Application of Finite Markov Chain Model to the ... The Markov Chain Process is stochastic: that in which an event is probabilistically dependent on preceding event. It uses a lithology

393 Opara et al., Application of Finite…

FUTOJNLS 2018 VOLUME- 4, ISSUE- 1. PP- 393 - 408

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Volume-4, Issue-1, pp- 393 - 408

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Research Paper July 2018

Application of Finite Markov Chain Model to the Lithofacies

Analysis of Successions of Mamu Formation Sediments in Parts of

Southeastern Nigeria

Onyekuru S.O., Iwuagwu, C.J. and *Opara K.D.

Department of Geology, Federal University of Technology, Owerri, Imo State, Nigeria

*Corresponding Author’s Email: [email protected]

Abstract

Finite Markovian Chain stochastic process was used in this study as an appropriate quantitative method for defining lithofacies succession trends, analysis and correlation. The model was used to distil lithofacies successions of the Mamu Formation sediments in the study area for a more realistic determination of environment of deposition (EOD). Six (6) lithofacies (F1-F6) were recognized from the six (6) outcropping profiles of the formation in the study area. A coarsening upward transition pattern was established from the depositional model generated from the composite Facies Relationship Diagram indicating prograding shoreline lithofacies assemblages. The transition patterns have memory functions (markovian) because computed Chi-Square (X2) value at 250 of freedom (85.62) is higher than the X2 value of 37.65 at 95 % confidence limit and 250 of freedom. This condition thus established a genetic link between successive lithofacies and consequently the environments of deposition of the deposits. The inferred environment of deposition (EOD) ranged from open marine through lower delta front (distal bar) to nearshore with occasional fluvial influences.

Keywords: Chain, environment, facies, markovian, memory, succession

1. Introduction

The key to the interpretation of depositional environments using sedimentary facies is

to combine observations on spatial relationships and internal characteristics of sedimentary

units (lithology and sedimentary structures) with information from other well-studied

stratigraphic units and particularly, studies of modern sedimentary environments (Walker,

1984; Wilson &Schieber, 2015). Traditionally, sedimentary environments and models are

reconstructed based on visual description of these variables which are subjective. The

interpretations of environment of depositions (EOD) from environmental models derived from

such subjective data are sometimes erroneous. Hence the need for semi-quantitative and

quantitative techniques for data analysis to complement descriptive stratigraphic analysis

(Krumbien and Dacey, 1969; Selley, 1969; Miall, 1973; Perimutter and DeazambujaFilho,

2005). Techniques, such as finite markov chain, time series and time spectral analyses

provide multiple working hypotheses to detect and define stratigraphic succession trends.



The Markov Chain Process is stochastic: that in which an event is probabilistically

dependent on preceding event. It uses a lithology versus order relationship to structure a

Markov model for stratigraphic analysis.

The Markov model therefore, assumes that a lithofacies state is influenced by that of the

underlying lithofacies. This historical link between the events is called “memory function”

(Feng, 2018; Selley, 1969; Krumbien and Dacey, 1969). Selley (1969) described three

models of structuring lithofacies states observed in the field for finite Markov analysis. These

are:

- The regular Markov Matrix where stratigraphic intervals are sampled at fixed or

regular vertical intervals of equal thickness. This method gives an accurate measure

of the relative frequencies of the lithofacies types in the sections but does not

emphasize changes in the depositional environment;

The embedded Markov Matrix which emphasizes every lithological change and a

good method for understanding the evolution of depositional environments and

processes, and

The multistory Markov Matrix which is a variant of the embedded Markov Matrix in

which lithofacies change is important but multistorylithologies (where similar

lithofacies but with different texture or sedimentary structures) are also counted.

In numerous previous studies, markov chain analyses have shown that conditional

probabilistic relationships (knowledge of an event/facies defines the probability of a following

event/facies) exist among vertical facies successions (Iwuagwu, 1993; Lehrmann and

Rankey, 1999; Parks et al., 2000; Davis, 2002; Wilkinson et al., 2003). The analysis has

therefore emerged as useful quantitative tool for investigations into potential linkages

between lateral, temporal and sedimentological dynamics in the tradition of comparative

sedimentology (Ginsburg, 1974).The application of markov chain analysis in the study of

Campano-MaastrichtianMamu Formation sediments in the Okigwe-Uturu area was aimed to

introduce objectivity and precision in the analysis of lithofacies for the interpretation of EOD

from generated stratigraphic summaries (models).

