Download Controlled Electromagnetic Sources for Measuring Electrical

JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 87, NO. B7, PAGES 5327-5338, JULY 10, 1982
Controlled ElectromagneticSourcesfor Measuring Electrical
ConductivityBeneaththe Oceans
1. Forward Problem and Model Study
ALAN D. CHAVE
Instituteof Geophysics
and PlanetaryPhysics,ScrippsInstitutionof Oceanography,Universityof California,San Diego
La Jolla, California 92093
CHARLES S. COX
OceanResearchDivision,ScrippsInstitutionof Oceanography,Universityof California, San Diego
La Jolla, California 92093
Exact closed-formexpressionsfor the electromagneticinduction fields producedby vertical and
horizontal current sourcesin the conductingocean overlying a one-dimensional earth are derived
from the Maxwell equations. Numerical methods for the evaluation of the solutions are given,
includingcorrectionfor the finite size of real sources. Simple models of the electricalconductivity
structureof the oceancrust and lithosphereare deducedfrom geologic,petrologic,and laboratory
data, and their electromagneticresponse is modeled. Horizontal electric dipole sources produce
much larger field amplitudesthan their vertical counterpartsfor a given frequencyand range, and
the horizontalelectricfield offers superiorreceivedsignalperformance. Reflectionsof electromagnetic waves from the sea surfaceand thermoclinemust be consideredfor low enough frequencies
or long ranges. Estimatesof the ambient noise level from natural electromagneticsourcesin the
frequencyrange0.01--10 Hz are presented. The ability of controlledsourcesto determinefeatures
of the conductivity of the ocean crust and upper mantle, especiallylow conductivity zones, is
demonstrated. If the mantle conductivityis low enough, horizontal ranges of 50 km and conductivity estimatesto over 20 km depth can be achieved.
INTRODUCTION
strain earth models and complement the data available
from other geophysicaldisciplines. The existence of a
Over the past two decades, natural electromagnetic low-conductivity zone in the lithosphere is suggestedby
fields at the floor of the deep ocean have been measured laboratory measurements of minerals. Such a region
and used to infer the electricalconductivity of the upper would exert a profound influence on the interpretation of
mantle by the magnetotelluricmethod [Cox et al., 1970; both the motional induction fields caused by large scale
Filloux, 1980, 1981; Law and Greenhouse,1981; Chave et oceanic flow and magnetotelluric experiments in the
al., 1981]. All of thesestudiesindicaterisingconductivity ocean-continenttransition zone [Cox, 1980, 1981; RanganyakiandMadden,1980].
Cox et al. [1978] measuredthe natural background
at depths of 60--200 km, but the data are unable to
resolve either the conductivity of the crust and lithosphere
or the thicknessof the high-conductivityregion underlying
the lithosphere. This is due in large part to the band limited nature of the natural electromagnetic spectrum
beneath the ocean. At frequencies above a few cph the
ionosphericsignals at the seafloor are attenuated by the
conductingseawater and masked by various forms of oce-
noise in the seafloor electric field at frequencies around 1
Hz and obtained a level near 1 pV/m. This very low
value suggeststhat the weak electromagneticfields that
propagate in the underlying sediments, crust, and lithosphere from a sea bottom mounted artificial source are
measurable at significant source-receiver separations.
anic backgroundnoise (e.g., induction from turbulent There are four fundamental electromagneticsource types:
watermotions,surface,and internalwaves). The magne- vertical and horizontalelectricdipoles (VED and HED),
totelluric method is a valuable tool for probingdeep earth
structure, but it is not likely that improvements in data
quality will allow it to be used for lithospheric studies.
The oceanic lithosphere is a region of changing temperature and composition whose detailed structure is
governed by the dynamic processesof plate accretion.
Electrical conductivity is sensitive mainly to temperature,
consistingof an insulated,current-carryingwire with bared
ends, and vertical and horizontalmagneticdipoles (VMD
and HMD), consistingof closed loops of insulated,
current-carryingwire. VED and HED oceanicmethods
have been proposed[Edwardset al., 1981; Cox et al.,
1981], and an experimenton the EastPacificRise yielded
measurements
of the horizontal
electric field from
an
composition,and the degreeof partialmelting [cf. Shank- HED sourceat a rangeof 19 km [Youngand Cox, 1981].
It is the purpose of this paper to develop the necessary
land, 1975]; thus indirect measurementsof it would contheory to calculate the electromagnetic induction fields
from controlled sources in the ocean and apply it to
models of the oceanic crust and mantle. Equations for
induction by horizontal and vertical current elements are
Copyright1982 by the AmericanGeophysicalUnion
Paper number 2B0583.
0148-0227/82/002B-0583 $05.00
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CHAVE AND COX:CONTROLLED
EM SOURCEIN OCEAN:FORWARDPROBLEM
derived from the Maxwell equations, and exact solutions allows the decompositionof any vector field into a combisuitable for numerical evaluation are obtained. Any of the nation of three scalar fields [cf. Morse and Feschbach,
four artificial source types can be constructedfrom these 1953,ch.13]:
expressions by suitable integrations. Electromagnetic
T=Vg•+Vx
(02)+Vx
Vx (x•)
(4)
waves generated at the ocean-crustinterface damp out
rapidly with range, in contrastto the terrestrialsituation. Using (1) and (4), the magneticinductionmay be written
Combined with the low ambient noise level, this yields
much greatersensitivityof electromagnetic
methodsin the
ocean compared to land. We illustrate this by modeling
some simple oceaniccrustal conductivitystructures.
The emphasisin this paper is on the forward geophysical problem and the ability to detect changesof conductivity at depth in the earth. In a subsequentpaper,we will
examine the inverse problem for an artificial source and
investigatethe separatequestionof resolutioncapability.
as
= v x
+ v x v x
(5)
where H and W represent TM and TE modes, respectively.
It should be noted that the formal Hertz vector represen-
tation is obtainedby replacingH with crH in (5); the
simpler form shown will be used.
Applying
(4)tothesource
current
j0 andseparating
out
the vertical component explicitly yields
f = Jzø}
+ VhT+ V x (Y})
GOVERNINGEQUATIONS
(6)
The fundamental equations for electromagneticinduc- where Tand Y are functions that satisfy the Poissonequation are the Maxwell equations with the displacement tions
current neglected
vr=
ß
(7)
v.
t=0
W•Y=--(X7hx •h)' }
(2)
(3)
(8)
with appropriate boundary conditions in the presenceof
vertical
discontinuities
in •h. The subscript
in (6)--(8)
refers to the transverse components.
Differential equationsfor H and W are derived by sub-
where•' is the electricfield, • is the magneticinduction,
tx is the magneticpermeability,cr is the electricalconduc- stituting(5) and (6)into (2) and (3)
tivity, a singleFourier componentproportionalto ei'"t is V•H + CrOz
(OzH/o')-iooixcrH= -ixJzø+ IxO'Oz
(T/or) (9)
considered,
andthesource
current
f istreated
explicitly.
