Aggregation and sampling in deterministic chaos: implications for chaos identification in hydrological processes

A review of the literature reveals conflicting re- sults regarding the existence and inherent nature of chaos in hydrological processes such as precipitation and streamflow, i.e. whether they are low dimensional chaotic or stochastic. This issue is examined further in this paper, particularly the effect that certain types of transformations, such as aggre- gation and sampling, may have on the identification of the dynamics of the underlying system. First, we investigate the dynamics of daily streamflows for two rivers in Florida, one with strong surface and groundwater storage contributions and the other with a lesser basin storage contribution. Based on estimates of the delay time, the delay time window, and the correlation integral, our results suggest that the river with the stronger basin storage contribution departs significantly from the behavior of a chaotic system, while the departure is less significant for the river with the smaller basin stor- age contribution. We pose the hypothesis that the chaotic behavior depicted on continuous precipitation fields or small time-step precipitation series becomes less identifiable as the aggregation (or sampling) time step increases. Similarly, be- cause streamflows result from a complex transformation of precipitation that involves accumulating and routing excess rainfall throughout the basin and adding surface and ground- water flows, the end result may be that streamflows at the outlet of the basin depart from low dimensional chaotic be- havior. We also investigate the effect of aggregation and sampling using series derived from the Lorenz equations and show that, as the aggregation and sampling scales increase, the chaotic behavior deteriorates and eventually ceases to show evidence of low dimensional determinism.

[1]  P. Cowpertwait,et al.  A Neyman-Scott shot noise model for the generation of daily streamflow time series , 1992 .

[2]  B. LeBaron,et al.  A test for independence based on the correlation dimension , 1996 .

[3]  Shie-Yui Liong,et al.  Singapore Rainfall Behavior: Chaotic? , 1999 .

[4]  Henry D. I. Abarbanel,et al.  Analysis of Observed Chaotic Data , 1995 .

[5]  P. Xu,et al.  Neighbourhood selection for local modelling and prediction of hydrological time series , 2002 .

[6]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[7]  Lachun Wang,et al.  Chaotic dynamics of the flood series in the Huaihe River Basin for the last 500 years , 2002 .

[8]  Ronny Berndtsson,et al.  Evidence of chaos in the rainfall-runoff process , 2001 .

[9]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[10]  Shaun Lovejoy,et al.  DISCUSSION of “Evidence of chaos in the rainfall-runoff process” Which chaos in the rainfall-runoff process? , 2002 .

[11]  Bellie Sivakumar,et al.  Rainfall dynamics at different temporal scales: A chaotic perspective , 2001 .

[12]  Bellie Sivakumar,et al.  Chaos theory in hydrology: important issues and interpretations , 2000 .

[13]  A. Jayawardena,et al.  Noise reduction and prediction of hydrometeorological time series: dynamical systems approach vs. stochastic approach , 2000 .

[14]  Jose D. Salas,et al.  Delay time window and plateau onset of the correlation dimension for small data sets , 1998 .

[15]  H. S. Kim,et al.  Nonlinear dynamics , delay times , and embedding windows , 1999 .

[16]  Soroosh Sorooshian,et al.  A chaotic approach to rainfall disaggregation , 2001 .

[17]  Konstantine P. Georgakakos,et al.  Chaos in rainfall , 1989 .

[18]  K. D. Barnett,et al.  On the estimation of the correlation dimension and its application to radar reflector discrimination , 1993 .

[19]  M. Rosenstein,et al.  Reconstruction expansion as a geometry-based framework for choosing proper delay times , 1994 .

[20]  H. Tong Non-linear time series. A dynamical system approach , 1990 .

[21]  Mark S. Seyfried,et al.  Searching for chaotic dynamics in snowmelt runoff , 1991 .

[22]  Nelson Obregón,et al.  A Deterministic Geometric Representation of Temporal Rainfall: Results for a Storm in Boston , 1996 .

[23]  P. Grassberger,et al.  A simple noise-reduction method for real data , 1991 .

[24]  A. Jayawardena,et al.  Analysis and prediction of chaos in rainfall and stream flow time series , 1994 .

[25]  Upmanu Lall,et al.  Nonlinear Dynamics of the Great Salt Lake: Dimension Estimation , 1996 .

[26]  Claire G. Gilmore,et al.  A new test for chaos , 1993 .

[27]  Konstantine P. Georgakakos,et al.  Evidence of Deterministic Chaos in the Pulse of Storm Rainfall. , 1990 .

[28]  Mees,et al.  Mutual information, strange attractors, and the optimal estimation of dimension. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[29]  Ronny Berndtsson,et al.  Reply to “Which chaos in the rainfall-runoff process?” , 2002 .

[30]  Hung Soo Kim,et al.  The BDS statistic and residual test , 2003 .

[31]  H. S. Kim,et al.  Searching for strange attractor in wastewater flow , 2001 .

[32]  Shie-Yui Liong,et al.  A systematic approach to noise reduction in chaotic hydrological time series , 1999 .

[33]  Upmanu Lall,et al.  Nonlinear Dynamics of the Great Salt Lake: Nonparametric Short-Term Forecasting , 1996 .

[34]  P. Grassberger An optimized box-assisted algorithm for fractal dimensions , 1990 .

[35]  Akira Kawamura,et al.  Chaotic characteristics of the Southern Oscillation Index time series , 1998 .

[36]  Luca Ridolfi,et al.  Nonlinear analysis of river flow time sequences , 1997 .

[37]  Franklin A. Graybill,et al.  Introduction to The theory , 1974 .

[38]  B. LeBaron,et al.  Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence , 1991 .

[39]  F. Takens Detecting strange attractors in turbulence , 1981 .

[40]  Exploring complexity in the structure of rainfall , 1991 .

[41]  Schreiber,et al.  Extremely simple nonlinear noise-reduction method. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  Gregory B. Pasternack,et al.  Does the river run wild? Assessing chaos in hydrological systems , 1999 .

[43]  Slobodan P. Simonovic,et al.  Noise reduction in chaotic hydrologic time series: facts and doubts , 2002 .

[44]  Klaus Fraedrich,et al.  Estimating Weather and Climate Predictability on Attractors , 1987 .

[45]  G. D. Jeong,et al.  Chaos characteristics of tree ring series , 1996 .

[46]  P. Ghilardi Comment on "Chaos in rainfall" by I. Rodriquez-Iturbe et al, WRR, July 1989 , 1990 .

[47]  Ignacio Rodriguez-Iturbe,et al.  A Possible Explanation for Low Correlation Dimension Estimates for the Atmosphere , 1993 .

[48]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[49]  Magnus Persson,et al.  Is correlation dimension a reliable indicator of low‐dimensional chaos in short hydrological time series? , 2002 .

[50]  Ignacio Rodriguez-Iturbe,et al.  Phase-space analysis of daily streamflow : characterization and prediction , 1998 .

[51]  Demetris Koutsoyiannis,et al.  Deterministic chaos versus stochasticity in analysis and modeling of point rainfall series , 1996 .

[52]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[53]  Renzo Rosso,et al.  Comment on “Chaos in rainfall” by I. Rodriguez‐Iturbe et al. , 1990 .