the use of geostatistics in the evaluation of commercial ground risk …ags-hk.org/notes/14/04_angus...

Quantitative Geological Modelling

The Use of Geostatistics in the Evaluation of Commercial Ground Risk in Tunnel and

Excavation Projects

Dr Angus MaxwellMaxwell Geosystems Ltd.


Objective of GBRs

• Fair apportioning of ground risk


Objectives of GBRs

• The risk is mainly due to uncertainty


Objectives of GBRs

• The objective is to quantify uncertainty


How to do this?

• Deterministic Models– We pretend we know everything

• Stochastic Models– We assume we know nothing and that

variations are represented only by data variations


Deterministic Models

We know everything.


Problems with Deterministic Models

• Subjective• One of many which fit the data• Experience or pre-conception• Not transparent (based on a database of

information which is not available to others)• Not defendable


Standard Statistical ModelsWe know nothing.

nszxUCL

Where: x = the mean

Z = the confidence interval (ie 0.95 for 95%)

s i= the standard deviation

n = the number of data values

N

X


Problems with Basic Statistical Models• Ideal Data Set

– Random Samples

– Unclustered Data

– Uncorrelated Data

• Typical Field Data

– Biased towards construction zones

– Clustered around critical spots

– Correlated due to geological processes


Basic Statistics are InadequateThey do not assess spatial distribution

N

X

N

X

Same statistical distribution – different impact


What normally happens – Neither!


Problems with typical “Join the dots” sections

• Geological input? ………….. Little• Statistical relevance? ………... None• Confidence? …………………. Not quantified• Aliasing ………………………… Unavoidable

??

? ?


Problems with typical “Join the dots” sections

• Geological input? ………….. Little• Statistical relevance? ………... None• Confidence? …………………. Not quantified• Aliasing ………………………… Unavoidable


Reality


What do we need?

• A defendable geological interpretation that is:– True to the data– Geologically sensible– Allows confidence, uncertainty and error to be

quantified– Allows the user to choose the risk profile he/she

wishes to adopt

= GEOSTATISTICS


Geostatistics• Describes the variance between two samples as a function of

distance.• Is appropriate for data that is biased, clustered and correlated.

Correlated Un-correlated

Geological surfaces are expected to be correlated because they have been formed by geological processes


Why do we need it?

• Quantitative models have application in:– Geotechnical baseline reports– Commercial project ground risk evaluation– Insurance ground risk evaluation– Assessment of ground related claims– Assessment of sufficiency of ground investigations– Planning of equipment and methods


How do we do it?• Exploratory data analysis

– Confirm that the data can be analysed by geostats.– Identify trends– Understand variations

• Variography• Estimation

– Kriging– Co-Kriging– Confidence, error and reliability


Exploratory Data Analysis

• Geological phenomena have the characteristics of continuous random variables

• Represented by probability distribution functions (normal, log normal)

• From these we can examine:– Central tendency (Mean, Mode, Median)– Spread (Variance, standard deviation)– Shape (Skewness)


Test suitability for Geostatistics• Test that the data is symmetrical about a mean

Normal (Gaussian Distribution)

Normal Probability Plot (symmetrical)

Probability Plot showing data skewed (non-symmetrical)

Interval

No.

Rank

Valu

e

Rank

Valu

e


Remove trend• Fit a trend surface to the data.

– Is there structure?– Is there a regional dip?

• Calculate residuals• Re-test residuals

Residuals


Experimental Variography• Measures spatial

correlation• Models differences

between measured values as a factor of distance

• Usually differences increase as separation increase

Range

Sill

Nugget

DistanceSe

mi-V

aria

nce


Calculating Semi-variance

interval

Zx

Zx+h

21

21

n hxxh zz

n

n = the number of pairs in the interval x to x+h

NB. Squared so always positive

is recorded so that direction control on variance can be identified

Repeat for all Zx

h


Result = Omni-directional semi-variogram

Distance

Sem

i-Var

ianc

e

Distance

Sem

i-Var

ianc

e

Distance Distance

Sem

i-Var

ianc

e

Sem

i-Var

ianc

e

Spherical

Hole

Trend

Pure Nugget


If there is trend separate by • Trends may be produced

by folding or faulting or by preferential weathering along foliation or cleavage.

h

Zx

Zx+h

1

2

Range 1 1-2

Distance

Sem

i-Var

ianc

e

Range 2 2-1


Estimation

• Variography identifies a set of weights that defines how a parameter can be expected to vary between neighbours.

• Estimation is then done by linear regression to minimise the variance between points.

• Termed Kriging – after South African Mining Engineer - Danny Krige who was

looking for a better way to estimate gold reserves from point samples. He found that averaging methods such as nearest neighbour tended to over estimate reserves.


Point Kriging

Z1 Z2

Z3

Z4Z5

Z0

n

i ii zwZ1

*0

Where:

Z0* = Estimated (kriged) value at x0

Zi = Measured value at xi

Wi = Estimation (Kriging) weight of Zi derived from the variogram analysis


Point Kriging: Danny’s Gold

Z1 Z2

Z3

Z4Z5

Z0

n

i ii zwZ1

*0

A

A’

High value

A A’

Over estimates


Don’t forget to add the trend back in here!


Benefits of Kriging• Unbiased• Minimum variance• We have a Kriging Standard Deviation (taken

from the variogram)• Automatic declustering of data

– In averaging methods eg. Nearest neighbour clusters have a higher effect on the value at an estimated point. Not so in kriging.


Benefits of Kriging• Because the Kriging Standard

Deviation for each distance class falls straight from the Variogram we can use this to draw a defendable Upper and Lower confidence limit.

95% UCL

95% LCL

0*00 KSDtZZ

Neat rules:

75% of any population fall within 2 SD of the mean

95% of any population with a normal distribution fall within 2 SD of the mean


Examples

95% UCL

95% LCL

Tunnel

95% UCL

95% LCL

Excavation or Foundation

Use UCL for 95% chance of avoiding rock or..

LCL for converse

Use UCL for 95% confidence in rock volumes.


Reliability• How accurate is my model?• Remove each known point in turn and estimate its value

from the model.• Resulting cross validation highlights areas where

samples are anomalous (sampling error, mis-identification, core stones etc)

• Plot Standard Error Maps (standard deviation of the variance). Where the standard error is greater than the sample standard deviation then the estimation is unreliable. More data may be needed.


Does this make the Geologist Redundant in favour of the statistician?

95% UCL

95% LCL

No. The best estimate may not be the right estimate.

Between the UCL and LCL any interpretation can be drawn eg..


Conclusions• Quantitative modelling of geological phenomenon is

possible in 2D and in 3D.• Modelling methodology focuses on spatial variance.• Estimation is based on weights derived from model(s) of

spatial variance.• The process provides statistics which can help the

decision process:– Confidence/Uncertainty, Reliability, Significance

• Interpretations can be geologically sensible anddefendable


Further Resources

• Isobel Clark – Practical Geostatisticshttp://www.kriging.com/pg1979_download.html

• What the heck is a semi variogram?http://www.youtube.com/watch?v=SJLDlasDLEU

• Register for the course in Practical Geostatistics using MissionOS in December


Thankyou

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