Spatial modeling principles in earth sciences

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Zekai Sen Spatial Modeling Principles in Earth Sciences Second Edition Spatial Modeling Principles in Earth Sciences ThiS is a FM Blank Page Zekai Sen Spatial Modeling Principles in Earth Sciences Second Edition Zekai Sen Faculty of Civil Engineering Istanbul Technical University Istanbul, Turkey ISBN 978-3-319-41756-1 ISBN 978-3-319-41758-5 DOI 10.1007/978-3-319-41758-5 (eBook) Library of Congress Control Number: 2016951726 1st edition: © Springer Science+Business Media B.V 2009 © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland Our earth is the only place where we can live in harmony, peace, and cooperation for the betterment of humanity It is the duty of each individual to try with utmost ambition to care for the earth environmental issues so that a sustainable future can be handed over to new generations This is possible only through the scientific principles, where the earth systems and sciences are the major branches This book is dedicated to those who care for such a balance by logical, rational, scientific, and ethical considerations for the sake of other living creatures’ rights ThiS is a FM Blank Page Preface Earth systems and sciences cover a vast amount of disciplines that are effective in the daily life of human beings Among these are the atmospheric sciences, meteorology, hydrogeology, hydrology, petroleum geology, engineering geology, geophysics, and marine and planary systems Their subtopics such as floods, groundwater and surface water resources, droughts, rock fractures, and earthquakes have basic measurements in the form of records of their past events and need to have suitable models for their spatio-temporal predictions There are various software for the solution of problems related to earth systems and sciences concerning the environment, but unfortunately they are ready-made models, most often without any basic information It is the main purpose of this book to present the basic knowledge and information about the evolution of earth sciences events on rational, logical, and at places scientific and philosophical features so that the reader can grasp the underlying principles in a simple and applicable manner The book also directs the reader to proper basic references for further reading and enlarging the background information I have gained almost all of the field, laboratory and theoretical as well actual applications, during my stay at the Faculty of Earth Sciences, Hydrogeology Department, King Abdulaziz University (KAU), and the Saudi Geological Survey (SGS), which are in Jeddah, Kingdom of Saudi Arabia Additionally, especially on the atmospheric sciences, meteorology and surface water resources aspects are developed at Istanbul Technical University ă ), Turkey (ITU It is well sought to adapt spatial modeling and simulation methodologies in actual earth sciences problem solutions for exploring the inherent variability such as in fracture frequencies, spacing, rock quality designation, grain size distribution, groundwater exploration and quality variations, and many similar random behaviors of the rock and porous medium The book includes many innovative spatial modeling methodologies with actual application examples from real-life problems Most of such innovative approaches appear for the first time in this book with the necessary interpretations and recommendations of their use in real life vii viii Preface The second print of the book indicates the need for spatial modeling in earth sciences The book has an additional chapter and also some recent methodological procedures in some chapters, which cannot be found in the first print I wish that the content will be beneficial to anyone interested in spatial earth system modeling and simulation Throughout the first and this second edition preparation process, my wife Fatma Hanim has encouraged me to think that such a work will be beneficial to all humans in the world and those who are interested in the topics of this book I appreciate the encouragement by Springer Publishing Company for the second printing of this book Istanbul, Turkey 18 May 2016 Zekai Sen Contents Introduction 1.1 General 1.2 Earth Sciences Phenomena 1.3 Variability 1.3.1 Temporal 1.3.2 Point 1.3.3 Regional 1.3.4 Spatial 1.4 Determinism 1.5 Uncertainty 1.5.1 Probabilistic 1.5.2 Statistical 1.5.3 Stochastic 1.5.4 Fuzzy 1.5.5 Chaotic Uncertainty 1.6 Random Field (RF) References 1 12 12 13 13 13 14 15 16 17 18 19 21 22 Sampling and Deterministic Modeling Methods 2.1 General 2.2 Observations 2.3 Sampling 2.4 Numerical Data 2.5 Number of Data 2.5.1 Small Sample Length of Independent Models 2.5.2 Small Sample Length of Dependent