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 spaceR2 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 ∈ [xmin,xmax] is now expressed as
s(x) =
K
X
k=1
dJk ΨkJ(x). (2.13)
Ψ identifies a linearly independent set of 1-D scaling functions Ψ1, . . . ,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 φJkφ(x) = Nk,mJφ (x) as 1-D basis functions for representing the signal within a bounded interval. Nk,mJφ are usual normalized B-splines of ordermwith
Kφ
X
k=1
Nk,mJφ (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 Jφ, the finer the signal structures that can be resolved.
The total number of B-splines is computed from Kφ = 2Jφ + m − 1. The basis is deployed by an increasing sequence of Kφ + m so-called knot points νkJφ ∈ {ν1Jφ, ν2Jφ, . . . , νKJφφ+m} where, at the boundaries, multiple knots may be linked to a specific coordinate point. The knot intervalνkJφ+1−νkJφ 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
Nk,1Jφ(x) =
1 νkJφ ≤ x < νkJφ+1
0 otherwise , k = m, . . . ,Kφ (2.15)
Nk,mJφ (x) = x−νkJφ
νkJφ+m−1−νkJφNk,m−1Jφ (x)+ νkJφ+m− x
νkJ+φm −νkJφ+1NkJ+1,m−1φ (x), m ≥ 2. (2.16) A uniform knot sequence is established as
0= ν1Jφ = · · ·= νmJφ
| {z }
Boundary multiplicity
< νmJφ+1< · · · < νKJφφ <
| {z }
Internal sequence
νKJφφ+1= · · ·= νKJφφ+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 distanceνkJφ+1−νkJφ yields consequently Kφ−m+1−1
on the unit interval.
Normalized quadratic polynomial B-spline functions φJkφ(x) = Nk,3Jφ(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, Kφ = 2Jφ +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 Jφ = 0,1,2,3 and accordingly different number of B-splinesKφJ =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→ Kφ0=3 (top-left), Jφ = 1 → Kφ1 = 4 (top-right), Jφ = 2 → Kφ2 = 6 (bottom-left) and Jφ = 3 → K3φ = 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 Jφ. 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= simax−simin
Kφ−1 (2.18)
2.3. B-splines 35
with the sampling si on the interval [simin,simax]. This formulation can be transformed to Jφ <log2 simax−simin
∆si −m+2
!
(2.19) under consideration ofKφ = 2Jφ+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
Jφ <log2 simax−simin−∆si
∆si
!
. (2.20)
2.3.2 Normalized periodic trigonometric B-splines
As a second base function type, normalized periodic trigonometric B-splines Tk,mJT (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 = 3·2JT +m−1 where altogether KT0 = KT −m+1 = 3·2JT 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
Tk,1JT(x) =
1 νkJT ≤ x < νkJT+1
0 otherwise , k = 1, . . . ,KT (2.21)
Tk,mJT (x) =
sin x−ν
JT k
2
!
sin ν
JT k+m−1−νkJT
2
! ·Tk,m−1JT (x)+
sin ν
k+mJT −x 2
!
sin ν
JT k+m−νkJT+1
2
! ·TkJ+1,m−1T (x), m≥ 2 (2.22) with the non-decreasing sequence of distinct knots
0= νmJT < νmJT+1 < . . . < νKJT
T < νKJT
T+1=2π. (2.23)
Additional constraints on the knot placement νiJT =νKJT0
T+i−2π νKJT0
T+m+i = νmJT+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 functionsTk,3JT can be determined asKT =3·2JT +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 < log2 simax−simin+∆si
3∆si
!
(2.25) by substitutingKT0 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 RSΩ in a 2-D spaceR2. The representation of a subsurface of SΩ0 defined withinϕ ∈ [ϕmin, ϕmax] andλ ∈ [λmin, λmax] in the rectangular domain can be obtained with
RSΩ0 := {(ϕ, λ) :ϕmin ≤ ϕ ≤ ϕmaxandλmin ≤ λ ≤ λmax}. (2.27) In order to represent multidimensional information on the rectangular modeling surfaces RSΩ or RSΩ0
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(x1,x2) =
K1
X
k1=1 K2
X
k2=1
dkJ1,J2
1,k2 Ψ1J1
k1
(x1) Ψ2J2
k2
(x2). (2.28)
Here, tensor products of two linearly independent 1-D B-spline functions Ψ1J1
k1
and Ψ2J2
k2
have been introduced together with the corresponding series coefficientsd. It should be noticed thatΨ1andΨ2 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, Jφ, Kφ).
From now on, the identification will be based on indices (J1, K1, J2, K2,. . . ) 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 x1and x2interval, with
x1→ Ψ1 =φ2k
1 and x2 →Ψ2= φ3k
2 (2.29)
for the levels J1 = 2 and J2 = 3. The corresponding basis is depicted in Fig. 2.4. Both plots show
Figure 2.4: Polynomial B-splinesφ2k
1(x1) andφ3
k2(x2) of orderm =3 with different levels J1 =2 andJ2 =3. φ23(x1) (left),φ21(x1) (right) andφ35(x2) 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 byφ23(x1) (red) andφ35(x2) (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 φ21(x1) (blue) and φ35(x2) (magenta) constraint by endpoint-interpolation on the x1axis.
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
x1→ Ψ1 =Tk2
1 and x2 →Ψ2= φ3k
2 (2.30)
for different levels J1 = 2 and J2 = 3 with endpoint-interpolation in the x2and continuity in the x1 direction. Figure 2.5 shows the support area forT32(x1) andφ35(x2) 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 J1 = 2 and polynomial B-splines with level J2 = 3. A specific spline combination identified byk1 =5 andk2 =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(x1,x2,x3)=
K1
X
k1=1 K2
X
k2=1 K3
X
k3=1
dkJ1,J2,J3
1,k2,k3 Ψ1J1
k1
(x1) Ψ2J2
k2
(x2)Ψ3J3
k3
(x3). (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.