Vertical Resolution of Two-Dimensional

8/18/2019 Vertical Resolution of Two-Dimensional

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Vertical Resolution of Two-Dimensional Dipole-Dipole Resistivity nversion

SatyendraNarayan, Univ. of Waterloo, Canad a

EM2.3

SUMMARY

4 practical wo-dimensional 2-D) algorithm or invertingdipole-

lipole res istivity data has been developed and applied to various

synthetic nd field da ta (Naray an, 1990). The theoreticalbasisof

nverse formulation s ba sedon adjoint solution and reciprocity.

3e has shown that the algo rithm is stable and capab le of

ielineating multiple con ductors embedded in a homog eneous

lalfspace. In this paper an attem pt is made to study vertical

.esolution f the dipole-dipolesurface esistivitymethodon a 2-D

:onductivebody embedded n a homogeneous alfspace.This is

accomplishedy inverting syntheticdataover a setof 2-D models.

I%e results obtained from this stud y are discussedherein. This

;hows that a conductivebody at a certain depth relative to its

lengthwill not producea resolvab le esponse n the dipole-dipole

surface esistivity method.

[NTRODUCTION

Until recently, direct interpretation of resistivity data using

inversion methods was com mon only for horizontally layered

stmctures. owe ver, layeredmodelsare nadequaten applications

such s mineralexploration;studyof dikes, valleys,contactzones,

and geotherm al ields; monitoring of steam, water or chemical

flooding for enhanced oil recovery; mapping of ground water

contam ination; and monitoring of in-situ mining methods.

Num erical modelling techniques or surface electrode arrays as

well as for subsurface lectrode(s)have been extensivelyusedon

a trial-and-errorbasis o interpret esistivity data n terms of two-

dimensional2-D) and hreedimensional 3-D) g eologic tructures .

Trial-and-errormodelling (i.e. o ptimizationof a model basedon

a forward solution) or interpreting esistivity ield d ata is rather

difficult and timeco nsuming.At the same ime forwardmodelling

doesnot yield informationon resolution.

The problem of 2-D resistivity inversion has been studied by

various investigators. Pelton et al. (197 8) developed an

inexpensive omputeralgorithm or the inversionof 2-D res istivity

and induced polarization IP) data. This m ethod involves spline

interpolation f the stored esponsesor a rangeof models n order

to ma tch the field data. This algorithm is n ot well suited to

complex cases , because interpolation of model response is

extremely difficult. Smith and Vozoff (198 4) and Tripp et al.

(198 4) propose d a 2-D resistivity inversion using a finite

difference technique, and transmission surface analogy w ith

Cohn’s sensitivity heorem, espectively. heir schemeswere quite

similar and suitable for com plex 2-D models. They did no t

incorporate he effects of top ographic eatureson resistivitydata

in their inversion scheme.Tong and Yang (1990) developedan

algorithm for the 2-D resistivity inversion where topograph y s

considered n the model, Thus, it allows inversion of resistivity

data obtained rom a rou gh terrain directly without applying any

externalcorrectionsn advance.McGillivray andOldenburg 1990)

described a comp arative study of several methods of 2-D

resistivity nversion.

A 3 -D resistivity nversionapproach singalpha centershas been

reported by Peaick et al. (1979 ). In this method, the forward

solution s accomplished y the alph a centersmethod,and a 3-D

inverse s algorithmdevelopedusing he ridge regressionmethod.

This algorithm requires less than 15CO Owords of computer

memoryand can bc used on sm all comp uters. h is alpha centers

method without m odification (as proposed by Shima, 1990),

however, is no t valid for a complex conductivity distribution.

Thus , the metho d is useful for field data interpretation o guide

drilling site choice and to o btain a good initial guess or more

sophisticated nd co stly inversion schemes.Recently, Park and

Van (199 1) developed an inverse algorithm to invert pole-pole

resistivity data over 3-D resistivity structureusing an approach

very similar to that of Narayan 1990). However, they were able

to map ateral esistivityvariationmore accurately han he vertical

resistivityvariation.

