UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

FINAL EXAMINATION: April 25, 2017

DURATION: 2 and 1/2 hrs

Third Year - Mining

M1N301 - Mineral Resource and Mineral Reserve Estimation

Calculator Type: 2

Exam Type: A

Examiner: 0, Leuangthong

1 Complete all questions on examination booklet(s), write your name on all booklets, make sure they

are stapled together and turn in at the close of the examination. This is a closed book examination. Please do not use any supplementary materials and adhere to

all rules on the examination booklet cover. Please review all questions before starting the examination and request clarification from the

examination proctor. Each of the five questions will be worth 20 percent of the total exam value.

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Question One: True and False Questions (answer in exam booklet with T or F for each question below). No penalty for a wrong answer.

Technical reports are required for all mineral properties of a company, if the company is listed

in the Toronto Stock Exchange.

There can be more than one Qualified Person for any section of a Technical Report.

Declustering changes the statistics of a population by changing the data values themselves.

Conditional simulation allows for uncertainty quantification at both local and global scales.

Change of support models refers to the analytical prediction of grade distributions at scales

other than the data support.

All Qualified Persons in a Technical Report must have visited the project site.

One possible outcome of non-compliance with National Instrument 43-101 is criminal code

charges (up to 14 years in prison). Only two people have been jailed for this in the last 5 years.

The two biggest variogram parameters that impact grade estimation is the nugget effect and

the shape of the variogram.

Precision refers to the reproducibility of a result, while accuracy refers to the relationship

between an analytical result and the expected result.

One problem with reverse circulation drilling is that samples result in inaccurate geological

logging.

Question Two: Collection of Short Answer Questions

Give Krige's relation or the "additivity of variance" equation. Explain all terms in the equation,

and why is it important?

What is the equation for determining weights to assign to a data value for inverse distance

estimation? Define all terms. What is the impact of the choice of the exponent value?

Draw the four main types of variogram structures used for variogram modeling. Properly label

all axes and types.

What four factors must be considered for resource classification?

What is a swath plot? How is it calculated and what is it used for? Show an example, with all

the relevant labels.

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Question Three: Covariance Calculation Question The next question (Question Four) asks you to undertake kriging with two data separated by 100 m and 50 m respectively; see sketch below:

Data 1 o

150 m Data 2 0

- Unsampled

100 > Location

The following data is available for calculating the required variogram values:

2.09 2.05 120 1.66 o 0 0 0

1.31 1.34 1.54 1.72 o 0 0 0

1.15 165 087 1.59 o 0 0 0

1,72 1.71 152 1.53 o o 0 0

Write the equation for the semivariogram for an arbitrary lag distance h. Describe the three semivariogram values that you need to calculate. Calculate the semivariogram values.

Question Four: Kriging by Hand Question Write the simple kriging equations for estimation with two data. Consider the data configuration shown below (same as in Question Three):

Data 1

Data 2 150

0 Unsampled

100 m > Location

The global mean is 1.5 %Cu and the variance is 0.1 %Cu2. Calculate the simple kriging estimate at the unsampled location using the semivariograms you calculated in Question Three. Show all steps clearly.

No double jeopardy; in case you did not get Question Three, consider an isotropic covariance function C(h)=1.5-h/300 with no loss of marks.

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Question Five: Theory Question Modern geostatistics depends on a few key theoretical results.

The variogram, covariance and variance are all related to each other. Derive this relationship, stating any assumptions.

The linear estimate in kriging is constructed such that the data values are reproduced. Prove that the kriging estimate is an unbiased estimator.

The estimation variance of a linear estimate at an unsampled location can be calculated using the variogram or covariance with no need for the data values or the true value to be known. Derive the estimation variance.

The minimized estimation variance leads to a set of simultaneous linear equations that are easily solved. Derive the kriging equations.

The covariance between the kriging estimate and one of the data used in kriging is the correct model covariance. Prove that kriging estimates have the correct covariance.

An indicator or binary variable, i(u), can have two outcomes, 1 or 0, with probability p and (1-p), respectively. Prove that the mean of the indicator is equal to the probability p, and the variance of the indicator is the product of the probability for both outcomes.

Choose four of these results and demonstrate their truth with as much mathematical rigor as possible.

Have a great summer!

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