Regressive and transgressive sequences in a raised Holocene gravelly beach, southwestern Crete

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Sedimentary Geology 17

Statistical analysis of bed-thickness variation in a Tortonian

succession of biocalcarenitic tidal dunes, Amantea Basin,

Calabria, southern Italy

Sergio G. Longhitano a,*, W. Nemec b

aDipartimento di Scienze Geologiche, Universita degli Studi della Basilicata, Campus di Macchia Romana, 85100 Potenza, ItalybDepartment of Earth Science, University of Bergen, 5007 Bergen, Norway

Received 2 February 2005; received in revised form 15 April 2005; accepted 3 May 2005

Abstract

The Tortonian biocalcarenitic succession in Amantea Basin was deposited in a ramp setting of a peri-Tyrrhenian shelf

embayment, whose northern part was linked with the southern part by a tectonic graben referred to as the Monte Pellegrino strait.

The graben was 1.5–2 km wide and 4 km long, had shallowly submerged margins and a south-sloping sublittoral floor dominated

by strong, asymmetrical tidal currents. The water depth fluctuated due to episodes of rapid subsidence driven by extensional

tectonics. The strait-fill biocalcarenites are up to 120 m thick and consist of planar cross-stratified beds that range in thickness

from b0.2 m to N20 m and represent 2-D dunes accumulated as a mounded longitudinal complex. Bed thicknesses have been

measured in four vertical logs along an outcrop section oblique to the palaeostrait. The stratigraphic pattern of bed-thickness

variation has been analyzed by using two versions of the runs test, the Moore–Wallis test, the median-crossing test, the mean

squared successive difference test, the Hurst statistic, the Spearman rank-correlation test and the Meacham rank-difference test.

The statistical tests indicate that the axial zone and inner flank of the tidal complex show alternating clusters, 5–15 m thick,

of thinner and thicker beds, whereas the outer flank and marginal zone show thinning-upward bed packages, 15–25 m thick. An

overall thinning-upward trend characterizes the transitional flank zone. The bed packages are considered to form aggradational

parasequences, whose varied lateral development is attributed to a differential response of the subtidal system to bathymetric

changes. The data set indicates that the bed thicknesses are self-similar, but beds thinner than 250 cm have a different fractal

dimension than the thicker beds. The pattern of bed-thickness variation is considered to be a result of internal forcing of a

sedimentary system in a state of self-organized criticality, perturbed by bathymetric changes. Fractal property bears important

implications for the spatial characteristics of sedimentary succession, allowing bed geometries and volumes to be assessed on

the basis of thickness data from isolated logs or drilling cores.

D 2005 Published by Elsevier B.V.

Keywords: Tyrrhenian Sea; Shelf embayment; Sublittoral strait; Parasequences; Geostatistics; Power law

0037-0738/$ - s

doi:10.1016/j.se

* Correspondi

E-mail addre

9 (2005) 195–224

ee front matter D 2005 Published by Elsevier B.V.

dgeo.2005.05.006

ng author. Fax: +39 971 206 077.

ss: [email protected] (S.G. Longhitano).

S.G. Longhitano, W. Nemec / Sedimentary Geology 179 (2005) 195–224196

1. Introduction

Sedimentary successions are commonlywell-bedded,

and the bedding as such indicates incremental sediment

deposition. The frequency distribution and stratigraphic

variation of bed thicknesses have long attracted research

interest, primarily because of their importance to strati-

graphic analysis and modelling of sedimentary basins.

An understanding of the bedding patterns produced in

particular sedimentary environments may give insights

into the dynamics of depositional systems and may also

aid the characterization and modelling of petroleum

reservoirs, where bedding heterogeneity and its spatial

variation are of crucial importance. Therefore, many

Fig. 1. Location of the Amantea Basin at the eastern periphery of Tyrrheni

study area in the south-trending Monte Pellegrino palaeostrait in the basin

statistical attempts have been made to assess the relative

role of random and deterministic factors in bed-thickness

variation and to develop predictive models.

Numerous quantitative studies have focused on

the bedding characteristics of turbidite successions,

and the bed thicknesses of debris-flow deposits, flu-

vial and nearshore deposits, platform carbonates and

hemipelagic deposits have also been analyzed. Little

is known thus far about the spatial patterns of bed-

thickness variation in inner-shelf environments,

where the record of incremental sand deposition by

tidal or storm-generated currents is often amalgam-

ated and not easy to measure in terms of discrete

beds.

an Sea (inset) and a simplified geological map of the basin. Note the

centre.

S.G. Longhitano, W. Nemec / Sedimentary Geology 179 (2005) 195–224 197

The present paper reports on the spatial pattern of

bed-thickness variation in a succession of Tortonian

biocalcarenites in the Amantea Basin, southern Italy

(Fig. 1), deposited in a peri-Tyrrhenian shelf embay-

ment dominated by strong tidal currents. The succes-

sion originated by vertical stacking of tidal dunes, with

the consecutive sets of planar cross-strata forming

natural depositional units (beds). Bed-thickness data

from outcrop logs have been analyzed with a range of

statistical tests to gain an insight into the spatial orga-

nization of dune cross-set thicknesses in this subtidal

sedimentary system. The analysis focuses on the dis-

tinction between random and preferential patterns of

bed-thickness variation, with a discussion of the con-

trolling factors. The study also evaluates the usefulness

of various statistical tests for this particular purpose.

2. Geological setting

TheAmanteaBasin is located at the eastern periphery

of the Tyrrhenian Sea (Fig. 1, inset) in western Medi-

terranean region. The Tyrrhenian Sea is a late Neogene

backarc basin, whose opening was due to the westward

subduction of the Ionian Sea plate under the Calabrian

Arc and involved eastward shifts of the axis of rifting

and crustal separation (Patacca et al., 1990; Spandini et

al., 1995; Malinverno and Ryan, 1997). The backarc

tectonic extension led to the structural collapse of the

Calabrian accretionary wedge, involving low-angle lis-

tric detachments, high-anglenormal faults and associated

strike-slip rotation of crustal blocks (Wallis et al., 1993;

Speranza et al., 2000).

The Amantea Basin is a triangular-shaped, peri-

Tyrrhenian Miocene shelf embayment, presently ex-

posed at the western coast of Calabria (Fig. 1). The

basin formed in Serravalian time by the coalescence

and marine inundation of fault-bounded coastal depres-

sions (Argentieri et al., 1998). During the middle Tor-

tonian to early Messinian time, the northern part of the

basin was linked with its southern part by a marine

graben referred to as the Monte Pellegrino palaeostrait,

where the present study area is located (Fig. 1).

2.1. Basin-fill stratigraphy

The basin-fill succession of late Serravalian to

Messinian deposits (Fig. 2A) has been extensively

studied (Di Nocera et al., 1974; Ortolani et al., 1979;

Tansi, 1991; Argentieri et al., 1998; Colella and Long-

hitano, 1998; Longhitano and Colella, 1998; Mattei et

al., 1999; Speranza et al., 2000; Martini et al., 2001;

Muto and Perri, 2002) and divided into three main

sequences bounded by unconformities (Colella, 1995;

Mattei et al., 2002).

The lower sequence (Fig. 2A) consists of middle to

late Serravalian siliciclastic deposits, up to 280 m

thick, overlying unconformably the bedrock suite of

Kabilo-Calabride terrain (Van Dijk et al., 2000;

Bonardi et al., 2001), known also as the Sila Piccola

Massif (Colonna and Piccarreta, 1975). The deformed

bedrock comprises an ophiolitic footwall overridden

by the collapsed slabs of former nappes, composed of

ophiolites and Variscan metamorphic rocks. The over-

lying deposits form a transgressive succession of poly-

mict alluvial-fan conglomerates, marine fan-deltaic

deposits, shoreface sandstones and offshore-transition

facies. The latter contain a marker bed of volcanic

tephra and locally bear palaeosol features at the top.

The middle sequence (Fig. 2A) is underlain by a

subaerial unconformity and consists of siliciclastic to

calciclastic deposits of middle Tortonian to early Mes-

sinian age. These deposits form a transgressive suc-

cession up to 260 m thick. The basal, polymict

conglomeratic alluvium and fan-deltaic deposits pass

upwards into shallow-marine sandstones, mainly mas-

sive. A transgressive lag horizon separates these

deposits from the overlying biocalcarenites of a

basin-wide ramp system, whose peritidal inner part

includes bioherms and wave-worked facies. The tidal

biocalcarenites, discussed in the present paper, are of

middle to late Tortonian age and up to 120 m thick.

They are overlain by the latest Tortonian to early

Messinian mudstones, nearly 100 m thick, inter-

bedded with calcarenite sheets in the lowermost and

the uppermost part. Outliers of these deposits at the

NE periphery of the basin show a thin relic cover of

evaporites. Relic Messinian evaporitic limestones

with gypsum, up to 3.5 m thick, occur also on an

intrabasinal bedrock block west of the Monte Pelle-

grino palaeostrait, in outer part of the basin (see

Mattei et al., 2002, their Fig. 3). We concur with

Ortolani et al. (1979) in interpreting the evaporites

as topmost deposits of the middle sequence (Fig. 2A),

although Mattei et al. (2002) linked them with the top

of the upper one.

