An alternative model has been adapted in this work. The main requirements of this approach are primarily defined by the disadvantages of the previously presented parametrization techniques:

1. The base functions shall only be different to zero in a local environment on the sphere to allow for the modification of present data or incorporation of new data into the model without causing a global effect.

2. The representation shall be continuous within the modeling region.

For this purpose, polynomial and trigonometric B-splines with local support have been selected as
appropriate base function candidates for representing the ionospheric information derived from
space-geodetic observations in the model space. In contrast to SH, which are defined on a sphere and
related to latitude and longitude via the associated Legendre polynomials, B-spline base functions are
commonly defined in the Euclidean spaceR^{2} and therefore need to be constrained in case of global
modeling. However, since space observations for ionosphere modeling are typically available with
heterogeneous distribution and quality, the feature of compact support clearly outweighs the issue of
applying global constraints. Furthermore, the selected B-spline functions allow to generate a
Multi-Resolution Representation (MRR) as shown by Lyche and Schumaker (2000) and Schmidt (2007)
where the target parameters such as VTEC or Necan be decomposed by successive low-pass filtering
into a certain number of detail signals. Non-significant information can then be neglected facilitate
data compression. A detailed comparison between the spherical harmonic and B-splines for modeling
VTEC has for instance been performed by Schmidt et al. (2011) and shall not be the repeated here. For

2.3. B-splines 33 the mathematical background of B-splines, it shall be referenced to Schumaker (1981), Schumaker and Traas (1990), Lyche and Schumaker (2000), Jekeli (2005) and Zeilhofer (2008).

The following sections focus on the use of B-splines for regional ionosphere modeling in accordance to P-I: Limberger et al. (2013) and P-II: Limberger et al. (2014) but besides, guidance for an ap-plication in the global domain is provided. In a first step, B-spline functions are introduced from a generally valid perspective before adapting the basis for ionosphere modeling in the following Section 2.4.

For the case of a 1-D representation, the approximation function s(x) with x ∈ [x_{min},xmax] is now
expressed as

s(x) =

K

X

k=1

d^{J}_{k} Ψ_{k}^{J}(x). (2.13)

Ψ identifies a linearly independent set of 1-D scaling functions Ψ_{1}, . . . ,ΨK of level J and dk are
associated series coefficients.

Two different kinds of B-spline base functions are introduced in the following Sections, namely the normalized quadratic polynomial B-splines and normalized periodic trigonometric B-splines. Both offer excellent features for ionosphere modeling in the regional and global domain.

2.3.1 Normalized quadratic polynomial B-splines

For regional modeling applications, we intend to consider normalized quadratic polynomial B-splines
denoted by φ^{J}_{k}^{φ}(x) = N_{k,m}^{J}^{φ} (x) as 1-D basis functions for representing the signal within a bounded
interval. N_{k,m}^{J}^{φ} are usual normalized B-splines of ordermwith

Kφ

X

k=1

N_{k,m}^{J}^{φ} (x) =1 where x ∈[0,1] (2.14)

where Jφ ∈ N0 defines the B-spline resolution level and k ∈ {1,2. . .Kφ} identifies a specific spline
function. k is sometimes denoted as shift, e.g., in Schmidt et al. (2015). The model resolution
is controlled by the level, i.e., the higher J_{φ}, the finer the signal structures that can be resolved.

The total number of B-splines is computed from K_{φ} = 2^{J}^{φ} + m − 1. The basis is deployed by
an increasing sequence of Kφ + m so-called knot points ν_{k}^{J}^{φ} ∈ {ν_{1}^{J}^{φ}, ν_{2}^{J}^{φ}, . . . , ν_{K}^{J}^{φ}_{φ}_{+}_{m}} where, at the
boundaries, multiple knots may be linked to a specific coordinate point. The knot intervalν_{k}^{J}^{φ}_{+1}−ν_{k}^{J}^{φ}
must not mandatory be constant. The basis for normalized quadratic polynomial B-splines is defined
recursively (Schumaker and Traas, 1990; Stollnitz et al., 1995) with

N_{k,1}^{J}^{φ}(x) =

1 ν_{k}^{J}^{φ} ≤ x < ν_{k}^{J}^{φ}_{+1}

0 otherwise , k = m, . . . ,Kφ (2.15)

N_{k,m}^{J}^{φ} (x) = x−ν_{k}^{J}^{φ}

ν_{k}^{J}^{φ}_{+}_{m−1}−ν_{k}^{J}^{φ}N_{k,m−1}^{J}^{φ} (x)+ ν_{k}^{J}^{φ}_{+}_{m}− x

