Electrical conductivity varies over 5 orders of magnitude in common Earth materials, and provides the ideal mechanism for studying fluids, water, and melt in the crust and mantle
Conductivity, S/m 10 1 0.1 0.01 10-3 10-4
20°C 0°C water seawater fresh water
sediments marine land 1400°C 1200°C 1100°C mantle lower mantle peridotite
basalt, salt, carbonates 10% 3% 1% basaltic melt partial melt
ice, hydrate hydrocarbons
0.1 1 10 100 1,000 10,000 Resistivity, Ωm CSEM:
Controlled Source ElectroMagnetic sounding has been used since the 1930’s. It is used for mining exploration, groundwater, archeology, and geological structure for oil exploration. Depths to a few km.
Transmitters Receivers
Bz
Bx I By grounded wire (E) V I Ey V horizontal loop (B) Ex MT:
MagnetoTelluric (MT) sounding, which uses similar receivers to measure Earth’s natural magnetic field variations and the induced electric currents. Depths to a few hundred km.
Natural-source Magnetotelluric fields
Transmitters Receivers
Bz
Bx I By grounded wire (E) V I Ey V horizontal loop (B) Ex Primary field
Secondary field
Transmitter coil
B secondary
B total
B primary
Induced currents Conductive body
Secondary fields are a) In a different direction b) Are a different magnitude c) Are a different phase A timeline. 1820: Oersted. Currents deflect magnets. Ampere.
1830: Fox maps earth potentials.
1831: Faraday makes current from moving magnet.
1838: Magnetic storm seen on telegraph cables.
1839: Gauss develops the spherical harmonic expansion.
1864: Maxwell’s equations.
1889: Schuster noted the relationship between magnetic field variations and earth potentials.
1939: Lahiri and Price model earth conductivity to 1,000 km depth. Barlow, 1849, currents in UK telegraph cables. Earth’s electromagnetic environment
Earth’s magnetic field varies on all time scales: Grand Spectrum
Reversals
Cryptochrons?
Secular variation
? ? Annual and semi-annual Solar rotation (27 days) Daily variation
Amplitude, T/√Hz Amplitude, Storm activity Quiet days
Powerline noise Radio
Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz
Frequency, Hz The high frequencies come from external sources ...
Grand Spectrum
Reversals
Cryptochrons?
Secular variation
? ? Annual and semi-annual Solar rotation (27 days) Daily variation external origin
Amplitude, T/√Hz Amplitude, Storm activity Quiet days
Powerline noise Radio
Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz
Frequency, Hz and the low frequencies are internal. Both can be used to estimate conductivity.
Grand Spectrum
Reversals
Cryptochrons?
Secular variation
? ? Annual and semi-annual Solar rotation (27 days) Daily variation external origin
Amplitude, T/√Hz Amplitude, Storm activity Quiet days internal origin
Powerline noise Radio
Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz
Frequency, Hz bearing in mind that between about one year and ten years there can be overlap between internal and external fields. Grand Spectrum
Reversals
Cryptochrons?
Secular variation
? ? Annual and semi-annual Solar rotation (27 days) Daily variation external origin
Amplitude, T/√Hz Amplitude, Storm activity Quiet days internal origin
Powerline noise Radio
Schumann resonances 1 year 10 million years 1 thousand years 1 month 1 day 1 hour 1 minute 1 second 10 kHz
Frequency, Hz A spatial view of Earth’s magnetic field: A spatial view of Earth’s magnetic field: The solar wind drives changes in Earth’s magnetic field
1.2
1 Magnetic Field (N-S)
0.8 pT 0.6
0.4
0.2 1 pT = 10-12 Tesla : Earth’s field is 30-60 µT (10-6 T) 0 So, this signal is one hundred millionth of the main field.
