『CONTENTS

1.	Introduction: Mineral resources and mineral resource analysis			1
	1.1.	Perspective on mineral resources		1
	1.2.	A conceptual framework for resources and resource analysis		2
		1.2.1.	Definitions of resource terms and identification of useful concepts	2
		1.2.2.	The appraisal of resource adequacy, a means for examining resource relations and issues	5
	1.3.	An overview of resource models and estimation procedures
		1.3.1.	Classification of resource models	6
		1.3.2.	Economic resource models	7
		1.3.3.	Quantity-quality models	7
		1.3.4.	Geological resource models	8
		1.3.5.	Geostatistical models	9
		1.3.6.	Compound models	9
	1.4.	A brief description of two compound resource models
		1.4.1.	Perspective	10
		1.4.2.	The mineral endowment of the Canadian North-west	10
		1.4.3.	Infrastructure and base and precious metal resources of Sonora, Mexico	12
	1.5.	A summary of resource definitions and concepts		13
	1.6.	Appendix: A formal statement of definitions and some concepts		14
2.	Resource appraisal by models of economic activities			18
	2.1.	Overview		18
	2.2.	Supply and demand−econometric models		18
	2.3.	Simple life-cycle (time series trend) models		21
		2.3.1.	Concepts and modelling issues	21
		2.3.2.	Hubbert's analysis of domestic crude oil resources	24
		2.3.3.	Other life-cycle studies and comments on application	26
	2.4.	A life-cycle model with price and technology		31
	2.5.	Discovery-rate models and oil resources (Hubbert)		32
		2.5.1.	Overview	32
		2.5.2.	Theory and data conformability	33
		2.5.3.	Estimation of parameters	34
		2.5.4.	Calculation of the relative yield factor, R: the ratio of the yield of undrilled, favourable sediments to the yield of sediments already derilled	38
		2.5.5.	Conceptual issues with respect to (ds/dh) and its use	39
		2.5.6.	Conclusions	42
	2.6.	Economic issues of Lieberman's discovery-rate analysis of uranium		43
		2.6.1.	A general description	43
		2.6.2.	Some economic issues	43
		2.6.3.	Two approaches to analysis	45
		2.6.4.	Productivity and inflation effects on discovery rates	46
		2.6.5.	An approximate adjustment of Lieberman's analysis for inflation effects	47
		2.6.6.	What adjustment for future reserve additions?	48
		2.6.7.	The ratio	49
		2.6.8.	Summary	50
	2.7.	A discovery-process model and the estimation of future oil and gas supply		52
		2.7.1.	Perspective	52
		2.7.2.	The discovery-process model	52
		2.7.3.	Predicted discoveries	53
		2.7.4.	The estimation of potential supply	53
	2.8.	Appendices		54
		2.8.1.	Derivation of equation for first-derivative logistic with price and technology variables	54
		2.8.2.	Demonstration of the effects of mixing drilling yields from two regions	55
3.	Quantity-quality relations (models)			59
	3.1.	Perspective		59
	3.2.	Lasky's initial treatise: a benchmark		59
	3.3.	Lasky on the appraisal of metal resources		62
	3.4.	Musgrove's exposition of exponential relations		62
	3.5.	Uraniu and resources: a departure from exponential relationsm reserve		64
	3.6.	Cargill, Root, and Bailey's use of production data		65
	3.7.	Singer and DeYoung−A Lasky relation across deposits		68
	3.8.	Model structure		69
		3.8.1.	The influence of support	69
		3.8.2.	The influence of correlation	71
		3.8.3.	The need for a more general approach to model structure	72
	3.9.	Issues in the use of quantity-quality relations for prediction		75
		3.9.1.	Perspective	75
		3.9.2.	Selection of the function to be fitted to tonnage-grade data	75
		3.9.3.	Determining the limit to extrapolation	76
		3.9.4.	Estimation of endowment or resources of a region	79
4.	Deterministic geological methods			93
	4.1.	Introduction		93
	4.2.	Crustal abundance		93
	4.3.	The abundance-reserve relationship		94
	4.4.	Estimation by analogy		95
		4.4.1.	Perspective	95
		4.4.2.	Simple density	96
		4.4.3.	Compound density	96
		4.4.4.	Some examples of the methodology and criticisms	97
	4.5.	World uranium resource estimates		106
		4.5.1.	Perspective	106
		4.5.2.	Estimation	106
		4.5.3.	Summary of total resource estimate	111
		4.5.4.	An international assessment by experts	112
		4.5.5.	Some criticisms and speculations	112
