The data that was used for developing the fuzzy inference system was limited by the need to have rainfall events that contained all the four variables that were used for the fuzzy model namely rainfall, soil type, soil moisture and runoff measured at field scale as described in chapter 5 section 5.4.
Rainfall events in the semi-arid areas of Zimbabwe are rare thus the number of rainfall events in field experiments where both soil moisture and field scale runoff were measured were limited. Furthermore measurement of soil moisture and runoff on the two catchments where the data was obtained was constrained by lack of automatic measuring instruments due to resource constraints further limiting data availability. Data was available from two different semi arid catchments of Zimbabwe described in chapter 5 section 5.4.2. These data gave 52 data points for the modelling that matched the criteria.
This was considered to be reasonably adequate for the fuzzy logic modelling. The minimum and maximum values of each of the variables from the 52 data points used for the modelling are shown in Table 7-1.
Variable Minimum value Maximum value
Runoff coefficient (%) 0.0 100.0
Rainfall duration (hours) 0.0 19.86
Rainfall intensity (mm/hour) 0.0 6.35
Soil moisture (mm) 3.0 35.0
Soil type (index) 1.68 3.73
Following subtractive data clustering, five cluster centres shown in Table 7-2 were established, each cluster centre forming the focal point of a sub model of the inference system. The value for each variable of a cluster centre in Table 7-2 indicates the relative position of the cluster centre in the range of possible values of the variable. For example rainfall duration can range from all the rainfall falling in one hour to rainfall spread over the whole 24 hour period of the day. Cluster centre number 1 implies that it is applicable to rainfall events in which rain falls during short durations clustered around 1.7 hours
Table 7-1: Minimum and maximum variable values used for normalising and denormalising data
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(0.072 of 24 hours = 1.7 hours) while cluster centre number 3 is for rainfall events that fall over very long periods clustered around18.5 hours. Similarly cluster 1 is applicable to events where rain falls on soil that is very wet while cluster 5 is for rainfall events falling on soil that is very dry. The values of the cluster centres on the scale of 0 to 1 could have been influenced by the maximum values of the variables that were used for normalising the data (Table 7-1). The cluster values for runoff coefficient and rainfall duration are based on a well established range as both the minimum value and maximum value are known with certainty. Both have a minimum value of zero when the rainfall event produces no runoff and when the rainfall occurs for a few seconds which becomes zero on a time scale of 1 hour. The maximum theoretical value of runoff coefficient is 100 (%) which occurs when all the rainfall is turned into runoff while the maximum theoretical value of rainfall duration is 24 (hours) which occurs when rainfall is received throughout the day.
However the maximum theoretical value of rainfall intensity is the maximum value that was obtained from the data range used for data clustering and there is a probability that such a rainfall intensity can be exceeded. The maximum value of soil moisture was that obtained from the data range and could be exceeded in reality.
Cluster number
Figure 7-1 illustrates the clustering results graphically clearly showing the range of runoff coefficients that can be generated. It shows location of the runoff coefficient and the four input variables for each sub model represented by a cluster centre on a scale of 0 to 1.
Table 7-2: Normalised cluster centre values after data clustering
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The five cluster centres can be interpreted in linguistic fuzzy terms to establish the corresponding five fuzzy rules as presented in the following paragraphs.
The spread of data points for cluster centre 1 indicates the physical characteristics of the variables that form the focal point for fuzzy sub model 1. This cluster centre represents heavy textured soils as the value of soil type is high. It also represents a situation where both the soil moisture and rainfall intensity are both high as their corresponding values are equally high. However despite representing high conditions of soil moisture and rainfall intensity falling on heavy textured soil that favours high runoff generation the cluster centre represents a cluster of events in which rain falls for a short duration as the Figure 7-1: The variation of the input variables for the cluster centres
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value of rainfall duration is small. This means that during that short rainfall period the infiltration rate would still be high given that infiltration rate exponentially decreases with the rainfall period. This result in low proportion of rainfall that is converted to runoff and hence the runoff coefficient for this cluster is nearly average (39%). Therefore rule 1 of the fuzzy model can be defined as follows: fuzzy rule 1: A heavy textured (clay loam) soil that receives high rainfall intensity falling for a very short duration on soil with high soil moisture generates slightly below average runoff.
The values of variables constituting cluster centre 2 are much lower when compared to those of cluster centre 1 except for the rainfall duration value which is relatively higher.
