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Scaling Properties of Topographic Indices and Crop Yield

Multifractal and Joint Multifractal Approaches

Dep. of Soil Sci., Univ. of Saskatchewan, Saskatoon, SK, Canada, S7N 5A8

View larger version (40K): [in a new window]   Fig. 1. The spatial distribution of the six variables along the transect: (a) measured variables (RE = relative elevation, YL = yield, and BM = biomass) and (b) calculated variables (USL = upslope length, WI = wetness index, and CR = curvature). The distance was measured from south to north.

View larger version (40K): [in a new window]   Fig. 2. Natural logarithms of the normalized µ measures (partition functions) plotted against natural logarithms of the measurement scales for q values of 15 (filled triangle), 10 (open diamond), 5 (filled diamond), 2 (open square), 0 (filled square), –2 (open and inverted triangle), –5 (filled and inverted triangle), –10 (open circle), and –15 (filled circle). YL = yield, WI = wetness index, USL = upslope length, RE = relative elevation, BM = biomass, and CR = curvature.

View larger version (22K): [in a new window]   Fig. 3. The mass exponent (of the qth moment) evaluated at 0.5 increments in q values for (a) yield, (b) upslope length, (c) wetness index, and (d) biomass.

View larger version (16K): [in a new window]   Fig. 4. The generalized dimensions (Dq) at –10 to 10 ranges of q evaluated at 0.5 increments. YL = yield, USL = upslope length, WI = wetness index, and BM = aboveground biomass.

View larger version (17K): [in a new window]   Fig. 5. The multifractal spectrum [f()] for yield (YL), upslope length (USL), wetness index (WI), and biomass (BM) for q values of –10 to 10, evaluated at 0.5 increments.

View larger version (31K): [in a new window]   Fig. 6. Joint multifractal spectrum, f(, ß), for the joint distribution of (a) yield (YL) vs. upslope length (USL), (b) YL vs. wetness index (WI), (c) YL vs. aboveground biomass (BM), and (d) USL vs. WI evaluated for q and t values (order of exponents) ranging from –20 to 20.