	4.6.	Composite deterministic models		114
5.	Geostatistical models of metal endowment−a conceptual framework			118
	5.1.	The evolution of geostatistical models		118
	5.2.	Conditions for a probabilistic model for metal endowment		118
	5.3.	Geostatistical theory of metal endowment		119
	5.4.	Taxonomy of models		121
	5.5.	Geostatistical deposit models		121
		5.5.1.	The basic model	121
		5.5.2.	Metal density	122
		5.5.3.	Multivariate models	122
		5.5.4.	Spatial models	122
		5.5.5.	Trend models	123
		5.5.6.	Composite multivariate and trend models	123
	5.6.	The crustal-abundance (element distribution) model		123
		5.6.1.	Concepts and theory	123
		5.6.2.	The case for the lognormal distribution	126
	5.7.	Simplified summary of concepts		128
	5.8.	Appendix: Derivation of the probability density for metal m from the densities for n, t, and q		129

6.	A multivariate model for wealth (a value aggregate of metals)			132
	6.1.	Perspective and theory		132
	6.2.	The Harris model		133
		6.2.1.	The basic proposition	133
		6.2.2.	The geological model	133
		6.2.3.	Usable variables	134
		6.2.4.	The value measure	135
		6.2.5.	Incomplete geological information	137
		6.2.6.	The use of the expanded information	139
		6.2.7.	Relating probability, mineral wealth, and geology	140
		6.2.8.	Multiple discriminant and Bayesian probability analysis	144
		6.2.9.	A word on discriminant analysis	146
		6.2.10.	The analysis	147
		6.2.11.	Test of the model on Utah	151
	6.3	A model of the conditional expectation for mineral wealth		152
		6.3.1.	Theory	152
		6.3.2.	The mineral-wealth equation for Terrace, British Columbia	153
	6.4.	A probabilistic appraisal of the mineral wealth of a portion of the Grenville Province of the Canadian Shield		154
		6.4.1.	Procedure	154
		6.4.2.	Probability analysis	155
	6.5.	The models of Agterberg		155
	6.6.	Some issues about mineral-wealth models		156
	6.7.	Appendix		158
		6.7.1.	Derivation of the aggregate mineral-wealth probability distribution	158
		6.7.2.	Siscriminant analysis	158
7.	Occurrence models			164
	7.1.	Perspective on occurrence models		164
	7.2.	Spatial models		164
		7.2.1.	Concepts of fitting of function	164
		7.2.2.	Allais's study	165
		7.2.3.	Fitting the Poisson	166
		7.2.4.	Implications of the Poisson	167
		7.2.5.	Distributions for mines	168
		7.2.6.	Slichter's work	168
		7.2.7.	Clustering and the negative binomial	173
		7.2.8.	Issues regarding spatial models	174
	7.3.	Multivariate models		177
		7.3.1.	Theory	177
		7.3.2.	A number equation	177
		7.3.3.	A probability model for number of copper deposits for the Abitibi Area, Ontario and Quebec	178
	7.4.	A compound probability model for number of deposits		181
		7.4.1.	Theory	181
		7.4.2.	Application possibilities	182
8.	The crustal abundance geostatistical (CAG) approach of Brinck			184
	8.1.	Relationship to geostatistical theory		184
	8.2.	The geostatistical relations of DeWijs−a foundation		184
	8.3.	The extensions made by Brinck		189
		8.3.1.	Perspective	189
		8.3.2.	Basis for a probabilistic model	190
		8.3.3.	Use of the normal probability law for estimation of tonnages	191
	8.4.	Estimation of average grade		194
	8.5.	Example of calculations of tonnage and average grades using the lognormal distribution		195
	8.6.	Importance of block (deposit) size		196
	8.7.	The variance-volume relationship of DeWijs		199
	8.8.	The Matheron-DeWijs formula for differing shapes of environment and deposit		200
		8.8.1.	Perspective	200
		8.8.2.	The variogram	200
		8.8.3.	The DeWijsian variogram	203
	8.9.	Application−an exercise in statistical inference		205
		8.9.1.	Procedure recapitulated	205
		8.9.2.	Demonstration on Oslo	205
		8.9.3.	The case of no usable geochemical data	206
	8.10.	A comparison of methods−New Mexico uranium		208
	8.11.	The logbinomial model		209
	8.12.	Issues of the CAG approach of Brinck		209
		8.12.1.	Continuity of grade distribution	209
		8.12.2.	Tonnage-grade relations and Brinck's calculations	210
		8.12.3.	Estimation of parameters from production and reserve data	210
		8.12.4.	Crustal abundance and geology	214
	8.13.	The economic model		215
		8.13.1.	Overview and recapitulation	215