2. Study Area Description

The study was carried out using successions enclosed within Longitudes 7°20' and 7°35'E

and Latitudes 5°45'and 5°55' N, covering Uturu, Okigwe, Ihube and Leru towns, in

southeastern Nigeria (Fig. 1).



Figure 1: Location map of the study area

The origin of the Anambra Basin is intimately related to the development of the

Benue Rift. The Benue Rift was installed as the failed arm of a triple fracture (rift) system,

during the breakup of the Gondwana supercontinent and the opening up of the southern

Atlantic and Indian Oceans in the Jurassic (Burke et al., 1972; Benkhelil, 1989; Hoque and

Nwajide, 1984; Fairhead, 1988, Nwajide, 2013). The initial synrift sedimentation in the

embryonic trough occurred during the Aptian to early Albian and comprised of alluvial fans

and lacustrine sediments of the Mamfe Formation in the southern Benue Trough. Two cycles

of marine transgressions and regressions from the middle Albian to the Coniacianinfilled the

ancestral trough with mudrocks, sandstones and limestones with an estimated thickness of

3,500 m (Murat, 1972; Hoque, 1977). These sediments belong to the Asu River Group

(Albian), the Odukpani Formation (Cenomanian), the Ezeaku Group (Turonian) and the

Awgu Shale (Coniacian). During the Santonianepeirogenic tectonics, these sediments

underwent folding and uplifted into the Abakaliki - Benue Anticlinorium (Murat, 1972) with

simultaneous subsidence creating the Anambra Basin and Afikpo Sub- basins to the

northwest and southeast of the folded belt respectively (Murat, 1972; Burke, 1972; Obi,

2000; Mode and Onuoha, 2001). The Abakaliki Anticlinorium later served as a sediment

dispersal center from which sediments were shifted into the Anambra Basin and Afikpo

Syncline (Nwajide, 2005). The Oban Masif, southwestern Nigeria Basement Craton and the

Cameroon Basement Complex also served as sources of the sediments in the Anambra

Basin (Hoque and Ezepue, 1977; Amajor, 1987; Nwajide an Reijers, 1996; Onu, 2017).

Sedimentary events that occurred in the Anambra Basin after the Santonianepeirogeny

included: the Campanian - Early Maastrichtian transgression that deposited the Nkporo

Group (i.e. the Enugu Formation, Owelli Sandstone and Nkporo Shale) as basal units,

followed by the Maastrichtian regressive event during which the coal measures (i.e. the

Mamu, Ajali and Nsukka Formations) were deposited (Fig. 2; Table 1).

OKIGWE

Ihube

Km75Km74

Leru

Lekwesi

Lokpanta

Lokpaukwu

Isuochi

Amuda

km65

E7020I E7030I

E7020I E7030I

N5047I N5047I

N5056I N5056I

Towns

Sample pointsMinor roadsMajor road

Ihube

Leru

Ezinachi

Lokpaukwu

Lekwesi

Lokpanta

Agbogugu

Isuochi

7020’E 7025’E 7030’E

Km 65

50 45’N

50 50’N

50 55’N

50 45’N

50 50’N

50 55’N

7020’E 7025’E 7030’E

Sample Location

Other towns

Major Road

Other Roads

1:200,000

LEGEND

Km 74

Km 75

1:200,000

LEGEND



Figure 2. Geologic map of the study area (modified from NGSA, 2004)

Table 2.1: Correlation Chart for Early Cretaceous Tertiary Strata in the Southeastern Nigeria. After Nwajide, 2013.

