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### Lecture 6: Stochastic Hydrology (Estimation Problem-Kriging-, Conditional Simulations and Reduction of Uncertainties)

1. Lecture (6)Lecture (6) Estimation Problem (Kriging), Conditional Simulations and Reduction of Uncertainties
2. The Estimation ProblemThe Estimation Problem The estimation problem is to determine the value of the quantity Zo for point (xo,yo) which has not been measured. Z2 *(x2,y2) Z1 Z3 *(x1,y1) *(x3,y3) Zo??? *(xo,yo) By continuously modifying the position (xo, yo) we shall estimate the whole field of Z.
3. Estimation MethodsEstimation Methods Usual Estimation Methods: 1. Linear interpolation (by hand). 2. Trend surface analysis. 3. Least squares fitting. The drawbacks of the methods: 1. They cannot give the confidence interval of the estimation. 2. They do not take into account the spatial structure of the phenomenon.
4. Estimation byEstimation by KrigingKriging Assumptions: • B.L.U.E.= Best Linear Unbiased Estimator. Best = minimizing the estimation (error) variance. Linear = linear combination of weights. Unbiased means E{Z*o}=E{Zo} • Stationary = no trend assumed in the data. • Normal distribution of the data.
5. SimpleSimple KrigingKriging ModelModel The estimation of Z*o is a linear combination of all available measurements of Z, [ ]* 1 n o o i i i Z Z = = λ∑ Z*o is the estimator, Zi measurements at the n-points xi (i=1,2,3,…n). λi is optimal weights to be computed.
6. Unbiased ConditionUnbiased Condition { } { } { } { } { } { } * 1 1 ( ) , o o n o i i o i n o i i o i E Z E Z assume m E Z substitution E Z E Z Linearity E Z E Z = = = = ∀ ⎧ ⎫ λ =⎨ ⎬ ⎩ ⎭ λ = ∑ ∑ x x
7. Unbiased Condition (cont.)Unbiased Condition (cont.) { } { } { } { } 1 1 1 1 but from our assumption, 1 n o i i o i i o n o i i n o i i n o i i E Z E Z E Z E Z m m m m m = = = = λ = = = λ = λ = λ = ∑ ∑ ∑ ∑ Condition of unbiasedness
8. The Variance of the EstimatorThe Variance of the Estimator ( ) * o oError of estimation,(Z -Z ) should be small 2 *2 o oSK E Z Z⎧ ⎫σ = −⎨ ⎬ ⎩ ⎭ •SK=Simple Kriging variance •σ2=Variance about estimated point Z is “true” value Z* is the estimate produced by kriging •Seek to minimize to determine weights (λ ’s)
