GeolCarp_Vol60_No4_295

www.geologicacarpathica.sk

GEOLOGICA CARPATHICA, AUGUST 2009, 60, 4, 295—305 doi: 10.2478/v10096-009-0021-4

Introduction

Sedimentologists in their field work usually provide records
of  sedimentary  rocks.  One  of  the  routine  ways  to  make
a sedimentary record is to plot the studied sedimentary rocks
as  a vertical  section  of  successive  facies.  Sediment  facies
represent sediment or sedimentary rock that display distinc-
tive  physical,  chemical  and/or  biological  characteristics
(Stow 2005). There are descriptive and genetic facies. While
descriptive facies are defined purely on sedimentary charac-
teristics,  such  as  muddy  sandstone,  laminated  mudstone,
etc.,  genetic  facies  are  determined  by  the  comparison  with
standard  facies  models  and  interpreted  in  terms  of  deposi-
tional processes, such as turbidite, contourite, lahar deposit,
etc. The key to the interpretation of the sedimentary record is
to study facies in association, in particular their relative ver-
tical and lateral positions (Graham 1988). Comparison of de-
scriptive  facies  from  the  study  area  with  standard  facies
models for different depositional systems is a useful practice
(Stow 2005). However, the balance of data being interpreted
against models tempts us to look for some kind of arrange-
ment,  even  if  there  is  no  order  (Carr  1982).  Application  of
statistical  techniques  provides  an  opportunity  to  the  reduce
subjective impact on interpretations of the sedimentological
record. Some simple statistical methods have been applied to
detect the possible presence of vertical order in sedimentary
successions. Almost all use the probability matrices and em-
ploy  the  idea  of  Markov  Chains.  In  this  study,  the  Markov
property was applied to validate the presence of order in the
sequence  of  structures  or  descriptive  facies  in  the  turbiditic
formation  from  the  Outer  Western  Carpathians.  The  main
goal of this study was to find statistically significant succes-
sions of descriptive facies in the investigated strata and com-
pare  them  to  the  facies  models  of  deep-water  deposits  with
the  purpose  of  supporting  sedimentological  interpretations

Markov Chain analysis of turbiditic facies and flow dynamics

(Magura Zone, Outer Western Carpathians, NW Slovakia)

SIDÓNIA STAŇOVÁ, JÁN SOTÁK and NORBERT HUDEC

Geological Institute of the Slovak Academy of Sciences, Severná 5, 974 01 Banská Bystrica, Slovak Republic;

stanova@savbb.sk; sotak@savbb.sk

(Manuscript received April 17, 2008; accepted in revised form December 18, 2008)

Abstract: Methods based on the Markov Chains can be easily applied in the evaluation of order in sedimentary se-
quences. In this contribution Markov Chain analysis was applied to analysis of turbiditic formation of the Outer West-
ern Carpathians in NW Slovakia, although it also has broader utilization in the interpretation of sedimentary sequences
from other depositional environments. Non-random facies transitions were determined in the investigated strata and
compared to the standard deep-water facies models to provide statistical evidence for the sedimentological interpreta-
tion of depositional processes. As a result, six genetic facies types, interpreted in terms of depositional processes, were
identified. They comprise deposits of density flows, turbidity flows, suspension fallout as well as units which resulted
from syn- or post-depositional deformation.

Key words: Outer Western Carpathians, facies, density flows, Markov Chains, turbidite.

of  depositional  processes.  The  Markov  Chain  analysis  has
commonly been used in the past for the analysis of ordered
sequences  from  a wide  range  of  depositional  environments.
However, the calculations in the analysis of ordered succes-
sion of facies with the assistance of the Markov Chains are
time consuming. As a consequence, a simple computer pro-
gram  entitled  “phpSedistat”  was  devised  for  this  study  to
make the calculations easier.

