Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 88

1,0) and (0,1) the canonical cubics each obey four condi-
tions that determine them uniquely, and we have

On the subintervals ( −

⎩

1) 2 (1

L 3 2

η

0 (ξ)

=− 2ξ − 3ξ + 1 = (ξ + − 2ξ), − 1 ≤ ξ ≤ 0

1) 2 (2ξ +

η 0 (ξ) =

0 (ξ) 3 2

(1.570)

η = 2ξ − 3ξ + 1 = (ξ − 1), 0 ≤ ξ ≤ 1

0, otherwise,

⎩

L 3 2

1) 2 ψ

0 (ξ)

= ξ + 2ξ + ξ = ξ(ξ + , − 1 ≤ ξ ≤ 0

1) 2

ψ 0 (ξ) =

0 (ξ) 3 2

(1.571)

ψ = ξ − 2ξ + ξ = ξ(ξ − , 0 ≤ ξ ≤ 1

0, otherwise.

The four cubics used in the approximating function (1.569) are then defined by

R R L L

η

i ( x )

= η

0 (ξ i ), η i + 1 ( x )

= η

0 (ξ i + 1 ),

(1.572)

1
ξ i ψ

1
ξ i + 1 ψ

i ( x ) 0 (ξ i ),ψ L

i +

0 (ξ i + 1 ),

= =

ψ

1 ( x )

(1.573)

with

x i
x i + 1 − x i , i + 1

x − x i + 1

x i + 1 − x i ,

x −

ξ i = = (1.574)

1
ξ i =

1
ξ i + 1 =

x i + 1 − x i . (1.575)

The term ξ i maps the interval ( x i , x i + 1 ) onto the interval (0,1) of the ξ-axis, while
ξ i + 1 maps the interval ( x i , x i + 1 ) onto the interval ( − 1,0) of the ξ-axis.

1.6.3 Even or odd local basis functions
Although, traditionally, spherical harmonics have been used to represent scalar
and vector fields in geophysics because of Earth's nearly spherical shape, they are
global functions that are sometimes ine cient as basis functions. Local basis func-

tions, closely related to Hermite splines , can be constructed that are much more
e cient, and which can have globally even or odd properties directly incorpor-

ated. They are found to be particularly useful in the calculation of long-period core
modes, where Coriolis coupling severely limits the e ciency of spherical harmon-

ics (see Chapter 8). In that application, the modes are known to be either purely
even or purely odd in the equatorial plane, and these properties can be directly
incorporated in the basis functions.
If z cosθ, where θ is the co-latitude, we wish to represent a function f ( z )on

the z -axis. On the interval ( z i , z i + 1 ), if the polynomial approximating function s ( z )
is to match the function and derivative values, f i , f i + 1 , f i , f i + 1 , at the end points,

=