		8.13.2.	The cost model	216
		8.13.3.	Comment on the exploration model	218
		8.13.4.	Long-term metal price	218
	8.14.	Brinck's analysis of resources and potential supply		219
9.	Univariate lognormal crustal abundance geostatistical models of mineral endowment			224
	9.1.	Perspective and scope		224
	9.2.	The analysis by Agterberg and Divi of the mineral endowment of the Canadian Appalachian Region		224
		9.2.1.	General description of approach	224
		9.2.2.	Specific relations	224
		9.2.3.	Estimates	224
	9.3.	US uranium endowment		226
		9.3.1.	Estimates by the approach of Agterberg and Divi	226
		9.3.2.	Estimation of an asymptotic σ, a modification of the approach of Agterberg and Divi	228
		9.3.3.	A comparison of estimates	229
	9.4.	An important qualification		230
	9.5.	Endowment is not resources or potential supply		231
10.	The bivariate lognormal deposit model of PAU−a crustal abundance geostatistical model			233
	10.1.	General perspective		233
	10.2.	Demonstration of concepts		234
		10.2.1.	A simpler model	234
		10.2.2.	Mathematical expectation	234
		10.2.3.	Truncation by a cost surface and expectations	235
		10.2.4.	The solution for α and β	236
		10.2.5.	A numerical example using the simplified hypothetical model	236
	10.3.	The PAU model		237
		10.3.1	The mathematical form	237
		10.3.2.	Demonstration by PAU on US uranium	238
		10.3.3.	PAU's African model	240
	10.4.	The assumption of independence of grade and tonnage in crustal-abundance models		242
	10.5.	Appendix A: The mathematics of a solution for α and β		242
		10.5.1.	The problem	242
		10.5.2.	Evaluating the denominator D1 of (10.39)	242
		10.5.3.	Evaluating the numerator N1 of (10.39)	242
		10.5.4.	Putting the parts together for E[X1]*	243
		10.5.5.	Evaluating the denominator D2 of (10.40)	243
		10.5.6.	Evaluating the numerator N2 of (10.40)	244
		10.5.7.	Putting the parts together for E[X2]*	245
	10.6.	Appendix B: Computer program		245

11.	The statistical relationship of deposit size to grade−a grade-tonnage relationship			253
	11.1.	Perspective		253
	11.2.	Why the concern about this issue?		253
	11.3.	Perceptions and beliefs		254
	11.4.	Empirical studies		254
	11.5.	Difficulties in the statistical analysis of ore-deposit data		256
		11.5.1.	Contamination of statistical data by economics	256
		11.5.2.	Truncation	258
		11.5.3.	Translation	258
		11.5.4.	Possible remedies	259
	11.6.	Grade-tonnage relations and crustal abundance		259
		11.6.1.	Perspective	259
		11.6.2.	Singer and DeYoung's analysis	260
		11.6.3.	The PAU model	262
	11.7.	Final comment		263
12.	Size and grade dependency and an explicit treatment of economic truncation: theory, method of analtsis, demonstration, and a case study (New Mwxico uranium)			265
	12.1.	Overview		265
	12.2.	Two hypotheses		265
	12.3.	Theory and model form		265
		12.3.1.	Theory	265
		12.3.2.	Specification of the model	266
	12.4.	A compromise of theory to facilitate estimation		267
	12.5.	Estimation methods		268
		12.5.1.	Perspective	268
		12.5.2.	Cohen's solution	269
		12.5.3.	The Newton algorithm applied to Cohen's solution	270
		12.5.4.	Estimation of the parameters μy and σy by computer search	272
	12.6.	Demonstration on a synthetic truncated normal distribution		273
		12.6.1.	Pverview	273
		12.6.2.	Generating synthetic sample data from a model	274
		12.6.3.	Estimating the parameters of the conditional distributions for (X|Yi)	278
		12.6.4.	Investigating the dependency relationship	279
		12.6.5.	Estimating the parameters of the marginal distribution (Y)	280
	12.7.	Analysis of uranium deposits of New Mexico		281
		12.7.1.	The data	281
		12.7.2.	Costs: components, relations, and estimates	282
		12.7.3.	Parameters of the grade distribution	285
		12.7.4.	The dependency relationship	288
		12.7.5.	The parameters of the tonnage distribution	289
		12.7.6.	The model	290
	12.8.	Conclusions ans some criticisms		291
	12.9.	Appendices		293
		12.9.1.	Deivation of equations for Newton's algorithm for Cohen's solution	293
		12.9.2.	Differentiation of the integrals I', I''	295
		12.9.3.	Computer programs SIGMU and SEARCH	296
