Journal of Petroleum Science and Engineering 31 Ž2001.

135–155
www.elsevier.comrlocaterjpetscieng

Stratigraphic simulations using fuzzy logic to model

sediment dispersal
Robert V. Demicco a,) , George J. Klir b
a
Department of Geological Sciences and EnÕironmental Studies, Binghamton UniÕersity, Binghamton, NY 13902-6000, USA
b
Center for Intelligent Systems, Watson School of Engineering and Applied Science, Binghamton UniÕersity,
Binghamton, NY 13902-6000, USA

Abstract

The purpose of this paper is to report on our preliminary two- and three-dimensional stratigraphic simulations that use
fuzzy logic to model sediment production, sediment erosion, sediment transport and sediment deposition. Fuzzy logic offers
a robust, easily adaptable, and computationally efficient alternative to the traditional numerical solution of complex, coupled
differential equations commonly used to model sediment dispersal in stratigraphic models. Fuzzy logic is based on the
concept of fuzzy sets, and, since the 1980s, fuzzy logic has been successfully applied in virtually all areas of engineering and
computer sciences, as well as in areas of decision making, optimization, management, and operations research. Fuzzy logic
is also rapidly being assimilated into the sciences and, since it is capable of utilizing both AhardB data and AsoftB qualitative
statements, fuzzy logic naturally lends itself to applications in the Earth Sciences. Here we first compare two-dimensional
simulations of reef growth: one based on step-wise solution of a partial differential equation and one in which an elementary
fuzzy logic system is employed. The two simulations produce identical results. We then present three fully three-dimensional
models: Ž1. a simulation of the last 200,000 of sedimentation in Death Valley, CA; Ž2. a simulation of sedimentation on the
Great Bahama Banks west of Andros Island during the latest 10,000 years of sea level rise; and Ž3. a hypothetical delta and
floodplain under varying regimes of sea level change. The results of the first two models match surface and subsurface data
from Death Valley and the Great Bahama Bank to a remarkable degree even though the models are in preliminary stages.
Moreover, the hypothetical deltaic simulations also produce remarkably complex and realistic cross-sections. Thus, our
preliminary modeling suggest that the utility of fuzzy logic in stratigraphic simulations may be profound. q 2001 Published
by Elsevier Science B.V.

Keywords: Stratigraphic models; Fuzzy logic; Sediment dispersal

1. Introduction geological science, both applied and theoretical ŽBur-

ton et al., 1987; Tetzlaff and Harbaugh, 1989;
In recent years, two- and three-dimensional com- Angevine et al., 1990; Bosence and Waltham, 1990;
puter-based models of sedimentary basin-filling have Franseen et al., 1991; Bosscher and Schlager, 1992;
become increasingly important tools for research in Flint and Bryant, 1993; Bosence et al., 1994;
Slingerland et al., 1994; Mackey and Bridge, 1995;
Foster and Merriam, 1996; Nittrourer and Kravitz,
)
Corresponding author. Tel.: q1-607-777-2604. 1996; Nordlund, 1996; Leederet al., 1996; Wende-
E-mail address: demicco@binghamton.edu ŽR.V. Demicco.. bourg and Harbaugh, 1996; Whitaker et al., 1997;

0920-4105r01r$ - see front matter q 2001 Published by Elsevier Science B.V.

PII: S 0 9 2 0 - 4 1 0 5 Ž 0 1 . 0 0 1 2 6 - 7
136 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

Harff et al., 1999.. Such models commonly produce Schlager, 1992., and bank-interior sediments Žc.f.
synthetic stratigraphic cross-sections that are of great Broecker and Takahashi, 1966; Morse et al., 1984..
value because they give us a predictive picture of the Coastal oceanographic modelers have made great
subsurface distribution of rocks Žsedimentary facies. strides in dealing with the complexities of coupled
whose petrophysical properties are useful in oil ex- solutions and wave and current dynamics and sedi-
ploration, gas exploration, groundwater exploitation, ment transport. However, finite difference and finite
groundwater remediation, and even naval warfare. element numerical simulations such as those in Aci-
From a theoretical point of view, synthetic strati- nas and Brebbia Ž1997. and Harff et al. Ž1999. have
graphic models increase our understanding of how two drawbacks when applied to stratigraphic models.
sediment accumulation varies in time and space in First, they are site specific and depend on rigorous
response to external driving factors Žsuch as eustasy application of boundary conditions, initial condi-
and tectonics. and internal driving factors Žsuch as tions, and wave and tidal forcing functions over a
compaction, isostatic adjustments, and crustal flexu- discrete domain. Secondly, these process-response
ral adjustments. made in response to tectonic loading models operate at tens to hundreds of year time
and sedimentary accumulation Žc.f. Angevine et al., scales, which are very short in comparison to basin-
1990.. These basin filling models have been particu- filling models. As a result, the effects of large,
larly successful in elucidating the effects of isostasy complex storm events, which are suspected of being
and crustal flexure during sedimentation. This is important agents in ancient depositional systems, are
nowhere more obvious than in models of foreland only rarely included in coastal models. Indeed, such
basin fill Žc.f. Beaumont et al., 1992; Flemings and complexities lead Pilkey and Theiler Ž1996. to ques-
Jordan, 1989; Jordan and Flemings, 1991; Chalaron tion the applicability of even short-term coastal mod-
et al., 1996.. els built around dynamic sedimentary process simu-
Perhaps the thorniest problem faced by strati- lators.
graphic molders is simulating sediment erosion, sedi- A number of approaches toward modeling sedi-
ment transportation, and sediment accumulation ment dynamics have been used in basin-filling strati-
within a forward model Žwhat Wendebourg and Har- graphic models. At one extreme are models that have
baugh, 1996 refer to as Asedimentary process simula- no sediment erosion or transportation, only deposi-
torsB .. For example, in coastal and shallow marine tion. These models tended to be early one- and
systems, waves, wave-induced currents, tidal cur- two-dimensional models of carbonate platform depo-
rents and storm-induced Ži.e. ‘event’. waves and sition. Early siliciclastic sedimentary process simula-
currents lead to ever-changing patterns of sediment tors employed the diffusion equation to represent
erosion, transportation, and accumulation. Modeling sediment dispersal Žsee discussion in Wendebourg
such events entails handling physical laws and em- and Harbaugh, 1996, p. 4.. Another way to treat
pirically derived relationships Žc.f. Slingerland et al., sediment dispersal is to use linear approximations of
1994.. These physical laws and empirical relation- more complicated sediment dispersal. Such models
ships are generally described by nonlinear, complex as the two-dimensional code of Bosence and Waltham
sets of partial differential equations ŽSlingerland, Ž1990. and Bosence et al. Ž1994., the ADr. Sedi-
1986; Wendebourg and Harbaugh, 1996; Li and mentB code of Dunn Ž1991., the two-dimensional
Amos, 1995; collected papers in Acinas and Brebbia, alluvial architecture code of Bridge and Leeder
1997; Harff et al., 1999.. Moreover, these equations Ž1979., the three-dimensional update of that code by
must be coupled during solution. Furthermore, some Mackey and Bridge Ž1995., and the ACYCOPATH
parameters that cannot be easily formalized, such as 2DB code of Demicco Ž1998. all use such an ap-
antecedent topography and changing boundary con- proach. Finally, there exist a number of sophisti-
ditions need to be taken into account. When we cated, sedimentary process simulators that employ
consider carbonate depositional systems, we are also numerical solutions of the fundamental, dynamical,
confronted by the in situ formation of the sediments physical equations coupled with empirical and semi-
themselves both as reefs Žc.f. Smith and Kinsey, empirical equations. Such integrated flow and sedi-
1976; Buddemeier and Smith, 1988; Bosscher and ment transport models involve calculations of bed
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 137