ν_{k}^{J}_{+}^{φ}_{m} −ν_{k}^{J}^{φ}_{+1}N_{k}^{J}_{+1,m−1}^{φ} (x), m ≥ 2. (2.16)
A uniform knot sequence is established as

0= ν_{1}^{J}^{φ} = · · ·= ν_{m}^{J}^{φ}

| {z }

Boundary multiplicity

< ν_{m}^{J}^{φ}_{+1}< · · · < ν_{K}^{J}^{φ}_{φ} <

| {z }

Internal sequence

ν_{K}^{J}^{φ}_{φ}_{+1}= · · ·= ν_{K}^{J}^{φ}_{φ}_{+}_{m} =1

| {z }

Boundary multiplicity

(2.17)

34 Chapter 2. Parametrization
whereKφ−m+2 distinct knots are taken into account. At the boundaries there is a multiplicity ofm
knots each that allow for the endpoint-interpolation. The knot distanceν_{k}^{J}^{φ}_{+1}−ν_{k}^{J}^{φ} yields consequently
Kφ−m+1−1

on the unit interval.

Normalized quadratic polynomial B-spline functions φ^{J}_{k}^{φ}(x) = N_{k,3}^{J}^{φ}(x) of order m = 3 have been
considered by P-I: Limberger et al. (2013); P-II: Limberger et al. (2014) and P-V: Liang et al. (2014);

P-VI: Liang et al. (2015). In case ofm =3, K_{φ} = 2^{J}^{φ} +2 normalized B-spline functions anchored at
Kφ+3 knot points are distributed on the unit interval.

0.0 0.2 0.4 0.6 0.8 1.0

x 0.0

0.2 0.4 0.6 0.8 1.0

Φ(x)

0.0 0.2 0.4 0.6 0.8 1.0

x 0.0

0.2 0.4 0.6 0.8 1.0

Φ(x)

0.0 0.2 0.4 0.6 0.8 1.0

x 0.0

0.2 0.4 0.6 0.8 1.0

Φ(x)

0.0 0.2 0.4 0.6 0.8 1.0

x 0.0

0.2 0.4 0.6 0.8 1.0

Φ(x)

Figure 2.2: Normalized quadratic polynomial B-splines with different levels J_{φ} = 0,1,2,3 and accordingly different
number of B-splinesK_{φ}^{J} =3,4,6,10.

Figure 2.2 provides examples of polynomial B-spline functions of m = 3 regarding different levels
Jφ = 0,1,2,3. The number of splines varies withJφwhere the subplots are related toJφ =0→ K_{φ}^{0}=3
(top-left), Jφ = 1 → K_{φ}^{1} = 4 (top-right), Jφ = 2 → K_{φ}^{2} = 6 (bottom-left) and Jφ = 3 → K^{3}_{φ} = 10
(bottom-right). Special features of polynomial B-splines are in particular given by the

endpoint-interpolation, i.e., adaptation of the splines to a bounded interval, and localization, i.e., compact support only within a restricted interval.

It can be clearly seen from the plots, that the first two and last splines are different with respect to
the interior spline functions contributing to the endpoint-interpolation in the bounded interval. Each
spline differs from zero only within a certain interval according to the level J_{φ}. A comparison of
all four figures demonstrates how the B-spline support interval changes with Jφ, i.e., the higher the
level the smaller the influenced area and the sharper the peaks. For further details about the support
interval, it shall be referenced to Mößmer (2009) and Schmidt et al. (2011).

From Fig. 2.2 it becomes visible that always three adjacent splines are overlapping, i.e., a single data point contributes to the determination of exactly three B-spline coefficients. The model resolution is defined by Jφ should be adapted to the input data density to overcome data gaps. For the case of a homogeneous data sampling, Schmidt et al. (2011) derived the relation

∆si= si_{max}−si_{min}

K_{φ}−1 (2.18)

2.3. B-splines 35

with the sampling si on the interval [si_{min},si_{max}]. This formulation can be transformed to
J_{φ} <log_{2} si_{max}−si_{min}

∆si −m+2

!

(2.19)
under consideration ofKφ = 2^{J}^{φ}+m−1. The level needs to be adapted to the observations density to
minimize the data gaps and avoid the loss of significant information. Form= 3, the level can thus be
derived from

J_{φ} <log_{2} si_{max}−si_{min}−∆si

∆si

!