Fig. 2. (A) Interpreted stratigraphy of the Amantea Basin. Letter symbols: APS—aggradational parasequence set; BSPS—back-stepping

(transgressive) parasequence set; FS—marine flooding surface; FSPS—forward-stepping (regressive) parasequence set; HST—highstand

systems tract; LST—lowstand systems tract; MFS—maximum flooding surface; SFR—surface of forced regression; TST—transgressive

systems tract. Note that the Tortonian succession of tidal biocalcarenites comprises of a wave-influenced FSPS in the basin’s main ramp

system (Mattei et al., 2002), but consists of a sublittoral APS in the intra-basinal Monte Pellegrino palaeostrait (as discussed in the present

paper). (B) Schematic map of southern Italy, showing some of the tidally influenced late Cenozoic basins of Calabria and northern Sicily.

Fig. 3. Outcrop photographs of the cross-stratified biocalcarenites in Monte Pellegrino palaeostrait. (A) Cliff outcrop in the vicinity of log 1. (B)

Cross-stratified biocalcarenites backlapping local relief of the southwards down-stepping floor of the palaeostrait in the vicinity of log 2. (C)

Thick set of tangential cross-strata in the southern, left-hand part of the cliff in photograph A. (D) Cliff outcrop in the vicinity of log 4, showing

dune cross-set 5 m thick. (E) Close-up detail of the tangential lower part of a thick dune cross-set, with lenticular interbeds of massive massflow

deposits attributed to dune-front collapses. (F) Close-up detail of the planar upper part of a thick dune cross-set, showing bioturbation horizons

and bundles of alternating coarse- and finer-grained strata. (G) Strongly truncated dune cross-sets stacked upon one another.


Giano

Giano

Giano

Giano


The upper sequence (Fig. 2A) comprises late Mes-

sinian siliciclast ic deposi ts that form another trans -

gressive succession, underlain by a subaerial

unconformity representing the well-known Mediterra-

nean salinity crisis (Cita, 1982). This sequence is only

locally preserved and consists of polymict alluvial tofan-deltaic conglomerates and transitional marine

sandstones, up to 25 m thick, overl ain by shallow -

marine sandstones interbedded with mudstones and

up to 30 m in preserved thickness.

The thick basin-fill succession, together with an

evidence of growth folds, progressive unconformities

and buried normal faults, indicates active basin-floor

subsidence driven by WNW–ESE tectonic extension

(Mattei et al., 1999, 2002). The unconformable bases

of the middle and upper sequence and the ultimate

emergence of the basin are attributed to episodes of

regional uplift due to the arc tectonics (Rossetti et al.,

2001). The basin development involved also subordi-

nate strike-slip deformation (Argentieri et al., 1998),

with local inferred compressional effects (Colella,

1995; Colella and Longhitano, 1998).

The tectonic activity had strong effect on the ba-

thymetry of peri-Tyrrhenian shelf. The large changes

in relative sea level that occurred in the Serravalian–

Pliocene time, recorded by the three sequences in the

Amantea Basin, were major events superimposed on a

whole series of shorter-term and lower-magnitude

fluctuations induced by local tectonics (Ortolani et

al., 1979; Colella, 1995; Longhitano and Colella,

1998; Mattei et al., 2002).

The late Cenozoic tidal conditions in the western

Mediterranean Sea were much like today: microtidal,

but with the tidal currents enhanced by the out-of-

phase setup of tidal prism in the Tyrrhenian and Ionian

sectors and further amplified in the local straits and

coastal embayments (Colella and D’Alessandro, 1988;

Colella, 1995). An impressive example of this phe-

nomenon is afforded by the present-day Messina

Strait, whose southward-sloping sublittoral to bathyal

floor is dominated by descending tidal currents with

velocities of 1–3 m/s, forming 2-D dunes of medium

to very coarse sand and up 12 m in height (Montenat

et al., 1987); dune cross-sets up to 20 m thick occur in

the strait’s shallower, Plio-Pleistocene deposits ex-

posed on its Calabrian side. Virtually all significant

accumulations of late Cenozoic tidal deposits in on-

shore Calabria and northern Sicily owe their origin to

such specific settings, including the Serravalian–Mes-

sinian Amantea embayment, the late Pliocene Castro-

villari embayment and the Plio-Pleistocene Monte

Torre, Catanzaro and Messina straits and Castroreale

embayment (Fig. 2B).

2.2. The biocalcarenite succession

The Tortonian biocalcarenites occur throughout the

Amantea Basin, but are best developed and exposed

in its south-trending central palaeostrait, where they

accumulated as a mound-shaped longitudinal complex

of tidal dunes (Colella, 1995). The Monte Pellegrino

strait formed as a tectonic graben, 1.5–2 km wide and

4 km long, with shallowly submerged margins and a

southward-sloping sublittoral floor cut by local

escarpments (Fig. 3B). The strait acted as a corridor

through which abundant skeletal sand was transferred

from the northern to the southern part of the shelf

embayment. The calcarenites here are nearly 120 m

thick and characterized by large-scale planar cross-

stratification (Fig. 3), representing straight- to sinu-

ous-crested 2-D dunes (sensu Harms et al., 1982) with

predominantly southward transport direction. The

cross-strata sets range from less than 0.2 m to more

than 10 m in thickness, sporadically exceeding 20 m.

The cross-strata sets have sharp, erosional tops,

which indicate that the vast majority of dunes were

to some degree truncated during erosive phases of

tidal currents, when sediment bypass occurred. The

outcrops show no cross-sets with sigmoidal, upward-

flattening strata or well-preserved dune forms, and

most of the thin cross-sets are bbottom-setQ relics of

originally thicker dunes (Fig. 3G). The sand consists

of shell detritus and is medium- to very coarse-

grained, with some strata rich in shell hash of granule

to pebble grade. Most cross-sets show rhythmic alter-

nation of thicker/coarser and thinner/finer strata (Figs.

3C–F and 4A), considered to be the record of diurnal

and monthly tidal cycles (Fig. 4C). No obvious record

of lunar apsides and nodal cycles (Pugh, 1987; Archer

et al., 1991) has been recognized.

The lack of interstratal mud drapes (Fig. 4C)

implies currents capable of keeping the finest-grained

sediment perennially in suspension, although the com-

mon presence of burrows–isolated or forming foreset

horizons (Figs. 3F and 4A)–indicates considerable

pauses in sand deposition (Pollard et al., 1993). Iso-

Fig. 4. (A) Bundles of alternating coarse- and finer-grained strata in dune cross-sets with bioturbation horizons; note the erosional cross-set

boundary and hammer (encircled) for scale. (B) Isolated, convex-upward truncation surfaces and concave-upward slump scar within a dune

cross-set. (C) Interpretation of foreset strata bundles in terms of diurnal and monthly tidal cycles.


lated convex-upward truncation surfaces (Fig. 4B) are

attributed to episodic erosion of the dune brink by

abnormally strong reversing currents, enhanced by

storm (Hannah et al., 1991; Amos et al., 1995;

Keene and Harris, 1995) or equinoctial cycle (Berne

et al., 1989). Local scoop-shaped, concave-upward

scours are thought to be slump scars representing

dune-front failures (Fig. 4B; see also Fig. 3E), possi-

bly triggered by earthquakes.

The data used in the present study are from four

logs measured in the Monte Pellegrino cliff section

(Fig. 5), which extends obliquely to the palaeostrait

axis (Fig. 1). The logs are 150–170 m apart, with logs

1 and 4 representing, respectively, the axial part and

the eastern margin of the tidal dune complex. The

measured bed thicknesses range from less than 0.2 m

to nearly 8 m (Fig. 6). More than 90% of the beds

show palaeocurrent azimuths in the range of 190–

2308. The other beds, mainly thin and isolated, repre-

sent reversed palaeocurrents in an azimuth range of

40–508 (Longhitano, 2003). The tidal currents were

parallel to the strait axis, for there is little evidence of

oblique or transverse flow. There is also no evidence

of subaerial exposure or deposition above wave-base

level, albeit the fairweather wave base in the strait was

probably no deeper than 2 m, and also the depth of

Fig. 5. Stratigraphic logs 1–4 from the Monte Pellegrino cliff section (see locality map in Fig. 1). The logs show the thickness and mean grain

size of the successive calcarenite beds, which all are dune cross-sets (internal cross-stratification not drawn, for the simplicity). The letter

symbols in mean grain-size scale are: m=mud, s=very coarse sand and g=pebble gravel.


Fig. 6. Bed-thickness frequency distribution of the whole data set (combined data from logs 1–4 in Fig. 5). Note the strong asymmetry of the

distribution.


storm wave base was likely of a few metres, as in

small modern straits.

The lowest part of the tidal complex is poorly

exposed and only slightly covered by the logs, but

seems to abound in thinner cross-sets, many with

northward palaeocurrent directions, which differ

from the main part of the succession (Longhitano,

2003). This upward change may reflect an increase

in the relative role and power of southward currents

in the evolving graben. The logs show highly varied

cross-set thicknesses, but no obvious pattern of

variation (Fig. 5).

