Case formalization is the process of extracting and describing each individual case in a formal way, so that the case can be retrieved by a corresponding case-based reasoning method. Among the parts of a case, the case problem consists of a set of factors describing the contextual information associated with the case. This set of factors is quantified using a set of quantitative attributes that are directly involved in case-based reasoning. It is of crucial importance to design and quantify these factors properly for case-based reasoning. The solution part of a case records the problem's solution used for this case, which could be provided as the result of the case-based reasoning and does not participate in the reasoning procedure. The case output is an optional part of the description that is used to record the status of factors describing the case problem after the case occurred (Kolodner, 1993). Therefore, the key to designing a case-based formalization of DTA application-context knowledge is choosing and quantifying a set of factors influencing DTA algorithm selection and parameter setting to describe the case problem appropriately.

According to the characteristics of DTA application modeling, the case problem can be described based on three groups of factors that influence DTA algorithm selection and parameter setting (Table 1): application purpose, data characteristics, and study area characteristics. For example, a single-flow-direction algorithm (e.g., the classic D8 algorithm) is suitable for deriving flow accumulation from a SRTM DEM (with a resolution of 90 m) for drainage network extraction in high-relief areas, whereas a multiple-flow-direction algorithm should be used with a 10 m DEM created from a contour map for estimating detailed spatial distribution of flow accumulation and other related regional topographic attributes (such as topographic wetness index) in a low-relief area. In this example, the choice between a single-flow-direction algorithm and a multiple-flow-direction algorithm is influenced by the application purpose (i.e., the DTA task of drainage network extraction or deriving the spatial distribution of regional topographic attributes), data characteristics (i.e., a SRTM DEM with 90 m resolution or a contour-originated DEM with fine resolution), and study area characteristics (mainly terrain condition, e.g., high or low relief). This example shows the typical content of application-context knowledge in DTA application modeling.

Among these three groups of factors, the application purpose can be formalized by an enumeration-type variable. Data characteristics can be mainly described by the spatial resolution of the DEM, the type of data source, etc. In particular, the spatial resolution, which is often indicated by the grid-cell size for the widely used grid-based DTA, is the most important factor among the data characteristics. The group of factors describing the study area characteristics related to DTA application-context knowledge could include location, area, terrain condition, and other environmental conditions (such as climate, geology, etc.). Generally, terrain condition in a study area comprehensively reflects the influence of all geographical processes on the landforms in the area. This means that terrain condition might be one of the most important factors influencing the DTA algorithm selection and parameter settings. Because of its comprehensiveness, the terrain condition factor should be quantified by multiple attributes during case-based formalization of DTA application-context knowledge. Different designs of the quantitative attributes will result in different case-based methods.

In a case-based formalization of DTA application-context knowledge, the solution part of a case can be formalized by recording the name of the DTA algorithm and the corresponding parameter values used in this case, which is much simpler than describing the case problem. The output part of a case, which is optional in the case-based formalization (Kolodner, 1993), is set to be null because normally there is no change in the application context of a DTA application problem when the solution of this case is applied to the application problem.

Case-based reasoning is based on the principle that solutions for similar problems are often similar, even identical. Therefore, a new DTA application problem can be formalized in the same way as the case problem part in a prepared DTA case base and then be used in case-based reasoning by calculating the similarity between this new application problem and the problem part of each case in the case base. The solution of the case with the highest similarity (i.e., the most similar application context considered) is retrieved as the solution for the new DTA application problem. Note that in the conceptual framework of a case-based reasoning method, the solution of the retrieved case with the highest similarity might be further revised to adapt to the new application problem when the final solution for the new application problem is retained in the case base (Watson and Marir, 1994). However, the method developed in this preliminary study currently considers neither the revision nor the retention process.

Calculating the similarity between a new DTA application problem in case format and the problem part of each case in the DTA case base consists of the following two steps:

Step 1 requires calculating the similarity of each individual attribute between the new application problem and the problem description of an existing case. As usual, the range of the similarity value is [0, 1]; the larger the value, the more similar are the two cases. As mentioned above, the attributes used to formalize the problem part of a DTA application case may have different value types, such as enumeration type (e.g., application purpose), single-value type (e.g., spatial resolution and area), or even a frequency distribution (e.g., hypsometric curve). For each attribute, a similarity function should be designed correspondingly to quantify the deviation on this attribute between the new application problem and an existing case. The design is generated in an empirical way and should match the domain knowledge.

The set of quantitative attributes should be designed to effectively reflect the contextual information related to DTA application modeling, and be fit for the case-based reasoning to follow. The purpose of a DTA application case is naturally described by an enumeration-type attribute, i.e., the name of the target task. Here, cell size has been chosen as the attribute to quantify the data characteristics of a DTA application case (Table 2); other potential factors (such as type of data source) for describing data characteristics are not currently considered.

To describe the study area characteristics of a DTA application case, the area and the terrain condition of the case are considered in the current method (Table 2). Like cell size, area is an attribute with a single numeric value. Terrain condition is an important and comprehensive factor indicating the difference in study area characteristics between a new DTA application problem and an existing case.

