GSS710S - Geo Statistics 314 - 2nd Opp - June 2022 :: NUST past examination papers between 2020 and 2024

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GSS710S - Geo Statistics 314 - 2nd Opp - June 2022

1 Pages 1-10

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2 Pages 11-20

2.1 Page 11

2.2 Page 12

2.3 Page 13

2.4 Page 14

2.5 Page 15

2.6 Page 16

2.7 Page 17

2.8 Page 18

2.9 Page 19

2.10 Page 20

GSS710S - Geo Statistics 314 - 2nd Opp - June 2022

1 Pages 1-10

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1.1 Page 1

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nAmlBIA unlVERSITY

OF SCIEnCE Ano TECHnOLOGY

FA CUL TY OF ENGINEERING AND SPATIAL SCIENCE

DEPARTMENT OF MINING AND PROCESS ENGINEERING

QUALIFICATION : BACHELORS OF ENGINEERING IN MINING ENGINEERING

QUALIFICATION CODE: BEMIN LEVEL: 6

COURSECODE: GS~l~

COURSE NAME: GEOSTATISTICS

SESSION: JUNE 2022

PAPER: THEORY

DURATION: 3 HOURS

MARKS: 100

SECOND 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)

1.2 Page 2

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1.3 Page 3

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This EXAM has two sections. Section A and B.

Time allowed: 3 hours

SECTION A

Instructions: Answer any 2 questions. Excess questions will not be marked.

Question 1 Answer the following questions as succinctly as possible

a) Discuss the revenue factors involved in operating a mining venture

(8)

b) Discuss the main factors involved in the valuation ofan ore body

(8)

c) Distinguish the main differences between Geostatistics and statistics. Discuss the (9)

differences that are apparent in a data set with a statistical and a geostatistical variance.

Explain the effects of these two variances on a mine scenario where block grades are being

evaluated.

Question 2

a) What are the limitations of statistical data m solving geological and mining (JO)

problems of grade-tonnage relationships?

b) Discuss the following modes of exploration indicating what happens and what steps ( I 5)

follow afterwards: Geochemistry, Geological and Geophysics.

Question 3

a) Discuss the differences between resources and reserves

(4)

b) What are the purposes of ore reserves evaluation?

(4)

c) What is regression effect and how can it be overcome?

(4)

d) What is data optimisation? In what ways can data be optimised?

(6)

e) What are simulations and how do we use them on a mine setting?

(7)

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1.5 Page 5

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SECTION B

Instructions: Answer Question 1 and any 2 other questions. Excess questions will not be marked.

Question 1 is compulsory.

Question 1

a) What is suppo1i?

( J)

b) Which statistical quantity represents reliability of an estimation method?

(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 (I, b) gives?

(I)

e) What 'D' matrix in Kriging system of equations represents?

(I)

f) Which of spherical, gaussian and exponential variograms is more continues at the

(I)

origin?

g) What is the difference between extension variance and Kriging variance?

(I)

h) What is the difference between zonal anisotropy and geometric anisotropy?

(2)

i) What is screen effect in Kriging? [I]

(I)

Question 2

a) Briefly discuss the effects of scale, nugget effect and range on Kriging weights

( I0)

b) Spatial continuity of Zinc grades in the ore body is following spherical variogram model ( I0)

with a sill value of2.5 (%) 2 and range of300 m. Determine the model values for the given

lags.

l[ l(h) C LS-h--

r(h)= C ::~:

ifh~a

Lags:

100

200

300

400

Question 3 In a copper deposit, to excavate a stope having a size of 30 m x 30 m, two level of 15 m (20)

apart were driven as shown in Figure I. The average grade of level I was found to be 5.4%

and level 2 was 6.7%. If the grade of the stope is taken as average grade of both levels,

estimate the extension variance. Spatial continuity in the deposit is best described with

spherical varigoram having range of influence of 90 m and a sill of0.6 (%)2

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1.7 Page 7

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15 Ill

Level~

30111

som

Figure 1

Question 4 Copper grade at three borehole locations A, B, and Care found to be 7.8%, 5.2% and 6.2% (20)

respectively. Grade at a location X to be estimated. The inter distance between (A, 8, C,

and X) is given in the form of a distance matrix. Estimate the grade at location 'X' with

96% confidence using Kriging method of estimation?