-0.2 0 400 800 1200 1600 2000 seconds These changes generate an electric field in the ground through induction
1.2
1 Magnetic Field (N-S)
0.8 pT 0.6
0.4 Electric Field (E-W) 0.2
0 x10 µV/m
-0.2 0 400 800 1200 1600 2000 seconds In practice we measure both north and south fields, and maybe Bz Particles from the sun and magnetic storms inject particles into the ring current. The size of the ring current generated by solar activity is measured using the Dst index (“disturbed storm time”). Because the effect of the ring current is to reduce Earth’s main field, storms go negative. Magnetic storms are tied to the 11-year solar cycle, but not too strongly.
Dst index (blue, nT) and 10.7 cm radio flux (red, 10−22 J/s/m2/Hz) 400
300
200
100
0
−100
−200
−300
−400
−500
−600 1960 1970 1980 1990 2000 2010 Year The ionosphere is electrically conductive.
200
160
Ionosphere (anisotropic) 120
80 Atmosphere Altitude, km 40
Earth’s surface 0 -14 -10 -6 -2 log(conductivity, S/m) Thermally induced currents generated on the dayside create a daily variation, or Sq, in Earth’s magnetic field.
USGS via BGS Schumann resonance at 1/8 second and harmonics. Electrical conductivity of rocks and minerals.
Rock conductivity (and resistivity):
1 m
1 m
current = I 1 m
Electrical resistivity has units of Ωm.
A 1m x 1m x 1m cube of 2 Ωm rock would have a series resistance of 2 Ω across the faces: L RA R = = A L where L is the length of any prism and A is the cross-sectional area.
Conductivity is just the reciprocal of resistivity. Units are S/m. Approximate conductivities and resistivities of various Earth materials. Ionic conduction through water
1 seawater at 20 C 5 Sm 0.2 Wm 1 seawater at 0 C 3 Sm 0.3 Wm 1 marine sediments 1–0.1 Sm 1–10 Wm 1 land sediments 0.001–0.1 Sm 10–1000 Wm 5 2 1 igneous rocks 10 –10 Sm 100–100,000 Wm Good semiconductors
1 4 6 graphite 10,000–1,000,000 Sm 10 –10 Wm 1 1 5 galena 10–100,000 Sm 10 –10 Wm 1 3 pyrite 1–1,000 Sm 10 –1 Wm 1 5 magnetite 20,000 Sm 5 10 Wm ⇥ Thermally activated conduction
1 olivine at 1000 C 0.0001 Sm 10,000 Wm 1 olivine at 1400 C 0.01 Sm 100 Wm 1 theoleite melt 3 Sm 0.3 Wm 1 silicate perovskite 1 Sm 1 Wm Resistivity of Rocks and Minerals (Data from Telford et al.)
basalts/lavas limestones
OIL SANDS sandstones, quartzites wet dry clays graphite evaporites pyrrhotite intrusive igneous rocks seawater groundwater wet dry quartz
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Log 10 (Resistivity, Ωm) In semiconductors, charge carriers can be a product of doping, as well as Intrinsic thermal activation. regime
The same is true of electrically charged Extrinsic regime defects in mineral crystals. Conductivity
Impurity Dislocation Temperature
Frenkel defect
Schottky defect Now the theory...
Some basic electromagnetic theory. Let’s start with
E is the electric field, measured in volts/meter.
B is the magnetic field, measured in tesla.
H is sometimes also called the magnetic field, and has units of A/m, and then B is called magnetic induction vector, or the magnetic flux density. But here we call B the magnetic field, and H the magnetizing field, in recognition of its relationship to magnetic polarization.
B and H are related through the permeability µ of a material.