None of these investigations e scribesvertical resolutionof the

inversealgorithm. The ob jective of this paper s to studyvertical

resolutionof a 2-D inverse algorithm i.e. at w hat depth a 2-D

conductiveheterogeneity elative to its length will not producea

resolvable response in the dipole-dipole surface resistivity

measurements.

INVERSE FORMULATION

The mostcommonapproachn solving e sistivity nverseproblems

is to linearize the problem and then perform a least squares

minimizationon a systemof linear equations o solve for changes

in resistivity. t is very well describedby m any workersand their

namesare mentioned n the previoussection. have developeda

practical 2-D algorithm which is based on adjoint solution and

reciprocity.This approachs similar o thatof MaddenandMackie

(198 9). The detail of this inverse orm ulation,matrix formulation,

leastsquares ptimization,and resolution f model param eters re

described n Narayan (1990). The m ethod of inverse ormulation

is entirely different from those that have been used so far in the

resistivity nverse problems.This inverse algorithm s also tested

on numerous yntheticmodels as well as on field data (Narayan,

1990).

The advantag es f this approac h re that t gives an efficient way

of calculating he partial derivatives of data with respect o the

model parametersas it involves linearized form of non-linear

problem, and the se nsitivity of surface measu rementsare

proportional o the power dissipated n the anoma lous one.

RESULTS

The inverse algorithm has been thoroughly ested with several

models.The theoreticaldata computedby the forward modelling

(Madde n, 1971) over the realistic-geologiceaturesand field da ta

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Vertical resolution of 2-D resistivity inversion

have been

nvertedby

this algorithm Narayan, 1990).This shows

measurementsar delmeahng esistive zones within the crust, I

that the algorithm s stableeven for poor initial guess nd capable

the structure and physical properties of the earth crust: J.G

of reso lving multiple conductors mbedd ed n the homogeneou s

Heacock, Ed., Am . Geophys. Union, Geophys. Monogr., 14

halfspace.

95-105.

Herein, vertical resolutionof 2-D inverse algorithm studiedon a

conductiveheterogeneity mbedded n a homogeneous alfspace

is only described.This is accomplished y generatingsynthetic

data for the dipole-dipole surface resistivity method using a

forward algorithm Madden, 1971) over a 2-D conductive ody of

a given dimension 3 d ipole units ength and 1 dipole unit width)

and given resistivity contrastembedded n the resistivehostrock

at

various

depths (i unit, 2 units, 3 units, and 5 units), and

inverting theses data by a 2-D inverse algorithm. The result

obtained s discussed elow.

McCillivray, P.R., an d Oldenburg, D.W., 1990, Methods fol

calculating rechet’derivativesand sensitivities or the non-lineo

inversion problem: A comp arative study: Geophy s. Prose.. 38

499-524.

Naray an, S., 1 990 , Two-dime nsional esistivity nversion:M.Sc

Thesis,University of Ca lifornia, Riverside,California.

Park, S.K., and Van, G.P., 1991, nversionof pole-poledata or 3.

D resistivitystructurebeneatharrays of electrodes:Geophysics

56,95 l-960.

Figures 1,2, 3, and 4 show the computedsyntheticdata over a 10

ohm-m conductive body embedded n a 100 ohm-m halfspace

situated t 1 u nit, 2 units, 3 units, and 5 units depth respectively.

Thesedata are inverted with a initial guess onsisting f 100 ohm-

m five different blocks n a fixed 100 ohm-m host rock (Figures

5, 6, 7, and 8). T he top and bottom of the conductive body

situated t 1 unit depth are very well resolved F igure 5). Fo r this

model, RMS error was 50% at the beginning and dropped to

0.25% in 30 iterations. When the conductivebody is at 2 units

depth,only the top of con ductivebody s resolvedwell (Figu re 6).

