5.2. Observation equations for the discrete and integrated electron density 81

82 Chapter 5. Modeling the ionosphere profile. Equation (5.1) can then be reformulated to

Ne = N mF2 exp 0.5 1− h−hmF2

HF2 −exp −h−hmF2 HF2

! ! !

| {z }

α-Chapman for F2 layer description

+N0P exp −|h−hmF2|

HP

!

| {z }

Transition between F2 layer and plasmasphere

.

(5.2)
The set of original Chapman parameters is extended by two more quantities only: the plasmasphere
basis density N0P and the plasmasphere scale height parameter HP. The extended Chapman profile
is a simple but efficient formulation to indirectly account for the necessity of a second or altitude
dependent scale height, although the physical meaning of the plasmasphere parameters is rather
ques-tionable. Following Jakowski (2005), reasonable values for HP and N0P can be introduced with
HP = 10^{4} km and N0P = γN mF2 where proportionality between N mF2 andN_{0}^{P}, controlled by the
scaling factorγ, is supposed. Here, the F2 scale height is still invariant with height but the function
shape nevertheless benefits from the slowly decaying plasmaspheric exponential term.

From ionospheric radio occultations, discrete electron density observations can be retrieved as de-scribed in Chapter 4. Consequently, the Chapman equation can directly be considered to formulate the relation between measurements and unknowns. GNSS, DORIS and altimetry on the contrary pro-vide TEC, i.e., the integrated electron density as introduced with Eq. (1.28), leading to the demand for an integration function to solve

STEC=

x^{s}

Z

x_{r}

Neds=

x^{s}

Z

x_{r}

N mF2 exp 1

2 1− h−hmF2

HF2 −exp −h−hmF2 HF2

! ! !

+N0P exp −|h−hmF2| HP

! !

ds. (5.3)

The integral boundaries are defined by the satellite transmitter x^{s} and receiver xr and the integration
has to be performed along the (curved) raypath. For altimetric VTEC,xr identifies the water surface

Figure 5.2: Layer composition for the extended Gauß-Legendre integration of the electron density for the computation of TEC.

5.2. Observation equations for the discrete and integrated electron density 83
point on Earth in nadir direction of the satellite. To integrate numerically in the 3-D space, an extended
Gauß^{37}-Legendre^{38}Quadrature is applied. A similar approach has been considered by Zeilhofer et al.

(2008).

Figure 5.3: Geometrical ray path distribution including five Sistema de Referencia Geocéntrico para Las Américas (SIRGAS) network stations (red triangles) tracking five GPS satellites passes (blue).

Therefore, the Earth ionosphere is firstly approximated by three different layers as depicted in Fig. 5.2. The layer boundaries are chosen to sepa-rate the lower atmosphere with less ionization (e.g. between 80 km and 200 km) from the F2 and plasmas-phere transition region (e.g. 200 km - 1,000 km) and the upper atmosphere including mainly the less ionized plas-maspheric layers (e.g. 1,000 km -2,000 km). The lowermost and upper-most boundaries are chosen at heights where the electron content is negli-gible for the TEC derivation. After-wards, each synthetic layer is decom-posed into a set of integration inter-vals defined by a given integration step width. The selection of an adequate step width orients on the ionospheric gradients that are expected within the

three synthetic layers, i.e., typically a small step width for layer 2 and larger ones for layer 1 and 3 are chosen. In this work, the input step width refers to a vertical integration path and is transformed to the slant by

SW_{s} =

SW_{v}/sinel if 30^{◦} ≤ el < 90^{◦}

SWv/sin 30 ifel <30^{◦} (5.4)

where SW_{s} and SW_{v}are the slant and vertical stepwidth, respectively, and el is the elevation angle
of the ray. Each integration interval is finally integrated by the classical Gauß-Legendre method of
order N. This method yields the approximation of the integration interval as a sum of given sample
values at N so-called knot points (or nodal points) ni ∈ {n_{1}, . . . ,nN} with positive-valued weights
w ∈ {w_{1}, . . . ,wN}. The knot points are derived from the zeros of the Legendre polynomials LN. With
the given knot points, the individual weights are then calculated from

wi = Z1

−1 N

Y

j=1,j,i

n−nj

ni−nj

dn for 1 ≤i ≤ N (5.5)

leading to the approximation of the integral as If =

P2

Z

P1

f(n)w(n)dn≈

N

X

i=1

f(ni)wi (5.6)

where P1 and P2 are the boundaries defined by the integration interval.

In order to assess the performance of this integration method at varying step widths and quadrature orders, different test scenarios have been analyzed. For instance, the positions of GPS satellites to-gether with five GPS tracking stations in South America belonging to the SIRGAS network have been

37Johann Carl Friedrich Gauß (30.04.1777 - 23.02.1855) was a German mathematician, astronomer, geodesist and physicist.

38Adrien-Marie Legendre (18.09.1752 - 10.01.1833) was a French mathematician.

84 Chapter 5. Modeling the ionosphere considered to simulate a realistic geometrical ray path distribution. For one hour data between 06:00 UT and 07:00 UT at 1 July 2002, the geometry of five GPS satellite passes is shown by Fig. 5.3.

Satellite passes are depicted as blue lines and receivers as red triangles where pass start- and end-points are connected by dashed lines. From these stations, totally 2,404 synthetic TEC observations according to the epochs provided by their RINEX data are generated with 30 s sampling according to Eq. (5.3). The Chapman parameters are derived from IRI data while the integration is adjusted with N = 9 and 1 km step width fixed for all three layers to obtain a synthetic reference solution for the validation of the internal accuracy. The Root Mean Square (RMS) of the reference solution for this selected testbed yields 21.85 TECU.

Afterwards, integrations are performed with varying vertical step widths of 20 km, 40 km, 60 km and 80 km and different quadrature order N ∈ {3,5,7,9}. Figure 5.4 depicts the comparison of the RMS values related to the TEC differences (green bars) and the processing time (blue bars). The darker the color, the smaller the step width and consequently, the computation time decreases with decreasing quadrature order and increased step width. The RMS on the contrary increases with the step width and decreasing order. In this internal validation, it can be seen that the RMS values are permanently small while the processing time must be considered as the limiting factor, in particular since the results here are based on only 2,404 observations. Typically, a maximum quadrature level of N = 6 was considered in this work. A vertical step width of 20 km was assigned for the second layer to account for increased ionospheric gradient and larger step widths between 60 km and 80 km are commonly chosen for the first and third layer.

Figure 5.4: Comparison of processing effort in computation time (blue bars) with the RMS of TEC differences (green bars) for different quadrature orders and integration step widths in a simulated scenario.