The following is a summary of replies for my question: Risk assessment with

Gaussian simulation. Thanks for all the thoughtful replies.

Generally speaking, there are three ways to produce a probability map for

the purpose of risk assessment:

(1) Gaussian simulation with many (e.g.: 100) realisations/maps;

(2) MultiGaussian or Indicator Kriging;

(3) Ordinary Kriging + Kriging Variance.

The first way is traditional, and will not receive much criticism if you use

it. The second method is recommended by many people as it saves CPU time and

disk space.The third method may have some advantages as it is direct and

provides a way to incorporate errors from other sources, but there are some

criticisms. Generally, the results are quite similar, but some theoretical

and practical research are necessary. Well, the following are the replies

and you may digest them by yourselves.

Regards,

Chaosheng Zhang

====================================

Original question:

Dear list,

First, I would like to say thank you to Gregoire for keeping this list

alive.

I'm trying to do "risk assessment", and I have some questions about risk

assessment with Gaussian Simulation:

(1) How to produce a probability map?

With Gaussian simulation, we can produce many maps/realisations, e.g., 100.

Based on the 100 maps, a probability map of higher than a threshold can be

produced. I wonder how to produce such a probability map? My understanding

is that for each pixel, we just count how many values out of the 100 are>threshold, and the number is regarded as the "probability". Am I right? It

seems that this is a time consuming procedure with GIS map algebra. Are

there any suggestions for a quick calculation?

(2) Is a probability map better than a Kriging interpolated map for the

purpose of risk assessment?

(3) Is "PCLASS" function in IDRISI 32 Release 2 better/easier than the

probability map from Gaussian simulation?

From the online help of IDRISI 32 R2, Section "Kriging and Simulation

Notes", it says "If the final goal of simulated surfaces will be to directly

reclassify the surfaces by a threshold value, and calculate a probability of

occurrence for a process based on that threshold, conditional simulation may

be unnecessary. Instead kriging and variance images may be created and then

used together with PCLASS." Any comments?

(4) How to carry out "PCLASS"?

Following the above question, I have a problem in doing PCLASS. I cannot

input the file name of Kriging variance to the field of "Value error" of the

documentation file. It seems that this field only accepts a "value", not an

"image file name" or anything in text. Anyone has the experience?

Cheers,

Chaosheng Zhang

=================================================

Dr. Chaosheng Zhang

Lecturer in GIS

Department of Geography

National University of Ireland

Galway

IRELAND

Tel: +353-91-524411 ext. 2375

Fax: +353-91-525700

Email: Chaosheng.Zhang@...

ChaoshengZhang@...

Web: http://www.nuigalway.ie/geography/zhang.html

=================================================

Replies:

---------------------------------------------------------------------------

Hello,

In the past few years stochastic simulation has

been increasingly used to produce probability maps.

To my opinion it's generally a waste of CPU time since

similar information can be retrieved using kriging,

either in a multiGaussian framework or applied to

indicator transforms.

The issue of when using simulation vs kriging

is further discussed in:

Goovaerts, P. 2001.

Geostatistical modelling of uncertainty in soil science.

Geoderma, 103: 3-26.

I take this opportunity to thank Gregoire

for a remarkable and often challenging job

of keeping this e-mail list alive through the years.

Pierre><><><><><><><><><><><><><><><><><><><><><><><><><><>

________ ________

| \ / | Pierre Goovaerts

|_ \ / _| Assistant professor

__|________\/________|__ Dept of Civil & Environmental Engineering

| | The University of Michigan

| M I C H I G A N | EWRE Building, Room 117

|________________________| Ann Arbor, Michigan, 48109-2125, U.S.A

_| |_\ /_| |_

| |\ /| | E-mail: goovaert@...

|________| \/ |________| Phone: (734) 936-0141

Fax: (734) 763-2275

http://www-personal.engin.umich.edu/~goovaert/

><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

Dr Zhang:

If you are interested in assessing uncertainty, simulation offers some

advantages over kriging. Use of the kriging variance to model uncertainty

at a point or over a block requires the assumption of constant variance

(homoscedasticity): the kriging variance does not consider where the point

or block is on the map. For example, assume you have two exposure units

that

you are interested in; one is located in an area characterized by low

contaminant concentrations and low variance and the other is located in an

area with high concentrations and high variance (commonly observed in

environmental contamination data). You wish to estimate the probability

that the mean concentration within each of these exposure units exceeds a

threshold. If

the data that are used to predict the concentration for each of the exposure

units have the same or similar configurations, the uncertainty in the

predictions

will be the same or similar, even though the variability in concentrations

in the two areas are very different. The values of the samples used in the

prediction (not only geometric arrangement of the locations) are considered

in the simulation approach to modeling uncertainty. Indicator kriging

offers

some advantages to ordinary kriging, although it requires modeling more than

one variogram and modeling the variogram for high concentration levels can

be problematic.

