Surface Harmonics

Exercise 3.5 Graphically describe the potential and attraction of a uniform, thick-walled shell (inner radius a\ and outer radius CI2) along a line

6.3 Surface Harmonics

(a)

(b)

Spherical Harmonic Analysis f(6)

-/ -

o -

1 -

• ' • • • • p,

p

5

V

/ ^{y * \}

6 = 0

f_k (6) = 3/2 7/8P3 + 66/96

\

f(d) -J \\

-/ 1 \

Fig. 6.5. Approximation of the discontinuous function shown in Figure 6.4 with the sum of three Legendre polynomials, (a) Unweighted Legendre func- tions of degree 1, 2, and 3; (b) the weighted sum of the three Legendre func- tions.

d^r

(6.11) Notice that associated Legendre polynomials reduce to Legendre poly- nomials when m = 0. A few examples of the associated Legendre poly- nomials are as follows:

Pi,i =sin<9, P2 j l = § sin 20, P3A = f sin 0 (5 cos 2(9 + 3),

P2,2 = 3 sin² (9, P3j2 = M gin 6 sin 20, (6.12) P3,3 = 15 sin³ 6.

As promised, the associated Legendre polynomials are orthogonal over the interval - 1 < // < 1 and with respect to degree n, that is,

fO, *„*„';

(6.13)

2 ( n + m ), _{, II 77- — 72 .}

6.3 Surface Harmonics 111 Exercise 6.2 Try equation 6.13 for n = 4 and m — 1. Try n = 4 and m = 4.

The results of Exercise 6.2 illustrate the large differences in the mean values (i.e., when integrated over the interval — 1 < \x < 1) of the squares of associated Legendre polynomials for any particular degree n. Later, we will normalize these functions in order to make their relative importance more alike in any given series.

In the last section, Legendre polynomials were shown to be a suitable building block for functions that are independent of longitude, that is,

= CoPo(0) + CiPi(0) + C2P2(O)

The associated Legendre polynomials are more powerful in general be- cause they also depend on order ra, and this allows /(#, </>) to remain a function of (j) in equation 6.2,

f{O,<t>) = Y, (am{6) cosm<j>+ bm(6) sinm<j>) .

ra=O

In a later section of this chapter, we will see another important reason for switching to the associated Legendre functions.

Now we are in a position to rewrite equation 6.2, the original expansion of /(#, 0), using the associated Legendre polynomials. Similar to the derivation for the zonal harmonic expansion, we let

ao(0) =

bl T

^Bi,i

B²,2 b,o(6

-f2,2 P2,2

') + ' (6)-\

(0)^

(0)-

CiPi,o(O) h^2>1P2,i hS2,iP2,i

hB3 2P32

+ c

^{s!^2,0}

• ^ , :

•B4. (0)-

LP3,

1^3,

2P4

((X\

1 1 (7 1

and substitute these equations into equation 6.2 to get f{9, <f>) = C_oPo,o(0) + CiPi,_Q{6)

+ [Ai,iPu(0) + A2,iP2,i(9) + A3>1P3tl(0)

+ [Bi,iPi,i(e) + B₂,iP₂,i(J) + B3.1P3.1W + • • •] sin^

+ [^2,2^2,2(0) + A3,2P3,2(0) + ^4,2^4,2(0) + • • •] COS 20 + [B2fiP2,2(^e) + ^B3,2Ps,2(°) + Bi,2P4,2(°) + ' ' '] ^{s i n 2(}P -\ •

Rearranging terms provides

COS 20 + ^2,2^2,2^) Sin 20]

which can be written

f(0,0) = J j CnPn,0(6>) + X ! (^An,m cosm0 + 5n,m sinm

n=0 •- m=l

(6.14) Hence, f(0, 0) is represented by an infinite sum of functions, each func- tion composed of associated Legendre polynomials, sines, and cosines.

For reasons that will become clear in Section 6.4, equation 6.14 is called a spherical surface harmonic expansion, and the functions Pn,m(6) cosm0 and Pn,m(0) sinm0 are called spherical surface harmon- ics. Notice that when m = 0, the spherical harmonic expansion reduces to a zonal harmonic expansion, as in equation 6.8. As we should expect, surface harmonics are orthogonal over the sphere; unless any two surface

6.3 Surface Harmonics 113 harmonics are identical, their product will average to zero over the sur- face of any sphere. For example,

o Pnm (0) cos vn(\)Pni m/ {&) cos ml'(p r2 sin 6 dO d(f) 47rrz J J

0 0

' 0 , if n ^ n' or m ^ m'\

2(2n%7n-my. > if n = n'and m = m ' ^ 0; (6.15) if n = n' and m = m! = 0,

and similarly if cos ra0 is replaced with sin mcj) amd cos TTI'C/) is replaced with sinra^ in equation 6.15.