Models 2.6 Regional Representation 2.6.1 Variability Range 2.6.2 Inverse Distance Models 25 25 26 29 34 36 37 40 46 46 49 ix 398 Spatial Simulation Table 7.8 Input data for fracture network generation Property Orientation Trace length (m) Intact length (m) Location Distribution Normal Lognormal Negative exponential Bristol Set Mean 30 0.2 Variance 0.05 Set Mean 130* 0.18 Variance 0.04 Set Mean 0.16 Variance 0.04 0.013  10À4 0.013  10À4 0.013  10À4 0.5 0.5 0.5 0.5 0.5 0.5 pattern Three different orientations are adopted for the fracture orientation, each of which is assumed to be normally distributed about a given value Table 7.8 shows the relevant parameters taken for the simulation of fractures in this paper However, in any future study, they may be changed according to the field conditions This program can typically handle up to any number of sets, each containing any desired number of discontinuities The program finds the beginning and ending points’ coordinates of any trace The results of this program appear as a numerical representation of a 2D discontinuity network Such a network is one of the example samples of random discontinuity geometry generated according to the input data shown in Table 7.8 The sample area size in these generations is considered as  However, its transformation into any desired size is possible by the multiplication with any desired scale Some sample realization of generated discontinuity sets are shown in Fig 7.34 in detail as individual and combined fracture sets It is important to notice in the individual sets, i.e., Fig 7.34a, that some fracture traces intersect with the general frame of the area These fractures are referred to as censored trace lengths The generation scheme gives their total trace lengths, but in practice they are never known On the other hand, some short length traces appear entirely within the sample area Consideration of the whole set on one sample area, as in Fig 7.34c, makes the picture rather complex as it is the case in nature In order to assess the effect of different fracture sets individually or collectively on the RQD calculations, the sample area is overlain with scanlines in the east-west direction at different levels between and Of course, it is possible to take a scanline at any level, but the discussion here will concentrate on five different levels which are at 0.1, 0.3, 0.5, 0.7, and 0.9 The computer program lays these scanlines, at these levels, on the sample area of a generated fracture network; subsequently, it finds individually and collectively the intact lengths along them, ordering them in an ascending order, and attaches to each intact length an order to be used in Eq 7.101 The results are presented on semilogarithmic paper as shown in Fig 7.35 A visual inspection of these figures indicates that there appear wide ranges of data scatter, and with the consideration of more than one set of fracture, the scatter becomes less The best straight lines are fitted through the scatter of points; their slopes are found by considering a full cycle on the logarithmic axis The 7.7 Multi-directional RQD Simulation 399 a b 1.0 1.0 0.8 0.8 Y-Axis Y-Axis 0.6 0.4 0.6 0.4 0.2 0.2 0.0 0.0 0.2 0.4 0.6 0.8 0.0 0.0 1.0 X-Axis 0.2 0.4 0.6 X-Axis 0.8 1.0 c 1.0 0.8 Y-Axis 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 X-Axis Fig 7.34 (a) A fracture set with 30 orientation from the north, (b) two fractures sets with 0 and 30 orientation from the north, (c) three fractures sets with 0 , 30 , and 130 orientation from the north corresponding length on the horizontal axis is equal to the slope, i.e., 1/λ parameter This slope calculation procedure is presented in Fig 7.35c The results are summarized in Table 7.9 As is obvious from Table 7.9, the best rock quality is obtained when only one fracture set is considered As expected, an increase in the number of fracture sets can cause a decrease in the RQD value There appears a practically significant difference between the RQD values based on the single set of fracture and those on two or three fracture sets The rock quality is of an excellent type when one set of fractures is considered; whereas the quality deteriorates and becomes very good and fair for two and three fracture sets, respectively The last row in Table 7.9 gives average RQD values for all the scanlines considered in this study It is obvious from the comparison of these values that when two sets are considered, the RQD deterioration relative to one set is almost 15 %, whereas for three sets consideration is almost 30 % 400 Spatial Simulation Fig 7.35 Cumulative probability plots of intact lengths at the 0.1 level (a) Single set at 30 , (b) two sets at 0 and 30 , (c) three sets at 0 , 30 , and 130 Table 