shear stress along the bottom of a circulation model. include descriptions of rock types, interpretations of
The bed shear stress from that model would then be depositional settings, and their positions within Asys-
used as input to solve the temporal and spatial terms tem tractsB Žc.f. Vail et al., 1977; Wilgus et al.,
in bedload and suspended load sediment transport 1989; Schlager, 1992, in press; Loucks and Sarg,
equations. Examples of such models are the 1993; Emery and Myers, 1996..
STRATAFORM family of models ŽSyvitski and Al- Fuzzy logic allows us to formalize and treat such
cott, 1995; Nittrourer and Kravitz, 1996., the SED- information in a rigorous, mathematical way. It also
SIM models of Wendebourg and Harbaugh Ž1996, allows qualitative information to be treated in a more
see p. 11. Žsee also Tetzlaff and Harbaugh, 1989., natural, continuous fashion. The purpose of this pa-
and the river avulsion model of Slingerland and per is to present the results of three simulations
Smith Ž1998.. Although these models have been where we have used fuzzy logic to model sediment
successful, they can be computationally quite com- dispersal in three-dimensional stratigraphic models
plex. wherein sea level changes, subsidence, isostasy, and
We present here a different approach in modeling crustal flexure are modeled using conventional math-
sediment erosion, sediment transport, and sediment ematical representations ŽTurcotte and Schubert,
deposition, namely using qualitatively and quantita- 1982; Angevine et al., 1990; Slingerland et al., 1994..
tively defined observational rules as the basis of Our preliminary results along with the model FLU-
fuzzy logic models of sediment production, erosion, VSIM ŽEdington et al., 1998. and the modeling of
transportation and deposition. Recently, Nordlund the Smackover Formation described by Parcell et al.
Ž1996. and Fang Ž1997. suggested that, fuzzy logic Ž1998. suggest that fuzzy logic may be a powerful
could be used to overcome some of the difficulties and computationally efficient alternative technique to
inherent in modeling sediment dispersion. There is a numerical modeling for the basis of a sedimentary
wealth of observational data on flow and sediment process simulator. It has the distinct advantage in
transport in the coastal zone, in river systems, on that models based on fuzzy logic are robust, easily
carbonate platforms, and in closed basin settings. adaptable, computationally efficient, and can be eas-
Nordlund Ž1996, p. 286. refers to this as AsoftB or ily altered internally allowing many different combi-
qualitative information on sedimentary dynamics. nations of input parameters to be run in sensitivity
However, we also have a fair amount of quantitative analyses in a quick and efficient way.
information on some sedimentary processes Že.g. the
volumetric production of lime sediment per year on
different areas on carbonate platforms—see Broecker 2. Basic principles of fuzzy logic
and Takahashi, 1966; Morse et al., 1984.. Examples
of qualitative information would be Abeach sands 2.1. Fuzzy sets
tend to be well sorted and are coarser than offshore
sandsB, or Acarbonate sediment is produced in an Since the late 1980s, many surprisingly successful
offshore carbonate ‘factory’ and is transported on- applications of fuzzy logic have been developed in
shore and deposited in tidal flatsB. Such statements virtually all areas of engineering ŽDubois et al.,
carry information, but are not easily quantified. In- 1997; Yager and Filev, 1994; McNeill and Freiberger,
deed, these types of qualitative statements are com- 1993; Passino and Yurkovich, 1998; Sebastian and
monly the exact kind of information that is obtained Antonsson, 1996. and computer science ŽPetry, 1996;
by studies of ancient sedimentary sequences. More- Kandel and Langholz, 1998.. In addition, fuzzy logic
over, with the development of Aseismic stratigraphyB has been successfully applied in the areas of robotic
and Asequence stratigraphyB, applied and academic control, decision making, optimization, management,
geologists have both moved into an arena where business, and operations research ŽChen and Hwang,
there is a commonly a complex blend of AhardB and 1992; Kacprzyk and Fedrizzi, 1990; Cox, 1995; Del-
AsoftB information. Hard data might include seismic gado, 1994; Ostaszewski, 1993; Sakawa, 1993.. Al-
Žor outcrop-scale. geometric patterns of reflectors or though applications of fuzzy logic in sciences are
bedding geometries, whereas soft information would less developed than those in engineering and busi-
138 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

