Groundwater.Groundwater.Notes on geostatisticsNotes on geostatistics

Monica Riva, Alberto Guadagnini

Politecnico di Milano, Italy

Key reference:

de Marsily, G. (1986), Quantitative Hydrogeology. Academic Press, New York, 440 pp

In practiceIn practice: : random spatial variabilityrandom spatial variability of hydrogeologic medium of hydrogeologic medium properties, and properties, and stochastic naturestochastic nature of corresponding flow of corresponding flow (hydraulic (hydraulic head, fluid flux and velocity)head, fluid flux and velocity) and transport and transport (solute concentration, (solute concentration, solute flux and velocity)solute flux and velocity) variables, variables, are often ignoredare often ignored..

Instead, the Instead, the common approachcommon approach has been to analyse flow and has been to analyse flow and transport in multiscale, randomly heterogeneous soils and rocks transport in multiscale, randomly heterogeneous soils and rocks deterministicallydeterministically..

Yet with increasing frequency, the popular Yet with increasing frequency, the popular deterministicdeterministic approach approach to hydrogeologic analysis is proving to be to hydrogeologic analysis is proving to be inadequateinadequate. .

Modelling Modelling flow and transport in heterogenous mediaflow and transport in heterogenous mediamotivation and motivation and general ideageneral idea

Understanding the role of heterogeneityUnderstanding the role of heterogeneity

Jan 2000 editorial "It's the Heterogeneity!“ (Wood, W.W., It’s the Jan 2000 editorial "It's the Heterogeneity!“ (Wood, W.W., It’s the Heterogeneity!, Editorial, Heterogeneity!, Editorial, Ground WaterGround Water, 38(1), 1, 2000), 38(1), 1, 2000): : heterogeneity of chemical, biological, and flow conditions should be a heterogeneity of chemical, biological, and flow conditions should be a major concern in any remediation scenario.major concern in any remediation scenario.

Many in the groundwater community either failed to "get" the message or Many in the groundwater community either failed to "get" the message or were forced by political considerations to provide rapid, untested, site-were forced by political considerations to provide rapid, untested, site-specific active remediation technology.specific active remediation technology.

"It's the heterogeneity," and it is the Editor's guess that the natural "It's the heterogeneity," and it is the Editor's guess that the natural system is so complex that it will be many years before one can effectively system is so complex that it will be many years before one can effectively deal with heterogeneity on societally important scales.deal with heterogeneity on societally important scales.

Panel of expertsPanel of experts (DOE/RL-97-49, April 1997): (DOE/RL-97-49, April 1997): As flow and transport are As flow and transport are poorly understood, previous and poorly understood, previous and ongoing computer modellingongoing computer modelling efforts are efforts are inadequate and based on unrealistic and sometimes optimistic inadequate and based on unrealistic and sometimes optimistic assumptions, which render their assumptions, which render their output unreliableoutput unreliable..

Flow and Transport in Multiscale Fields (conceptual)

Field & laboratory-derive conductivities & dispersivitiesconductivities & dispersivities appear to varyvary continuously with the scale of observationscale of observation (conductivity support, plume travel distance). Anomalous transport.

Recent theoriestheories attempt to linklink such scale-dependencescale-dependence to multiscale multiscale structurestructure of Y = ln K.

PredictPredict observed effect of domain sizeeffect of domain size on apparent variance and integral scale of Y.

PredictPredict observed supra linear growth rate of dispersivitysupra linear growth rate of dispersivity with mean travel distance (time).

Major challengeMajor challenge: develop more powerful/general stochastic theories/models for multiscale random media, and back them with lab/field observation.

Shed some light

Conceptual difficulty:

Data deduced by means of deterministic Fickian models from laboratory and field tracer tests in a variety of porous and fractured media, under varied flow and transport regimes.

Linear regression:

aLa 0.017 s1.5

Supra-linear growth

Neuman S.P., On advective transport in fractal permeability and velocity fields, Water Res. Res., 31(6), 1455-1460, 1995.

Natural Variability. Geostatistics revisited

• Introduction: Few field findings about spatial variability

• Regionalized variables

• Interpolation methods

• Simulation methods

AVRA VALLEY

Clifton and Neuman, 1982Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

Regional Scale

Columbus Air Force [Adams and Gelhar, 1992]

Aquifer Scale

Mt. Simon aquifer

Bakr, 1976

Local Scale

Summary: Variability is present at all scales

But, what happens if we ignore it? We will see in this class that this would lead to interpretation problems in both groundwater flow and solute transport phenomena

Examples in transport: - Scale effects in dispersion - New processes arising

Heterogeneous parameters: ALL (T, K, , S, v (q), BC, ...)