9. Derivation of the Error VarianceDerivation of the Error Variance ( ) { } { }2 1 2 *2 2 1 2 2 1 1 2 2 1 2 n o o i o i i o oSK n o o i i i n n o o E o o i i i i i i n o E Z E Z Z E i i i E Z Z E Z Z Z Z Z Z Z = = = = = = − λ + = ⎧ ⎫σ = −⎨ ⎬ ⎩ ⎭ ⎧ ⎫⎛ ⎞⎪ ⎪= − λ∑⎜ ⎟⎨ ⎬ ⎝ ⎠⎪ ⎪⎩ ⎭ ⎧ ⎫⎛ ⎞⎪ ⎪− λ + λ∑ ∑⎜ ⎟⎨ ⎬ ⎝ ⎠⎪ ⎪⎩ ⎭ ⎧ ⎫⎛ ⎞⎪ ⎪λ∑∑ ⎜ ⎟⎨ ⎬ ⎝ ⎠⎪ ⎪⎩ ⎭
10. Derivation of the Error Variance (cont.)Derivation of the Error Variance (cont.) 2 2 1 1 2 2 1 1 1 1 2 1 1 1 ( ) ( )( ) ( ) ( ) ... n n n n o o o o o o o i i i i i i n n i i i i i i n n n o o o i i j j i i i j i a b a b a b a a b b a b aa ab ba bb Z Z Z Z Z Z Z Z Z Z = = = = = = = + = + + = + + + = + + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ λ =λ λ +λ λ λ λ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ λ = λ λ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 1 1 n n o o j i j i j i Z Z = = = λ λ∑ ∑
11. Derivation of the Error Variance (cont.)Derivation of the Error Variance (cont.) { } { } { } { } { } { } { } 2 2 1 2 2 1 2 2 1 1 1 2 2 1 1 2 2 1 2 1 1 2 2 ( , ) ( , n o SK o i o i i n o SK o i o i i n n n o o o SK o i o i j i j i i j i n n o o o SK o i o i j i i j i j n o E Z E Z Z E i i i n n o o E Z E Z Z E j i j i j i E Z E Z Z E Z Z Cov Z Z Cov Z Z Z Z Z = = = = = = = σ = − λ + = σ = − λ + = = σ = − λ + λ λ σ =σ − λ + λ λ ⎧ ⎫⎛ ⎞⎪ ⎪λ∑∑ ⎜ ⎟⎨ ⎬ ⎝ ⎠⎪ ⎪⎩ ⎭ ⎧ ⎫ λ λ∑ ∑∑ ⎨ ⎬ ⎩ ⎭ ∑ ∑ ∑ ∑ ∑ 1 ) n i= ∑
12. Minimizing The Error VarianceMinimizing The Error Variance ( ) 2 2 1 1 1 1 11 1 1 0 2 ( , ) ( , ) 0 ( , )2 ( , ) 0 0 ( , ) ... ( , ) ... ( , ) 2 SK o k n n n o o o o i o i j i i jo i i jk n nn o oo j i i ji o i i ji o o k k o o o o k o k n o n Cov Z Z Cov Z Z Cov Z ZCov Z Z Cov Z Z Cov Z Z Cov Z Z = = = = == ∂σ = ∂λ ⎛ ⎞∂ σ − λ + λ λ =⎜ ⎟ ∂λ ⎝ ⎠ ⎛ ⎞⎛ ⎞ ∂ λ λ∂ λ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠− + = ∂λ ∂λ ∂ λ + + λ + + λ − ∂λ ∑ ∑ ∑ ∑ ∑∑ 1 1 1 1 1 1 1 ( , ) ... ( , ) ... 0 2 ( , ) ( , ) o k n n o o o o i i k i i ko i ik o n o o k i io ik Cov Z Z Cov Z Z Cov Z Z Cov Z Z = = = ∂ ⎛ ⎞ + λ λ + + λ λ + =⎜ ⎟∂λ ⎝ ⎠ ∂λ ⎛ ⎞ − + λ⎜ ⎟∂λ ⎝ ⎠ ∑ ∑ ∑ 1 1 1 1 1 ( , ) ... ( , ) ( , ) ... 0 n o o i io ik o n n o o ok i i k k i i ko o i ik k Cov Z Z Cov Z Z Cov Z Z = = = ∂ ⎛ ⎞ + λ λ +⎜ ⎟∂λ ⎝ ⎠ ∂λ ∂⎛ ⎞ ⎛ ⎞ + λ + λ λ + =⎜ ⎟ ⎜ ⎟∂λ ∂λ⎝ ⎠ ⎝ ⎠ ∑ ∑ ∑ We seek the weights λi that minimize the error variance.