1.s[ y(h)= 1- exp(- ; )] [Variogram model]

Distance Matrix

29.2 53.9 26.9

29.2

25.5 50.2

53.9 25.5

75.7

26.9 50.2 75.7

C-'

[-191.71 0.19

1.71 -3.67 1.96

0.19 1.96 -2.15

0.46 0.09 0.45

l0.4069

0.45

-0.23

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1.9 Page 9

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Standard Normal Probabilities

.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

Table entry for z 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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2 Pages 11-20

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2.1 Page 11

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Standard Normal Probabilities

z Table entry for is the area under the standard normal curve

to the left of 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

·o.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 .75fi°0 .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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2.3 Page 13

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Auxiliary function y(L,B) for Spherical model with range I .0 and sill I .0

L .1

.:l

.!J 1.0

L 1.2 u

1.6 1.8 2.0 2.5 3.0 3.-5 4.0 5.0

.05 .094 .132 .175 .2l!l .263 .306 .348 .:l88 .426 .461 .05 .524 .,,75 .617 .652 .681 .737 .777 .SOG .828 .861

.10 .!Gl .188 .223 .261 .300 .340 .379 .416 .452 .486 .JO .545 .m4 .634 .667 .G95 .748 .786 .814 .836 .867

.15 .231 .252 .280 .312 .3H .38:l .4l!l .-!53 .486 .518

.20 .302 .318 .341 .36n .400 .432 .-!6-1 .-!95 .526 .555

.25 .372 .3Sf, .4(14 .428 .45.1 .-!S3 .512 .,,,11 .568 .59-1

.15 .573 .GI() .656 .687 .714 .764 .709 .825 .8-16 .87!j

.20 .605 .648 .682 .711 .7.35 .782 .814 .s:is .857 .884

.25 :641 .679 .711 .7:37 .75n .SOI .831 .853 .870 .894

.:lO .440 .451 .467 .-188 .fill .536 .562 .588 .613 .63G .30 .G78 .712 .741 .764 .784 .822 .848 .868 .883 .n05

.3f> .507 .51G .529 .547 .568 .WO .612 .63:3 .G57 .G78 .:15 .llf.i .,4() .771 .7!)2 .809 .8,n .866 .S84 .8!17 .017

AO .571 .578 .5!Jll .605 .62:l .G42 .662 .6S3 .702 .i21

.40 .,,:,3 .780 .SOI .820 .8:15 .86-1 .884 .89') .fJJI .!J28

.-If, .632 .638 .G4S .661 .677 .G!l3 .711 .729 .HG .762 .45 .,!JO .812 .831 .847 .SGO .884 .no2 .!Jlfi .n24 .n3!l

.50 .689 .695 .703 .715 .728 .742 .7f>8 .773 .781 .SOI .50 .82;, .844 .860 .872 .883 .004 .nIS .n2!J .037 .n.1n

.55 .743 .748 .75f, .765 .776 .7Sn .802 .S14 .827 .8:18

.GO .793 .797 .SlXl .811 .821 .8:ll .842 .853 .8il3 .872

.55 .858 .87:J .886 .8!l7 .906 .n22 .n34 .94:l .049 .%!)

