It is Normal
Not to be Gaussian
J. Jaime Gómez-Hernández
Group of Hydrogeology
Universitat Politècnica de València, Spain
Disjunctive Kriging:
The Untold Story
Un
pu
blis
h ed
Disjunctive Kriging:
The Untold Story
To be or not to be
multiGaussian?
The danger of parsimony
To be or not to be
multiGaussian?
The danger of parsimony
by
J. Jaime Gómez-Hernández
To be or not to be
multiGaussian?
A reflection on stochastic
hydrogeology
Advances in Water Resources, 1998
(136 cites)
Training image
2000
y(m)
sand
clay
0.0
0.0
x(m)
2500
Reference lnK with high variance
1.00
400
-1.00
y(m)
-3.00
-5.00
-7.00
0
-9.00
0
x(m)
500
Reference lnK with high variance
1.00
400
-1.00
y(m)
-3.00
-5.00
-7.00
0
-9.00
0
500
x(m)
0.0600
Reference lnK with high variance
Number of Data 8000
mean -4.43
std. dev. 3.15
coef. of var undefined
0.0500
maximum
upper quartile
median
lower quartile
minimum
Frequency
0.0400
0.0300
0.0200
0.0100
0.0000
-12.0
-8.0
-4.0
lnK
0.0
2.85
-1.31
-5.58
-7.12
-11.53
Reference lnK with high variance
1.00
400
Scenario 3:
400
lnK r1
Scenario 4:
-2.00
-1.00
lnK r550
-2.00
400
-3.00
-3.00
y(m)
-3.00
-4.00
-5.00
y(m)
-4.00
-5.00
y(m)
-7.00
0
-5.00
-6.00
-9.00
0
500
x(m)
0
-6.00
-7.00
0
500
x(m)
-7.00
0
0
x(m)
500
Ensemble Kalman Filter
with non-Gaussian prior ensemble
Initial lnK
Number of Data 8000000
mean -4.51
std. dev. 1.28
coef. of var undefined
0.0700
0.0600
maximum
upper quartile
median
lower quartile
minimum
Frequency
0.0500
0.0400
0.0300
0.0200
0.0100
0.0000
-8.00
-7.00
-6.00
-5.00
-4.00
lnK
-3.00
-2.00
-1.00
-0.47
-3.08
-4.90
-5.52
-7.98
Scenario 1
Number of Data 8000000
mean -4.39
std. dev. 1.24
coef. of var undefined
0.0700
0.0600
maximum
upper quartile
median
lower quartile
minimum
Frequency
0.0500
0.0400
0.0300
0.0200
0.0100
0.0000
-8.00
-7.00
-6.00
-5.00
-4.00
lnK
-3.00
-2.00
-1.00
2.71
-3.25
-4.49
-5.34
-16.83
State Vector
Flow Model
Conductivity Ensemble
Simulation Head Ensemble
TF1
Observation Head
TF2
TF2
Normal Score Transforms
Transformed
Conductivity Ensemble
Transformed
Observation Head
Transformed Simulation
Head Ensemble
Ensemble Kalman Filter
Updated Transformed
Conductivity Ensemble
TF1
Updated Transformed
Head Ensemble
TF2
Back Transform
Updated
Conductivity Ensemble
Updated
Head Ensemble
Updated State Vector
Initial lnK
Number of Data 8000000
mean -4.51
std. dev. 1.28
coef. of var undefined
0.0700
0.0600
maximum
upper quartile
median
lower quartile
minimum
0.0400
0.0300
0.0200
0.0100
0.0000
-8.00
-7.00
-6.00
-5.00
-4.00
-3.00
-2.00
-1.00
Scenario 2
lnK
Number of Data 8000000
mean -4.45
std. dev. 1.23
coef. of var undefined
0.0700
0.0600
maximum
upper quartile
median
lower quartile
minimum
0.0500
Frequency
Frequency
0.0500
-0.47
-3.08
-4.90
-5.52
-7.98
0.0400
0.0300
0.0200
0.0100
0.0000
-8.00
-7.00
-6.00
-5.00
-4.00
lnK
-3.00
-2.00
-1.00