Note that •' is the inducedelectricfield;the totalelectric
fieldisgivenby•' + •'Jø/o-)
. Equations
(1)--(3)canbe
solvednumericallyfor arbitraryIx and or, but closedforms
exist when only one-dimensional variations of these
parameters are allowed. Since magnetizable materials are
not common in the earth outside of certain types of ore
bodies, Ix is taken to be the free spacevalue Ix0 everywhere. Extension of the solution for varying permeability
is straightforward.
In a homogeneous material any electromagneticfield
may be separatedinto two modes about an arbitrary direc-
tion in space,each of which satisfies(1)--(3) independently. The separationaxis is taken as th• direction along
which conductivity varies, labeled the z axis. The
V2•-
iroixcr•=-IxY
(10)
where the electric field is
R= 1 VxVx(II})-l(Jzø2+Vhr)-iroVx(*2)
The sourcecurrent has been decomposedin (6) into
exactlythosepartswhichproduceTM modes,Jz
ø and
V hT, and that part which produces a TE mode,
V x (Y}).
At any horizontal boundary, the usual conditionson the
tangential electric and magneticfields, the normal magnetic field, and the normal current must be satisfied.
transverseelectric(TE) mode is markedby currentloops
Using (5) and (11), it is easyto showthat continuityof
circling the z axis and coupled to each other through the
•P, Oz•, H, and (1/cr)(OzH- T) meets these requireconducting medium by induction. The transverse mag- ments.
netic (TM) mode consistsof currentloopsin the planeof
the z axis which cut acrossthe changing medium proper-
ties. TE and TM modesproduceno verticalelectricand
POINT CURRENT SOLUTIONS
magnetic fields, respectively.Owing to the nonconducting
Solutions of (7)--(10) for either vertical or horizontal
atmosphere, ionospheric current systemscan induce only
TE modes in the earth. Sources within the conducting current sources serve the purpose of Green functions
earth; whether artificial or natural, may produce a combi- since the fields from an arbitrary source can be connation of the two modes. Note that TE modes vanish in
structed from them by summation or integration. From
the limit of zero frequencybecauseinductiveeffectsdisap- (9) it is clear that a VED source producesonly TM
modes, while (9) and (10) indicatethat an HED source
pear, while DC currents are the limiting case of TM
modes.
A Cartesian coordinate system with z positive upward,
oriented in the direction of changing electrical conductivity, is used. A standard theorem of vector analysis
produces both TE and TM modes. It can be shown that
VMD sourcesinduce only TE modes, while HMD sources
induce only TM modes. The HED is the most general
case and will be consideredin detail in this paper, but the
CHAVEANDCOX:CONTROLLED
EM SOURCE
IN OCEAN:FORWARD
PROBLEM
expressions
givenareapplicable
to anyo• thefoursource
AIR
5329
o'=0
z=H
typesß
An artificialpoint sourceat the coordinates(0, 0, z')
with magnitudep -- Idl, where I is the sourcecurrentand
OCEAN
dl is its infinitesimal length is representedby
jo = pS(x)8(y)8(z-z')
(12)
I
where 8 is the Dirac delta function. General horizontal
location of the source will be considered later.
Equations(9) and (10) are mosteasilysolvedby using
the Besseltransform pair
z
o'= O'o
LITHOSPHERE
:}z
Z= 0
o'= o'(z)
Fig. 1. Geometry used in the mathematicalderivations. The
sourceis locateda distancez' above the seafloor (z = 0), while
the r•ceiveris at a horizontal
distance
p anda heightz. The
oceanhasa uniformconductivity'
of rr0, and the lithospheric
•(k,z)
=•odp
Jo(kp)pf
(p,z)
f (p,z)
=•odk
Jo(kp)
k•(k,z)
conductivityor(z) is dependentonly on depth. The oceandepth
(13)
where J0 is the Besselfunction of the first kind of order
zero. The parameter k is analogousto the wave number
in the Fourier integral,and (13) is formally equivalentto
a double Fourier transform.
H is taken as infinite in the text, and the finite depth case is
treated in Appendix A.
Since (13) separatesthe vertical and horizontalcoordinate dependenceof the solution,generalizationof (16)-(18) for a sourceat (p', {b',z') followsfrom the geometry
by replacingp in (16)--(18) with
Applying(13) to (9) with (12) for a verticalcurrent
• -- •p2+p'2-2pp'cos(•b-•b')
source(T-- Y = 0) yields
O'Oz(Ozflv/o')
- fi2fIv= - •p 8(z-z')
(14)
where
fi(z) -' •/k2+irotxo'(z)
(15)
(19)
It should be noted that a Bessel function with argument
(19) can be expandedinto a seriesof productsof Bessel
functions and trigonometricfunctions [Watson, 1962].
This yields the formal eigenfunctionexpansionsolution of
(9) and (10).
This is solved by the usual variation of parameters method
to give
THE ELECTROMAGNETIC FIELDS
The electricalconductivityof seawateris almost entirely
a function of temperature and salinity, with a very small
nv(p,z,•o)
=•-P-•o
dkJo(kp)k
o'(z)
Ui(•>)U2(z<)
(16)pressureeffect [Horneand Frysinger,1963]. In the ocean
2rt
o'(z')
the gross thermal and salinity structure consists of a thin
where U• and U2 satisfythe homogeneous
form of (14),
surface layer of warm water separatedfrom a thick, cold
the boundary conditions U• - 0 as z --, oo, U2 = 0 as
water mass by the main thermocline. The subthermocline
z--,-oo, and the Wronskiancondition W(U•,U2)= 1.
ocean has a nearly uniform electricalconductivityof 3,2
The subscriptson z refer to the larger or smaller of the
field point z and the sourcepoint z'.
The horizontalcurrent sourcerequiressolutionsof (7)
S/m.
In this section the ocean will be modeled as a unit
form half space. This is a good approximation except in
shallowwater or at very low frequencies(<0.05 Hz) when
and (8). We considera point currentorientedalongthe y
the source is located near the seafloor. The complete
axis ({b'-0); the conjugatesolution is straightforward.
solution for an HED source, including the effect of interSubstituting(12) into (7), solvingfor T, using a Green
nal reflections from a thermocline or the sea surface, is
function for (9), and integratingby parts yields the TM
contained in Appendix A. For simplicity the treatment
mode solution
here is limited to a point source; again the complete solution is in the appendix. Figure 1 illustratesthe geometry.
Point sourcesolutionsfor the electromagneticpotentials
Ilh(p,z,to)
=--txP
Ox'odkJo(kp)
k
1o'(z')
o'(z)
ßOz,[U•(z>)U2(z<)l
(17)
II and xp are found by combining (16)--(18) for a uniform whole spacewith solutionsof (9) and (10) without
sources and including the boundary conditions at the
where U• and U2 weredefinedfor (16). Substituting(11)
into (8), findingY, and solvinggives
seafloor. Two electromagnetic
responsefunctions, analo-
T1VI(Z>)
V2(z<)(18)
•(p,z,to)
--- 2rr
.ixp
Oy
J'o
dk
Jo
(kp)
defined as
where Vl and V2 are analogousto U1 and U2. Note that
(17) and (18) are azimuth dependent, while (16) is
independentof azimuth, as would be expectedfrom symmetry considerations. Note also that the VED and HED
producefunctionallydifferent TM modes.
gous to the magnetotelluricresponsefunction, contain all
of the information on suboceanicconductivity. They are
A= 0z•
• Iz=0
(20)
K=
(21)
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CHAVEANDCOX:CONTROLLED
EM SOURCE
INOCEAN:
FORWARD
PROBLEM
where A correspondsto TE modesand K to TM modes.