13. Minimizing The Error Variance (cont.)Minimizing The Error Variance (cont.) ( )1 1 1 2 ( , ) ( , ) o n o o k i io i k Cov Z Z Cov Z Z = ∂λ − + λ ∂λ ∑ ( ) ( ) ( ) 1 1 1 1 1 1 1 2 2 1 ( , ) ... ( , ) ( , ) ... 0 2 ( , ) ( , ) 2 ( , ) ( , ) ( , ) ... ( , ) ( , ) ... 0 n o o i io i k o n n o o ok i i k k i i ko o i i k k o o k i i k o o o k k k n o o i i k k k k i Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z = = = = ∂ + λ λ + ∂λ ∂λ ∂ + λ + λ λ + = ∂λ ∂λ − + λ − + λ + λ + + λ + λ + = ∑ ∑ ∑ ∑ 1 1 1 1 ( , ) 0 because ( , ) ( , ),so, 2 ( , ) 2 ( , ) 0 ( , ) ( , ) n n o i k i i i i k k i n o o i ik k i n o i i ok k i Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z = = = = + λ = = − + λ = λ = ∑ ∑ ∑ ∑
14. KrigingKriging Matrix FormMatrix Form 1 1 1 2 1 1 1 2 1 1 ** ( , ) ( , ) ... ( , ) ( , ) ( , ) . ... . . . . . ... . . . ( , ) . ... ( , ) ( , ) Once this system is solved we get the estimate, n o n n n n o n o o Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Cov Z Z Y m Z λ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ λ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ = + = 1 2 * 0 1 var( ) (0) ( , ) n o i i i n o SK o o i i i m Z Z Z Cov Cov Z Z = = + λ σ = − = − λ ∑ ∑
15. Some Useful Remarks RegardingSome Useful Remarks Regarding KrigingKriging * * = (value measured) I.e. =1, =0, ( ) 0 (no Uncertainty in a measured Point) If the error is assumed to be normally distributed, then 1. Exact Interpolator: 2. Confidence Intervals: k k k k k i k Z Z i k Var Z Z λ λ ≠ − = ( )* * 1 we can estimate 95% confidence intervals of the estimation as, = = 2 3. The Kriging system does not depend explicitly on the measurement values . Only the locations of t o o n o o i i i i i Var Z Z Z Z Z x = σ − λ ± σ∑ he measurement points are needed to compute the weights. 4. Drawing contour maps: by solving the Kriging system, one can estimate at any point . 5. Kriging estimator is generally much smoother tha o oZ x n the actual spatially variable field.
16. SimpleSimple KrigingKriging ExampleExample Computer code SIMKRIG.exe Data files: SIMKRIG.dat, Lx, Ly 100., 50. Dx,Dy, 1., 1. λx , λy 20., 10. SKFIELD.dat No. of Measurements mean_k sd_k 10 10 50 X_COR Y_COR Y 0 0 10 100 0 15 0 50 12 100 50 20 30 25 15 60 25 10 80 10 20 10 5 13 50 50 16 50 0 16 10 15 12 20 15 10 20 13 16 16 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 Data Kriged Map Error variance 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 5 30 55 80 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
17. LinearLinear GeostatisticsGeostatistics • Simple Kriging.(mean is known and const.) • Ordinary Kriging.(mean is unknown and const.) • Non-stationary Kriging. Use moving neighborhood or Universal Kriging. • Co-Kriging (Coregionalization): If more than one variable is sampled. • Ref. Deutsch, C. V. and Journel, A. G. 1992, GSLIB; Geostatistics Software Library and user’s guide: Oxford Univ. Press, New York, 340p. ' * 1 1 n n o i i j j i j Z Z = = = λ + ν ϕ∑ ∑
18. Conditional SimulationsConditional Simulations From a practical point of view, it is desirable that the random fields not only reproduce the spatial structure of the field but also honour the measured data and their locations. This requires an implementation of some kind of conditioning, so that the generated realizations are constrained to the available field measurements.
19. Representation of a Conditional SimulationRepresentation of a Conditional Simulation
20. Methods of ConditioningMethods of Conditioning Direct Methods "Metrical Methods" Indirect Methods "Kriging Method" Methods of Conditioning
21. Indirect Conditioning byIndirect Conditioning by KrigingKriging (1) A kriged map is generated from the field data with the sampled locations which will be smoother than reality. (2) An unconditional simulated field is generated by TBM from the data which reproduces the spatial structure of the underlying random function. (3) Allocation of the unconditional values (pseudo measurements) at the sites of measurements is done on the simulated map in step 2. (4) Another kriged map is generated from the pseudo measurements. (5) A pseudo error is calculated by subtracting the kriged map in step 4 from the unconditional simulation in step 2. (6) The conditional simulation map is generated by adding the pseudo error in step 5 to the kriged map in step 1. So, ( - )cs kd us kusZ Z Z Z= + Zcs is the required conditional simulation, Zkd is the kriged map from the real data, Zus is the unconditional simulation, Zkus is the kriged map with the pseudo measurements.
