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GSS710S - Geo Statistics 314 - 1st Opp - June 2022 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FACUL TY OF ENGINEERING AND SPATIAL SCIENCE
DEPARTMENT OF MINING AND PROCESS ENGINEERING
QUALIFICATION : BACHELORS OF ENGINEERING IN MINING ENGINEERING
QUALIFICATION CODE: BEMIN LEVEL: 6
COURSE CODE: GSS710S
COURSE NAME: GEOSTATISTICS
SESSION: JUNE 2022
PAPER: THEORY
DURATION: 3 HOURS
MARKS: 100
FIRST OPPORTUNITY EXAM
EXAMINER(S) Mallikarjun Rao Pillalamarry
MODERATOR: Lawrence Madziwa
INSTRUCTIONS
1. Answer all questions.
2. Read all the questions carefully before answering.
3. Marks for each question are indicated at the end of each question.
4. Please ensure that your writing is legible, neat and presentable.
PERMISSIBLE MATERIALS
I. Examination paper.
THIS QUESTION PAPER CONSISTS OF 10 PAGES (Including this front page)
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This EXAM has two sections. Section A and B.
Time allowed: 3 hours
SECTION A
Instructions: Answer Question 1 and any 2 other questions. Excess questions will not be marked.
Question 1 is compulsory.
Question 1
a) With illustrations, explain the differences between quantitative and qualitative data (8)
interpretation and evaluation. For a mining Engineer on an operating mine, how is
qualitative data used to convey the positive or negative aspects of mining operations
especially concerning the ore body?
b) State the advantages and disadvantages of each approach in interpreting qualitative and (4)
quantitative data interpretation.
c) Given the following data set:
SET A: 4 3 7 5 3 2 4 5 6 2 6
SET B: 2 2 4 4 5 5 6 6 7 3 3
i) From a statistical and geostatistical point of view; discuss whether this data set is (2)
the same or different population.
ii) Calculate the means of Data set A and 8.
(4)
iii) Find the statistical variance { o-2 = [ x; -µ]2/ (n-1)} for both populations. Where pis the (4)
mean, and x; are the individual values in data set.
iv) Find the Geostatistical variance { o-2 = [ x; -X(i+h) ]2I (n-1)} where i and i+h denote next
(4)
neighbour values in a data set.
v) Discuss the significance of these differences in the variance and what they imply in case (4)
of blocks ofan ore body.
Question 2
State the three main stages required to move from exploration to mining an ore deposit.
( I 0)
For each stage state the financial implications associated with that particular stage and the
most probable source of funding.
Question 3
a) Discuss the revenue factors involved in operating a mining venture.
(2)
b) Discuss the main factors involved in the valuation ofan ore body.
(4)
c) In an ore body, discuss the best way to deal with grade outliers (high grades and very low (2)
grades).
d) In the country of Vietnam, there is a correlation coefficient of0.9 between GDP and Foreign (2)
Direct Investment (FDI). Explain this correlation coefficient and state what variables drive
this economy to such levels?
Question 4 Discuss with examples the difference between measured, indicated and inferred reserves ( I 0)
on a mine setting.
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SECTION B
Instructions: Answer Question I and any 2 other questions. Excess questions will not be marked.
Question I is compulsory.
Question I
a) How the sample mean and variance changes with support?
(2)
b) Why Kriging method of estimation is more reliable than other estimation methods?
(I)
c) What is the necessary condition for a variogram model to be used in Kriging estimation
(I)
method?
d) What is auxiliary function F(l) gives?
(I)
e) What 'C' matrix in Kriging system of equations represents?
(I)
d) Why we have Lagrangian Multiplier in Kriging system of equations?
(2)
f) What is zonal anisotropy and what is reason for a variogram exhibiting zonal anisotropy? (2)
Question 2
a) The Zn grade for 2 m long core sections has modeled with a spherical variogram with a ( I 0)
vertical range of 15 m, a sill of 10(%)2. As per the mine planning the bench height will be
IO m and the data needs to be regularized over this height. Calculate the variogram values
for 10 m, 15 m, and 20 m lags for IO m core sections. [Assume that the point variogram
parameters derived are appropriate]
b) Chip samples are collected from an underground tunnel at regularly spaced for every 5 m, (I 0)
and grades are available for 11 out of 13 samples as shown in Figure below. Calculate the
experimental variogram for the first two lags for the given data.
