1Since our landscapes are very divers and complicated, soft computing is only useful for land evaluation, if it can take into account several kinds of uncertainty. The assessment frame is often uncertain itself. The selected data are of a stochastic nature and they are variable in space and time. Additionally, the local connections of different parameters sometimes change. For all these reasons, the results of landscape analysis are a range of values, rather than a single one. In order to combine solutions of the problems mentioned, a special evaluation methodology for heterogeneous land units was developed.
2The favoured way of evaluation is to view landscape as a whole. Following the holistic approach of Smuths (1926), the landscape complex is more than a sum of its parts. Therefore, it’s not the best way to assess landscape elements, factors or grid cells and put their values together at the end. Instead of this, the evaluation feature is viewed as a spatially limited landscape complex including its internal relations. The basic scale hierarchy of land units corresponds with the horizontal aspect of this holistic idea.
3Land units should be homogeneous with regard to several characteristics. Therefore it’s possible to assess them by more than just one factor and to characterise the connections between different partial complexes (e. g. water and soil). But such a homogeneity is related to statistical or abstract features only. Most measures change from one point to another, because they are defined by the locality. With regard to these local defined measures, a continuous land patch has to be heterogeneous. However, these differences are not at random. There are regularities though, in the way some values vary over space and how they are connected with others. Therefore, a method is necessary, which can tolerate the natural heterogeneity and consider the internal regularities. The following fuzzy approach seems to be suitable to do this, particularly when applied to medium scales. Many uncertain evaluating instructions already exist in literature (Bastian & Schreiber 1994, AG Boden 1994 a. o.). Their technical implementation demands a soft method like fuzzy logic.
4The project team of our Working Group on Natural Balance and Regional Characteristics is engaged in mapping mediumscaled land units for the whole German state Saxony. The desired map covers an area of more than 18,400 km², graphically presented on 55 sheets of the Topographic Map 1:50,000 of the Federal Republic of Germany (figure 1). The map units are socalled microchores (from Greek word „chora" for land and space) with an average size of about 12 km². Their rank is defined by the second aggregation stage of ecotopes. Each stage has a simple structure: a combination of several core areas with an outward transition area around and some smaller singular patches inside.
5Approximately more than 2000 microchores are to be selected at the end of mapping program. Each full map sheet shows between 25 and 60 units. Microchores are further characterised by five pages of documentation for each individual. The main features of all geocomponents (soil, geology a. o.) are characterised there by types, values or verbal descriptions. Finally, an evaluation and leitmotivs for the future utilisation of each unit will be given. We are currently developing the appropriate methodologies, and we are applying them in a selected test area.
6This area of application is the mesochore „Westlausitzer Hügel und Bergland", a land unit of the nexthigher level (figure 2). Eightythree microchores cover this area. Each microchore should be evaluated individually and as a whole, but in consideration to its internal structure and the chosen level of scale.
7Application of a fuzzy decision algorithm requires a set of uncertain or certain rules. These rules are to be derived from the used methods. A methodological simplification of procedures is useful because the computing of uncertain data is very time consuming. Therefore, algorithms are transformed into flow plans or matrices and quantities are classified. Since classifications are designed according to the applied models or algorithms, many calculations can be done beforehand and only once. Indicators and criteria get ordinal scales if possible. Results will be scaled into approximately five classes.
8In order to demonstrate the methodology, an example of the transformations and their results will be given. The landscape function of resistance against soil erosion was chosen as an example. Erosion can be very different, even in small areas. Therefore, if microchores will be evaluated, the procedure must be uncertain. Table 1 shows the evaluation matrix of potential erosion tendency, depending on soil, slope degree and precipitation characteristics. Each field represents a rule for the high rain factor class (in bold types) and the low rain factor class (in standard types). Sometimes only a single evaluation value occurs. Other combinations have uncertain results with several valid classes. In this case, an interval and a main evaluation class is given. Since flood plains have no high slope, the corresponding matrix fields are grey.