In this study, the three following attributes were designed to describe the terrain condition factor empirically (Table 2):

Total relief attribute, which is calculated as the maximum minus minimum elevation within the study area, is a commonly used value to describe the overall terrain condition of a study area.

Table 2 lists the attributes used to formalize a case problem in this method.

The design of the similarity function for an individual attribute should be compatible with the value type of the attribute and in accord with domain knowledge regarding the level of similarity due to the difference in the attribute value between the new application problem and an existing case. Currently, the similarity function on individual attribute is designed to be with a simpler form before more detailed research could be conducted to improve it. For an attribute of the enumeration type, its similarity value between a new application problem and an existing case can be calculated by a Boolean function (Fig. 2a). When the attribute values are matched, the similarity value is 1, otherwise it is 0.

For an attribute of the single-numeric-value type, two commonly used kinds of basic similarity function are considered in this study: the linear function and the bell-shaped function (Fig. 2). Both kinds of similarity function are in accord with common sense in that the similarity is 1 for the minimum difference (i.e., zero) of attribute value, and the greater the difference in attribute value, the lower the similarity is. With the linear function, the similarity value is set to 0 or 1 when the absolute difference of the attribute between a new application problem and an existing case reaches its maximum or minimum value. The similarity can be calculated for other difference values by linear interpolation (Fig. 2b). The similarity function based on a linear function fits the specification that the maximum difference in attribute values can be preset.

With the bell-shaped function, the maximum difference in attribute values is not easy to preset and does not need to be. A simplified version of the commonly used bell-shaped function (Shi et al., 2005; Qin et al., 2009; Fig. 2c) is S=e-0.693×(|vnew-vcase|/w)0.5, where S is the similarity between a new application problem and an existing case; vnew and vcase are attribute values of the new application problem and the existing case, respectively; and w is the shape-adjusting parameter of the function. When the difference between vnew and vcase is equal to w, the similarity S = 0.5 (Fig. 2c). Some sort of numerical transformation on the attribute value could be necessary for the similarity calculation to yield a reasonable reflection of the similarity level due to differences in the attribute.

For an attribute of a more complex type (such as a frequency distribution), a quantitative index should be designed to quantify the difference in an attribute between a new application problem and an existing case. Then the similarity on this attribute can be calculated based on this index, similarly to the single-numeric-value type.

Based on these kinds of basic similarity functions, similarity functions for each individual attribute used for case-based reasoning in this paper were designed as shown in Table 2. The following discussion introduces them one by one.

The overall similarity between a new application problem and an existing case is calculated as the minimum of all similarity values for every individual attribute between the new application problem and the existing case. The use of a minimum operator means synthesizing the similarity values on every attribute in a cautious manner. On the one hand, the overall similarity result by these means is lower (i.e., higher uncertainty of reasoning result) than those from other synthesis means such as weighted average. On the other hand, a case with a low similarity value for any individual attribute will not get a higher overall similarity result by the minimum operator. This can prevent the proposed method from some unreasonable performance. For example, two cases with similar values of total relief and very different area sizes will have a low overall similarity, because of their low similarity on the area attribute and the overall similarity calculation by the minimum operator. This means that these two cases would not be credible solution provider for each other, which is reasonable. Another example is that because of using the minimum operator, a low similarity of cell size between two cases will prevent a fake high similarity on an attribute due to the DEM resolution effect (such as the attribute of elevation–slope cumulative frequency distribution) driving the overall similarity up. Therefore, the overall similarity calculation by a minimum operator should be more effective than that by a weighted-average operator.

The extraction of a drainage network, one of the most important DTA applications, was taken as an example to evaluate the proposed method. The commonly used workflow of river network extraction based on a gridded DEM includes the following three DTA tasks in sequence: (1) preparing a DEM by filling in the artificial pits and removing absolutely flat areas; (2) using a flow direction algorithm to derive the spatial distribution of flow accumulation; and (3) setting a catchment area (CA) threshold to extract those positions with a flow accumulation larger than the CA threshold to be the drainage network. Although there are some variants of this workflow based on new algorithms (e.g., Metz et al., 2011), it does not influence the following experimental design for evaluating the proposed method.

In this DTA workflow, proper selection of the DTA algorithms (such as the DEM preparation algorithm and the flow direction algorithm) and parameter values (e.g., the CA threshold) is based on DTA application-context knowledge. In many geographical information systems (such as ArcGIS), the DTA algorithm used for drainage network extraction has often been set to a default selection (e.g., the D8 algorithm as the default flow direction algorithm) in such a way that the user cannot choose the DTA algorithm. The CA threshold is an empirical parameter which varies with the study area characteristics and affects the extraction results directly. Current DTA-assisted tools often leave the choice of CA threshold for drainage network extraction to the user. However, it is difficult for users, especially non-expert users, to determine the appropriate threshold for their applications.

Therefore, this experiment was designed to focus on using the proposed method to determine the CA threshold for drainage network extraction. This means that the cases used in this experiment have the same name as the target task, i.e., drainage network extraction. The core of the solution part of the cases is the parameter value, i.e., the CA threshold. Although this experiment is somewhat simplified, we believe that it can evaluate the proposed method as effectively as an experiment with a more complex design.