864
65
289563
Question 3 A 160 m x 150 m panel of Zn ore was intersected by two exploration drill holes as shown (20)
in Figure. The average Zn grade for the drill hole I and drill hole 2 for the intersected length
is 4.5% and 3.4% respectively. Spatial continuity of Zinc grades in the orebody is following
spherical variogram with a sill value of2.5 (%) 2 and range of300 m. Estimate the grade of
the panel using Kriging.
30m
100m
30m
1150m
-0.65 0.65
C- 1 = [ 0.65 -0.65
0.5 ]
0.5
0.5 0.5 -0.99
2
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Question 4 Average grade of panel in a copper deposit to be estimated, and the panel was intersected (20)
two exploration drill holes as shown in Figure. The average grades at the intersected pa1t
of drill holes are given in the Figure as g1 and g2. If the average grade g1 and g2 is used as
the grade of the panel, estimate the extension variance. Spatial continuity of the grade in
the deposit can be described using a spherical form with a range of influence of 90 m and
a sill of0.6(%)2.
--· 20m-••
80m
.
i
j 2om
'
g 1 = 7.68% Cu
80m
~------------g
.. I
=4.3% Cu
2
3
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Standard Normal Probabilities
z
z
.00
.01
.02
-3.4 .0003 .0003 .0003
-3.3 .0005 .0005 .0005
-3.2 .0007 .0007 .0006
-3.1 .0010 .0009 .0009
-3.0 .0013 .0013 .0013
-2.9 .0019 .0018 .0018
-2.8 .0026 .0025 .0024
-2.7 .0035 .0034 .0033
-2.6 .0047 .0045 .0044
-2.5 .0062 .0060 .0059
-2.4 .0082 .0080 .0078
-2.3 .0107 .0104 .0102
-2.2 .0139 .0136 .0132
-2.1 .0179 .0174 .0170
-2.0 .0228 .0222 .0217
-1.9 .0287 .0281 .0274
-1.8 .0359 .0351 .0344
-1.7 .0446 .0436 .0427
-1.6 .0548 .0537 .0526
-1.5 .0668 .0655 .0643
-1.4 .0808 .0793 .0778
-1.3 .0968 .0951 .0934
-1.2 .1151 .1131 .1112
-1.1 .1357 .1335 .1314
-1.0 .1587 .1562 .1539
-0.9 .1841 .1814 .1788
-0.8 .2119 .2090 .2061
-0.7 .2420 .2389 .2358
-0.6 .2743 .2709 .2676
-0.5 .3085 .3050 .3015
-0.4 .3446 .3409 .3372
-0.3 .3821 .3783 .3745
-0.2 .4207 .4168 .4129
-0.1 .4602 .4562 .4522
-0.0 .5000 .4960 .4920
z Table entry for is the area under the standard normal curve
to the left of z.
.03
.0003
.0004
.0006
.0009
.0012
.0017
.0023
.0032
.0043
.0057
.0075
.0099
.0129
.0166
.0212
.0268
.0336
.0418
.0516
.0630
.0764
.0918
.1093
.1292
.1515
.1762
.2033
.2327
.2643
.2981
.3336
.3707
.4090
.4483
.4880
.04
.0003
.0004
.0006
.0008
.0012
.0016
.0023
.0031
.0041
.0055
.0073
.0096
.0125
.0162
.0207
.0262
.0329
.0409
.0505
.0618
.0749
.0901
.1075
.1271
.1492
.1736
.2005
.2296
.2611
.2946
.3300
.3669
.4052
.4443
.4840
.OS
.0003
.0004
.0006
.0008
.0011
.0016
.0022
.0030
.0040
.0054
.0071
.0094
.0122
.0158
.0202
.0256
.0322
.0401
.0495
.0606
.0735
.0885
.1056
.1251
.1469
.1711
.1977
.2266
.2578
.2912
.3264
.3632
.4013
.4404
.4801
.06
.0003
.0004
.0006
.0008
.0011
.0015
.0021
.0029
.0039
.0052
.0069
.0091
.0119
.0154
.0197
.0250
.0314
.0392
.0485
.0594
.0721
.0869
.1038
.1230
.1446
.1685
.1949
.2236
.2546
.2877
.3228
.3594
.3974
.4364
.4761
.07
.0003
.0004
.0005
.0008
.0011
.0015
.0021
.0028
.0038
.0051
.0068
.0089
.0116
.0150
.0192
.0244
.0307
.0384
.0475
.0582
.0708
.0853
.1020
.1210
.1423
.1660
.1922
.2206
.2514
.2843
.3192
.3557
.3936
.4325
.4721
.08
.0003
.0004
.0005
.0007
.0010
.0014
.0020
.0027
.0037
.0049
.0066
.0087
.0113
.0146
.0188
.0239
.0301
.0375
.0465
.0571
.0694
.0838
.1003
.1190
.1401
.1635
.1894
.2177
.2483
.2810
.3156
.3520
.3897
.4286
.4681
.09
.0002
.0003
.0005
.0007
.0010
.0014
.0019
.0026
.0036
.0048
.0064
.0084
.0110
.0143
.0183
.0233
.0294
.0367
.0455
.0559
.0681
.0823
.0985
.1170
.1379
.1611
.1867
.2148
.2451
.2776
.3121
.3483
.3859
.4247
.4641
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Standard Normal Probabilities
z Table entry for is the area under the standard normal curve
z
to the left of z.