If you are interested, I have put a few pages on our web site that compare

ordinary krigng and simulation. http://esc.syrres.com/geosem

Bill Thayer

----------------------------------------------------------------------------

Pierre,

Thanks for the comments. It's my first time to use Gaussian simulation to do

something possibly useful, and I have also found the calculation quite slow

even though the speed of my computer is not so bad. I'm using Idrisi 32

(with GStat), and the grid is about 500*500.

What I worry about is that how useful these realizations are? Obviously they

are not "realistic" even though some people say they want to produce a more

realistic map, instead of the smoothed Kriging map. Another concern is that

the probability map produced based on these realisations may not be so good

as the PCLASS (available in Idrisi), as PCLASS may have a better probability

background or clearer assumption. In PCLASS, the square root (not sure

yet???) of Kriging variances can be used as the RMS (root mean square) or

standard deviation of the pixel corresponding to the Kriging map, and the

probability > a threshold can be calculated based on the normal assumption.

More comments and suggestions will give me more confidence in doing the risk

assessment (heavy metal pollution in soils of a mine area).

Cheers,

Chaosheng

----------------------------------------------------------------------------

Dear Chaosheng

I have the same opinion. This is a paper that is published at Geostats Rio

2000 (Dr Margaret Armstrong Kluwer book). I think if your realizations honor

the

conditional data you can associate uncertainity measures for your dataset. I

have beeing working with this with Isatis, using the Simulation Post

Processing to calculate error for a confidence interval

At this paper I use probability map, mean of simulations map and IQR as a

uncertainity measure.

All the Best

Alessandro Henrique Medeiros Silva

Geologist - Anglogold Brasil

alessandro@... <mailto:alessandro@...>

+55-31-3589-1687

+55-31-9953-0759

----------------------------------------------------------------------------

I am curious about the use of 100 realizations to generate a probability

map. is this a standard approach? if so, is a "small" p-value (such as

.05) used? if so, it would seem like 100 iterations might be a smallish

sample size for distinguishing, say, .05 (ie 5 outcomes out of 100) from,

say, .01. is 100 used because it seems like it is a reasonable number or

because of the computer time restrictions?

also, do geostat folks treat these as realizations or as

pseudo-realizations?

brian

****************************************************************

Brian Gray

USGS Upper Midwest Environmental Sciences Center

575 Lester Avenue, Onalaska, WI 54650

ph 608-783-7550 ext 19, FAX 608-783-8058

brgray@...

*****************************************************************

----------------------------------------------------------------------------

Chaosheng, I agree with Pierre that if your only goal is to generate a

probability map, then IK is faster and more straightforward than simulation

and that MG kriging will give the same results, faster, than MG simulation.

However, we have found a couple of practical reasons where it may be

advantageous to use simulation for soil contamination studies, so I'll add

my two cents worth to this discussion:

1) When trying to explain the concepts of spatial variability and

uncertainty, we have found that showing example realizations of what the

possible distribution of contaminants could look like provides the groups

involved to get a more intuitive understanding of these ideas. People

understand the idea of flipping a coin 100 times to get the probability of

heads or tails, but have a hard time visualizing in their mind what a "coin

flip" looks like in a 2-D soil contamination problem. Showing some example

conditional realizations gives them a stronger feel for the nature of the

answers geostats is providing to their questions.

2) A number of sites are in the process of designing chemical and/or

mechanical treatment systems for the soil that will be removed from the site

while the remediation map is being determined. One set of design parameters

for these treatment systems is the best and worst case estimates of the

total amount of contamination (curies, grams, etc.) contained in the soil at

the site. These best/worst case estimates depend on the joint estimate of

the contamination at all locations across the site. This is something

simulation provides, but kriging doesn't.

3) For soils with radioactive contaminants, there are a number of different

sensors (e.g., a gamma detector mounted several meters off the ground) being

deployed at field sites that integrate the activity of the contaminant over

a larger area/volume. Simulation of the fine scale distribution of the

activity can be useful in looking at how these sensors scale up the activity

values to the integrated measurement.

