## Phase Path and Group Path Variation

The change in the phase path, AL, at VHF and above, is given by the integrated difference between the refractive index in the ionosphere and unity:

The negative sign indicates that the impact of the ionosphere is to reduce the phase path length relative to free space.

The group path length ASg — AS1 at VHF and above. To obtain the time delay ATK, we simply divide by the speed of radio propagation (i.e., c = 3 x 108 m/sec) to obtain:

Al\, (seconds) = A S/c = ~ 1.34 (10 7)f '2 (EC) (4.3)

The ionospheric time delay is an important parameter in connection with satellite navigation. Figure 4-10 illustrates how ATg varies with radio frequency, using the electron content as parameter. Note that the TEC may range between ~ 1016 and 1018 electrons/m2. The EC may be ~ 3 times larger for highly oblique paths. Taking the EC to be 3 x 10'8, and f = 1.6 GHz, we find ATg = 157 nanoseconds.

To obtain the phase change, we recognize that it is simply the phase path length AS times 2%/X where X is the wavelength. This leads to the result:

The range error associated with the ionospheric path is obtained by integrating (nK -1) along the refracted path, where ng is the group refractive index. This is just the change in group path length, and it is a positive number. At 100 MHz and above, ng ~ 1/n, where n is the real part of the index of refraction. Figure 4-11 gives the representative limiting values for range error (i.e., troposphere + ionosphere) at 100 and 200 MHz. In Figure 4-12 there is a display of the diurnal variation of elevation error and range error is given at 400 MHz for a typical midlatitude profile of electron density. To go from one frequency to another, recall that the scaling factor will be/ 2.

1018

SlO17

E1016

100 200 500 1000

Frequency (MHz)

2000 3000

Figure 4-tO: Ionospheric time delay as a function of radio frequency with electron content as parameter. From ITU-R [1996].

104 8 6

tT 2

cr 4

102 8 6

4 2 101

 - 1 1 1 I I I I - - - - _ 100 Mc/s - Dav - - - - - ^ 100 Mc/s - Niqht 200 Mc/s - Day — - ~ - - 1 100 Mcy 's - Nigh it __ - \ \ - lonoi spheric error snhfirio error - - \> trnnr - \ > RH = relative humidity - N > 000% RH 1 I "** -Mi -- I

0 10 20 30 40 50 Elevation Angle (deg)

Figure 4-11: limiting values for ionospheric and tropospheric range errors From Millman 119671.

Figure 4-12: Ionospheric refraction and range errors through a real ionosphere. The profile was derived using the Millstone Hill incoherent (Thomson) scatter radar. The 400 MHz elevation error is given in the top curve, and the range error is given in the bottom curve. From Evans and Wand [1975).
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