7.9 Slope and RQD values for t ¼ 0.1 m Level Slope RQD % 0.3 Slope RQD % 0.5 Slope RQD % 0.7 Slope RQD % 0.9 Slope RQD % Average RQD % 0.1 One set 0.267 94.5 0.214 92.0 0.227 92.7 0.212 91.8 0.188 90.0 92.2 Two sets 0.120 79.8 0.135 83.0 0.117 79.0 0.122 80.0 0.092 70.4 78.4 Three sets 0.106 75.6 0.128 81.0 0.079 63.9 0.050 40.6 0.076 62.0 64.6 References 401 References Agterberg F (1975) New problems at the interface between geostatistics and geology Geostatistics 75:403–421 Baczynski NRP (1980) Rock mass characterization and its application assessment of unsupported underground openings Ph.D Thesis University of Melbourne, Australia, 233 p Baecher GB, Lanney NA, Einstein HH (1977) Statistical description of rock properties and sampling 18th U.S symposium on Rock Mechanics, PP 5C1 Barton CM (1977) Geotechnical analysis of rock structure and fabric in C.S.A Mine, Cobar, New South Wales Applied Geomechanics, Technical paper 24, CSIRO, Australia Barton N, Lien F, Lunde J (1974) Engineering classification of rock masses for the design of tunnel support Rock Mech 6(4):189–236 Bieniawski ZT (1974) Geomechanic classification of rock masses and its application in tunneling International Society of Rock Mechanics, 3rd Denver Colorado September 1974 Proceedings, vol 2A pp 2732 Box GEP, Jenkins GM (1970) Time series analysis, forecasting and control Holden Day, San Francisco, 475 pp Brady BHG, Brown ET (1985) Rock mechanics for underground mining Allen and Unwin, London Bridges MC (1975) Presentation of fracture data for rock mechanics Second Australia New Zealand Conference on geomechanics, Brisborne, Australia, pp 144–148 Brown ET (1981) Rock characterization testing and monitoring Pergamon Press, Oxford, 221 pp Call RB, Savely J, Nicholas DE (1976) Estimation of joint set characteristics from surface mapping data 17th U.S symposium on Rock Mechanics, pp 2B21–2B29 Cliff AD, Ord JK (1973) Spatial autocorrelation Pion, London Coates DF (1964) Classification of rocks for rock mechanics Int J Rock Mech Min Sci Geomech Abstr 1(3):421–429 Cording EJ, Mahar JW (1978) Index properties and observations for design of chambers in rock Eng Geol 12:113–142 Cramer (1938) Specific surface? 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Journel AG (1974) Simulation conditionelle de gisements miniers-theorie et pratique These de Doctor Ingenieur, Universite de Noncy Kazi A, Sen Z (1985) 1Volumetric RQD; an index of rock quality In: Proceedings of the international symposium on fundamentals of Rock Joints Bjorkliden pp 99102 Krige DG (1951) A statistical approach to some basic mine valuation problems on the Witwaterstrand J Chem Metall Min Soc S Afr 52:119–139 Krumbein WC (1970) Geological models in transition to geostatistics, in geostatistics Plenum Press, New York, pp 143–161 Kulhawy FH (1978) Geomechanical model for rock foundation settlement J Geotech Eng Div ASCE 104(GT2, Proc Paper 13547):211–228 Long JCS (1983) Investigation of equivalent porous medium permeability in networks of discontinuous fractures Ph D Thesis, University of California, Berkeley Louis C, Pernot M (1972) Three-dimensional investigation of flow conditions of Grand Maison Dam Site In: Proceedings of the symposium on international society for 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scanline surveys Int J Rock Mech Min Sci Geomech Abstr 18(3):183–197 Rice SO (1945) Mathematical analysis of random noise Bell Sys Tech J 24:46–156 Roach (1968) For independent processes simulation Roulean A, Gale JE (1985) Statistical characterization of the fracture system in the Stripa Granite, Sweden Int J Rock Mech Min Sci Geomech Abstr 22(6):353–367 Rouleau A (1984) Statistical characterization and numerical simulation of a fracture system — application to groundwater flow in the Stripa granite Ph.D thesis, University of Waterloo, Ontario Ryckes KD (1984) A routine method to evaluate the modes of failure and the stability of rock slopes Unpublished thesis, Imperial College of Science, Technology and Medicine, 246 pp S¸en Z (1978) Autorun analysis of hydrologic time series J Hydrol 36:75–85 S¸en Z (1979a) Application of autorun test to hydrologic data J Hydrol 42:1–7 S¸en Z (1979b) Effect of periodic parameters on the autocorrelation structure of hydrologic series Water Resour Res 15(6):1639–1642 References 403 S¸en Z (1980) Statistical analysis of hydrologic critical droughts J Hydraul Div ASCE Proc Pap 14 134, 106(HY1):99–115 S¸en Z (1984) RQD models and fracture spacing J Geotech Eng Am Soc Civ Eng 110(2):203–216 S¸en Z (1985) Autorun model for synthetic flow generation J Hydrol 81:157–170 S¸en Z (1989a) Cumulative semivariogram models of regionalized variables Int J Math Geol 21 (3):891–903 S¸en Z (1989b) Comment