Orthogonality of Harmonic Potentials and Fields in Spheroidal

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Orthogonality of Harmonic Potentials and Fields in Spheroidal Coordinates Frank Lowes Newcastle Upon Tyne,

Orthogonality of Harmonic Potentials and Fields in Spheroidal Coordinates Frank Lowes Newcastle Upon Tyne, UK With help from Denis Winch, University of Sydney 1

OUTLINE OF TALK Spherical coordinates Spherical harmonics Orthogonal over the sphere Magnetic Power Spectra

OUTLINE OF TALK Spherical coordinates Spherical harmonics Orthogonal over the sphere Magnetic Power Spectra Spheroidal coordinates Spheroidal harmonics NON-orthogonal over the spheroid Use a geometrical weighting factor to restore orthogonality 2

SEPARATION OF VARIABLES If we write down the Laplace equation 2 V=0 in spherical

SEPARATION OF VARIABLES If we write down the Laplace equation 2 V=0 in spherical polar coordinates (r, , ), it turns out that the solutions can be written in the form Vnm(r, , ) = Rn(r) nm ( ) m( ) Here the 'degree' n is an integer, which specifies the smallest length-scale of the Vnm, and the 'order' m is another integer (with |m|<=n) that specifies how many zeros there as we go from 0° to 360° in longitude . 3

Separation of Variables (b) Vnm(r, , ) = Rn(r) nm ( ) m( )

Separation of Variables (b) Vnm(r, , ) = Rn(r) nm ( ) m( ) = Rn(r) Snm ( , ) Here Rn(r) = (r/r 0)n or Rn(r)=(r/r 0)-(n+1) is a simple function only of r/r 0 nm( ) = Pnm(cos ) is a simple function only of m( ) = cos m or sin m is a simple function only of and Snm( , ), the usual surface harmonic, is just a shorthand notation for the product nm( ). 4

SPHERICAL HARMONIC REPRESENTATION OF POTENTIAL Each of these Vnm(r, , ) is called a

SPHERICAL HARMONIC REPRESENTATION OF POTENTIAL Each of these Vnm(r, , ) is called a 'spherical harmonic', and we can represent (almost) any real Laplacian potential V(r, , ) by means of the (infinite) sum V(r, , ) = r 0 n, m gnm Vnm(r, , ) = r 0 n, m gnm Rn(r) Snm( , ). Here r 0 is the radius for which the Gauss coefficients gnm are defined. 5

IGRF CALCULATOR 6

IGRF CALCULATOR 6

DETERMINATION OF THE gnm For a set of measurements of a given potential V(r,

DETERMINATION OF THE gnm For a set of measurements of a given potential V(r, , ), how do we estimate these gnm? In practice we use some sort of Least Squares' fitting, solving a large set of simultaneous equations for all the gnm together. In this case, if we truncate the series (make some coefficients zero) then the values we get for the coefficients we do solve for might well be changed. But if V(r, , ) is known over a sphere about the origin, we can determine ALL the gnm INDIVIDUALLY! 7

ORTHOGONALITY OF SPHERICAL SURFACE HARMONICS OVER THE SPHERE We can do this because the

ORTHOGONALITY OF SPHERICAL SURFACE HARMONICS OVER THE SPHERE We can do this because the SPHERICAL surface harmonics Snm( , ) are mutually ORTHOGONAL over the sphere: sphere. Snm( , ) SNM( , ) d. Asphere = 0 unless n=N and m=M 8

SPHERICAL HARMONIC ANALYSIS Multiply the observed potential V(r, , ) by one particular surface

SPHERICAL HARMONIC ANALYSIS Multiply the observed potential V(r, , ) by one particular surface harmonic, SNM( , ) say, and integrate over a sphere, say r=r 0. Then V(r 0, , ) SNM( , ) d. A = [r 0 n, m gnm Snm( , )] SNM( , ) d. A. Because of the orthogonality, only the g. NM term is non-zero, and we find For Schmidt semi-normalised Legendre functions [Snm( , )]2 d. A = 4 r 02/(2 n+1) 9

This Spherical Harmonic Analysis (SHA) is the 2 -dimensional spherical equivalent of a 1

This Spherical Harmonic Analysis (SHA) is the 2 -dimensional spherical equivalent of a 1 dimensional Fourier Analysis round a circle. Of course except in theoretical problems we never know V(r 0, , ) over the whole surface of the sphere, but given values at discrete points we can use some numerical approximation to integration. The better our approximation is to a uniform distribution over the sphere, then the more nearly will our estimates of the g. NM be independent of those for other n, m. In practice the usual Least Squares approach is very similar to this numerical-approximation 10 classical SH analysis approach.