-0.41
-3.07
-4.84
-5.43
-7.79
Valencia - The Opera House
t= 0
t=5
t = 60
Mean
4
3
2
1
0
-1
-2
-3
-4
Var
7
6
5
4
3
2
1
0
Reference
t= 0
t=5
t = 60
Mean
4
3
2
1
0
-1
-2
-3
-4
Var
7
6
5
4
3
2
1
0
Prior
With updated model
0
(m)
-0.5
Well 19
-1
-1.5
-2
100
200
300
400
500 (day)
100
200
300
400
500 (day)
0
(m)
-2
-4
Well 44
-6
-8
-10
Control plane A
Particle
injection
Prior
Updated
NS-EnKF (Control plane A)
1
1
0,8
0,8
Normalized concentration
Control
plane
A
Normalized concentration
Prior (Control plane A)
0,6
0,4
0,6
0,4
0,2
0,2
0
0
1
10
100
1000
1
10000
10
10000
NS-EnKF (Control plane B)
Prior (Control plane B)
1
1
0,8
0,8
Normalized concentration
Normalized concentration
1000
Time (day)
Time (day)
Control
plane
B
100
0,6
0,4
0,6
0,4
Reference
th
th
5 and 95 percentiles
0,2
0,2
Median
0
0
1
10
100
Time (day)
1000
10000
1
10
100
Time (day)
1000
10000
B
Valencia - The Opera House
Non-inflation
Inflation
S1: t60
240.000
3.000
S2: t60
240.000
3.000
Northing
111 obs
2.000
Northing
2.000
1.000
.0
a
.0
Easting
300.000
S3: t60
240.000
.0
1.000
b
3.000
.0
.0
Easting
300.000
S4: t60
240.000
3.000
Northing
2.000
Northing
2.000
56 obs
1.000
.0
c
.0
Easting
300.000
S5: t60
240.000
.0
1.000
d
3.000
.0
.0
Easting
300.000
S6: t60
240.000
Northing
Northing
2.000
1.000
e
.0
.0
3.000
2.000
24 obs
.0
.0
Easting
300.000
.0
1.000
f
.0
.0
Easting
300.000
.0
Conclusions
Conclusions
• A marginal transformation helps a lot
Conclusions
• A marginal transformation helps a lot
• Even if there are not hard data
Conclusions
• A marginal transformation helps a lot
• Even if there are not hard data
• There is lot of info in the transient heads
Conclusions
• A marginal transformation helps a lot
• Even if there are not hard data
• There is lot of info in the transient heads
• I am not a normal guy
Marginally non-Gaussian
Inverse Stochastic Modeling
Zu, Gómez-Hernández, Hendricks, Li
Adv. Water Res. 34(7), 2011
http://jgomez.webs.upv.es
It is Normal
Not to be Gaussian
J. Jaime Gómez-Hernández
Group of Hydrogeology
Universitat Politècnica de València, Spain
Fully non-Gaussian Inverse
Stochastic Modeling
J. Jaime Gómez-Hernández
Universitat Politècnica de València - Spain
jgomez@upv.es
http://jgomez.webs.upv.es
A pattern-search-based
inverse method
Zhou, Gómez-Hernández, Li
Water Resour. Res., vol. 38, 2012
http://jgomez.webs.upv.es
Replace cokriging
• Try to use multiple point geostatistics to capture higherorder statistics
• We do not have a training image of forecast errors
K ensemble
P ensemble
......
Multiple small
training images
Look for a data event including both
permeability and pressure in the ensemble
of realizations
K3
K1
P1
Ki =?
P2
K2
Ki = ?
K ensemble
P ensemble
Starting training image
t=0
New training images
Flow Model
Permeability ensemble
Simulation press ensemble
Observation press.
