a new joint inversion approach in conjunction with cluster ... · introduction synthetic model...

Introduction Synthetic model Joint inversion Cluster analysis Field data Conclusions

A new joint inversion approach in conjunction withcluster analysis to improve the reliability of

hydrogeophysical models

Thomas Günther1 & Carsten Rücker2

1Leibniz Institute for Applied Geosciences, Hannover (Germany)

2Institute of Geophysics and Geology, University of Leipzig (Germany)

Acapulco, 24.05.2007

AGU 2007 (Acapulco): Günther & Rücker Joint inversion & cluster analysis 1/16


Introduction

Objective of geophysical investigations

obtain an simple model with reliable structures and parameters that isable to fit the measured and a-priori data

Probleminversion is ambiguous

automated data analysis usually produces smooth models withuncertain parameters

by-hand forward modeling is time-intense

statistical parameter distribution is often distorted by artifacts



IntroductionOutline

Problems1 geophysical data are

non-unique2 models look different3 parameters not coupled4 models are smooth5 cluster model does not fit

data

Solution1 use different data to

decrease ambiguity2 apply joint inversion3 structural coupling4 cluster analysis5 parameter improvement

use fuzziness to improvemodel



ExampleThe synthetic model

Three-layered model

unsaturated - partly saturated- saturated (Φ = 30%)

water content distribution

statistical distribution

Experimental setup

two 10m deep boreholes,10m distance

0.5m electrode/geophonedistance

hole-to-hole tomography

Water content model0 2 4 6 x/m 10

0

2

4

6

z/m

10

0

2

4

6

z/m

20 40 60 80



ExampleThe synthetic model

DC Resistivity model

0 2 4 6 x/m 100

2

4

6

z/m

10

0

2

4

6

z/m

39.8 63.1 100 158 Ohmm 398

Archie (ρw =10Ωm,ρm=200Ωm)

GPR Velocity model

0 2 4 6 x/m 100

2

4

6

z/m

10

0

2

4

6

z/m

0.03 0.04 0.05 0.06 0.08 m/ns 0.13

CRIM (εm = 5,εw = 81)



Joint inversionGeneralized inversion approach

Objective function

‖D(d− f(m))‖22 +‖WbCWm(m−mref )‖2

2 →min

d . . .data, m . . .model, f(m) . . .model responseD . . .data error weighting, C . . . derivative matrixWm = diag(wm

i ) . . .model control: strength of reference model mref

Wb = diag(wbi ) . . .boundary control: strength of parameter contrasts

Structural coupling

Structure=gradients=roughness vector CmIdea: use roughness of one parameter for boundary control of other

wb1 = g(Cm2) and wb

2 = g(Cm1)



Joint inversionMethod

Joint inversion scheme

resistivity

rho0

data

ρ0

ρn

ρ2

ρ1

Crho Cv

v1

v0

v2

vn

data

derivativematrix C

mesh

cluster model

velocityExample Resistivity+Seismics

identical parameter mesh

electrodes/geophones arenodes

smoothness constraints

1. Iteration separate

v controls weight of ρ

ρ controls weight of v

finally: cluster analysis ofboth



ExampleResult of separate inversion


0 2 4 6 x/m 100

2

4

6

z/m

10

0

2

4

6

z/m

39.8 63.1 100 158 Ohmm 398

GPR Velocity model

0 2 4 6 x/m 100

2

4

6

z/m

10

0

2

4

6

z/m

0.03 0.04 0.05 0.06 0.08 m/ns 0.13



Cluster analysis

Cluster analysis

arrange all parameters

divide into classes

minimize distances to classcenter

here: c-means clustering

membership function to eachcluster⇒ reliability

Cluster cross-plot

102

10−1

resistivity in Ohmm

velo

city

in m

/ns



ExampleThe cluster view

Separate inversion

102

10−1

resistivity in Ohmm

velo

city

in m

/ns

Joint inversion

102

10−1

resistivity in Ohmm

velo

city

in m

/ns



ExampleCluster value optimization

Improvement

reduce cluster to centervaluesχ2

DC = 41.7, χ2GPR = 7.1

optimize values byleast-squares inversionχ2

DC = 21.6, χ2GPR = 5.2

use cluster model asreference, membership asmodel control and start over

Joint inversion0 2 4 6 x/m 10

0

1

2

3

4

5

6

7

8

z/m

10




Improvement


DC = 41.7, χ2GPR = 7.1


DC = 21.6, χ2GPR = 5.2


Joint inversion

102

10−1

resistivity in Ohmm

velo

city

in m

/ns




Improvement


DC = 41.7, χ2GPR = 7.1


DC = 21.6, χ2GPR = 5.2


Cluster guided inversion

102

10−1

resistivity in Ohmm

velo

city

in m

/ns




Improvement


DC = 41.7, χ2GPR = 7.1


DC = 21.6, χ2GPR = 5.2


inversion0 2 4 6 x/m 10

0

1

2

3

4

5

6

7

8

z/m

10



Field data

Pine creek study site

Bow river near CalgarySand, gravel and fine sediments overshaly sandstone bedrock (2.5-8.5m)uneven due to paleochannelswater table between gravel and bedrock

Survey data

obtained by M. Hirsch

three 2-d profiles

DC resistivity data(56 electrodes witha=2/4m roll-along)

refraction seismics(60 channels, d=2m,shots every 30m)

GPR measurements

boreholes



Field dataInversion results - profile 1 left

Seperate Inversion

0 20 40 60 80 100 120 140 x/m 180

−20−15z/m−5

Ωm

30 60 122 245 493 993 2000

0 20 40 60 80 100 120 140 x/m 180

−20−15z/m−5

m/s

500 917 1333 1750 2167 2583 3000

low-resistivity, high-velocity bedrocktransition fine silts - gravel at x=35m



Field dataInversion results - profile 1 left

Joint Inversion

0 20 40 60 80 100 120 140 x/m 180

−20−15z/m−5

Ωm

30 60 122 245 493 993 2000

0 20 40 60 80 100 120 140 x/m 180

−20−15z/m−5

m/s

500 917 1333 1750 2167 2583 3000

sharper bondary shows paleochannelsborehole: gravel - saturated - bedrock



Field dataFinal result: cluster model

Separate inversion

shaly bedrock

102

103

104

103

resistivity

velo

city

silts sands gravel

Joint inversionshaly bedrock

102

103

103

resistivity

velo

city

silts sands gravel



Field dataFinal result: cluster model

4-cluster modelbased on joint inversion

0 20 40 60 80 100 120 140 x/m 180

−20−15z/m−5

v=556m/s,ρ=2157Ωm

v=3546m/s,ρ=39Ωm

v=521m/s,ρ=629Ωmv=359m/s,ρ=140Ωm

after cluster value optimization0 20 40 60 80 100 120 140 x/m 180

−20−15z/m−5

v=723m/s,ρ=2084Ωm

v=5156m/s,ρ=10Ωm

v=528m/s,ρ=629Ωmv=356m/s,ρ=138Ωm

most changes in shaly bedrock: resistivity is decreased, velocity isincreased



Conclusions

Conclusionsstructural joint inversion decreases ambiguity

cluster analysis yields cooperative model even if boundaries aredistinct and provides reliability

cluster value optimization decreases data misfitcluster-guided inversion is able to

1 improve cluster model2 yield structures within clusters

Outlookoptimize/change cluster number in inversion

use for time-lapse measurements


a new joint inversion approach in conjunction with cluster ... · introduction synthetic model...

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