Methods for their computationare derived in the next section. The three potential functionsfor vertical and hor-
Ez
=-c
•odk
Jl(kp)k
2[RTMe-•O(Z+Z')
*e-fiOlz-z"
(32)
izontal current sources are
(kp)
H,,(p,z,to)
p
RTM
--l(kp)
RTE]
e-•ø(z+z')
(22)
=a dk
Jo
(kp)
•oo
RTMe-•O(Z+z')+
(33)
-+fodk
Jo
(kP
)ke
-•ølz-z'l
Hh(P,Z,to)
l( kp) RTM- •oo
l [RTMe_/•O(Z+D•Ze_/•01•_•,i]
=aOXfo
dk
Jo
(kp)
-•
Jl(kp)
RTE]e-t•0
(z+z')
(23)
p
•:fodk
Jo(kp)
k?
=-aOy
• 1[/{TEe
q(24)
•odkJo(kp)
[- -/•ø(•+•')
e-/•øl•-z'l]
wherea--p,p/4rr and fi0 is givenby (15) with rr = fro,
the conductivityof the oceanhalf space. The upper sign
in (23) holds for z > z' and vice versa. RTE and RIM are
(34)
oo
k2[RTEe-/•ø(x+x)
,+e_/S01z_•,l]
B•=e
fo
dk
J,(kP)
•oo
(35)
where c=bcos{b,
d= bsin{b, e--asin{b,
f=acos{b,
and
analogsof the reflectioncoefficients
of electromagneticy• = ito/xcr0
wave theory, where
fioA-1
l(kp)
- kJo(kp)
- J•(kp)
(25)
RTE=
fi0A+I
P
In each of (27)--(35)
(36)
the last terms may be evaluated
analytically(Appendix B). Separateexpressions
for the
(26) TE and TM mode partsof the horizontalsourcefieldscan
fioK/cr0
-1
RrM
= fi0K/cr0
+1
also be obtained.
Each of (22)--(24) holds for a point sourcelocatedat the
The mathematicalcomplexityof (27)--(35) maskstheir
conceptualsimplicity. In each casethe first terms, involving the coefficientsRTE and RTM, representsthe electromagneticfield due to an image current inducedin the
conducting earth, while the second terms represent the
primary field in the ocean. In contrast to the terrestrial
case, the primary fields attenuate rapidly with range from
coordinatesystemorigin, and the generalcaseis obtained
by substituting(19) for p in the Besselfunctions.
The electromagnetic
fieldsare obtainedby applying(5)
and (11) to (22)--(24). For the VED only three nonzero
electromagneticcomponentsexist.
the sourceor with increasingfrequency(seeAppendixB).
oo
+e-t•ø
]
E,=
b•odkJl(kp)k2[RTMe-/•ø(z+z')_
Iz-z'l
(27)
Note that the vertical source fields are independent of
azimuth, while the horizontal source fields display simple
trigonometricdependence. The vertical field expressions
] (28) (32) and (35) representonly TM
E•=boo
f dk
Jo
(k,)k3[RTMe-/•Ot•+z"
+e_t•olz_z,i
or TE modes,as they
must by definition. Note also that increasing vertical
separation of either the source or receiver from the sea
floor results in both rapid attenuation of the fields and
decreasedspatial resolution of its features due to the wave
k2
,+e_/S01•_z,i]
B•=a•oo
dk
Jo
(kP)
•o
ø[RTM
e-fi0(x+x)
(29)
where b--p/4rrcro. The horizontal source current producesall six electromagneticfield components
number dependenceof the exponential.
Edwardset al. [1981] considereda VED source consist-
ing of an insulatedwire stretchingfrom the seafloorto the
sea surface. For this situation the sea surface is impor-
Ep--c
I(kp)fioRTM
p
•0
RTEe-fio(Z+Z') tant, and (22) and (27)--(29) must be modifiedto include
?olz-z'l
I
(30)
p
E• = -d
(kp)
p
reflections will influence the received fields.
e-fio(z+z')
fi0RTM--I(kp)
'•0RTE
k
Jo
( )
rio
a reflected wave and integrated over the extent of the
sourceto model it. For an HED, the finite length of real
sources is important for observation points near the
source. At very low frequencies, surface and thermocline
+
Jl (kP) k2
p
,So
e-fiolz-z'l(31)
NUMERICAL
METHODS
Approximate or asymptoticsolutionsof electromagnetic
problems in a half space over certain ranges of p, z, to,
and tr are availablein the literature[cf. Wait, 1961; Keller,
1968]. The treatmentof a more generalconductivestruc-
CHAVE AND COX: CONTROLLEDEM SOURCEIN OCEAN:FORWARDPROBLEM
ture cannot be accomplished with analytic methods, and
numericalevaluationof the integralsmust be applied.
5331
fields in the presenceof conductive structure and to place
some broad constraints on the experimental variables--
Integralslike (27)--(35), whoseintegrandconsistsof a sourcefrequencyand source-receiverrange--that are used
Bessel function Jv(kp) multiplying a kernel function in actual measurements. In all cases the earth is modeled
K(k), are ubiquitousin physicalproblems involving as a stack of layers for computational and conceptualsimcylindrical symmetry. Owing to the diffusion nature of
plicity.
The
(1)--(3), the kernel functionsfor electromagnetic
induc-
VED
source will not be considered
further
since
it in depth. It should
tion are smoothly varying with wave number k, in con- Edwardset al. [1981] have discussed
trast to the nearly discontinuous kernels that occur in be noted that the electromagneticfields generated by a
seismologicalproblems. This property has been exploited VED are both weaker by a factor of at least 100 for a
in the digitalconvolutionmethodof evaluation[Anderson, given rangeand frequencyand attenuatemore rapidly with
1979], whichyieldsadequateresultsfor a restrictedrange range when compared to those from an HED with the
of horizontal distance and frequency. The TE and TM same source strength. This is causedby reduced coupling
modes for the HED are of similar amplitude but are of the VED electromagneticfields to the low dissipation
asymptoticallyout of phaseas frequencyincreases.If they rock and suggeststhat much deeper penetration will be
are to be calculatedexplicitly, an algorithm with more pre- obtained using the HED source.
cision than that of Anderson[1979] must be used.