22. Graphical Illustration of Conditioning byGraphical Illustration of Conditioning by KrigingKriging (1) A kriged map is generated from the data: (2) Unconditional simulation is generated from the data: (3) Allocation pseudo measurements. (4) Kriged map is generated from the pseudo measurements: (5) A pseudo error =kriged map(step 4) - the unconditional simulation in step 2. (6) Conditional simulation = the pseudo error in step 5 + the kriged map in step 1: kdZ usZ kusZ csZ ( - )cs kd us kusZ Z Z Z= +
23. Program to perform Conditional SimulationProgram to perform Conditional Simulation byby KrigingKriging 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 3.5 10 15 12 20 15 10 20 13 16 16 1614 13 2010 15 20 3.5 10 15 12 20 15 10 20 13 16 16 1614 13 2010 15 20 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 3.5 10 15 12 20 15 10 20 13 16 16 1614 13 2010 15 20 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 Data Kriged Map Single Realization of Conditional Simulation CSTBM.DAT 20, 20, 1 corr.length x, corr.length y, MC 100, 1, 0 No. of TB lines, Dz, RLD (0 OR 1) 0.25, 100 Dk, Max. Harmonics 19999, 20 Seed, ib Csfield.dat 200 100 Lx, ly 1 1 dx,dy 17 no. of measurements X_coor Y_coor Y 0 0 10 100 0 15 0 50 12 100 50 20 30 25 15 60 25 10 80 10 20 10 5 13 50 50 16 50 0 16 200 100 16 0 100 14 200 0 13 200 25 20 0 25 10 100 100 15 80 40 20
24. Conditioning in OneConditioning in One--dimensional Markovdimensional Markov ChainChain i0 1 i+1i-1 N2 SkSl Sq )1( )( )1( )( 1 11 11 1 1 1 1 1 11 1 1 1 1 1 )(Pr )Pr().|(Pr )(Pr).|(Pr).|(Pr )(Pr ),(Pr ),(Pr).|(Pr )(Pr ),(Pr ),(Pr),|(Pr )(Pr ),(Pr ),,Pr( )(Pr )(Pr +− − +− − − −− −− − − − − − −− − − − − − = ==== === ===== ==== == ==== ==== == ===== ==== == === ==== === iN lq iN kqlk qlk iN lq lk iN kq qNliki liliqN lilikikiqN qNliki qNli kilikiqN qNliki qNli kilikiliqN qNliki qNli qNkili qNliki qNliki p pp p p pp SZ,SZ|SZ SZSZSZ SZSZSZSZSZ SZ,SZ|SZ SZSZ SZSZSZSZ SZ,SZ|SZ SZSZ SZSZSZSZSZ SZ,SZ|SZ SZSZ SZSZSZ SZ,SZ|SZ SZ,SZ|SZ
25. Coupled Markov Chain “CMC” in 2DCoupled Markov Chain “CMC” in 2D Dark Grey (Boundary Cells) Light Grey (Previously Generated Cells) White (Unknown Cells) i-1,j i,j i,j-1 1,1 Nx,Ny Nx,1 1,Ny Nx,j , , 1, , 1 , 1, , 1 ,, Unconditioinal Coupled Markov Chains : Pr( | , ) . 1,... Conditioinal Coupled Markov Chains : Pr( | , , )x h v lk mk lm k i j k i j l i j m h v lf mf f i j k i j l i j m N j qlm k q h lk .p p p Z S Z S Z S k n .p p p Z S Z S Z S Z S .p − − − − = = = = = = = = = = = = ∑ ( ) ( ) , 1,... . x x h N i v kq mk h h N i v lf fq mf f .p p k n . .p p p − − = ∑
26. Coupled Markov Chain (application)Coupled Markov Chain (application) Two-dimensional Cross-sectional Panel of the Fluvial Succession of the Medial Area of the Tόrtola Fluvial System, Spain Length of The Section (m) = 648. Depth of The Section (m) = 115. Sampling interval in X-axis (m) = 9. Sampling interval in Y-axis (m) = 2.5 Horizontal Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.893 0.009 0.005 0.000 0.000 0.000 0.000 0.093 2 0.000 0.796 