Most relevant one: T (2D), or K (3D), as they have been shown to vary orders of magnitude in an apparently homogeneous aquifer

Variability in T and/or K

Summary of data from many different places in the world. Careful though! Data are not always obtained with rigorous procedures, and moreover, as we will see throughout the course, data depend on interpretation method and scale of regularization

Data given in terms of mean and variance (dispersion around the mean value)

Variability in T and/or K

Almost always σlnT (or σlnK ) < 2 (and in most cases <1) This can be questioned, but OK by now Correlation scales (very important concept later!!)

But, what is the correct treatment for natural heterogeneity?

First of all, what do we know?

- real data at (few) selected points

- Statistical parameters

- A huge uncertainty related to the lack of data in most part of the aquifer. If parameter continuous (of course they are), then the number of locations without data is infinity

Note: The value of K at any point DOES EXIST. The problem is we do not know it (we could if we measured it, but we could never be exhaustive anyway)

Stochastic approach: K at any given point is RANDOM, coming from a predefined (maybe known, maybe not) pdf, and spatially correlated ------ REGIONALIZED VARIABLE

Regionalized Variables

T(x,ω) is a Spatial Random Function iif:

- If ω = ω0 then T(x,ω0) is a spatial function (continuity?, differentiability?)

- If x = x0 then T(x0) (actually T(x0, ω)) is a random function

Thus, as a random function, T(x0) has a univariate distribution (log-normal according to Law, 1944; Freeze, 1975)

2 21

( ) exp( ( ) /(2 ))2

x

D x d

Hoeksema and Kitanidis, 1985

Hoeksema & Kitanidis, 1985

Log-T normal, log-K normal

Both consolidated and unconsolidated deposits

Now we look at T(x), so we are interested in the multivariate distribution of T(x1), T(x2), ... T(xn):

Most frequent hypothesis:

Y=(Y(x1), Y(x2), ... Y(xn))=(ln T(x1), ln T(x2), ... ln T(xn)) Is multinormal

with

But most important: NO INDEPENDENCE

1/ 21 1/ 2

1 1( ) exp ( ) ( )

2(2 )t

nf

Y C Y μ C Y μ

1

ij

( ( ( )),..., ( ( )))

variance-covariance matrix

C ( ( ), ( )) (( ( ) ) ( ( ) ))

n

i j i i j j

E Y E Y

Cov Y Y E Y Y

μ x x

C

x x x x

What if independent?

and then we are in classical statistics

But here we are not, so we need some way to characterize dependency of one variable at some point with the SAME variable at a DIFFERENT point. This is the concept of the SEMIVARIOGRAM (or VARIOGRAM)

2ijC (( ( ) ) ( ( ) ))i i j j ijE Y Y x x

Classification of SRF• Second order stationary

E[Z(x)]=constC(x, y) is not a function of location (only of separation distance,

h)Particular case: isotropic RSF; C(h) = C(h)Anisotropic covariance: different correlation scales along

different directions

Most important property: if multinormal distribution, first and second order moments are enough to fully characterize the SRF multivariate distribution

Relaxing the stationary assumption

1. The assumption of second-order stationarity with finite variance, C(0), might not be satisfied (The experimental variance tends to increase with domain size)

2. Less stringent assumption: INTRINSIC HYPOTHESISThe variance of the first-order increments is finite AND these increments are themselves second-order stationary. Very simple example: hydraulic heads ARE non intrinsic SRF

E[Y(x + h) – Y(x)] = m(h)

var[Y(x + h) – Y(x)] = (h)Independent of x; only function of h

Usually: m(h) = 0; if not, just define a new function, Y(x) – m(x), which satisfies this consition

Definition of variogram, (h)

E[Y(x + h) – Y(x)] = 0

(h) = (1/2) var[Y(x + h) – Y(x)] = (1/2) E[(Y(x + h) – Y(x))2]

Variogram v. Covariance

1. The variogram is the mean quadratic increment of Y between two points separated by h.

2. Compare the INTRINSIC HYPOTHESIS with SECOND-ORDER STATIONARITY

E[Y(x)] = m = constant

(h) = (1/2) E[(Y(x + h) – Y(x))2] = = (1/2) ( E[Y(x + h)2] + E[Y(x)2] – 2 m2 – 2 E[Y(x + h) Y(x)] + 2 m2) = = C(0) – C(h)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

covariance

variogram

h

The variogram

The definition of the Semi-Variogram is usually given by the following probabilistic formula

When dealing with real data the semi-variogram is estimated by the Experimental Semi-Variogram.