Also when looking at IK vs MG kriging (or simulation) keep in mind that

rarely do the client, stakeholder(s) and regulator(s) have a single action

level or threshold that they have all agreed to for application at the site.

There are usually multiple thresholds corresponding to different future-land

use scenarios and different health risk models. If creating the probabilty

maps through IK then each different threshold requires a new set of

indicator variograms. If you use MG kriging or simulation, you only need do

the variography once-keep in mind that the MG assumption does have other

problems with connectivity of extreme values that may or may not be

important in your application (this is generally a bigger concern in fluid

flow problems than in soil contamination problems).

I'll add my thanks to Gregoire for 7 years of superb work!

Sean

Sean A. McKenna Ph.D.

Geohydrology Department

Sandia National Laboratories

PO Box 5800 MS 0735

Albuquerque, NM 87185-0735

ph: 505 844-2450

----------------------------------------------------------------------------

Hi Brian,

One hundred realizations are typically generated

mainly for CPU reasons.

You are perfectly right that this number is

too small when looking at small probabilities

like 0.05 or 0.01. It's why I wouldn't recommend

using stochastic simulation to derive probability of occurrence

of events at pixel locations. Just use kriging to build

your local probability distributions.

Use simulation if you have a transfer function, such as flow

simulator, that requires a model of spatial uncertainty,

or if you need to derive block probability distributions

(upscaling or aggregation problems).

More generally, there is more research to be done on the

use of stochastic simulation for probabilistic assessment,

including the question of equally-probability of realizatiuons

being generated.

Pierre

----------------------------------------------------------------------------

My tuppence worth.

The major advantages of simulation as a risk

assessment tool lie in the cases where you are trying

to derive some conclusion from the data rather than

just look at the values themselves.

For example, see Bill and my papers at Battelle

Conference 1987 or the paper at the Geostat Avignon in

1988. There are oters. All of these are available in

Word format for download at my page

http://uk.geocities.com/drisobelclark/resume/Publications.html

We were trying to derive the travel path of a particle

given the pressure of fluid in an aquifer. Not a

linear transform by anyone's standards.

Isobel Clark

----------------------------------------------------------------------------

>1) When trying to explain the concepts of spatial variability and

Taking this a step further, there was a paper in the AAPG Stochastic

>uncertainty, we have found that showing example realizations of what the

>possible distribution of contaminants could look like provides the groups

>involved to get a more intuitive understanding of these ideas.

Modeling and Geostatistics Volume entitled "The Visualization

of Spatial Uncertainty" (R Mohan Srivastava) which proposes the use

of probability field simulation to generate dynamic animations

of different realizations. I have yet to see it being implemented in

commercial software, although in concept I can see the benefit

of having something like this to illustrate the "equiprobable"

realizations. The idea was to generate smooth transitions of

successive "frames" by sampling from adjacent columns of a set of

probability values, for a movie-like effect.

Syed

----------------------------------------------------------------------------

Dear all,

Thanks for so many interesting replies and thoughtful discussion. This is

not a summary yet, as I am expecting more to come.

Just to express my feeling about Indicator Kriging. To produce a probability

map, IK might be one of the choices. However, I always feel that too much

information is lost when doing the indicator transformation. When I see so

many "0"s in a dataset, I just feel the data quality is too poor.

Well, the other method of combination of Kriging and Kriging variance for

risk assessment has not been well discussed yet, and I would like to read

more comments.

My last question "(4) how to carry out "PCLASS" " is now answered by the

developer of Idrisi. The fact that the file name of Kriging variance cannot

be entered (with Metadata command) is a bug of the program, which will be

corrected soon. At present time, a text editor may be used to modify the

image documentation file.

Now, let me discuss how I would like to make a probability map based on

Kriging and Kriging variance. For each pixel of the Kriging interpolated

map, there is a value of Kriging variance. The Kriging variance is a measure

of uncertainty (which is related to sampling density and spatial variation,

etc.???). If we assume that the value of the Kriging pixel follow a normal

distribution and the standard deviation is equal to the SQRT of Kriging

variance, the probability of any threshold can be calculated. Furthermore,

to make the risk assessment more realistic, I would like to include other

errors, such as sampling error and laboratory analysis error into risk

assessment. These errors can hardly be quantified, but if we say 10% or 20%

of the pixel value (for soil samples), perhaps there is no objection.

Therefore, the standard deviation of the pixel is increased by adding this

kind of errors.

