FUZZIM: forward stratigraphic modeling made simple

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1 PERGAMON Computers & Geosciences 25 (1999) 449±456 FUZZIM: forward stratigraphic modeling made simple Ulf Nordlund * Department of Earth Sciences, Historical Geology and Paleontology, Uppsala University, Norbyv. 22, S Uppsala, Sweden Received 29 June 1998; received in revised form 17 August 1998; accepted 17 August 1998 Abstract FUZZIM is a simple but powerful 3-D forward stratigraphic modeling tool developed at Uppsala University, Sweden. It di ers from other stratigraphic modeling programs in that it uses fuzzy logic instead of conventional mathematics for controlling the distribution of sediments in space and time. This allows for incorporation of qualitative data in the modeling, and permits a high degree of exibility as regards modeling strategy and variables used. It also makes the operation of the program simple and intuitive. FUZZIM is particularly well suited for teaching and for experimental stratigraphic research work, but it can also be used for prediction in exploration. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: FUZZIM; Forward stratigraphic modeling; Fuzzy logic; Fuzzy rules; Fuzzy sets; Computer simulation 1. Introduction FUZZIM is a program for simulating large-scale marginal deposition and erosion over geologic time spans. The program was written with the purpose to investigate the possibility of using `fuzzy logic' (Zadeh, 1965) for controlling the spatial and temporal distribution of sediments in forward stratigraphic models. The fuzzy approach proved to have several advantages (Nordlund, 1996). The most important is that qualitative data can be easily incorporated into the models, and that complex nonlinear functions can be modelled without using more precision than is necessary to solve the problem. The following is a brief presentation of the program. 2. Background FUZZIM was originally written in 1994 by U. Nordlund and M. Silfversparre, Department of Earth * ulf.nordlund@pal.uu.se Sciences at Uppsala University, Sweden. The purpose was to explore the possibilities of using fuzzy logic to formalise qualitative information used in stratigraphic modelling. The need for a simple forward modelling program at the department became obvious during work involving predictive methods based on sequence stratigraphic concepts. Since the modelling programs available from commercial or academic sources were either too expensive or too limited (two-dimensional; few available sediment types), we decided to write our own program. The required general knowledge about the depositional systems, as well as the necessary programming skills, were all available at the institute. However, the knowledge available was qualitative rather than quantitative. That is, we knew how the program should work and we could express this easily in words, but we did not seem to be able to express it in strict mathematical form suitable for the computer. Eventually, the idea of using fuzzy logic came up as a possible solution. The results from a test program were promising and a few weeks later the rst version of FUZZIM was ready. Di erent versions of FUZZIM have since then been successfully used in teaching, in experimental stratigraphic work and as a predictive tool in hydrocarbon exploration /99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S (98)

2 450 U. Nordlund / Computers & Geosciences 25 (1999) 449± Fuzzy logic Fuzzy logic was rst time formally described in the 1960s by Zadeh (1965), but it was not until the 1980's that it became widely known and applied. Unlike conventional logic, fuzzy logic can handle nonrandom uncertainty (vagueness, ambiguity, imprecision) and is thus useful for quanti cation of qualitative data. It also provides e ective methods for modelling complex nonlinear functions. These features make it suitable in stratigraphic and sedimentologic modeling applications. The fuzzy system in FUZZIM acts as a process controller. Control is an area in which fuzzy methods have been particularly successful. There are numerous commercial applications of fuzzy process controllers, ranging from video cameras and washing machines to trains and industrial processes (Marks, 1994). Despite its power, fuzzy logic is simple and can be easily described without resorting to mathematical jargon. No special knowledge apart from basic mathematical logic is required. The fundamental idea is that an object can be partially, as well as completely a member of a logic set. In conventional `crisp' logic, a water depth of 1 m (the object in this situation) may, for instance, be regarded as de nitely being a member of the logic set `shallow', whereas a depth of 1000 m in a similar manner may be regarded as a member of the set `deep'. But what about a depth of 100 m? Is that shallow? Is it deep? In a sense it is both, but not completely. If we assume that a degree of 1.0 means completely belonging to a set and a degree of 0.0 means not belonging, then we could say that a depth of 100 m belongs to the set `shallow' to a degree of, say, 0.7 and to `deep' to a degree of 0.3 (Fig. 1). We can thus refer to an object as having a `degree of membership' in a fuzzy set. The degree of membership can be regarded as the truth of statements like `100 m water depth is shallow' (which according to the above would be true to