z
.00
.01
.02
.03
.04
.OS
.06
.07
.08
.09
0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998
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Auxiliary function y(L,B) for Spherical model with range 1.0 and sill 1.0
l
.I
.2
.05 .O!l-1 .l:l2
.10 .!GI .188
.15 .231 .252
.20 .302 .318
.25 .372 .38;,
.30 .440 .451
.35 .507 .516
.40 .571 .578
.45 .G32 .638
.50 .68!) .605
.i15 .743 .748
.GO .7!l3 ,7!)7
.65 .839 .842
.70 .87!) .882
.7f, .!ll5 .917
.80 .945 .!l46
.85 .968 .970
.!lll .986 .!J87
.!lf> .!l9G .9!l7
1.00 1.000 1.000
.:l
.17[,
.223
.280
.:l-11
.404
.467
.52!)
.590
.648
.703
.755
.80:1
.8H
.886
.920
.!l4!l
.!lit
.!l88
.!l'l7
1.000
.4
.2HJ
.261
.312
.36!)
.428
.488
.547
.605
.661
.715
.765
.811
.854
.S!l2
.925
.952
.974
.98!)
.!J!l8
1.000
n
.5 _r, .7
.263 .306 .3-18
.:100 .340 .:rnl
.3H .38:l ,41!)
.400 .432 .464
.45'\\ .483 .512
.511 .r,~J6 .562
.568 .5!l0 .612
.623 .642 .G62
.677 .693 .711
.728 .742 .758
.776 .78') .802
.821 .8:ll .8-12
.862 .870 .87'l
.sns .!lOf, .912
.!l30 .9:l5 .9-IO
.%6 .960 .96:l
.!l76 .978 .981
.9!)0 .!l!ll .!l!l2
.!l!lS .998 .!l!l8
1.000 1.000 1.000
.8
.9
.388 .42G
.416 .4:i2
.453 ..!86
A!l5 .52G
,.;41 .568
.588 .613
.635 .G57
.G83 .702
.72!) .746
.773 .787
.814 .827
.853 .863
.888 .S'lG
.!ll!l .!l25
.945 .!l49
.9611 .Oll!l
.982 .984
.!)!l3 .!J94
.9'l!l .999
1.000 1.000
1.0
.461
.486
.f,18
.555
.5!l4
.6:l6
.678
.721
.762
.801
.838
.872
.!lO:l
.!l30
.%3
.971
.085
.!l94
.99!)