ness, the utility of fuzzy logic has already been by the membership function of a fuzzy set is inter-
demonstrated in chemistry ŽRouvray, 1996., quan- preted as the degree of membership of the individual
tum physics ŽPykacz, 1992., biology ŽVon Sternberg, in the standard fuzzy set. Other nonstandard types of
1998., ecology ŽSalski, 1992; Meesters et al., 1998., fuzzy sets have been introduced in the literature ŽKlir
psychology ŽZetenyi, 1988; Smithson, 1987., and and Yuan, 1995.. In this paper, however, we con-
economics ŽBillot, 1992.. According to our prelimi- sider only standard fuzzy sets. The adjective Astan-
nary results, and those of Edinton et al. Ž1998. and dardB may thus be omitted. Observe that, contrary to
Parcell et al. Ž1998., the role of fuzzy logic in the symbolic role of numbers 1 and 0 in characteris-
geological stratigraphic modeling is quite profound. tic functions of crisp sets, numbers assigned to indi-
Fuzzy logic is based on the concept of fuzzy sets viduals by membership functions of standard fuzzy
in the sense that the various components of fuzzy sets have clearly a numerical significance. This sig-
propositions Žpredicates, truth qualifiers, probability nificance is preserved when crisp sets are viewed
qualifiers, quantifiers, etc.. are represented by appro- Žfrom the standpoint of fuzzy set theory. as special
priate fuzzy sets ŽZadeh, 1965; Klir and Yuan, 1995; fuzzy sets.
Bardossy and Duckstein, 1995.. In conventional, Geology has many examples of Apigeon holeB
crisp sets, an individual is either included in a given classifications built around crisp sets. For example,
set or not included in it. This distinction is often as Nordlund Ž1996, p. 689. points out, a grain with
described by a characteristic function. The value of diameter of 1.9999 mm would be classified as coarse
either 1 or 0 is assigned by this function to each sand, whereas a grain with diameter of 2.0001 mm
individual of concern, thereby discriminating be- would be classified as gravel. If fuzzy sets are used
tween individuals that either are members of the set instead of crisp sets, than the artificial classification
Žthe assigned value is 1. or are not members of the boundaries are replaced by gradational boundaries
set Žthe assigned value is 0.. In standard fuzzy sets, and the two grains described would share member-
the characteristic function is generalized by allowing ship in both sets, described by the linguistic terms
us to assign not only 0 or 1 to each individual of Acoarse sandB and AgravelB. With increasing diame-
concern, but also any value between 0 and 1. This ter of grains, the membership in AgravelB will in-
generalized characteristic function is called a mem- crease and the membership in Acoarse sandB will
bership function. The value assigned to an individual decrease in some way that depends on the applica-

Fig. 1. Example of using fuzzy sets describe tidal range Žthe 0 point is Amean sea levelB .. The shapes of the membership functions are
arbitrary, and can be adjusted to better fine tune the modeling. Such a Afuzzy setB representation better captures natural variations due to
periodic tidal curve changes Žsuch as the ebb–neap–ebb cycle., and non-periodic, random variations such as storm flooding.
R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 139
140 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 141

tion context. The basic idea is that the membership Using fuzzy sets, there can be a complete gradation
in a fuzzy set is not a matter of affirmation or denial, between all these depth ranges. Each membership
as it is in a classical set, but a matter of degree. function in these figures is represented by a curve
Many of the variables contributing to sediment that indicates the assignment of a membership de-
production Žin carbonate systems. and sediment dis- gree in a fuzzy set to each variable within the
persal Žin all systems. naturally lend themselves to domain of the variable involved Že.g. the variable
fuzzy set descriptions. An example would be water Atidal rangeB .. The membership degree may also be
depth. In tidal flat systems we have definitions of interpreted as the degree of compatibility of each
supratidal, intertidal and subtidal based on mean tidal value of the variable with the concept represented by
ranges, with the intertidal being further subdivided the fuzzy set Že.g. subtidal, low-intertidal, etc.. curves
into high-tidal, mid-tidal, and low-tidal areas ŽRead- of the membership functions can be simple triangles,
ing and Collinson, 1996, p. 213.. However, on trapezoids, bell-shaped curves, or have more compli-
modern tidal flats, these boundaries are constantly cated shape.
changing due to periodic variations in over a dozen So far, we have described what might be consid-
principle tidal harmonic components Žc.f. Table 11.1 ered the ‘input’ variables of a sediment production
in Knauss, 1978.. More importantly, it is commonly or dispersal system. However, not only ‘input’ vari-
flooding due to anomalous Awind tidesB and Abaro- ables but the dependent or ‘output’ variables can also
metric tidesB ŽKnauss, 1978. that is important for be described by fuzzy sets. An obvious example is
deposition in tidal flats. A possible fuzzy set descrip- grain size or sediment type. The thickness of sedi-
tion of tidal range is given in Fig. 1 and better ments eroded and deposited can also be described by
captures the essence of the gradations between loca- fuzzy sets. In this context, terms such as Aproduce
tions on a tidal flat. Similarly, 1 to 2 m below sea someB, Aerode a littleB, or Adeposit a lotB have
level is certainly shallow, but where does a carbonate meaning. There are two distinct advantages in this
platform Žor a siliciclastic shelf. become AdeepB or approach. First, fuzzy sets describe systems in ‘natu-
AopenB Žc.f. Nordlund, 1996, his Fig. 2, p. 701.? ral language’. More importantly, the shapes of the