OKIGWE

Ihube

Km75Km74

Leru

Lekwesi

Lokpanta

Lokpaukwu

Isuochi

Amuda

km65

E7020I E7030I

E7020I E7030I

N5047I N5047I

N5056I N5056I

Imo Formation

Nsukka Formation

Ajali Sandstone

Mamu Formation

Nkporo Formation

Ezeaku Formation

Asu River Group

Sample points

Ihube

Leru

Ezinachi

Lokpaukwu

Lekwesi

Lokpanta

Agbogugu

Isuochi

7020’E 7025’E 7030’E

Km 65

50 45’N

50 50’N

50 55’N

50 45’N

50 50’N

50 55’N

7020’E 7025’E 7030’E

Sample Location

Other towns

Major Road

Other Roads

1:200,000

LEGEND

Km 74

Km 75

LEGEND

Awgu Shale

Imo Formation

Nsukka Formation

Ajali Formation

Mamu Formation

NkporoFm

NkporoShale

EnuguFm

OwelliSs

AfikpoSs

OtobiSs

LafiaSs

Agwu Formation

Niger Delta

Anambra Basin

SouthernBenueTrough

Thanetian

Danian

Maastrichtian

Campanian

Santonian

Akata Formation

Eocene Ameki/Nanka Fm/Nsugbe Sandstone (Ameki Group)

Agbada Formation

Oligocene-Recent

Benin FormationOgwashi-Asaba Fm

Age Basin Stratigraphic Units

Imo Formation

Nsukka Formation

Ajali Formation

Mamu Formation

NkporoFm

NkporoShale

EnuguFm

OwelliSs

AfikpoSs

OtobiSs

LafiaSs

Agwu Formation

Niger Delta

Anambra Basin

SouthernBenueTrough

Thanetian

Danian

Maastrichtian

Campanian

Santonian



3. Methodology

Lithostratigraphic mapping was carried out by identifying and describing outcropping profiles

in the study area. Simple traverses were made along major and minor roads in the mapped

area. Described outcrop sections covered parts of Okigwe, Ihube and Leru. Four sections of

the outcropping profiles of Mamu Formation at the Okigwe-Leru Axis carefully logged,

described and referenced to the base map, revealed six lithofacies. Each lithofacies was

delimited on the basis of gross lithology, texture, sedimentary structures and nature of

bedding surface discontinuity.

Lithologs were drawn to capture lithology, bed thickness, sedimentary structures and

bedding contacts. Outcrop locations were geo-referenced with the aid of Global Positioning

System (GPS). Essential sedimentary and structural features were also taken.

Facies were identified, described and subjected to statistical Markov Chain analysis. In the

first order Markov event, a lithofacies state, Fj observed at point n, depends on the

lithofacies state, Fi observed at point (n-1). The transition probability of a lithofacies state, Fj

observed at point n, given that the lithofacies state is now in state Fi, at point (n-1) is

denoted by equation 1:

Pij (n-1 , n) = P(Fjn/Fin-1) (Selley, 1969) 1

It is the multistory Markov Chain process that is applied in this study. The transition count

matrix (TCM) will be such that along the principal diagonal of the erected matrix, the values

will be zero (Fij = 0).

4. Results

Field Observations

Identified lithofacies and their distinguishing features are described thus:

Facies 1 (F1): Dark blue-grey laminated shale facies

This facies is comprised of bluish to greyish shale with a thickness range of 1 to 3 m. The

grain size of this facies is clay. The shales are fissile and highly cleaved, argillaceous in

nature and post depositional joints are present. The shale facies contains pyritic nodules or

spheroidal secondary structures (concretions). The deep bluish variety is often calcareous

with fossilized body parts of marine organisms.

Facies 2 (F2): Siltstone/claystonefacies

The facies comprises of light grey to brown variegated and mottled siltstone and claystone.

The facies is rippled or parallel laminated in places. It is slightly sandy and carbonaceous

with comminuted plant remains. Individual occurrences of this facies are averagely about

2m in thick.

Facies 3 (F3): Poorly stratified sandstone facies

The lithology of this facies is a very fine to medium grained sandstone. The facies is rippled,

mainly with asymmetrical ripples, however, they assume symmetrical occasionally. The

ripple laminations sometimes show bidirectional orientations indicating reversal of the

depositing current. The units are also often bioturbated. Recognized burrows include the

straight and branched forms of Rhizocorallium and Ophiomorphanodosa.



Facies 4 (F4): Planar cross-bedded sandstone facies

The Planar cross-stratified sandstone facies is characterized by sets of planar-tabular cross-

beds that occur as single sets or co-sets. The sandstone of this facieshave cementing

materials that are ferruginous and argillaceous. Height of the planar cross-strata varies from

0.10 to 0.80m. The grainsize varies from medium to coarse-grained sand.