M. S. POTENTIAL OVER SPHERE What is the mean-square potential <V 2> over the

M. S. POTENTIAL OVER SPHERE What is the mean-square potential <V 2> over the sphere of reference radius r 0 ? V 2 d. A= ( n, m r 0 gnm Vnm)( n, mr 0 gnm Vnm)d. A. Because of the orthogonality all the cross terms vanish, and we have <V 2> = n, m(r 0 gnm)2<Vnm. Vnm> = n, m(r 0 gnm)2 <[Snm( , )]2> at r=r 0 = n, m(r 0 gnm)2/(2 n+1). for Schmidt semi-normalised Legendre functions 11

M. S. Potential over Sphere (b) <V 2> = n, m(r 0 gnm)2/(2 n+1)

M. S. Potential over Sphere (b) <V 2> = n, m(r 0 gnm)2/(2 n+1) However, for terms of a given degree n, the division between the cos m and sin m terms depends on the arbitrary choice of the Greenwich meridian for the origin of longitude. And in this spherical case, for a given physical potential the TOTAL potential of degree n is INDEPENDENT of our choice of the rotation axis as the =0 axis. Also, all the harmonics of a given degree n have the same minimum wavelength, and vary with radius r in the same way. 12

M. S. Potential over Sphere (c) <V 2> = n, m(r 0 gnm)2/(2 n+1)

M. S. Potential over Sphere (c) <V 2> = n, m(r 0 gnm)2/(2 n+1) So it is usual to collect together all the terms of degree n to give the (spatial) degree "Spectrum": Rn = m(r 0 gnm)2/(2 n+1). This is essentially the 'Degree Variance' of geodesy. 13

OUR DATA ARE MAGNETIC FIELD B, NOT POTENTIAL V In geodesy the data are

OUR DATA ARE MAGNETIC FIELD B, NOT POTENTIAL V In geodesy the data are effectively of the potential. But in geomagnetism our data are in the form of values of the VECTOR FIELD B=-grad V, rather than of the potential itself. However, if we put Bnm(r, , ) = -r 0 grad Vnm(r, , ), then these individual spherical VECTOR harmonics Bnm are also orthogonal over any sphere: sphere. Bnm(r, , ) ·BNM(r, , ) d. Asphere = 0 unless n=N and m=M. 14

"Mean-Square Values on Sphere of Spherical Harmonic Vector Fields" J. Geophys. Res. 1966 <Bnm

"Mean-Square Values on Sphere of Spherical Harmonic Vector Fields" J. Geophys. Res. 1966 <Bnm ·Bnm> = (n+1) 15

SPHERICAL HARMONIC ANALYSIS OF MAGNETIC FIELD We can therefore extend all the results we

SPHERICAL HARMONIC ANALYSIS OF MAGNETIC FIELD We can therefore extend all the results we had for potential to our magnetic field. If we put B(r 0, , ) = n, m gnm Bnm(r 0, , ), multiply by BNM(r 0, , ), and integrate over the surface of the sphere r=r 0 then because of the orthogonality we find For Schmidt semi-normalised Legendre functions [Bnm(r 0, , )]2 d. A = 4 r 02(n+1) 16

MEAN-SQUARE MAGNETIC FIELD What is the mean-square vector field <B·B> over the sphere of

MEAN-SQUARE MAGNETIC FIELD What is the mean-square vector field <B·B> over the sphere of reference radius r 0 ? As for potential, because of the orthogonality all the cross terms vanish, and we have <B·B> = n, m(gnm)2<Bnm·Bnm> = n, m(gnm)2(n+1)(2 n+1) <[Snm( , )]2> = n, m(gnm)2(n+1) for Schmidt semi-normalised Legendre functions. 17