Fig. 2. Application of fuzzy logic to a two-dimensional simulation of reef growth. Ža. Measured growth rates of the main Caribbean
reef-building coral M. annularis Žfrom Bosscher and Schlager, 1992, their Fig. 1.. The two blue lines are solutions of G s
Gm tanhŽ I0 eyk z 4rIk . where z is water depth, G is growth rate at a given depth Ž z ., Gm is maximum growth rate Ž G at z s 0., I0 is surface
light intensity, Ik is saturation light intensity, and k is the extinction coefficient given in the Beer–Lambert law I z s I0 eyk z. The red line is
a fuzzy logic system approach to this data set. Žb. The upper plot shows two membership functions for the fuzzy sets AshallowB and AdeepB
for the input variable depth over the domain range 0 to 50 m. The lower plot shows two membership functions for the fuzzy sets AfastB and
AslowB for the variable growth rate over the domain 0 to 15 mmryear. Žc. A standard ŽAMamdaniB . interpretation of the if–then rules ŽAif
the water is shallow, then coral growth rate is fast; if the water is deep, then the coral growth rate is slowB . cases, a water depth of 2 m
Župper plot. and a water depth of 35 m Žlower plot.. The input variable is evaluated for each water depth and a truth Õalue s degree of
membership of the input variable in each of the potential input sets ŽAshallowB and AdeepB . is calculated. These truth values truncate the
membership functions of the appropriate output variable. For each water depth, the truncated membership functions of the output variable
are summed, and the centroid of the appropriate curve is taken as the AdefuzzifiedB output value. Žd. Comparison of two-dimensional
numerical models of the geologic history of coral reefs growing on the Atlantic shelf-slope break of Belize. Right panel is a step wise
solution of the differential equation d hŽ t .rd t s Gm tanhŽ I0 expŽyk w h 0 q hŽ t .x y w s0 q sŽ t .x4rIk . developed by Bosscher and Schlager
Ž1992.. Here d hŽ t .rdŽ t . is the change in the height of the coral surface with time, h 0 is the initial height of the surface at the start of a time
step, hŽ t . is the growth increment in that time step, s0 is the initial sea level position for a time step and sŽ t . is the variation in sea level for
that time step. The forward model solution for coral reef growth assumes an initial starting slope, initial values of Gm Žmaximum growth
rate., I0 Žinitial surface light intensity., k Žextinction coefficient., and the variable sea level curve of the past 80,000 years Žshown in the
inset in the right panel.. Left panel models reef growth based on the same sea level curve, same starting slope, and same initial value of Gm
are the same, but with the fuzzy inference system described above replacing the differential equation for coral growth production. Že. Curve
fit to the coral reef growth data using a Sugeno interpretation of the rules Aif the water is shallow, then coral growth rate is fast; if the water
is deep, then the coral growth rate is slowB. In this case, the appropriate membership functions have been selected by the computer to
provide a Abest fitB to the data.
142 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

membership functions can easily be changed by small intensity, and k is the extinction coefficient given in
increments, thereby allowing rapid ‘sensitivity analy- the Beer–Lambert law:
sis’ of the effects of changing the boundaries of the
fuzzy sets. In robot control algorithms, where fuzzy Iz s I0 eyk z .
logic was first developed, systems could self-adjust In Fig. 2a, the two blue dot–dash curves are fit using
the shapes of the membership functions and set different values of the parameters listed above Žsee
boundaries, until the required task was flawlessly Bosscher and Schlager, 1992 for details..
performed. This aspect of fuzzy systems, commonly A fuzzy logic system approach to this data set
facilitated via the learning capabilities of appropriate would use natural language to capture the essence of
neural networks ŽKosko, 1992; Klir and Yuan, 1995; the data: AIf the water is shallow, then coral growth
Nauck and Klawonn, 1997. or by genetic algorithms rate is fast. If the water is deep, then the coral
ŽSanchez et al., 1998. is one of their great advan- growth rate is slowB. The input parameter here is
tages to numerical solution approaches. water depth, whereas the output or dependent vari-
able is coral growth rate. Both of these variables can
2.2. Fuzzy logic systems be represented by fuzzy sets. The upper plot in Fig.
2b shows two possible membership functions for the
A fuzzy logic system is a set of fuzzy inference fuzzy sets AshallowB and AdeepB for the input vari-
rules Žif–then rules.. These are conditional fuzzy able depth over the domain range 0 to 50 m. The
propositions that describe dependence of one or more lower plot in Fig. 2b shows two possible member-
output-variable fuzzy sets to one or more input-vari- ship functions for the fuzzy sets AfastB and AslowB
able fuzzy sets. Logical connections employed in for the variable growth rate over the domain 0 to 15
fuzzy inference rules are adjustable in a similar way mmryear. The standard ŽAMamdaniB . interpretation
as the membership functions, which offers another of the if then rules ŽAif the water is shallow, then
way to improve the performance of the model. Be- coral growth rate is fast; if the water is deep, then the
low we offer two examples of fuzzy logic systems: coral growth rate is slowB . is shown in Fig. 2c for
Ž1. their use in a simple, two-dimensional model of two cases, a water depth of 2 m Župper plot. and a
coral reef development; and Ž2. their use in predict- water depth of 35 m Žlower plot.. The input variable
ing where carbonate precipitation will occur on a is evaluated for each water depth and a truth Õalue
platform. s degree of membership of the input variable in
each of the potential input sets ŽAshallowB and
AdeepB . is calculated. These truth values truncate the
2.2.1. Coral reef growth
membership functions of the appropriate output vari-
Coral animals are capable of rapid fixation of
able. For each water depth, the truncated member-
limestone from sea water because of symbiotic pho-
ship functions of the output variable are summed,
tosynthetic algae within their tissues. Thus, the lime-
and the centroid of the appropriate curve is taken as
stone production of these animals is, in some way,
the AdefuzzifiedB output value Žsee captions of Fig.
related to light penetration into the shallow ocean.
2c for further details.. The solid red curve in Fig. 2a
Fig. 2a shows data on growth rates of the main
shows this fuzzy inference system evaluated over the
Caribbean reef-building coral Montastrea annularis
Žfrom Bosscher and Schlager, 1992, their Fig. 1, p. depth 0 to 50 m.
Bosscher and Schlager Ž1992. developed a two-
503.. Bosscher and Schlager Ž1992., following
dimensional numerical model of the geologic history
Chalker Ž1981., fit the following equation to this
of coral reefs growing on the Atlantic shelf-slope
data:
break of Belize by step-wise solution of the follow-
G s Gm tanh Ž I0 eyk zrIk . . ing differential equation:

Here z is water depth, G is growth rate at a given d h Ž t . rdt s Gm tanh Ž I0 exp  yk h 0 q h Ž t .

depth Ž z ., Gm is maximum growth rate Ž G at z s 0.,
I0 is surface light intensity, Ik is saturation light y s0 q s Ž t . 4 rIk . .
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 143

Here d hŽ t .rdŽ t . is the change in the height of the It is important to note that fuzzy logic systems are
coral surface with time, h 0 is the initial height of the very versatile and, indeed, can be more versatile than
surface at the start of a time step, hŽ t . is the growth deterministic equations. Fig. 2e shows a curve fit to
increment in that time step, s0 is the initial sea level the coral growth data using a different ŽASugenoB .
position for a time step and sŽ t . is the variation in interpretation of the rules Aif the water is shallow,
sea level for that time step. The forward model then coral growth rate is fast; if the water is deep,
simulation for coral reef growth assumes an initial then the coral growth rate is slowB. In this case, the
starting slope, initial values of Gm Žmaximum growth appropriate membership functions have been se-
rate., I0 Žinitial surface light intensity., k Žextinction lected by the computer to provide a Abest fitB to the
coefficient., and the variable sea level curve of data. So far we have been using ordinary fuzzy sets
80,000 years shown in inset in Fig. 2d. Fig. 2d wherein for a given input value there is one output
compares the simulations of reef growth based on value. In general, although we will not use them in
the above differential equation Žright panel. with a this paper, we can generalize ordinary fuzzy sets into
simulation wherein the sea level curve, starting slope, interÕal-Õalued fuzzy sets ŽKlir and Yuan, 1995.,
and initial value of Gm are the same, but the fuzzy where the membership function does not assign to
inference system described above replaces the differ- each element of the universal set one real number,
ential equation for coral growth production Žleft but a closed interval of real numbers between the
panel.. identified upper and lower bound. Clearly, this ap-

Fig. 3. Application of fuzzy logic to carbonate production. The four Aif –thenB rules above are evaluated for a point on the platform with a
water depth of 1.5 m Žleft column. at a distance of 100 km from the platform margin Žmiddle column.. Sediment production Žthe right
column. is the output variable and ranges from 0 to 100 mgrcm2 ryear Žbased on AexpertB information, Broecker and Takahashi, 1966, their
Fig. 10, p. 1585.. The domain of the two Depth membership functions is 0 to 50 m depth. The two membership functions are: Amaximum
production depthB Žrows 1–3. and Anot maximum production depthB Žrow 4. which is 1-maximum production depth. The domain of the
Distance membership functions in 0 to 150 km. The three membership functions are, Aclose to platform edgeB Žrow 1., Afar from platform
edgeB Žrow 2., and Ainterior of platformB Žrow 3.. The three output Production functions are Aproduce lotsB Žrow 1., Aproduce someB Žrow
2. and Aproduce littleB Žrows 3 and 4.. The value of 1.5 m depth has a Atruth valueB of approximately 0.6 in the membership function
Amaximum production depthB and 0.4 in the membership function Anot maximum production depthB. The value of 100 km from the margin
has a truth value of 0 in the membership function Aclose to platform edgeB, 0.3 for Afar from platform edgeB, and 0.75 for Ainterior of
platformB. In this standard application of fuzzy inference, the conjunction AandB implies that the membership function of the output variable
is truncated at the lowest truth value of the two input variables. So, in row 2, the membership function Aproduce someB is truncated at 0.4.
To evaluate the four rules, all of the truncated production membership functions are added. This produces the last plot in column three.
Finally, the centroid of this function is taken. For 1.5 m at 100 km from the shelf edge, this process yields are final value of 39.4
mgrcm2ryear carbonate production. This value is in excellent agreement with the range of the values given in Broecker and Takahashi
Ž1966..
144 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 145

Fig. 4. Ža. Isometric, Abird’s eye viewB of the Death Valley model, view toward north. Model is 65 cells long Žeach 1 km in length. by 30
cells wide Žeach 0.5 km in width.. Maximum subsidence is at block 32-8 along the eastern margin of the basin. Color indicates sediment
type. The bright red on the basin floor represents the deposits of a saline pan, whereas the colors around the margins of the basin represent
alluvial fan gravels Žred. to mud flat Ždark blue.. Žb. Synthetic stratigraphic sections from the model are across Žtop. and along Žbottom. the
strike of the basin. Blue tones continuous with alluvial sediments are mudflats, whereas blue tones that lap up on the fans are freshwater
lake muds. Note change in scale between sections. See text for details. Žc. Examples of membership functions used in the Death Valley
simulation. The two input variables are Temperature Žtop panel. and Rainfall Žmiddle panel. both of which are described by two simple,
Gaussian membership functions. The output variable Basin-Floor-Sediment type comprises four triangular membership functions.
146 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