on “The generation of multi-dimensional autoregressive series by herringbone method” Math Geol 21(2):267–268.22 S¸en Z (1990a) RQP RQR and fracture spacing Technical note Int J Rock Mech Min Sci Geomech Abstr 22:135–137 (in print) S¸en Z (1990b) Spatial simulation of geologic variables J Math Geol 22(2):175–188 S¸en Z (1991) The profile-area method for statistical evaluation of rock mass quality Bull Assoc Eng Geol XXVIII:351–357 S¸en Z (1974) Small sample properties of stationary stochastic models and the hurst phenomenon in hydrology Unpublished Ph.D thesis, Imperial College of Science and Technology, University of London, 286 pp S¸en Z, Kazi A (1984) Discontinuity spacing and RQD estimates from finite length scanlines Int J Rock Mech Min Sci Geomech Abstr 21(4):203–212 Sharp WE, Aroian LA (1985) The generation of multidimensional autoregressive series by herringbone method Math Geol 17:67–79 Smith L, Schwartz FW (1984) An analysis on the influence of fracture geometry on mass transport in fractured media Water Resour Res 20:1241–1252 Steffen O (1975) Recent developments in the interpretation of data from joint surveys in rock masses 6th regional conference of Africa on oil Mechanical and Foundation, II, pp1726 Subyani A, S¸en Z (1990) A comparative study between cumulative semivariogram and semivariogram for regular data J Hydrol (in print) Switzer P (1965) A random set process with a Markovian property Ann Math Stat 36:1859–1863 Terzaghi KC (1946) Introduction to tunnel geology: rock defects and loads on tunnel supports In proctor R.V and Terzaghi RD (1965) Sources of error in joint surveys Geotechnique 15:237–304 Wallis PF, King MS (1980) Discontinuity spacing in a crystalline rock Int J Rock Mech Min Sci Geomech Abstr 17:63–66 Watson (1975) Correlation slope at the origin?? Index A Aboufirassi, M., 185, 311 Agterberg, F., 330 Akin, J.E., 52 Al-Jerash, M., 309 Anisotropy model auto- and cross-correlation in, 333, 350 ratio and angle, 180, 181 Area of influence methods, 178 Areal average estimation (AAE), 52, 54, 58, 59, 64–68, 71 Areal coverage probability (ACP), 71–73, 75 ARIMA Model See Autoregressive integrated moving average (ARIMA) models Aroian, L.A., 331, 333 Atmosphere, 5, 7, 40, 159, 179, 182 Atwater, M.A., 59 Autocorrelation structures, 40, 41, 43, 124, 361, 364, 386 Autoregressive integrated moving average (ARIMA) models, 40, 41, 43–45, 123–124, 303, 361, 363, 364, 366, 381, 386, 387, 392 RQD–persistence relation, 365–366 Autoregressive models, 331, 339 Autorun coefficient, 351 vs autorun function, 359, 362, 364 discontinuity vs scanline length, 354 Autorun function vs autorun coefficient, 154 of dependent process, 361 for discrete processes, 371 of independent process, 372 Autorun simulation of porous material analysis of sandstone, 371–375 line characteristic function of porous, 370–371 medium, 370–372 modeling of porous media, 375–380 B Baczynski, N.R.P., 394 Baecher, G.B., 382 Barchet, W.R., 190 Barnes, R.J., 208 Barnes, S.L., 150–152, 183, 256 Barros, V.R., 189, 190 Barton, C.M., 381 Barton, N., 339, 341 Bayer, A.E., 183 Bayraktar, H., 67 Benjamin, J.R., 121, 123, 140, 143, 186 Bergman, K., 179 Bergthorsson, P., 183 Bernoulli distribution theory, 77, 86 Bieniawski, Z.T., 339, 341 Block Kriging, 282, 301, 302 Box, G.E.P., 17, 330, 384, 386 Box, G.F.D., 124, 185 Box, G.F.P., 303 Brady, B.H.G., 360 Branstator, G.W., 270 Bras, R.L., 52 Bratseth, A.M., 184 Bridges, M.C., 381 Brown, E.T., 358 Brown, R.G., 159 Bruce, J.P., 65 Bucy, R., 159 © Springer International Publishing Switzerland 2016 Z Sen, Spatial Modeling Principles in Earth Sciences, DOI 10.1007/978-3-319-41758-5 405 406 Buell, C.E., 256 Buzzi, A., 184 C Call, R.B., 382 Carr, J.R., 185, 276 Chow, V.T., 52 Chu, P.S., 40 Clark, I., 195, 198, 203, 239, 245, 256, 276, 280, 294 Clark, R.H., 65 Classical spatial variation models cluster sampling, 145–146 multisite Kalman filter methodology, 159–175 KF application, 163–175 KF iterative procedure, 159–160 one-dimensional KF, 137 ongoing discrete KF cycle, 162 nearest neighbor analysis, 146–148 random field, 141–144 ratio statistics, 149 search algorithms geometric weighting functions, 150–153 quadrant and octant, 148, 150 weighting functions, 151–152 spatial data analysis needs directional variations, 140 easting, northing, and ReV, 134 relative frequency diagram, 135 representative ReV values, 137 ReV sampling locations, 137–138 spatial pattern, 134, 136 univariate statistical parameters, 136–137 spatial pattern search data variation, 131–132 systematic and unsystematic components, 132–133 spatio-temporal characteristics, 130 trend surface analysis (global interpolator) calculation table, 156–157 model parameter estimations, 155–158 uniformity