SIO229: Lecture 5Geomagnetism

Constructing Geomagnetic Field Models

Geomagnetic Field ModelingIntroduction

• Observations are measurements of the field at specific times and places.

• What about locations where we have no measurements?

• We would like to combine our observations to provide a model, mathematical description of the GLOBAL GEOMAGNETIC FIELD

• This could give us the “best” estimate of the magnetic field at any desired altitude, latitude, and longitude.

Geomagnetic Elements and Uniqueness

X, Y, Z - orthogonal components of the geomagnetic field in local coordinate system

Br, Bθ, Bφ – orthogonal components in geocentric reference frame

X =−Bθ

Y =Bφ

Z=−Br

H =(Bθ2+ Bφ

2)1 /2

F=(Bθ2+Bφ

2+ Br

2)1 /2

I =arctan [ −Br

( Bθ2+Bφ

2)1/2 ] where− π2⩽I ⩽ π

2

D=arctan [ Bφ

−Bθ] where−π⩽D⩽π

If we assume Earth is a sphere, then

Uniqueness?Suppose we specialize to case of internal field only

• If we know Br exactly everywhere on a spherical surface, we can find a unique representation of the field everywhere Laplace’s equation holds (exterior Neumann BVP, see page 18, Gravity)

• But |B| is not enough - unlike the gravity case ambiguity about the sign is enough to cause problems (see Backus, 1968 for one example)

• Nor does knowing the direction of B everywhere determine B to within a scalar multiple

• In any case we do not know B everywhere or exactly - this produces fundamental non-uniqueness in the results of any modeling activity.

Mathematical Properties Relevant to Geomagnetic Field Modeling

J (r > c)

r=a

r=c

r=b

r=R

J (r < a)

JpolJs (r=b)

Observatories No sources

Satellite

⊡ Fig. Uniqueness of a magnetic field recovered from partial informationwithin a current-carrying shell.In this special case relevant to geomagnetism, it is assumed that any source can lie below r = a(internalJ(r < a) sources), and above r = c (externalJ(r > c) sources), no sources can lie withinthe lower subshell (a < r < b, the neutral atmosphere), a spherical sheet current can lie at r = b(the E-region Js(r = b) sources), and only poloidal sources can lie within the upper subshell (b <r < c, the F-region ionosphere). The knowledge of B on a sphere r = R in the upper subshell (asprovided by, e.g., a satellite) and of enough components of B on the sphere r = a (as provided by,e.g., observatories at the Earth’s surface), is then enough to recover the field produced by mostsources in many places (see text for details)

neutral atmosphere and is therefore source-free, while the spherical sheet current describesthe ionospheric E-region (which is again considered as infinitely thin). Finally assume that theupper subshell describes the ionospheric current-carrying F-region within which near-Earthsatellites orbit. Since those currents are known to be mainly the so-called field-aligned cur-rents at polar latitudes (i.e., aligned with the dominant poloidal field, see, e.g., Olsen ), it isnot unreasonable to further assume that those currents have no toroidal components (they aremainly in radial direction). In other words, assume that no Jshtor sources, but only Jshpol sourceslie in the b < r < c upper subshell.These assumptions, together with the previous uniquenesstheorem can then be combined in a very powerful way.

Indeed, since B(R, θ, ϕ) is already assumed to be known on a sphere r = R in the uppersubshell, the Gauss coefficients (gmn (R), hm

n (R), qmn (R), smn (R)) can be inferred from Eqs. –.The (qmn (R), smn (R))Gauss coefficients then describe the field produced below r = R by allsources above r =R. But since between r =R and r = c only Jshpol sources are to be found whichonly produce local toroidal fields, the only sources contributing to the field described by the(qmn (R), smn (R)) coefficients are the J(r > c) sources. It thus follows that the (qmn (R), smn (R))are the (qmn , smn ) Gauss coefficients describing the potential field produced below r = c by theJ(r > c) sources. A similar use of the Gauss coefficients (gmn (R), hm

n (R)) recovered fromEqs. and can then also be made. Those describe the field produced above r = Rby all sources below r = R. But no sources between r = b and r = R contribute. OnlyJ(r < a) and Js(r = b) sources do. It then also follows that the (gmn (R), hm

n (R)) are the sums(gmn + amn , hmn + bmn ) of the Gauss coefficients describing the field produced above r = a by the

J(r < a) sources and above r = b by the Js(r = b) sources. This then brings one back to a

Construction of Field Models

● Spherical harmonics can be used to define a global geomagnetic field model.

● From this model we can determine any magnetic component at any location of interest (outside source regions).

● But, how do we determine the Gauss coefficients defining the spherical harmonic model best fitting observations?

MODELb

Spherical Harmonic Field Model

DATAd

Magnetic Observations

X i ,Y i , Z i ,

H i , F i ,Di , I i

{g lm, hlm

}

Forward problem

Inverse problem

Construction of Field Models

d=G b+n

Total number of parameters: K=L(L+2), L is the truncation level.