General approach

A Markov process describes a sequence of states, or

events, for which the occurrence of one state may exhibit a
dependence on a previous state or states (Powers & Easter-
ling  1982).  Development  of  methods  based  on  the  Markov
property which can be used in geology started at the end of
the 60’s. They have been used by a number of investigators
(e.g.  Potter  &  Blakey  1968;  Gingerinch  1969;  Read  1969;
Doveton 1971; Miall 1973; Ethier 1975) to validate the pres-
ence of ordered and cyclic successions of facies. Stratigraph-
ic  “order”  may  be  indicated  by  repeated  patterns  of  either
thickness and/or lithological variation, but it should also be
noted that the presence of lithological order alone makes no
compelling  argument  for  periodic  accumulation  (Wilkinson
et al. 1997). On that account, the Markov property proved to
be  of  little  use  for  characterizing  sedimentary  cyclicity
(Weedon  2005).  However,  it  was  considered  useful,  when
for  example,  the  particular  order  of  lithologies  helps  in  the
description  of  sedimentological  processes  (Wilkinson  et  al.
1997). This is why the Markov property has been used in this
study,  which  is  focused  on  the  analysis  of  facies  and  flow
dynamics of turbiditic formation.

The presence of Markov process implies a degree of order

in a system (Carr 1982). In most studies, the raw data consist

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STAŇOVÁ, SOTÁK and HUDEC

of  an  observed  number  of  upward  facies  transitions  which
are plotted in matrix form (transition matrix) and compared
to  one  generated  by  random  methods  (Graham  1988).  The
initial works in this field proceeded from a mistaken mathe-
matical  technique  of  generating  an  independent  trial  matrix
and resulted in inappropriate conlusions. The transition ma-
trix  can  be  determined  in  two  ways  (Powers  &  Easterling
1982). One method is to make observations at regular inter-
vals of thickness. The advantage of this method is that the di-
agonal  frequencies  of  the  transitional  matrix  carry
information  about  thickness,  but  a  disadvantage  is  that  ob-
served  transition  rates  depend  on  the  sampling  interval.  An
alternative  method  is  to  construct  a  sequence  of  lithologies
by  recording  the  transitions  of  lithologies  as  they  occur,
withouth regard to thickness. Since the transition from a bed
of one lithology (facies) to another case of the same type is
assumed  to  be  objectively  non-recordable,  the  diagonal  fre-
quencies  are  constrained  zero.  Transition  matrices  contain-
ing  structural  zeroes  are  called  embedded  matrices.  Then
observed  facies  transitions  are  compared  to  wholly  random
facies transitions. It was pointed out by Schwarzacher (1975)
that  the  presence  of  previously  defined  zeroes  was  a  major
obstacle for accurate analysis (Graham 1988). Attempting to
overcome  this  problem,  both  Carr  (1982)  and  Powers  &
Easterling (1982) independently recommended a method (ef-
fectively the same one) for the estimation of wholly random
facies transitions, termed quasi-independence, by employing
a technique initially developed by Goodman (1968). The ex-
pected  transition  frequencies  under  a model  of  quasi-inde-
pendence  are  generated  by  an  iterative  method  (Powers  &
Easterling  1982).  In  the  identification  of  significant  transi-
tions,  the  observed  and  expected  cells  are  used  to  derive  a
difference matrix (Gingerich 1969; Carr 1982). The positive
values  in  the  difference  matrix  were  regarded  as  non-ran-
dom, while negative values were regarded as random, which
led to some questionable conclusions in initial works. Con-
sequently it has been suggested that all positive values in the
difference matrix have to be statistically tested for random-
ness (Harper 1984).

Geological settings

The research area extends in the NW, Slovak part of the

Outer Western Carpathians (Fig. 1), contouring the northerly
convex  arcuate  shape  of  the  Western  Carpathian  orocline.
The  Western  Carpathians  create  the  northernmost,  W-E
trending  orocline  of  the  European  Alpides,  linked  through
the Vienna Basin to the Eastern Alps in the West and to the
Eastern Carpathians in the East (Plašienka et al. 1997). The
present  structural  pattern  of  the  Western  Carpathians  was
formed  by  the  Late  Jurassic-Tertiary  subduction-collision
orogenic  processes  in  the  Tethyan  mobile  belt  between  the
stable  North  European  Platform  and  drifting  Apulia/Adria
related  continental  fragments.  The  Outer  Western  Car-
pathians  contain  sediments  of  the  late  Tertiary  Carpathian
Foredeep deposited on the southern flanks of the North Eu-
ropean Platform and a broad Flysch Belt. Several thousands
of meters thick Upper Jurassic to Lower Miocene, mostly si-

liciclastic,  flysch  deposits  built  up  several  nappes,  subhori-
zontally  overthrusted  onto  Miocene  deposits  of  the  Car-
pathian Foredeep or directly onto Precambrian, Paleozoic or
Mesozoic rocks of the Carpathian foreland (Oszczypko et al.
2003).  They  are  completely  uprooted  from  their  basement
and  separated  from  the  Central  Carpathians  by  the  Pieniny
Klippen Belt suture zone. Little is known about the nature of
the  Flysch  Belt  basement  substratum  which  was  shortened
and underthrust below the Central Western Carpathians dur-
ing the Tertiary (Plašienka et al. 1997).