Find a conditioning data pattern around current node
Calculate distance between candidate pattern and conditioning pattern
No
t ++
n ++
r ++
d < threshold
Yes
Accept the value
Yes
n < Nn
r < Nr
No
Postprocessing and update of the conductivity ensemble
Example
Training image
400
80
Reference facies
lnK (m/d)
sand
1
North
North
#45
#19
-4
shale
0
0
East
500
0
East
Permeability measurement
Pressure observation
100
-4
0
0
Reference facies
80
lnK (m/d)
100
Prior
Updated
80
1
R1
-4
0
0
0
Prior
80
80
100
Updated
R2
R1
Realizations
0
80
0
Prior vs. updated
80
R3
0
R2
80
0
R4
80
0
R3
0
100
0
100
Reference facies
lnK (m/d)
80
1
-4
0
0
100
Prior
Updated
1.0
80
0.0
500 realizations
After 20 time steps
-1.0
EA
-2.0
-3.0
0
-4.0
80
2.0
1.5
1.0
Std.dev.
0.5
0
0.0
80
2.0
1.5
1.0
RMSE
0.5
0
0.0
0
100
0
100
Piezometric head (m)
0
-0.5
-1
-1.5
-2
500 realizations after 20 time steps
Piezometric head (m)
0
0
5
10
15
20
Time (day)
25
-1.5
-2
Prior piezometer #45
0
-10
-15
-20
-1
30
-5
0
5
10
15
20
Time (day)
25
30
Updated piezometer #19
-0.5
Piezometric head (m)
Pressure
Piezometric head (m)
0
Prior piezometer #19
0
5
10
15
20
25
30
20
25
30
Time (day)
Updated piezometer #45
-5
-10
-15
-20
0
5
10
15
Time (day)
RMSE
500 realizations after 20 time steps
Average RMSE (m)
4
prior
updated
3
2
1
0
0
5
10
15
Time (day)
20
25
30
0
Reference head
80
100
Prior
h(m)
Updated
80
0.0
0.0
-2.0
-2.0
-4.0
EA
-4.0
-6.0
-6.0
-8.0
-8.0
0
Prior
0
-10.0
80
2.0
-10.0
0
100
Updated
1.6
80
0.0
1.2
Std.dev.
-2.0
0.8
Pressure
-4.0
-6.0
0
500 realizations after 20 time steps
0.4
80
0.0
-8.0
2.0
0
-10.0
1.6
80
2.0
1.2
1.6
0.8
1.2
0.4
v.
0
RMSE
0
0.0
0.8
0
100
0.4
0.0
0
100
Normalized concentration
0.6
0.4
0.2
101
102
Time (day)
Prior (Control plane B)
0.6
0.4
0.2
0
1
101
102
Time (day)
0.6
0.4
0.2
Time (day)
0.4
0.2
103
101
102
103
Time (day)
Updated (Control plane B)
0.8
0.6
0.4
0.2
1
0.8
102
0.6
0
103
Prior (Control plane C)
0 1
10
0.8
1
0.8
Updated (Control plane A)
0 0
10
103
Normalized concentration
1
Normalized concentration
95% confidence intervals, median and
reference
prior vs. upated
0.8
0 0
10
Normalized concentration
Fractional flow at 3
cross-sections
1
Normalized concentration
Normalized concentration
1
Prior (Control plane A)
101
102
103
Time (day)
Updated (Control plane C)
0.8
0.6
0.4
0.2
0 1
10
102
Time (day)
103
Conclusions
• Method to include transient state data into multiple point
simulation for history matching
• Including pressure in the data event requires considering
the location in the updating
• Replace a large single training images by many smaller
ones associated with a flow solution with wells and
boundary conditions
Pattern-search-based
inverse method
J. Jaime Gómez-Hernández
Universitat Politècnica de València - Spain
jgomez@upv.es
http://jgomez.webs.upv.es
(A) 250
(B)100
K(m/d)
7
2
North
North
4
1
10
6
9
3
10
8
5
0
0
0
East
250
0
East
100
-3
Prior
Updated
100
10
10
-3
0
100
1.0
0.8
0.6
Prob (sand)
0.4
0.2
0.0
0
100
0.25
0.20
0.15
Var.