Table 1

Statistical characteristics of the bed-thickness data sets from the four

outcrop logs (Fig. 5)

Characteristics Log 1 Log 2 Log 3 Log 4

Number of

data (n)

52 35 23 58

Median (Md) 106.5 cm 40 cm 40 cm 50 cm

Mean (x) 148.9 cm 46.8 cm 94.3 cm 87.0 cm

Standard

deviation (sx)

161.7 cm 30.2 cm 119.0 cm 107.1 cm

Skewness (Sk) 2.11 1.83 2.20 3.47

Kurtosis (K) 4.75 3.68 4.47 15.97

3. Bed-thickness data

The depositional architecture of the cross-stratified

biocalcarenites involves a hierarchy of elements, from

the tangential foreset strata with isolated thicker mass-

flow interlayers to strata bundles, tabular sets and

cosets. The individual cross-strata sets, defined as

beds, are bounded by planar to gently undulatory

surfaces. Bed thicknesses have been systematically

measured in the local vertical logs (Fig. 5), which

provided four sample data series from the succession.

The number of beds (n) per log is between 23 and 58,

which renders the data sets larger than bsmallQ (n N20)and suitable for statistical analysis (Milenkovic,

1989). The data set as a whole can be regarded as

bspatialQ, because it represents more than one dimen-

sion of the geological space sampled. The measured

bed thicknesses range from 0.1 to 7.7 m, with a

distinct mode of 0.2–0.6 m (Fig. 6). Basic statistics

of the indi vidual data sets are given in Table 1.In statistical methodology, a data set is considered

to be a bsample populationQ representing the bgeneralpopulationQ (physical space) from which it has been

derived. Conclusions about the general population are

drawn on the basis of the sample population, with a

specified probability (or acceptable confidence level).

However, the parametric methods of classical statis-

tics rely on the distribution normality of the studied

variable, which is not the case here.

The frequency distribution of the bulk data set

(Fig. 6) is strongly asymmetrical, positively skewed

and extremely leptokurtic, apparently non-normal.

The same pattern is shown by the four individual

data sets, although their statistics are somewhat varied

(Table 1). An exceedence frequency plot (Drummond

and Wilkinson, 1996) of the bulk data set, constructed

with the use of logarithmic bed-thickness values and a

Fig. 7. Exceedence frequency plot of the bulk data set (Fig. 6), with the frequency in probability scale and the bed thicknesses given as

logarithmic values. Note that the plot can be approximated as a straight line, which indicates a lognormal frequency distribution.


probability scale for frequency (Fig. 7), appears to be

approximately linear, suggesting that the density dis-

tribution here is lognormal (Nemec, 2003). The irreg-

ular departures from linearity are attributed to

undersampling, as the data set is limited.

To obviate the methodological problem of non-

normal distribution, the more convenient non-para-

metric, distribution-independent methods have been

applied, instead of using logarithmically transformed

and normalized data sets. Logarithmic conversion has

been used in only one test.

4. Statistical methods

A range of statistical tests have been applied to

evaluate as to whether the bed-thickness data series

are random or non-random, bearing a deterministic

component of preferential organization. It is a com-

mon practice in applied statistics to use more than one

test, because different tests often complement and

verify one another and also because some tests may

be more sensitive than others with respect to particular

data variation. The non-parametric methods used in

the present case are: the runs tests, the Moore–Wallis

test, the median-crossing test, the mean squared suc-

cessive difference test, the Hurst statistic, the Spear-

man rank-correlation test and the Meacham rank-

difference test. In all tests, the null hypothesis (H0)

to be tested and its conjunct alternative (H1) are

basically the same:

H0 : The data series is random:H1 : The data series is non-random:

�

and it is only their exact formulation and the test

function to be calculated that are different. If the

character of a data series, random or otherwise, is

sufficiently well pronounced, the results from differ-

ent tests are theoretically expected to be the same. In

reality, the results may differ, depending on the test

strength and the amount of randomness in a given data

series.


4.1. The runs tests

The non-parametric runs test evaluates random-

ness of a binary (two-state) data series, with brunsQdefined as uninterrupted occurrences of each state of

the variable. For this purpose, the bed-thickness

data sets are converted into binary series, as is

explained further below. Two varieties of this sta-

tistical method have been used, known as the runs

up-and-down (RUD) and runs about median (RAM)

tests (Wald and Wolfowitz, 1944; Chakraborty et

al., 2002).

In the RUD test, the thickness of each consecutive

bed is compared with that of the overlying bed, and

the observed difference is coded as bplusQ (+) if thisnext bed is thicker or as bminusQ (�) if it is thinner.

All bzeroQ differences, or ties (cases of equal thick-

ness), are disregarded and given arbitrarily either of

the adjacent signs (Davis, 1986). In this way, the

original series of n data is converted into a binary

series of (n�1) bplusQ and bminusQ signs, with all

continuous occurrences of one sign regarded as runs.

The number of observed runs (Uo) is counted and

compared with their average number (U e) and its

standard deviation (SU ) expected for all possible ran-

dom series composed of a similar number of signs

(Davis, 1986):

Ue ¼ 1þ 2mp

Nand SU¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mp 2mp� Nð ÞN2 N � 1ð Þ

sð1Þ

where m and p are the tallies of the minus and plus

signs, respectively, and N =m +p. The test function is:

Z ¼ jUo � Uej=SU ð2Þ

and refers to standard normal distribution. The RUD

test effectively focuses on the occurrence of local

thickening- or thinning-upward trends within the

bed-thickness data series.

The procedure in the RAM test is similar, except

that the thickness of every consecutive bed is com-

pared with the median value of the data series. The

resulting binary series then consists of n bplusQ andbminusQ signs, and the observed number of runs (Uo)

is evaluated in an analogous way. This test focuses on

the occurrence of clusters of beds that are thicker or

thinner than the median thickness. The arithmetic

mean of data series is sometimes used, instead of

median, but the latter renders the test less biased by

non-normal data frequency distribution.

The two runs tests thus serve not only to discrim-

inate between random and non-random data series,

but also to recognize the pattern of preferential orga-

nization that may characterize particular non-random

series. For each test, a one-tail hypothesis is formu-

lated for the general population, with H1 reflecting the

observed relationship between the runs tally and their

expected mean number for random series (whether

UoN U e or Uob U e), and with H0 postulating bnullQdifference (Uo= U e). The test critical Z value for an

acceptable significance level a is taken from the table

of standard normal distribution, and the test confi-

dence level is (1�a).

4.2. The Moore–Wallis test

This non-parametric test similarly requires that the

bed-thickness data sets be converted into binary data

series, which is done in the same way as for the RUD

test, but with the numbers of bminusQ signs (m), bplusQsigns ( p) and bzeroQ ties (t) counted. For all possiblerandom series composed of m, p and t elements and

their sum n N12, the m values are expected to be

normally distributed (Moore and Wallis, 1943), with

a mean:

m e ¼ N � 1ð Þ=2 ð3Þ

and standard deviation:

Sm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN þ 1ð Þ=12

pð4Þ

where N =m +p� t. The corresponding test function

is:

Z ¼ jmc � mej=Sm ð5Þ

where mc is a corrected m tally. The tally requires a

continuity correction because the m values are con-

strained to be integers, whereas the notion of normal

distribution implicitly assumes a continuous variable.

The corrected value is mc=m�0.5 when mz me, and

is mc=m +0.5 when m b me (Moore and Wallis, 1943).

A one-tail hypothesis is similarly formulated on the

basis of the calculated mc and me values, with H1

stating either mcN me (thinning-upward trends) or

mcb me (thickening-upward trends) and the null hy-


pothesis H0 postulating random series (mc= me). The

critical Z value is taken from statistical table for

significance level a, and the test confidence level is

(1�a).

4.3. The median-crossing test

This non-parametric test (Fisz, 1964) is similar to

the RAM test, using the series median value, but with

the number of sign changes (i.e., cases of a plus

followed by a minus or vice versa) tallied as Mo.

The runs transitions are thus counted, instead of the

runs tally. For all possible random series of n signs,

the M values are expected to be normally distributed

with a mean:

M¯¯¯ e ¼ n� 1ð Þ=2 ð6Þ

and standard deviation:

SM¯¯¯ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� 1ð Þ=4

p: ð7Þ

A one-tail hypothesis is similarly formulated on the

basis of the Mo and Me values, and is evaluated by the

following test function:

Z ¼ jMo �M¯¯¯ ej=SM¯¯¯ : ð8Þ

The critical Z value is taken from statistical table

for significance level a, and the test confidence level

is (1�a).

4.4. The mean squared successive difference test

This non-parametric test (Wonnacott and Wonna-

cott, 1977) is for a data series of continuous variable,

and hence requires no transformation of bed-thickness

data sets. A series with n N20 data is required, and the

test employs the following function approximating

standard normal distribution:

Z ¼ j1� 0:5kjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� 2ð Þ= n2 � 1ð Þ

p ð9Þ

where:

k ¼Xn�1

i¼1

xi � xiþ1ð Þ2� Xn

i¼1

ðxi � x Þ 2 ð10Þ

and n is the number data in the bed-thickness series;

(xi�xi+1)2 is the squared difference between a given

bed thickness and the successive one; and (xi� x)2 is

the squared difference between the given bed thick-

ness and the mean thickness value of the whole series.

The k value is a ratio of the averaged bed-to-bed

variation and the series bulk variance. Instead of the

arithmetic mean (x), the series median value (Md) is

used as a measure of data set central tendency in the

present case, which renders the test less biased by

non-normal data frequency distribution.