Table 3 lists the results of 50 evaluation cases solved by the proposed method using the case base presented in the previous section. For six evaluation cases, the proposed method arrived at the CA threshold result same as that originally recorded in the evaluation case. The counts of evaluation cases which got shorter and longer drainage networks (i.e., larger and smaller CA threshold, respectively) from the proposed method are 16 and 28, respectively. The similarities between every evaluation case and its most similar case as reasoned by the proposed method were found in this experiment to lie within a value range from 0.47 to 0.9. A larger overall similarity value from the proposed method often corresponds to a smaller relative deviation of river density (E) (Table 3). Note that the higher the similarity, the lower the uncertainty of the result is from the proposed method. This shows that the proposed method performs reasonably.

Table 4 summarizes the distribution of the similarity results of the evaluation cases from the proposed method among the deviation levels of the drainage network results using the solved CA thresholds. The counts of evaluation results with E ∈ [0, 0.1), E ∈ [0.1, 0.25), E ∈ [0.25, 0.5), and E ∈ [0.5, +∞) are 26, 16, 3, and 5, respectively (Table 4). For most of the evaluation cases, the results from the proposed method are with lower deviation level of E, which means that the proposed method performs effectively. All solution cases with higher similarity (above 0.7) to the evaluation cases produced drainage network results with smaller E values, whereas solution cases with lower similarity (below 0.7) often produced the drainage network results with larger E values. This shows the effectiveness with which similarity reflects uncertainty in the proposed method.

Taking the results of two evaluation cases, Godavari (1053) (the “(1053)” means that the original CA threshold recorded in the Godavari case was 1053 km2) and Burdekin (502) (“(502)” defined similarly) as examples, their most similar cases in the case base as reasoned by the proposed method were KrishnaRiver (908.08) and MahanadiRiver (891), respectively (Table 3). The CA threshold values from the solution of the most similar cases (908.08 and 891 km2) were applied, respectively, to the Godavari and Burdekin evaluation cases. The extracted drainage networks are with close spatial distribution as those extracted with the original CA thresholds of the evaluation cases (Fig. 4). Their values of relative deviation of river density are smaller (i.e., 0.07 and 0.24, respectively).

The evaluation results with larger E values also have lower similarities. This means that there is no case in the current case base that has an application context highly similar to that of the evaluation case. Hence, the solution from the proposed method has higher uncertainty and might lead to questionable or even unreasonable application results for new application problems. Taking the result for the YbbsRiver (1.01) evaluation case (E = 0.43) as an example, the similarities between this evaluation case and other cases in the case base depend mostly on the similarities on the cell-size attribute during the case-based reasoning process proposed in this paper (Table 5). Because the cell size of the YbbsRiver case is 10 m, which is relatively unlike cell size (30 or 90 m) of most other cases in the case base, the overall similarities between this evaluation case and these cases in the case base are mainly limited by the individual similarity of cell size when synthesizing the similarities on individual attributes by the proposed method. Furthermore, Table 5 shows that the CA threshold values of the cases with the top 10 highest similarity values to the YbbsRiver evaluation case would make a large E value of the application result for the evaluation case (E: 0.33–21.73). The solution selected by the proposed method achieved a relatively better application result.

As for the reasoning results on the Kasilian (0.08) evaluation case (E = 0.63) using the proposed method, no individual attribute has a controlling effect on the overall similarity between the Kasilian evaluation case and the other cases in the case base (Table 6). The CA threshold values of the cases with the top 10 highest similarity values to the Kasilian evaluation case would almost always lead to a larger E value of the application result for the evaluation case (E: 0.48–0.92). The similarities between this evaluation case and the cases in the case base are lower (Table 6). This problem could be mitigated by extending the case base to contain cases with more combinations of data characteristics and study area characteristics.

The effect of calculating the overall similarity by a simple average operator instead of the minimum operator used in the proposed method was also evaluated (Table 3). When the minimum operator was replaced by the simple average operator, the overall similarity for every case increased and the lowest overall similarity among results for 50 evaluation cases increased from 0.47 to 0.68. Among 50 evaluation cases, the solutions for 13 evaluation cases from the proposed method changed because the cases with the highest similarity resulted by the simple average operator were different from those resulted by the minimum operator. Due to the synthesis by the simple average operator instead of the minimum operator, the relative deviation of river density (E) increased for 10 of these 13 evaluation cases with different solutions, when E slightly decreased for other 3 evaluation cases. The increase of E even reached 20–80 times for some cases (e.g., the evaluation cases YbbsRiver (1.01) and Batchawana (0.75)) with the overall similarity values larger than 0.8 (see Table 3). Because the overall similarity values by the simple average operator were larger than 0.8 for most of evaluation cases, there is no reasonable relationship between the overall similarity value and the E as the proposed method with the minimum operator achieved. This shows that the proposed method performed poorly when the simple average operator was used instead of the minimum operator. Therefore, the synthesis by a minimum operator is proper for the proposed method.