proach would be warranted by the spread in the Examples of other sediment dispersal mechanisms
initial data on coral growth rates versus depth in Fig. that lend themselves to fuzzy sets are degree of
2a. windward versus degree of leeward, AprotectedB ver-
sus AopenB shores, even variables such as member-
2.2.2. Carbonate production on shallow platforms ship in long-shore circulation cells can be modeled
Another example of a fuzzy logic system is pro- using fuzzy sets. Thus, virtually any sediment pro-
duction of sediment on a carbonate platform. In duction and dispersal system that has been described
two-dimensional models, carbonate sediment produc- or quantified can be adapted for a sediment disper-
tion is commonly modeled as being linearly depth- sion model using fuzzy logic. The next question to
dependent as in the preceding example. However, evaluate is Ado fuzzy logic models have the potential
the pioneering work of Broecker and Takahashi to reproduce realistic sedimentation systems?B We
Ž1966. showed that sediment production on shallow think that they do, and hence their value to three-di-
carbonate platforms is not only dependent on shal- mensional stratigraphic models. Below we describe
low depths, but on the residence time of the water, two simulations of modern sedimentary systems
which, practically speaking, translates into distance ŽDeath Valley and western Andros Island and the
away from the nearest shelf margin Žsee Fig. 3, p. adjacent Great Bahama Banks. and one hypothetical
1579 and Fig. 10, p. 1585 of Broecker and Taka- deltaic system. It is important to stress that these are
hashi, 1966.. Thus, for a carbonate platform, dis- preliminary models, but we hope that they will
tance from margin Ža fuzzy set. and water depth demonstrate that fuzzy logic can be used for the
Žalso a fuzzy set. would be the input variables that sedimentary process simulator portion of a basin-fill-
control sediment production ŽFig. 3.. In order to ing stratigraphic model.
properly model carbonate production on a platform,
these two fuzzy sets would be linked in a fuzzy
system to an output variable, in this case amount of 3. Examples of stratigraphic models that employ
sediment produced, by the ArulesB of the system. fuzzy logic to model sediment production and
Again, these rules are almost natural language and dispersion
are intuitively easy to grasp. The rules here are:
Our modeling produces: Ž1. an isometric Abird’s
eye viewB of the topography and sediment type of
1. Aif close to platform edge and at maximum
the basin surface at each time step; Ž2. synthetic
production depth, then produce lots;B
cross-sections through the final thickness of deposits;
2. Aif far from platform edge and at maximum
and Ž3. synthetic stratigraphic columns for predeter-
production depth, then produce some;B
mined locations in the simulation. The isometric,
3. Aif interior of platform and maximum produc-
Abird’s eye viewsB of the sediment surface through
tion depth, then produce littleB;
time can be assembled into animated video clips
4. Aif not at maximum production depth, then
ŽDemicco and Hardie, 1998.. The video clips are
produce little.B
available on CD ROM from R.V.D. Here we present
thumbnail views of the basin topography and sedi-
According to the standard interpretation of these
ment distribution for selected time steps. It is impor-
rules, the input variables are evaluated for each cell
tant to note that the simulated stratigraphies com-
in the model Žsee Fig. 3. and a truth Õalues degree
prise rock Žor sediment. type, i.e. facies are not
of membership of the input variable in each of the
simply depth ranges of the deposits, but are the
potential input sets is calculated. These truth values
sediment Žor rock. type deposited at that site for that
truncate the membership functions of the appropriate
time step.
output variable. For each cell in the model, the
truncated membership functions of the output vari- 3.1. Death Valley, CA
able are summed, and the centroid of the appropriate
curve is taken as the AdefuzzifiedB output value Žsee The Death Valley model simulates 191,000 years
Fig. 3 for further details.. of sedimentation. The model is 15 km across and 65
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 147

km long and is represented by 1950 cells each of yellow and orange. Playa mud flats develop in the
1 = 0.5 km in size ŽFig. 4a.. Subsidence is y0.2 floor of the basin when the chemical or lacustrine
mrka along the edges of the model and increases to sediments are minimal. Fig. 5 is a direct comparison
1 mrka along the steep Žeastern. margin of the basin between the sedimentary record of cores DVSC-1
halfway down the axis of the basin. In this prelimi- and our simulation. It is important to note that we
nary model, the current topography is a starting wrote the rules for this system based on the AexpertB
point, the locations of alluvial fans around the basin initial description of the system that is recorded and
is held constant, and there are only about 1r2 of the measured in the core of these deposits. The sedi-
number of current alluvial fans around the edges of ments produced in this simple untuned fuzzy logic
the model. Synthetic cross-sections across the width, model are remarkably like the Aground truthB recov-
and along the axis of the deposit are shown in Fig. ered in the core. With further variations in the shapes
4b. In this model, both alluvial fan input along the of the membership functions and the boundaries of
sides of the basin and sediment type and deposition the fuzzy input and output sets Žperforming a sensi-
rate on the valley floor are determined by fuzzy tivity analysis. in the modeling, a much better match
inference systems. The input variables for the flank- could be obtained. The point here is that a very
ing alluvial fans are slope and distance from a simple fuzzy logic system has been able to reproduce
number of discrete alluvial fan heads along the edges chemical and detrital sedimentation within this well-
of the model. The results can be seen in the cross- know closed basin system.
valley synthetic section where the short steep fans on
the eastern side of the basin comprise coarser gravels
and contrast to the long, lower gradient fans on the 3.2. Western Andros Island, Bahama Bank
western side of the basin comprised of finer sedi-
ment. The alluvial input into the basin ultimately The tidal flats found on the western side of An-
leads to the deposition of playa muds in the floor of dros Island have developed over the last 10,000
the basin. The basin floor sediments are calibrated years as sea levels rose and flooded the platform. A
using the results of a deep sediment core taken long-standing problem has been that the tidal flats of
through the central portion of the basin ŽLi et al., northwestern Andros Island show an erosive mor-
1996; Roberts and Spencer, 1995.. The fuzzy logic phology, whereas the tidal flats on the southern
system producing the basin floor sediments is based portion of Andros Island have apparently continued
on an input temperature and rainfall signal deter- to prograde seaward ŽHardie, 1977; Gebelein, 1974;
mined from the core by Lowenstein Žpers. comm.. Gebelein et al., 1980.. We have developed a fuzzy
The ArulesB of the basin floor system are: logic model of sedimentation on the western side of
the Great Bahama Bank that assumes a simple start-
1. Aif rainfall is high and temperature is low then ing geometry but has complicated sediment produc-
basin floor sediment is freshwater lakeB; tion, erosion, and deposition elements all determined
2. Aif rainfall is low and temperature is high then by fuzzy logic systems. Indeed, this model has been
basin floor sediment is playa mud flatB; able reproduce the asymmetry of tidal flat prograda-
3. Aif rainfall is high and temperature is high then tion ŽFig. 6.. The model simplifies the geometry of
basin floor sediment is salt panB; the platform into a cellular mesh 300 cells long by
4. Aif rainfall is low and temperature is low then 150 cells wide Žeach cell is 1 km2 . and uses 101
basin floor sediment is saline lakeB. time steps each 100 years long. No subsidence is
input in the model, initial topography is a simplifica-
Fig. 4c shows the input and output membership tion of the geometry described in the references
functions associated with these rules. The sedimenta- above and is in Bathurst Ž1971.. Sea level rise was
tion rates also follow a similar set of fuzzy rules. In digitized from data from Ginsburg Žpers. comm..
the synthetic cross-sections, basin floor sediment is with rapid initial rise till 4500 years B.P. when sea
color-coded: deep freshwater lake and playa mud level was approximately 2.0 m below present. There
flats are blue, salt pan is red and saline lake is shades are three sets of fuzzy logic rules: Ž1. sediment
148 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