SPATIAL POWER SPECTRUM <B ·B> = n, m(gnm)2(n+1) Again it is usual to collect

SPATIAL POWER SPECTRUM <B ·B> = n, m(gnm)2(n+1) Again it is usual to collect together all the terms of degree n, to give the degree "Spatial Power Spectrum": <B ·B> = n Rn where Rn = m(gnm)2(n+1)(2 n+1) <[Snm( , )]2> = m(gnm)2 (n+1) for Schmidt semi-normalised Legendre functions. 18

Geophys. J. R. astr. Soc. 1974 19 "Fig. 5 Idealised surface power spectrum for

Geophys. J. R. astr. Soc. 1974 19 "Fig. 5 Idealised surface power spectrum for 1965"

Magnetic Power Spectrum (d) Olsen et al. 2010 20

Magnetic Power Spectrum (d) Olsen et al. 2010 20

THE EARTH IS SPHEROIDAL NOT SPHERICAL 21

THE EARTH IS SPHEROIDAL NOT SPHERICAL 21

The Earth is Spheroidal (b) From icsm. gov. au However the Earth is NOT

The Earth is Spheroidal (b) From icsm. gov. au However the Earth is NOT a sphere, but is much more nearly an oblate SPHEROID. Because of its spin the distance to the centre increases by 21 km as we go from pole to equator. 22

The Earth is Spheroidal (c) For the long-wavelength field from the core, or from

The Earth is Spheroidal (c) For the long-wavelength field from the core, or from the magnetosphere, this 21 km change from pole to equator does not matter. We just use spherical harmonics to calculate the field at the correct 23 location.

The Earth is Spheroidal (d) But if we are interested in the short wave-length

The Earth is Spheroidal (d) But if we are interested in the short wave-length field near the surface produced by the crustal rocks, this 21 km variation in radius is NOT small compared with the shorter wavelengths. If we try to use the techniques devised for spherical harmonic analysis, we have first to extrapolate the (imperfect) observations outward on to a sphere; for short-wavelength fields this can add large errors. In fact our colleagues in geodesy have had this problem 24 for a long time.

The Earth is Spheroidal (e) 1°=111 km 25

The Earth is Spheroidal (e) 1°=111 km 25

The Earth is Spheroidal (f) Workers in geodesy have used increasingly complicated methods to

The Earth is Spheroidal (f) Workers in geodesy have used increasingly complicated methods to extrapolate from the spheroid on to a sphere, but to avoid loss of accuracy they really needed a method of analysis in which the SPHEROIDAL surface corresponds to a CONSTANT value of the 'size' coordinate. This means using SPHEROIDAL geometry, and hence the spheroidal harmonics that are the spheroidal equivalent of the spherical harmonics. 26

The Earth is Spheroidal (g) The problem is that these SPHEROIDAL harmonics are MUCH

The Earth is Spheroidal (g) The problem is that these SPHEROIDAL harmonics are MUCH more difficult to compute. And each harmonic involved an iteration in which there were serious difficulties with convergence for large degree n. Eventually, Jekeli's geodesy group at AFGL found a re-scaling that gave convergence, and pioneered the use of SPHEROIDAL harmonics in 1988. 27

The Earth is Spheroidal (h) 28

The Earth is Spheroidal (h) 28

The Earth is Spheroidal (i) n. T Maus G 3 2010 We are now

The Earth is Spheroidal (i) n. T Maus G 3 2010 We are now having the same problem in geomagnetism. So Maus (2010) has analysed the crustal field in terms of SPHEROIDAL harmonics, up to degree n=719, and it is these harmonics that I 29 want to talk about now.