Facies 5 (F5): Pebbly to conglomeritic sandstone facies

The facies is composed of clasts of pebbles in a predominantly sandstone lithology. The

clasts are moderately sorted and subangular to subrounded in shape. The size of the clast

ranges from 1 to 3 mm. Beds of this facies maintain slightly erosional contacts with the

underlying beds and are often ferruginized.

Facies 6 (F6): Trough cross-bedded sandstone facies

This facies consists of distinct sets of cross-beds with troughs. The facies is characterized by

light yellow to yellowish brown color, medium to coarse grained, The sandstone is

moderately hard and compact and resistant. The grains are sub-angular to sub-rounded and

moderately sorted.

Figure 3: The lithofacies section at km 65 Uturu-Afikpo road

Thickness

(m) Lithology Cl Si Fs Ms Cs Gr

Description Facies Environment

10

8

7

6

5

4

3

2

1

0Cl Si Fs Ms Cs Gr

Light grey mottled

claystone/siltstone capped

with lateritic sandstone

Dark grey laminated shale

F1

F2

F1

Reddish brown claystone

greyish brown mottled

claystone/siltstone


with pyritic concretions


with thin band of coaly

shale

F2

F2

F1

Shale

Siltstone

Claystone

Coal



Figure 4: The lithofacies section at km 75 Enugu-Port Harcourt express road

Figure 5: The lithofacies section at km 74 Enugu-Port Harcourt express road

Thickness



20

18

16

14

12

10

8

6

4

2

0

Cl Si Fs Ms Cs Gr

F3

F1

F3Vf-m grained ripple laminated

and bioturbated sandstone

Dark grey laminated pyritic

shale

Medium grained planar

cross bedded sandstone

Vf grained silty sandstone

Medium to coarse grained

trough cross bedded sandstone

Vf grained silty sandstone



trough cross bedded sandstone

Vf grained silty sandstone and

slightly bioturbated


F1

F6

F3

F1

F6

F4

F3

Shale

Siltstone

Sandstone

Claystone

Cross bed

Bioturbation

Trough Cross beds

Pebbly sandstone band F5

Thickness



7

6

5

4

3

2

1

0Cl Si Fs Ms Cs Gr

Siltstone/mottled claystone

Brownish mottled

claystone and siltstone F2

F2


planar cross-bedded

sandstone

Coarse grained trough


Fine to medium grained

ripple laminted silty

sandstone


F4

F6

F3

F1

Siltstone/mottled claystone


F2

F1

Shale

Siltstone

Sandstone

Claystone

Cross bed

Bioturbation

Trough Cross beds



Figure 6: The lithofacies section at NkwoaguUmunneoche off Leru junction

Figure 7: The lithofacies section at km 52 near Leru junction on the Port Harcourt bound

side of Enugu-Port Harcourt Expressway

Thickness



6

5

4

3

2

1

0Cl Si Fs Ms Cs Gr

Light grey mottled

claystone/siltstone

Medium grained planar



Bioturbated sandstone

Dark grey laminated shaleF1

F3

F4

F2

F2Reddish brown claystone

Shale

Siltstone

Sandstone

Claystone

Cross bed

Bioturbation

Ferruginized pebbly sandstone

bandF5

Thickness



10

8

6

4

2

0Cl Si Fs Ms Cs Gr

Dark grey laminated shaleF1

F3

F6

F2Ripple laminated sitlstone

Shale

Siltstone

Sandstone

Claystone

Cross bed

Bioturbation

F3Fine to medium grained

ligt grey sandstone

Bioturbated siltstone


Medium-coarse grained

ripple bedded sandstone

conglomeritic sandstone

F2

F1



Figure 8: The lithofacies section near Total filling station along Okigwe-University road,

Okigwe town

Figure 9. Planar cross bedding at km 74 Enugu-Port Harcourt express road.

Thickness



12

11

10

8

6

4

2

0Cl Si Fs Ms Cs Gr

Clayey feldspathic

sandstone

F3

F2

F5

Medium grained plannar


Shale

Siltstone

Sandstone

Claystone

Cross bed

Bioturbation

F3Fine to medium grained

ligt grey sandstone

Deep blue to dark grey

carbonaceous and

laminated shale with fish

teeth fossil

F1

Conglomeritic sandstone

Greyish mudstone

F4



Figure 10. The Road cut exposure at NkwoaguUmunneoche showing the basal shale unit

and overlying sandstone and mudstone units.