The electrical conductivity of seawater varies between
Direct numerical integration was employed to solve 3.2 and 5 S/m, with the higher value occurring in the
(27)--(35) numerically. The integral was evaluated warm, near-surfacewaters. The ocean will be treated as a
between the zero crossingsof the Bessel functions by uniform half spacewith a conductivityof 3.2 S/m in most
adaptivequadrature[Patterson,1973]. The seriesof par- of the models of this section. By contrast, the electrical
tial integrationsis, in general, slowly convergentor even conductivity of the crust and mantle has not been
divergent, and the Pad6 approximantsto the sum were measured extensively, particularly the less accessiblezone
used to force convergence. The result yields double preci- below seismic layer 2. A model for the crust will be
sion computer accuracyat a slight cost in computer time. presented which suggeststhat the highest electrical conDetails and a computer program are available from the ductivity normally occursnear the surface. The large-scale
first author.
resistivity experiment on DSDP leg 70 indicates a mean
Computationof the electromagnetic
fields (27)--(35) value of 0.1 S/m for the layer 2 pillow basalts on very
requiresthe evaluationof the response
functions(20) and young lithosphere[yon Herzeneta!., 1981]. Downhole
(21). These may be obtainedin two ways. A suitable electricallogs in this hole yield a similar value when proptransformationexiststo convert the homogeneousform of erly averaged(AppendixC). Logsin other holessuggest
the Besseltransformsof (9) and (10) into Ricatti equa- a decreasein conductivityby a factor of 3 as the crust ages
tions for the responsefunctions
OzA+ f12A2-- 1
(37)
•rOzK + f12Ii
a = •r2
(38)
from 0 to 10 m.y. In this section the entire lithosphere
will be modeled as a half spaceof conductivity 0.05 S/m,
near the upper crustal value. This should be regardedas
an upper limit and is useful for estimatingminimum field
amplitudes and in demonstrating sensitivity to buried
Theseare easilysolvedfor any o-(z) by standardnumeri-
structure.
Figure 2 illustrates the general behavior of the eleccal methods. It is common in geophysicsto model the
tromagnetic
fields generated by artificial sources in the
earth as a stack of layers of varying thicknessand conducocean.
The
plot shows the phase and amplitude of the
tivity. Recursionrelations for (20) and (21) can be
horizontal electric field as a function of source frequency
derived and expressedas continued fractions
from 0.01 to !00 Hz and range from 1 to 5 km. The point
source is assumedto have unit amplitude (i.e., a 1 a-M
A= Q•+
q•2_Q•
QI+Q2+ qt-Q22
(39)dipole moment) and is located1 m above the seafloor.
Measurements
are made at the seafloor at various hor-
izontalranges.The azimuthis 0ø (the electricfield vector
in the direction of the wire source is measured). The
K= R•+
r•-Riz
R •+ R2+ra2
- R22
;
RN-I+RN
(40)
experimentalapparatusdiscussed
by Cox eta!. [1981] consistsof a 0.8-km wire carrying75 A, so that actual field
amplitudes
wouldbe 6 x 104timeslargerthanshown.
The electromagneticskin depths at 1 Hz are 280 m in
seawaterand 2.3 km in the lithospherefor the parameters
used. Propagationthrough the ocean is significantonly
where qj: 1//3j, Qj: coth(/3jhj)//Sj, rj = o-j/flj, for the lowest frequenciesand rangesin Figure 2, and
Rj = o-jcoth(fljhj)/flj. The layeredstructure
terminates most of the attenuation is controlled by the lithospheric
in a half space.
conductivity. At ranges much smaller than a skin depth,
the field amplitudesand phasesare nearly independentof
ELECTROMAGNETIC RESPONSE OF THE OCEAN LITHOSPHERE
frequency. This is commonly called the quasi-staticzone.
At rangesmuch larger than a skin depth the electricfield
In this section the electromagneticresponseof simple phase and amplitude decreaserapidly with both frequency
models of the ocean and ocean lithosphere will be and range. This is the far-field zone; note that most of
explored. The purpose of the analysis is both to gain the values shown in Figure 2 are located in the very far
some insight into the behavior of the electromagnetic field for propagation in seawater. In between these
5332
CHAVEANDCOX:CONTROLLED
EM SOURCE
IN OCEAN:FORWARDPROBLEM
lithosphere(2.3 km), and the amplitudedecreases
nearly
-100
-200
-,500
\
-400
IIII
I I I IIIIII
10-2
I I I IIIIII
10-1
I I I Illill
100
\
\
\! •1I•1111
101
102
FREQUENCY(Hz)
10-4
exponentiallyin the far field. In the quasi-staticzone both
the phases and amplitudes are nearly independent of
range. From Figures 2--4 it is clear that measurementof
the horizontal field componentsis preferred using an HED
source. By the reciprocity theorem of electromagnetism,
the dashed curves in Figure 4 are equivalent to the horizontal fields produced by a VED, illustrating the advantage of a horizontal source.
Since any practical transmitter antenna will have an
appreciablelength, the point sourcemodel has errors associated with it. Figure 5 displaysthe fractionaldifference
for a point source and a 1-km-long wire with an identical
moment. The azimuth is 0ø. Since the observation point
and wire are closest for this azimuth, this should be
regardedas a worst case calculation. For both the electric
and magnetic field components, the point source is an
excellent approximation at ranges greater than a skin
depthin the lithosphere(2.3 km for the caseshown). The
10-6
error rises very rapidly at shorter ranges, especiallyin the
electric field, and is infinite at the origin. Figure 5 suggests that Figures 2 and 3 are seriouslyin error at their
10 -8
much smaller.
0 -lø
Figure 6 showsthe effect that a finite depth oceanhas
on the horizontal electric and magnetic fields. In both
cases the ocean half-space model is compared to a 1-km-
respectivelow-frequencylimits, but the phase effect is
0-12
•,
IIII
10-2
I ] III IIII
I I I IIIIII
10-1
I I I IIIIII
10 0
101
I I I IIIII
10 2
FREQUENCY(Hz)
Fig. 2. The radial electric field phase and amplitude at an
azimuth of 0ø for a point sourceas a function of frequencyand
range. The oceanhalf spaceis assumedto have a conductivityof
3.2 S/m, while the lithospherehalf space has a conductivityof
50
•'
0
<T --50
a. -100
q -so
\ \\
-200
0.05 S/m. The source is located 1 m above the seafloor, where
\\ \\[
-250
the observations
are made. The units of E are • V/m and of B
-300
are nT and the sourcestrength is 1 A m.
-350
extremes at ranges of the order of one skin depth in the
lithosphere the near field is encounteredand is the region
where significant attenuation and dispersion of the electromagneticfields beginsto be manifest.
Figure 3 shows the phase and amplitude for the horizontal magnetic field as a function of source frequency
from 0.01 to 10 Hz and range from 1 to 5 km. The
remaining model parameters are as for Figure 2, although
the magneticfield componentis oriented at right anglesto
the electric field component. Qualitative similarity is
apparent, but the decrease in field amplitude with frequency and range is slower, as is expected from the
,,1
,
,
, ,,,1,,
10-2
,
,
, ,,,,,,
10-1
,
,
, ,,,,,q
10 0
101
FREQUENCY
(Hz)
10-3
c 10-4
cu 10_ 5
......
I
.............
..j 10 -6
asymptotic
theory [ Wait, 1961].