1.000
11
l
1.2 u
1.6 1.8 2.0 2.5 3.ll 3.,5 4.0 5.0
.05 .524 .575 .617 .652 .681 .737 .Ti7 .so11 .828 .861
.JO .54:) .5'l4 .fl34 .667 .G!l5 .748 .786 .814 .8:l6 .Sfl7
.nm .15 .573 .61D .656 .G87 .714 .7Cvl
.825 .8-16 .875
.20 .1105 .6-18 .682 .711 .735 .7S2 .814 .838 .857 .884
.25 .6'11 .679 .711 .7:J7 .75!l .SOI .831 .853 .870 .8!J4
.30 .G7S .712 .741 .764 .784 .822 .848 .868 .88:l .905
.35 .i15 .7411 .771 .7!l2 .SO!l .8-1:J .S66 .884 .8!)7 .!Jl7
.40 .753 .780 .SOI .820 .835 .86-1
.899 .011 .!l28
.45 .7!l0 .812 .831 .847 .860 .884 .!l02 .915 .!l2-l .939
.50 .82f, .844 .86() .872 .883 .904 .!l!S .92!) .!!37 .!l4!l
.5G .858 .87:3 .SSG .8!)7 .906 .922 .934 .943 .'l49 .%9
.Oil .888 .'lOl .!lll .!lHJ ,!)26 .939 .948 .955 .960 .968
.65 .915 .925 .933 .939 .944 .!15-1 .961 .!l66 .970 .!l76
.70 .9:l'l .946 .9fi2 .%6 .960 .%7 .'l72 .976 .!J7!l .083
.75 .959 .!l64 .968 .!lil .97-1 .!J78 .9S2 .9S4 .986 .989
.80 .975 .978 .!lSl .983 .984 .987 .98!) .!l!ll .!)!)2 .'l!J3
.85 .'l87 .'l8!l .!J!lO .!JOI .992 .99:l .!l!l4 .!l!l5 .9% .'l97
.!JO .995 .!)% .996 .!l'l7 ,!Jf)7 .!J!l7 .!l'l8 .!l!lS .9!l8 .!l!l!)
.05 .9'l9 .!l!J!l .!l9'l .!l!)!) .!J'l9 1.000 1.000 1.000 1.000 J.000
1.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
Auxiliary function F(L; 8) for Spherical model with range 1.0 and sill 1.0
L .J .2
.10 .078 .120
.20 .120 .155
.30 .165 .196
.-!O .211 .237
.50 .256 .280
.60 .300 .321
.70 .342 .362
.80 .383 .-101
.90 .-122 .438
1.00 .457 .47a
1.20 .520 .534
1.40 .572 .584
1.60 .61..J .625
I.SO .6:",0 .659
2.00 .67!) .688
2.50 .735 .743
:mo .775 .781
3.50 .804 ,81[)
-1.00 .827 .832
5.00 .860 .864
.3
.165
.196
.2:H
.270
.309
.349
.387
.424
.460
.493
.551
.600
.639
.672
.700
.752
.789
.817
.838
.86!)
.4
.211
.2:37
.270
.305
.342
.:379
.415
.451
.48-1
.516
.572
.618
.655
.687
.713
.763
.79!)
.82.:,
.845
.87-1
13
.5 .6 .7
.256 .300 .342
.280 .321 .:162
.:10!) .349 .:187
.3-12 .379 .4lfi
.376 .411 .-145
.411 443 .476
.445 .476 .506
.-179 .507 .5:16
.511 .5:38 .565
.541 .566 .591
.593 .616 .638
.637 .657 .677
.673 .6!ll .70!)
.703 .719 .736
.728 .7-1:3 .758
.775 .788 .800
.80!) .820 .830
.834 .843 .852
.853 .861 .870
.881 .887 .894
.8 .!)
.38:1 .422
.-:101 .438
.424 .460
.451 .48-1
.479 .511
.507 .538
.516 .565
.564 .591
.591 .616
.616 .640
.660 .682
.697 .716
.727 .744
.752 .767
.7n .787
.813 .824
.8-11 .851
.861 .870
.878 .885
.901 .!J07
1.0
.457
.473
.4!l3
.fil6
.541
.566
.591
.616
.640
.662
.701
.733
.760
.782
.800
.835
.860
.878
.892
.!Jl:3
L
.10
.20
.30
.40
.50
.60
.70
.80
.!JO
1.00
1.20
1.40
um
1.80
2.00
2.50
3.00
3.50
4.00
5.00
1.2 1.4 1.6 1.8
.520 .572 .614 .650
.534 .584 .625 .659
.551 .GOO .6:1!1 .672
.572 .618 .655 .687
.593 .637 .673 .701
.616 .657 .691 .719
.6:38 .677 .709 .736
.660 .697 .727 .752
.682 .716 .744 .767
.701 .7:33 .760 .782
.736 .764 .788 .807
.764 .790 .811 .828
.788 .811 .829 .845
.807 .828 .845 .859
.823 .842 .858 .871
.854 .870 .883 .894
.876 .890 .901 .910
.892 .904 .914 .!J21
.905 .915 .!J24 .931
.!J2:3 .!J31 .!J:38 .944
D
2.0 2.5
.67!) .735
.688 .743
.700 .752
.713 .763
.728 .775
.743 .788
.758 .800
.773 .813
.787 .82-1
.800 .835
.82:1 .8.54
.842 .870
.858 .88:1
.871 .894
.882 .903
.903 .920
.917 .9:32
.928 .fJ4J
.936 .948
.948 .!J5i
3.0
.775
.781
.789
.79!)