0.011 0.000 0.000 0.000 0.000 0.194 3 0.000 0.000 0.989 0.000 0.000 0.000 0.000 0.011 4 0.006 0.000 0.013 0.885 0.000 0.000 0.000 0.096 5 0.074 0.000 0.000 0.074 0.593 0.037 0.000 0.222 6 0.000 0.013 0.000 0.000 0.000 0.946 0.000 0.040 7 0.040 0.000 0.000 0.000 0.000 0.000 0.940 0.020 8 0.007 0.006 0.002 0.007 0.005 0.005 0.001 0.968 Vertical Transition Probability Matrix State 1 2 3 4 5 6 7 8 1 0.591 0.000 0.000 0.000 0.014 0.000 0.042 0.353 2 0.011 0.753 0.097 0.000 0.000 0.000 0.000 0.140 3 0.032 0.000 0.623 0.000 0.000 0.238 0.000 0.107 3 0.000 0.025 0.000 0.662 0.013 0.000 0.000 0.299 5 0.111 0.000 0.000 0.074 0.519 0.000 0.000 0.296 6 0.000 0.000 0.026 0.032 0.006 0.084 0.000 0.851 7 0.120 0.000 0.000 0.100 0.000 0.000 0.360 0.420 8 0.029 0.008 0.039 0.017 0.003 0.031 0.010 0.863 1 2 3 4 5 6 7 8
27. Coupled Markov Chain (application cont.)Coupled Markov Chain (application cont.) 0 50 100 150 200 250 300 -80 -60 -40 -20 0 0 50 100 150 200 250 300 -80 -60 -40 -20 0 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 -80 -60 -40 -20 0
28. Application of C_CMC Single RealizationsApplication of C_CMC Single Realizations 1 2 3 4 5 6 7 8 0 50 100 150 200 250 300 -80 -60 -40 -20 0 0 50 100 150 200 250 300 -80 -60 -40 -20 0 0 50 100 150 200 250 300 -80 -60 -40 -20 0 0 50 100 150 200 250 300 -80 -60 -40 -20 0 0 50 100 150 200 250 300 -80 -60 -40 -20 0 0 50 100 150 200 250 300 -80 -60 -40 -20 0
29. Application of C_CMC Single RealizationsApplication of C_CMC Single Realizations 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 0 50 100 150 200 -10 -5 0 1 2 3 4 Lithology Coding 0 80 160 240 -10 -5 0 1 2 3 4 2 boreholes 9 boreholes 25 boreholes 31 boreholes
30. Many Realizations: Hypothetical ExampleMany Realizations: Hypothetical Example 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 1 2 3 4 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0
31. Probability MapsProbability Maps ⎪ ⎩ ⎪ ⎨ ⎧ = = .0 1 )( otherwise SZif ZI kij ijk Let the realizations be numbered 1,…, MC, and let Zij (R) be the lithology of cell (i,j) in the Rth realization. The empirical relative frequency of lithology Sk at cell (i,j) is: 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 1 2 3 4 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0 20 40 60 80 100 120 140 160 180 200 -40 -20 0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 ( )∑= = = = MC R R k k R k ij ji ji ZI MCMC SZ 1 )( )( , , 1}{# π k ijπPlots of
32. Procedure for Extracting a Final GeologicalProcedure for Extracting a Final Geological ImageImage ⎪ ⎩ ⎪ ⎨ ⎧ = = .0 1 )( otherwise SZif ZI kij ijk ( )∑= = = = MC R R k k R k ij ji ji ZI MCMC SZ 1 )( )( , , 1}{# π Let the realizations be numbered 1,…, MC, and let Zij (R) be the lithology of cell (i,j) in the Rth realization. The empirical relative frequency of lithology Sk at cell (i,j) is: }...,,max{ 21 n ijijij l ij ππππ = In the final image Z* the lithology at cell (i, j) will be the lithology which occurs most frequently in the MC realizations. So, if Sl is such that Zij *= Sl.