a degree of 0.7). A fuzzy set is de ned by its membership function (Fig. 1). A membership function is a curve showing the truth of membership in a particular fuzzy set for di erent values of the domain variable on which the set is based. In the example of the sets `shallow' and `deep' the domain variable is water depth. Note that fuzzy sets are context-dependent. A depth of, for instance, 10 m may be shallow in the context of an ocean, but deep for a swimming pool. A fuzzy rule is a logic if±then rule using fuzzy sets instead of conventional crisp sets in the premise and conclusion. The general form of a fuzzy rule with two conditions in the premise is: `if condition 1 and condition 2 then conclusion'. A fuzzy rule may be formulated on the basis of, for instance, personal experience, empirical data or expert opinions. An example of a Fig. 1. Possible membership functions for sets `shallow' and `deep'. Using these curves, depth of 100 m would be `shallow' to degree of 0.3 and `deep' to a degree of 0.7. simple fuzzy rule potentially useful in stratigraphic simulations would be: `if depth is not too shallow and distance to river is short then accum. rate is high', where `too shallow', `short' and `high' all are fuzzy sets, and `depth', `distance' and `rate' are the measured domain variables (numeric values). Since the righthand side of the rule can only be as true as the lefthand side, and since the operator between the two conditions is `and' (conjunction), the lower truth value derived from the conditions will determine the maximum truth of the conclusion (Fig. 2A). If the measured depth is `not too shallow' to a degree of 0.6, and the measured distance is `short' to a degree of 0.3, then the conclusion set `high' cannot be more true than 0.3, i.e. it's membership function cannot exceed y = 0.3. In practice, this can be achieved by multiplying each value in the conclusion set's membership function with 0.3 (as in Fig. 2A), or by truncating the membership function at y = 0.3. The modi ed membership function is the nal result from evaluating the fuzzy rule. One rule in isolation is of little use. There must also be other, competing rules forming a fuzzy system. Suppose that we have two rules applying to the same point in a model: `if depth is not too shallow and distance to river is short then rate is high' and `if slope is high then rate is zero' (Fig. 2A and B). The rst rule simply says that, provided there is accommodation space available, we should have high accumulation rates near the river mouth, whereas the second rule prevents accumulation on steep slopes. If we assume that the rst rule is true to a degree of 0.3 and the second to a degree of 0.8 (i.e. the depositional surface has a slope that quali es as `high' to a degree of 0.8), then we will end up with two fuzzy sets, `high' and `zero', both of which are valid and true to some degree, and both of which are de ned on basis of the same domain variable, accumulation rate. However, we cannot deposit at both `high' and `zero' rate at the same time! The answer to this problem is nding a compromise. Usually, this is done by simply overlaying the member-

3 U. Nordlund / Computers & Geosciences 25 (1999) 449± Fig. 2. Process of evaluating fuzzy system consisting of two rules (see text for details). ship functions of the conclusion sets and let the resulting composite membership function represent the answer (Fig. 2C). This would, however, be a fuzzy answer in the form of a new fuzzy set. In order to be useful in a computer simulation we need a single number. This is achieved through `defuzzi cation'. A frequently used method is to compute the centroid (center of gravity) for the area below the membership function and let this represent the set. Note that the value derived through defuzzi cation is not a truth value, but a value referring to the domain variable (in our situation a rate). A popular `beginners mistake' when applying fuzzy methods is confusing truth values with domain values. A fuzzy set such as `deep' does not become deeper with increasing truth value. It is still deep, only we are more sure of it. In this discussion, we have brie y described what a fuzzy set is and how it is used in a fuzzy rule. We have also shown how to evaluate competing fuzzy rules in a fuzzy system and how to extract a nonfuzzy number as a nal answer. This should be su cient to understand the basic mechanism of the control system used in FUZZIM. Fuzzy logic is by its nature easy to understand and to work with. At the same time it provides powerful tools for modelling complex multidimensional systems. Both the sets and the rules of fuzzy systems can often be de ned directly from published data. An example of a useful source for the de nition of fuzzy sets in stratigraphic applications is the compilation of papers from the conference on sedimentary modeling held in Lawrence, Kansas, in 1989 (see Franseen et al., 1991) FUZZIM FUZZIM is a Power Macintosh program which largely conforms to Apple Graphical User Interface standards. This means that most of the windows, menus and dialog boxes behave as in any ordinary Mac application. FUZZIM was written in C/C ++. A compiled version, free of charge, is available for downloading (please contact the author for more information. (Note: a Java version is currently being prepared and should be available at the time of publication).

