Discharge-Stage Relationship on Urban Streams Evaluated at USGS Gauging Stations, St. Louis, Missouri

The earth in the state of dissipation conformation undergoes several geological actions, heterogeneous and nonlinear. The underground lithology and physical properties are characterized by the very strong heterogeneity and nonlinearity.

The natural systems including biological, metallogenetic systems are continuously fighting against and compromising with their circumstances to adapt to the changeable environment for their survival and development.

The concept of chaos dynamics has been developed and applied to geology (Yu, 1999;, 1998; Ma et al., 1998; Turcotte, 1997). Many recently studies show that in the crust the element contents (Cheng, 1995; Bolviken et al., 1992), ore grades (Turcotte, 1997, 1986; Sheng and Sheng, 1995), ore reserves (Scholz and Barton, 1991) and spatial distribution of ore deposits (Barton and Scholz, 1991; Carlson, 1991) all follow the principle of fractal statistics (Mandelbrot, 1983).

This paper studies the chaos dynamic mechanism of the migration, enrichment and mineralization of elements in the crust. The research shows that the interaction of the nonlinear process in the geological environment is an essential factor for the uneven distribution of elements and the mineralization in the crust, determining the element contents and the fractal structure of the distribution of the large- and small-sized mineral deposits. The logistic map is a better mathematical model describing the behavior of the chaos dynamics. The parameter μ, i.e., the mineralizing potential, is employed to divide the region into non-mineralization region or mineralization region. The value of the parameter μ in model (3) with true data (in Xinjiang Automatic Region, China) is obtained by statisticnd method. The forecasting results are generally in accordance with those obtained with other methods, for example, with the characteristic analysis.

CHAOS DYNAMICS MODEL OF MINERALIZATION ENRICHMENT

In geological environment, the geological substances are always moving, and thus elements are enriched to a certain extent and at the same time are inevitably dispersed. We consider a region in the crust and suppose that migration enrichment rate and outward dispersion rate of elements in the region are constant. The enriched elements in turn participate in their enrichment or dispersion. Let X(t) be the content of element in the region at time t. We consider the following equation

where positive a (>0) is the enrichment rate, positiveb (>0) is outward dispersion rate, and Δt is time increment.

Equation (1) may be rewritten as an iteration equation

where μ=a+1. Substitution of x=bX/μ into (2) gives

Equation (3) is actually the logistic map. x represents the concentration of element and μ the mineralizing potential.

The solution to iteration equation (3) is sensitive to parameter μ. In the range 0 < x_n < 1, the gradual changes of x_n are summarized as follows with the change of μ

(1) In the range 0≤μ< 1, the value of x_n decreases regularly to 0 as n→∞.

(2) In the range 1≤μ< 2, the value of x_n increases regularly to 1-1/ as n→∞.

(3) In the range 2≤μ< 3, the value of x_n fluctuates decreasingly to 1-1/μ as n→∞.

(4) In the range 3 < μ≤1+6=3.449, the value of x_n gradually approaches the cycle 2 as n→∞.

(5) In the range 1+6 < μ≤4, the balanced value of x_n varies in a very complicated way (see Fig. 1 below). As μ increases, cycle 2, cycle 4, cycle 8, …, cycle 2ⁿ occur, respectively. Cycle 2^∞ occurs at approximately μ=3.569 945 672….

Figure  1.  Asymptotic behavior for large n of the logistic map x_n+1=μx_n (1-x_n) as a function of μ.

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If equation (3) is valid, the range of x must be 0 < x_n < 1 since the contents of elements should not be negative. Hence the range of parameterμ must be 0 < μ< 4.

The iteration equation (3) has two fixed points, x*=0 or x*=1-1/μ. For positive values of μ we find the fixed point at x*=0 is stable for 0 < μ< 1 and unstable for μ>1. The fixed point x_*=1-1/μ is unstable for 0 < μ< 1, stable for 1 < μ≤3, and unstable for μ>3. In the range 3 < μ< 3.569 9…, the iteration process produces the cycle-doubling flip bifurcations, i.e., cycle T=2ⁿ (n=1, 2, 3, …, …). In the range μ≥3.569 9…, this phenomenon interrupts abruptly, since the iteration process evokes the chaotic character, i.e., an infinite sequence of cycle-doubling flip bifurcations occurs as n→∞.