The Flysch Belt represents the Tertiary accretionary wedge

of the Carpathian orogen. The flysch nappes are generally ar-
ranged into the Silesian—Moldavian (or “Krosno-Menilite”)
group of nappes in the North, and the Magura thrust system in
the South (Plašienka et al. 1997). In the research area, there
are several large thrust units of the Magura Group, consisting
of the Rača, Bystrica and Oravská Magura (Krynica) Units.

All the studied sedimentary sections expose the Kýčera

Member of Rača Unit, deposited in the Magura Basin during
the  Late  Eocene.  On  the  basis  of  the  field  sedimentary  re-
search, the Kýčera Member represents a sandstone-dominat-
ed  sequence  alternating  with  mudstone  rich  intervals.  The
sandstones  mostly  form  thin  to  thick  beds,  although  some
very  thick  beds  are  present  as  well.  Most  sandstone  beds
have parallel or slightly wavy bases. Amalgamation of beds
is common. It is sometimes indicated by clustered amalgam-
ation clasts (Fig. 2D). Generally, mud clasts float in the up-
per part of a bed (Fig. 2H,J). Medium to very thick beds of
structureless sandstones and pebbly sandstones with normal
gradation  of  pebbles  are  present.  A lot  of  them  have  T

b—d

Bouma divisions in the upper part of beds (Fig. 2A,B) or
most frequently they have only a thin laminated mudstone
top directly above T

Bouma interval. These thin laminated

tops are often bioturbated (Fig. 2C). The upper parts of sand-
stone  beds  are  frequently  eroded  by  the  next  current  or  re-
worked by bottom currents. Mudstone rich intervals contain
thick claystones/silty claystones or thick beds of claystone/
silty claystone with mm to cm silt/very fine sandstone lami-
nae (Fig. 2K). The Kýčera Member represents either channel
systems (or upper parts of lobes) alternating with interchan-
nel areas (Starek & Pivko 2001) or depositional lobes alter-
nating with interlobe deposits (Staňová & Soták 2007). The
Kýčera Member belongs to the Zlín Formation, which termi-
nated the deposition in the Rača Unit. The Kýčera Member is
considered  equivalent  to  the  muscovite  sandstone  facies  of
the  “Magura  Sandstone”  of  the  Polish  Western  Carpathians
(Ksiazkiewicz 1953; Pivko 2002).

Methodology

Detailed field sedimentary research was carried out in

a series of quarries and road-cut exposures in the NW Slovak
part of the Magura Group of the Outer Western Carpathians
(Figs. 1, 2). In the field research extended attention was giv-
en to detailed study of sedimentary structures. Visual com-
parators were used in the field to determine grain sizes. The
starting point for data analysis and interpretation is the verti-
cal section of sedimentary sequence. The main members of

297

MARKOV CHAIN ANALYSIS OF TURBIDITIC FACIES AND FLOW DYNAMICS (SLOVAKIA)

sedimentary sequences – descriptive facies were defined on
the basis of sedimentary characteristics recorded in the field
and marked with facies codes.

The presence of order in the studied successions of facies

was  validated  using  the  statistical  methods  based  on  the
Markov property. The procedures for the calculation of sig-
nificant  facies  transitions  were  incorporated  into  a  self-de-
signed  computer  program,  entitled  “phpSedistat”,  allowing
automation of the calculations (see Appendix for details). As
the  calculations  are  practically  provided  using  a  computer

program, attention is given to the input data for calculation
and to the form and interpretation of the output data. The in-
put data for the calculation is a vertical sequence of facies,
expressed as a column of facies codes representing descrip-
tive facies and saved in a .txt file. Interruptions or absence of
sedimentary record is marked with a dividing symbol (/).