4 452 U. Nordlund / Computers & Geosciences 25 (1999) 449±456 FUZZIM is basically a tool for writing fuzzy control systems for controlling the spatial and temporal distribution of sediments in stratigraphic simulations. It does not contain a ready-made control system with xed strategies for deposition and erosion. The control system must instead be provided by the user. This means that the main responsibility for the quality of the models produced lies with the user rather than with the programmer. (The fuzzy systems that accompany the program are examples supplied only for pedagogic reasons). All input data necessary for running a model, including the fuzzy system, is stored in a project le. Among the parameters de ned in the project le are: the initial surface (a xyz-grid), curves for eustacy, subsidence, temperature and sediment input, and options for lobe switching, turbidites, isostatic compensation, etc. Also plotting parameters such as viewing angle, etc. are de ned in the project le. Many of the parameters in the project le can be changed interactively from within the program. In particular, the fuzzy system can, and should, be edited from within the program. A simulation can be temporarily halted at any time, changes made and the simulation restarted using the modi ed system. The results of the changes are thus immediately visible through the continuously updated graphical display, making it easy to adjust fuzzy systems and to experiment with di erent modelling strategies Domain variables All fuzzy sets in FUZZIM are de ned on the basis of numerical domain variables describing the current status of the model. Many of these are also input variables, or are derived directly from the input variables. Some of the domain variables, such as depth, are local in scope (i.e. de ned for each individual cell on the depositional surface) whereas others, such as temperature, are global. Examples of domain variables that are used in the premise of fuzzy rules are: depth (m), distance to source (m), distance to shore (m), slope (8), concavity/convexity (ratio) and temperature (8C). Domain variables used in the conclusion are: deposition rate (mm/year) and sediment type (e.g. % sand) Fuzzy sets in FUZZIM In FUZZIM, a fuzzy set consists of (1) a name, (2) a reference to a domain variable and (3) a series of points de ning the membership function. A complete description of a set in the project le may look like: shallow 2 3 0:0 0:0 0:0 1:0 300:0 0:0, where `shallow' is the name of the set, `2' is the domain reference (domain variable No. 2 is depth), `3' is the number of points in the membership function (max. ve points) and ` ' are the xy-coordinates for the three points. (This particular example is a simple pyramidal shape). Adding new fuzzy sets, or changing existing sets, is done in a dialog box (Fig. 3A) containing the data for each set together with a graphic display of the membership function. Also the membership functions for other sets de ned on the basis of the same domain variable are shown Fuzzy rules in FUZZIM The left-hand side of a fuzzy rule, i.e. the premise, begins with an `IF', and is separated from the righthand side, the conclusion, by `THEN'. The premise and conclusion contain the names of one or more fuzzy sets separated by the logic operator `AND'. In the project le this may look like: IF shallow AND near river THEN much AND coarse, where `shallow', `near river ', `much ' and `coarse' are all names of fuzzy sets de ned previously. Similar to the sets, fuzzy rules can at any time be added, removed or modi ed in a dialog box (Fig. 3B). Sets used in the rules are selected from pop-up menus The river and the shore At present, uvial deposition is not speci cally modelled in FUZZIM. It is however possible to produce a uvial equilibrium pro le using suitable fuzzy rules referring to, for instance, surface slope and distance to the shore and/or elevation. During simulation, the position of the point source (the river mouth) at the shore will vary as the morphology of the model surface changes due to deposition and erosion. Soon after starting a simulation, the river mouth starts to change position in a manner super cially similar to lobe switching. This is particularly obvious when accommodation space is constant or decreasing (falling sea level). The switching is caused by the river's attempt to follow the shortest route to the sea. Although this behaviour may appear similar to real delta lobe switching, it should not be over-interpreted. Especially not when it comes to the small scale variations, which are dependant on the temporal and spatial resolution used. On a large scale, however, the geometries are probably reasonably realistic. The `lobe switching' in FUZZIM can be switched o if so desired. The distance to the point source (the river mouth) and to the shore, are domain variables in FUZZIM, computed at each time step, for each point on the surface. For points on land, the shortest straight line is

5 U. Nordlund / Computers & Geosciences 25 (1999) 449± Fig. 3. (A) Fuzzy sets dialog box. (B) Fuzzy rules dialog box. used, whereas for points below the water surface the shortest path through water is computed. Irregularities in the coast line, and islands, are thus taken into account when computing the distances. If the whole of the modelled surface in FUZZIM should become completely covered with water or completely subaerially exposed, the simulation does not automatically stop. Instead the point source and the shoreline move out of the model, o the proximal or distal edge, respectively. Distances are then computed assuming a xed general slope (presently 18) of the surface outside of the model Tectonic subsidence, isostasy and compaction The total subsidence for each cell consists of three di erent components: (1) tectonic subsidence, (2) iso-

