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Inference for a Generaliaed Gibbsian Distribution Channel Networks
We introduce a two‐parameter (β0 and β1) Gibbsian probability model to characterize the spatial behavior of channel networks. This model is defined for trees draining basins on a square lattice. The probability of a tree s is proportional to exp [−β0H(s, β1)[, where H(s,β1) is taken to be a summatio...
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| Published in: | Water resources research 1994-07, Vol.30 (7), p.2325-2338 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | We introduce a two‐parameter (β0 and β1) Gibbsian probability model to characterize the spatial behavior of channel networks. This model is defined for trees draining basins on a square lattice. The probability of a tree s is proportional to exp [−β0H(s, β1)[, where H(s,β1) is taken to be a summation over lattice points ν in the basin of A(ν, s)β1, letting A(ν, s) be the area upstream from (but not including) point ν. Procedures are developed for estimating the parameters given an actual network obtained, for example, from digital elevation data and for using bootstrapping to obtain confidence intervals for these estimates. We also develop several goodness of fit tests for the model. These procedures are applied to a set of data for 50 subnetworks from Willow Creek in Montana. The estimation algorithm converged successfully for 34 of these subnetworks. For these 34 subnetworks the two‐parameter model gives in most cases a fit significantly better than a one‐parameter model (with β1constrained to be 1) studied in a previous paper. There is still, however, statistically significant lack of fit for as many as 13 subnetworks (depending on which test is applied). The parameter β1 averages 0.75 for the 34 subnetworks. The function H(s, β1) in the Gibbsian model may be given a statistical mechanical interpretation of “energy” associated with a configuration s. Other researchers have employed a similar definition of energy expenditure in drainage systems but have given physical arguments that the exponent β1 should be 0.50. We finally discuss the idea of defining “optimal channel networks” as those which minimize free energy in the Gibbsian model; it is shown that minimization of energy (rather than free energy) as other researchers have proposed is valid only under the assumption of a “low temperature” (i.e., large β0) system. Our results indicate that this assumption would not be justified for Willow Creek. We devote some discussion to the discrepancies between our approach and results and those of other researchers. |
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| ISSN: | 0043-1397 1944-7973 |
| DOI: | 10.1029/94WR00765 |