Figure 11: Sharp and near concordant contact between Mamu Formation and Ajali

Sandstone at km 65 section Uturu-Afikpo road

The six lithofacies were structured into a multistory Markov Model in the form of a tally matrix

to indicate the observed cumulative number of times which a particular lithofacies state

overlies another.

Table 1 shows the Transition Count Matrix (TCM) for the lithofacies observed in the study

area. An Independent Trial Matrix (ITM) was computed from the TCM using the formula:

( )

2

AJALI SANDSTONE

MAMU FORMATION

Coal Seam



Where Ct = Column total, Rt = Row Total, T = Total number of transitions

In the Independent Trial Matrix, (ITM), the frequency with which each lithofacies underlies

another is proportional to its relative abundance, although no lithofacies transition pattern is

obvious. Table 2 shows the Independent Trial Matrix (ITM) for structured lithofacies.

However, when the Difference Matrix (DM) is calculated, lithofacies transition patterns and

their frequencies become visible. The Difference Matrix (DM) is calculated for each

corresponding cell using:

3

Table 3 is the Difference Matrix computed from the structured TCM and ITM for the logged

sections in the Okigwe-Uturu axis. The positive values in the Difference Matrix (DM) illustrate

the Markovian property and reflect the transitions that have greater than random

frequencies.

Table 1: Transition Count Matrix (TCM)

F1 F2 F3 F4 F5 F6 Rt

F1 0 8 4 0 0 0 12

F2 5 0 3 0 0 0 8

F3 0 0 0 3 0 4 7

F4 0 1 0 0 3 0 4

F5 1 2 0 0 0 0 3

F6 1 0 2 1 0 0 4

Ct 7 11 9 4 3 4 38

Table 2: Independent Trial Matrix (ITM)

F1 F2 F3 F4 F5 F6

F1 2.21 3.47 2.84 1.26 0.95 1.26

F2 1.47 2.32 1.89 0.84 0.63 0.84

F3 1.3 2.03 1.66 0.74 0.55 0.74

F4 0.74 0.74 0.95 0.42 0.32 0.42

F5 0.55 0.87 0.71 0.32 0.24 0.32

F6 0.74 1.16 0.95 0.42 0.32 0.42



Table 3: Difference Matrix (DM)

F1 F2 F3 F4 F5 F6

F1 -2.21 4.53 1.16 -1.26 -0.95 -1.26

F2 3.53 -2.32 1.11 -0.84 -0.63 -0.84

F3 -1.3 -2.03 -1.66 2.26 -0.55 3.26

F4 -0.74 0.26 -0.95 -0.42 2.68 -0.42

F5 0.45 1.13 -0.71 -0.32 -0.24 -0.32

F6 0.26 -1.16 1.29 0.58 -0.32 -0.42

To determine if this transition pattern is a product of chance events (i.e. random) or of if the

sedimentation process had a “memory function” (i.eMarkovian), a test of significance was

applied to the result.

The Chi-Square (X2) Test has been used by several authors for this purpose (Miall, 1973;

Bernajee, 1979). In this test, a null hypothesis, HO is proposed which assumes that the

transition pattern generated by the sedimentary process was random (i.e non Markovian).

The X2 test is used to reject or uphold this hypothesis.

The Chi-Square value is calculated with the formula

∑( )

( ) 5

Where DM = Difference Matrix and ITM = Independent Trial Matrix. Table 4 shows the result

of the computed X2 value from the lithofacies transition patterns of the sections of Mamu

Formation studied showing a Chi-Square (X2) value of 85.62.

The number of Degree of Freedom is given as the square of the difference between the

number of facies with non-zero entries in the Independent Trial Matrix minus one.

DF = (N-1)2= (6-1)2 = 250 6

Where N = number of lithofacies in ITM with non-zero entries.