The output data are: (1) a list of facies codes and short de-

scription of associated facies, (2) a matrix of observed facies
transitions and (3) a matrix of significant facies transitions.
The matrix of observed facies transitions contain the number

Fig. 1. A – Geological map of investigated area showing the locations of the outcrops of the Kýčera Member. 1—5 – Magura Group: 1 – Vsetín
Member, 2 – Kýčera Member, Babia Góra Sandstone, 3 – Bystrica Member, 4 – Beloveža Formation (“Variegated beds”), 5 – Soláň Forma-
tion; 6—8 – Silesia Group: 6 – Istebná Member, 7 – Krosno Formation, 8 – Hieroglyph Formation; 9 – Pieniny Klippen Belt; 10 – geo-
logical boundaries: determined, expected; 11 – fractures: determined, expected; 12 – lines of the nappes and thrusts: determined, expected;
13 – localities; 14 – state boundary (modified from Biely et al. 1996). B – Location of the research area on the tectonic scheme of the Outer
Western Carpathians (modified from Żytko K. et al. in Poprawa & Nemčok 1989).

298

STAŇOVÁ, SOTÁK and HUDEC

Fig. 2. Photographs showing some general views and structures of investigated outcrops of the Kýčera Member. See the outcrop locations
on Fig. 1A. A – Detailed view from Krásno nad Kysucou outcrop showing the turbidite deposit, described by the Bouma model, consisting
of massive sandstone, T

at the base passing to the lower parallel lamination, T

, wave lamination, T

and upper parallel lamination, T

the top. B – Detailed view from the outcrop in Lysá pod Makytou quarry showing massive sandstone followed by laminated sandstone
(T

Bouma sequence). C – Field photograph of Taphrhelminthopsis traces in laminated siltstones on the top of bed from the outcrop be-

tween Klubina and Zborov nad Bystricou villages. D – Detailed view from the Krásno nad Kysucou outcrop showing clustered amalgam-
ation clasts. E – Detailed view from the quarry between Klubina and Zborov nad Bystricou villages showing post-sedimentary
deformation of claystone/silty claystone with mm to cm silt/very fine sandstone laminae. F – Detailed view from the Krásno nad Kysucou
outcrop showing submarine slump. G – General view of the quarry between Klubina and Zborov nad Bystricou villages. H – Detailed
view from the outcrop in Lysá pod Makytou quarry showing imprints of shale clasts at the top of a sandstone bed. I – General view of a
part of the quarry near Ve ké Rovné village. J – Detailed view from the quarry between Klubina and Zborov nad Bystricou villages show-
ing imprint of shale clasts at the top of sandstone beds. K – Detailed view from the quarry between Klubina and Zborov nad Bystricou vil-
lages showing claystone/silty claystone with mm to cm silt/very fine sandstone laminae (Facies A7).

300

STAŇOVÁ, SOTÁK and HUDEC

sions (Fig. 3) during the field research, tested for ordered ar-
rangement using the computer calculation of Markov Chains.
The  facies  transitions  A1—A3,  A1—A5,  A1—A7,  A1—A8,
A2—A3,  A2—A5,  A3—A4,  A4—A5,  A5—A1,  A5—A2,  A5—A6,
A5—A7, A6—A1, A6—A2, A6—A8, A7—A1, A7—A2 reached the
probability higher than 95 %, that is they are non-random with
the  probability  higher  than  95 %  (Table 3).  These  transitons
were considered regular at significance level of

α=0.5. All of

these significant facies transitions are illustrated in the scheme
of significant facies transitions (Fig. 4).

Discussion

Analysis of facies and flow dynamics

In the interpretation of significant facies transitions it is

important  to  keep  in  mind  that  the  calculated  significant
transitions represent the most probable facies transitions, but
not  their  frequency  in  the  studied  sedimentary  sequences.
The real frequencies of facies transitions are written down in
the  matrix  of  observed  facies  transitions  (Table 2).  There-

Table 1: Lithofacies.

Facies Mean

Code Thickness

(cm)

Description

structureless (massive) sandstone

118

pebbly sandstone with normally graded pebbles

parallel laminated sandstone

rippled, wavy laminated very fine sandstone to siltstone

laminated, very fine sandstone to siltstone

mudstone

claystone/silty claystone with mm to cm silt/very fine sandstone laminae

262

chaotic, deformed heterolithic units

Table 2: Matrix of observed facies transitions.