33. Program WELLLOGProgram WELLLOG Input Data for preInput Data for pre--processingprocessing Boreho01.dat 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 . . Wells 16 no. of boreholes 1, 0.0 well no. and X_Coor. 2, 21 . 3, 34 . 4, 39 . 5, 46 6, 56 7, 76 8, 96 9, 126 10, 152 11, 172 12, 194 13, 214 14, 234 15, 256 16, 276 borehol 1 borehol 2 borehol 3 borehol 4 borehol 4 borehol 6 borehol 7 borehol 8 borehol 9 borehol 10 borehol 11 borehol 12 borehol 13 borehol 14 borehol 15 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 Boreho02.dat Boreho02.dat Boreho03.dat Boreho04.dat . 0 50 100 150 200 250 -10 -5 0 0 1 2 3 4 5 T T=p v lq n q v lkv lk ∑=1GEO.DAT 7 No. of states 276, 14.2 Lx, Ly 3, 0.1 Dx, Dy
34. Program WELLLOG Output Data for preProgram WELLLOG Output Data for pre-- processingprocessing Vertical Sampling Interval=0.1 m 0 50 100 150 200 250 -10 -5 0 0 1 2 3 4 5 S. 1 2 3 4 5 6 7 1 .879 .103 .009 .000 .009 .000 .000 2 .026 .911 .046 .009 .003 .000 .005 3 .003 .030 .897 .044 .010 .000 .016 4 .000 .006 .094 .869 .031 .000 .000 5 .000 .000 .003 .010 .961 .000 .026 6 .009 .014 .009 .005 .000 .963 .000 7 .000 .000 .000 .000 .000 .000 1.00 Well.res Well.grd
35. Program F_CMC2 Input DataProgram F_CMC2 Input Data GEOSIM 7, 1, 30 No. of sataes,Seed , MC 276.00, 14.20 Lx, Ly 3.0, 0.10 Dx, Dy Hor. Transitions 0.922 0.015 0.015 0.015 0.015 0.015 0.001 0.015 0.922 0.015 0.015 0.015 0.015 0.001 0.015 0.015 0.922 0.015 0.015 0.015 0.001 0.015 0.015 0.015 0.922 0.015 0.015 0.001 0.015 0.015 0.015 0.015 0.922 0.015 0.001 0.015 0.015 0.015 0.015 0.015 0.922 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.992 Vert. Transitions 0.879 0.103 0.009 0.000 0.009 0.000 0.000 0.026 0.911 0.046 0.009 0.003 0.000 0.006 0.003 0.031 0.897 0.044 0.010 0.000 0.016 0.000 0.005 0.094 0.869 0.031 0.000 0.000 0.000 0.000 0.003 0.010 0.961 0.000 0.026 0.009 0.014 0.009 0.005 0.000 0.962 0.000 0.001 0.001 0.001 0.001 0.001 0.001 0.994 WELLSIM 16 no. of boreholes 1, 0.0 well no. and X_Coor. 2, 21 . 3, 34 . 4, 39 . 5, 46 6, 56 7, 76 8, 96 9, 126 10, 152 11, 172 12, 194 13, 214 14, 234 15, 256 16, 276 WELLS 16 no. of boreholes 1, 0.0 well no. and X_Coor. 2, 21 . 3, 34 . 4, 39 . 5, 46 6, 56 7, 76 8, 96 9, 126 10, 152 11, 172 12, 194 13, 214 14, 234 15, 256 16, 276 Well.grd
36. Horizontal Transition Probability MatricesHorizontal Transition Probability Matrices S. 1 2 3 4 5 6 7 1 .500 .100 .100 .100 .100 .100 .000 2 .100 .500 .100 .100 .100 .000 .100 3 .100 .100 .500 .100 .100 .000 .100 4 .100 .100 .100 .500 .100 .000 .100 5 .100 .100 .100 .100 .500 .100 .000 6 .001 .001 .001 .001 .001 .994 .001 7 .001 .001 .001 .001 .001 .001 .994 S. 1 2 3 4 5 6 7 1 .879 .103 .009 .000 .009 .000 .000 2 .026 .911 .046 .009 .003 .000 .005 3 .003 .030 .897 .044 .010 .000 .016 4 .000 .006 .094 .869 .031 .000 .000 5 .000 .000 .003 .010 .961 .000 .026 6 .009 .014 .009 .005 .000 .963 .000 7 .001 .001 .001 .001 .001 .001 .994 S. 1 2 3 4 5 6 7 1 .922 .015 .015 .015 .015 .015 .003 2 .015 .922 .015 .015 .015 .015 .003 3 .015 .015 .922 .015 .015 .015 .003 4 .015 .015 .015 .922 .015 .015 .003 5 .015 .015 .015 .015 .922 .015 .003 6 .015 .015 .015 .015 .015 .922 .003 7 .001 .001 .001 .001 .001 .001 .994
37. Program FCMC2 Output DataProgram FCMC2 Output Data 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 - 1 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 - 1 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 - 1 0 - 5 0 1 2 3 4 5 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 - 1 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 - 1 0 - 5 0 a b c d e