9The evaluation of actual erosion danger is much more complicated. In addition to the natural features, it regards the slope length and the land use. As an example, table 2 shows the evaluation matrix for arable land and main slope length around 500m. Different schemes exist for other slope lengths of arable land and meadow or forest land use.
Table 1 : Evaluation Scheme of potential erosion tendency
Table 2 : Evaluation Scheme of actual erosion danger (cutting for the special slope length of 500 m)
10Although the Universal Soil Loss Equation (Wischmeier & Smith 1978) and some simplified derivations (especially Marks et al. 1992) are the bases of the presented scheme, the results are not necessarily related to real amounts of soil loss. Because of the spatial heterogeneity and the several kinds of vagueness to be regarded, the results can state the relative degree of danger, the requirement for changes, or the need for protection. The medium scale is particularly suitable for regional and state planning. In order to make the required decisions, at this level of administration qualitative judgements are especially needed.
11UNIXarc/info is the applied GIS software. In this vectorbased system each land unit is a polygon with its own attribute table. This PATfile contains the individual identification of all patches and additionally items with some special crisp values and evaluation results. Most feature data are stored in separate files, which can be referenced to the concerned land units using their identification numbers. In general, the information of the mentioned 5page documentation is managed by the GISdata bank INFO. Due to the special requests of the evaluation procedure these data had to be completed. The most additions were made using of the original data, created in the microchore determination process. Precise spatial proportions of several features and a finer slope classification could be gathered from the original material by this way.
12Additional crisp data for evaluation purposes were put into a new item in the PATcopy (table 3). First the slope length in meter is derived from detailed topographic maps. Second precipitation data are added, in order to estimate the rain factor. Such rain statistics are available from the valid climate station, mentioned in the documentation list. Sometimes this estimation had to be corrected by means of verbal climate description, for example if the unit is abnormal rich of thunderstorms.
Table 3 : Polygon attribute table (cutting) of evaluated microchores
13Always the first step to work with fuzzy sets is fuzzification. There are several possibilities to fuzzificate crisp data. In order to characterise spatial data, the fuzzification approach is to estimate the areal proportions of all existing quantities. Thus, all features get the appropriate membership values to assigning them to the according uncertain feature classes.
14Additionally, a code was to be defined in order to relate the features to its position inside the microchore. Therefore, all nouns of the verbal relief description get letters in alphabetic order. A feature class gets the same letter, if it occurs in the according relief element of the microchore. By help of these letters, several assessed features can be related together. Thus, if two features don’t have an equal letter in their codes the combination is not valid.
15The fuzzy data are stored and classified with regard to evaluating algorithms. For each parameter, a special file was created. Such a file includes at first the identification item to relate each row to a basic land unit. After that, the essential fuzzy data are saved as discrete membership values. Each column contains the membership value of one feature class, according to the areal proportion approach. Finally, the position letter codes are added as strings in a separate column to each feature class (table 4).
Table 4 : Fuzzy data file for soil parameters (cutting, see explanation of soil classes in table 1).
16The results will be new fuzzy sets with true values, assigning the land units to classes of erosion resistance.
17If a decision by fuzzy logic is necessary, several operators have to be combined and optimised into an „inference system" on basis of a set of rules. Each rule has the general form:
18IF X_{1} IS A_{1 }AND/OR X_{2} IS A_{2} ... THEN IS Y := B
19The most simple fuzzy inference system covers the following fife steps (Tilli 1991):