Fig. 5. Direct comparison between sedimentary record of Death Valley salt core ŽA. ŽLowenstein, pers. comm.. and synthetic column from
fuzzy inference model described in text Žcenter panel.. Although this model is AuntunedB there is general agreement between the Aground
truthB of the core and the synthetic stratigraphy produced by the model. Lower panel is the temperature Žred. and rainfall Žblue. records
inferred from the core ŽLowenstein, pers. comm., and see references in text.. These data were used as the input for the fuzzy inference
system that produced basin floor sediment type and thickness Žsee Fig. 4c..

production; Ž2. sediment erosion; and Ž3. sediment from shelf edge is easily calculated. Moreover, the
deposition. These rules are given in Table 1. erosion and deposition terms AopenB, AsemiopenB
The sediment production, erosion and deposition and ArestrictedB are a quantified version of the sur-
patterns in this model are based on AexpertB knowl- face distribution of pine pollen ŽFig. 6 of Traverse
edge of the system. For example, the sediment pro- and Ginsburg, 1966, p. 433, and see this paper for a
duction functions are a slightly more complicated discussion of using the pollen as a AproxyB for
version of the system discussed in Fig. 3. Distance circulation.. In addition, and significantly, we have
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 149

Fig. 6. Isometric views of the sediment surface topography and sediment type of model of western Andros Island and adjacent Great
Bahama Bank, view is to the southeast. The simulation steps are at 4500 B.P. Žtop., 3500 B.P. Žmiddle., and today Žbottom.. Note how the
tidal flats continue to prograde in the south and east of the island Žlarger arrows., whereas shorelines remain in the same position in the
northern portion of the island Žsmaller arrows.. Total differential progradation here is approximately 5 km, slightly less than on Andros
Island.

built a sediment budget into this model. The model show the topography and the sediment type. The
keeps track of the amount of sediment eroded in the color on the sediment surface reflects sediment type
offshore. After tidal flats are built, offshore sediment with reds as packstonergrainstones, oranges and yel-
is deposited according to a windward leeward lows as wackestones, light blues as tidal flat mud-
scheme. Fig. 6 shows three AsnapshotsB of the sedi- stones, and dark blues as freshwater marsh deposits.
ment surface at 4500 year B.P., 3500 year B.P., and Note that the tidal flats in the northern end of the
today. These simulation steps are during a generally platform develop in one place and erode back Žstee-
constant - 1 mmryear sea level rise. These views pen up., whereas the tidal flats to the south continue
150 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

Table 1
Fuzzy inference rules for Andros Island simulation
Ž1. If Ždistance is edge. and Ždepth is shallow., then Žproduction is little.
Ž2. If Ždistance is edge. and Ždepth is middepth., then Žproduction is lots.
Ž3. If Ždistance is edge. and Ždepth is deep., then Žproduction is some.
Ž4. If Ždistance is midplatform. and Ždepth is middepth., then Žproduction is some.
Ž5. If Ždistance is midplatform. and Ždepth is not middepth., then Žproduction is little.
Ž6. If Ždistance is interior. and Ždepth is shallow., then Žproduction is very little.
Ž7. If Ždistance is interior. and Ždepth is middepth., then Žproduction is little.
Ž8. If Žexposure is open. and Ždepth is shallow., then Žerosion is some.
Ž9. If Žexposure is open. and Ždepth is not shallow., then Žerosion is lots.
Ž10. If Žexposure is semiopen. and Ždepth is middepth., then Žerosion is some.
Ž11. If Žexposure is semiopen. and Ždepth is not middepth., then Žerosion is little.
Ž12. If Žexposure is protected. and Ždepth is middepth., then Žerosion is little.
Ž13. If Žexposure is protected. and Ždepth is not middepth., then Žerosion is very little.
Ž14. If Žshoredistance is nearshore – neg., then Ždeposit is lots. . . . then Žsedtype is mudstone.
Ž15. If Žshoredistance is nearshore – pos. and Žexposure is protected., then Ždeposit is some. . . . then Žsedtype is wacke.
Ž16. If Žshoredistance is nearshore – pos. and Žexposure is semiopen., then Ždeposit is little. . . . then Žsedtype is pack.
Ž17. If Žshoredistance is offshore. and Žexposure is open., then Ždeposit is very little. . . . then Žsedtype is wacke.
Ž18. If Žshoredistance is offshore. and Žexposure is semiopen., then Ždeposit is little. . . . then Žsedtype is pack.
Ž19. If Žshoredistance is offshore. and Žexposure is open., then Ždeposit is some. . . . then Žsedtype is grain.