CONFOCAL SPHEROIDS z x, y u (u 2+E 2) If the symmetry axis is

CONFOCAL SPHEROIDS z x, y u (u 2+E 2) If the symmetry axis is the z-axis, the equation of an oblate spheroid is where u is the semi-minor axis, and E is the focal distance. 30

SPHEROIDAL COORDINATES (u, , ) Semi-minor axis of spheroid u (u=u 0 is the

SPHEROIDAL COORDINATES (u, , ) Semi-minor axis of spheroid u (u=u 0 is the reference spheroid) Reduced colatitude (on sphere) Longitude Auxiliary (circumscribing) sphere P P' Spheroid 31 The point P is the axial projection of the point P'

SPHEROIDAL HARMONICS In these particular SPHEROIDAL coordinates, solutions of Laplace's equation take the form

SPHEROIDAL HARMONICS In these particular SPHEROIDAL coordinates, solutions of Laplace's equation take the form V'nm = Unm(u) Snm( , ), where Unm(u) is a known function of u, with Unm(u) = 1 for u = u 0, Snm( , ) = nm( ) has the same and form as for spherical surface harmonics. But note that now the angle is not the colatitude of the actual point on the spheroid, but the colatitude when that point is PROJECTED ON TO THE AUXILIARY SPHERE. 32

Spheroidal Harmonics (b) Note that in V'nm = Unm(u) Snm( , ) (1) The

Spheroidal Harmonics (b) Note that in V'nm = Unm(u) Snm( , ) (1) The prime on V'nm shows it refers to the SPHEROIDAL situation. (2) The angle is not the colatitude of the actual point on the spheroid, but the colatitude when that point is PROJECTED axially on to the auxiliary sphere. (3) The 'radial' function Unm(u) depends on the order m as well as the degree n. It is MUCH 33 more complicated to compute!

Spheroidal Harmonics (c) As for spherical harmonics, we can specify an arbitrary potential V(u,

Spheroidal Harmonics (c) As for spherical harmonics, we can specify an arbitrary potential V(u, , ) in terms of the sum V(u, , ) = r 0 n, m g'nm V'nm(u, , ) = r 0 n, m g'nm Unm(u) Snm( , ). The coefficients g'nm can be estimated by least squares, and this is what Maus (2010) did. His data were ON the spheroid u=u 0 ; no extrapolation in 'radius' was needed, so there was no loss of accuracy in the fitting. 34

NON-ORTHOGONALITY OF SPHEROIDAL HARMONICS But these Snm( , ), and hence the V'nm, are

NON-ORTHOGONALITY OF SPHEROIDAL HARMONICS But these Snm( , ), and hence the V'nm, are NOT orthogonal over the SPHEROID: spheroid Snm( , )SNM( , ) d. Aspheroid 0 This is because the element of area d. Aspheroid introduces an extra function of . 35

Non-orthogonality (b) So many of the things we take for granted with spherical harmonics,

Non-orthogonality (b) So many of the things we take for granted with spherical harmonics, are NOT valid for SPHEROIDAL harmonics. For example: It is NOT possible to determine the spheroidal harmonic coefficients g'nm separately. It is NOT possible to separate the meansquare potential, or the mean-square vector field, as the sum of contributions from each degree. For the Earth, having only small eccentricity, the effect is not large, only about 1% for low degree harmonics, but its effects can become important 36 at high degrees.

USE WEIGHTING TO REGAIN ORTHOGONALITY For potential, orthogonality CAN BE RESTORED if we weight

USE WEIGHTING TO REGAIN ORTHOGONALITY For potential, orthogonality CAN BE RESTORED if we weight the integration over the spheroidal surface by a simple geometrical factor, W'(u, ), of reduced colatitude: W'(u, ) = [(u 2+E 2)/(u 2+E 2 cos 2 )]1/2. This weighting factor gives unit weight at the poles, and (slightly) more weight at the equator, in such a way as to restore orthogonality: spheroid W'(u, ) Snm( , )SNM( , ) d. Aspheroid = 0 unless n=N and m=M 37

GEOMETRICAL INTERPRETATION OF WEIGHTING In this case of potential there is a simple geometrical

GEOMETRICAL INTERPRETATION OF WEIGHTING In this case of potential there is a simple geometrical interpretation of the effect of the weight W'(u, ): d. A ( , ) sphere d. A'spheroid( , ) W'(u, ) d. A'spheroid( , ) = d. Asphere( , ). 38

PROJECTING ON TO THE AUXILIARY SPHERE So when dealing with potential on the spheroid

PROJECTING ON TO THE AUXILIARY SPHERE So when dealing with potential on the spheroid we can use a simple trick: spheroid W'(u, )spheroid F'nm( , ) d. A'spheroid = spheroid F'nm( , ) spheroid d. Asphere = sphere F'nm( , )projected d. Asphere. If we project the potential values F'nm( , ) axially from the spheroid on to the sphere, and then integrate over the sphere, this is equivalent to doing the WEIGHTED integration over the 39 spheroid.