B = µH
7 In a vacuum, µ = µ =4⇡ 10 H/m o ⇥ Integrating over surfaces: S
A surface vector, s, has a direction that is the outward normal to the surface and a magnitude proportional to the area of the surface. An infinitesimal surface element, ds, is useful for integrating things over surfaces. In particular, if we have a vector field A, then the integral
A ds = A. cos ✓.ds · ZW ZW over surface W is the Flux through the surface. ✓ is the angle between A and ds. We are going to need a little calculus, and the use of the operator. r @ @ @ = , , r @x @y @z ✓ ◆ so A is just a gradient: r
@Ax @Ay @Az A = , , r @x @y @z ✓ ◆ A is the divergence: r·
@Ax @Ay @Az A = + + r· @x @y @z
A is the curl, r⇥
@Az @Ay @Ax @Az @Ay @Ax A = , , r⇥ @y @z @z @x @x @y ✓ ◆ and A is the Laplacian. r·r 2 2 2 @ Ax @ Ay @ Az A = 2A = + + r·r r @x2 @y2 @z2 Gauss’ Law:
Q ⇢ E ds = E = · ✏ r· ✏ ZW o o
Q is charge, C ⇢ is charge density, C/m3 12 ✏ is permittivity of free space, = 8.895 10 F/m. o ⇥ Gauss’ Law says that the electric field leaving a volume is proportional to the enclosed charge. E
Q Faraday’s Law:
dFB @B E dl = E = · dt r⇥ @t Ic
FB is magnetic flux
Faraday’s Law says that the electric field integrated around a loop (i.e. the voltage) is given by the time rate of change of the enclosed magnetic flux.
E
dB dt Gauss’ Law (magnetism):
B ds =0 B =0 · r· ZW
Gauss’ Law for magnetism says that there are no magnetic monopoles. Any flux entering a volume has to leave it.
B Ampere’s` Law:
B dl = µoI B = µoJ · r⇥ Ic
I is electric current, A J is current density, A/m2 8 µ is permeability of free space, = 4⇡ 10 H/m. o ⇥ Ampere’s` Law says that an electric current will generate a circulating magnetic field.
B
I Maxwell’s equations (in a vacuum):
@B ⇢ E = E = r⇥ @t r· ✏o @E B = µo J + ✏o B =0 r⇥ @t r· ✓ ◆
The extra term in Ampere’s` Law was added by Maxwell. It allows fields to exist without charges or currents, and allows electromagnetic radiation 2 to propagate in a vacuum at speed c, where c =1/(µo✏o).
λ
E M
distance
"Light-wave". Licensed under CC BY-SA 3.0 via Wikimedia Commons - http://commons.wikimedia.org/wiki/File:Light-wave.svg#/media/File:Light-wave.svg In matter: P = ✏o EE
M M = B µo J = E
E is electric susceptibility M is magnetic susceptibility electrical conductivity, S/m
The last equation is, of course, Ohm’s Law, but these are approximations! Matter doesn’t have to be linear and isotropic. Clearly, there will be saturation phenomena. E or B + + - - + + - + - - + + - + - - + - polarization Maxwell’s equations in matter:
@B E = D = ⇢ r⇥ @t r·
@D H = J + B =0 r⇥ @t r· where
D = ✏oE + P is the electric displacement field, and H = B/µo M is the magnetizing field. @D/@t is called the displacement current, and can be ignored at the fre- quencies, length scales, and conductivities that are relevant to geomag- netic induction. Similarly, we won’t concern ourselves with polarizable media, so E = D/✏o and B = µoH. H has units of A/m. Electromagnetic induction in pictures and equations:
Both MT and GDS sounding use electromagnetic induction, which primary magnetic describes what happens around a field time-varying primary magnetic field: Electromagnetic induction in pictures and equations:
Faraday’s Law says that a time varying (or moving) magnetic field will induce electric fields: primary magnetic d field E dl = · dt electric field C ( is magnetic flux). Electromagnetic induction in pictures and equations:
Faraday’s Law says that a time varying (or moving) magnetic field will induce electric fields: primary magnetic d field E dl = electric field C · dt and current ( is magnetic flux).
