Cumulants-Based Analysis of Concentration Data from Soil-Column Studies for System Identification
Publication: Journal of Hydrologic Engineering
Volume 1, Issue 1
Abstract
The problem of system identification from input-output data continues to be a challenge in all aspects of hydrology. In this study, cumulants are proposed as tools for determining the relevant information contained in output data. A method for computing cumulants in terms of moments is proposed using ideas from combinatorics theory. This method was applied to effluent concentration data from soil-column studies. Analyzing effluent concentration data from soil columns is the most common laboratory method for calculating solute transport properties. The ratios of cumulants to moments for various orders were computed using physically realistic examples and actual experimental results. The use of cumulants in determining deviation from Gaussianity and providing an estimate of the useful orders of moments was also investigated. It was concluded that cumulants provide a useful analytical tool for guiding the determination of model choice.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Aris, R.(1958). “On the dispersion of a solute in a fluid flowing through a tube.”Proc. Roy. Soc. London, London, U.K., Ser. A, 235, 67–77.
2.
Chu, S.-Y., and Sposito, G.(1980). “A derivation of the macroscopic solute transport equation for homogeneous saturated porous media.”Water Resour. Res., 16, 542–546.
3.
Fox, R. F.(1975). “A generalized theory of multiplicative stochastic processes using cumulant techniques.”J. Math. Phys., 16(2), 289–297.
4.
Fox, R. F.(1976). “Critique of the generalized cumulant expansion method.”J. Math. Phys., 17(7), 1148–1153.
5.
Gardiner, C. W. (1985). Handbook of stochastic methods, Springer Verlag, New York, N.Y.
6.
Govindaraju, R. S., and Kavvas, M. L.(1991). “Stochastic overland flows. 2: Numerical solutions of evolutionary probability density functions.”Stochastic Hydrol. Hydraul., 5, 105–124.
7.
Jacquez, J. A., and Greif, P.(1985). “Numerical parameter identifiability and estimability: integrating identifiability, estimability and optimal sampling design.”Math. Biosci., 77, 201–227.
8.
Jin, M., and Duffy, C. J.(1994). “Spectral and bispectral analysis for single- and multiple-input nonlinear phreatic aquifer systems.”Water Resour. Res., 30(7), 2073–2095.
9.
Jury, W. A., Gardner, W. R., and Gardner, W. H. (1991). Soil physics . John Wiley & Sons, New York, N.Y., 218–267.
10.
Jury, W. A., and Sposito, G.(1985). “Field calibration and validation of solute transport models for the unsaturated zone.”Soil Sci. Soc. Am. J., 49, 1331–1341.
11.
Kavvas, M. L., and Govindaraju, R. S.(1991). “Stochastic overland flows. 1: Physics based evolutionary probability distribution distributions.”Stochastic. Hydrol. Hydraul., 5, 89–104.
12.
Kavvas, M. L., Govindaraju, R. S., Rolston, D. E., Or, D., and Biggar, J. (1992). “On the stochastic pollution transport equations.”Heat and mass transfer in porous media, M. Quintard and M. Todorovic, eds., Elsevier, New York, N.Y., 137–142.
13.
Kreft, A., and Zuber, A.(1978). “On the physical meaning of dispersion equation and its solutions for different initial and boundary conditions.”Chem. Engrg. Sci., 33, 1471–1480.
14.
Kubo, R. (1962a). “A stochastic theory of line-shape and relaxation.”Fluctuation relaxation and resonance in magnetic systems, ter Haar, ed., Oliver and Boyd, Edinburgh, U.K., 174–183.
15.
Kubo, R.(1962b). “Stochastic Liouvile equations.”J. Math. Phys., 4(2), 174–183.
16.
Kubo, R.(1962c). “Generalized cumulant expansion method.”J. Phys. Soc. Japan, Japan, 17(7), 1100–1120.
17.
Meeron, E.(1957). “Series expansion of distribution functions in multicomponent fluid systems.”J. Chem. Phys., 27(6), 1238–1246.
18.
Nkedi-Kizza, P.(1984). “On the equivalence of two conceptual models for describing ion exchange during transport through an aggregated oxisol.”Water Resour. Res., 20, 1123–1130.
19.
Parker, J. C., and van Genuchten, M. Th. (1984). “Determining transport parameters from laboratory and field tracer experiments.”Bull. 84-3, Virginia Agric. Experiment Station, Blacksburg, Va.
20.
Roerdink, J. B. T. M. (1981). “Inhomogeneous linear random differential equations with mutual correlations between multiplicative, additive and initial value terms.”Physica, 109A, 23–57.
21.
Sposito, G., and Barry, D. A.(1987). “On the Dagan model of solute transport in groundwater: foundational aspects.”Water Resour. Res., 23(10), 1867–1875.
22.
Terweil, R. H.(1974). “Projection operator method applied to stochastic linear differential equations.”Physica, 74, 248–265.
23.
Toride, N., Leij, F. J., and van Genuchten, M. Th.(1993). “A comprehensive set of analytical solutions for nonequilibrium solute transport with first-order decay and zero-order production.”Water Resour. Res., 29, 2167–2182.
24.
Unlu, K., Kavvas, M. L., and Nielsen, D. R.(1989). “Stochastic analysis of field measured unsaturated hydraulic conductivity.”Water Resour. Res., 25(12), 2511–2519.
25.
van Genuchten, M. Th., and Wagenet, R. J.(1989). “Two-site/two-region models for pesticide transport and degradation: Theoretical development and analytical solutions.”Soil Sci. Soc. Am. J., 53, 1303–1310.
26.
Van Kampen, N. G.(1974a). “A cumulant expansion method for stochastic linear differential equations. I.”Physica, 74, 215–238.
27.
Van Kampen, N. G.(1974b). “A cumulant expansion method for stochastic linear differential equations. II.”Physica, 74, 239–247.
28.
Van Kampen, N. G.(1976). “Stochastic differential equations.”Phys. Reports C, 24(3), 171–228.
Information & Authors
Information
Published In
Journal of Hydrologic Engineering
Volume 1 • Issue 1 • January 1996
Pages: 41 - 48
Copyright
Copyright © 1996 American Society of Civil Engineers.
History
Published online: Jan 1, 1996
Published in print: Jan 1996
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.