test, 138–141 Cliff, A.D., 331 Cluster sampling, 29 Coates, D.F., 339 Co-Kriging, 278, 282 Cooley, R.L., 185 Cording, E.J., 341 Cornell, C.A., 121, 123, 140, 143, 186 Covariogram analysis, 138 Cramer, H., 38, 375 Index Cressie, N.A.C., 304 Cressman, G.P., 150, 151, 183, 256 Cross-validation technique, 239, 273 Cruden, D.M., 341 Cumulative semivariogram (CSV) ARIMA model, 386, 387 attributes and advantages, 218 vs empirical method/simulation technique, 382 experimental, 237–239 graphs comparison, 392 independent model, 384 intact length correlation structure, 381–393 intact length CSV, 383–388 Markov model, 385 model for ARIMA intact lengths, 387 obtaining from ReV data, 216 parameters, 392 practicalities and limitations of, 199 theoretical CSV model, 220–226 D Daley, R., 179, 184 Data collection, 9, 21, 131, 133–135, 281 Data types and logical processing methods areal coverage probability, 71–73 dry durations and areas, 72 extreme value probabilities, 76–77 theoretical treatment, 73–76 time and spatial units, 72 number of data dependent/independent models, sample, 37–46 sample points scatter diagrams, 25 standard Gaussian pdf, 38–39 numerical data types pixel sampling, 34 point sampling, 34 observations, 26–28 polygonizations Delaney, Varoni, and Thiessen polygons, 57–59 percentage-weighted polygon (PWP) method, 59–71 triangular coordinate paper, 60 regional representation inverse distance models, 49–51 variability range, 46–49 sampling areal patterns, 29–30 calcium, sodium, and chloride measurements, 32–33 elevation data, 31–32 Index irregular sample points scatter, 31–32 quadrangle, 29, 31 spatio-temporal drought theory and analysis areal drought variance percentage, 84 average areal drought coverage, 84 average variance, 85 drought area probability, 80 drought parameters, 80–84 drought percentage areal coverage, 81 drought time change, 82 probability of drought area, 79 sub-areal partition monthly averages, 56 seasonal averages, 56 triangularization, 51–55 triangular mesh, 55 values of i, j, k, 56 Dee, D.P., 159 Dee, P.D., 184 Deere, D.U., 339–341, 347, 359, 360 Delhomme, J.P., 280 Demaree, G., 303 Determinism versus uncertainty cause, environment, and effect, 15 triplicate, 255 diurnal temperature variation, 304 estimation, identification, and, 17, 19 prediction, 14–16 Deutsch, C.V., 311 DeWijs, H.J., 225 Dirichlet, G.L., 57 Discontinuities average number of, 353, 357, 393 scanline length for autorun coefficients, 354 average number per unit length, 353 occurrences along scanline, 354, 395 probability, 396 scanline survey of, 340 Discontinuity network properties, 394 Dixon, R., 256 Donnelly, K.P., 147 Drought occurrences, 3, 12, 78, 79, 83, 98, 130 Droughts, 3, 12, 37, 71, 77–83, 85, 86, 90, 99, 113, 130 E Earth, environment, and atmospheric researches, 2, 8, 11, 130 Earth sciences events, 8, 13, 14, 21, 22, 100, 114, 145 Earth sciences phenomena, 3–6, 8, 13, 25, 129 data-processing technique, geological phenomenon, characteristics, 407 rock subunit map, stratigraphic sequence, structural non-determinacy, thin-section spatial variability, 11 three rock type map, 6, Eberthardt, A.K., 303 Eddy, A., 179, 257 Ege, J.R., 339 Einstein, H.H., 382 Eissa, E.A., 382–384 Eliassen, A., 182, 257 Empirical linear interpolations, 256 Empirical technique, 382 Error theory, 178 Estevan, E.A., 189, 190 Expected analysis errors, 258, 261, 264, 269 zero intercept value, 269 F Feller, W., 77, 376 Field quantity, 112, 113, 117 First-order Markov model, 40 Fisher, R.A., 26, 255 Flattery, T.W., 256 Fracture network, 393–395 Franke, R., 184 Fritsch, J.M., 256 G Gale, J.E., 350 Gandin, L.S., 151, 179, 182, 256, 257, 272 Gauss, K.F., 26 General circulation models (GCM), 35, 40 Geometric weighting functions, 150–153 drawbacks, 151, 152 Geometric weights, 150–153 Geometry, 2, 3, 7, 8, 13, 20, 46, 57, 59, 97, 130, 131, 150–153, 340, 355, 357, 376, 381, 393 Geostatistical approaches, 199, 256 Geostatistics, 21, 52, 135, 191, 195, 198, 224, 275–276, 278–281, 323 defined, 276 Gilbert, S., 311 Gilchrist, B., 150, 183 Glass, C.E., 276 Global interpolator, 153 Goodin, W.R., 151, 183 Goodman, R.E., 339 Goodman, R.R., 341, 394 Gustafsson, N., 179 408 H Habib, Z.Z., 186, 227, 257, 262–264 Hammersley, J.M., 343 Handscomb, D.C., 343 Harrison, P.J., 159 Hevesi, J.A., 52, 311 Higgs, N.B., 350, 382, 383 Hoeksema, R.J., 185 Hohn, M.E., 203, 204 Homogeneity, 5, 7, 8, 84, 179, 198, 228, 254, 278 Hubert, P., 303 Hudson, J.A., 341, 351, 353, 355, 356, 358, 382, 384, 388, 395, 396 Huff, F.A., 59 Huijbregts, C.I., 185, 200, 204, 219, 223, 256, 280 Hwang, Y.C., 159 Index Journel, A.G., 185, 200, 219, 223, 256, 280, 311, 331 Journel, A.J., 185 I Independent processes, 43, 117–122, 124, 222, 364, 371, 372, 374–376, 