Figure 4 compares the horizontal and vertical electric
and magneticfields as a function of range at a frequency
of 1 Hz with an azimuth of 0 ø for the electric field and 90 ø
::3 10-7
\
N
10-8
,,,
,
,
,
,
,,,,,
,
,
,
,
,,,,,
,
,
,
\
,
,,,,,
for the magnetic field. In both cases the vertical field
10-2
10-1
10 0
101
amplitudesare smaller by at least an order of magnitude,
FREQUENCY
(Hz)
and the differenceincreaseswith range as the vertical field
Fig. 3. The azimuthal magneticfield phaseand amplitudeat an
attenuatesmore rapidly. In the near to far field the phase azimuth of 0ø as a function of sourcefrequencyand range. See
decreasesat a rate of about 1 rad per skin depth in the Figure 2 caption for details.
CHAVEANDCOX:CONTROLLED
EM SOURCE
IN OCEAN:
FORWARD
PROBLEM
.•
o
lOO
-•
0
ß -100
,., -200
o
?• -100
(n -300
-100
-200
• -200
•n -200
a. -400
-300
• -300
•. -300
I
o
4
8
12
0
RANGE(krn)
10-4
,•
10-7
1::3
c• 10-8
'
I
'"
4
8
12
>•.10-5
\\\
•
8
,
I
l
101
10-2 10-1
12
RANGE(kin)
100
101
FREQUENCY
(Hz)
- .__.
• 10-5
• 10-6
,-• 10-6 :'::--..•.
•
,'7 10-7
--
0
I
FREQUENCY
(Hz)
,T 10 -10
I
03
,
I
10-2 10-1 100
R•N•œ (•)
E 10-4 '
•. 10-6
v
,.,
• 10-10
5333
10-7
I
"' 10-8r
,.I,
m 10_
8
10-2
10-1
10 o
FREQUENCY
(Hz)
101
10-2
10-1
10 o
101
FREQUENCY
(Hz)
Fig.4. The radial(solidline)andvertical(dashed
line)electric
Fig. 6. The electromagneticresponsefor an ocean of depth H
fieldphase
andamplitude
at anazimuthof 0ø (leftpanels)
andthe - 1 km comparedto the infinite depth model for the radial elec-
radial (solid line) and vertical (dashedline) magneticfield phase tric field (left panels)and azimuthalmagneticfield (right panels)
for an azimuth of 0ø. The half-spaceoceanat 2 km (solid line)
range. SeeFigure2 captionfor detailsof the otherparameters.
and 5 km (dashed line) and the 1 km ocean at 2 km (dot-dashed
line) and 5 km (doubledot-dashedline) are shown. See Figure2
andamplitude
at an azimuthof 90ø (rightpanels)asa functionof
caption for other details.
deep ocean at source-receiverranges of 2 and 5 km.
Differences are observed at frequencies where the water
column representsa few skin depthsor less,and the effect
increaseswith source-receiverseparation. The influence
of destructive and constructive interference
from surface
reflectionsis seenand is mostseverefor the electricfield.
Inclusion of a higher-conductivityregion above the main
thermoclineproducesresponsecurvesqualitativelysimilar
to those shown.
Figures 7 and 8 illustrate the sensitivity of the controlled source electromagneticfields to a thin conductive
or resistive zone at depth. For the first figure a 100,mthick, 0.25-S/m layer is buried at I km in a 0.05-S/m half
space, while for the second case a low conductivity of
0.005 S/m is used. The choice of layer thicknessesand
depthsis arbitrarybut servesto showthe different effects
of high- and low-conductivitymaterial. Since electromagnetic waves in conducting media attenuate and disperse
with distance,the qualitative effect of deeper conductivity
changeswill be similar to that shown but with reduced
amplitude.
It is quite apparentthat the fieldsare preferentiallysensitive to low-conductivity material. The amplitude change
shown in Figure 8 is an order of magnitude larger than
that of Figure 7, and the phaseshifts are also bigger. At a
fixed source-receiverrange an initial decreasein amplitude
is followed by a larger increaseas frequency rises for the
high-conductivitycase. The sense of the field change is
reversed for the low-conductivity material. This pattern
repeatsin an oscillatorymanner at higher frequenciesand
is the result of internal trappingof electromagneticenergy.
Note that a larger range is required to produce amplitude
change for the low-conductivity zone, but the phase does
changesignificantlyat shorter separations.The magnetic
field is only slightlylesssensitiveto structureas compared
to the electric field.
CONDUCTIVITY MODEL OF THE OCEAN CRUST
As noted previously, the ocean crust must exhibit a
decrease in electrical conductivity beneath seismic layer 2
to explainthe resultsobtainedby Youngand Cox [1981],
and more detailedconductivitymodelswill be considered
basedon field and laboratorydata. Our knowledgeof typi-
2.5
cal ocean crustal and mantle structure comes from terrestrial ophiolite exposures and deep ocean seismic studies.
2.0
The most thoroughly studied ophiolitesare those in Oman
'"
[Coleman, 1981] and Newfoundland [Salisburyand
Christensen,
1978]. The petrological
and seismicstructures
1.0
<• 0.5
0.0
o
4
8
RANGE(km)
Fig. 5. The fractionaldifferencebetweenthe finite and point
source calculationsfor the radial electric (solid line) and azimuthal
magnetic(dashedline) componentsas a functionof range at a
of these rocks are in grossagreementwith those obtained
in the presentday ocean if allowancefor weatheringof the
ophiolites is made. From the seafloor down the ocean
crustal structure exhibits (1) sediments whose thickness
depends on plate age, tectonic setting, biological produc-
tivity, and chemicaldissolution,(2) basaltsof both fractured flow and pillowtypeswith a total thicknessof' 0.5 to
1.5 km, (3) feederdykesshowingextensivefracturingin
the uppermost part, grading into a solid sheeted dyke layer
frequency of 1 Hz. See Figure 2 caption for other parameters.
The fractional difference is the quantity (finite field--point severalhundredmetersthick, (4) a gabbrolayer3--,5 km
field)/point field.
thick showing a petrologicalgradation diagnosticof multi-
5334
CHAVE AND COX: CONTROLLEDEM SOURCEIN OCEAN: FORWARD PROBLEM
pie plutonicevents [Pallisterand Hopson,1981], and (5)
an abrupt transition to peridotite depleted of basaltforming minerals. This interface is widely believed to
coincidewith the seismicMohorovicic discontinuity.
The basalts from ophiolite sequenceshave undergone
profound hydrothermal alteration, especially in the pillow
layers, and fractures are often filled with clay minerals and
calcite. This process probably occurs at the seafloor and
causesthe gradual conductivity decreasewith age in[erred
from DSDP electric logs as the closingof cracksreduces
the effect of fluid conduction. The deeper gabbro layers
display both low and moderately high temperature
hydrothermal alteration, the latter associatedwith interaction of the rock with a small volume (around twice the
200
-200
•
10-2
10-1
100
200
-200
10-2
101
u,.
0.2
•
0.0
r:-.
10 o
101
0.2
i
-
m -0.2
• '" "•'x•x,'xx
\,'x '
•. -0.4
;
\!