I am not clear how to calculate the total standard deviation of the two

sources, is it:

Total standard deviation =

SQRT (Kriging Variance + SQUARE (Sampling Errors) ) ?

Any ideas and comments on this method?

Chaosheng Zhang

----------------------------------------------------------------------------

Chaosheng,

The kriging equations themselves are non-parametric. Without a

transformation of the data, the kriging estimate and the kriging variance do

not define a Gaussian distribution. If you want to make the Gaussian

assumption, as you state below, then using multiGaussian (MG) Kriging as

mentioned a few times in the replys to your question yesterday, is the way

to go. MG Kriging requires the transform of the raw data to standard normal

space and that the variograms be calculated using the transformed data.

Good luck

Sean

Sean A. McKenna Ph.D.

Geohydrology Department

Sandia National Laboratories

PO Box 5800 MS 0735

Albuquerque, NM 87185-0735

ph: 505 844-2450

----------------------------------------------------------------------------

>

I believe it is useful to compare maps of predicted concentrations or maps

>Well, the other method of combination of Kriging and Kriging variance for

>risk assessment has not been well discussed yet, and I would like to read

>more comments.

of the probabilityof exceeding threshold(s) that have been prepared with

different methods (e.g., SGS and OK) and make some judgments/decisions

about which maps appear reasonable given your knowledge of the data and

processes that created, transported and possibly transformed the

contaminants. My experience with kriging is it is very useful for central

tendency estimates but should be viewed with a high level of suspicion if

used to model uncertainty.

>

measure

>Now, let me discuss how I would like to make a probability map based on

>Kriging and Kriging variance. For each pixel of the Kriging interpolated

>map, there is a value of Kriging variance. The Kriging variance is a

>of uncertainty (which is related to sampling density and spatial variation,

In general, you can add the variances if you believe the two sources of

>etc.???). If we assume that the value of the Kriging pixel follow a normal

>distribution and the standard deviation is equal to the SQRT of Kriging

>variance, the probability of any threshold can be calculated. Furthermore,

>to make the risk assessment more realistic, I would like to include other

>errors, such as sampling error and laboratory analysis error into risk

>assessment. These errors can hardly be quantified, but if we say 10% or 20%

>of the pixel value (for soil samples), perhaps there is no objection.

>Therefore, the standard deviation of the pixel is increased by adding this

>kind of errors.

>

>I am not clear how to calculate the total standard deviation of the two

>sources, is it:

> Total standard deviation =

> SQRT (Kriging Variance + SQUARE (Sampling Errors) ) ?

>

>Any ideas and comments on this method?

error are independent. However, the data you use to develop a model of

spatial autocorrelation already includes measurement error (it is

incorporated in the 'nugget'), unless you have 'filtered it out' by

estimating their magnitudes and variances with quality control samples.

**************************************************

William C. Thayer, P.E.

Environmental Science Center

Syracuse Research Corporation

301 Plainfield Road, Suite 350

Syracuse, NY 13212

phone: (315) 452-8424

fax: (315) 452-8440

email: thayer@...

web: http://esc.syrres.com/

http://esc.syrres.com/geosem/

**************************************************

The kriging variance at short distances will increase with increasing

nugget effect. In any case, the kriging variance can not be less than the

nugget, so if you ensure that the nugget includes the measurement error

(which you can set based on your knowledge of the sampling and measurement

protocols, as well as the site-specific data and quality control data), the

minimum kriging variance (i.e., at short distances) will be valid.

A related point: What do risk estimates over small areas represent? I

think it is important to keep in mind the exposure unit or the area over

which exposure occurs. It seems to me that one first has to estimate the

likelihood that the risk from exposure to contamination exceeds some

threshold, which should be done at the exposure unit scale (i.e., over an

area that exposure occurs, such as an industrial site or perhpas a

residential yard). If the risk is unacceptable, calculating the

probability of exceeding a threshold at a smaller scale may be appropriate

for determining what areas of the site should be cleaned up. In my

opinion, one of the shortcomings of the risk assessment/geostatistics

programs that the U.S. EPA has developed is its inability to assess the

uncertainty in the mean on the exposure unit scale. So they have set about

answering the second question (where to clean up or where to collect more

samples) without answering the first sufficiently (is there a problem?);

again, in my opinion.

I apologize if you are already aware of these issues. Its hard to know

sometimes with e-mail!

Best regards,

Bill

----------------------------------------------------------------------------

For some resources classification (measured, indicated, inferred)

it's possible to find some kriging variance values that have correlation

with sample density. When you have more samples, you have small values for

KV...