Facies A1 A2 A3 A4 A5 A6 A7 A8

Sum

row

–

160

318

–

144

–

276

126

–

190

–

Sum of column

313 77 81 85 285

188 21 3

Fig. 4. Scheme of significant facies transitions. Facies codes are described in Table 1.

301

MARKOV CHAIN ANALYSIS OF TURBIDITIC FACIES AND FLOW DYNAMICS (SLOVAKIA)

fore, when interpreting sedimentary successions it is useful
to consider both statistically significant and real facies tran-
sitions in order to better understand their significance and
real occurrence in the studied sedimentary record.

The calculated significant successions of facies A1—A3—

A4—A5—A6 and A2—A3—A4—A5—A6 (Fig. 4) remarkably re-
semble a Bouma sequence (Bouma 1962). This succession of
lithofacies is considered a typical sequence deposited by tur-
bidity currents. It comprises a massive or graded sandstone in-
terval T

(facies A1 or A2) at the base, followed upwards by

laminated sandstone interval T

(facies A3), then passing to a

rippled/wavy interval T

(equivalent to A4 facies), upper lam-

inated siltstone interval T

(facies A5) and terminating with

mud interval T

(facies A6). The real occurrence of this suc-

cession of facies in the sedimentary record (Fig. 2A,B) was
considered with reference to the matrix of observed facies
transitions (Table 2). There are 396 A1/A2—AX facies transi-
tions in the matrix of observed facies transitions, but only 78
of them (= 19.70 %) are A1/A2—A3 transitions resembling T

a—b

intervals of the Bouma sequence (Fig. 2B). There are also 74
A3—A4 facies transitions (equivalent to T

b—c

Bouma intervals),

65 A4—A5 facies transitions (equivalent to T

c—d

Bouma inter-

vals) and 97 A5—A6 facies transitions (equivalent to T

d—e

Bou-

ma  intervals)  in  the  matrix  of  observed  facies  transitions
(Table 2). This means that only 19.70 % of the sandy intervals
beginning  with  the  A1  or  A2  facies  constitute  either  a  com-
plete or a basal part of the classical Bouma sequence. In addi-
tion to A1/A2—A3 facies transitions, the A1 or A2 facies can
also be followed by the A5, A6, A7, A8 facies as well as by
the A1 or A2 facies. The A1 or A2 facies are most frequently
followed by facies A5. There are 193 transitions of A1/A2—A5
facies in the matrix of observed facies transitions (Table 2). It
is 48.74 % of the whole 396 transitions of A1/A2—AX facies
in the matrix of observed facies transitions. While the previ-
ously  discussed  transitions  of  facies  A1/A2—A3  can  be  com-
pared  to  the  T

Bouma divisions, the A1/A2—A5 transitions

resemble rather the T

Bouma divisions. According to the

field  record,  the  units  of  facies A5  are  usually  much  thinner
than those of the A1 or A2 facies. This may suggests that apart
from the deposits of turbidity currents (Bouma sequence), per-
haps the most significant part of the massive or graded sand-
stone  beds  with  thin,  fine  laminated  tops  in  the  investigated
sections  are  products  of  probably  the  most  disscussed  deep-
water  sediment  gravity  flows,  frequently  interpreted  as  high-
density  turbidity  currents  (Lowe  1982),  concentrated  density
flows  (comp.  Mulder  &  Alexander  2001)  or  sandy  debris
flows  (Shanmugam  1997),  while  their  upper  laminated  tops

were deposited from dilute current. These tops contain pieces
of  mica  and  fossil  plants  and  they  are  frequently  bioturbated
(Fig. 2C).  They  could  be  reworked  by  bottom  currents.  The
A1/A2—A6  facies  transitions  have  a  similar  explanation.  At
the top of the A1 or A2 facies there are sometimes shale clasts.
They apparently represent rip-down clasts (Fig. 2H,J) that in-
dicate  post-depositional  liquefaction  (Stow  &  Johansson
2000). Sometimes, amalgamated intervals in the investigated
sedimentary section are indicated by the presence of clustered
amalgamation  clasts  (Fig. 2D)  formed  by  erosional  break-up
of a thin intervening shale layer between successive sand de-
posits (see Stow & Johansson 2000).