,., -0.6
•" -0.8
10-1
FREQUENCY
(Hz)
FREQUENCY
(Hz)
I
,
I
10_2 10_1 10o
rockvolume)of seawater[Gregoryand Taylor,1981]. The
low-temperature alteration, in which olivine is transformed
to serpentine, is problematical and may be either subaerial
or a consequence of the obduction process. The subcrustal peridotite undergoes very limited high-temperature
=
•
101
FREQUENCY
(Hz)
m 0.0
m -0.2
\,'\ \.
• -0.4
,
,., -0.6
--
,
•" -0.8
m
•
'0_
2 10• -1 100
•
1
101
FREQUENCY
(HZ)
Fig. 8. Sameas for Figure 7 with the 0.25-S/m layerreplaced
by a low conductivity0.005 S/m layer.
alteration[Gregoryand Taylor,1981], but low-temperature At an age near 10 m.y., weathering lowers this to 0.035
serpentinization is frequently observed.
Conductivity in rocks at subsolidus temperatures is
dominated by the effects of interstitial fluids at low tem-
S/m by the closingof cracksand pores.
3. Feeder dykes and gabbros(layer 3)--The conduc-
tivity probablydecreasesto a low value of 0.003 S/m. The
peratures [Shanklandand Waft, 1974] and thermally estimates of Youngand Cox [1981] show that counteractivated mineral conduction at higher temperatures balancing effects tend to keep the conductivity constant
[Shankland,1975]. By combiningthe field observations with plate age. As the gabbro cools, crackingpermits the
with laboratorydata on basaltsand ultramaficrocks [Rai entry of seawater, keeping the upper part of this layer
and Manghnani,1977, 1978], the tollowingconductivity more highly conducting. In the lower parts of newly
model of the ocean crust is proposed:
formed layer 3 gabbrosthis mechanismis less important
1. Sediments--The conductivity is estimated from
due to temperaturesin excessof 400øC, where fluid ionic
Archie's law as a [unction of porosityand varies from 0.01 mobility is reduced, but mineral conductivity mechanisms
S/m in indurated materials to 2.5 S/m for fresh,
do becomeimportant. If the temperatureis low enough,
seawater-saturatedsediments. The sediment layer is rarely the lower part of the gabbrounit could have a conductivity
more than a few hundred meters thick.
2. Basalt (layer 2)--The bulk conductivityof young,
fracturedbasaltis near 0.1 S/m [yonHerzenet al., 1981].
of 0.001 S/m, and highertemperature,as on youngcrust,
may raise this by a factor of 10. The possiblepresenceof
serpentine minerals from low-temperature alteration
makes for even greater uncertainty,as this material can be
as conductiveas 0.01 S/m or more [Steskyand Brace,
1973].
ß
'o
200 •'L--•
"'""•
"'
'--...
"o
0
'"
:r -200
<:J
4. Peridotite -- Below the Moho
200
0
:r -200
il
I
10-2
10-1
I
i
100
101
FREQUENCY
(Hz)
i
:z:
"'
0.02
i
10-2
10-1
i
i
100
101
FREQUENCY
(Hz)
1979;Souriau,1981];this wouldincrease
the conductivity
0.02
/%.
0.00
markedly.
Figure 9 summarizesthe ocean crust model.
The electromagneticresponseof two oceancrust models
0.00
'•',
-0.02
I
10-2
10-1
10 o
"•,,
•,•/¾•
-0.02
\
I
101
the electrical conduc-
tivity is probablylower than in the gabbrolayer becauseof
reduced hydrothermalactivity, lower water content, and
the absenceof conductive mineral phases. The bulk conductivity may be around 0.0004 S/m in the top 20 km of
the mantle, but this is quite uncertain. For example,
aseismicridgesexhibit low upper mantle seismicvelocities
which may result from extensiveserpentinization[Chave,
I
10-2
10-1
'L,' "/
'
•'/
I
10 o
101
were examinedand appearas Figures 10 and 11. The left
panel of both figurescorrespondsto a model with a 0.05-
S/m basaltlayer1 km thick,a 0.003-S/mdykeandgabbro
layer
2.5 km thick, and a 0.001-S/m lower crustallayer 2
Fig. 7. The phasedifferenceand fractionalamplitudechange
for a conductivity
modelconsisting
of a halfspaceof conductivity km thick, underlainby a mantlehalf spaceof conductivity
0.05 S/m containinga 0.25-S/m layer 100 m thick buriedat 1 km 0.0005 S/m. The right panel model differs only in the
overlain by a 3.2 S/m ocean half space. The radial electric field lower crustlayer, where the conductivityis 0.01 S/m, and
(left panels)and azimuthalmagnetic
field (rightpanels)at an representsyoung, hot crust, while the first caserepresents
r•rovrucY
(•z)
FREQUENCY
(Hz)
azimuthof 0ø are shownas a functionof sourcefrequencyand
ranges
of 1 km (solidline), 2 km (shortdashed
line),3 km (long older material. Figure 10 shows the horizontal electric
field at ranges of 4, 10, and 16 km from the source.
dashed line).
Departure from the half-spacebehavior of Figure 2 is
dashed line), 4 km (dot-dashedline), and 5 km (doubledot-
CHAVEANDCOX:CONTROLLED
EM SOURCEIN OCEAN:FORWARDPROBLEM
'
'
'
tactere
too•
1
I
•
,oo
cn
5335
1 O0
v
o
o
..........
.
2.0
-
• - 1O0
•
-r
-100
10-210-•100 10•
ßo
10-2
10-1
100
101
FREOUENCY
(Hz)
E 10-6
v
6.0
• 10-7......
•
• 10-7
.....
--
la..
10-4
i0-•
S/m
10-2
i0-•
•
10
-2 10
-• 10
ø 10•
,
,.I, 10-8
10-2
rRrOU[.CV
(.z)
,
•
,
10-•
10 0
101
FREQUENCY
(Hz)
Fig. 9. Electrical conductivity model ot the ocean crust and
Fig. 11. Th0,modelsof Figure10 are shownat rangesof 3 km
uppermantlederivedfrom petrological,geological,and laboratory (shortdashedline), 4 km (solidline), 5 km (longdashedline),
data.
See text for details.
and8 km (dot-dashed
line). SeeFigure10captionfor details.
obvious, and marked phase differences between the two
crust models are observed at long source-receiverseparations. Figure 11 showsthe same two models at rangesof
3, 4, 5, and 8 km. Amplitude and phase differencesare
large, especiallyat 4 km and high frequencies. This variation is caused by the deep crustal layer, and the active
source method is capable of detecting differences in
structure at such depths.
DISCUSSION
finite size of real sourcesis important at short sourcereceiverseparations,(4) the effect of reflectionsoff of the
sea surface will be important at large source-receiver
separations, low frequencies, or in relatively shallow
water, and (5) the choiceof transmitterand receiverlocations and source frequencieswill affect the resolution ability of actual data. Of these points, 3 and 4 are easily
incorporated into the theory if both water depth and
source location are accurately determined. The former is
accomplishedby acousticranging on transponderslocated
at the ends of the transmitter
The models presentedin the last two sectionsshow that
antenna.