.Gil .888 .!lOI .!Jll .nm .n26 .!J39 .n48 .%5 ,!)60 .n6S

.of, .839 .842 .SH .854 .8(i2 .870 .S7n .888 .S!lG .!JO:l .65 .n15 .925 .!l:l3 .!)3n .944 .!l5-l .961 .!JGG .!l70 .!J76

.70 .879 .882 .886 .892 .8!lS .nor, .n12 .91!) .n25 .!J30

.75 .n15 .917 .920 .925 .930 .9:35 .9-IO .945 .n49 .%3

.70 .930 .046 .!)52 .956 .!JGO .967 .072 .!l76 .!l7!l .98:l

.75 .%n .964 .!JGS .n71 .n74 .978 .ns2 .!l84 .DSG ,989

.80 .!J45 .!l46 .!J4n .952 .956 .960 .963 .!l66 .!l6!J .!J71

.85 .968 .970 .!l7! .974 .976 .9,8 .981 .982 .!J84 .985

.80 .975 .n78 .981 ,!}83 .nS4 .987 .!l8!J .9nl .n!l2 .n!J3

.85 .987 .OS!J .9nO .!J(lj .nn2 .993 .!J94 .!J!l5 .!lfJ6 .!Jn7

.no .986 .987 .!JSS .989 .n!JO .!J9! .!J92 .9!J3 .!l94 .994

.!J5 .!J!lG .997 .!Jn7 .9!J8 .998 .99S .998 .99n .9!J!J .nnn

.no .n05 .n96 .!lnG .!J97 .nn7 .9n7 .!J!J8 .!J98 .9n8 .nnn

.95 .nnn .nnn .nnn .non .!ln!l 1.000 1.()()0 1.000 1.1100 1.000

I.Oil 1.000 I.ODO 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.01111 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; B) for Spherical model with range I .0 and sill I .0