Projecting on to the Auxiliary Sphere (b) spheroid W'(u, ) Snm( , )SNM( ,

Projecting on to the Auxiliary Sphere (b) spheroid W'(u, ) Snm( , )SNM( , ) d. A'spheroid = sphere Snm( , ) SNM( , ) d. Asphere= 0 unless n=N and m=M This orthgonality leads to which uses only SPHERICAL calculations to produce the Gauss coefficients for SPHEROIDAL harmonics! 40

Projecting on to the Auxiliary Sphere (c) This geometrical projection on to the sphere

Projecting on to the Auxiliary Sphere (c) This geometrical projection on to the sphere is in effect what some geodesists have done for a long time, but I suspect they did not know why! It was in 1988 that the geodesists Jekeli and Gleeson sorted out the difference between spheroidal and spherical Gauss coefficients, and even now I am not convinced that all the users realise that what they are doing is in effect using weighting! 41

SPHEROIDAL VECTOR HARMONICS But what we measure and analyse in geomagnetism is NOT the

SPHEROIDAL VECTOR HARMONICS But what we measure and analyse in geomagnetism is NOT the potential V(u, , ), but the vector magnetic field B(u, , ) that is the GRADIENT of this potential. Taking this gradient introduces yet more functions of (u, , ), which complicates the situation even further! As with the spheroidal potential harmonics V'nm(u, , ), it turns out that the corresponding spheroidal vector harmonics B'nm(u, , = -r 0 grad V'nm( , ) are also NOT orthogonal over the spheroid! 42

WEIGHTED ORTHOGONALITY OF MAGNETIC FIELD HARMONICS But, as with the spheroidal potentials, the orthogonality

WEIGHTED ORTHOGONALITY OF MAGNETIC FIELD HARMONICS But, as with the spheroidal potentials, the orthogonality CAN BE RESTORED by using an appropriate geometrical weighting factor. But now this weighting factor is 1/W'(u, ), the INVERSE of the weighting function needed for potential: spheroid(1/W') B'nm( , ) ·B'NM( , ) d. Aspheroid = 0 unless n=N and m=M. This (different) weighting gives unit weight at the poles, and 0. 9933 at the equator. 43

In this case there is NO geometrical trick to help out; the integration has

In this case there is NO geometrical trick to help out; the integration has to be done over the SPHEROIDAL surface, with the weighting applied EXPLICITLY. (It is probably not at all useful in applications, but the spheroidal vector field harmonics ARE orthogonal when integrated over the VOLUME of the ANNULUS between two confocal spheroids!) 44

CONVERSION FROM SPHEROIDAL to SPHERICAL GAUSS COEFFICIENTS The ANALYSIS is made using SPHEROIDAL harmonics,

CONVERSION FROM SPHEROIDAL to SPHERICAL GAUSS COEFFICIENTS The ANALYSIS is made using SPHEROIDAL harmonics, so as to avoid losing accuracy in the high degree coefficients. These spheroidal coefficients g'nm completely define the potential/field in the external region. This external field can equally well be specified by a set of SPHERICAL harmonics, having the appropriate spherical coefficients gnm : where w is the largest integer (n-m)/2. 45

Conversion (b) In practice, for the small ellipticity of the Earth, only 10 terms

Conversion (b) In practice, for the small ellipticity of the Earth, only 10 terms are needed for each series, and the coefficients Knmk are known. So in both geodesy and geomagnetism, AFTER analysis the conversion is made from the set of spheroidal coefficients to the equivalent set of spherical coefficients; all subsequent SYNTHESIS of fields is done using this set of SPHERICAL Gauss coefficients. 46