379, 385, 388, 390 Indicator Kriging, 280, 282 Infinite sampling line, 376 Inman, R.L., 183 Intact length characteristics, 357, 358 correlation structure, 381–388 CSV model, 384–393 CSV technique, 381–388 dependent applications, 357–358 correlation measurement, 350–352 expectations of, 352, 353, 355 of independent, 340–347 simulation of, 358–360 International Society for Rock Mechanics, 340 Interpolation point value, 263, 264 Intrinsic hypothesis, 279 Intrinsic property, 277–279 Iruine, K.N., 303 Isaaks, E.H., 273, 302, 304 Isotropic models, 207, 335 Isotropy, 5, 7, 8, 179, 191, 198, 199, 254, 312, 335, 337, 370, 392 K Kalman filter annual precipitation, 164, 170 annual rainfall values, 164, 166, 167, 169, 175 application, 163–175 areal rainfall values, 174 difficulties, 159, 175 estimation execution steps, 163 geographic locations of precipitation, 164 iterative procedure, 159, 160 measurement update equations, 162 percentage error of annual rainfall, 164, 175 rainfall distribution, 163, 164, 167, 169, 170, 175 stations, 163–165, 169, 175 time update equations, 162 Kalman filtering techniques, 184 Kalman gain matrix, 174 Kalman, R.E., 159, 256 Karz, R.W., 40 Kazi, A., 339, 341, 347, 381, 395, 396 Kedem, B., 52 Kendall, M.G., 40 Kinematics variability, King, M.S., 382 Kitanidis, P.K., 185, 216, 304 Koch, S.E., 60, 136, 151, 183 Kolmogorov, A.N., 26 Krige, D.G., 185, 280, 330 Kriging advantages, 281–283 application of algorithm, 198, 301, 309 based on simple linear models, 280 characteristics and advantages, 280 estimation variances, 294, 297, 316, 320 intrinsic property, 277–279 methodology, 281–283 techniques, 276–277 types, 282 Kruger, H.B., 179 Krumbein, W.C., 330 Kulhawy, F.H., 341 J Jackson, I.J., 59 Jenkins, G.M., 17, 124, 185, 303, 330, 363, 372, 384, 386 L Lag-k autorun coefficient, 371 Lag-one Markov process, 337, 339, 361 Latif, A.M., 159 Index Ledimet, F., 256 Liebhold, A.M., 201 Line characteristic functions, 377, 380 number of voids on, 380 schematic representation of, 369 Link, R.F., 60, 136 Long, J.C.S., 394 Lorenc, A.C., 179, 184, 270 Loucks, F.D., 303 Louis, C., 341 M Mahar, J.W., 341 Maps, 6, 7, 13, 54, 66–70, 101, 131, 148, 153, 164, 172, 177, 178, 180, 227, 231, 233, 239, 245, 247, 270, 275, 277, 282, 297, 304, 306, 308, 314, 316, 317, 319, 321, 323, 325 Mari~no, M.A., 311 Markov processes, 40–42, 122, 123, 308, 330, 331, 337, 339, 361–364, 366, 368, 385, 392 first-order, 40–42, 123, 337, 363 Marsily, G.D., 280 Martinez, C.A., 282 Matern, B., 330 Matheron, G., 129, 185, 189, 195, 216, 230, 276, 278, 280, 330, 383 Mathier, L., 303 Meleschko, V.P., 270 Mercalli scale, 28 Modeling, 25–94, 97–127, 254–324 Mood, A.M., 374 Multidirectional RQD simulation fracture network model, 394–395 RQD analysis, 395–397 RQD simulation results, 397–400 Myers, D.E., 185, 200, 204, 304 N Nearest neighbors, 148, 149, 209 Negative exponential distribution, 145, 146, 364, 381, 390, 396 Neill, J.C., 59 Nugget (zero distance jump), 197 Nugget effect, 192, 201, 202, 208, 213, 222, 224, 228, 231, 281, 283, 319 Numerical data, 25–27, 34–36 409 O Objective modeling, 256 Observations, 2, 5, 8, 11, 16, 19, 26–27, 40, 51, 58, 79, 117, 148, 150, 159, 174, 179, 180, 182–184, 186, 190, 197, 254–261, 263, 267, 270, 275, 276, 280, 281, 283, 301, 302, 304, 332 Octant search, 148 Olea, R.A., 294, 302 Operational objective analysis, 150 Optimal interpolation model (OIM) assumptions, 257, 258 execution steps, 265 points to remember, 265 steps necessary for practical applications, 264, 265 Optimum interpolation cross-validation of model, 272–273 data and application, 262–264 data search and selection procedure, 270–272 equal correlation lines, 271 estimation of zero intercept value, 269 execution steps, 265 expected error, 269–270 location of grid point and measurement site, 258, 259 number of influencing stations, 259, 272 observed and predicted monthly values, 274, 275 observed versus estimated values, 274 spatial correlation function, 264–269 spatial distributions, 263 univariate statistics, 273 Ord, J.K., 331 Ordinary Kriging, 282, 291, 293, 294, 297, 298, 301, 302 Otaibi, A., 357 P Pannatier, Y., 210, 212, 215 Panofsky, H.A., 182, 256 Pedder, M.A., 179, 183, 184, 186 Percentage-weighted polygon (PWP) method, 59–71 basic tasks, 60 step-by-step algorithm, 61 Pernot, M., 341 Perrie, W., 217 Persistence, 38, 40, 42, 86, 303, 308, 358, 359, 367, 387 410 Pham, T.D., 239, 242, 243, 245 Piteau, D.R., 341 Pixel sampling, 34, 35 Point cumulative semivariogram (PCSV), 189, 227, 236, 276 Point sampling, 34 Polygonalization, 57 Popper, K., 1, 10 Powers, R.W., 220 Priest, S.D., 341, 350, 353, 355, 356, 358, 382, 384, 388, 395, 396 Prigodich, A.E., 270 Privalsky, V., 303 The Profile area method, 382 