Table 4: Result of the Chi-Square (X2) Test

F1 F2 F3 F4 F5 F6

F1 2.21 5.91 0.474 1.26 0.95 1.26

F2 8.48 2.32 0.652 0.84 0.63 0.84

F3 1.3 2.03 1.66 6.9 0.55 14.36

F4 0.74 0.09 0.95 0.42 22.4 0.42

F5 0.37 1.47 0.71 0.32 0.24 0.32

F6 0.09 1.16 1.752 0.8 0.32 0.42

X2 = ∑ 85.62



5. Discussion

In this study, the computed X2 value at 250 of Freedom from the Chi-Square table is 85.62

which is higher than X2 value at 95% confidence limit at 250 of Freedom from the Chi-Square

Table given as 37.65. If the computed X2 value is higher than the limiting X2 value at 95%

confidence interval from the Chi-Square Table, then null hypothesis (HO) is rejected (i.e

sedimentary process is Markovian and not random) (Miall, 1973; Bernajee, 1979; Selly,

1969). If on the other hand X2 value is lower than the limiting X2 value at 95% confidence

interval, then null hypothesis is accepted (i.e sedimentary process is non-Markovian and

random.

The null hypothesis, (HO) is rejected; therefore the Lithofacies transition pattern is Markovian

with “memory function”.

To visualize the facies transition pattern, a composite Facies Relationship Diagram (FRD)

(Fig. 12) is constructed to represent lithofacies path lines using the positive entries in the

Difference Matrix (Miall, 1973). A model (Fig. 13) was also generated from the FRD pattern

depicting a coarsening upward trend typical of a regressive shoreline.

The thick, dark grey to black shalesfacies are interpreted as shallow to open marine

deposits.The succession exhibits “sandier (coarsening) upward” motif characterized by

poorly stratified sandstone, cross stratified sandstone (planar and trough) and pebbly

sandstone facies often interpreted as “shoaling upward” or progradation (Reading, 1986).

These can be interpreted as the sedimentary deposits of the lower delta front (distal bar) to

near-shore environment with occasional fluvial influence. These facies are thus assigned to

marginal marine (near-shore to shoreface) environment. The trough cross bedded

sandstone may be the result of scour and channel fill features. The facies show a cut and fill

events during the initial stage of deposition at higher hydrodynamic condition and high

sediment (Walker and Cant, 1984). The planar cross bedded sandstone represents

transverse bar, point bar and sand wave in active part of channel supply. The rippled and

mottled claystone/siltstone facies are interpreted as tidal/fluvial/ flood plain deposits in marsh

and coastal swamp setting.

(Cant and Walker, 1984; Miall, 1978).The poorly stratified sandstone facies represents the

tendency to form at the final stage of channel in filling when stream power reaches a

minimum traction level (Harms, et al, 1982). This type of facies can be originated when wave

or current ripples are produced and migrate on a sediment surface (Reineck and

Wunderlich, 1970). It is most abundantly developed in sandy intertidal flat, part of channel,

bar and inactive part channel (Smith, 1974; Miall, 1978). In almost any environment with

non-cohesive sediment small current ripple bedding is produced. This may also be

generated when wave interacts with reversal of flow in shallow tidal environment.



Figure 12: Composite Facies Relationship Diagram for the Lithofacies

Figure 13: Lithofacies model for sections in the study area using the probability Difference

Matrix

6. Conclusion

Owing to unavailability of subcrop data that could ensure continuity in sedimentary

succession, the finite Markov Chain stochastic process has been utilized in this paper to

distill the lithofacies pattern on outcrops of the Mamu Formation exposed around the

Okigwe-Uturu axis Anambra Basin SE Nigeria which was masked severely by several

erosion truncation surfaces and road construction activities. A coarsening upward transition

pattern is documented in the logged sections. The transition pattern was also found to

Markovian, indicating historical links between the underlying lithofacies, and consequently

the environments of deposition. The environment of deposition ranged from open marine

through lower delta front (distal bar) to near-shore environment with occasional fluvial

influence. Finite Markovian Chain Stochastic Process has been shown in this study to be

appropriate quantitative method for defining Lithofacies trends. This method can be utilized

in oil well lithofacies data analysis and local and field-wide correlation especially in the light

of increased interest in the Nigeria’s frontier basins.

F1 F3 F4 F6

F5

F2

0.26

1.29

0.58

2.68

0.45

1.16

4.53

3.531.11

3.26

1.13

Facies 1

Facies 2

Facies 3

Facies 4

Facies 5

Facies 6



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