MAGNETIC SPECTRUM FOR POTENTIAL To properly use the concept of a spectrum we need

MAGNETIC SPECTRUM FOR POTENTIAL To properly use the concept of a spectrum we need orthogonality - so we need to use WEIGHTED mean-squares. For potential this is easy. At u=u 0 we have, for the weighted m. s OVER THE SPHEROID u=u 0: =1/(2 n+1) 47. for Schmidt semi-normalised Legendre functions

Magnetic Spectrum for Potential (b) So for the potential on the spheroid u=u 0

Magnetic Spectrum for Potential (b) So for the potential on the spheroid u=u 0 we can define the degree spatial spectrum R'n (u 0) = m(r 0 g'nm)2/(2 n+1). Again, all the degree-n harmonics have a common minimum wavelength of about 2 r 0/n. And the cos/sin m difference is not fundamental. BUT for these spheroidal harmonics, even within a given degree n, the 'radial' variation is different for different m; this degree n potential will have a DIFFERENT geometry at different values of u. So such a 'degree power spectrum' is not so useful in the spheroidal case. 48

MAGNETIC SPATIAL POWER SPECTRUM FOR FIELD This is more complicated. Because the weighting is

MAGNETIC SPATIAL POWER SPECTRUM FOR FIELD This is more complicated. Because the weighting is different, there is no geometrical simplification, and the weighted m. s. vector field is <(1/W')B·B>spheroid = n, m(g'nm)2 Dnm(n+1)2/(2 n+1). The constants Dnm (about unity for the Earth) are known algebraically, but have yet to be calculated. However the equivalent constants for the VERTICAL component Z have been calculated by Maus (2010), so I have been able to calculate the weighted Z degree power spectrum for the Maus 49 spheroidal analysis of the crustal field:

Spheroidal Spectrum over Spheroid 50

Spheroidal Spectrum over Spheroid 50

Spherical Spectra One can also use the SAME external field, but now given in

Spherical Spectra One can also use the SAME external field, but now given in terms of SPHERICAL harmonics, to produce the UNWEIGHTED Z (RADIAL) degree spectra over SPHERES of various radii. (Formally the sphere should be physically completely OUTSIDE the spheroid. I will ignore this restriction, and simply extrapolate the known EXTERNAL field inwards. ) First I add the spatial spectrum at the geomagnetic reference radius r 0. 51

Spheroidal/Spherical Spectra 52

Spheroidal/Spherical Spectra 52

SPHERICAL SPECTRA (b) This geomagnetic reference radius is very nearly the same as the

SPHERICAL SPECTRA (b) This geomagnetic reference radius is very nearly the same as the geodesists' mean radius. In this spherical situation it is very easy to go to different radii - now I add the spectra for the polar and equatorial radii. That for the polar radius is completely valid, as the surface everywhere outside the sources, but the surface for the polar radius is entirely within the source region! 53

Spheroidal/Spherical Spectra 54

Spheroidal/Spherical Spectra 54

Spheroidal/Spherical Spectra I would argue that the spherical spectra are NOT relevant for short-wavelength

Spheroidal/Spherical Spectra I would argue that the spherical spectra are NOT relevant for short-wavelength sources near the Earth's surface; for these sources only a spheroidal spectrum is appropriate. (Once the set of constants (equivalent to Dnm) appropriate to the total vector field has been calculated, it will be valid for all future spheroidal analyses for the Earth. ) 55

SUMMARY Using an appropriate coordinate system, one can still get separation of variables in

SUMMARY Using an appropriate coordinate system, one can still get separation of variables in the spheroidal situation, and estimate the values of the g'nm. NEITHER the V'nm(u, , ) NOR the B'nm(u, , ) are orthogonal over the SPHEROIDAL surface. HOWEVER one CAN use a geometrical weighting function, different in each case, to restore orthogonality. So one CAN make a spheroidal harmonic analysis, accurate to high degree, of data on the spheroid 56 AND produce a power spectrum on the spheroid.

Any questions? 57

Any questions? 57