介質非均勻性之空間結構在水文地質場中扮演著重要的角色，其顯著地影響地下水流及污染傳輸之行為。許多現地和實驗室量測資料顯示土壤性質之空間分佈並非平滑變化而呈現出不均勻的分佈，且此非均勻性存在於不同大小之空間尺度中。傳統序率分析多使用多變量高斯模式，此模式適用於描述中度非均勻性之土壤資料，但現地的資料可能顯示出高度變異之非高斯行為，因此傳統之高斯模式並不適用於許多的案例。近來一些研究顯示地球物理資料的增量值可以Levy-stable機率密度函數分佈予以適當地描述。Levy-stable統計分佈具有自我相似的特性，其機率密度函數具有冪指數(Power-law)的特徵，此緩慢衰退之統計特徵乃由無限二階動差所導致，使其具有模擬高度非均勻性介質性質之能力。本研究利用卡方合適度檢定來檢定高斯分佈之適用性，接著使用濁水溪沖積扇正長距電阻井測、正短距電阻井測、自然伽瑪射線井測及自然電位井測四種地球物理井測來分析Levy-stable分佈之存適用性。研究結果顯示Levy-stable分佈較傳統之高斯模式能適當地描述地球物理井測資料之機率密度函數。本研究並利用條件模擬建構一隨機物理地球場，將其轉換為水力傳導係數場。所求得隨機場保留原有之Levy-stable統計特性，有助提高地下水流及污染傳輸行為預測之精確性。

　　The spatial variability of hydrogeological properties plays an important role in understanding and predicting the behavior of the groundwater flow. Many field and laboratory measurements show that the soil properties of geological formations do not smoothly vary in spatial domain and the heterogeneity may exist over a wide range of spatial scales. The classical Gaussian-based model can be used only for mildly heterogeneous formations while the field data usually display highly non-Gaussian behaviors. Thus, the Gaussian-based model may not be appropriate for most geophysical data. Several researchers have shown that the incremental values of various geophysical data can be accurately modeled by using the Levy-stable distribution. The Levy-stable distribution has a canonical role in mathematical statistics similar to the Gaussian distribution but with a power-law tail. The character of slowly decaying tails in probability density function is due to the infinite variance which makes the Levy-stable distribution useful for modeling a system with a strongly spatial variability. In this study, four geophysical well logging measurements of Chou-Shui-Shi alluvial fan including long normal electrical resistivity logging, short normal electrical resistivity logging, natural gamma-ray logging and spontaneous potential logging were analyzed to examine the existence of the Levy-stable distribution. Our results show that the Levy-stable distribution is a more appropriate model than the Gaussian distribution for the analyzed data. The Levy-stable distribution does exist in geophysical data of Chou-Shui-Shi alluvial fan. A geophysical random field was constructed by conditional simulation of trend-removed residuals. This random field was transformed to the hydraulic conductivity random field based on the relation between electrical resistivity and hydraulic conductivity in Chou-Shui-Shi alluvial fan. The constructed random field preserves fLm characteristic of original data and provides a high-resolution model for ground water model.