Ohm’s Law says that E will generate a current J in a conductor: J = E Electromagnetic induction in pictures and equations:
Faraday’s Law says that a time varying (or moving) magnetic field will induce electric fields: primary magnetic d field E dl = electric field C · dt and current ( is magnetic flux). Ohm’s Law says that E will generate a current J in a conductor: secondary J = E magnetic field Ampere’s Law says that the current will generate a secondary magnetic field:
B dl = µI · C Electromagnetic induction in pictures and equations: The secondary field opposes the changes in the primary field. The consequence of this is that conductive primary magnetic rocks absorb variations in EM fields field more than resistive rocks. electric field This absorption is exponential: and current
z/zs E(z)=Eoe
secondary The rate of absorption is given by the magnetic skin depth, which depends on rock resistivity and period: High resistivity, field long periods = large skin depths, greater penetration. “Skin effect” describes the tendency for current to flow in the skin of a conductor. I used to think that it was the reason that the high voltage from an RF Tesla coil didn’t kill people, but that’s not the case.
Wikipedia commons Let’s derive the skin depth equation:
Substitute Ohm’s Law into Ampere’s` Law:
J = E B = µoJ r⇥ to get B = µo E r⇥ Take the curl of this and use Faraday’s Law ( E = @B/@t): r⇥ @B B = µo E B = µo r⇥r⇥ r⇥ !r⇥r⇥ @t
Now we need the vector identity ( A)= ( A) 2A to get r⇥ r⇥ r r· r
2 @B ( B) B = µo r r· r @t
But, the no-monopoles law says that B = 0, so ... r· 2 @B B = µo r @t Similarly, we can take the curl of Faraday’s Law and substitute Ampere’s` and Ohm’s Laws to get
@ @E E = ( B) E = µo r⇥r⇥ @t r⇥ !r⇥r⇥ @t
To pull the same vector identity trick we need to use ( A) = 0 on r· r⇥ Ampere’s` Law ( B = µoJ) to get r⇥
J = 0 which for constant o gives E =0 r· r· so 2 @E E = µo o r @t These equations in B and E are diffusion equations. The resolution of EM induction sits between wave propagation and potential fields:
Wave equation: Resolution ~ wavelength High frequency ⌅E ⌅2E @u 1 @2u (megahertz) Radar 2E = µ⇤ + µ Seismics 2u = ✏ + ⌅t ⌅t2 r @t c2 @t2
Diffusion equation: Resolution ~ size/depth
Mid frequency ⇤E (0.001 - 1000 Hz) Inductive EM 2E = µ⇥ ⇤t
Laplace equation: Resolution ~ bounds only Gravity/ Zero frequency Resistivity 2E =0 2U =0 Magnetism r
6 ⇤ = electrical conductivity ~ 3 10 S/m 4 6 µ = magnetic permeability ~ 10 10 H/m 9 11 = electric permittivity ~ 10 10 F/m Now it is time to consider a single frequency !, so
@B B(t)=Bei!t and = i!B @t and the same for E, so our diffusion equations become
2 2 E = i!µo oE and B = i!µo oB r r For external sources of B, at Earth’s surface B is purely horizontal and uniform. Then
d2B B = B eiwt and our diffusion equation is = k2B(z) o dz2
2 where we have defined a complex wavenumber k = i!µo o. d2B 2 dz2 = k B(z) is a second order linear ODE with solutions of the form
kz kz B(z)=c1e + c2e
The first term grows with depth so c1 = 0, and for z = 0 we can infer that i!t c2 = Boe . We can write k as
1+i 2 k = where zo = zo s!µo o to get i!t z(1+i)/zo B(z)=Boe e
So B(z) falls off exponentially with zo, which is the skin depth. z 500p⇢T m where ⇢ =1/ and T is period in seconds. o ⇡ In uniform conductors, the fields fall off exponentially as 1/e, with one radian of phase shift every skin depth. phase B/Bo -0.5 0 0.5 1 -300 -250 -200 -150 -100 -50 0
Re (B) Im (B) 1 1
|B| 2 2 z/z z/zo o
3 3
4 4
5 5 To a good approximation skin depth is given by z 500 ⇢T meters s ⇡ where T is period in seconds. Somep typical skin depths:
material , S/m 1 day 1 hour 1 sec 1 ms seawater 3 85 km 17 km 290 m 9 m sediments 0.1 460 km 95 km 1.6 km 50 m 5 igneous rock 1 10 50000 km 9500 km 160 km 5 km ⇥
Skin depth is not a measure of resolution, but is a guide to the maximum distance that EM energy can propagate. The MT equation: We just showed
i!t z(1+i)/zo B(z)=Boe e and recall B = µo E r⇥ so if B is in the x direction, the only non-zero component of the curl is @Bx/@z 1 dBx 1+i k Ey = = Bx = Bx µo o dz µo ozo µo o because
@Az @Ay @Ax @Az @Ay @Ax A = , , r⇥ @y @z @z @x @x @y ✓ ◆ Similarly, 1 dBy 1+i k Ex = = By = By µo o dz µo ozo µo o
This is valid for any depth z, but in practice we are only interested in the iwt surface where z = 0 and Bx,y = Boe . The MT equation continued...