This is a two-tail test, which means that–for a

significance level a–the critical Z value from statisti-

cal table is taken for Oa, while the test confidence

level remains (1�a). If H0 is rejected, the ratio k

bears further information: k N2 indicates short-scale

oscillations or local trends within the data series,

whereas k b2 indicates an overall trend or series-

scale oscillation.

4.5. The Hurst statistic

This method (Hurst, 1951), known also as the

rescaled range analysis (Feder, 1988), is a useful

means for determining the degree of clustering of

low and high values within a data series, or for the

recognition of anticlustering (regular spacing of data

values), which may indicate an increasing or decreas-

ing trend. The method requires a series length of

n N100. This condition is not met here, and hence

the present application is little more than tentative.

The method resembles the RAM test in its purpose,

but is not quite independent of the data distribution

normality. Since the data sets in the present case have

lognormal distribution, they need to be normalized by

logarithmic transformation (Bhattacharyya and John-

son, 1977). To calculate the Hurst K-statistic for a

series of n data, the consecutive values xi are replaced

with their logarithms, xi(L)= logxi, and the mean value

(xL) and standard deviation (SL) are calculated; a

cumulative plot of unit deviations (i.e., departures of

the consecutive values xi(L) from their mean xL) is

made; and the maximum range of cumulative devia-

tions (R) is read off from the plot. The K-statistic is

calculated as follows:

K ¼ log R=SLð Þlog n=2ð Þ : ð11Þ

A random data series is expected to have K =0.5,

whereas K p 0.5 will signify some serial dependence:


K N0.5 indicates data clustering (i.e., a tendency for

the data values higher and lower than the mean to

occur in clusters) and K b0.5 indicates anticlustering.

The K value serves to recognize and quantify the

degree of serial dependence. Furthermore, the approx-

imate relationship R/SLcnB allows the exponent B

to be calculated and the fractal dimension (D) of the

data series to be estimated as (Feder, 1988): D =2�B.

4.6. The Spearman rank-correlation test

The Spearman rank-correlation coefficient (Leh-

mann and D’Abrera, 1975) is a distribution-indepen-

dent alternative to the Pearson coefficient of linear

correlation and can be used to recognize a thinning- or

thickening-upward trend in a bed-thickness data se-

ries. For this purpose, the data in the series are given

separately their position ranks, PR (i.e., consecutive

indices 1, 2, 3, . . . n), and value ranks, VR (i.e., the

smallest value is ranked as 1, the next larger value as

2, the next larger as 3, etc.). Identical data are assigned

a common mean VR, which may not necessarily be an

integer. For example, if the data series contains two

identical values that are the next larger than, say, a

value with VR=7, their own VRs would be 8 and 9;

instead of assigning the two ranks arbitrarily, both

values are given a mean VR=8.5.

The rank difference RD=PR�VR is then calcu-

lated for every element of the data series, and the sum

of squared differences is used to calculate the Spear-

man correlation coefficient (rs):

rs ¼ 1�6Xni¼i

RD2i

n n2 � 1ð Þ : ð12Þ

The coefficient assumes values in the range of

�1V rsV+1, indicating either a negative or a positive

linear correlation between the data VRs and PRs;

rs=0 means a lack of correlation. Because the calcu-

lated rs value is most often intermediate, not obvious-

ly close to +1 or �1, its statistical significance needs

to be tested. For this purpose, the Fisher test for

significance of linear correlation is used, with the

following function:

t ¼ jrsjffiffiffiffiffiffiffiffiffiffiffiffiffin� 2

1� r2s

sð13Þ

that is based on the Student’s t-distribution and has

(n�2) degrees of freedom. In the test one-tail

hypothesis, H0 postulates a lack of correlation in

the general population (qs=0), whereas H1 claims

the existence of correlation as suggested by the

calculated rs value (i.e., qsN0 or qsb0). The critical

t value from statistical table is taken for a significance

level a, and the test confidence level is (1�a).If the PRs for the data series have been set in an

ascending stratigraphic order, they make for an ideal

increasing-upward reference series. A significant

positive correlation will then imply that also the

data series itself has a thickening-upward trend,

whereas a significant negative correlation will indi-

cate an opposite, thinning-upward trend.

If there are ties in VR indices within the data series,

such that mean VRs have been used, the formula for

the Spearman coefficient becomes (Kendall and Gib-

bons, 1990):

rs ¼EPR þ EVR �

Xni¼1

RD2i

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEPRd EVR

p ð14Þ

with

EPR ¼ n3 � n� �

=12 and

EVR ¼ n3 � n� �

=12

� w3 � w� �

=12

ð15Þ

where n is the total number of data in the series and

w is the number of data tied by VR indices.

4.7. The Meacham rank-difference test

This non-parametric test (Meacham, 1968)

resembles the previous one at the outset, with

the value ranks (VR) given similarly to all elements

of the data series. The following statistic is then

calculated:

SD ¼Xni¼2

jVRi � VRi�1j ð16Þ

as the sum of the absolute differences between the

VR indices of the successive elements. For all

possible random series composed of similar ele-

ments, the SD values are expected to be normally

Table 2

Results of statistical tests for the four log-derived data series

Test Log 1 Log 2 Log 3 Log 4

The RAM test

Number of signs m =26, p =26 m =17, p =18 m =11, p =12 m =29, p =29

N =m +p N =52 N =35 N =23 N =58

Series median (Md) Md=106.5 cm Md=40 cm Md=40 cm Md=50 cm

Number of runs (Uo) Uo=22 Uo=14 Uo=9 Uo=23

Ue and SU for random

series

Ue=27, SU=3.57 Ue=18.5, SU=2.91 Ue=12.5, SU=2.33 Ue=30, SU=3.77

Calculated Z-value Z =1.40 Z =1.54 Z =1.50 Z =1.85

Result and confidence

level

H0 rejected with

92% confidence

H0 rejected with

93% confidence

H0 rejected with

93% confidence

H0 rejected with

96% confidence

The RUD test

Number of signs m =27, p =24 m =16, p =18 m =12, p =10 m =28, p =29

N =m +p N =51 N =34 N =22 N =57

Number of runs (Uo) Uo=30 Uo=21 Uo=15 Uo=35

Ue and SU for

random series

Ue=26.4, SU=3.5 Ue=17.9, SU=2.8 Ue=11.9, SU=2.2 Ue=29.2, SU=3.7



level

H0 cannot be rejected

(V85% confidence)


(V86% confidence)

H0 rejected with

92% confidence

H0 rejected with

94% confidence

The Moore–Wallis test

Number of signs m =27, t =0 m =16, t =1 m =12, t =2 m =28, t =6

N =m +p� t N =51 N =33 N =20 N =51

Corrected mc-value mc=26.5 mc=16.5 mc=11.5 mc=27.5

me and Smj for random

series

me=25, Sm=2.08 me=16, Sm=1.68 me=9.5, Sm=1.32 me=25, Sm=2.02



level


(V76% confidence)


(V61% confidence)

H0 rejected with

93% confidence

H0 rejected with

90% confidence

The Meacham test

Calculated SD-value SD=656 SD=328 SD=124.5 SD=937

SDe and SSDP for random

series

SDe=901, SSDP =76.93 SDe=408, SSD

P =41.89 SDe=176, SSDP =21.81 SDe=1121, SSD

P =90.88



level

H0 rejected with

99.9% confidence

H0 rejected with

97% confidence

H0 rejected with

99% confidence

H0 rejected with

98% confidence

The median-crossing test

Number of sign changes (Mo) Mo=21 Mo= 13 Mo=8 Mo=22

Me and SM for random series Me=25.5, SM=3.57 Me=17, SM=2.91 Me= 11, SM=2.34 EM=28.5, SM=3.77


Result and confidence level H0 rejected with

90% confidence

H0 rejected with

91% confidence

H0 rejected with

90% confidence

H0 rejected with

95% confidence

The mean difference test

Calculated k-value k =1.55 k =1.29 k =1.34 k =1.59



level

H0 rejected with

90% confidence

H0 rejected with

96% confidence

H0 rejected with

90% confidence

H0 rejected with

90% confidence


Table 2 (continued)

Test Log 1 Log 2 Log 3 Log 4

The Spearman rank test

Correlation coefficient (rs) rs=�0.09 rs=�0.56 rs=�0.34 rs=�0.15

Calculated t-test value t =0.65 t =4.84 t =1.77 t =1.20


level

H0 cannot be

rejected

H0 rejected with

99.9% confidence

H0 rejected with

95% confidence

H0 cannot be

rejected

The Hurst statistic

Series mean log-value (xL) xL=1.95 xL=1.60 xL=1.74 xL=1.75

Series standard deviation (SL) SL=0.44 SL=0.24 SL=0.43 SL=0.38

Range value (R) R =1.88 R =0.89 R =1.46 R =1.83

Calculated K-value K =0.44 K =0.46 Kc0.50 K =0.47

Result Weak

anticlustering

Weak

anticlustering

Random series Weak

anticlustering

Estimated D -value D = 1.63 D = 1.63 D = 1.61 D = 1.54


distributed with the following mean and standard

deviation (Meacham, 1968):

SDP

e ¼nþ 1ð Þ n� 1ð Þ

3and

SSDP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� 2ð Þ nþ 1ð Þ 4n� 7ð Þ

p90

: ð17Þ

The corresponding test function is:

Z ¼ jSD� SDP

e jS PSD

ð18Þ

and the test one-tail hypothesis is formulated accord-

ing the calculated SD and SDP

e , with H1 claimin g a

non-ra ndom pattern (whether SD N SDP

e or SD b SDP

e )

and H0 p ostulating random series (SD = SDe). Thecritical Z value from statistical tabl e is taken for a

signifi cance level a , and the test confidence level is(1�a).

