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Stochastic Modeling of Hydraulic Geometry Using an Extremal Hypothesis

This research investigates the generation of theoretical distributions of hydraulic geometry using an extremal hypothesis and compares them with field and laboratory data. The study utilizes stochastic modeling techniques to simulate hydraulic exponents and validate them against real-world data.

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Stochastic Modeling of Hydraulic Geometry Using an Extremal Hypothesis

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  1. Stochastic Modeling of Hydraulic Geometry Using an Extremal Hypothesis • Nicholas A. Coleman • Leong Lee • Department of Computer Science • Gregory S. Ridenour • Department of Geosciences • Austin Peay State University • Clarksville, Tennessee, USA Department of Computer Science & IT

  2. Introduction • Hydraulic geometry equations express relationships between the flow (velocity and discharge) and geometry (width and depth) of a stream. • Variables may change in time at a single cross section (at-a-station) or in space along a stream channel for a given frequency of discharge (downstream). Department of Computer Science & IT

  3. Introduction • The purpose of this investigation is to generate theoretical distributions of hydraulic geometry using an extremal hypothesis subject to specific constraints and then compare the computer simulated distributions with field and laboratory (flume) data obtained from the literature. Department of Computer Science & IT

  4. Mass Continuity Equation w [ft] × d [ft] × v [ft/sec] = Q [ft3/sec]

  5. Hydraulic Geometry Width (w) w = aQb mean depth (d) d = cQf mean velocity (v) x v = kQm discharge (Q) Q = ackQb+f+m Thus, it is evident that a × c × k = 1 b + f + m = 1 Hydraulic exponents are compositional (unit-sum constrained) data. Department of Computer Science & IT

  6. Linearized Power Functions • log w = b log(Q) + log(a) • log d = f log(Q) + log(c) • log v = m log(Q) + log(k) • Linear regression of log-transformed data is used to determine a, c, and k (antilog of the y-intercept) b, f, and m (slope) Department of Computer Science & IT

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  10. Extremal Hypotheses • An extremal hypothesis is an assumption that a channel will adjust its geometry so that some stream characteristic (e.g., rate of energy expenditure or sediment transport) will be minimized or maximized within certain constraints on the system. Department of Computer Science & IT

  11. Theory of Minimum Variance Langbein(1964) • The most probable values for at-a-station hydraulic exponents are postulated as that condition which represents a minimal and equable adjustment to a change in stream power. This tendency towards the most probable condition is consistent with the system tending towards minimum work. Department of Computer Science & IT

  12. Algorithm for Analytical Solution • Express model parameters as power functions of discharge, Q; • Define variances as the squares of exponents of the power functions, using same unknowns if possible; • Express the objective function (to be minimized) as the sum of the variances; • Expand and collect like terms; • Designate one or more parameters as variables, set the first derivative (or partial derivatives) equal to zero, then solve for unknown(s). Department of Computer Science & IT

  13. Parameters for This Example (Step 1) Width (w), depth(d), velocity(v), slope (s), bed shear (τ), and frictional resistance (ff). w= k1Qb d= k2Q f v= k3Qm s= k4Qz τ= k5ds = k5Q fQz = k5Q f+z ff= k6ds/v2 = k6Q fQz/Q2m = k6Q f+z-2m Department of Computer Science & IT

  14. Objective Function (Steps 2 and 3) • b2+ f 2 + m2 → minimum • subject to the constraint that b + f + m = 1. • Without further constraints, b = f = m = 1/3. Department of Computer Science & IT

  15. Collecting Like Terms (Step 4) 6m2+ 4f 2 + 2z2 + 1 – 2m – 2f – 2mf + 4fz – 4mz → min Department of Computer Science & IT

  16. Setting Partial Derivatives to Zero (Step 5) • 12m – 2f – 4z – 2 = 0 • - 2m + 8f + 4z – 2 = 0 • - 4m + 4f + 4z = 0 • The solution to this system of equations is • f = 0.429 • z = -0.287 • m = 0.142 Department of Computer Science & IT

  17. Stochastic Modeling • Thus far comparisons of theoretically derived values have been compared with means of natural data. Our model iteratively creates a distribution of hydraulic exponents by designating one or more parameters as a random variable and then extends comparison to natural and laboratory data to sampling distributions. Department of Computer Science & IT