comparison of facts „Xi" with premises „Ai" by distance operator to get a correspondence value

combination of correspondences of the left rule side by a tnorm (if the rule has an „and") or an snorm (if the rule has an „or") to get the satisfaction value „B"

conclusion from satisfaction value by an inference operator to get the result „Y"

consideration of uncertainty of the rule by a weight factor

aggregation of results if several rules exist by snorm
20The main task is to select the operators, which will fit a problem best. Each field of matrix is regarded as a rule. Because of a crisp premise, a special compatibility operator is not necessary. Therefore, all rules with input values above 0 will be executed. The combination operator is a tnorm, that is the possibility of result should be less than both of the input variables. In this case, the minimum operator was selected. By a sharp conclusion, its value  the minimum of all input variables  will become the membership grade of the result class, specified by the rule. If a matrix field gives more than one result, the main class gets the full „load" as well. Half of this load is assigned to all other valid classes.
21If several rules lead to equal result classes, an aggregation of their membership values follows. Since all rules are valid independently, aggregation has to be an snorm. A good way to choose an optimal operator, is to check two extreme cases before a decision. The different results of algebraic sum and maximum operators were compared while evaluating the erosion resistance. Finally, the sum operator has been selected.
22The key problem while evaluating of heterogeneous units is to prevent the assessment of several features, which all occur inside of the evaluated unit admittedly, but never at the same location. For example, a microchore with arable land almost may even have woods on steep escarpments along their valleys; high slope and crop area don’t match there. Therefore, the position codes have to be analysed by combination of the left rule side. With the help of string functions the computer can find out, if any letters match in all codes. Otherwise, the evaluation of this combination must be suppressed. Optionally, the degree of matching can be considered.
23According the request above, a special inference operator combination is now needed. Since each combination in each unit may change, the operator must be a dynamic one. Since it controls the spatial overlap, it is called a „dynamic incidence operator". At least, it has to work like a sign function, i. e. it should give „full load" if an overlap exist and zero otherwise. This is realised as logical casedifferentiation for the multidimensional procedure. Here an arithmetic version was more desirable, especially for the twodimensional procedure. Since Arc/info doesn’t offer a sign function, the formula
24I = 2√ x/(x+1) is used as a substitute, additionally taking into account the degree of incidence.
25Here x is the product of matching positions in both the addressed letter codes.
26The necessary computations were done by selfwritten programs using the GIS macro language AML. Thus, not only mathematical but also geometrical operations like buffer and intersection functions can be used without data export or interface problems. Although complicated fuzzy algorithms run better on native fuzzy software, interfaced with the GIS, the suitable preselection of data by describing the microchores allowed a direct computation here. Such a computation needs no additional reduction of the data sets, and it consumes about one hour, depending on operator combination.
27Evaluation programs have a similar basic structure. Some differences exist in operator selection and among the methods of validation. At first, all units are selected, matching a special rule, respectively one matrix field. Using the selected units, a loop is processed, computing the inference values for each unit individually. Those values are aggregated to get the possibilities for each result class, which is assigned by the actual rule. After that, the same procedure begins with the next matrix field. At the end of program, some operations are executed that round, normalise, and defuzzificate the results.
28Table 5 shows and explains the main steps of a selected loop for evaluating the potential erosion tendency.
Table 5 : program section for potential erosion evaluation by maximumaggregation
29With the help of some microchores, the results of several procedure variants will be compared. These four units are situated at the extreme northwest and southeast edges of the test area (see figure 2). They represent different types of land use, soils, relief and climate.
30The following tables show the results of the four units. The membership values of all six result classes (from EROS1: „no erosion" to EROS6: „extreme high erosion") appear after the microchore numbers. At last, a suitable defuzzification is done by the centre of gravity rule. All units are ordered with the help of defuzzificated values (EROS). Therefore, the best feature to compare with, is the record number („position") at the beginning of rows.
31At first, the results of evaluation for potential erosion tendency are compared and computed without regard to the feature positions. Tables 6 and 7 confront the evaluation results determined by maximumaggregation with the ones aggregated by the arithmetic sum operator. While the most representative combinations define the maximum results, some smaller values can affect the summarised grades, if they occur together. Thus, the greatest difference can be stated between the values of the fine structured „Kleinkuppengebiet" (494627).
Table 6 : valuation without position regard, algebraic sum aggregation
Table 7 : evaluation without position regard, maximum aggregation
32The comparison of evaluation results, computed by dynamic incidence operator (tables 8 and 9), shows that they are generally similar. However, some systematic changes occur. Particularly the northwest microchores get clear better places. When comparing the new maximum with the new sum results, the advantage becomes evident: differences between the unit positions are smaller, than the ones in the previous run. Besides the more plausible way of evaluation, the good match of different computed results is a statistical confirmation of quality.
Table 8 : evaluation with dynamic incidence, maximum aggregation
Table 9 : evaluation with dynamic incidence, algebraic sum aggregation
33Because of its better suitability, only the algebraic sum operator was used to aggregate the actual erosion values (table 10). The influence of land use is obvious by a clear change in the defuzzificated results. Particularly, the southeast situated hilly microchores swapped their positions. Now the full forested SebnitzTal (515045) has a better place than the more flat, agricultural used GranitRiedelgebiet (515046), which is the most endangered unit of the whole test area. Just as serious is the relative improvement of the Kleinkuppengebiet (494627). Because the land use structure agrees with the natural conditions there, the actual soil loss will not essentially exceed the natural erosion.
Table 10 : evaluation with dynamic incidence, algebraic sum aggregation
34In order to map all results clearly, the defuzzificated values were classified. This cartographic presentation (figure 3) shows the differences more completely. The two versions, evaluated by dynamic incidence, are quite similar. The influences of precipitation, slope degree, and loess cover are clearly visible. On the other hand, the results without position control seem more extreme and less comprehensive. On the map of „actual erosion danger" only the large forests and even areas between very critical evaluated units stand out.
35Certainly, fuzzy results can also be mapped directly. Therefore, a rough segmentation of membership values and a suitable space dividing each signature field were applied to make it readable. The essential results stand out better, if sharp evaluated plots get a flat colour. A small cardinality (i.e. the sum of all membership values) is favourable.
36A new designed methodology allows the evaluation of medium scaled land units, based on the fuzzy set theory. In order to solve the main problems connected with space heterogeneity and landscape diversity, some appropriate approaches were tested successfully. Application of an uncertain evaluation procedure for the landscape function of erosion resistance was carried out by fuzzy inference system. Utilisation of uncertain information as well as spatial stochastic data, requires a special fuzzification approach, combining the determination of spatial proportions with an estimation of class memberships. The key problem with evaluation of heterogeneous land units is consideration of local feature overlap. It is solved by the implementation of a dynamic incidence operator into the fuzzy inference system. Several variants of presentation have been explored to map the defuzzificated values as well as the fuzzy results.
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