5. The results of statistical tests

The results of the statistical tests for the four series

of bed-thickness data are summarized in Table 2. The

critical values for the test functions Z and t have been

taken from standard statistical tables (Davis, 1986). In

all tests, the significance level (i.e., acceptable risk of

error) has been taken as no higher than a =0.10, whichmeans that a rejection of the null hypothesis of series

randomness required a confidence level no lower than

90%.

The RAM test indicates with 92–96% confidence

that the data series are non-random, which suggests

significant clustering of beds thicker and thinner than

the median. The RUD test, which is stronger by

taking account of bed-to-bed differences, indicates

local systematic trends in the data series from logs

3 and 4 only (Table 2). A similar result is yielded by

the Moore–Wallis test (Table 2), which indicates

further that the local trends in these two series are

thinning-upward bcyclesQ (because mcN me in each

case). Consequently, the pattern in the data series

from logs 1 and 2 can be regarded as mere clusters

of relatively thick and thinner beds, lacking system-

atic internal trend. As to the data series from logs 3

and 4, it should be noted that the results of the RAM

and RUD tests are mutually compatible, rather than

contradictory, because–as far as the median is

concerned–the beds in the lower and the upper part

of a thinning-upward cycle will obviously tend to

deviate oppositely, much like clusters.

The Meacham test, the median-crossing test and

the mean squared successive difference test (Table

2) confirm with 90% to 99.9% confidence that the

data series are non-random, and hence bearing a

deterministic component of preferential stratigraphic

organization.

The Spearman rank-correlation test (Table 2) indi-

cates with 95–99.9% confidence that the data series

from logs 2 and 3 show a systematic overall change,

which is identified as a general thinning-upward trend

Giano


on the basis of the negative rs values. Such a general

trend is also implied by the ratio k b2 in the mean

squared successive difference test (Table 2), although

the k values themselves cannot readily be tested. In the

light of the Spearman test and their departure from 2,

the lower k values from log 2 (k =1.29) and log 3

(k=1.34) would appear to be more significant than

the higher values from log 1 (k =1.55) and log 4

(k=1.59).

The calculation of Hurst statistic (Fig. 8 and Table

2) has yielded results inconsistent with the other tests,

suggesting series randomness or slight anticlustering.

The fractal dimensions estimated for the data series

are around 1.6 (Table 2, bottom row). However, the

present application of the Hurst statistic has been a

tentative trial, for the data series are too short. This

application confirms that the method relies heavily on

the assumption of n N100, and its use for smaller data

sets should thus be avoided.

In summary, the results of the statistical tests

appear to be consistent. The bed series in logs 1

Fig. 8. Bed-thickness data sets from logs 1–4 (Fig. 5) transformed into serie

The maximum range of cumulative deviations (R) is used to calculate the

and 2 are characterized by clustering (i.e., alternating

packages of relatively thick and thinner beds), with

an overall thinning-upward trend in log 2, but no

similar significant trend in log 1. The bed series in

logs 3 and 4 are characterized by local thinning-

upward trends, or bcyclesQ, with an overall thin-

ning-upward trend in log 3, but no similar trend in

log 4. On the account of the log locations, the

overall thinning-upward trend would appear to char-

acterize the transitional flank zone of the tidal dune

complex.

6. Analysis of bed-thickness distribution

6.1. Population fractality

For a variable x to be regarded as fractal, its data

population should conform to a power-law distribu-

tion. The simplest way to assess this property is to

make an exceedence frequency plot of the data set

s of cumulative unit deviations from the corresponding mean values.

Hurst K-statistic.


using logarithms of the data (logx) and also logarith-

mic frequency values, logEF(x). An ideal power-law

distribution in such a plot should be a straight line

(Turcotte, 1992; Middleton et al., 1995):

logEF xið Þ ¼ logC � Dlogxi ð19Þ

where: EF(xi) is the number frequency of data ex-

ceeding the xi value; C is a constant defining the

ordinate of the line’s intersection point with the fre-

quency axis, such that logC is the logEF(x) value

extrapolated for logx =0; and D is a scaling coefficient

referred to as the fractal dimension, or practically the

line gradient. This relationship is more commonly

written as:

EF xið Þ ¼ Cx�Di ð20Þ

which defines self-similarity of the variable and

means that the changes in its higher values resemble

changes in its small values.

Fig. 9. Exceedence frequency plot of the bulk data set (Fig. 6), with both fre

set appears to be bipartite, with each data subpopulation approximated by a

the trend line’s goodness-of-fit coefficient.

The bulk data set in the present case, when

plotted in this manner, appears to break down into

two subpopulations, each approximately linear and

characterized by a different fractal dimension (Fig.

9). The irregular departures from linearity can be

attributed to undersampling, since the data set is

far from exhaustive. The Pearson coefficient of linear

correlation is r=�0.974 for subpopulation 1 and

r=�0.969 for subpopulation 2, which means that

the goodness of fit of the trend lines exceeds 94%

(see the R2 values in Fig. 9). The fractal dimension,

estimated as the trend line gradient, is D =0.82 for

beds in the thickness range of 10–250 cm and

D =2.38 for the thicker beds, in the range of 250–

780 cm. Similar bipartition characterizes the compo-

nent data sets from the individual logs, which

implies that the bimodal fractality is stationary, per-

sisting laterally in the sedimentary succession. This

property is also reflected in the consistency of fractal

quency and bed thicknesses as logarithmic values. Note that the data

straight trend line (for further discussion, see text). The R2 value is


dimensions estimated for the four data sets (Table 2,

bottom), although these bulk D values would now

appear to be pseudo-averages with little practical

significance for the bimodal population.

It is now widely recognized that the majority of

natural variables, or geological objects, are fractal

only over a limited range of scales (Middleton et al.,

1995; Beattie and Dade, 1996). A similar bipartition

of bed-thickness data sets has been reported, for ex-

ample, from turbidite successions (Rothman and Grot-

zinger, 1995; Chen and Hiscott, 1999; Talling, 2001).

However, the bimodal fractality in the present case

deserves closer consideration, not least because no

such property has thus far been reported from tidal

dune complexes and because there is also no universal

physical explanation of this property with respect to

sedimentary systems. As pointed out by Malinverno

(1997, p. 270), bthere are many realistic ways to

generate a [power-law] distribution of bed thicknesses

that has multiple exponents.QFor a univariate data population, fractality means

that the values of the measured variable are self-

similar and scaling, in a power-law manner, with

respect to some reference bgrandQ measure. Fractal

dimension (D) is a coefficient linking the reference

bgrandQ measure (L) with the scaling factor (F)

according to the following equation (Middleton et

al., 1995, p. 5):

L ¼ 1=FD�1 ð21Þ

The F value specifies the fraction of bgrandQ L repre-

sented by data element x, with the L considered to be

a unity by convention. The data population is said to

be self-similar when the F values are such that the

measured x values appear to be a fractional power-law

function of the reference unit L. The previous equa-

tion can also be written in a linear form as:

logL ¼ 1� Dð ÞlogF ð22Þ

where (1�D) is the gradient of the line defining the

relationship between logL and logF.

It is argued below that, since tidal dunes are hy-

draulic bedforms, the reference bgrandQ measure in

this case could be the scale of flow, or the water depth

at which the dunes formed and migrated. Accordingly,

for a dune of height h =5 m formed at water depth

L=100 m, for example, the scaling factor would be

F =5/100=1/20.

The data set (Fig. 9) indicates that the dune cross-

set thicknesses are self-similar, but show a different

degree of self-similarity over two different thickness

ranges. It is thus worth noting from Eq. (22) the

implication of the two different D values: since

D b1 for subpopulation 1, the F in this lower thick-

ness range increases as the L increases; and since

D N1 for subpopulation 2, the F in this higher thick-

ness range decreases as the L increases. These rela-

tionships may seem puzzling, especially with the L

interpreted as water depth, but are possible to explain

when the effects of both deposition and erosion are

taken into consideration.

6.2. Interpretation

Tidal dunes are hydraulic bedforms, and their

heights can be expected to scale with the flow thick-

ness, or water depth. The relationship between the

height of large dunes (h) and the water depth (H) is

defined most explicitly by Allen’s (1984b) semi-em-

pirical equation:

h ¼ 0:086H1:19: ð23Þ

In stricter terms, the control on dune height

involves also such variable factors as the flow strength

and boundary layer conditions, as specified by the Gill

equation (Allen, 1984a):

h ¼ H1� Fr2� �

2eu1� scr

s0

� �ð24Þ

where: Fr is the Froude number, or a ratio of inertial to

gravitational forces in the current; scr is the threshold

bottom shear stress for sediment particle motion; s0 isthe mean bottom shear stress; e is the exponent of the

Meyer-Peter and Muller equation of bedload transport;

and u is a coefficient related to the dune cross-section

shape (OVuVI). Since the bottom shear stress in

free-surface flow is proportional to Fr2 for constant

flow depth and resistance, the relative dune height h/H

can have a maximum in either Fr or s0. Therefore, thesimplified Eq. (23) does not necessarily hold in all

natural settings (e.g., see Lanckneus and De Moor,

1995). The relationship between dune height and

water depth is nonlinear, and the relative dune height

can vary between 0.05 and 0.5, typically around 0.17

(Allen, 1984a, his Figs. 8–20 and 11–25C).