.10

.20

.30

.40

.50

.60

.70

.80

.!JO

1.00

1.20

1.40

1.60

I.SO

2.00

2.50

3.00

3.50

4.00

5.00

.l .2

.078 .120

.120 .155

.165 .196

.211 .237

.256 .280

.:JOO .321

.342 .362

.383 .-IOI

.422 .438

.4!i7 .473

.520 .53-1

.572 .584

.614 .625

.650 .659

.679 .688

.735 .743

.775 .781

.804 .810

.827 .832

.860 .864

.165

.196

.231

.270

.309

.:149

.387

.424

.460

.493

.551

.600

.639

.672

.700

.752

.789

.817

.838

.869

.211

.2:37

.270

.305

.342

.379

.415

.451

.484

.516

.572

.618

.655

.687

.713

.763

.799

.825

.845

.87-1

.5 .6

.256 .:300

.280 .321

.309 .349

.342 .379

.376 .411

.411 -143

.445 .476

.479 .507

.511 .538

.541 .566

.593 .616

.637 .657

.673 .691

.703 .719

.728 .743

.775 .788

.809 .820

.834 .843

.853 .861

.881 .887

.342

.362

.387

.415

.445

.476

.506

.5:16

.565

.591

.638

.677

.709

.736

.7-58

.800

.830

.852

.870

.894

.383

.401

.424

.451

.479

.507

.536

.564

.591

.616

.660

.697

.727

.752

.773

.81:3

.841

.861

.878

.901

.422

.438

.460

.484

.511

.538

.565

.591

.616

.6-10

.682

.716

.744

.767

.787

.824

.851

.870

.885

.907

1.0

.457

.47:1

.493

)il6

.541

.!i66

.591

.616

.(J-10

.662

.701

.733

.760

.782

.800

.835

.860

.878

.892

.9U

L 1.2 1.4 1.6

.10 .520 .572 .614

.20 .534 .fi84 .625

.30 .551 .GOO .639

.40 .572 .618 .655

.50 .;,9;3 .637 .673

.60 .616 .657 .691

.70 .638 .677 .709

.80 .GGO .697 .727

.90 .682 .716 .744

1.00 .701 .733 .7(i0

1.20 .736 .764 .788

1.40 .764 .790 .811

1.60 .788 .811 .829

1.80 .807 .828 .845

2.00 .823 .842 .858

2.50 .854 .870 .883

3.00 .876 .890 .901

3.50 .892 .904 .914

-1.00 .905 .915 .924

5.00 .923 .931 .938

1.8

.650

.6fi!l

.672

.687

.70:3

.719

.736

.752

.767

.782

.807

.828

.845

.859

.871

.894

.910

.921

.931

.944

2.0 2.5 :to 3.5 4.0

.679 .735 .775 .804 .827

.688 .743 .781 .810 .832

.700 .752 .789 .817 .838

.713 .763 .799 .825 .845

.728 .775 .809 .8:H .85:3

.743 .788 .820 .84:3 .861

.758 .800 .830 .8f>2 .870

.77:1 .813 .841 .861 .878

.787 .82-1 .851 .870 .885

.800 .835 .860 .878 .892

.823 .Sf,4 .876 .892 .905

.842 .870 .890 .904 .915

.858 .88:3 .901 .914 .924

.871 .894 .910 .921 .931

.882 .903 .917 .928 .936

.903 .920 .932 .941 .948

.917 .!J:12 D-12 .%0 .%5

.928 .941 .950 .956 .961

.936 .948 .955 .9Gl .966

.948 .957 .064 .969 .972

5.0

.860

.864

.869

.874

.881

.887

.894

.901

.907

.913

.923

.9:31

.9:~8

.9-14

.948

.%i

.964

.!JG9

.972

.977

2.4 Page 14

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2.5 Page 15

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Auxiliary function H(L,B) for Spherical model with range 1.0 and sill 1.0

.10

.20

.30

.40

.50

.GO

.70

.80

.!JO

1.00

1.20

1.40

I.Gil

I.SO

2.00

2.50

3.00

3.50

4.00

5.00

.11-l

.177

.243

.310

.:374

.436

.494

.546

.593

.633

.694

.738

.771

.796

.817

.853

.878

.895

.!l08

.927

.2 .3 .4

.177 .243 .:310

.227 .285 .346

.280 .336 .3!JO

.346 .390 .439

.406 .445 .48!)

.464 .499 .539

.518 .5SO .586

.568 .597 .629

.613 .63!) .6GS

.651 .674 .701

.70() .72!) .751

.751 .767 .786

.782 .797 .81:l

.806 .819 .834

.826 .837 .850

.860 .870 .880

.884 .801 .000

.!JOO .907 .914

.913 _!)l!) .925

.930 .!)35 .940

.5 .6 .7 .8 .9 .JO

.374 .4:1G .494 .546 .59:l .6:J:1

.406 .-164 .:jl8 .568 .613 .651

.445 .<J!J() .550 .G!l7 .G3!J .674

.489 .539 .58G .G29 .668 .701

.535 .580 .623 .GG3 .698 .728

.580 .621 .660 .697 .728 .755

.623 .660 .696 .72!) .7[>7 .781

.663 .697 .72!) .758 .783 .805

.G!J8 .728 .757 .783 .806 .82G

.728 .755 .781 .SOE, .826 .843

.774 .796 .818 .837 .85-5 .86!)

.806 .825 .844 .8Gl .875 .888

.830 .847 .86:l .878 .891 .902

.8-19 .864 .87!) .892 .903 .!Jl:l

.864 .878 .891 .002 .013 .921

.891 .902 .913 .922 .!J30 .!J:l7

.909 .918 .927 .!J3G .942 .948

.922 .930 .!l38 .944 .950 .955

.932 .939 .!J.15 .951 .956 .961

.946 .951 .956 .!JGl .965 .96!l

.10

.20

.30

.40

.50

.60

.70

.80

.!JO

1.00

1.20

1.40

I.GO

1.80

2.00

2.50

3.00

3.50

4.00

5.00

1.2 1.4 1.6

.G!J4 .738 .771

.70!) .751 .782

.72!) .767 .797

.751 .786 .813

.774 .806 .8:lO

.796 .825 .847

.818 .844 .8G3

.837 .861 .878

.855 .875 .801

.86!) .888 .002

.891 .907 .!Jl8

.!l07 .920 .930

.918 .n:m .93!l

.927 .938 .945

.935 .944 .951

.948 .955 .961

.956 .963 .!l67

.963 .968 .!l72

.!l67 .!l72 .975

.974 .078 .080

1.8

.7!JG

.806

.81!)