Facies A7 most probably follows the A1 or A5 facies

(Fig. 2K).  Facies A7  is  considered  to  represent  repetitive
A5—A6 facies transitions and it is interpreted as representing
deposition  from  low-concentration  turbidity  currents.  Ac-
cording  to  the  matrix  of  observed  facies  transitions  (Ta-
ble 2), the facies A6 follows all the defined facies. It is most
likely to follow A2, A5 or A8 facies (Table 3). Thick inter-
vals  of  A6  facies  were  deposited  from  suspension  fallout.
Facies A8  represents  chaotic,  contorted  heterolithic  units,
which  commonly  overlie  A1  or  A6  facies  (Table 3,  Fig. 4).
The  A8  facies  is  apparently  formed  by  submarine  slumps
(Fig. 2F). In some cases, they could be the result of post-sed-
imentary deformation (Fig. 2E).

Genetic facies definition

The examination of both the calculated significant facies

transitions  and  the  original  sedimentary  records  allowed  us
to  define  six  genetic  facies  types,  which  were  compared  to
the standard deep-water facies models and schemes of deep-
water  gravity-driven  deposits.  The  genetic  facies,  when  ar-
ranged  laterally  according  to  the  declining  grain  size,
provide the evidence of deposition from increasingly diluted
and more mature flows (Fig. 5). Such facies arrangement re-
flects depositional settings from proximal to distal. The most
proximal and usually the coarsest facies are chaotic and de-
formed  units  (Facies  no. 1  in  Fig. 5),  which  apparently  de-
veloped  on  the  slope  of  depositional  lobes.  However,  some
of these units may developed as post-sedimentary deforma-
tions.  The  next  facies  represent  thick-  to  medium-bedded
sandstones  with  normally  graded  granules  and  pebbles  (Fa-
cies no. 2 in Fig. 5) and thick- to medium-bedded structure-
less sandstones (Facies no. 3 in Fig. 5). These facies reflect
the deposits of density flows (Alexander & Mulder 2001) or
dense flows (sensu Mutti et al. 2003) or high-concentration
turbidity  currents  (Lowe  1982;  Pickering  et  al.  1986).  Both
these  facies  types  may  have  thin  laminated  tops,  which  de-
veloped  as  the  flows  were  progressively  diluted  by  the  en-
trainment of surrounding water. Normal gradation developed
if  the  dilution  of  water  was  sufficient  to  allow  particle  fall
out within the flow. The next facies (Facies no. 4 in Fig. 5)
are characterized by the Bouma sequence of structures, rep-
resenting  deposits  from  turbidity  currents.  The  next  facies
(Facies  no. 5  in  Fig. 5)  are  claystone/silty  claystone  with
mm  to  cm  silt  to  very  fine  sandstone  laminae.  These  facies
are  compared  to  the  deposits  of  low-concentration  turbidity
currents. The last and the most distal facies are claystones or

Table 3: Matrix of significant facies transitions.

Facies

A1 A2 A3 A4 A5 A6 A7 A8

– – 99.99 – 99.99 – 99.99

99.99

– – 99.82 – 98.38

52.02

– –

– – –

99.99

– – – –

99.99

– – –

99.99

– – –

99.99

–

99.99

– – – – –

99.99

99.36

96.02

– – – – – –

– – – –

36.46

58.74

– –

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STAŇOVÁ, SOTÁK and HUDEC

silty claystones (Facies no. 6 in Fig. 5), which are compared
to the deposits of low-concentration turbidity currents or sus-
pension fallout.

Conclusions

The application of Markov Chain analysis allowed us to

identify ordered succession of descriptive facies or structures
in a turbiditic formation from the Magura Zone of the Outer
Western Carpathians. Significant facies transitions were de-
termined providing the evidence that classical turbidites con-
stitute  the  important  part  of  the  studied  sedimentary
sequences  of  the  Kýčera  Member.  In  addition  to  turbidite
currents,  the  Kýčera  Member  facies  were  also  deposited  by
density  flows,  suspension  fallout  or  they  were  reworked  by
bottom  currents  or  resulted  from  syn-  or  post-depositional
deformation.