● Least squares estimation:

d j= s j⋅B(r j)=∑l=1

L

∑m=−l

l

blmal+2s j⋅∇ [Y l

m( r j)

r jl+1 ]+ϵ j

b=(GTG )−1

GTd

E [∥n∥2]=(N−K )σ2

Ψ(r ,θ ,φ)=a∑l=1

L

∑m=−l

l

blm ( ar )

l+1

Y lm(θ ,φ) B=−∇Ψ

If we measure orthogonal components of the field Br , Bθ , Bφ

we have a set of observations atN sites designated r j :

∥n∥2=∥d−G⋅b∥2

● If the misfits are due to random, uncorrelated perturbations with zero mean, and have a common variance, then the least squares solution is the best linear unbiased estimate (BLUE) available, in the sense that they have the smallest variance amongst such estimates:

Regularization – an Alternative to Least Squares

• In many geophysical inverse problems:

(i) Models can be very complex, especially those arising from the discretization of continuous physical fields, and may possess large ‘null spaces’ not constrained by data.

(ii) Data are contaminated by large errors or parts of the model are very sensitive to data noise.

Then model non-uniqueness and instability are problems:

i.e. Many models can fit the data within the error estimates and minimizing the least squares criteria alone will not necessarily yield the most plausible model.

Regularization – an Alternative to Least Squares

Example: cubic spline interpolation

Figs. 3.4.4.1 and 3.4.4.2 notes

U=∑j=1

N ( f ( x j)− y j)2

σ j

2+λ∫

x1

x N

[ d2f

dx2 ]

2

dx

subject to ∑j=1

N ( f (x j)− y j)2

σ j

2=T

smoothest curve connecting the points (RMS second derivative)

Interpolation Smoothing Spline - allows misfit

Regularization – an Alternative to Least Squares

Penalty functions:

∫r>a∣B∣2

d3r wl=( l+1)

∫S (a)B⋅Bd 2

r wl=(2l+1)(l+1)

∫S (a) [∇ l r⋅B ]2d

2r wl=l (l+1)2( l+

1

2)

∫S (a)

[∇ l

2r⋅B ]

2

d2r wl=l

2( l+1)4

∫r<aJ T⋅J T d

3r wl=a

3μ0

2( l+1)(2l+1)2(2l+3)

∥B∥w2 =∑

l=1

∞

wl ∑m=−l

l

∣blm∣2 , wl>0

Results: Gauss Coefficients

∫S (1 )

(Y lm)2d

2r=

4π2l+1

● In geomagnetism, basis functions (spherical harmonics) are normalized so that:

Ψ(r ,θ ,φ)=a∑l=1

∞

( ar )l+1

∑m=0

l

N lm( glm

cos (mφ)+hlm

sin (m φ))Plm (cosθ)

N lm=1, m=0

N lm=√ (l−m)!

(l+m)!, m>0

● The normalization constant is:Schmidt normalization

● In geomagnetism, basis functions (spherical harmonics) are normalized so that:

Recall from Gravity:

Y lm(θ ,φ)=N lm e

imφPlm(cos θ)

N lm=(−1)m( 2 l+1

4 π )1/2

( (l−m) !

(l+m)! )1/2

Regularization Example of core surface field trade-off curve

Regularization Solution on the trade-off curve

Regularization Solution on the trade-off curve

Regularization Solution on the trade-off curve

Regularization Solution on the trade-off curve

Regularization Solution on the trade-off curve

Fig. 3.4.5 notes

Various views of the radial component of the magnetic field from IGRF 2000: (top) Br at Earth's surface;

(middle) the nondipole contribution to Br at Earth's surface;

(bottom) Br downward continued to the core-mantle boundary;

The Geomagnetic Spectrum

Figs. 3.4.6.1 and 3.4.6.2 notes

CORE

CRUST

Rl=(2 l+1)(l+1)

4 π∑m=−l

l

∣blm∣2=( l+1)∑

m= 0

l

[ (glm)2+(hlm)2 ]

Rl(r )=( ar )2 l+4

Rl(a) Sources of the Geomagnetic Field and the Modern Data That Enable Their Investigation

0 10 20 30 40 50 60 70 80100

102

104

106

108

100

102

104

106

108

Core Crust

Degree n

Rn

[(nT

)2 ]

40,000 4,000 2,000 1,500 1,000 800 700 600 500

ln (km)

⊡ Fig.��������Spatial power spectrum of the geomagnetic field at the Earth’s surface. Black dots represent thespectrum of a recent field model (Olsen et al. ; Maus et al. ). Also shown are theoreticalspectra (Voorhies et al., ) for the core (blue) and crustal (magenta) part of the field, as well astheir superposition (red curve)

which have been fitted to the observed spectra; their sum (red curve) provides a remarkablegood fit to the observed spectrum. There is a sharp “knee” at about degree n = which indi-cates that contributions from the core field are dominant at large scales (n < ) while those ofthe crustal field dominate for the smaller scales (n > ).

A more detailed overview of the various field sources and their characteristics is nowpresented.

. Internal Field Sources: Core and Crust

.. Core Field

Although the Earth’s magnetic field has been known for at least several thousands of years (see,e.g., Merrill et al. ), the nature of its sources has eluded scientific understanding for a verylong time. It was not until the nineteenth century that its main source was finally proven tobe internal to the Earth. It is now known that this main source is most likely a self-sustainingdynamo within the Earth’s core (see, e.g., Roberts ; the > Chap. “Theory and modelingof planetary dynamos” by Wicht et al. in this handbook)

The Geomagnetic Spectrum

Figs. 3.4.6.1 and 3.4.6.2 notes

CORE

CRUST

Rl=(2 l+1)(l+1)

4 π∑m=−l

l

∣blm∣2=( l+1)∑

m= 0

l

[ (glm)2+(hlm)2 ]

Rl(r )=( ar )2 l+4

Rl(a)