.834

.849

.864

.879

.892

.003

.913

.927

.!l38

.945

.952

.956

.965

.971

.975

.078

.983

2.0 2.5 3.0

.817 .85:l .878

.826 .860 .884

.837 .870 .891

.850 .880 .900

.864 .801 .90!)

.878 .902 .!ll8

.891 .913 .927

.!J02 .!J22 .!J35

.013 .9:lO .!J42

.921 !)37 .948

.935 .948 .!J5G

.94-l .!l55 .963

.951 .!JG! 967

.956 .965 .971

.961 .!JG!J .974

.969 .!l75 .979

.!J74 .979 .983

.978 .982 .985

.980 .984 .987

.!J84 .!l87 .990

3.5

.895

.900

.!J07

.914

.!J22

.030

.938

.944

.950

.955

.D6:l

.!l68

.972

.975

.!l78

.982

.085

.987

.989

.091

4.0 5.0

.908 .!J27

.01;1 .fJ:3()

.910 .9:l5

.025 .940

.n:12 .946

.939 .051

.945 .906

.9fil .961

.056 .!J65

.061 .9Ci!l

.967 .974

.972 .!l78

.!l75 .980

.!l78 .!l83

.!JSO .984

.984 .987

.987 .990

.989 .991

.990 .9!l2

.!l!l2 .!J!l4

Auxiliary function F(L; L; B) for Spherical model with range 1.0 and sill 1.0

.JO

.20

.30

.40

.50

.60

.70

.80

.!JO

1.00

1.20

!AO

1.60

1.80

2.00

2.50

3.00

3.50

4.00

5.00

.099

.168

.239

.:lll

.380

.445

.507

.565

.616

.662

.735

.78!)

.828

.858

.880

.918

.940

.!l54

.963

.!l74

.136

.l!)G

.262

.32!)

.395

..!5!)

.51!)

.574

.625

.669

.741

.794

.832

.861

.883

.!l20

.941

.955

.964

.!J75

.178

.231

.2!ll

.353

.416

.477

.5:15

.588

.637

.680

.750

.800

.838

.866

.887

.!l23

.944

.957

.965

.976

.222

.269

.324

.382

.441

.4!l!l

.554

.606

.652

.G!J4

.760

.80!)

.845

.872

.8!)2

.926

.946

.!l5!l

.967

.!l78

.5 .6 .7 .8

.266 .30!) .350 .390

.308 .347 .:385 .423

.358 .3!14 .429 .463

.413 .445 .476 .508

.468 .497 .526 .554

.523 .549 .574 .600

.576 .5!l8 .622 .644

.625 .645 .666 .686

.669 .687 .70(i .724

.709 .725 .741 .757

.772 .785 .797 .810

.818 .828 .83!) .849

.852 .861 .869 .877

.878 .88.'.1 .892 .8!J!)

.897 .903 .!)09 .915

.930 .934 .938 .!l42

.949 .952 .Dfi5 .958

.!lfil .063 .!JGG .968

.!JG!J .!170 .972 .!l74

.979 .980 .081 .9S3

.9 1.0

.428 .464

.458 .49 I

.496 .527

.538 .566

.581 .607

.G24 .648

.66G .687

.705 .724

.741 .757

.772 .786

.822 .833

.858 .867

.885 .8!l2

.!JOG .DI 1

.!l20 .925

.946 .949

.960 .!JG3

.!l70 .!l72

.976 .977

.9S-1 .!J85

.10

.20

.30

.40

.50

.GO

.70

.80

.!JO

1.00

1.20

1.40

1.60

1.80

2.00

2.50

3.00

3.50

4.00

5.00

1.2 1.4

.526 .577

.550 .598

.581 .626

.616 .657

.652 .689

.688 .722

.723 .753

.756 .782

.785 .809

.811 .832

.853 .86!)