In this paper, the Markov Chain analysis has been applied to

calculate the arrangement of facies in turbiditic formation, al-
though  it  has  been  utilized  in  the  past  for  the  analysis  of  or-
dered  sequences  of  facies  in  other  environments  (e.g.
meandering  river,  shallow-marine,  beach-barrier  etc.).  The
calculation  aspects  of  Markov  analysis  are  reviewed  in
a concise and instructive form, therefore this study can serve
as a case study for the application of the Markov Chains in
the  interpretations  of  sedimentary  sequences  in  the  field  or
from boreholes. A subsequent contribution of this study was
the  creation  of  a  self-designed  computer  program,  named
phpSedistat.  Computer-based  calculation  of  regular  facies
transitions noticeably simplified and hastened the utilization
of Markov Chains.

Acknowledgments: This study was supported by the Slovak
Research and Development Agency (APVV 51-011305
Project) and by VEGA Grant GA 6093. Official reviews by
Prof. Dr. Nestor Oszczypko, Dr. Ivan Baráth and Prof. Dr.
Szczepan Porębski provided insightful comments that helped
improve the original manuscript.

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305

MARKOV CHAIN ANALYSIS OF TURBIDITIC FACIES AND FLOW DYNAMICS (SLOVAKIA)

Step-by-step list of procedures incorporated into the

phpSedistat program:

1. The matrix of observed facies transitions is created.
As the quasi-independence principle was applied (see e.g.

Carr 1982; Powers & Easterling 1982 for details), the facies
transitions between the same facies were excluded. They are
usually hardly identified in the field.

2. The matrix of random transitions is calculated.
Row/column sums of the matrix of observed facies transi-

tions and parameters a

, b

are utilized to calculate the matrix

of random transitions. Parameters a

, b

are estimated using

an iterative solution (see Powers & Easterling 1982).

The first iteration of a

and b

is calculated as follows:

, i = 1, 2, ..., m

(1)

, j = 1, 2, ..., m

(2)

where m is a number of facies members, n

is an i

row

sum and n

is a sum of j

column.

Then the i

iteration is calculated as follows (3, 4):

, i = 1, 2, ..., m

(3)

, j = 1, 2, ..., m

(4)

The iteration is continued until (5, 6):

(I)

— a

(I—1)

| < 0.01 a

(I)

, for i = 1, ..., m (5)

and

(I)

—b

(I—1)

| < 0.01 b

(I)

, for j = 1, ..., m (6)

Let a’ and b’ be the last iterations of a

(I)

and b

(I)

, then the

estimated random transitional frequencies are calculated as
follows (7):

= a

’

· b

’

i = 1, 2, ..., m; j = 1, 2, ..., m; i

≠j

(7)

3. The matrix of observed possibilities of facies transitions

and the matrix of random possibilities of facies transitions
are calculated.

The matrix of observed transition frequencies and the ma-

trix of random transition frequencies are recalculated through
their row sums, e.g. the first row is recalculated as follows:

, a

, ... a

(8)

The result of these recalculations is the creation of the ma-

trix of observed possibilities of facies transitions and the ma-
trix of random possibilities of facies transitions.

4. Calculation of a difference matrix.
The difference matrix is calculated as the matrix of ran-

dom possibilities of facies transitions is substracted from the
matrix of observed possibilities of facies transitions.

5. Positive values in a difference matrix are tested.
All positive values in the difference matrix are tested for

randomness  at  a  selected  significance  level.  Testing  proce-
dures within the phpSedistat computer program are provided
at  significance  level  0.5.  It  means  that  all  computed  values
greater  than  0.05  are  non-random  with  the  possibility  less
than 5 %, i.e. random with the possibility greater than 95 %
and all computed values less than 0.05 are non-random with
the  possibility  greater  than  5 %.  The  testing  criterion  is  as
follows (see Harper 1984):

(9)

where N = relevant row sum in the matrix of observed tran-

sition frequencies, n

cont

= observed number of a specific tran-

sition, p = the possibility of a specific transition in the matrix
of random possibilities of transition frequencies and q = 1—p.

6. The matrix of regular facies transitions is created.
The values computed in the testing procedure are multiplied

by 100 and the matrix of regular facies transitions is created,
where the regular facies transitions are calculated at the signif-
icance level 0.5 and expressed in the number of percent. The
formulation of the resulting possibilities of facies transition in
the number of percent simplifies their evaluation.

The program phpSediStat was programmed under a GNU

lincence as an open source software product. It can be down-
loaded from http://phpsedistat.sourceforge.net.

Appendix

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