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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
431
8/18/2019 Vertical Resolution of Two-Dimensional
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2
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
/ ’ sf’ 90 mp’ rpr
ID
aa
.0 y.0 p PO
i
--yiz &.r
XC. 1. Crosssectionof 2-D resistivitymodel (top) and
.esistivitypseudosection ver the mode l (bottom).
synthetic
432
8/18/2019 Vertical Resolution of Two-Dimensional
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Vertical resolution of 2-D resistivity inversion
3
IO
I
7 .0 p y.0
9 .0 9 .D
=“-Y
p
y.0
12
P’ P 4’
?IG.
2.
Cross ection f
2-D
resistivitymodel top)andsynthetic
.esistivity seudosectionver the model bottom).
12
~.?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.
433
8/18/2019 Vertical Resolution of Two-Dimensional
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4
Vertical resolution of 2-D resistivity inversion
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
the parenthesis. esistivity out side the parenthesiss the starting
model parameter.Symbo l f is used o fix the resistivityparameter
during nversion.
FIG. 8. Inversionof synthetic esistivity data shown n Figure 4.
The resistivity param eterobtainedafter inversion s indicated n
the parenthesis. esistivity out side the parenthesiss the starting
model parameter.Symbo l f is used o fix the resistivityparameter
during nversion.
434