.809
.820
.830
.841
.851
.860
.876
.890
.901
.910
.917
.932
9-12
.fJ50
.955
.!J64
3.5 4.0
.804 .827
.810 .832
.817 .838
.825 .845
.834 .853
.843 .861
.852 .870
.861 .878
.870 .885
.878 .892
.892 .905
.904 .915
.914 .924
.921 .931
.928 .936
.9-11 .!J48
.950 .955
.956 .!J61
.961 .966
.!JG!) .972
5.0
.860
.864
.86!)
.874
.881
.887
.894
.901
.!l07
.9l~l
.923
.fl31
.9:38
.944
.948
.957
.964
.96!)
.972
.977
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8 Page 8 |
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Auxiliary functionH(L,B) for Sphericalmodel with range 1.0 and sill 1.0
n
L .1 .2 .3 .4 .5 .6 .7 .8 .9 .10
.10 .114 .li7 .243 .310 .374 .436 .494 .5-16 .5!J:3 .633
.20 .177 .227 .285 .346 .406 .46-1 .518 .568 .61:3 .651
.30 .243 .28G .336 .3!)0 .-145 .4!J!) .f>50 .rm .63!) .674
.40 .310 .346 .3!)0 .439 .489 .5:l!J .586 .62!) .668 .701
.50 .:374 .40(i .H5 .48!) .5:Vi .580 .623 .663 .698 .728
.GO .-136 .464 .499 .539 .580 .621 .660 .6!)7 .728 .7G5
.70 .404 .518 .5fi0 .586 .623 .660 .696 .72~) .7fi7 .781
.80 .546 .568 .597 .629 .663 .6!J7 .72!) .758 .783 .805
.90 .593 .613 .639 .668 .698 .728 .757 .783 .806 .826
1.00 .633 .651 .674 .701 .728 .755 .781 .805 .826 .843
1.20 .6!)4 .70!) .729 .751 .774 .7!JG .818 .837 .855 .86!)
1.-10 .738 .751 .767 .786 .806 .825 .844 .8Gl .875 .888
1.60 .771 .782 .797 .81:l .830 .847 .86:l .8i8 .891 .902
1.80 .796 .806 .819 .8:34 .849 .864 .879 .892 .903 .[)13
2.00 .817 .826 .837 .850 .86-1 .878 .891 .!102 .913 .921
2.50 .853 .860 .870 .880 .891 .902 .!113 .922 .!130 .937
3.00 .878 .884 .891 .!JOO .909 .918 .!)27 .93G .!J-12 .948
3.50 .8!J5 .!JOO .907 .91-1 .922 .930 .!138 .944 .950 .95-5
4.00 .!JOS .!Jl3 .!)19 .925 .932 .!J3!) .!J-15 .051 .956 .961
5.00 .927 .!J30 .935 .040 .946 .!)51 .95Cl .961 .965 .!JG!)
fl
L 1.2 1.4 l.G 1.8 2.0 2.5 3.0 3.5 4.0 5.0
.10 .Cl!J4 .738 .771 .7!)6 .8li .Sf,:l .878 .8!)5 .908 .927
.20 .70!) .751 .782 .806 .826 .860 .88.J .900 .!113 .!J30
.30 .72!) .767 .797 .81!) .837 .870 .891 .!J07 .91!) .935
.40 .751 .786 .813 .834 .8511 .880 .900 .914 .925 .940
.50 .774 .806 .830 .849 .864 .8!11 .909 .922 .!132 .946
.60 .796 .825 .847 .864 .878 .!102 .!)JS .!)30 .9:19 .951
.70 .818 .844 .8G3 .879 .891 .913 .927 .!)38 .!J45 .%6
.80 .837 .861 .878 .8!J2 .902 .!122 .!135 .9-14 .051 .961
.!JO .855 .875 .891 .!)03 .013 .!J:lO .!)42 .950 .956 .!)65
1.00 .86!) .888 .!)02 .!Jl3 .921 .937 .948 .!)55 .!JGl .969
1.20 .8!)1 .907 .918 .927 .935 .948 .956 .96:l .067 .974
1.40 .007 .020 .030
I.GO .018 .D:JO .03!)
