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Fractional calculus in hydrologic modeling: A numerical perspective.

ABSTRACT: Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.

SUBMITTER: Benson DA

PROVIDER: S-EPMC3603590 | biostudies-other | 2013 Jan

REPOSITORIES: biostudies-other

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Fractional calculus in hydrologic modeling: A numerical perspective.

Benson David A DA Meerschaert Mark M MM Revielle Jordan J

Advances in water resources 20120504

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensiona ...[more]

PMID: 23524449

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Project description:The Soil Conservation Service Curve Number (SCS-CN, or CN) is a widely used method to estimate runoff from rainfall events. It has been adapted to many parts of the world with different land uses, land cover types, and climatic conditions and successfully applied to situations ranging from simple runoff calculations and land use change assessment to comprehensive hydrologic/water quality simulations. However, the CN method lacks the ability to incorporate seasonal variations in vegetated surface conditions, and unnoticed landuse/landcover (LULC) change that shape infiltration and storm runoff. Plant phenology is a main determinant of changes in hydrologic processes and water balances across seasons through its influence on surface roughness and evapotranspiration. This study used regression analysis to develop a dynamic CN (CNNDVI) based on seasonal variations in the remotely-sensed Normalized Difference Vegetation Index (NDVI) to monitor intra-annual plant phenological development. A time series of 16-day MODIS NDVI (MOD13Q1 Collection 5) images were used to monitor vegetation development and provide NDVI data necessary for CNNDVI model calibration and validation. Twelve years of rainfall and runoff data (2001-2012) from four small watersheds located in the Konza Prairie Biological Station, Kansas were used to develop, calibrate, and validate the method. Results showed CNNDVI performed significantly better in predicting runoff with calibrated CNNDVI runoff increasing by approximately 0.74 for every unit increase in observed runoff compared to 0.46 for SCS-CN runoff and was more highly correlated to observed runoff (r?=?0.78 vs. r?=?0.38). In addition, CNNDVI runoff had better NSE (0.53) and PBIAS (4.22) compared to the SCS-CN runoff (-0.87 and -94.86 respectively). In the validated model, CNNDVI runoff increased by approximately 0.96 for every unit of observed runoff, while SCS-CN runoff increased by 0.49. Validated runoff was also better correlated to observed runoff than SCS-CN runoff (r?=?0.52 vs. r?=?0.33). These findings suggest that the CNNDVI can yield improved estimates of surface runoff from precipitation events, leading to more informed water and land management decisions.

Dataset Information

Fractional calculus in hydrologic modeling: A numerical perspective.

Publications

Fractional calculus in hydrologic modeling: A numerical perspective.

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