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The structure of the atmosphere can be described, for most practical purposes, as a set of concentric spherical shells with different physical and chemical properties. Various subdivisions are possible, that in most cases follow the main characteristic feature of interest. Fig. 2.18 gives a simplified schematic representation.

With respect to signal propagation a subdivision into troposphere and ionosphere is advisable, because the particular propagation conditions are quite different.

− The troposphere is the lower part of Earth’s atmosphere which extends from the surface to about 40 km. Signal propagation depends mainly on the water vapor content and on temperature.

− The ionosphere is the upper part of Earth’s atmosphere between approximately 70 and 1000 km. Signal propagation is mainly affected by free charged particles.

Figure 2.18. Possible subdivision schemes of Earth’s atmosphere

The troposphere is the gaseous atmosphere where the daily weather takes place.

The temperature decreases with height by 6.^◦5 C/km. Horizontal temperature gradients are only a few degrees/100 km. Charged particles are virtually absent. The uncharged atoms and molecules are well mixed, and thus the troposphere is practically a neutral gas. The index of refraction is slightly greater than 1. It decreases with increasing height and becomes nearly 1 at the upper limit of the troposphere, corresponding to the continuously decreasing density of the medium. Nearly 90% of the atmospheric mass is below 16 km altitude, and nearly 99% is below 30 km (Lutgens, Tarbuck, 1998).

For electromagnetic waves in the radio-frequency spectrum the troposphere is not a dispersive medium. The index of refraction does not depend on the frequency; it de-pends on air pressure, temperature, and water vapor pressure. Because of the dynamic behavior of tropospheric conditions it is difficult to model the index of refraction.

The ionosphere can be defined as that part of the high atmosphere where sufficient electrons and ions are present to affect the propagation of radio waves (Davies, 1990;

Langley, 1998b). The generation of ions and electrons is proportional to the radiation intensity of the sun, and to the gas density. A diagram indicating the number of ions produced as a function of height shows a maximum in ion production rate. Such a

Height

Intensity of solar radiation

Ion production rate Density of ionized gas hmax

n_e

Figure 2.19. Chapman curve of ionization diagram is called the Chapman-profile;

the general behavior of this profile is il-lustrated in Fig. 2.19. The exact shape of the curve and the related numerical val-ues are not given in the graph because they depend on several parameters, and they are highly variable functions (see later). The spatial distribution of elec-trons and ions is mainly determined by two processes:

− photo-chemical processes that de-pend on the insolation of the sun, and govern the production and de-composition rate of ionized parti-cles, and

− transportation processes that cause a motion of the ionized layers.

Both processes create different layers of ionized gas at different heights. The main layers are known as theD-, E-, F1-, andF2-layers. In particular, theF1-layer, located directly below the F2-layer, shows large variations that correlate with the relative sun spot number. Geomagnetic influences also play an important role. Hence, signal propagation in the ionosphere is severely affected by solar activity, near the geomagnetic equator, and at high latitudes (cf. [7.4.4.1]).

The state of the ionosphere is described by the electron densityne with the unit [number of electrons/m³] or [number of electrons/cm³]. The four principal layers are designated in Table 2.5.

Table 2.5. Characteristic features of the main ionospheric layers

layer D E F1 F2

height domain [km] 60–90 85–140 140–200 200–1000 electron density at day 10²–10⁴ 10⁵ 5· 10⁵ 10⁶ n_e[el/cm³] at night — 2· 10³ 5· 10⁴ 3· 10⁵

Due to variable insolation of the Sun the spatial distribution of the layers varies during the day. The D-layer is only generated over the daylight side of Earth. The

impact of the state of the ionosphere on the propagation of waves is characterized by the Total Electron Content TEC, where

TEC=  _R

S n_e(s) ds. (2.88)

The integral contains the total number of electrons that are included in a column with a cross-sectional area of 1 m², counted along the signal paths between the satellite S and the receiver R. For comparison purposes among sets of TEC data the vertical electron content VTEC is formed as