I.SO .027 .938 .945
2.00 .035 .944 .051
2.50 .948 .!J55 .961
3.00 .D5G .963 .067
.!)38
.!J45
.%2
.!J56
.965
.!J71
.%.I .055 .063
.051 .DG! .967
.056 .!JG5 .!)71
.DGl .96!) .!l74
.960 .075 .979
.074 .9i9 .083
.968 .072
.972 .075
.975 .078
.!)78 .080
.!)82 .084
.08::i .087
.078
.080
.983
.DS.J
.987
.!J!JO
3.50 .96:l .068 .072 .!J75 .078 .082 .085 .087 .!JS9 .!J!Jl
4.00 .!167 .072 .!175 .078 .080 .98.J .087 .98!) .D!JO .902
5.00 .974 .978 .980 .!)83 .984 .987 .990 .991 .9!J2 .094
Auxiliary function F(L; L; B) for Sphericalmodel with range 1.0 and sill 1.0
n
L .l .2 .:l ..:J .5 .G .7 .8 .9 1.0
.10 .099 .136 .178 .222 .266 .30!) .350 .390 .428 .464
.20 .168 .196 .231 .269 .:308 .347 .:l85 .423 .458 .491
.:m .23D .262 .291 .:l24 .358 .3!J4 .420 .4(i;3 .4!JG .527
.40 .311 .32!) .35:l .382 .41;3 .4.J5 .476 .508 .538 .566
.5D .380 .395 ..JIG .441 .-.168 .-.197 .526 .554 .581 .607
.GD .-145 .45!J .477 .49!) .523 .549 .574 .GOO .624 .G4S
.70 .507 .51!) .535 .554 .576 .508 .622 .6-1.J .GG6 .687
.so .565 .574 .58S .606 .G2f, .6-15 .GG6 .686 .705 .724
.90 .GIG .625 .637 .652 .GGD .687 .706 .724 .741 .757
1.00 .662 .660 .680 .694 .709 .725 .741 .757 .772 .786
1.20 .735 .741 .750 .760 .772 .78-5 .rn7 .810 .822 .833
1.40 .i89 .7!J4 .800 .809 .818 .82S .83!) .840 .858 .867
1.60 .828 .832 .838 .845 .852 .861 .86!) .877 .885 .892
1.80 .858 .861 .866 .872 .878 .88f> .802 .8!J9 .!)05 .Dll
2.DO .880 .883 .887 .802 .8!)7 .003 .009 .915 .920 .025
2.50 .918 .020 .!)23 .926 .930 .934 .938 .942 .9-16 .049
3.00 .!l40 .041 .D-14 .!J4G .940 .!)52 .%5 .958 .!160 .DG3
3.50 .954 .955 .957 .950 .!JG! .06:l .!JGG .068 .970 .972
4.00 .!)63 .964 .065 .967 .!JGD .070 .072 .074 .976 .977
5.00 .974 .075 .976 .!J78 .97!) .080 .!J81 .983 .!JS-I .985
L
.10
.20
.30
.40
.50
.GO
.70
.80
.DO
1.00
1.20
1.40
I.GO
1.80
2.00
2.50
3.00
:1.50
4.00
5.00
B
1.2 1.4 l.G 1.8 2.0 '_J_ ;•J 3.0 3.5 4.0 5.0
.526 .:j77 .Gl!J .653 .683 .738 .777 .807 .820 .861
.550 .508 .638 .671 .699 .751 .78!) .SlG .837 .868
.581 .626 .663 .603 .710 .768 .80:l .820 .84!) .877
.GIG .6G7 .GDl .71!) .742 .787 .Sl!J .843 .861 .887
.652 .GSD .720 .745 .767 .808 .836 .858 .874 .808
.688 .722 .740 .772 .791 828 .854 .873 .887 .900
.723 .753 .777 .798 .815 .847 .870 .887 .900 .919
.756 .782 .804 .822 .837 .865 .886 .901 .012 .029
.785 .SOD .828 .84:3 .857 .882 .900 .013 .923 .937
.811 .832 .8.JD .862 .874 .806 .!Jl2 .023 .9:32 .045
.853 .869 .882 .803 .902 .Dl!J .031 .040 .947 .057
.883 .896 .DOG .9F, .022 .036 .945 .952 .958 .966