Importantly, the measured bed thicknesses are not

exactly dune heights. The dune cross-sets have ero-

sional tops and were clearly truncated to a variable

degree by the tidal currents, which probably also

destroyed many dunes in the course of sedimentation.

In other words, the sedimentary succession is a result

of the episodes of vertical stacking of dunes (incre-

mental aggradation) alternating with episodes of in-

tervening erosion (incremental degradation). The

measured bed thickness (xi) can thus be regarded as

the difference between an increment of deposition, or

primary dune height (hi), and an increment of subse-

quent erosion (ei):

xi ¼ hi � ei ð25Þ

When ei N0, the dune thickness is truncated; and

when eizhi, the dune will be totally removed by

erosion. As pointed out by Allen (1980), an inevitable

aspect of tidal dune development is the combination

of formative and destructive processes.

Accordingly, it is worth noting that the bed-thick-

ness density distribution (Fig. 6) resembles a strongly

skewed and left-side truncated normal distribution, as

would be expected from a Kolmogorov-type random

process of alternating incremental deposition and

erosion (Dacey, 1979; Thompson, 1984). The Kol-

mogorov coefficient for the data set distribution,

calculated by the procedure of Mizutani and Hattori

(1972), is k1=0.73, which implies that in, say, a 100

alternations of dune formation and erosion, about 73

dunes were preserved and 27 were eroded. The

corresponding Kolmogorov ratio is k2=0.58, which

implies that about 58% of the total thickness of

sediment deposited in the system was preserved,

while about 42% was removed by the erosional

action of tidal currents. Erosion thus played an

important role in this subtidal depositional system,

perhaps partly because carbonate sand requires

lower bottom-shear stresses for transport initiation

than quartz sand (Prager et al., 1996). The lesser

proportion of dunes that were totally erased, com-

pared to the high net amount of erosion, could be

due to their early cementation, as the strong currents

and porous skeletal sand would allow substantial

volumes of seawater to flow through the sediment.

In the modern Torres Strait, for example, the ce-

mentation of skeletal sand dunes commences at a

critical burial depth of ca. 2 m ( Keene and Har ris,

1995).

The bed thicknesses and their frequency distribu-

tion are thus considered to be a result of mediation

by incremental deposition and erosion. The incre-

ments of deposition (hi) and erosion (ei) are

expected to be nonlinear and mutually independent

functions of water depth and flow power, character-

ized by different density distributions. Combinations

of nonlinear controls result in phenomena of nonlin-

ear dynamics, which would explain the fractal pat-

tern (Turcotte, 1992; Middleton et al., 1995).

Fractality implies that the sedimentary system was

controlled by nonlinear processes, at least two of

which–such as the deposition and erosion–were cou-

pled (Ortoleva et al., 1987). The dune-forming tidal

system would remain in perennial disequilibrium, as

a consequence of nonlinear dynamics, but have an

intrinsic tenden cy for self -organization, passing from

an unpatterned to a bcriticalQ patterned state without

the intervention of an external regulating factor. An

external factor, such as relative sea-level changes,

could be responsible for perturbing the system and

forcing the two different modes of its nonlinear

operation (cf. Fig. 9).

There is little textural difference between the thick

and the thinner cross-sets, and the latter indicate

stronger truncation (Fig. 3G). The bed subpopulation

1 (Fig. 9) might then represent conditions–local or

temporal–in which dunes were more prone to ero-

sion, whereas subpopulation 2 would represent con-

ditions in which dunes were much less eroded and

better preserved. If this interpretation is correct, an

increase in water depth (or L in Eq. (22)) under the

first conditions would reduce erosion and increase the

preserved dune thicknesses (x), resulting in an in-

crease in F ; and a decrease in L would, like wise,

lead to a decrease in F. Under the second conditions,

in contr ast, an increase in wat er d epth mig ht have little

effect on the modest erosion inherent in the process of

dune superposition, while the weaker currents would

produce smaller dunes and result in a decrease in F;

similarly, a decrease in L would cause an increase in

F. The two subpopulations of beds with different

fractal dimensions and L–F relationship might then

be attributed to the tidal system’s differential

responses to changes in relative sea level, as discussed

in the next section. Bimodal fractality would simply


reflect the summation of these lognormal subpopula-

tions (Talling, 2001).

Bed-thickness fractality may have important

implications for the 3-D characteristics of sedimen-

tary succession. For example, Malinverno (1997)

derived the following approximate relationship (no-

tation modified):

xbc0:371=cxmax ð26Þ

where: xb is the bed thickness corresponding to the

break point of exceedence frequency of the bimodal

fractal population (xbc250 cm in the present case, cf.

Fig. 9); xmax is the largest measured bed thickness

(xmax=678 cm in Fig. 9); and c is the exponent relatingbed thickness and bed bdiameterQ (breadth). The coef-ficient would appear to be cc1 in the present case,

which implies that the bed breadths can be considered

to scale linearly with the bed thicknesses (see Malin-

verno, 1997, his Fig. 6). This means that, for example,

a twice thicker bed can also be expected to have an

approximately twice greater lateral extent.

Based on the power-law property of data set, Roth-

man et al. (1994a,b) and Malinverno (1997) also

demonstrated a scaling between bed thicknesses (xi)

and bed volumes (vi), which means that Eq. (20) can

translate into a corresponding equation:

EF við Þcv�Gi ð27Þ

where the exponent G for a bimodal population can be

derived from the following relationship:

G ¼ D2

1þ D2 � D1

ð28Þ

in which D1 and D2 are, respectively, the smaller and

the larger fractal dimension of the bimodal population.

The bed-volume fractal dimension in the present case

is G =0.93 and can serve as a single-parameter char-

acterization of the frequency distribution of bed

volumes, since a linear exceedence frequency distri-

bution is almost completely described by the line

gradient.

Quantitative estimates of this kind are particular-

ly valuable when it comes to the characterization

and modelling of petroleum reservoirs, where the

assessment of 3-D bed geometries and volumes on

the basis of 1-D drilling core logs is a formidable

task.

7. Discussion

The ensuing discussion focuses first on the rates

and time scale of the sedimentation phenomena and

then on the dynamics of tidal dune populations, there-

by providing a conceptual framework for the subse-

quent interpretation of the spatial pattern of bed-

thickness variation.

7.1. Possible rates and time scale of processes

Tidal conditions are changing on several periodic-

ity scales: the semi-diurnal (or diurnal) flood–ebb

cycles, the half-monthly neap–spring cycles, the

semi-annual equinoctial cycles, the 8.8-year lunar

apside cycles and the 18.6-year nodal cycles (Pugh,

1987; Archer et al., 1991). All these equilibrium

cycles are expected to occur on the migration time

scale of a single tidal dune, since large dunes are

relatively long-lived features. The spring tides are

strongest and neap tides weakest during the spring

and autumn equinoxes, which may accelerate the dune

advance or increase its erosion under the prevalent

current, or may cause the dune-brink scour or form

subordinate new dune under the reversing current.

Although the record of apside and nodal changes is

not easy to recognize, the cross-strata architecture in

the present case reflects tidal cyclicity (Fig. 4) and is

consistent with observations from other tidal deposits

(Allen, 1980, 1984a,b; Dalrymple, 1984; Dalrymple

and Makino, 1989; Archer et al., 1991; Kvale and

Archer, 1991; Nio and Yang, 1991; Shi, 1991; Archer,

1995, 1998; Tessier, 1998). The large-scale variation

of bed thicknesses recognized in the sedimentary

succession would thus indicate major hydrodynamic

changes on a time scale much longer than that of the

tidal equilibrium cycles.

The subtidal biocalcarenites were deposited below

the storm wave base on a sublittoral strait floor.

Extrapolation from the flow depth/velocity diagram

of Costello and Southard (1981) suggests possible

flow velocities in the range of 1–2 m/s for the sand

dunes, comparable to the bottom velocities in the

modern Messina Strait (Montenat et al., 1987). A

threshold flow velocity for the transport of coarse

skeletal sand is estimated at ca. 0.6 m/s (Miller et

al., 1977), which might also be the upper limit for the

majority of the reversed northward currents in the


strait. The reversing currents were probably capable of

transporting sand only when enhanced by storms and/

or equinoctial cycle.

The migration rate of large and coarse-grained

dunes is much lower than that of small and finer-

grained ones under the same flow velocity, but can

vary greatly. For example, Keene and Harris (1995)

reported on coarse-grained biocalcarenitic dunes with

a mean height of 5.4 m and mean wavelength of 41

m, formed in the Torres Strait by strongly asymmet-

rical tidal currents with ebb flow velocities of up to

1.15 m/s and flood flow below the sand transport

threshold; and the estimated mean rate of dune mi-

gration was ca. 52 m/year, reaching 121 m/year

during the peak spring ebb flow. Bartholdy (1992)

reported on straight-crested sand dunes more than 5

m in height and up to 200 m in wavelength, formed

in the Danish Wadden Sea by tidal currents with peak

velocities of ca. 1.5 m/s, migrating at a rate of 40 m/

year. Based on the foreset bioturbation, or bcoloniza-tion windowsQ, Pollard et al. (1993) estimated a

migration rate of 20–70 m/year for tidal sand

dunes. In some settings, tidal dunes are stagnating

between the equinoctial phases of spring tide peaks

or remain dormant for periods longer than a year

(e.g., Lanckneus and De Moor, 1995). In the present

case of microtidal environment, the thicknesses and

cyclic organization of foreset strata (Fig. 4C) suggest

a migration rate of only 2–4 m/year for most of the

tidal dunes.