.883 .896

.905 .!l15

.922 .!l30

.934 .941

.055 .960

.!JG7 .971

.975 .978

.980 .!l82

.987 .988

1.6 1.8

.61!) .653

.638 .671

.66:3 .69:l

.691 .719

.720 .745

.74!l .772

.777 .798

.804 .822

.828 .843

.8.1!) .862

.882 .893

.!JOG .9lf,

.924 .!l31

.!J:37 .943

.947 .052

.964 .967

.fl74 .976

.980 .982

.984 .986

.989 .990

2.0 2.5

.683 .7:38

.69!) .751

.71!) .768

.742 .787

.767 .808

.791 .828

.815 .847

.837 .8G5

.857 .882

.874 .896

.902 .91!l

.922 .936

.937 .948

.948 .957

.!156 .964

.970 .!l75

.978 .982

.!l83 .!J86

.987 .989

.!l91 .9!l3

3.0 3.5 4.0 5.0

.777 .807 .829 .861

.789 .SIG .837 .SGS

.803 .82!) .84!) .877

.819 .843 .861 .887

.836 .sr,s .874 .898

.854 .873 .887 .909

.870 .887 .!JOO .919

.886 .!JOI .912 .929

.900 .913 .923 .937

.912 .923 .!J:l2 .945

.931 .940 .!l47 .957

.945 .9E,2 .958 .966

.956 .!l61 .!JGG .!l72

.96:3 .968 .!li"2 .!l77

.!JG!J .973 .976 .981

.979 .982 .984 .987

.985 .!l87 .988 .!J!ll

.988 .!l90 .991 .993

.991 .992 .993 .994

.994 .!J!l5 .!l95 .9!l6

2.6 Page 16

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2.7 Page 17

▲back to top

Auxiliary function x(L;B) for Spherical model with range 1.0 and sill 1.0

L .I .2 .3 .4

.10 .098 . t:311 .1'"i8 .222

.20 .164 .194 .229 .268

.:10 .23:1 .2fi7 .288 .321

.40 .302 .322 .:HS .378

.50 .368 .385 .408 .434

.GO .430 .4-15 .-166 .489

.70 .488 .G02 .520 .5-ll

.80 .542 .554 .570 .589

.DO .589 .600 .614 .632

1.00 .62!) .630 .653 .668

1.20 .601 .6!JD .711 .723

1.-10 .735 .742 .752 .763

1.60 .768 .775 .783 .793

1.80 .794 .800 .807 .816

2.00 .815 .820 .826 .834

2.50 .852 .856 .861 .86,

3.00 .876 .880 .884 .889

3.50 .8!)4 .807 .!JOI .905

4.00 .90i .910 .!ll3 .917

5.00 .926 .928 _!);3[ .93-l

.5 .11

.266 .:109

.:307 .346

.356 .392

.409 .441

.462 .492

.515 .541

.564 .588

.610 .6:31

.G50 .670

.685 .70;1

.737 . -J-•;),)_

.775 .788

81);3 .814

.82f, .835

.8-12 .851

.874 .881

.805 .!lOl

.!llO .915

.921 .!l26

.9:1i .941

./ .8 .9 1.0

.350 .390 .428 .464

.385 .422 .458 .491

.427 .-162 .495 .526

.474 .505 .-535 .564

.521 .550 .577 .603

.568 .594 .619 .642

.612 .6:111 .6f,8 .680

.65:.3 .6i.J .605 .714

.680 .708 .727 .i.J-l

.7211 .7:17 .75.J .760

.76, .781 .795 .808

.800 .812 .82•1 .835

.825 .836 .846 .856

.845 .85.J .86:3 .872

.860 .869 .877 .8s;,

.888 .8!l5 !)02 .908

.907 .012 .918 .923

.!l20 .025 _!);)(] .!)34

.!l30 .93<-! .!l:38 .942

.944 .94, .951 .9G4

.ID

.20

.30

.50

.60

.iO

.80

.!JO

1.00

1.20

1.40

1.60

l.80

2.00

2.50

:l.00

:l.50

.J.00

5.00

1.2 1.4 1.6 1.8

.526 .577 .619 .653

.550 .598 .638 .671

.580 .625 .662 .6!)3

.614 .655 .689 .718

.6-19 .687 .718 .i43

