )

P,

Pl "cos fll cos q~'cos fll + s i n q~. sin fll

Given Eqs. A2 and A3 in A p p e n d i x 1, it is possible to d e m o n s t r a t e that if p a = p b , then a l = 90 ° - ~b/2, and a 2 = q~/2. I f Pa = Pb, then Pr = 1. Thus Eq. 1 can be rewritten as:

c o s fll

tan at = sin 4~/(1 - cos ~b)

= c o s th+ sin q~.tan fix

= sin(~b/2 + ~b/2)/[1 - cos(~b/2 + q5/2)]

tan fll = (Pb/Pa- COS ~b)/sin q~

= 2 sin ~b/2 cos q~/2/(1 - cos z q~/2 + sin 2 ~b/2)

The ratio o f the two a p p a r e n t densities Pb/P~ is referred to as the a p p a r e n t density ratio, and is replaced by the symbol p,. Angle al will be measured from line A A ' in the direction o f the obtuse angle. As fll = 9 0 ° - a x , tan a~ = 1/tan fit, so that:

= 2 sin ~b/2 cos ~/2/2 sin 2 ~b/2

Therefore, al = 90 ° - ~b/2 Similarly, Eq. A3 can be rewritten as:

tan a l - - s i n q~/(Pr- COS ~b)

tan al = s i n ~b/(l + c o s q~)

(A2)

= cot ~b/2

The absolute density o f the linear features, p~, is found by rearranging Eq. A l :

= sin(q~/2 + ~b/2)/[1 + cos(~b/2 + ~b/2)]

Pl = p j c o s fl~

= 2 sin q~/2 cos q~/2/2 cos 2 q~/2

= pa/sin al

(A3)

The second solution, az, is found by a process similar to that used to find a t. T h e only difference is that flz is the angle between line OPz and line AA', while the angle between OP2 and BB' is 180 ° - q~ - fiE" Consequently: P a = P 2 " C O S f12

Pb = PZ "COS[180° -- (q~ + flz)] = -- P2" cos(~b + flz) Calculating the a p p a r e n t density ratio (p, = Pb/Pa), and bringing all o f the k n o w n terms to the right side, we find: tan t2 = (P~ + cos ~b)/sin ~b

= 2 sin ~b/2 cos q~/2/(1 + c o s 2 q~/2- sin 2 q~/2)

= tan q~/2 Therefore, a2 = 4~/2.

References Bidgood, D.E.T., 1974. A deep towed sea bottom profiling system. In: Oceans '74: Proc. Int. Conf. Engineering in the Ocean Environment. (Halifax, Nova Scotia, Aug. 21-23, 1974.) IEEE, New York, 2: 96-107. Dredge, L.A., 1982. Relict ice-scour marks and late phases of Lake Agassiz in northernmost Manitoba. Can. J. Earth Sci., 19: 1079-1087. Gipp, M.R. and Piper, D.J.W., 1989. Chronology of Late Wisconsinan glaciation, Emerald Basin, Scotian Shelf. Can. J. Earth Sci., 26: 333-335. Hutchins, R.W., McKeown, D.L. and King, L.H., 1976. A deep tow high resolution seismic system for continental shelf mapping. Geosci. Can., 3: 95-100.

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