If the latter rates and a mean dune wavelength of

100 m are assumed, an average dune reconstitution

time (i.e., the time required for a bedform to advance

over its own wavelength) would be between 25 and

50 years. The primary dune heights were mainly in the

range of 1 to 20 m; their vertical form index (i.e., the

wavelength to height ratio) could theoretically be

between 10 and 160 (see Allen, 1984a, his Fig. 11–

25A); and their reconstitution time could thus vary

from a few years for small dunes to a few hundred

years for the largest ones. For an average dune, it

might have taken 250 to 500 years to migrate over a

distance of 1 km. The available biostratigraphic data

(Mattei et al., 2002) suggest a time-span probably no

greater than 300 ky for the biocalcarenite succession,

which would imply a mean net accumulation rate of

ca. 0.4 m/ky, similar as reported by Keene and Harris

(1995) from the Torres Strait. These estimates are in

keeping with the inferred high net amount of erosion

involved in the tidal system.

The clusters and thinning-upward packages of

vertically stacked dune cross-sets (Fig. 10A) would

thus likely represent morphodynamic changes on a

time scale of several thousand years, which we attri-

bute to changes in relative sea level. In this interpre-

tation, we concur with Mattei et al. (2002, p. 154),

who recognized–on a visual basis–the clustering of

thinner and thicker beds in the northern part of the

Monte Pellegrino cliff section and interpreted these

stratigraphic changes as parasequences representing

transgressive–regressive cycles. As pointed out earli-

er in the text, the relative sea level in the Amantea

Basin fluctuated due to tectonic activity (Colella,

1995; Mattei et al., 2002) and these changes could

have a considerable impact on the strait-confined

tidal system, as discussed in the next section. If

eustatic sea-level changes in the Milankovitch time-

frequency band (20–100 ky) also occurred, forced by

astronomical cycles (Goldhammer et al., 1990;

Jacobs and Sahagian, 1993), these fluctuations

would combine with the tectonically forced changes

and be difficult to distinguish in subtidal sedimentary

record.

7.2. Dune population dynamics

The development of dunes signifies a state of

instability involving a sediment-transporting current

and erodible substrate. Dunes grow up to a limit

imposed by the general flow power and sediment

grain size, but are self-regulating features, adjusting

their heights and wavelengths to changes in flow

conditions. For a given grain size, dune height

depends upon the scale of flow (water depth) and

the boundary layer conditions. Since tidal dunes

form and migrate under cyclically changing condi-

tions, the dune heights are also a stochastic function

of time and distance. Dune wavelength decreases with

the increasing grain size and bedload to suspended-

load ratio, and increases with flow depth, but may

lack the latter relation or even correlate negatively

with depth, depending on the degree of flow unstead-

iness and local boundary-layer conditions. These latter

control the exchange of sediment between substrate

and current, which can alter the shape of individual

dunes, form new dunes, or truncate and possibly

Fig. 10. A summary of the spatial pattern of bed-thickness variation in the biocalcarenite succession in Monte Pellegrino palaeostrait, as

revealed by the statistical tests (see results in Table 2).



destroy previous ones. Tidal currents are characteris-

tically unsteady and non-uniform, whereby the sedi-

ment transport rate is a strongly nonlinear function of

mean local flow velocity (Allen, 1984a), with the

deposition and erosion acting as two alternative

modes of substrate regulation.

Dune populations are sensitive to changes in the

scale and power of flow, and hence different hydraulic

conditions will favour different dune sizes and involve

different amount of erosion. The development of a

dune population in any bfavouredQ size range involvescreation of short-wavelength forms, their growth to

greater heights and wavelengths, and their ultimate

demise; this phenomenon of stochastic variation is

known as a variance cascade (Allen, 1984a). As

pointed out by Allen (1984a), dunes are deterministic

and stable as a bedform type, but are an unstable

random phenomenon as a dimensional population. A

range of dune sizes is formed by equilibrium flow, and

the populatio ns of long- lived dunes are inevitabl ypolymodal.

For the tide-driven seawater passing through a

strait confinement, the following equation of flow

continuity will hold:

Q ¼ W dHdU ð29Þ

where: Q is the discharge volume of tidal current

(m3/s); W is the strait width (m); H is the strait water

depth (m); and U is the mean flow rate (m/s). The

volume of water forced into the strait will dependent

upon the local tidal prism, but can be regarded as

roughly constant. With the strait width remaining con-

stant, any significant change in the water depth (H)

will thus have to be compensated by a corresponding

inverse change in flow power (U). This relationship

points to the importance of relative sea-level changes

in narrow seaways, even if microtidal.

A change in flow depth will affect the flow power

and induce morphodynamic adjustments in the tidal

system. The population of dunes in changing flow

conditions is said to be blagging behind the flowQ,because of the substrate resistance to change—mea-

surable in terms of sediment flux or morphodynamic

relaxation time (i.e., the time required for bedforms to

equilibrate to the new flow conditions, analogous to

the bequilibrium timeQ of Paola et al., 1992). As the

mean flow strength and bottom shear stress increase,

the favoured dune height increases, but only until a

threshold of sediment flux is reached and erosion

prevails, which reduces the dune heights and may

destroy some dunes. The ratio of dune height (h) to

wavelength (E) is a bell-shaped random function of

bottom shear stress (Allen, 1984a):

h

k¼ 0:12 1� 0:6

r� 0:4r

� �2

ð30Þ

where r is the Shields–Bagnold dimensionless mean

bottom shear stress (or density-adjusted grain Froude

number). The relative dune heights thus reach a max-

imum for the intermediate stress range, and decrease

when the current either exceeds its boptimalQ power,as during a sea-level fall, or becomes weaker, as

during a relative sea-level rise (cf. Allen, 1984a, his

Figs. 8–21).

These hydraulic notions provide a conceptual

framework for our interpretation of the observed spa-

tial pattern of bed-thickness variation, discussed in the

next section. The notion of equilibrium conditions

invoked below refers to the morphodynamic state of

an evolving sedimentary system which–after the re-

laxation time following bathymetric change–acquired

a new stable set of the coupled density distributions of

bfavouredQ incremental deposition and erosion. This

notion is broadly analogous to the morphodynamic

concept of a bgeometric profile of equilibriumQ usedin stratigraphic modelling (Ross et al., 1994) and the

concept of bdepositional regimeQ (Thorne and Swift,

1991, p. 35) introduced to bdenote a dynamic equi-

librium occurring at geological time and space scalesQin shelf systems.

7.3. Spatial pattern of bed-thickness variation

The stratigraphic pattern of bed-thickness variation

revealed by statistical tests, though visually not ap-

parent at the logging sites, is recognizable in some

other parts of the outcrop section. The calcarenite

succession in the axial zone of the palaeostrait

shows alternating clusters, 5–15 m thick, of thinner

and thicker beds (Figs. 10A and 11A,B), passing

laterally into thinning-upward bed packages, 15–25

m thick, in the marginal zone (Figs. 10A and 11C,D).

The transition zone is characterized by an overall

thinning-upward trend (Fig. 10A). The individual

packages are difficult to trace throughout the cliff

Fig. 11. (A) Cluster packages of thinner and thicker beds recognizable in the axial part of the biocalcarenite complex. (B) Close-up view of the

lower right-hand part of the same outcrop. (C) Thinning-upward bed packages recognizable in the marginal eastern part of the biocalcarenite

complex. (D) Close-up view of a thinning-upward bed package in the latter area.


section, because of faults (Fig. 5) and uneven expo-

sure, but lateral correlations suggest that (Fig. 10B):

(1) the thinner-bedded clusters in the axial zone are

thinning towards the palaeostrait margin, where they

correspond to the thinnest-bedded upper parts of the

thinning-upward packages; (2) the thickest-bedded

lower parts of the latter packages in the marginal

zone correspond to the topmost parts of the thinner-

bedded clusters in the axial zone, or virtually pinch

out in this direction; and (3) the thicker-bedded clus-

ters in the axial zone correspond to the middle por-

tions of the thinning-upward packages in the marginal


zone. This tentative correlation is assumed to approx-

imate chronostratigraphic relationships.