  18. Algorithm for Computer Solution • An iterative process defined by the equation • xk+1= xk – αkgk • gkis the transpose of the gradient vector f (x)' of objective function f (x) • αkis a non-negative scalar minimizing the function f (xk – αkgk). • αkis computed by line minimization along a gradient segment using the golden section search Department of Computer Science & IT

  19. Method of Steepest Descent (Gradient Method) Department of Computer Science & IT

  20. Golden Section Search • A modification of the Fibonacci search, whose objective is to successively select N measurement points (x1,x2,...xN) for a unimodal function f of a scalar x on an interval [c1, c2] that determine the smallest possible region of uncertainty in which the minimum must lie. Department of Computer Science & IT

  21. Golden Section Search Department of Computer Science & IT

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  25. Model with 1 Stochastic Variable Department of Computer Science & IT

  26. Model with 1 Stochastic Variable Department of Computer Science & IT

  27. Univariate normality of hydraulic exponents • Univariate distributions of hydraulic exponents were tested for normality using Chi-square analysis for goodness of fit: • where Oi are the observed values, Eiare the expected values, and k is the number of observations. Department of Computer Science & IT

  28. Univariate normality of hydraulic exponents • Chi-square statistic (6.11) < critical value (9.49) • simulated values of b are normally distributed. • Alternatively, • p-value (0.19) > significance level (0.05) • simulated values of b are normally distributed. Department of Computer Science & IT

  29. Model with 2 Stochastic Variables Department of Computer Science & IT

  30. Compositional Data Analysis • Compositions belong to a restricted part of space known as the simplex. In 1986, statistician John Aitchison warned that "...even today uncritical applications of inappropriate statistical methods to compositional data with dubious inferences are regularly reported. One of the reasons for such misguided effort is undoubtedly the slowness of the emergence of statistical methods appropriate to the special nature of the simplex sample space." Department of Computer Science & IT

  31. Compositional Data Analysis • Aitchison (1986) developedtransformations from simplex space to Cartesian space which make available the whole range of procedures based on multivariate normality. The additive logistic transformation of a composition (x1,x2,x3…xD) with D components that sum to 1 is • yi= ln(xi/xd) for i = 1,…,d • where d = D – 1, leading to an additive logistic normal distribution with a logratio mean vector and logratio covariance matrix. Department of Computer Science & IT

  32. Compositional Data Analysis • Inferences regarding significant differences between means and/or variances can then be conducted with standard multivariate methods. • Permutation Invariance – to prove the order of columns does not matter. Department of Computer Science & IT

  33. Online Calculator • Transforms a matrix of raw data into row vectors of compositions; performs logratio transformations of the compositions • Computes measures of dispersion: a) logratiocovariance matrix, b) centered logratio covariance matrix, and c) variation matrix • Computes measures of central tendency: a) logratiomean vector, and b) centered logratio mean vector Department of Computer Science & IT

  34. Laboratory Flume Department of Computer Science & IT

  35. Computed Test Statistics for Logistic Normality of the Hydraulic Geometry of Flume Data and Computer Simulations. • boldface = not logistic normal Computed Test Statistics for Logistic Normality of Hydraulic Geometry Exponents redfont= not logistic normal Department of Computer Science & IT

  36. Compositional Measures of Central Tendency and Dispersion Department of Computer Science & IT

  37. Significance Probabilities for Comparisons of Flume Data and Model 2 Simulation. Department of Computer Science & IT

  38. Conclusions • Validation of extremal hypotheses for hydraulic geometry by comparison of theoretically derived values with the meansof natural observations or experimental data can be expanded to include the variability of distributions generated by stochastic modeling. • Because exponents b, f, and m are unit-sum constrained, statistical comparisons must utilize compositional data analysis. Department of Computer Science & IT

  39. Conclusions • Virtual, controlled experiments and sensitivity analyses can be conducted by randomizing one variable in an extremal model, which produces variation in one dimension on a ternary diagram. • Simulation of natural distributions requires randomizing at least two variables, which produces variation in two dimensions on a ternary diagram. Department of Computer Science & IT

  40. Conclusions • Computer generated distributions of hydraulic geometry facilitate assessment of the probabilistic impact of environmental variables on hydraulic geometry and identification of extremal hypotheses that best explain their natural distributions. Department of Computer Science & IT

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