But The best way to consider a error measure is calculate based on a set of

simulations

(My opinion...). What's you geostats area? environment? mining? oil?

All the best

Alessandro Henrique Medeiros Silva

Geologist - Anglogold Brasil

alessandro@... <mailto:alessandro@...>

+55-31-3589-1687

+55-31-9953-0759

----------------------------------------------------------------------------

Three comments on your ideas below

1. I would be very careful about using the kriging variance to generate

confidence intervals, with or without the assumption that the "errors" are

normally distributed. The kriging variance is a useful measure of the

relative reliability of the kriging estimator but it is certainly not an

absolute measure. If you multiply the variogram model by a postive number (I

know this will make the model appear to not fit the graph of the

experimental variogram) then the kriging variance will change by that

multiplier BUT the kriging weights will not change and hence the estimated

values will not change. Thus the kriging variance can not be treated as an

absolute measure of reliability.

2. If you really want to incorporate the measurement or instrument

variability then it should be done another way. You should use the smoothing

form of the kriging equations, if you want details send me an email. This is

also discussed in several but not all books on geostatistics.

3. The probability distribution that you would generate , i.e., estimate,

by using indicator kriging is very different from the distribution you would

generate by using confidence intervals. There is a reason why the term is

"confidence" and not "probability" and that is at the heart of the

distinction between the two approaches (indicator kriging vs confidence

intervals).

Donald E. Myers

http://www.u.arizona.edu/~donaldm

----------------------------------------------------------------------------

Dear Dr. Zhang,

I understand from the discussion on the ai-geostats forum that your are

dealing with producing a probability map for the purposes of risk

assesment. I am interested in agricultural hydrology. In this context, I

would like to apply IK technique to precipitation fields and produce

probability maps. As you know the difficulty comes from the lack of

literature explaining the subject clearly with examples. I would like to

learn if you have any kind of documents such as reports, class solved

examples, books,..etc. on the subject.

Thanks for your attention in advance.

Yours Sincerely,

Dr. Mahmut CETIN

---

Dr. Mahmut CETIN

Department of Agricultural Structures and Irrigation

Faculty of Agriculture

University of Cukurova

01330 Balcali-Adana/TURKEY

Tel: ++90 (322)338 6877 (Direct)

++90 (322) 338 6318 (Switchboard, ext. : 25)

Fax: ++90 (322) 338 6386

E-Mail: mcet64@...

----------------------------------------------------------------------------

Dear Syed, et al.,

I did much of what you described in the GRASS GIS a while back. (GRASS

is public domain, not commercial, but it is a very good GIS.) The title

of the paper is "Visualizing Spatial Data Uncertainty Using Animation"

and a copy of it is located at:

http://www.geo.hunter.cuny.edu/~chuck/CGFinal/paper.htm

The special issue of Computers & Geosciences (Vol. 23, No. 4, pp.

387-395, 1997) included a CD-ROM that contained some of the animations

in MPEG form. My web site includes the animations and instructions on

how to construct them.

I used spherical interpolation to generate smooth transitions between

realizations in order to keep the interpolations valid statistically.

I have a more recent work that studies user perception of animated maps

representing data and application uncertainty. An outline of that work

from a conference presentation (with all equations and animations) is

available at:

http://www.geo.hunter.cuny.edu/~chuck/GIScience2000/paper.html

The full paper is about to head out for peer review.

sincerely, chuck

----------------------------------------------------------------------------

Dear Syed, et al.,

I did much of what you described in the GRASS GIS a while back. (GRASS

is public domain, not commercial, but it is a very good GIS.) The title

of the paper is "Visualizing Spatial Data Uncertainty Using Animation"

and a copy of it is located at:

http://www.geo.hunter.cuny.edu/~chuck/CGFinal/paper.htm

The special issue of Computers & Geosciences (Vol. 23, No. 4, pp.

387-395, 1997) included a CD-ROM that contained some of the animations

in MPEG form. My web site includes the animations and instructions on

how to construct them.

I used spherical interpolation to generate smooth transitions between

realizations in order to keep the interpolations valid statistically.

I have a more recent work that studies user perception of animated maps

representing data and application uncertainty. An outline of that work

from a conference presentation (with all equations and animations) is

available at:

http://www.geo.hunter.cuny.edu/~chuck/GIScience2000/paper.html

The full paper is about to head out for peer review.

sincerely, chuck

===========================================================

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