We can take the ratio of the electric to magnetic field at any particular frequency to obtain the half-space resistivity:
E 2 k 2 !µ ! y = = o o = B µ (µ )2 µ x ✓ o o ◆ o o o o µ E 2 ⇢ = o y ! B x This is the MT equation made famous in Cagniard’s 1953 paper. The phase between E and B is given by the (1 + i) term, which is -45 . time series MT in a nutshell: Magnetic Bx
Magnetic By s13 2 10 MT response functions Processing Electric Ex 1 10 m - m h
Electric Ey O 0 10
-1 10 0 1 2 3 4 10 10 10 10 10 Time Series
90 Inversion 75 s
e 60 e r 45 g resistivity models e 30 0 4 D
1000 ♦ ♦ ♦♦♦♦♦♦♦♦♦♦♦ ♦ ♦ ♦ 15 2000 3 0 ) 0 1 2 3 4
m 10 10 10 10 10 3000 - ) m m h Period (s) (
4000 o ( h t
2 p y t e
5000 i v D i t
6000 s i s
1 e 7000 R
8000
9000 0 -4000 -2000 0 2000 4000 6000 8000 10000 12000 14000 Horizontal Position (m) Half-space Here is what the MT fields look like in a uniform conductor. The secondary field opposed the changes in the primary field with the consequence that conductive rocks absorb the field variations exponentially.
The induced electric field is 45° out of phase with the primary magnetic field. Half-space We can compute a half-space equivalent electrical resistivity (apparent resistivity) at each frequency: µ E(!) 2 ⇢ (!)= o a ! B(!) We can also compute the phase difference between E and B. These
2 10 become the MT sounding curves.
0 10 Apparent
Resistivity −2 10 80
60
40 Phase 20
0 −2 0 2 4 10 10 10 10 Period (s) Half-space Conductive Layer
We can add a conductive layer at depth and things change at the surface
2 2 10 10
0 0 10 10 Apparent Apparent −2 Resistivity −2 10 Resistivity 10 80 80
60 60
40 40 Phase Phase 20 20
0 0 −2 0 2 4 −2 0 2 4 10 10 10 10 10 10 10 10 Period (s) Period (s) Half-space Conductive Layer Resistive Layer
Same for a resistive layer
2 2 2 10 10 10
0 0 0 10 10 10 Apparent Apparent −2 Apparent −2 Resistivity −2 Resistivity 10 10 Resistivity 10 80 80 80
60 60 60
40 40 40 Phase Phase Phase 20 20 20
0 0 0 −2 0 2 4 −2 0 2 4 −2 0 2 4 10 10 10 10 10 10 10 10 10 10 10 10 Period (s) Period (s) Period (s) CSEM field obey the same physics but are more complicated because of the 3D source (transmitter).
Air
Sea Tx Seafloor
Amplitude and phase of the magnetic/electric fields on the seafloor can be used to infer geological structure to depths of several km.