This cluster centre represents rainfall falling relatively on dry soil with rainfall intensity that is not very high. This favours high infiltration which explains why the runoff coefficient associated with this cluster is low (17.1%). This suggests that rule 2 of the fuzzy model can be stated as follows: fuzzy rule 2: A loamy soil that receives above average rainfall intensity falling for a short duration on soil with low soil moisture generates low runoff.
The spread of values of variables making up cluster centre 3 indicate that while the rainfall duration increased to a level where infiltration capacity would be expected to reduce thereby allowing more runoff to be generated the rainfall intensity was low compared to cluster 1 and 2. This means the low rate of rainfall intensity allowed for drainage of water from top soil horizon thereby creating reduced runoff generation than that suggested by the long rainfall period. Despite this disadvantage of the location of the cluster centre the runoff coefficient (41.2%) that results from this cluster suggest average runoff is generated under these conditions. The corresponding rule for the fuzzy model is suggested as follows: fuzzy rule 3: A clay loam soil that receives low rainfall intensity falling for a long period on soil with high soil moisture generates nearly average runoff.
A typical situation where high runoff is generated is shown under conditions of cluster 4.
While rainfall duration for cluster 4 is average compared to all the five clusters the runoff coefficient (61.6%) is highest owing to the high values of rainfall intensity, soil moisture and a heavy textured soil that have low hydraulic conductivity. The rainfall intensity, soil moisture and soil texture of the soil for cluster 4 are comparable to cluster 1. However the higher rainfall duration in cluster 4 suggests that the infiltration capacity of the soil reduced
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as the rainfall event progresses resulting in more runoff being generated compared to that which is generated under cluster 1. Therefore rule 4 for the fuzzy model could be described as follows: fuzzy rule 4: A heavy textured soil (clay loam soil) that receives high rainfall intensity falling for nearly average period on soil with high soil moisture generates high runoff.
The influence of soil conditions and properties on the runoff amount generated is demonstrated by the distribution of input variables for cluster centre 5. The rainfall duration is fairly high which would suggest sufficient time to reduce infiltration capacity while rainfall intensity is above that of cluster 3. However the rainfall with such characteristics that would suggest average runoff generation falls on dry light textured soil. A soil with such conditions has a very high infiltration capacity that remains high during the rainfall event as the infiltrating water quickly drains away resulting in very low runoff being generated. This results in a very low runoff coefficient value (2%). Thus the last rule for the fuzzy model is described as follows: fuzzy rule 5: A sandy soil that receives average rainfall intensity falling for an average duration on soil with low soil moisture generates very low runoff.
The SCE optimization guidelines provided by Duan et al. (1994) were successfully applied in identifying the consequent parameters of the fuzzy model (see Equation 5-18 reproduced in Equation 7-1). All the five sub models met the convergence criterion that was set for this study of the average value of RMSE for best top half and the least bottom half falling within 5% (see section 5.4.4 in Chapter 5).
The consequent coefficient,𝑐𝑖,𝑚, for each of the sub models with focal points (cluster centres) presented in Table 7-2 are shown in Table 7-3. The coefficients (Table 7-3) were applied to the general model of Equation 7-1 for each of the corresponding cluster centres shown in Table 7-2. The corresponding runoff coefficients which were obtained from this calibrated fuzzy model were compared to the runoff coefficient that was established during data clustering for the corresponding cluster centre and the results are shown in Figure 7-2. The results show that the consequent parameters of the fuzzy model provide a very good estimate of the runoff coefficients. The model was therefore tested on the data set that was used for developing the model and from another data set obtained from independent sites, the results of which are presented in the next sub section.
7-7 𝑞𝑗,𝑚 = 𝑐0,𝑚+ 𝑐1,𝑚𝑇𝑗+ 𝑐2,𝑚𝑃𝑖,𝑗+ 𝑐3,𝑚𝜃𝑗+ 𝑐4,𝑚𝛼𝑁,𝑗
Where:
qj,m is normalised runoff coefficient contributed by partial model m during rainfall event j in; Tj is normalised mean rainfall duration (hours) for rainfall event j; Pi,j is normalised mean rainfall intensity(mm/hour) during rainfall event j; θj is normalised root zone soil moisture (mm) at the start of rainfall event j; αN,j is normalised soil parameter defining soil characteristic (type) for rainfall event j; c0,m is a constant for partial model m; c1,m, c2,m, c3,m and c4,m are coefficients to input variables for partial model m.
Table 7-3: Model consequent coefficients from shuffled complex Evolution calibration
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