If the conductiveheterogeneity s loca ted at 3 and 5 units depth,

it is not at all resolvable Figu res7 and 8). Therefore,a conductor

of 3 units length and 1 u nit thickness will not produce a

measurableesponse t the earth’s surface f it is situated t depth

of 3 units or greater. It is also obvious from the comp uted

syntheticdata for a 10 ohm-m conductivebody embedded n a

100 ohm-m halfspaceat the dep th of 3 units and 5 units that the

data do n ot con tain enough information to resolve the bo dy at

depth.A changeof about20% and 10% in the resistivitydata for

2-D conductivebody locatedat 3 units and 5 units respectively s

not adequate o invert them in terms of 2-D structure. hus, the

investigationproves that surface resistivity methodsare suitable

only for shallow geologic problems within a depth range of less

than 2 dipole units engths)and t doesnot give a better es olution

for deeperstructures.

Pelton,W.H., Rijo, L., and Swift, C.M.Jr., 1978 , nversionof twc

dimensional esistivityand nducedpolarizationdata:Geophysics,

43,788-803.

Petrick,W.R. Jr., Sill, W.R.,

Ward,

S.H., 1979,Three-dimensional

resistivity nversionusingalphacenters:University of U tah, Dept.

of Geology and Geophysics,Report no. DE-AC07-79E T/27002.

Shima,H., 1990, Two-dimensionalautom atic esistivity nversion

technique sing alpha centers:Geophysics, 5, 682-684.

Smith, NC., and Vozoff, K., 1984, Two dimensional DC

resistivity nversion for dipole- dipole data: Inst. of E lect and

Electron. Engineers,Tran. Geoscienceand R emote Sensing,22,

21-28.

Tong, L.T., and Yang, C.H., 1990, Incorporationof topography

into tw o-dimensional esistivity inversion:Geophysics, 5, 354-

361.

Tripp, AC., Hohmann, G.W., and Swift, C.M., 1984, Two.

dimensional esistivity nversion:Geophysics, 9, 708-171 7.

TRI’ ESlSTlvLTYHOOEL

CONCLUSIONS

The resolvingpower of 2-D resistivity nverse algorithmbasedon

adjoint solution and reciprocity is studied for the d ipole-dipole

surface esistivity method using variou s syntheticmodels.These

results ndicate that a conductivebody (3x1 u nits) with a given

resistivity contrast (1:lO) located a depth three units or greater

does not producea resolvable esponse n the surface esistivity

measurem ents. he information derived from this study may be

useful in the design of field experiments and mapping of 2-D

shallow geologicstructures.

REFERENCES

Madden, T.R., and Mackie, R.L., 1989, Three dimensional

magnetotelluric odellingand nversion:Proc. EEE, 77,318-333.

Madde n, T.R., 197 1, The resolving power of g eoelectric

I

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aa

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XC. 1. Crosssectionof 2-D resistivitymodel (top) and

.esistivitypseudosection ver the mode l (bottom).

synthetic

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7 .0 p y.0

9 .0 9 .D

=“-Y

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2.

Cross ection f

2-D

resistivitymodel top)andsynthetic

.esistivity seudosectionver the model bottom).

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~.?j2j?y

FIG. 3. Crosssection f 2-D resistivitymodel top)andsynthetic

resistivity seudosectionver the model bottom).

FIG. 4. Cros s ection f 2-D resistivitymodel top)and synthetic

resistivity seudo sectionver he model bottom).

FIG. 5. Inversion f syntheticesistivity atashownn Figure 1.

The resistivity aram eter btained fte r nversions indicatedn

theparenthesis.esistivity ut side heparenthesiss th e starting

model arameter.ymbol is used o fix the esistivity aram eter

during nversion.

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FIG. 6. Inversion of synthetic esistivity data shown n Figure 2.

The resistivity param eterobtained after inversion s ind icated n

the parenthesis. esistivity out side the parenthesiss the starting

model parameter.Symbo l f is used o fix the resistivityparameter

during nversion.

FIG. 7. Inversion of synthetic esistivity data shown n Figure 3.

The resistivity param eterobtained after inversion s indicated n



during nversion.

FIG. 8. Inversionof synthetic esistivity data shown n Figure 4.

The resistivity param eterobtainedafter inversion s indicated n



during nversion.

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Vertical Resolution of Two-Dimensional