6 454 U. Nordlund / Computers & Geosciences 25 (1999) 449±456 static compensation and (3) compaction. Any one of these can be switched o if so desired. The tectonic subsidence is of regional extent and is implemented using a model of a tilted plane in 3-D space. This is de ned in the project le by three subsidence curves, one for each of three geographically de ned reference points. Simple Airy isostasy is used in FUZZIM. The degree of isostatic compensation is controlled by an isostatic coe cient ranging between 0 and 1, where 0 means no isostasy and 1 means maximum isostasy. In the case of maximum isostasy, the compensation of sediment replacing water is ca. 0.37, and for water or sediment replacing air 0.5. (All sediment types are assumed to have the same initial density). Compaction is controlled by a separate fuzzy system which, in the present version of FUZZIM, cannot be edited by the user. Decompaction due to unloading (erosion) is not implemented. Nor is there any account made for early cementation or diagenetic changes. The domain variables used in the compaction system are: load, grain size and current degree of relative compaction. The system assures that ne-grained sediments are compacted earlier in the burial process, and that compaction ceases when the sediments approach their minimum porosity (prede ned for di erent sediment types) Gravity mass movement Gravity mass movement, or `turbidite', modelling in FUZZIM is still at an experimental stage. The position of the turbidites in space and time is probably at least in the neighbourhood of being realistic. The geometries may sometimes be less so. The lithologies are most likely not. A turbidite is released when the slope at a certain point exceeds a critical angle (`angle of repose') de ned by the user. Enough sediment is removed around the critical cell to make all angles smaller than a userde ned subcritical angle (`the residual angle after shearing'), which is less than the critical angle. The result of this procedure is a V-shaped scar being formed around the critical point. The sediment removed is then mixed, i.e. the sediment type in the transported volume will be a homogeneous mixture of the sediments removed. (Reefs are converted to coarse carbonate sand in this process). The sediment is then deposited down-slope according to a separate fuzzy system. The critical and subcritical angle, and the maximum number of turbidites per time step can be set by the user. In brief, these parameters a ect the results in the following ways: low critical angle will start turbidite activity at an earlier stage. A subcritical angle only slightly less than the critical angle will result in smaller turbidites (smaller amounts eroded) whereas a larger di erence will result in larger turbidites. Setting the maximum number of turbidites to a small value will produce few but large turbidites, whereas a high value will produce many small ones. Modelling of a turbidite in three dimensions constitutes a major problem. The way this is handled in FUZZIM must be regarded as a rather primitive and preliminary solution. The important parameters are the distance and the gradient relative to the source point. A `too high' gradient means that sediment cannot be deposited due to the high slope and (presumably) the high velocity, whereas a `too small' gradient means that the slope is too small for sediment transport. Deposition will occur between these extremes. The large-scale geometries produced using this approach appear to be reasonable, and the timing of turbidites (being more frequent at times of regression) seems valid. The turbidites can be shut o by the user if so desired. 4. Building a fuzzy system for controlling deposition and erosion The ability to control simulation of deposition and erosion through fuzzy sets and rules means that the user is given an almost unlimited freedom to choose a strategy for deposition and erosion. This freedom means that a certain measure of self-discipline is required Ð it is easy to produce wild and strange models that have little to do with reality. When constructing a fuzzy system based on knowledge, it is usually a good idea to rst sit down and really think the problem over. This may seem a trivial piece of advice, but it is easy otherwise to start creating erroneous or less e ective systems containing unrealistic and/or redundant rules. The next step is to express explicitly in words all the relevant knowledge about the system you want to model, and to reformulate this in the form of if±then rules. Next, in the program, the rules are added to the system, one at a time and starting with the most general ones. For each rule, the required fuzzy sets are de ned prior to adding the rule. In Fig. 4, the results from successively adding a few rules to a system are shown. Initially, the system consists of only one rule Rule 1 Ð IF deep THEN deposit very little AND very ne Such a one-rule system will produce only crisp results; either it is deep and we deposit little very ne material, or it is not deep and we do nothing (since