38. Effect of Number of Boreholes on SiteEffect of Number of Boreholes on Site CharacterizationCharacterization 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 0 -5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 0 -5 0 1 2 3 4 5 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 - 1 0 - 5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 0 -5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 0 -5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 0 -5 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 0 -5 0
39. Application of CS_CMC at MADE siteApplication of CS_CMC at MADE site 0 50 100 150 200 250 -10 -5 0 0 1 2 3 4 5
40. Simulation of the MADE1 ExperimentSimulation of the MADE1 Experiment 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 0.1 1 10 100 0 50 100 150 200 250 -10 -5 0 1 2 3 4 5 0 50 100 150 200 250 -10 -5 0
41. Effect of the Number of Boreholes on theEffect of the Number of Boreholes on the Simulated PlumeSimulated Plume 0 50 100 150 200 250 -10 -5 0 0 0.1 1 10 100 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 279 days 0 50 100 150 200 250 -10 -5 0 49 days 0 50 100 150 200 250 -10 -5 0 594 days 0 50 100 150 200 250 -10 -5 0 49 days 0 50 100 150 200 250 -10 -5 0 279 days 0 50 100 150 200 250 -10 -5 0 594 days 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0 0 50 100 150 200 250 -10 -5 0
42. Quantification of Uncertainties using MCQuantification of Uncertainties using MC Classifications of Uncertainties: Geological Uncertainty: Geological configuration. Parameter Uncertainty: Conductivity value of each unit. 0 50 100 150 200 250 300 -50 0 Single realization of the geological structure used in the experimentsFigure 1.
43. Geological and Parameter UncertaintiesGeological and Parameter Uncertainties 1 2 3 4 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 time = 1600 days 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -40 -20 0 0 50 100 150 200 250 300 -40 -20 0 Geology is Certain and Parameters are Uncertain Geology is Uncertain and Parameters are Certain 0 0.01 0.1 1
44. MonteMonte--Carlo Results for GeologicalCarlo Results for Geological UncertaintyUncertainty 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0.00 0.01 0.10 1.00 time = 200 days time = 1000 days time = 2000 days time = 3000 days Concentration in mg/l time = 1600 days 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0.00 0.01 0.10 1.00 time = 200 days time = 1000 days time = 2000 days time = 3000 days Concentration in mg/l time = 1600 days 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0 50 100 150 200 250 300 -50 0 0.00 0.01 0.10 1.00 time = 200 days time = 1000 days time = 2000 days time = 3000 days Concentration in mg/l time = 1600 days
45. MonteMonte--Carlo Method ResultsCarlo Method Results -50 0 -50 0 0 50 100 150 200 250 300 -50 0 -50 0 -50 0 0 50 100 150 200 250 300 -50 0 -50 0 -50 0 0 50 100 150 200 250 300 -50 0 -50 0 -50 0 0 50 100 150 200 250 300 -50 0 -50 0 -50 0 0 50 100 150 200 250 300 -50 0 Concentration in (mg/l) -50 0 -50 0 0 50 100 150 200 250 300 -50 0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 (a) (b) (c) (a) (b) (c) (c) (c) (c) (c) (b) (b) (b) (b) (a) (a) (a) (a) Concentration field in set 1L: (a) Single realization, (b) Ensemble concentration, (c) Standard deviation on concentration. Figure 2 Concentration field in set 1H: (a) Single realization, (b) Ensemble concentration, (c) Standard deviation on concentration. Figure 5 Concentration field in set 2L: (a) Single realization, (b) Ensemble concentration, (c) Standard deviation on concentration. Concentration field in set 2H: (a) Single realization, (b) Ensemble concentration, (c) Standard deviation on concentration. Concentration field in set 3L: (a) Single realization, (b) Ensemble concentration, (c) Standard deviation on concentration. Concentration field in set 3H: (a) Single realization, (b) Ensemble concentration, (c) Standard deviation on concentration. Figure 3 Figure 6 Figure 4 Figure 7
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