Punctual Kriging, 282, 301 Q Quadrant search, 143, 144, 146, 148 Quenouille, M.H., 40 R Random field (RF), 21–22, 48, 71, 73, 141–145, 148, 186, 207, 214, 222, 284, 369 statistically homogeneous and isotropic, 22 Random variables (RV), 40, 41, 123, 160, 189, 278, 337, 342, 343, 352, 356, 370, 371, 374, 390 Randomness, 3, 8, 12, 14, 16, 19, 29, 35, 36, 98, 116, 130, 143, 146 Range (radius of influence), 197 Regional dependence (correlation) function, 285 Regionalized variable (ReV) data scatter, 134 function, 195, 254, 369 theory of, 190, 218, 276 Regional rainfall pattern, 309–325 Rice, S.O., 374 Roach, S.A., 375, 377, 379 Rock quality designation (RQD) average number of discontinuity, 356, 396 classification, 344 correlated intact length simulation, 358–360 disadvantages, 342 formulation and discussion, 352–357 method adopted by Deere, 341 negative exponential pdf, 347, 393 persistence relation, 366–368 scanline and models, 359 threshold value discontinuity number chart, 345 Index Rock quality designation simulation dependent intact lengths, 347–358 independent intact lengths, 340–347 Rock quality percentage (RQP), 341, 362 Rock quality risk (RQR), 341, 362 Rodriguez-Iturbe, I., 52 Rouleau, A., 394 RQD and correlated intact length simulation proposed models of persistence, 361–363 simulation of intact lengths, 364–369 Rutherford, I.D., 179 Ryckes, K.D., 358, 360, 388 S Salas, J.D., 52 Salas, M., 311 Sample SV, 192, 196, 198, 200–205, 208, 218, 219, 281, 289, 290, 304, 313, 314 interpretation of, 201 Sampling, 4, 5, 10, 25–94, 131–138, 140–148, 154, 178, 184, 185, 195, 198, 200, 201, 218, 228, 231, 239, 254, 275, 280, 282, 284, 370, 374–377, 379, 381 Sasaki, Y., 152, 256 Scanlines frequency distribution functions, 389 mathematical modeling of, 352 rock qualities, 99, 100, 364 Schlatter, T.W., 179, 256, 270 Schubert, S.D., 40 Schwartz, F.W., 394 Seaman, R.S., 184 Second-order stationarity, 278, 284 Semivariogram (SV) development of, 207 information required, 197 models, 196, 198–201, 205, 207, 208, 210–216, 218, 219, 226, 279, 290, 297, 304, 312, 313 patterns, 383 practical difficulties, 196, 218, 383 reasoning, 196 technique, 185, 195, 196, 216, 383 S¸en, Z., 11, 18, 21, 22, 37, 38, 59, 71, 78, 114, 135, 153, 186, 189, 190, 196, 199, 216, 218, 219, 222, 227, 228, 231, 237, 256, 257, 304, 307, 331, 332, 336, 339, 341, 342, 348, 351, 355, 362, 369, 371, 374, 381–384, 386, 388, 390, 394, 396 Seo, D.J., 282 Sharp, W.E., 331, 333 Index Shenfield, L., 183 Similarity, 37, 180, 231–235 Simple iterative method, 184 Simple Kriging, 284–288, 291 Simulation model extension into 3D space, 339 stages, 332 Simulation of stochastic variables, 342 Skibin, D., 189 Slivitzky, M., 303 Smith, H.R., 341, 394 Sneyers, R., 303 Solid lengths, 376 Spatial correlation function average and theoretical, 188 empirical, 186, 187 model parameters, 266 negative-exponential model, 264 pertinent statistics, 268 Sivas city, 267 Sivas correlation contour lines, 268 Spatial data modeling techniques, 256 Spatial dependence function (SDF), 182–185, 236–250, 255, 266, 288, 302 obtaining weighting function from sample CSV, 236, 238 Spatial dependence measures anisotropy, 179–182 average annual temperature, 181 cumulative semivariogram, 216–219 function, 185–186 homogeneity, 179–182 isotropy, 179–182 representative, 200, 217 sample, 178, 180, 182, 185 theoretical models, 216 Spatial estimation of ReV elements, 255 objective analysis, 255, 256 Spatial modeling assumptions, 254 block Kriging, 301–302 block pieces, 301–302 covariance–distance graph, 286 data and application, 262–275 ordinary Kriging, 291–294 dataset, 239, 311 descriptive statistics and normality test, 312 empirical and theoretical SV, 305 estimation of ReV, 255–257 geostatistical analysis, 275, 279 estimation variance, 45 411 experimental and fitted SV models, 312, 313 geostatistical estimator (Kriging), 279–281 interpretations, 267, 304 isohyetal map, 256 Kriging methodologies and advantages, 281–283 Kriging technique, 256 Kriging variance, 312, 316, 317, 320 lag-one lake level prediction, 308 lag-one model verification, 307 lake level TDMs, 302–304, 306 location map, 303 observed and predicted lake levels, 308, 309 optimum interpolation model, 257–262 regional rainfall pattern description, 309–324 ReV sample sites and estimation site, 284 ReV theory and concepts, 276 spatiotemporal Kriging maps, 325 standardized ReV, 284 SV cross-validation, 313 theoretical Gaussian SV parameters, 305 topographic map, 178, 310 triple diagram model, 302–309 unbiasedness principle, 288 universal Kriging, 299 Spatial pattern, 21, 52, 60, 136, 148, 207 Spatial prediction study, steps, 179 Spatial simulation autorun analysis of sandstone, 371–375 autorun modeling