Figures 7--8 and 10--11 show that deep structuredoes
(1) the HED sourceis preferredto the VED sourcefor influencethe electromagnetic
fieldsat all rangesand over
soundingthe suboceaniclithosphere, (2) the horizontal a wide band of frequencies. The choiceof these parameelectromagnetic components rather than their vertical
counterpartsare the best measurementvariable, (3) the
'•
lOO
'•
v
will be deferredto a subsequentpaper, but some guidelines can be inferred from the models. Sinceelectromagneticinductionis a diffusiveprocess,datacollectedat long
rangeswill reflect an averageconductivityover a greater
depth zone than for shortersource-receiver
spacings.This
is clearlyreflectedin Figures10 and 11, where the sharp-
•oo
•
,.,
0
0
ß'r -100
-r -100
10-2 10-1 100
101
10-2
FREQUENCY
(Hz)
ters is structure dependent and will influence the resolution ability of the data. Quantitative assessmentof this
10-1 100
FREQUENCY
(Hz)
101
est changesin the electric field amplitude occur at 4 km
and the variation is progressivelysmoother at longer
ranges. This suggeststhat the result of Youngand Cox
[1981]is an averagevaluefor the crustanduppermantle.
10-6
•-•
10-7_
10_
8
....
-. -,•
10-9
X
10-6
The conductivityvalue they infer (0.004 S/m) is in the
10-7
middle of the range postulatedin the last section.
• 10_8
•
1
The selection of the measurement variable--horizontal
electric or magnetic field--must be based on sensitivity,
noise level, and instrumental considerations. It is useful
to estimate the ambient noise power at the seafloor pro10-210-• 100 10•
10-210-• 10o 10• duced by natural electromagneticsources. Three major
rR[OU[NCY(Hz)
•R[OU[•CY (Hz)
components--ionospheric currents, microseisms and the
Fig. 10. The radial electricfield responsefor two oceancrust
modelsshownas a functionof frequencyat rangesof 4 km (solid nonlinear interferenceof surface wind waves, and direct
line), 10 km (shortdashedline), and 16 km (longdashedline). production of electromagneticfields by surface wind waves
The left panelsrepresent
coldcrustconsisting
of a 0.05-S/mlayer and swell--can be identified. Ionospheric power levels
2 basaltlayer 1 km thick,a 0.003-S/mdike and gabbrolayer2.5 were measuredby Wertzand Campbell[1976] on land at
km thick,and a 0.001-S/mlowercrustlayer2 km thickoverlying frequenciesbetween 0.003--3 Hz and shown to displaya
a 0.0005-S/mhalf space.The right panelsrepresentyoung,hot
10-•ø
,
,
,
• 10-•ø
Attenuationof theseexternally
crust, and the lower crust layer has a conductivityof 0.01 S/m. roughf-3 dependence.
The sourceis located1 m above the seafloor,where measure- producedfieldsas they propagatethroughthe conducting
ments are made, and the point dipoleapproximationis used.
ocean follows an exponential law
5336
CHAVEAND COX: CONTROLLED
EM SOURCEIN OCEAN:FORWARDPROBLEM
S(o•)
= e- 2•.•---•
Ho
Osurface
seafloor
(o•)
(41)
10-4
where S is the power spectral density of the magnetic
field, ro is the angular frequency, rr is the conductivityof
seawater,and H is the water depth. Magnetic field values
are converted to electric field values using the usual electromagnetic impedance relation. Microseisms, the nonlinear interference of surface wind waves, and seismic
wavesare treatedby Cox et al. [1978] and Webband Cox
1 0 -8
10-12
_.J
[1982]. The latter two effectsare localphenomena
that
maybe transient,but microseisms
producea broad,steady
state increase of the electromagnetic power level centered
near 0.2 Hz. Oceanographicnoise is generatedby surface
wind waves' swell' and turbulence. The former two quantities can be estimated by combining the electromagnetic
10-16
10-20
I
--J
10-2
LI.I
I
I
I I IIii
10-1
I
I
10 0
I
I,I
III
101
FREOUENCY(Hz)
field amplitudesgiven by Larsen[1971] or Chave[1982]
with the Piers0n-Moskowitz
wind wavepowerspectrum
[Phillips,1977].
10-2
E(f) = (2rr)4f
ag25 e-0'74(f/fm)-4(42)
where E is the wave amplitude variance spectrum in
me/Hz, g is the acceleration
of gravity,f is the frequency,
fm is the peak frequency where the phase velocity of a
gravity wave equals that of the wind producingit, and c• is
a dimensionless semiempirical constant with a value near
0.001. Additional oceanographicnoise is produced by turbulence. The magneticeffect is very small and the electric
fieldis of order V x F, whereV is the turbulentvelocity
[Cox et al., 1970]. Sinceturbulenceis a localphenomena,
its electric field influence can be reduced arbitrarily by
increasingthe length of the receiver antenna.
Figure 12 showsestimatesof the noise level from ionospheric currents, microseisms, and wind waves and swell
in water of 100 m, 1 km, and 5 km depth. Ionospheric
noise is the most important contributor, and attenuation
10-6
• 10-10La 1 0 -18
-
•
III
z
10-2
E
I
I I I •111
10-1
I
I I I
II
10 0
I
I.
I
t01
FREQUENCY
(Hz)
Fig. ]2. The power spectraldensityof noise in the horizontal
electricfield (top) and the horizontalmagneticfield (bottom)at
theseafloor
in waterdepths
or ]00 m, 1 kin, and5 km asa function or frequencyfrom 0.01 to l0 Hz. The peakedfunctionscentered near 0.1 and 0.2 Hz are the contributions from ]0 s swell
(solid) and 5 s wind waves (dashed) in 100 m or water; these
by the oceanis apparentat highfrequencies
in deepwater. terms vanish in deeper water. The gently slopinglines are the atIt is unlikely that frequenciesbelow 0.05 Hz will be usable
due to ionospheric contamination. Microseisms are a
basin wide, depth-independent phenomenon that produce
the peak between 0.1 and 1 Hz in Figure 12 by the largescale, nonlinear interference of surface wind waves. The
same mechanism can produce peaks in the power spec-
trum at twice the frequencyof local wind waveswith an
amplitude that depends on the sea state. Swell is important only in shallow water at low frequencies,and the signal produced by it is more than 10 orders of magnitude
smaller in 1 km of water. Note that the magneticfield is
more affected by swell noise, while direct induction by
wind wavesis negligible.
Figures 7 and 8 show that the sensitivity of the electric
field to structure is marginally better than that of the magnetic field. Figures 2, 3, and 12 predict a higher signalto-noise ratio for the electric field. Cox et al. [1981] compared the performance of electrode-type electric field
instrumentsto fluxgate, inductioncoil, and cryogenicmagnetometers. Only the superconductingunit promises
better performance than silver-silver chloride electrodes,
but the cost and engineeringproblems make use of the
cryogenicmagnetometerdifficult. The horizontal electric
renuarealionosphericcontributionwith a broad peak, centered
near 0.2 Hz, representingthe microseisinsource.
field does offer slight advantagesover the magneticfield
for controlledsourceexperiments.
