The Garn Formation (Bajocian-Bathonian) in the Kristin Field, Halten Terrace: its origin, facies architecture and primary heterogeneity model - From Depositional Systems to Sedimentary Successions on the Norwegian Continental Margin - page 544

(A) (B) Cross-set thickness distribution

Cross-set volume distribution

100

100

Line pinning point for EF ( v max ) = EF ( x max )
and estimated v min

D 1

Best-fit lognormal curve

log C

Power-law (linear)
approximation

Line gradient ( G ) calculated
from the values of D 1 and D 2

G

D 2

Measured data range

Cut-off limit EF ( v min ) = EF ( x min )

0

0

log x

log v

0

Measured maximum
Possible 'absolute' maximum

Limit from
dune definition

Estimated v min Expected v max

Fig. A-7. Diagrams showing how of a lognormal bed-thickness frequency EF ( x ) approximated as a two-tier power-law
distribution (A) allows the bed-volume frequency distribution EF ( v ) to be determined and numerically constrained (B).

which, in a log-log plot, defines a straight line
(Fig. A-7B):

x max = 2.5 m, which gives γ = 0.56. From equation
A7, one can expect that the bed-thickness distri-
bution will thus have two exponents:

(A12)

log EF v ()= log C Gv − log

i i

DG =+− ( 12 2 forthe thinnerbeds

12 forthe thicke

γ ) γ ( xx ≤ )

1 b

The exponent G (negative value) defines the
distribution line gradient and the slant line is
constrained further as follows (Fig. A-7B):
The frequencies of the volumetrically smallest
and largest beds, EF ( v min ) and EF ( v min ), are assumed
to be the same as those of the thinnest and thickest
beds, EF ( x min ) and EF ( x min ), since the thinnest beds
are expected to be the smallest and the thickest
beds to be the largest.
The horizontal co-ordinate of the line's left-hand
pinning point is the lower cut-off limit log v min ,
calculated for an estimated v min (see below).
The line's coefficient log C is the value of
log EF ( v i ) corresponding to log v i = 0.
The v min value for dune cross-sets can be indi-
rectly estimated by using the Flemming empirical
formula (Ashley, 1990, Fig. 9A):

DG =+ ( γ ) rrbeds( xx b > )

2

(A9)

Malinverno (1997) had tested this observation
with a simple numerical experiment and rear-
ranged the two equations to obtain the unknown
exponent G from the known values of the expo-
nents D 1 and D 2 of bed-thickness distribution
(with D 2 > D 1 , Fig. A-7A):

D
DD

G = +−

2

(A10)

1

2 1

For the bed-thickness dataset in the present case,
this exponent is G = 0.93 and allows us to deter-
mine the frequency distribution of bed volumes,
EF ( v i ).

h = 0 0677 . L

0 8098 .

(A13)

to derive the dune wavelength ( L ) for dune min-
imum height ( x min ) and take this value to approx-
imate the dune breadth as w ≈ 2 L . The
wavelength for h = 0.07 m is L = 0.96 m and the
corresponding dune volume can be calculated
from the geometrical equations in Figure A-4B,
which yields v min = 0.16 m 3 . The right-hand limit

The derivation of bed-volume frequency
distribution

The exceedence frequency distribution of bed
volumes has the following general form:

− G

EFv v
i

()= (A11)

i