6.3.1 Normalized Functions

As illustrated in Exercise 6.3, the magnitude of an associated Legendre polynomial depends on its degree and order, so the magnitude of each coefficient must compensate accordingly. A spherical harmonic analysis would be more instructive if the magnitude of each coefficient reflected the relative significance of its respective term in the expansion.

This can be accomplished by normalizing the associated Legendre functions. Two normalizing schemes are in common usage. The fully normalized functions, commonly used in geodetic studies, are related to the unnormalized Legendre polynomials by

P™{6) =

In geomagnetic studies, the Schmidt functions are more typical, and these are given by

ifm = 0;

Rewriting equation 6.14 but using, for example, the Schmidt functions yields

OO r U

« W + E W cosm0 + B% sin m^P^(

m = l

(6.16)

Table 6.2. Surface harmonics of degree and order 0 through 3 expressed in terms of Schmidt functions.

n 0 1 1 2 2 2 3 3 3 3

m 0 0 1 0 1 2 0 1 2 3

Normalized Surface Harmonic 1

COS0

|(3cos20 + l) = §cos²6>- \

^- sin 2$ 1 ^ | 0 = v 3 sin 0 cos 0 | ^ | 0

^sin²0{-_n^s}20

|(5cos30+ 3cos0) = f cos³0- f cos0

^ sin 0(5 cos 20 + 3) { ™ } 4> = ^ (5 cos² 0

^sin0sin20{^}2(/>= ^psin²0cos0{

^P sin³ 0 { ^c°* } 30

- l ) s i n 0 { -^s} 0

" n } ²<t>

The magnitude of Schmidt surface harmonics, when squared and aver- aged over the sphere, are independent of their order, that is,

2n ir

ft™

0 0

{

0, if n / n' or m ^ m⁷:

•f / H / ^{( 6}-^{1 7 )}

2^pj , if n = n and m = m .

Hence, the magnitudes of the coefficients A™ and I?^¹ quickly indicate the relative "energy" of their respective terms in the series. Schmidt functions are commonly used in global representations of the geomag- netic field.

A few low-degree Schmidt functions are shown in Figure 6.6, and Ta- ble 6.2 shows several low-degree surface harmonics based on the Schmidt normalization. Subroutine B.4 in Appendix B provides a Fortran algo- rithm, modified from Press et al. [233], to calculate normalized associated Legendre functions.

6.3 Surface Harmonics 115

fi=-l

Fig. 6.6. Normalized (Schmidt) surface harmonics of degree 6 and order 0 through 6.

Just as for zonal harmonic functions, the coefficients A™ and B™ can be found from measurements of /(#, 0) using the orthogonality property:

0 0

In practice, this calculation can be done in two steps, by first numerically integrating the data over <j> to find the coefficients am(^) and &m(^), and then integrating over 0. The first step amounts to a Fourier series expansion. Equipped with A™ and B™, /(#, (j>) can be represented as the infinite sum of weighted surface harmonic functions by equation 6.16.

6.3.2 Tesseral and Sectoral Surface Harmonics The normalized surface harmonic

p™rm J

c o s m (

H

{

sin

vanishes along (n — m) circles of latitude that correspond to the zeroes of P™(6). It also vanishes along 2m meridian lines from 0 to 2?r due to the sin mcj) or cos ra0 term. The lines of latitude and meridian along which the normalized surface harmonics vanish divide the spherical surface into

116

cos

Pg COS 3<t>

Tesseral

Fig. 6.7. Specific examples of zonal, sectoral, and tesseral surface harmonics.

patches of alternating sign. If TTT, = 0, the surface harmonic only depends on latitude and is called a zonal harmonic. If n — m = 0, it depends only on longitude and is called a sectoral harmonic (like the "sectors" of an orange). If m > 0 and n — m > 0, the harmonic is termed a tesseral harmonic. Specific examples of each of these three types of normalized surface harmonics are shown in Figure 6.7.

As indicated by equations 6.14 and 6.16, any reasonably well-behaved function can be represented by an infinite sum of zonal, sectoral, and tesseral patterns, each weighted by an appropriate coefficient A™ or B™, as shown by equation 6.16. This sum is just a three-dimensional analog of Fourier series, in which f(t) also is represented by an infinite sum of patterns (sinusoids in the Fourier case) multiplied by appropriate coefficients.