References

1. 中央地調所,濁水溪沖積扇水文地質調查研究總報告「台灣地區地下水觀測網整體計劃」第一期計畫成果，1999。
2. 紀明輝，”濁水溪沖積扇之水文地質特性分析”，國立成功大學資源工程研究所碩士論文, 1997。
3. 楊哲一、陳冠志、徐國錦，Levy-stable統計分佈是否存在於自然環境資料，台灣水利季刊，52(4), 69-78, 2004。
4. Bartoli, F. G., J. J. Burtin, M. G., V. Gomendy, R. Philippy, T. Leviandier, and R. Gafrej, Spatial variability of topsoil characteristics within one silty soil type. Effects on clay migration, Geoderma, 68, 279-300, 1995.
5. Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36(6), 1403-1412, 2000.
6. Benson, D. A., R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft,
Fractional Dispersion, Levy motion, and the MADE tracer tests,
Transport in porous media, 42, 211-240, 2001.
7. Boufadel, M. C., S. Lu, F. J. Molz, and D. Lavallee, Multifractal scaling of the intrinsic permeability, Water Resour. Res., 36(11), 3211-3222, 2000.
8. Brax, P., and R. Peschanski, Levy stable law description of intermittent behavior and quark-gluon plasma phase transitions, Phys. Lett. B, 253, 225-230, 1991.
9. Carr, J. R., and W. B. Benzer, On the practice of estimating fractal dimension, Mathematical Geology, 23, 945-958, 1991.
10. Desbarats, A. J., and S. Bachu, Geostatistical analysis of aquifer heterogeneity from the core scale to the basin scale：A case study, Water Resources Research, 30(3), 673-684, 1994.
11. Deutsch, C. V., Geostatistical reservoir modeling, Oxford University
Press, New York, 2002
12. Fama, E., and R. Roll, Some properties of symmetric stable distributions, Jour. Am. Stata. Assoc., 63, 817-836, 1968.
13. Fama, E., and R. Roll, Parameter estimates for symmetric stable distribution: Jour. Am. Stata. Assoc., 66(334), 331-338, 1972.
14. Goovaerts, P., Geostatistics for natural resources evaluation︰Oxford University Press, New York, 483 p, 1997.
15. Hsu, K.-C., and C.-C. Tung, On estimating the earthquake-induced changes in hydrogeological properties of the Choshuishi Alluvial Fan, Taiwan, Hydrogeology Journal, accepted, 2004.
16. Herrick, M. G., D. A. Benson, M. M. Meerschaert, and K. R. McCall, Hydraulic conductivity, velocity, and the order of the fractional dispersion derivative in a highly heterogeneous system, Water Resour. Res, 38(11), 1227, 2002.
17. Hewett, T. A., Fractal distribution of reservoir heterogeneity and their influence on fluid transport, paper presented at the 61st Annual Technical Conference, Soc. of Pet. Eng., Richardson, Tex., 1986.
18. Hurst, H. E., A suggested statistical model for some time series that occur in nature, Nature, 180, 494-495, 1957.
19. Isaaks, E. H. and M. R. Srivastava, An introduction to applied geostatistics︰Oxford University Press, New York, 561 p, 1989.
20. Janicki, and Weron, simulation and chaotic behavior of α－stable stochastic processes, marecl dekker, 1994
21. Kida, S., Log-state distribution and intermittency of turbulence, J. Phys. Soc. Jpn., 60, 5-8, 1991.
22. Leuangthong, O., and C. V. Deutsch, Transformation of residuals to avoid artifacts in geostatistical modeling with a trend, Mathematical Geology, Vol. 36, No. 3. 287-305, 2004.
23. Liu, H. H. and F. J. Molz, Comment on“Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations”by Scott, Water Resour. Res., 33, 907-908, 1997.
24. Liu, H. H., G. S. Bodvarsson, F. J. Molz, and S. Lu, A corrected and Generalized successive random additions algorithm for simulating fractional Levy motions, Mathematical Geology., 36, 361-378, 2004.
25. Lu, S., F. J. Molz, and H. H. Liu, An efficient, three-dimensional, anisotropic, fractional Brownian motion and truncated fractional Levy motion simulation algorithm based on successive random additions Computers geosciences, 29, 15-25, 2003.
26. Mandelbrot, B. B., How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 155, 636-638, 1967.
27. Mandelbrot, B. B., The Fractal Geometry of Nature, 460 pp., W. H. Freeman, New York, 1982.
28. Mandelbrot, B. B., and J. W. Van Ness, Fractional Brownian motion, fractional noises, and applications, SIAM Rev., 10, 422-437, 1968.
29. Mandelbrot, B. B., and J. R. Wallis, Noah, Joseph, and operation Hydrology, Water Resour. Res., 4, 909-918, 1968.
30. Mandelbrot, B. B., and J. R. Wallis, Some long run properties of geophysical records, Water Resour. Res., 5, 321-340, 1969a.
31. Mandelbrot, B. B., and J. R. Wallis, Robustness of the rescaled range R/S in the measurement of noncyclic long-run statistical dependence, Water Resour. Res., 5, 967-988, 1969b.
32. Molz, F. J., and G. Boman, A fractal-based stochastic interpolation scheme in subsurface hydrology, Water Resour. Res., 29, 3769-3774, 1993.
33. Molz, F. J., and G. Boman, Further evidence of fractal structure in hydraulic conductivity distributions, Geophys. Res. Lett., 22, 2545-2548, 1995.
34. Molz, F. J., H. H. Liu, and J. Szulga, Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions, Water Resour. Res., 33, 2273-2286, 1997.
35. Neuman, S. P., Universal scaling of hydraulic conductivities and dispersivities in geologic media, Water Resour. Res., 26, 1794-1758, 1990.
36. Neuman, S. P., On advective transport in fractal permeability and velocity fields, Water Resour. Res., 31, 1455-1460, 1995.
37. Painter, S., Random fractal models of heterogeneity: The Levy-stable approach, Math. Geoll., 27, 813-830, 1995.
38. Painter, S., Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formation, Water Resour. Res., 32, 1183-1195, 1996a.
39. Painter, S., Stochastic interpolation of aquifer properties using fractional Levy motion, Water Resour. Res., 32, 1323-1332, 1996b.
40. Painter, S., Reply, Water Resour. Res., 33, 909-910, 1997.
41. Painter, S., Numerical method for conditional simulation of Levy random field, Mathematical Geology., 30, 163-179, 1998.
42. Painter, S., Flexible scaling model for use in random field simulation of hydraulic conductivity, Water Resour. Res., 37, 1155-1163, 2001.
43. Painter, S., and L. Paterson, Fractional Levy motion as a model for spatial variability in sedimentary rocks, Geophys. Res. Lett., 21, 2857-2860, 1994.
44. Press, S. J., Multivariate stable distributions︰Jour. Multivar. Anal., v. 2, 444-462 p, 1972a.
45. Press, S. J., Estimation in univariate and multivariate stable distributions︰Jour. Am. Stat. Assoc., v. 7, no. 340, 842-846 p, 1972b.
46. Schertzer, D., and S. Lovejoy, Physical modeling and analysis of rain and clounds by anisotropic scaling multiplicative processes, J. Geophys. Res., 92, 9693-9714, 1987.
47. Schertzer, D., S. Lovejoy, D. Lavallee, and F. Schmitt, Universal hard multifractal turbulence: Theory and observations, in Nonlinear Dynamics of Structures, edited by R. Z. Sagdeev et al., 213-235 p, North-Holland, New York, 1991.
48. Shen, W., and P. D. Zhao, Multidimensional self-affine distribution with application in geochemistry, Mathematical Geology, Vol. 34, No. 2, 2002.
49. Sung, Q., and D. Tsaan, Fractal dimension of topographic surface of Taiwan and its geomorphic implications, Journal of the Geological Society of China, 35(4), 389-406, 1992.
50. Taqqu, M. S., Random processes with long-range dependence and high variability, Journal of Geophysical research, 92, 9683-9686, 1987.
51. Tennekoon, L., M. C. Boufadel, D. Lavallee and J. Weaver, Multifractal anisotropic scaling of the hydraulic conductivity, Water Resour. Res., 39(7), 1193, 2003.