For a given separation vector, h, there is a set of observation pairs that are approximately separated by this distance. Let the number of pairs in this set be N(h).

The experimental semi-variogram is given by:

21ˆ2 i j

N

Y YN

h

hh

21( ) ( ) ( )

2E Y Y x h h h

Some comments on the variogramIf Z(x) and Z(x+h) are totally independent, then

If Z(x) and Z(x+h) are totally dependent, then

One particular case is when x = x+h. Therefore, by definition

2 2

1 2

2 2 2 22 2 21 2 1 2

1 1( ) ( )

2 21 1 1 1

2 2 2 2

E Z Z E Z Z

E Z E Z E

x x h x x h

x x h

210

2E Z Z

x x h

0 0

2( ; 0)ct In the stationary case:

Variogram Models

DEFINITIONS:

•Nugget

•Sill

•Range

•Integral distance or correlation scale

Models:

•Pure Nugget

•Spherical

•Exponential

•Gaussian

•Power

0( )h C

2 3( ) ((1.5 / ) 0.5( / ) )h h a h a 2( ) (1 exp( / ))h h a

2 2 2( ) (1 exp( / ))h h a

( ) bh ah

( )h

h

Correlation scales: Larger in T than in K. Larger in horizontal than in vertical. Fraction of the domain of interest

Additional comments• Second order stationary

E[Z(x)]=constant(h) is not a function of location

Particular case: isotropic RSF (h) = (h)Anisotropic variograms: two types of anisotropy depending

on correlation scale or sill value

Important property: (h) = 2 – C(h)Most important property: if multinormal distribution, first

and second order moments are enough to fully characterize the SRF multivariate distribution

Estimation vs. Simulation

Problem: Few data available, maybe we know mean, variance and variogram

Alternatives:

(1) Estimation (interpolation) problems: KRIGING

Kriging – BLUE

Extremely smooth

Many possible krigings Alternative: cokriging

http://www-sst.unil.ch/research/variowin/

The kriging equations - 1We want to predict the value, Z(x0), at an unsampled location, x0, using

a weighted average of the observed values at N neighboring locations, {Z(x1), Z(x2), ..., Z(xN)}. Let Z*(x0) represent the predicted value; a

weighted average estimator be written as

0 0 01 1

*( ) ( )N N

i ii i

i i

Z Z Z

x x

The associated estimation error is

*0 0 0 0 0*Z Z Z Z x x

In general, we do not know the (constant) mean, m, in the intrinsic hypothesis. We impose the additional condition of equivalence between the mathematical expectation of Z* and Z0.

*0 0E( ) 0Z Z

The kriging equations - 2

0 01

E EN

ii

i

Z Z m

This condition allows obtaining an unbiased estimator.

0 01 1

E[ ]N N

i ii

i i

Z m m

01

1N

i

i

Unknown mathematical expectation of the process Z.

The kriging equations - 3

2 2

2*0 0 0 0 0 0 0

1 1 1

E = E EN N N

i i ii i

i i i

Z Z Z Z Z Z

We wish to determine the set of weights. IMPOSE the condition

*0 0 0var( ) var minimumZ Z

2

0 01

EN

ii

i

Z Z

0 0 0 01 1

EN N

i ji j

i j

Z Z Z Z

0 0 0 01 1

EN N

i ji j

i j

Z Z Z Z

The kriging equations - 4We then use the definition of variogram

2 2

0 0

1 1E E ( ) ( )

2 2i j i j i jZ Z Z Z Z Z x x

2 2

0 0 0 0

1 1E E E

2 2i j i jZ Z Z Z Z Z Z Z

0 0 0 0Ei j i jZ Z Z Z x x x x

THEN: 0 0 0 0E i j i j i jZ Z Z Z x x x x x x

2*0 0 0 0 0 0

1 1

E = EN N

i ji j

i j

Z Z Z Z Z Z

Which I will use into:

The kriging equations - 5By substitution

We finally obtain:

2*0 0 0 0 0 0 0

1 1 1 1

0 0 01 1

E =N N N N

i j i ji j i

i j i j

N Ni j

ji j

Z Z

x x x x

x x

Noting that: 0 0 0 01 1

N Ni j

i ji j

x x x x

2*0 0 0 0 0 0

1 1 1

E = 2N N N

i j ii j i

i j i

Z Z

x x x x

The kriging equations - 6This is a constrained optimization problem. To solve it we use the method of Lagrange Multipliers from the calculus of variation. The Lagrangian objective function is

To minimize this we must take the partial derivative of the Lagrangian with respect to each of the weights and with respect to the Lagrange multiplier, and set the resulting expressions equal to zero, yielding a system of linear equations

10 0 0 0

1

0 0 01 1 1

, , , 2 ( ), ( )

( ), ( ) 2 1

NN i

ii

N N Ni j i

i ji j i

L Z x Z x

Z x Z x

The kriging equations - 7 1

0 0 0 01

0 0 01 1 1

, , , 2 ( ), ( )

( ), ( ) 2 1

NN i

ii

N N Ni j i

i ji j i

L Z x Z x

Z x Z x

0 1 1 01

0 2 2 01

0 01

01

( ), ( ) ( ), ( )

( ), ( ) ( ), ( )

( ), ( ) ( ), ( )

1

Nj

jj

Nj

jj

Nj

j N Nj

Nj

j

Z Z Z Z

Z Z Z Z

Z Z Z Z

x x x x

x x x x

x x x x

Minimize this:

and get (N+1) linear equations with (N+1) unknowns

The kriging equations - 8The complete system can be written as: A = b

1 2 1

2 1 2

1 2

11 00

00

0 , , 1

, 0 , 1

, , 0 1

1 1 1 0

,

,

1

N

N

N N

NN

Z Z Z Z

Z Z Z Z

Z Z Z Z

Z Z

Z Z

x x x x

x x x x

A

x x x x

x x

λ bx x

The kriging equations - 9We finally get the Variance of the Estimation Error

2* *0 0 0 0var =EZ Z Z Z

*0 0 0 0

1

var =N

ii

i

Z Z

x x

0TVar λ b

2*0 0 0 0 0 0

1 1 1

E = 2N N N

i j ii j i

i j i

Z Z

x x x x

0 01

( ), ( ) ( ), ( )N

jj i i

j

Z Z Z Z

x x x xHint: just replace

into

Estimation vs. Simulation (ii)

(2) Simulations: try to reproduce the “look” of the heterogeneous variable

Important when extreme values are important

Many (actually infinite) solutions, all of them equilikely (and with probability = 0 to be correct)

For each potential application we are interested in one or the other

Estimation. 1AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

Estimation. 2AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

Estimation. 3AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

Estimation. 4AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

Estimation. 5AVRA VALLEY. Regional Scale - Clifton, P.M., and S.P. Neuman, Effects of Kriging and Inverse Modeling on Conditional Simulation of the Avra Valley Aquifer in southern Arizona, Water Resour. Res., 18(4), 1215-1234, 1982.

Monte Carlo approach

2 2h th

hh11

hh22

hh20002000

..

. .

..

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

- 3

- 2

- 1

- 0 . 1

- 0 . 0 1

0 . 0 1

0 . 1

1

2

3

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

- 3

- 2

- 1

- 0 . 1

- 0 . 0 1

0 . 0 1

0 . 1

1

2

3

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

- 3

- 2

- 1

- 0 . 1

- 0 . 0 1

0 . 0 1

0 . 1

1

2

3

..

. .

..

2000 simulations2000 simulations

Statistical CONDITIONAL Statistical CONDITIONAL moments, first and second moments, first and second

orderorder

CONDITIONALCONDITIONAL CROSS- CROSS-CORRELATED FIELDS Y = lnTCORRELATED FIELDS Y = lnT

NUMERICAL ANALYSIS - MONTE CARLONUMERICAL ANALYSIS - MONTE CARLO

Simple to understand Applicable to a wide range of linear and nonlinear problems High heterogeneities Conditioning

Heavy calculations Fine computational grids Reliable convergence criteria (?)

Evaluation of key statistics of medium parameters (K, porosity, …)

Synthetic generation of an ensemble of equally likely fields

Solution of flow/transport problems on each one of these

Ensemble statistics

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000 10000 100000 1000000

Hyd

rau

lic

hea

d v

aria

nce

Number of Monte Carlo simulations

y

Q

r

L

ir

i

x

h = HL

Problems: reliable assessment of convergence – Problems: reliable assessment of convergence – Ballio and GuadagniniBallio and Guadagnini [2004] [2004]