.!J05 .915 .924 .!)31 .037 .048 .OGG .%1 .!JGG .072
.022 .030 .037 .04:3 .048 .057 .96:3 .068 .972 .077
.!J3.J .041 .047 .052 .956 .064 .!JG!J .073 .976 .981
.955 .DGO .064 .DG7 .970 .075 .!)79 .082 .084 .987
.067 .071 .!J74 .076 .078 .082 .085 .!J87 .088 .DDl
.97:"i .078 .080 .982 .DS'.l .986 .088 .D!JO .DDl .!JD3
.080 .982 .084 .DSG .087 .089 .DDl .092 .003 .904
.087 .088 .98!) .9!)0 .DDl .!JD3 .DD'1 .095 .005 .906
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9 Page 9 |
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Auxiliary function x(L;B) for Spherical model with range 1.0 and sill 1.0
L .l
.10 .098
.20 .164
.:lO .23:l
.40 .302
.50 .368
.GO .430
.70 .488
.so .542
.!JO .589
1.00 .629
1.20 .691
1.40 .735
I.GO .768
1.80 .7!J4
2.00 .815
2.50 .852
3.00 .87G
3.50 .894
,J.Q[) .!J07
5.00 .926
.2
.13(i
.194
,2;,7
.322
.385
.445
.502
.554
.600
.639
.69!)
.742
.775
.800
.820
.856
.880
.8D7
.!JlO
.928
.3
.178
.229
.288
.:l-18
.408
.466
.520
.570
.614
.653
.711
.752
.783
.807
.826
.8Gl
.884
.!JOI
.Dl3
.D31
.4
.222
.2Ci8
.321
.:378
.434
.489
.541
.589
.632
.668
.72:l
.763
.7!J:3
.816
.8:l4
.8G7
.88!)
.905
.917
.n:34
n
.5 .Ci
.266 .309
.307 .3.J(i
.:l5Ci .:392
.40() .441
.4(i2 .492
.515 .541
.564 .588
.610 .6:H
.650 .670
.685 .70:l
.737 .752
.775 .788
.803 .81,1
.825 .8:35
.8-12 .851
.874 .881
.89:i .!JOl
.910 .915
.921 .926
.937 .941
.7 .8 .9 1.0
.350 .300 .428 .464
.385 .422 .458 .491
.427 .462 .49:i .;>w
.-174 .505 .535 .564
,;:;21 .f>50 .:.>77 .603
.568 .594 .61() .642
.Cil2 .Ci36 .658 .680
.65:l .67-l .6()5 .714
.68!) .708 .727 .7-l-l
.720 .7:H .754 .769
.767 .781 .795 .808
.800 .812 .824 .8:15
.825 .836 .8'16 .85G
.845 .8:i-l .863 .872
.8GO .8GD .877 .885
.888 .8% .902 .908
.907 .!Jl2 .918 .9:2:3
.920 .925 .930 .9:l-l
.!J:10 .9:H .!J:38 .!J42
.!J44 .947 .!J51 .9;i4
n
L 1.2 1.4 l.G 1.8 2.0 2.5 3.0 3.5 4.0 5.0
.10 .526 .577 .61!) .653 .683 .738 .777 .807 .820 .861
.20 .550 .508 .638 .671 .6!J8 .751 .788 .SIG .837 .868
.30 .580 .62;, .GG2 .G!J3 .71D .768 .803 .828 .848 .877
.40 .614 .65~ .G89 .718 .741 .787 .81!) .842 .861 .887
.50 .G-l!J .G87 .718 .743 .7G5 .806 .8:lf> .857 .87:3 .897
.60 .684 .718 .746 .7G!J .788 .825 .852 .871 .886 .!J07
.70 .717 .747 .772 .793 .811 .844 .8G7 .885 .898 .917
.80 .747 .i74 .7!J7 .815 .831 .861 .881 .897 .DO!l .!J2G
.no .774 .798 .818 .8:lfi .849 .875 .894 .908 .Dl9 .934
1.00 .796 .818 .836 •. 851 .864 .888 .90fl .917 .927 .941
1.20 .830 .848 .8Ci-l .876 .886 .DOG .920 .!J:31 .93D .!J50