of porous media, 375–380 autorun simulation of porous material, 369–380 CSV technique, 381–393 dependent intact lengths, 347–358 extension to 3D, 339 fracture network model, 394–395 independent intact lengths, 340 intact length CSV, 340 line characteristic function of porous medium, 370–371 multidirectional RQD simulation, 393–399 parameters estimation, 332–335 proposed models of persistance, 361–363 rock quality designation simulation, 339–358 RQD analysis, 395–397 RQD and intact length simulation, 358–369 RQD simulation results, 397–399 simulation of intact lengths, 364, 366, 369 412 Spatial simulation (cont.) 3D autoregressive model, 331–358 theoretical CSV model, 384–393 2D uniform model parameters, 335–338 Spatial simulation models, 330 proposed, 331 Spatial variability, 4, 11, 51, 52, 60, 185, 189, 199, 201, 207, 227, 233, 256 Spatio-temporal depth variations application of Kriging modeling to ReV, 284–288 bicarbonate sample, 205, 225 characterizing spatial correlation, 198 chloride, 201, 205 continuous linear trend, 193 correlation coefficient drawback, 186–190 definition, 97 directional SV, 197, 202, 207 discontinuous trend surface, 193, 194 evaluating wind power, 189 exponential, 210 forms, 202, 213 Gaussian, 210–211 global SV and elements, 197 homogeneous and isotropic ReV SV, 191 homogeneous, isotropic, and uniform ReV, 191 independent spatial data, 192, 194 limitations, 199–201 linear, 208–209 linear trend surface SV, 193 logarithmic, 215–216 measure, 98 objective methods and drawbacks, 183 philosophy, 190–195 point cumulative semivariogram, 227 power, 212 quadratic, 211 random ReV SV, 192, 194 rational quadratic, 212, 213 sample SV, 198, 200–203 seismic categories, 228 seismic events, 228, 229 semivariogram regional dependence measure, 190–201 semivariogram (SV) technique, 180 similarity map, 231, 233–235 similarity measures at fixed level, 232 simple iterative method, 184 simple nugget, 212 spatial correlation function, 186–190 spatial dependence function, 236 spherical, 215 successive correction method, 183, 184 theoretical SV, 198, 200, 203–207 Index total dissolved solids, 203, 206, 219 triple surface of chloride, 203, 205 triple surface of TDS, 206 wave (hole effect), 213, 214 Spatiotemporal Kriging, 325 Specific surface, 372, 374 Srivastava, R.M., 273, 302, 304 Statistical interpolation, 261 Statistical objective analysis, 256 Steffen, O., 381 Stevens, C.F., 159 Stochastic process, 8, 15, 17–19, 21, 22, 40, 124, 361, 362, 369, 371, 384 Stout, G.E., 59 Stratigraphic variation, Student, 15, 255 Subyani, A.M., 219, 309, 311, 383 Successive correction methods, 150, 256 Summer, G., 52, 59 Surface fitting methods, 256 Switzer, P., 330 T Tabios, G.O., 52 Talagrand, O., 256 Tase, N., 78 Taylor, G.I., 195, 231 Temporal variations, 3, 11, 86, 130, 275, 320 Terzaghi, K.C., 339 Terzaghi, R.D., 381 Theoretical CSV model, 384–393 exponential, 224–225 Gaussian, 226 linear, 221–223 logarithmic, 225–226 power, 223–224 Thiebaux, H.J., 179, 183, 184 Thiessen, A.H., 59 Thiessen polygons, 52, 57–59, 65, 67, 68, 71 obtaining sub-polygons, 58 3D autoregressive model, 331–339 2D uniform model parameter, 335–338 extension to 3D, 339 parameters estimation, 332–335 Time series, 3, 12, 17, 19, 20, 98, 99, 102, 116–126, 129, 130, 138, 185, 190, 262, 276, 303, 304, 371, 384 Toulany, B., 217 Trend surface calculation table, 156, 157 variables, 154 Triangularization, 51–55 Triple diagram method (TDM), 304 Index 3D isotropic simulation model, 339 2D discontinuity network, 398 2D isotropic model parameter, 336 U Uncertainty techniques, 4, 8, 15 Uniformity test, 138–141 Universal Kriging, 279, 280, 297–301 procedural structure, 298 V Variability deterministic and stochastic variations, 350 geometric similarities, quantitative consideration, 3, spatial, 4, 11, 51, 52, 60 Variational techniques, 256 413 Void length, 376, 379, 380 Voronoi, G., 57 W Wallis, P.F., 382 Watt, D.G., 124, 372 Weighted-averaging analysis scheme, 150, 183 Wiener, N., 26 Wiesner, C.J., 65, 66 Wilson, J.W., 59 Y Yates, F., 255 Yevjevich, Y., 22 Z Zadeh, L.A., 18, 26, 135, 307 .. .Spatial Modeling Principles in Earth Sciences ThiS is a FM Blank Page Zekai Sen Spatial Modeling Principles in Earth Sciences Second Edition Zekai Sen Faculty of Civil Engineering Istanbul... must be deductive In this manner, the © Springer International Publishing Switzerland 2016 Z Sen, Spatial Modeling Principles in Earth Sciences, DOI 10.1007/978-3-319-41758-5_1 Introduction conclusion... uncertainty in earth and atmospheric sciences and uncertainty in physics which has, inevitably it seems, led to the question of determinism and indeterminism in nature (Leopold and Langbein 1963;
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