7 U. Nordlund / Computers & Geosciences 25 (1999) 449± Fig. 4. Models produced using fuzzy systems with increasing number of rules in fuzzy system (1±5 rules; A±E). All models were run for 100 kyr, corresponding to one sea-level cycle with amplitude of 50 m, and used time step of 5000 years. there are no other rules available) (Fig. 4A). Note that it does not matter in this situation how true `deep' is Ð also low truth values will result in deposition of little very ne material. Only if we add another rule, competing with the rst, can we get areas with a continuous di erentiation of the deposition. Such a rule may look like, for instance

8 456 U. Nordlund / Computers & Geosciences 25 (1999) 449±456 Rule 2: IF shallow THEN deposit some AND intermediate This simply says that we expect a higher sedimentation rate and coarser sediments at shallower depths (Fig. 4B). In those areas which are (to some degree) both `shallow' and `deep', the two rules will compete with each other and the result will be a compromise whereas in areas in which only one rule applies, only one result is possible. Already with only two rules de ned, we can produce simple depth-facies models. A higher level of sophistication is easily reached by adding some rules referring to, for instance, water energy and distance to source. An example of such a rule is Rule 3: IF at surface AND exposed to waves THEN deposit very little AND very coarse The idea behind this rule would be that only very slow accumulation of coarse material will occur in shallow high-energy areas (cf. Fig. 4C). (`Exposed to waves' is a fuzzy set de ned on basis of the slope towards the open ocean). Another possible rule would be Rule 4: IF near source AND NOT at surface THEN deposit much AND coarse Here, we assume that the area near the river has high deposition rates, but that sediment actually will accumulate only if there is accommodation space available. Since here for the rst time we refer to a point source, our model becomes a 3-D model rather than 2- D, i.e. we get di erentiated deposition also in the strike direction (Fig. 4D). Now assume we want to add carbonate reefs to our model Rule 5: IF within photic zone AND hot AND NOT sheltered AND far from river THEN deposit very much AND reef The reasoning behind this rule should be obvious. In order to get carbonate reef growth, we need shallow depths within the photic zone and a high temperature. We furthermore need good water circulation and a low clastic input (i.e. we need to be far from the river). The result of a simulation using this new rule is shown in Fig. 4E. At times when the temperature is high and we get shallow, not sheltered areas far from the river mouth we indeed get reef growth. This last rule clearly shows the power and exibility of using fuzzy methods. The whole operation of adding reefs to our model takes a couple of minutes, and the resulting reefs are realistic both as regards geometries and timing. To add reefs to a model using conventional mathematical methods would be di cult, both for the programmer to implement and for the user to understand. In this discussion, nothing has been said about membership functions. A membership function must be de ned for each new set that is used. The reason for not treating this here is simply that it is a less critical procedure. As long as we have a general idea of what `shallow' or `near source' is, the models we produce will look similar. This makes the fuzzy approach robust and emphasises the point that high precision often is unnecessary. We should use only the precision required to solve the problem Ð using more is a waste of time. Note that the previous example is merely a simple illustration, far from complete. We have, for instance, no sedimentation or erosion on land, and we are lacking carbonate sediments associated with the reefs. 5. Conclusions The exibility of FUZZIM makes it an excellent teaching tool. A student that has a fundamental understanding of the mechanisms of deposition and erosion, and some knowledge of rates and sediment compositions for di erent environments, will in a short time be able to produce realistic models `from scratch'. A student without this knowledge will most likely not. The exibility also encourages experiments. Rules are easily added, and the results become immediately available. Although simple and easy to work with, FUZZIM is also powerful. It has successfully been applied to real problems in, for instance, the area of petroleum exploration. Fuzzy logic, or related methods for incorporating qualitative data in our analyses, will most likely become important tools for solving real geological problems in the future. To a geologist, FUZZIM may be a good way of quickly getting an idea of how fuzzy methods work, and what they can be used for. References Franseen, E.K., Watney, W.L., Kendall, C.G.St.C., Ross, W., Sedimentary modeling: computer simulations and methods for improved parameter de nition. In: Kansas Geological Survey Bulletin, vol. 233, 524 pp. Marks, R.J. (Ed.), Fuzzy Logic Technology and Applications, IEEE Technology Update Series, IEEE, New York, 575 pp. Nordlund, U., Formalising geological knowledge: with an example of stratigraphic modeling using fuzzy logic. Journal of Sedimentary Research 66, 689±698. Zadeh, L.A., Fuzzy sets. Information and Control 8, 338±353.