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Determining Soil Hydraulic Properties from Tension Infiltrometer Measurements

Fuzzy Regression

Dep. of Soil Science, University of Saskatchewan, Saskatoon, SK, Canada

View larger version (27K): [in a new window]   Fig. 1. Illustration of fuzzy linear regression algorithm and degree of fitting of i to given fuzzy observation i. The upper panel indicates how the upper and lower bounds are constructed with a degree of belief (H); A0 is the intercept and A1 is the slope of the linear relationship between X and Y; m0 and m1 are the center of the fuzzy number A0 and A1. c0 and c1 are the half width of the fuzzy number A0 and A1. The lower panel illustrates how the fuzzy linear regression was constructed. cj·|Xj| is the zero-cut distance from the center and (1 – H)cj·|Xj| is the H-cut distance from the center. Since m0 + m1·X is the center line, m0 + m1·X ± cj·|Xj| are the 0-cut upper and lower boundaries and m0 + m1·X ± (1 – H)cj·|Xj| is the H-cut upper and lower boundaries. Fuzzy linear regression is to minimize the total fuzziness criterion (Eq. [5]), with constraints that all measurements falls between the H-cut upper and lower boundaries.

View larger version (20K): [in a new window]   Fig. 2. Measured and predicted steady-state infiltration rate q as a function of tension (h) and their 95% confidence interval, and fuzzy upper and lower limits with a degree of belief H = 0.75 for the grassland soil.

View larger version (20K): [in a new window]   Fig. 3. Measured and predicted steady-state infiltration rate q as a function of tension (h) and their 95% confidence interval, and fuzzy upper and lower limits with the degree of belief H = 0.75 for the cultivated soil.

View larger version (17K): [in a new window]   Fig. 4. Estimated fuzzy membership function for the saturated hydraulic conductivity of the grassland and cultivated soils.

View larger version (17K): [in a new window]   Fig. 5. Estimated fuzzy membership function for the effective porosity of the grassland and cultivated soils.