l.-10 .854 .Sin .883 .894 .903 .!J20 .932 .!J41 .948 .958
1.60 .873 .886 .898 .!J07 .915 .930 .940 .!J,18 .954 .963
1.80 .887 .8!J9 .!JO!J .917 .924 .!J:l8 .!l47 .95·1 .!J5!) .!JG7
2.00 .898 .!JO!J .!Jl8 .926 .932 .!J44 .!J52 ,!J,59 .!JG:l .!J70
2.50 .!)18 .!J27 .!J34 .!J40 .!J46 .fl5f> .%2 .067 .D71 .D7G
3.00 .!J32 .939 .94;, .!J50 .%f> .%3 .!J68 .!J72 .976 .980
:l.50 .942 .948 .95:l .!157 .961 .068 .973 .976 .07!) .983
-LOO .949 .!J5G .%!J .!JG:l .!J66 .!Ji2 .976 .979 .!J82 .!J85
5.00 .!J59 .964 .!J67 .!J70 .97:3 .!J78 .D81 .!JS:l .085 .088
Regularised semi-variogram y(h) for Spherical model with range a and sill 1.0 for various distances h
h/L
a/L 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
.50 .:300 .325 .32.J .325 .0•)-'D)~ .325 .325 .325 .325 .325
1.00 .450 .550 .550 .550 .550 .550 .550 .550 .550 .550
1.50 .463 .678 .681 .681 .681 .681 .681 .681 .681 .681
2.00 .412 .728 .756 .756 .756 .756 .756 .756 .756 .756
2.50 .355 .717 .802 .803 .803 .8D:3 .803 .803 .803 .803
3.00 .307 .669 .822 .835 .835 .835 .835 .835 .835 .83-5
:1.50 .269 .610 .812 .858 858 .858 .858 .858 .858 .858
4.00 .239 .555 .778 .868 .876 .876 .876 .876 .876 .876
4.50 .215 .507 .733 .861 .889 .889 .889 .889 .889 .889
5.00 .104 .464 .686 .836 .896 .900 .900 .900 .900 .900
5.50 .178 .428 .642 .802 .890 .909 .909 .909 .909 .909
6.00 .lG3 .396 .601 .764 .872 .914 .917 .917 .917 .917
6.50 .151 .368 .564 .726 .845 .909 .923 .92:3 .923 .923
7.00 .141 .:H4 .530 .690 .814 .895 .926 .929 .929 .929
7.,50 .132 .:323 .500 .655 .782 .874 .92:3 .933 .933 .933
8.00 .124 .304 .472 .623 .751 .849 .912 .936 .938 .938
8.50 .117 .287 .447 .593 .720 .822 .894 .933 .941 .941
9.00 .110 .272 .425 .566 .690 .794 .874 .924 .943 .945
9.50 .104 .258 .404 .541 .663 .767 .851 .!HO .941 .947
10.00 .099 .246 .386 .517 .636 .741 .827 .892 .9:33 .949
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10 Page 10 |
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Spherical Variogram Model
Additional Information {GSS710S)
[¾(~)-½(~f] y(h) = C
for h < a
=C
for h 2: a
Relationship between sill of regularized and point variograrn
CL = C [ l - 2La + 2~: 3] for L < a
Ca a] CL= l
[15
20 -
4
20
L
for L 2: a
Auxiliary functions for Spherical variograrn
2
l) = BCal ( 6 - alZ) when l $ a
a) XC
=
C
8
(
8
-
3y
when l > a
a l2) =
C
20
l ( 10 -
a2
T %2) F(l)
=
C(
20 20 -
15
+4
when l $ a
when l > a
Y ( l; b)
M
IM' X(l;b)
b
j
M
X M'
I
b
J
H ( l: b)
M
rv,'
X
1r M
I 1 M'