13.	Resource analyses which used geological analysis and conventional assessments of subjective probabilities for mineral occurrence and discovery: concepts, methods, and case studies			310
	13.1.	The concept of subjective probability		310
	13.2.	Scope		310
	13.3	Mineral endowment and potential supply of British Columbia and the Yukon Territory		311
		13.3.1.	Overview	311
		13.3,2.	The study design for mineral endowment	311
		13.3.3.	Economic analysis	314
	13.4.	Northern Sonora, Mexico		320
		13.4.1.	Estimation of copper endowment	320
		13.4.2.	The economic analysis of infrastructure development and potential copper supply	326
	13.5.	Mineral endowment of Manitoba, Canada		330
	13.6.	Estimation of uranium resources of New Mexico by Delphi procedures		330
		13.6.1.	Background	330
		13.6.2.	Survey design	330
		13.6.3.	Analysis by cell of initial information	333
		13.6.4.	Modified Delphi reassessment	335
		13.6.5.	Probability distributions for the State of New Mexico	338
	13.7.	The oil resource appraisal by the US Geological Survey (Circular 725): description, critique, and comparison with Hubbert		340
		13.7.1.	The nature of the appraisal	340
		13.7.2.	Methodology	343
		13.7.3.	Comments on methodology	347
		13.7.4.	The supposed unanimity of recent oil resources estimates (Hubbert versus the Survey)	349
	13.8.	Mineral resources of Alaska		354
		13.8.1.	Perspective	354
		13.8.2.	Methodology	354.
		13.8.3.	An estimate of expected copper in porphyry deposits	358
	13.9.	Uranium endowment estimates by NURE		359
		13.9.1.	General commentary on the NURE appraisal	359
		13.9.2.	Scope of this section	360
		13.9.3.	Logistics and approach in general	360
		13.9.4.	The methodology in perspective and in comparison with other subjective probability appraisal methodologies	361
		13.9.5.	Some details on design of elicitation	363
		13.9.6.	Critique of endowment estimation, particularly execution of the elicitation	364
		13.9.7.	Some possible modifications	369
		13.9.8.	Final comments	371
14.	Psychological, psychometric, and other issues and motivations in the perception of and the assessment of subjective probability			314
	14.1.	Perspective		314
	14.2.	The concept of bounded intelligence		314
	14.3.	Heuristics and biases		375
	14.4	Doubts about self-ratings of expertise and about Delphi		378
		14.4.1.	Expertise and self-ratings in other studies	378
		14.4.2.	A critique of Delphi methods	379
	14.5.	Purposeful hedging		380
	14.6.	Implications of psychometric issues to the appraisal of mineral resources		381
	14.7.	Preferred procedures		383
15.	Formalized geological inference and probability estimation			387
	15.1.	Perspective and scope		387
	15.2.	Formalized geoscience which supports active analysis of data by the geologist (approach A)−the uranium endowment appraisal system of Harris and Carrigan		388
		15.2.1.	Overview of appraisal system	388
		15.2.2.	The geological decision model−formalized geoscience	391
		15.2.3.	Preparation and calibration of the appraisal system	400
		15.2.4.	Selected comments on features of the appraisal system	403
		15.2.5.	An improved method for linking formalized geoscience to mineral endowment	407
		15.2.6.	An experiment on the effect of subjective probability methodology on estimates of uranium endowment of San Juan Basin, New Mexico	409
	15.3.	A comment on PROSPECTOR−a second example of approach A		422
		15.3.1.	Perspective and scope	422
		15.3.2.	Some specific features	423
		15.3.3.	A brief comment on the use of PROSPECTOR for regional analysis of uranium favourability	427
	15.4.	Genetic modelling, characteristic analysis, and decision analysis (approach B)−a methodology developed by the US Geological Survey		430
		15.4.1.	Perspective	430
		15.4.2.	Characteristic analysis	430
		15.4.3.	Genetic modelling for decision analysis	433
		15.4.4.	Decision analysis−the integration of characteristic analysis and genetic modelling	434
		15.4.5.	Some comments about this methodology and appraisal of endowment and resources	437
	15.5.	Selected comments		438
	15.6.	Appendix A−mathematical basis for characteristic weights		439
Index				443

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