The thick succession of subtidal biocalcarenites

testifies to active tectonic subsidence of the graben

floor, possibly combined with broader regional subsi-

dence (Mattei et al., 1999, 2002). Episodes of rapid

subsidence would cause relative sea-level rises in the

strait, affecting its tidal system. In our hypothetical

model for the strait’s responses to bathymetric

changes (Fig. 12), the subtidal system is considered

to be in a state of morphodynamic equilibrium during

the phases of deepest and shallowest water, and be

subject to disequilibrium during the intermediate

phases of water deepening and shallowing. The flow

Fig. 12. Interpretation of the differing stratigraphic pattern of bed-thicknes

system in Monte Pellegrino palaeostrait (for discussion, see text).

power will be at a minimum during the sea-level

highstand, resulting in smaller dunes and minimum

erosion in the lower-lying flank zone, while producing

optimal conditions for dune development in the axial

zone, where large dunes will form and minimum

erosion will occur. The conditions will change once

the aggrading system reaches a morphodynamic

threshold and becomes affected by the water shallow-

ing. The flow power will rise, favouring larger dunes,

but the increase in bottom shear stresses will cause

more erosion, such that the preserved dune cross-sets

will tend to be relatively thin.

In the axial zone, these conditions will persist

during the phase of shallowest water, when the local

s variation in the axial part and marginal zone of the Tortonian tidal


flow power will be at a maximum, tending to form

largest dunes, while also inflicting maximum erosion.

The system’s own morphodynamic accommodation

(i.e., its capacity to store sediment) will here be at a

minimum, with similar volumes of seawater forced to

flow through the shallowing strait and the large dunes

being cannibalized each time the bottom shear stress

exceeds its optimal middle range (see previous sec-

tion). Sediment bypass conditions will occur, as the

accommodation space provided by the overlying

water column cannot easily be exhausted in a strait

with strong tidal currents. More sand will concurrently

accumulate on the lower flank and margin, where the

reduced water depth will maximize sediment flux and

create optimal conditions for dune growth and pres-

ervation (Fig. 12).

The conditions will change again during the next

phase of water deepening (Fig. 12), when the prev-

alent currents in the axial zone may remain strong

and the population of large dunes will be blaggingbehind the flowQ, while erosion will be much reduced

due to the system’s increased morphodynamic ac-

commodation, resulting in considerably thicker dune

cross-sets. The interaction zone of flood and ebb

currents will also shift away from the margin.

Along the system outer flank and margin, the flow

power will decrease, reducing the size of new dunes,

but erosion may increase due to the extremely un-

steady flow conditions, forcing adjustments in dune

relief.

This conceptual model is obviously a simplifica-

tion, because the magnitude of tectonically forced

bathymetric changes could vary greatly and the

changes would likely be asymmetrical, with the deep-

ening events much faster than the shallowing phases.

Nevertheless, the model seems sufficient to explain

the general pattern of bed-thickness variation. The

asymmetry and variable magnitude of bathymetric

changes are probably reflected in the varied thickness

of successive bed packages. The model considers only

main bathymetric changes, with minor ones assumed

to have added brandom noiseQ to the bedding pattern.

The overall thinning-upward trend in the transition

zone (Fig. 10A) can be attributed to tidal current

rectification on the flank of the aggrading tidal com-

plex (Loder and Wright, 1985), where also the inter-

action between flood and ebb currents would likely

increase in a shallowing strait.

7.4. Implications for sequence stratigraphy

The Tortonian biocalcarenites constitute the main,

middle part of a transgressive succession (Fig. 2A),

where their deposition was initiated and terminated by

dramatic rises in relative sea level. These marine

flooding events were probably caused by rapid region-

al subsidence. The biocalcarenite succession can thus

be regarded as a parasequence comprising a set of

higher-order parasequences (cf. Fig. 12), chiefly

aggradational. They differ from the classical, progra-

dational parasequences related to shoreline advance

and readily recognizable by the shallowing-upward

signature of component facies (Van Wagoner et al.,

1988, 1990). The parasequences in the present case

resemble the bamalgamated subtidal cyclesQ of Gold-hammer et al. (1990) and the generic bgive-up cyclesQof Soreghan and Dickinson (1994). The aggradational

shallowing phases were accompanied by no obvious

facies change, with all deposits reflecting subtidal

conditions. The only recognizable signature of shal-

lowing would be the upward change in bed thick-

nesses, or a bthickness-incompleteQ bed succession

(sensu Soreghan and Dickinson, 1994).

As pointed out by Arnott (1995), the conventional

definition of a parasequence, based on shoreline pro-

gradation, gives little provision for significant

btransgressiveQ deposition during the phase of water

deepening. The marine flooding surfaces are also

assumed to be readily recognizable in parasequences,

as their very boundaries. Therefore, it is worth noting

that the episodes of water deepening in the present

case apparently involved significant deposition in the

subtidal system (Fig. 12). The marine flooding sur-

face underlying the transgressive deposits is easy to

recognize at the base of thick-bedded cluster in the

axial zone of the system, but difficult to pinpoint in

the marginal zone, where the correlative surface

would be in the lower mid-part of coeval thinning-

upward package. The upper boundary of transgres-

sive deposits, or the surface of maximum flooding, is

difficult to identify, as in majority of progradational

parasequences.

The remarks above point to the lateral variability

of parasequences and potential problems with their

recognition. An intuitive notion that the thick calcar-

enite succession in the Amantea Basin consists of

parasequences was raised by Mattei et al. (2002), but


their actual distinction proved to be difficult. The

recognition of parasequences in a subtidal aggrada-

tional succession is, indeed, a difficult task. The

present model (Fig. 12) may thus be a useful

guide, possibly also for other similar successions.

However, the model is hypothetical and requires

verification through a wider range of case studies,

as it is uncertain if this model is of local or broader

significance.

8. Conclusions

The Tortonian biocalcarenite succession in the

Monte Pellegrino palaeostrait was deposited in a

sublittoral environment dominated by strong and

highly asymmetrical tidal currents, with estimated

flow velocities of up to 1–2 m/s and dune migration

rates of 2–4 m/year. The 2-D dunes, composed of

medium to very coarse sand, formed a mounded

longitudinal complex up to 120 m thick. Episodic

erosion and sediment bypass played an important role

in this subtidal system. It is estimated that only about

58% of the total thickness of sediment deposited in

the system was preserved, and that in a 100 episodes

of dune formation and erosion, only about 73 dunes

were preserved. The mean rate of net sediment accu-

mulation is estimated at ca. 0.4 m/ky.

The thicknesses of dune cross-sets (beds), mea-

sured in four logs along an outcrop section oblique

to the palaeostrait, range from less than 0.2 m to

nearly 8 m. More than 90% of them show south-

ward palaeocurrent direction, which indicates subor-

dinate role of reversed northward currents. The

stratigraphic pattern of bed-thickness variation has

been evaluated by using two versions of the runs

test, the Moore–Wallis test, the Meacham test, the

median-crossing test, the mean squared successive

difference test, the Spearman rank-correlation test

and the Hurst statistic (Table 2). The latter method

apparently yields false results when applied to in-

sufficiently large data sets. The results of the other

statistical tests are consistent, but show that only the

application of a group of mutually supplementing

tests renders their results reliable and easy to inter-

pret. The tests indicate that the axial part and inner

flank zone of the tidal complex are characterized by

alternating clusters, 5–15 m thick, of thinner and

thicker beds, whereas the outer flank and marginal

zone are characterized by thinning-upward bed

packages, 15–25 m thick (Figs. 10A and 11). An

overall thinning-upward trend characterizes the tran-

sitional flank zone.

Based on tentative lateral correlations (Fig. 10B)

and the existing knowledge of dune population dy-

namics, the bed packages are considered to constitute

aggradational parasequences, whose varied lateral de-

velopment is attributed to the differential response of

the axial zone and lower-lying margin of the tidal

system to tectonically forced bathymetric changes.

The suggested model (Fig. 12) can serve as a guide

for the distinction and interpretation of parasequences

in this subtidal aggradational complex and similar

other successions, albeit it also needs to be verified

by a wider range of case studies.

The data set indicates that the dune cross-set

thicknesses are self-similar, but have different fractal

dimensions in different thickness ranges (Fig. 9).

The thinner beds (b250 cm) are attributed to the

system’s local or temporal conditions in which dunes

were prone to erosion, whereas the thicker beds

represent conditions in which dunes were less trun-

cated and better preserved. The bimodal fractality

derives from the summation of these two subpopula-

tions, each lognormally distributed (Talling, 2001).

The spatial pattern of bed-thickness variation is

considered to be a result of internal forcing of a

depositional system in a state of self-organized crit-

icality, combined with the system’s differential

responses to tectonically forced episodes of water

deepening.

Bed-thickness fractality bears important implica-

tions for the spatial heterogeneity of sedimentary

succession (Stølum, 1991) and allows the 3-D bed

geometries and volumes to be assessed on the basis of

1-D thickness data (Rothman et al., 1994a,b; Malin-

verno, 1997). Such quantitative estimates are of great

value to the characterization and modelling of petro-

leum reservoirs, where only local bed thicknesses can

be measured from the drilling cores.

Acknowledgements

This paper stems from the first author’s field

project funded by the University of Catania and


supervised by Albina Colella, and from his subse-

quent short stay at the University of Bergen, kindly

supported by Angiola Zanini (Catania University).

The second author’s trip to the study area was spon-

sored by Statoil. The manuscript was critically

reviewed by Fabrizio Felletti, Brian Sellwood and

an anonymous reviewer, and its earlier version was

also read by William Helland-Hansen, John Howell,

Ayhan Ilgar, Nils Janbu and Allard Martinius, whose

constructive comments are much appreciated by the

authors.

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Regressive and transgressive sequences in a raised Holocene gravelly beach, southwestern Crete

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