異質性介質為非單一同材料所組成，在大地材料中我們可將併構岩視為異質性介質，其力學性質與體積比(volumetric fraction,Vf)兩者呈現正相關或負相關的成長，因此，本文探討重點為量測體積比時的不確定性。透過立體量測學(stereology)的概念，以二維量測(2-D probes)來求取面積比，進而推估其三維形貌(3-D feature)下的體積比。本文針對三維形貌下以二維量測體積比之不確定性提出數值解與解析解。在解析解方面，利用球－立方體之表徵單元(representative volume element, RVE)作為數學模型，以應用統計學之方法，推導出在不同量測面積與體積比下，二維量測時所產生的不確定性，也提出如何處理多重粒徑的問題。在數值解方面，利用MATLAB撰寫三維空間中球顆粒隨機分佈的程式，以作為數值解分析所用模型，探討當隨機選取二維量測面下，各量測面上體積比的不確定性。兩者結果均顯示，當體積比越高時，以二維量測體積比的不確定性越低。最後，以磨石子地磚的案例，利用隨機抽樣方法對解析解、數值解比較。

This research presented analytical solution and numerical simulation to quantify the uncertainty of volumetric fraction (Vf) estimates in a heterogeneous material using 2-D probes. The analytical solution was derived based upon the concept of representative volume element (RVE).The numerical simulation was used random model to determine the uncertainty of volumetric fraction. The results show that the uncertainties of the estimates depend upon the size of blocks, measurement area, and volumetric fraction. Finally, using case of terrazzo tiles application examples to compare the analytical solution and numerical simulation at the end.

1.林惠玲、陳正倉，應用統計學，三版，雙葉書廊，台北市(2006)。
2.田永銘、盧育辰、鄔定樺、鐘翊展、林晉祥，「一維量測量測岩塊體積比之不確定性解析解」，2010全國岩盤工程研討會，高雄 (2010)。
3.田永銘、鐘翊展、盧育辰、張顥薰，「併構岩體積比與力學性質之不確定性(1/3)」，行政院國家科學委員會補助專題研究計畫期中進度報告，中壢 (2010)。
4.郭明傳，「複合岩體之岩塊體積比量測及其力學行為」，博士論文，國立中央大學土木研究所，中壢（2005）。
5.鄔定樺，「掃描線法量測體積比之不確定性-解析解及驗證」，碩士論文，國立中央大學土木工程研究所，中壢 (2010)。
6.蕭永燦，「以二維量測決定岩塊體積比之信賴區間」，碩士論文，國立中央大學土木研究所，中壢（2008）。
7.Kahraman, S., and Alber, M., “Estimating unconfined compressive strength and elastic modulus of a fault breccias mixture of weak blocks and strong matrix,” International Journal of Rock Mechanics & Mining Science, Vol. 43, pp. 1277-1287 (2006).
8.Kahraman, S., and Alber, M., “Triaxial strength of a fault breccias of weak rocks in a strong matrix,” Bulletin of Engineering Geology and the Environment, Vol. 67, pp. 435-441 (2008).
9.Lind,　D. A., Marchal, W. G., and Wathen, S. A., Statistical techniques in business and economics, 13th edition, The McGraw-Hill Companies, New York (2008).
10.Lindquist, E. S., “The strength and deformation properties of melange,” Ph.D. Dissertation, Department of Civil Engineering, University of California, Berkeley (1994).
11.Lindquist, E. S., and Goodman, R. E., “Strength and deformation properties of a physical model melange,” Proceedings of the 1th Congress of the North American Rock Mechanics Symposium, Austin, Texas, USA, pp. 851-858 (1994).
12.Medley, E. W., “Using stereological methods to estimate the volumetric proportions of blocks in melanges and similar block-in-matrix rocks (bimrocks),” Proceedings 7th International Congress, International Association of Engineering Geology, Lisbon, Portugal, pp. 1031-1040 (1994).
13.Medley, E. W., “Uncertainty in estimates of block volumetric proportions in melange bimrocks,” Proceedings International Symposium on Engineering Geology and the Environment, Athens, Greece, pp. 267-272 (1997).
14.Medley, E. W., and Goodman, R. E., “Estimating the block volumetric proportions of melanges and similar block-in-matrix rocks (bimrocks), ” Proceedings of the 1st North American Rock Mechanics Symposium, Austin, Texas, USA, pp. 851-858 (1994).
15.Medley, E. W., “Orderly characterization of chaotic Franciscan melanges,” Felsbau Rock and Soil Engineering Journal for Engineering Geology, Geomechanics and Tunneling, Vol. 19, No. 4, pp. 20-33 (2001).
16.Medley, E. W., “Estimating Block Size Distributions of Melanges and Similar Block-in-Matrix Rocks (Bimrocks),” Proceedings of 5th North American Rock Mechanics Symposium, ed. by Hammah, R., Bawden, W., Curran, J. and Telesnicki M., Toronto, Canada, pp. 509-606 (2002).
17.Russ, J. C., and Dehoff, R. T., Practical Stereology, 2nd edition, Plenum Press, New York (1999).
18.Sonmez, H., Gokeceoglu, C., Medley, E. W., Tuncay, E., and Nefeslioglu, H. A., “Relationships between volumetric block proportions and overall UCS of a volcanic bimrock,” International Journal of Rock Mechanics & Mining Science, Vol. 22, pp. 27-34 (2004).
19.Tien, Y. M., Kuo, M. C., Lu, Y.C., Chung, Y. J., Lin, J. S., Wu, T.H., and Lee, D. H., “Uncertainty in Estimation of Volumetric Block Proportion of Bimrocks by Using Scanline Method,” 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Symposium, paper No. 10-158, Salt Lake City, Utah, USA (2010).
20.Tien, Y.M., Lu, Y.C., Wu, T.H., Lin, J.S., and Lee, D.H., “Quantify uncertainty in scanline estimates of volumetric fraction of anisotropic bimrocks,” 45th U.S. Rock Mechanics Symposium, paper No. 11-345, San Francisco, California, USA (2011).
21.Tien, Y.M., Lu, Y.C., Chang, H.H., Chung, Y.C., Lin, J.S., and Lee, D.H., “Uncertainty of volumetric fraction estimates using 2-D measurements,” 46th U.S. Rock Mechanics Symposium, paper No. 12-494, Chicago, IL, USA (2012).