The calculationof the maximum source-receiverrange
for a given frequencydependson the conductivity-depth
relation. At 1 Hz with a lithosphericconductivityof 0.05
S/m in 1 km of waterand usinga 6 x 104 A m sourcea
signal level comparableto the noise level is achieved at a
range of 15 km using a receiver with a 10-m antenna and
a bandwidth of 10-4 Hz. Smaller bandwidths can be
achievedby increasingsignal integrationtime, raising the
maximum range. In practice, instrumental electronic
noise and attenuation of the fields by highly conductive
sediments will probably be the limiting factor. Since the
lithosphericconductivityis probablymuch lower than 0.05
S/m, longer rangescan be attained. If integrationtimes of
1 week are used, ranges in excess of 50 km can be
attained. The maximum horizontal range reflects the vertical region that is sampled by the electromagneticfields;
hence measurement of the lithospheric conductivity to
considerabledepths is feasible.
CHAVE
ANDCox:CONTROLLED
EM SOURCE
INOCEAN:
FORWARD
PROBLEM
APPENDIX A: ELECTROMAGNETIC FIELDS FOR A
FINITE HED IN SHALLOW WATER
5337
Oy{;
=[psinq•-p'sinqb
[
(A12)
The expressions (22) and (23) are valid when
reflectionsoff of the thermocline and sea surface are not
Calculation
of the finiteHED fieldsis accomplished
by
quadrature,with an integrandconsistingof successive
importantand for source-receiver
distances
muchlarger evaluations of the electromagneticfields for different
than the lengthof the sourcewire. Inclusionof upper source points.
conductive structure requires an additional pair of
responsefunctionsevaluatedat the upper boundary
APPENDIX B:
(z - H) givenby
ANALYTIC EVALUATIONOF THE SOURCETERMS
(A1)
z=H
In eachof (27)--(35) the sourcetermscorresponding
to the rapidly damped direct wave in the ocean can be
expressedanalytically,using the resultsof Watson[1962]
and the derivativerelationshipsfor Besselfunctions. The
integralsare
I
(A2)
t<= Oz
H z=H
and reflectioncoefficientsgiven by (25) and (26) with
(A1) and (A2) substituted
for (20) and (21). We considera finite wire of lengthd! orientedalongthe x' axis
with azimuth•b'. Replacing
p in (23) and (24) with Idx',
where I is the sourcecurrent,yields
J'o
dk
J1
(kP)
k2e-aølz-z'l
Iz-Z'{l•se-•ø•
(y•R
2+3yoR
+3)
(B1)
J'o
dkJ1
(k.)•k3
e-aølz-z'l
_yoe-YO
R
(A3)
R4
(yoRp2 + 3p2-- 2R2) (B2)
oo
R(3,oR+ 1)
J'o
dk
J,(kp)
•k2e-/sølz-z'l-- Pe-•ø
R3
(B3)
(A4)
'odk
Jo
(kp)
kl•oe
-/•ø1
z-z'l
where
e
r = e-/sølz-z'l
+ R•ER.•Ee-2t•øH
?olz-z'l (A5)
Z = 1- R•ER•'Ee-2t•øH
(A6)
Iz-z'l
E- e-aølz-z'lR•MR•Me-2aøH
7o
(A7)
A-' 1- R•/MR•Me-21•øH
(A8)
-yo R
- • g 5 {y•R2(z-z')2+ (yoR+ 1)[2(z-z')2-p2]}
(B4)
o•dk
k e-/Solz-z'l
J'o
Jo(kp)
•oo
- e-•OR
R
(B5)
- Iz-z'le-øR
R3 (yoR
+ 1)
oJ•
dk
Jo(kp)ke
-•ølz-z'l-
(B6)
where
and • is givenby (19). The denominators
(A6)--(AS)
may be expandedinto a powerseries,and the result
R
+ (z-z')
represents an infinite number of reflections in the
y0=-x/ioolXO'o.
Eachof (B1)--(B6)effectively
vanwaveguide
between
seafloor
andseasurface.Applicationand
isheswithin a few skin depthsof the sourcedue to the
of (5) and(11)to (A3) and(A4) is straightforward
but exponential
tediousandwill beomitted.Since• is dependent
on both
term.
p and•b,the derivatives
of Jo(k•) containtermslike
APPENDIX C:
AVERAGING
WELLLOGCONDUCTIVITY
VALUES
Op'-{Q--p'cOS(d•--d•')
]
(A9)
Electrical conductivitylogs from DSDP boreholes
- p'sin
1O•15
(•b-•b')
--
display
a widescatterfrompointto point,reflecting
the
(AlO)
inhomogeneous
nature of the basaltsas well as the effect
of drillingdisturbance.
For interpretation
purposes
it is
Ox• --
pcos,•-p'cos4,'
necessaryto averagethe data, but this should be done in a
(All)
mannerconsistent
withthe physical
mechanism
governing
the conduction
process.At lowtemperatures
in the pres-
5338
CHAVE AND COX: CONTROLLEDEM SOURCEIN OCEAN: FORWARDPROBLEM
ence of conductingfluids like seawaterthe electricalcon- Filloux, J. H., Magnetotelluricexploration of the North Pacific:
ductivityoften followsArchie'slaw [Shankland
and Waft,
1974].
cr = crwp
n
(C1)
where crwis the electricalconductivityof the fluid and p is
the mean porosity. The exponent n is estimated to lie
close
to
2.
For
the
ith
measurement
in
a well
the
Progress report and preliminary soundings near a spreading
ridge, Phys.Earth Planet.inter., 25, 187-- 195, 1981.
Gregory, R. T,, and M.P. Taylor, An oxygenisotopeprofile in a
section of Cretaceousocean crust, Samail ophiolite, Oman: Evi-
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Keller, G. V., Electricalprospectingfor oil, Q. Colo. Sch. Mines,
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= (rdO'w) .
(C2)
Larsen, J. C., The electromagneticfield of long and intermediate
water waves, J. Mar. Res., 29, 28--45, 1971.
Combining N measurements of conductivity by averaging
(C2) yieldsthe mean porosityfor the section. Combining Law, L. K., and J. H. Greenhouse, Geomagnetic variation sounding of the asthenospherebeneath the Juan de Fuca Ridge, J.
this with (C1) yields
Geophys.
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Morse, P.M., and H. Feshbach,Methodsof Theoretical
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rr
(C3)
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Pallister, J. S., and C. A. Hopson, Samall ophiolite plutonic suite:
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Field relations, phase variation, cryptic variation and layering,
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the square roots of the fluid and formation average conductivity from (C2). This brings well log results into Patterson, T. N. L., Algorithm for automatic numerical integration over a finite interval, Commun.A. C. M., 16, 694--699,
effectiveporosityPt is givenby rearranging(C1)
agreement with results from the large-scale resistivity
experiment.
1973.
Phillips,O. M., The Dynamicsof the UpperOcean,2nd ed., 336 pp.,
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sions about numerical methods. This work was supportedby the
Earth Physicsbranch of the Office of Naval Researchand the Rai, C. S., and M. H. Manghnani, Electrical conductivity of
ultramafic rocks to 1820 Kelvin, Phys. Earth Planet. Inter., 17,
ScrippsIndustrial Associates.
6--13, 1978.
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