.684 .il8 .i46 .769

.717 .74i .ii2 .793

.747 .i74 .79, .815

./1'1 .798 .818 .8:n

.796 .818 .836 •. 8Gl

.8:lO .848 .86-l .876

.854 .870 .883 .894

.873 .886 .898 .907

.887 .8!JD .!J09 .!Jl 7

.898 .909 .918 .92(i

.918 .927 .9:34 .940

.932 .9:l!) .945 .950

.!N2 .948 .95:l .957

.!H9 .955 .959 .963

.959 .964 .96i .970

2.0 2.5

.683 .738

.698 .,51

.7HJ .i68

.741 .787

.i65 .806

.i88 ..825

.811 .844

.831 .861

.840 .875

.864 .888

.886 .906

.903 .D20

.915 .930

.924 .9:38

.9:32 .944

.946 .055

.955 .9fi:l

.961 .968

.966 .972

.97:l .978

:3.0

.77i

.i88

.803

.819

.8:35

.852

.867

.881

.894

.905

.920

.932

.940

.947

.952

.962

.968

.973

.976

.981

3.5

.807

.816

.828

.842

.857

.871

.885

.897

.908

.917

.9:31

.941

.948

.954

.959

.!l67

.972

.976

.979

.!l83

4.0

.829

.837

.848

.861

.873

.886

.898

.909

.919

.927

.939

.948

.954

.9'.i9

.96:3

.971

.97(i

.979

.982

.985

5.0

.861

.868

.877

.887

.897

.907

.9li

.926

.934

.941

.950

.958

.!J63

.967

.9i0

.97(i

.980

.98:l

.!J8:j

.!)88

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 G.0 7.0 8.0 9.0 10.0

.50 .:300 .325 .325 .325 .325 .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 .7-56 .756

2.50 .~155 .717 .802 .803 .Sml .803 .803 .80:3 .8m .803

3.00 .307 .669 .822 .835 .835 .835 .835 .835 .835 .835

3.,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 .194 .464 .686 .836 .896 .900 .900 .900 .900 .900

5.50 .178 .428 .642 .802 .890 .909 .909 .909 .909 .909

6.00 .163 -~196 .601 .764 .872 .914 .917 .917 .917 .917

6..50 .151 .368 .564 .726 .845 .909 .923 .92;3 .923 .92:3

7.00 .141 .:344 .530 .690 .814 .895 .926 .929 .929 .929

7.50 .1:32 .:323 .500 .655 .782 .874 .92:3 .933 .933 .933

8.00 .124 .304 .472 .623 .751 .840 .912 .9:36 .938 .9:38

8.50 .117 .287 .447 .593 .720 .822 .894 .9:33 .941 .941

9.00 .llO .272 .425 .566 .690 .794 .874 .924 .943 .945

9.50 .104 .258 .404 .541 .663 .767 .851 .910 .941 .947

10.00 .099 .246 .386 .517 .636 .741 .827 .892 .933 .949

2.8 Page 18

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2.9 Page 19

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Additional Information (GSS710S)

Spherical Variograrn Model

G)-½(1f] [f y(h) = C

for h < a

for h 2': a

Relationship between sill of regularized and point variograrn

CL= C [1 - 2~ + 2~: 3] for L < a

Ca a] CL= L

[15

20 -

for L 2':a

Auxiliary functions for Spherical variogram

l) = BCal ( 6 - a12z ) when l $ a

a) (

X = 8C ( 8 - 3 T when l > a

a =

(

a122 )

F(l)

20 20-15y+4%

when l $ a

whenl>a

Y ( l; b)

H ( l: b)

1 M' X(l;b)

X tv1'

1r M

X M'

2.10 Page 20

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