. (2.20)

2.3.2 Normalized periodic trigonometric B-splines

As a second base function type, normalized periodic trigonometric B-splines T_{k,m}^{J}^{T} (x) of odd order
m shall be introduced (Schumaker, 1981; Schumaker and Traas, 1990). Trigonometric B-splines
with resolution level JT ∈ N0 are defined on a circle in the closed interval [0,2π] and have no knot
multiplicity but comply with the constraint of s(0) = s(2π). The number of spline functions is
computed from KT = 3·2^{J}^{T} +m−1 where altogether K_{T}^{0} = KT −m+1 = 3·2^{J}^{T} complete splines
are distributed on the basis interval, meaning that the interrupted boundary splines are completed by
the corresponding opposing sub-spline to enable periodicity. The basis definition follows from

T_{k,1}^{J}^{T}(x) =

1 ν_{k}^{J}^{T} ≤ x < ν_{k}^{J}^{T}_{+1}

0 otherwise , k = 1, . . . ,KT (2.21)

T_{k,m}^{J}^{T} (x) =

sin ^{x−ν}

JT k

2

!

sin ^{ν}

JT
k+m−1−ν_{k}^{JT}

2

! ·T_{k,m−1}^{J}^{T} (x)+

sin ^{ν}

k+mJT −x 2

!

sin ^{ν}

JT
k+m−ν_{k}^{JT}_{+}_{1}

2

! ·T_{k}^{J}_{+1,m−1}^{T} (x), m≥ 2 (2.22)
with the non-decreasing sequence of distinct knots

0= ν_{m}^{J}^{T} < ν_{m}^{J}^{T}_{+1} < . . . < ν_{K}^{J}^{T}

T < ν_{K}^{J}^{T}

T+1=2π. (2.23)

Additional constraints on the knot placement
ν_{i}^{J}^{T} =ν_{K}^{J}^{T}0

T+i−2π
ν_{K}^{J}^{T}0

T+m+i = ν_{m}^{J}^{T}_{+}_{i}+2π fori= 1, ...,m−1 (2.24)

are introduced for Eq. (2.23) to force periodicity.

Choosing the same orderm = 3 as for the polynomial B-splines the number of trigonometric spline
functionsT_{k,3}^{J}^{T} can be determined asKT =3·2^{J}^{T} +2. Periodic trigonometric B-splines are particularly
suitable for global modeling applications due to special properties in the

definition on the [0,2π] interval with "wrapping-around" condition and

localization, i.e., compact support only within a restricted interval as also provided by the poly-nomial B-splines.

A set of trigonometric B-splines of m = 3 for different levels JT = 0 → KT = 5 (top-left), JT =1→ KT =8 (top-right), JT =2→ KT =14 (bottom-left) and JT =3→ KT = 26 (bottom-right) is provided by Fig. 2.3. Similarities to the polynomial B-spline functions can be found in the local support and the overlapping of three splines in each point along x. The main difference is related

36 Chapter 2. Parametrization

0 1 2 3 4 5 6

x 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4

T(x)

0 1 2 3 4 5 6

x 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

T(x)

0 1 2 3 4 5 6

x 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

T(x)

0 1 2 3 4 5 6

x 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

T(x)

Figure 2.3: Trigonometric B-splines with different levels JT = 0,1,2,3 and accordingly different number of B-splines KT =5,8,14,26.

to the boundary splines where, in contrast to the endpoint-interpolation of polynomial splines, the periodicity is visible. Although different colors have been chosen for each function, it can be clearly seen that every sub-spline at one boundary is continued at the opposite boundary. For instance in the top-left illustration for JT =0, the magenta spline on the right can be connected with the green spline on the left without any discontinuation.

Similarly to Eq. (2.19), the relation of the data density to the B-spline level can be derived from
JT < log_{2} si_{max}−si_{min}+∆si

3∆si

!

(2.25)
by substitutingK_{T}^{0} as the number of complete splines into Eq. (2.18).

2.3.3 B-spline tensor products SΩshall identify a unit sphere as

RSΩ := {(ϕ, λ) :−π

2 ≤ ϕ ≤ π

2 and 0 ≤ λ ≤ 2π} (2.26)

with polar coordinates ϕand λ in an angular system mapped on a rectangle RS_{Ω} in a 2-D spaceR^{2}.
The representation of a subsurface of S_{Ω}^{0} defined withinϕ ∈ [ϕ_{min}, ϕ_{max}] andλ ∈ [λ_{min}, λ_{max}] in the
rectangular domain can be obtained with

RS_{Ω}^{0} := {(ϕ, λ) :ϕmin ≤ ϕ ≤ ϕmaxandλmin ≤ λ ≤ λmax}. (2.27)
In order to represent multidimensional information on the rectangular modeling surfaces RSΩ or RS_{Ω}^{0}

as introduced by Eq. (2.26) and Eq. (2.27), tensor products of B-spline base functions shall be applied in an orthogonal coordinate system. Details about tensor product techniques for surface modeling can, e.g., be found in Dierckx (1984), Gmelig Meyling and Pfluger (1987) or Zeilhofer (2008).