0.10
0.05
0.00
0
0
100
0
100
10
15
20
Time (day)
25
30
-2.5
0
Updated well 1
5
10
15
20
Time (day)
25
Piezometr
Updated
-8
-10
0
30
Updated well 2
0.5
0.5
0.5
0.5
0.5
0
-0.5
-1
-0.5
-1
-1.5
-2
5
10
15
20
Time (day)
25
30
Prior well 4
5
10
15
20
Time (day)
25
30
-1
Prior well 5
5
10
15
20
Time (day)
25
30
-1
Prior well 6
5
10
15
20
Time (day)
25
30
-1
Updated well 4
5
10
15
20
Time (day)
25
30
-2.5
0
Updated well 5
0.5
0.5
0.5
-1.5
-2
5
10
15
20
Time (day)
25
30
-2
-2.5
0
Prior well 7
5
10
15
20
Time (day)
25
30
0.5
0.5
Piezometric head (m)
1
0
-0.5
0
0
-0.5
-1
-1.5
-1
-1.5
-2
5
10
15
20
Time (day)
25
30
-2
-2.5
0
Updated well 1
5
10
15
20
Time (day)
25
0.5
-2.5
0
Piezometric head (m)
0.5
Piezometric head (m)
0.5
0
-0.5
5
10
15
20
Time (day)
Updated well 4
25
30
-1
5
10
15
20
Time (day)
Updated well 5
25
30
-1.5
-2
-2.5
0
5
10
15
20
Time (day)
25
30
1
0.5
0.5
0
-0.5
10
15
20
Time (day)
25
30
-1
-1
-2
5
10
15
20
Time (day)
Updated well 6
25
30
15
20
Time (day)
25
30
5
10
15
20
Time (day)
25
30
0
-1
-2
-2.5
0
20
25
30
20
25
30
20
25
30
0
-1
5
10
15
-2
-2.5
0
0
-1.5
-2
-2.5
0
10
-0.5
-1.5
5
5
Updated well 8
1
15
Time (day)
-1.5
-2
-2.5
0
10
-0.5
-1
Updated well 7
0
-2.5
0
-1
Prior well 9
-1.5
-2
-2.5
0
30
-0.5
-1.5
-2
Time (day)
25
Updated well 3
1
-1.5
20
-8
1
-1
15
-6
1
-0.5
10
-4
Updated well 2
0
5
-2
-10
0
30
-0.5
-1.5
-2
-2.5
0
Prior well 8
1
-2.5
0
-1.5
Piezometric head (m)
-2.5
0
-1
0
Piezometric head (m)
-1.5
-1
-0.5
Piezometric head (m)
-1
-0.5
Piezometric head (m)
-0.5
Piezometric head (m)
0.5
Piezometric head (m)
0.5
Piezometric head (m)
0.5
Piezometric head (m)
1
Piezometric head (m)
1
-0.5
5
Updated well 6
1
0
30
-2
1
0
25
-1
1
0
20
0
1
0
15
Time (day)
-1.5
-2
-2.5
0
10
-0.5
-1.5
-2
-2.5
0
0
-0.5
-1.5
-2
-2.5
0
0
-0.5
-1.5
-2
-2.5
0
0
Piezometric head (m)
0.5
Piezometric head (m)
1
Piezometric head (m)
1
Piezometric head (m)
1
0
5
Updated well 3
1
-2.5
0
Piezometric head (m)
Prior well 3
5
-2
-6
1
-1.5
Piezometric head (m)
-1.5
-2
-2.5
0
-1
1
-0.5
Piezometric head (m)
-1.5
Prior well 2
Piezometric head (m)
Piezometric head (m)
Prior well 1
Piezometr
Piezometr
Prior
-1
20
Time (day)
25
30
5
10
15
Time (day)
Updated well 9
-2
-4
-6
-8
-10
0
5
10
15
Time (day)