VTEC= 1

F · TEC, (2.89)

where

F = 1

cosz^I

is called the obliquity factor or mapping function. z^I is the zenith angle between the signal path and a horizontal plane in the mean altitudeh_i. The unit of measurement is the TECU (Total Electron Content Unit):

1 TECU= 1 · 10¹⁶el/m². (2.90)

Figure 2.20. Single layer model of the ionosphere A frequently used model for data

reduction in satellite geodesy is the single layer model. In this the total electron content is repre-sented by a spherical layer at the mean ionospheric heighth_I, usu-ally near 400 km (Fig. 2.20). On this layer, P_I is the ionospheric piercing point of the signal path to a satelliteS, P_S the subiono-spheric point, r_E Earth’s radius, andz the zenith angle of S for an observerR. The zenith angle z^I atP_I then is given by

F increases with increasing zenith angle z to a satellite target. Table 2.6 (Wanninger, 1994) shows that for small elevation angles TEC can reach at most three times the value of VTEC. This is also true for the effect of ionospheric path delay in satellite geodesy (see Table 2.6).

Values of TEC vary between 10¹⁶and 10¹⁹electrons per m²along the radio wave path. The electron density is highly variable and depends mainly on

Table 2.6. Obliquity factorF and distance d between observer and subionospheric point

E z z^I F d

[degree] [degree] [degree] [km]

90 0 0 1.00 0

60 30 28 1.13 215

30 60 55 1.73 603

20 70 62 2.14 873

10 80 68 2.66 1344

5 85 70 2.87 1712

− geographic location,

− time of the day,

− season of the year, and

− solar activity.

Regions of highest TEC are located approximately±15 to ±20 degrees each side of Earth’s magnetic equator (cf. Fig. 7.52, p. 313). The day to day variability has a standard deviation of±20% to 25% of monthly average conditions (Klobuchar, 1996).

Short term variations are travelling ionospheric disturbances (TID) with a period of minutes to about 1 hour, and ionospheric scintillation with a period of seconds. Of particular importance is the variance of the solar UV flux. The sun varies in its energy output over an approximate 11-year cycle (see Fig. 2.21). The last maximum was in the year 2000. In times of solar maximum the signals of operational GNSS systems can be heavily corrupted (cf. [7.4.4.1]).

1950 1960 1970 1980 1990 2000 1990 1995 2000 0

50 100 150 200 250

solar activity sunspot number

250

200

150

100

50

0

year Figure 2.21. 11-year cycles of solar activity

The high variability of the ionosphere makes modeling and prediction difficult.

Models of the electron density fall into two types: empirical models, derived from existing data, and physical models, derived from physical principles. Examples of

empirical models are the International Reference Ionosphere (IRI), the Bent model, and the Klobuchar model. Physical models are rather cumbersome and seldom used in satellite geodesy. Within the models the VTEC is described either by two-dimensional polynomials for local and regional applications, or by spherical harmonic expansion for continental and global representation. For details see Davies (1990); Wild (1994);

Klobuchar (1996); Wanninger (2000).

Since 1996 the International GPS Service (IGS) [7.8.1] has generated, on a regu-lar basis, global TEC models from GPS observations at selected globally distributed stations. Rapid products are available after several hours, and precise products after three days.

The ionosphere is a dispersive medium for radio waves. For an index of refraction n in ionized gas the formula of dispersion (e.g. Davies, 1990) is

n²= 1 − n_e C²e²

πf²m_e (2.92)

with e elementary mass, and m_e electron mass.

Rearranging and neglecting higher order terms gives n = 1 − C · n_e

f² , (2.93)

withC = 40.3. The coefficient C contains all constant parameters. An explicit deriva-tion of (2.93) can be found in Hartmann, Leitinger (1984) or in textbooks on geophysics.

Formula (2.93) indicates that the index of refraction, and thus the time delay of signal propagation, is proportional to the inverse of the squared frequency. Consequently, one part of the ionospheric delay can be modeled when two frequencies are used [2.3.3].

Furthermore (2.93) shows that higher frequencies are less affected by the ionosphere.