2.3. B-splines 37

For a 2-D case representation, the approximation functionscan be constructed from
s(x_{1},x_{2}) =

K_{1}

X

k_{1}=1
K_{2}

X

k_{2}=1

d_{k}^{J}^{1}^{,J}^{2}

1,k_{2} Ψ_{1}^{J}^{1}

k1

(x_{1}) Ψ_{2}^{J}^{2}

k2

(x_{2}). (2.28)

Here, tensor products of two linearly independent 1-D B-spline functions Ψ_{1}^{J}^{1}

k1

and Ψ_{2}^{J}^{2}

k2

have been
introduced together with the corresponding series coefficientsd. It should be noticed thatΨ_{1}andΨ_{2}
may differ but can also be of the same type.

So far, the B-spline levels and numbers were expressed in relation to the B-spline type (JT, KT, J_{φ}, K_{φ}).

From now on, the identification will be based on indices (J_{1}, K_{1}, J_{2}, K_{2},. . . ) to distinguish between
B-splines of the same kind in the tensor product notation. Furthermore, the order will permanently be
considered asm= 3.

At first, polynomial B-splines as defined in Section 2.3.1 are chosen on both, the x_{1}and x_{2}interval,
with

x_{1}→ Ψ_{1} =φ^{2}_{k}

1 and x_{2} →Ψ_{2}= φ^{3}_{k}

2 (2.29)

for the levels J_{1} = 2 and J_{2} = 3. The corresponding basis is depicted in Fig. 2.4. Both plots show

Figure 2.4: Polynomial B-splinesφ^{2}_{k}

1(x_{1}) andφ^{3}

k2(x_{2}) of orderm =3 with different levels J_{1} =2 andJ_{2} =3. φ^{2}_{3}(x_{1})
(left),φ^{2}_{1}(x_{1}) (right) andφ^{3}_{5}(x_{2}) are emphasized to show the support area.

the support area spanned by the tensor product of two polynomial B-splines which are emphasized by
thick lines. As can be clearly seen from the left illustration of Fig. 2.4, an ellipse shaped support area
is spanned byφ^{2}_{3}(x_{1}) (red) andφ^{3}_{5}(x_{2}) (magenta). Choosing the same level in both directions naturally
would result in a circle shaped area. The subfigure on the right, exemplarily depicts the support area
at the boundary for φ^{2}_{1}(x_{1}) (blue) and φ^{3}_{5}(x_{2}) (magenta) constraint by endpoint-interpolation on the
x_{1}axis.

In a next step, the 2-D basis is generated from the combination of polynomial and trigonometric B-spline functions. The basis is defined as

x_{1}→ Ψ_{1} =T_{k}^{2}

1 and x_{2} →Ψ_{2}= φ^{3}_{k}

2 (2.30)

for different levels J_{1} = 2 and J_{2} = 3 with endpoint-interpolation in the x_{2}and continuity in the x_{1}
direction. Figure 2.5 shows the support area forT_{3}^{2}(x_{1}) andφ^{3}_{5}(x_{2}) in this spline constellation. The
right subplot additionally shows the 3-D shape of a 2-D tensor B-spline product basis related to the
emphasized splines of the left graph.

For each additional dimension a new set of basis functions can be incorporated, i.e., the B-spline ex-pansion can easily be adapted to higher or lower dimensions by means of the tensor product technique.

38 Chapter 2. Parametrization

Figure 2.5: Combination of trigonometric B-splines with level J_{1} = 2 and polynomial B-splines with level J_{2} = 3. A
specific spline combination identified byk_{1} =5 andk_{2} =8 has been highlighted and plotted in the center part of the left
subplot. Accordingly, a 3-D representation of the tensor product is given on the right hand side.

A 3-D modeling basis is accordingly defined as
s(x_{1},x_{2},x_{3})=

K_{1}

X

k_{1}=1
K_{2}

X

k_{2}=1
K_{3}

X

k_{3}=1

d_{k}^{J}^{1}^{,J}^{2}^{,J}^{3}

1,k_{2},k_{3} Ψ_{1}^{J}^{1}

k1

(x_{1}) Ψ_{2}^{J}^{2}

k2

(x_{2})Ψ_{3}^{J}^{3}

k3

(x_{3}). (2.31)

Further details about B-spline expansions for multidimensional modeling are, for instance, published by Schmidt et al. (2015). These generally derived formulations for basis representations with normal-ized quadratic polynomial B-splines and normalnormal-ized periodic trigonometric B-splines, individual and combined in tensor products, will now be adapted for ionosphere modeling.