to prograde. This is simply accomplished by prefer- lateral differences in the tidal flat deposits. Finally,
ential deposition on the downdrift end of the Andros there is excellent agreement between the final dispo-
System. Fig. 7 is the cross-section through the de- sition of offshore sediments and the sediment map of
posit. Note the complicated internal geometry and Enos Ž1974.. Again, it is important to note that we

Fig. 7. Synthetic cross-sections through the Andros Island simulation in the center of the island, and 25 km to the south Žbackground. and
north Žforeground.. Colors here denote sediment type with deep blue as freshwater marsh deposits, light blue as tidal flat muds and
wackestones, and orange through red as subtidal packstones to grainstones. Note how the northerly deposits have built over subtidal
deposits, whereas the central and southerly tidal flat wedges are thicker.
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 151

used AexpertB information from cores, maps, and mentologists to construct the fuzzy logic systems
sedimentary observations, and the distribution of that model sediment dispersal in these cases. We
pollen ŽTraverse and Ginsburg, 1966. to assemble have tuned them to reproduce the deposition known
the fuzzy logic systems that model sedimentation in to haÕe occurred in these two examples. We suggest
this complex simulation. This model system has been that these preliminary examples hold the promise
able to reproduce in considerable detail the surface that fuzzy logic systems may in fact be good candi-
and subsurface geometries of this natural area. dates for sediment dispersal modeling. They hold the
The important point is that we used the AexpertB distinct advantage to more rigorous numerical calcu-
quantitative and qualitative knowledge of other sedi- lations in that they are more intuitive and can be

Fig. 8. Isometric view of hypothetical delta simulation at different time steps. Top view is sediment surface at the end of 20,000 years after
one sea level rise and fall. Dark blues represent floodplain muds and deeper marine muds, whereas reds denote clean sands in the river and
at the river mouth. Deltaic dispersion cone from clean sands through muddy sands to sandy muds of the deeper shelf. The middle and lower
views are from 28,000 and 30,000 years during the ensuing sea level fall. Note how the river deposits of abandoned channels sink into the
floodplain surface. Also note how the river AmeandersB across the floodplain as it seeks the lowest path to the shoreline. See text for further
details.
152 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155

more computationally efficient, yet retaining all the system begins with a simple geometry and a grid of
complexities of sediment dispersion and production cells 125 = 125 each 1 km2 ŽFig. 8.. The dark blue
that we have in the Pleistocene–Holocene record. In represent flood plain mud, whereas the simple deltaic
the simulation of the Andros Island tidal flats, we sediment cone and the river and its levee–crevasse–
have been able to incorporate a sediment budget splay system are indicated by the red through light
term. And, indeed, it is the sediment dispersal that blue hues. There are 50,000 years of sedimentation
accounts for the progradation of the tidal flats in one represented here at 200 years time steps in the video
area and their erosion in another. clip, selected frames of which are shown in Fig. 8.
The river system and its adjacent levee and
3.3. Hypothetical simple delta and riÕer flood plain crevasse–splay system are modeled by a fuzzy logic
system system as is the simple deltaic dispersive cone of
sediment using the same rules as Nordlund Ž1996..
Finally, we would like to show a hypothetical, There is also a simple sinusoidal oscillation of sea
greatly simplified river flood plain and deltaic sys- level in the model with amplitude of 10 m and
tem based on the modeling of Nordlund Ž1996.. The duration of 20,000 years. In addition, there are ran-

Fig. 9. Synthetic stratigraphic cross-sections down dip Žabove. and along the strike axis of the basin Žbelow. produced at the end of the
model. In the strike section, marine deltaic deposits Žreds through light blues. alternate with the fluvial muds and the aggrading channel
deposits. In particular, note the four avulsion events preserved in the lowest fluvial section in the strike section. The fluvial–marine tens of
meters cycles are allocyclic, dictated by the sea level driver. The other, smaller scales of cycles are driven by avulsion and kilometer-scale
shift of the mouth of the delta between time steps. See text for details.
 R.V. Demicco, G.J. Klir r Journal of Petroleum Science and Engineering 31 (2001) 135–155 153

dom Žin time. avulsions of the river upstream of the logic models hold the potential to accurately model
model. After an avulsion, the river enters the up- subsurface distribution of sedimentary facies Žnot
stream end of the model at the lowest point. The just water-depths of deposition. in terms of the na-
interplay of subsidence with floodplain aggradation tural variables of geology. As exploration moves
history and location of the prior channel belts will further into use of three-dimensional seismic data
determine this low point. From the lowest point at gathering, the utility of easy to use, flexible three-di-
the upstream end of the model, the river finds the mensional forward models is obvious. Such models
lowest set of adjacent cells to reach the shoreline. could be used to produce synthetic seismic sections.
When sea level is rising, the location of the river Moreover, the Alearning abilityB of fuzzy logic sys-
mouth simply backtracks up the river course, whereas tems coupled with neural networks offers the long-
when sea level is at a standstill or falling, the river term possibility of self-tuning sedimentary models
seeks the lowest adjacent cells in front of it until it that can match three-dimensional seismic in a nonhu-
reaches 0 elevation, i.e. sea level. Note that aban- man expert system. This method offers an alternative
doned channels sink into the floodplain and are to the statistical modeling of subsurface geology. It
buried by later flood basin deposits. In this model, is more computationally efficient and more intuitive
the subsidence is complex, with maximum subsi- for geologists than complicated models that solve
dence in the center of the model falling off to the coupled sets of differential equations.
edges of the model. Subsidence remains constant
through the simulation time.
Fig. 9 shows the synthetic cross-sections through References
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