Many actual distribution characteristics of the element contents and mineralization may well be explained by the aforementioned model. For example, the distribution of the element content in the crust and the distribution between the large and the small ore deposits have the self-similar structure and the solution to iteration equation (3) near the cycle-doubling flip bifurcation point also shows the self-similar structure. In the range 3.569 9… < μ< 4, the iterative process produces the chaos, i.e., x_n has an infinite set of different values. The set in geometrical distribution has an area of chaotic attractor characteristic of canton set, which is much analogous to the uneven distribution of element contents and mineralization in the crust. In the rangeμ≤1, the value of the stable fixed point of the iteration equation (3) is 0, which shows the non-mineralization area, i.e., the even out-of-order closure system. In the range 1 < μ< 4, the value indicates the mineralization area, i.e., the uneven order open system and nonlinear character. In the range 1 < μ≤2 the stable fixed point is a low value due to lower potential, indicating the formation conditions of the ore-bearing rock-for example, those of ferruginous rock, manganese rock and phosphatic rock. In the range 2 < μ≤3, the value of the stable fixed point increases, which shows the main mineralization cycle, for example, that of the formation of the stratabound deposits. In the range 3 < μ< 4, the value is characterized by the periodic stable orbit due to the increase of the nonlinearity, characteristic of the very main mineralization condition and the mineralization characteristics of the hydrothermal metasomatic deposits and stratum weathering mineralization. The super large-size or large-size deposits and metallogenic districts are possibly located at the edge of chaos (Yu, 1999).

The interaction of the nonlinear process in the geological environment is an essential factor for the uneven distribution of elements and the mineralization in the crust, determining the element contents and the fractal structure of the distribution of the large and small-size mineral deposits.

During the extremely complex evolutionary process of the earth, various mineralizations (such as the enriching process of mineralized elements) are likely to trace some "fixed" places on the crust, or some "fixed points (regions)" attract the loci of the mineralization.

The phenomena mentioned above are not accidental and their profound background causes are shown as follows: (1) The ore deposits occur under the chaos. (2) The reserves and spatial distribution of the ore deposits follow the fractal distribution. The chaotic attractor is the fractal set which is probably a spatial expression form of the attractor of this chaotic system evolution in the process of mineralization.

EXAMPLES

We now turn our attention to the value of the parameter μ in model (3) with true data by the statistical method.

The rule of evaluating parameter μ is given according to aforementioned analysis: There have no deposits in the cell for μ≤2; there is one or two small-size deposits in the cell for 2 < μ≤3; there have more than three small-size deposits or middle-size deposit in the cell for 3 < μ≤3.56; there have super large-sized or large-size deposits in the cell for 3.56 < μ≤4.

The research areas are divided into 16 deposit dense cells of Au, i.e., the model cells, and 14 anomaly dense cells of Au, i.e., the forecast cells. Two cells therein contain the large-size deposits.

The integrative information geological variables are extracted in line with the information on geology, geochemistry and geophysics. The mathematical model of the quantification theory Ⅰ is established using the synthesis information geological variables. Then fewer independent variables, much relative to the dependent variables, are chosen following the stepwise regression procedure, the regression coefficients are obtained and the regression equation as shown below is given

μ=1.842 0+0.000 4x₅-0.023 0x₁₀+0.114 4x₁₆+0.002 0x₂₃+0.029 1x₃₉-0.042 1x₄₀-0.082 8x₄₁+1.479 8x₄₈-0.229 4x₅₁+0.242 8x₅₉+0.992 5x₆₇-0.139 8x₇₆+0.019 7x₇₇-0. 430 9x₈₁

where x₅ is the anomaly intensity of Au element; x₁₀ is the anomaly intensity of associated element Mo in high temperature anomaly combination; x₁₆ is the anomaly area of associated element Mo in high temperature anomaly combination; x₂₃ is the anomaly intensity of associated element Zn in moderate temperature anomaly combination; x₃₉ is the anomaly area of associated element Ag in low temperature anomaly combination; x₄₀ is the anomaly area of associated element As in low temperature anomaly combination; x₄₁ is the anomaly area of associated element Hg in low temperature anomaly combination; x₄₈ is the concealed basement; x₅₁ is the volcanic lithology in Devonian period; x₅₉ is the ultrabasic and basic rock body; x₆₇ is the aeromagnetic structure in nonage; x₇₆ is the aeromagnetic north-westward; x₇₇ is the aeromagnetic west-eastward; x₈₁ is the gravitational and aeromagnetic fit north-southward.

The evaluations of parameter μ in 16 Au deposits cells are shown in Table 1.

Table  1.  VALUES OF PARAMETER μ IN Au DEPOSIT CELLS

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The estimated values of parameter μ in the forecast cells are shown in Table 2 following the regression equation mentioned above.

Table  2.  ESTIMATED VALUES OF PARAMETER μ IN FORECAST CELLS

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According to the estimated values of parameter μ in the forecast cells presented in Table 2, the following forecast results are shown here: The estimated values of parameter μ in the forecast cells numbered as 1, 11, 12, 16 and 28 are more than 3, indicating the occurrence of Au deposits in these cells, and therefore, the cells numbered as 16 and 12 possibly exist in the large-size Au deposits. The forecasting results are generally in accord with those obtained with other methods (Shen and Wang, 1999), for example, with the characteristic analysis.